Download Functional-Integral Representation of Quantum Field Theory {functint

H. Kleinert, PARTICLES AND QUANTUM FIELDS
November 19, 2016 ( /home/kleinert/kleinert/books/qft/nachspa1.tex)
Perfection is achieved, not when there is nothing more to add,
but when there is nothing left to take away.
Antoine de Saint-Exupèry (1900–1944)
14
Functional-Integral Representation
of Quantum Field Theory
{
functin
In Chapter 7 we have quantized various fields with the help of canonical commutation
rules between field variables and their canonical conjugate field momenta. From
these and the temporal behavior of the fields determined by the field equations
of motion we have derived the Green functions of the theory. These contain all
experimentally measurable informations on the quantum field theory. They can all
be derived from functional derivatives of certain generating functionals. For a real
scalar field this was typically an expectation value
Z[j] ≡ h0|T [j]|0i
(14.1) {5.254ab}
where T [j] was the time-ordered product in (7.827)
T [j] ≡ T̂ ei
R
d4 x j(x)φ(x)
.
(14.2) {5.250ab}
Here the Green functions can all be obtained from functional derivatives of Z[j] of
the type (7.841).
For complex scalar fields, the corresponding generating functional is given by
the expectation value (7.849). Now the Green functions can all be obtained from
functional derivatives of the type (7.850).
In theoretical physics, Fourier transformations have always played an important
role in yielding complementary insights into mathematical structures. Due to the
conjugate appearance of fields φ(x) and sources j(x) in expressions like (14.2), this is
also true for generating functionals and constitutes a basis for the functional-integral
formalism of quantum field theory.
14.1
Functional Fourier Transformations
An important observation is now that instead of calculating these generating functionals as done in Chapter 7 from a formalism of field theory, in which φ(x) is a field
926
927
14.1 Functional Fourier Transformations
operator, they can also be derived from a functional Fourier transform of another
functional Z̃[φ] that depends on a classical field φ(x):
Z[j] ≡
Z
DφZ̃[φ] ei
R
dD x j(x)φ(x)
(14.3) {12@fo51}
.
The symbol Dφ(x) in this expression is called a functional integral.
The mathematics of functional integration is an own discipline that is presented
in many textbooks [1,2]. Functional integrals were first introduced in ordinary quantum mechanics by R.P. Feynman [3], who used them to express physical amplitudes
without employing operators. The uncertainty relation that can be expressed by an
equal-time commutation relation [x(t), p(t)] = ih̄ between x(t) and the conjugate
variable p(t) of quantum mechanics is the consequence of quantum fluctuations of
the classical variables x(t) and p(t). In quantum mechanics, the functional integral
is merely a path integral of a fluctuation variable x(t). In field theory, there is a
fluctuating path for each space point x. Instead of a time-dependent variable x(t)
one deals with more general dynamical variables φx (t) = φ(x, t) = φ(x), one for
each spacepoint x. The path integrals over all x(t) go over into functional integrals
over all fluctuating fields φ(x).
Functional integrals may be defined most simply in a discretized approximation.
Spacetime is grated into a fine spacetime lattice. For every spacetime coordinate xµ
we introduce a discrete lattice point close to it
R
xµ → xµn ≡ nǫ,
n = 0, ± 1, ± 2, ± 3,
(14.4) {12@fo52}
where ǫ is a very small lattice spacing. Then we may approximate integrals by sums:
Z
dD x j(x)φ(x) ≈ ǫD
X
n
j(xn )φ(xn ) ≡ ǫD
X
n
jn φn
(14.5) {12@fo53}
where n is to be read as a D-dimensional index (n0 , n1 , . . . , nD ), one for each
R
component of the spacetime vector xµ . Now we define Dφ(x) as the infinite product
of integrals over φn at each point xn :
Z
Dφ(x) =
YZ
n
dφn
q
2πi/ǫD
(14.6) {12@fo54}
.
Operations with functional integrals are very similar to those with ordinary integrals. For example, the Fourier transform of (14.3) can be inverted, by analogy with
ordinary integrals, to obtain:
Z̃[φ] ≡
Z
Dj(x)Z[j]e−i
R
dD x j(x)φ(x)
.
(14.7) {12@fo51Y}
There are functional analogs of the Dirac δ-function:
Z
Z
Dj(x) e−i
Dφ(x) ei
R
R
dD x j(x)φ(x)
= δ[φ],
(14.8) {12@fo56}
dD x j(x)φ(x)
= δ[j],
(14.9) {12@fo56b}
928
14 Functional-Integral Representation of Quantum Field Theory
called δ-functionals. In the lattice approximation corresponding to (14.6), they are
defined as infinite products of ordinary δ-functions
δ[φ] =
Yq
2πi/ǫD δ(φn ),
δ[j] =
n
Yq
2πi/ǫD δ(jn ).
n
(14.10) {12@delfunio
They have the obvious property
Z
Z
Dφ δ[φ] = 1,
Dj δ[j] = 1.
(14.11) {12@fo57}
A commonly used notation for the measure (14.6) of functional integrals employs
continuously infinite product of integrals which must be imagined as the continuum
limit of the lattice product (14.6). In this notation one writes
Z
Dφ(x) =
YZ
x
dφ(x)
√
,
2πi
Z
and the associated δ-functionals as
Y√
2πi δ(φ(x)),
δ[φ] =
Dj(x) =
δ[j] =
x
Y√
2πi δ(j(x)).
x
x
14.2
dj(x)
√
,
2πi
YZ
(14.12) {12@fo54x}
(14.13) {12@delfunio
Gaussian Functional Integral
Only very few functional integrals can be solved explicitly. The simplest nontrivial
example is the Gaussian integral1
Z
i
Dj(x)e− 2
R
dD x dD x′ j(x)M (x,x′ )j(x′ )
(14.14) {12@fo58}
.
In the discretized form, this can be written as


YZ
n
djn
q
2πi/ǫD

i 2D
 e− 2 ǫ
P
j M
j
n,m n nm m
.
(14.15) {12@fo59}
We may assume M to be a real symmetric functional matrix, since its antisymmetric
part would not contribute to (14.14). Such a matrix may be diagonalized by a
rotation
jn → jn′ = Rnm jm ,
(14.16) {12@fo60}
which leaves the measure of integration invariant
∂ (j1 , . . . , jn )
= det R−1 = 1.
′
′
∂ (j1 , . . . , jn )
(14.17) {12@fo61}
1
Mathematically speaking, integrals with an imaginary quadratic exponent are more accurately
called Fresnel integrals, but field theorists do not make this distinction.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
929
14.2 Gaussian Functional Integral
In the diagonal form, the multiple integral (14.15) factorizes into a product of Gaussian integrals, which are easily calculated:
YZ
n
djn′
i 2D
q
2πi/ǫD
e− 2 ǫ
P
j′ M j′
n n n n
=
Y
n
√
1
= det −1/2 (−ǫD M).
D
−ǫ Mn
(14.18) {12@fo62}
On the right-hand side we have used the fact that the product of diagonal values
Mn is equal to the determinant of M. The final result

YZ

q
djn
2πi/ǫD
n

i 2D
 e− 2 ǫ
P
j M j
n,m n nm m
= det −1/2 (−ǫD M)
(14.19) {12@@DForm
is invariant under rotations, so that it holds also without diagonalizing the matrix.
This formula can be taken to the continuum limit of infinitely fine gratings ǫ → 0.
Recall the well-known matrix formula
det A = elog det A = etr log A ,
(14.20) {12@fo65}
and expand tr log A into power series as follows
∞
X
(−)k
tr log A = tr log [1 + (A − 1)] = −tr
(A − 1)k .
k=1 k
(14.21) {12@fo66}
The advantage of this expansion is that when approximating the functional matrix
A by the discrete matrix ǫD M, the traces of powers of ǫD M remain well-defined
objects in the continuum limit ǫ → 0:
tr ǫD M
D
tr ǫ M
2
= ǫD
X
n
2D
= ǫ
Mnn →
X
n,m
..
. .
Z
dD x M(x, x) ≡ TrM,
Mnm Mmm →
Z
dD x dD x′ M(x, x′ )M(x′ , x) ≡ TrM 2 , (14.22) {12@fo67b}
We therefore rewrite the right-hand side of (14.19) as exp[−(1/2)tr log(−ǫD M)], and
expand
h
tr log(−ǫD M) = tr log 1 + −ǫD M − 1
−
−−→ −
ǫ→0
∞
X
i
h
i
(−)k Z D
d x1 · · · dD xk −M(x1 , x2 ) − δ (D) (x1 − x2 ) × · · ·
k=1 k
h
ih
i
× −M(x2 , x3 ) − δ (D) (x2 − x3 ) −M(xk , x1 ) − δ (D) (xk − x1 ) . (14.23) {12@fo66fu}
The expansion on the right-hand side defines the trace of the logarithm of the
functional matrix −M(x, x′ ), and will be denoted by Tr log(−M). This, in turn,
930
14 Functional-Integral Representation of Quantum Field Theory
serves to define the functional determinant of −M(x, x′ ) by generalizing formula
(14.20) to functional matrices:
Det (−M) = elog det (−M ) = eTr log(−M ) .
(14.24) {12@fo65fu}
Thus we obtain for the functional integral (14.14) the result:
Z
i
Dj(x)e− 2
R
dD x dD x′ j(x)M (x,x′ )j(x′ )
1
= Det −1/2 (−M) = e− 2 Tr log(−M ) .
(14.25) {12@fo64new
This formula can be generalized to complex integration variables
√ by separating
the currents into real and imaginary parts, j(x) = [j1 (x) + ij2 (x)]/ 2. Each integral
gives the same functional determinant so that
Z
Dj ∗ (x)Dj(x)e−i
R
dD x dD x′ j ∗ (x)M (x,x′ )j(x′ )
= e−Tr log M .
(14.26) {12@fogaussi
Here M(x, x′ ) is an arbitrary Hermitian matrix, and the measure of integration
for complex variables j(x) is defined as the product of the measures for real and
imaginary parts: Dj ∗ (x)Dj(x) ≡ Dj1 (x)Dj2 (x).
14.3
Functional Formulation for Free Quantum Fields
Having calculated the Gaussian functional integrals (14.25) and (14.26) we are able
to perform the functional integrations over the generating functional (14.3) to derive
its Fourier transform Z̃[φ]. First we shall do so only for the free-field generating
functional (7.843), which we shall equip with a subscript 0 to emphasize the free
situation:
R 4 4
1
(14.27) {5.257aB}
Z0 [j] = e− 2 d y1 d y2 j(y1 )G0 (y1 ,y2 )j(y2 ) .
By writing M(x, x′ ) as
M(x, x′ ) = −iG0 (x, x′ ),
(14.28) {12@fo58BB}
the Gaussian functional integral (14.14) becomes
Z
− 21
Dj(x)e
R
dD x dD x′ j(x)G0 (x,x′ )j(x′ )
= Det −1/2 (−iG0 ).
(14.29) {12@fo58B}
This result can immediately be extended to calculate the functional Fourier tranform of the generating functional Z0 [j] defined in (14.3). Thus we want to form
Z̃0 [φ] =
Z
1
Dj(x)e− 2
R
dD x dD x′ j(x)G0 (x,x′ )j(x′ )+i
R
j(x)φ(x)
.
(14.30) {12@fo58BZ}
The extra term linear in j(x) does not change the harmonic nature of the exponent.
The integral can be reduced to the Gaussian form (14.29) by a simple quadratic
completion process. For this manipulation it is useful to omit the spacetime indices,
and use an obvious functional vector notation to rewrite (14.30) as
Z̃0 [φ] =
Z
i T 1
G j+ij T φ
i 0
Dje− 2 j
.
(14.31) {12@fo69}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
931
14.3 Functional Formulation for Free Quantum Fields
The exponent may be completed quadratically as
−
T 1
i
i T
−1
j + iG−1
φ
G
j
+
iG
φ
+ φT iG−1
0
0
0
0 φ.
2
i
2
(14.32) {12@fo70}
′
We now replace the variable j + iG−1
0 φ by j which, in each of the infinite integrals,
amounts only to a trivial shift of the center of integration. Thus
Z̃0 [φ] =
Z
′ − 2i j ′ 1i G0 j ′
Dj e
i
e2φ
T iG−1 φ
0
(14.33) {12@fo71}
.
We can now apply formula (14.26) and find
i
Z̃0 [φ] = Det (M)1/2 e 2
R
dD x dD x′ φ(x)M −1 (x,x′ )φ(x′ )
(14.34) {12@fo72}
,
Inserting for M(x, x′ ) the functional matrix (14.28) we see that the exponent contains the free-particle Green function:
M −1 (x, x′ ) = −iG0 (x, x′ ) = (−∂ 2 − m2 )δ (D) (x − x′ ).
(14.35) {12@fo73}
The determinant is a constant prefactor which does not depend on the field φ. We
shall abbreviate it as a normalization factor
N = Det −1/2 (−iG0 ) = Det −1/2 (−∂ 2 − m2 ).
(14.36) {12@fo75o}
With the help of Eq. (14.20), this can be rewritten as
N = exp
1
− Tr log(−∂ 2 − m2 ) .
2
(14.37) {12@fo75o2}
Inserting into (14.34) the differential operator (14.35), and performing a partial
integration in spacetime, we can rewrite it as
Z̃0 [φ] = N ei
R
2
dD x [ 12 (∂φ)2 − m2 φ2 ]
= N ei
R
dD x L0 (φ,∂φ)
.
(14.38) {12@fo74}
Thus the Fourier-transformed generating functional is, up to the normalization factor N , just the exponential of the classical free-field action under consideration.
By Fourier-transforming Z̃[φ] according to formula (14.3), we recover the initial
generating functional Z0 [j]. This procedure yields the famous functional integral
representation for the free-particle generating functional
Z0 [j] = N
Z
Dφ(x)ei
R
dD x [L0 (φ,∂φ)+j(x)φ(x)]
.
(14.39) {12@fo80new
In this representation, the field is no longer an operator but a real variable that
contains all quantum information via its field fluctuations. The functional integral
over the φ-field is defined in (14.6). By analogy with (14.25), we have for the real
field fluctuations the Gaussian formula
Z
i
Dφ(x)e 2
R
dD x dD x′ φ(x)M −1 (x,x′ )φ(x′ )
1
= (Det M) 2 ,
(14.40) {12@fo64new
932
14 Functional-Integral Representation of Quantum Field Theory
′
valid for real symmetric functional matrices
√ M(x, x ).
For complex fields φ = (φ1 + iφ2 )/ 2, there is a similar functional integral
formula
Z
Dφ∗ (x)Dφ(x)ei
R
dD x dD x′ φ∗ (x)M −1 (x,x′ )φ(x′ )
(14.41) {12@fogaussi
= Det M,
in which M(x, x′ ) is a Hermitian functional matrix. This follows directly from
(14.26).
Let us use the formula (14.40) to cross-check the proper normalization of (14.39).
At zero source, Z0 [j] has to be equal to unity [compare (14.27)]. This is ensured if
N −1 = Det 1/2 (−iG0 ) =
Z
Dφ(x)ei
R
dD x L0 (φ,∂φ)
(14.42) {12@fo75}
.
Indeed, after a spacetime integration by parts, the right-hand side may be written
as
Z
R
i dD x L0 (φ,∂φ)
Dφ e
Z
i dD x 12 [(∂φ)2 −m2 φ2 ]
= Dφ e
R
Z
R
i dD x φ(x)M −1 (x,x′ )φ(x′ )
= Dφ e
. (14.43) {12@fo75x}
Applying now formula (14.39), and using (14.36), we verify that Z0 [0] = 1.
Using the expression (14.42) for the normalization factor, we can rewrite
Eq. (14.38) for the Fourier transform Z̃0 [φ] as
Z̃0 [φ] = R
ei
dD x L0 (φ,∂φ)
Dφ(x)ei
This ratio is obviously normalized:
Z
R
R
dD x L0 (φ,∂φ)
(14.44) {12@fo76}
.
Dφ(x)Z̃0 [φ] = 1.
(14.45) {12@annorm
Note that the expression (14.44) has precisely the form of the quantum mechanical
version (1.491) of the thermodynamical Gibbs distribution (1.489):
−1
wn ≡ ZQM
(tb − ta )e−iEn (tb −ta )/h̄ .
(14.46) {[email protected]
There exists a useful formula for harmonically fluctuating fields which are encountered in many physical contexts that can be derived immediately from this.
Consider the correlation function of two exponentials of a free field φ(x). Inserting
into Eq. (14.59) the special current
j12 (x) = a
Z
dD x[aφ(x − x1 ) − bφ(x − x2 )],
(14.47) {12@@speccu
the partition function (14.27) reads
Z0 [j12 ] = Det −1/2 (−iG0 )
Z
Dφ(x) eiaφ(x1 ) e−ibφ(x2 ) ei
R
dD x [L0 (φ,∂φ)]
. (14.48) {12@fo80new
H. Kleinert, PARTICLES AND QUANTUM FIELDS
933
14.4 Interactions
As such it coincides with the harmonic expectation value
heiaφ(x1 ) e−ibφ(x2 ) i.
(14.49) {@}
Inserting (14.43) and performing the functional integral in (14.48) yields
heiaφ(x1 ) e−ibφ(x2 ) i = e− 2 [a
1
2 G(x
].
2
1 ,x1 )−2abG(x1 ,x2 )+b G(x2 ,x2 )
(14.50) {12@@wickex
In the brackets of the exponent we recognize the expectation values of pairs of fields
a2 hφ(x1 )φ(x1 )i − 2abhφ(x1 )φ(x2 )i + b2 hφ(x2 )φ(x2 )i,
(14.51) {@}
such that we may also write (14.50) as
1
2
heiaφ(x1 ) e−ibφ(x2 ) i = heiaφ(x1 )−ibφ(x2 ) i = e− 2 h[aφ(x1 )−bφ(x2 )] i .
(14.52) {12@@expexp
Of course, this result may also be derived by using field operators and Wick’s theorem
along the lines of Subsection 7.17.1.
14.4
Interactions
Let us now include interactions. We have seen in Eq. (10.24) that the generating functional in the interaction picture [more precisely the functional ZD [j] of
Eq. (10.22)] may simply be written as
Z[j] = ei
R
dD x Lint (−iδ/δj(x))
{INTACS}
(14.53) {12@fo77}
Z0 [j].
This can immediately be Fourier-transformed to
Z̃[φ] =
Z
Dj(x)e−i
R
dD x j(x) φ(x)
h R
ei
dD x Lint (φ)(−iδ/δj)
i
Z0 [j] .
(14.54) {12@fo78}
Removing the second exponential by a partial functional integration, we obtain
i
Z̃[φ] = e
= ei
= R
R
R
dD x Lint (φ)
Z
dD x Lint (φ,∂φ)
ei
R
Dj(x)e−i
N ei
dD x L(φ,∂φ)
Dφ(x)ei
R
R
dD x
dD x L0 (φ,∂φ)
R
1
2
dD x j(x)φ(x)
Z0 [j]
[(∂φ)2 −m2 φ2 ]
(14.55) {12@fo79}
.
The functional Fourier transform of this renders a generalization of (14.39) that
includes interactions. In this way we have derived functional integral representation
of the interacting theory:
Z[j] = N
=
R
Z
Dφ(x)e
Dφ(x)ei
R
R
dD x [L(φ,∂φ)+j(x)φ(x)]
R
dD x L0 (φ,∂φ)(φ)
i
R
dD x [L(φ,∂φ)+j(x)φ(x)]
Dφ(x)ei
.
(14.56) {12@fo80o}
934
14 Functional-Integral Representation of Quantum Field Theory
This representation may be compared with the perturbation theoretic formula of
operator quantum field theory
Z[j] = h0|T ei
R
dD x [Lint (φ)+j(x)φ(x)]
|0i,
(14.57) {12@fo81}
where the symbol φ(x) denotes free-field operators, and the vacuum expectation
value of products of these fields follow Wick’s theorem. In the functional integral
representations (14.54)–(14.56), on the other hand, φ(x) is a classical c-number field.
All quantum properties of Z[j] arise from the infinitely many integrals over φ(x),
one at each spacetime point x, rather than from field operators.
Note that in formula (14.56), the full action appears in the exponent, whereas
in (14.57), only the interacting part appears.
In contrast to Z̃0 [φ] of Eqs. (14.44) and (14.45), the amplitude for the interacting
theory is no longer properly normalized. In fact, we know from the perturbative
evaluation of (14.57) that it represents the sum of all vacuum diagrams displayed in
(10.81). Since the denominator does not normalize A[φ] anyhow, it is convenient to
drop it and work with the numerator only, using the unnormalized
Z[j] =
Z
Dφ(x)ei
R
dD x [L(φ,∂φ)+j(x)φ(x)]
(14.58) {12@fo80}
as the generating functional.
The normalization in (14.58) has an important advantage over the previous one
in (14.56). In the euclidean formulation of the theory to be discussed in Section 14.5,
it makes Z[0] equal to the thermodynamic partition function of the system.
For free fields, Z[0] is equal to the partition function of a set of harmonic oscillators of frequencies ω(k) for all momenta k. This statement can be proved only in
the lattice version of the theory. In the continuum limit the statement is nontrivial,
since the determinants on the right-hand sides are infinite. However, we shall see
in Section 14.7, Eqs. (14.123)–(14.133), that correct finite partition functions are
obtained if the infinities are removed by the method of dimensional regularization,
that was used in Section 11.5 to remove divergences from Feynman integrals.
Even though the operator formula (14.57) and the functional integral formula
(14.58) are completely equivalent, there are important advantages of the latter. In
some theories it may be difficult to find a canonically quantized set of free fields
on which to construct an interaction representation for Z[j] following Eq. (14.57).
The photon field is an important example where it was quite hard to interpret
the Hilbert space. In particular, we remind the reader of the problem that in the
Gupta-Bleuler quantization scheme, the vacuum energy contains the quanta of two
unphysical polarization states of the photon. Within the functional approach, this
problem can easily be avoided as will be explained in Chapter 17.
A second and very important advantage is the possibility of deriving the Feynman
rules, without any knowledge of the Hilbert space, directly from the representation
(14.58). For this we simply take the non-quadratic piece of the action, which defines
H. Kleinert, PARTICLES AND QUANTUM FIELDS
935
14.4 Interactions
the vertices of the perturbation expansion, outside the functional integral as in
(14.53), i.e., we rewrite (14.58) as:
Z[j] = ei
R
dD x Lint (−iδ/δj(x))
Z
Dφ(x)ei
R
dD x [L0 (φ,∂φ)+j(x)φ(x)]
.
(14.59) {12@fo77int}
Using (14.34), this reads more explicitly
i
Z[j] = e
R
dD x Lint (−iδ/δj(x))
Z
i
Dφ(x)e[ 2
R
′
′
dD x dD x′ φ(x)iG−1
0 (x,x )φ(x )+j(x)φ(x)]
. (14.60) {12@fo77int2}
A shift in the field variables to
φ′ (x) = φ(x) +
Z
dy G0 (x, x′ )j(x′ ),
(14.61) {@}
and a quadratic completion lead to
Z[j] = ei
R
×
dD x Lint (−iδ/δj(x)) − 12
e
Z
i
Dφ′(x)e 2
R
R
dD x dD x′ j(x)G0 (x,x′ )j(x′ )
′ ′ ′
dD x dD x′ φ′ (x)iG−1
0 (x,x )φ (x )
.
(14.62) {12@fo77int3a
The functional integral over the shifted field φ′ (x) can now be performed with the
help of formula (14.25), inserting there M(x, x′ ) = −iG0 (x, x′ ). The result is, recalling (14.36),
Z
i
Dφ′ (x)e 2
R
′ ′ ′
dD x dD x′ φ′ (x)iG−1
0 (x,x )φ (x )
= Det −1/2 (−iG0 ),
(14.63) {12@fo77int4}
such that we find
Z[j] = Det
−1/2
i
(−iG0 ) e
R
dD x Lint (−iδ/δj(x)) − 12
e
R
dD x dD x′ j(x)G0 (x,x′ )j(x′ )
. (14.64) {12@fo77int3x
Expanding the prefactor in (14.60) in a power series yields all terms of the
perturbation expansion (10.29). They correspond to the Wick contractions in Section 10.3.1, with the associated Feynman diagrams. The free-field propagators are
the functional inverse of the operators between the fields in the quadratic part of
the Lagrangian. If this is written as
i
2
Z
dD x dD x′ φ(x)D(x, x′ )φ(x′ ),
(14.65) {12@fo82a}
G0 (x, x′ ) = iD −1 (x, x′ ).
(14.66) {12@fo82b}
then
This formal advantage of obtaining perturbation expansions from the functional integral representations has far-reaching consequences. We have seen in Chapter 11
that the evaluation of the perturbation series proceeds most conveniently by Wickrotating all energy integrations to make them run along the imaginary axes in the
complex energy plane. In this way one avoids the singularities in the propagators
936
14 Functional-Integral Representation of Quantum Field Theory
that would be encountered at the physical particle energies. In Section 10.7, on the
other hand, these singularities were shown to be responsible for the fact that particles
leave a scattering region and form asymptotic states. Hence, Wick-rotated perturbation expansions describe a theory which does not possess any particles states. In
fact, they cannot be described by field operators creating particle states in a Hilbert
space. Such states can be obtained from a simple modification of the above functional integral representation of the generating functional Z[j]. We simply perform
the x-space version of the Wick-rotation that was illustrated before on page 495 in
Fig. 7.2.
14.5
Euclidean Quantum Field Theory
{12@eqft
In Eq. (7.136), we replaced the coordinates xµ (µ = 0, 1, 2, 3) in D = 4 spacetime
dimensions by the euclidean coordinates xµE = (x1 , . . . , x3 , x4 = −ix0 ). The same
operation may be done to the time x0 in any dimensions. Under this replacement,
the action
A≡
Z
D
d x L(φ, ∂E φ) ≡
Z
D
d x
("
m2 2
1
(∂φ)2 −
φ + Lint (φ)
2
2
#
)
(14.67) {@}
goes over into i times the euclidean action
AE =
Z
dD xE
("
1
m2 2
(∂E φ)2 +
φ + Lint (φ) .
2
2
#
)
(14.68) {@}
The euclidean versions of the Gaussian integral formulas (14.40) and (14.89) are
Z
1
Dφ(x)e− 2
and
Z
∗
R
dD x dD x′ φ(x)M (x,x′ )φ(x′ )
−
Dφ (x)Dφ(x)e
R
1
= (Det M)∓ 2 ,
dD x dD x′ φ∗ (x)M (x,x′ )φ(x′ )
= (Det M)∓1 ,
(14.69) {12@fo64new
(14.70) {12@fogaussi
√
for complex fields φ = (φ1 + iφ2 )/ 2, where Dφ∗ (x)Dφ(x) ≡ Dφ1 (x)Dφ2 (x).
The amplitude (14.55) becomes
wE [φ] = R
The normalized version of this,
w[φ] = R
e−
R
dD xE LE (φ,∂φ)
Dφ(x)e−
e−
R
R
dD xE LE 0 (φ,∂φ)
dD xE LE (φ,∂φ)
Dφ(x)e−
R
dD xE LE (φ,∂φ)
,
.
(14.71) {12@fo79r}
(14.72) {12@fo79n}
represents the functional version of the proper quantum statistical Gibbs distribution
corresponding to (14.46) [recall (1.489)]:
e−En /kB T
wn = P −En /k T .
B
ne
(14.73) {12@foboltz}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
937
14.6 Functional Integral Representation for Fermions
The functional integral representation for the unnormalized generating functional
of all Wick-rotated Green functions corresponding to (14.58) is then
Z
ZE [j] =
Dφ(x)e−
R
dD xE [LE (φ,∂E φ)−j(x)φ(x)]
.
(14.74) {12@fo80eu}
The euclidean action corresponds to an energy of a field configuration. The
integrand plays the role of a Boltzmann factor and gives the relative probability for
this configuration to occur in a thermodynamic ensemble.
We now understand the advantage of working with the unnormalized functional
integral: At zero external source, ZE [j] corresponds precisely to the thermodynamic
partition function of the system. This will be seen explicitly in the examples in
Section 14.7.
14.6
Functional Integral Representation for Fermions
If we want to use the functional technique to also describe the statistical properties of fermions, some modifications are necessary. Then the fields must be taken
to be anticommuting c-numbers. In mathematics, such objects form a so-called
Grassmann algebra G. If ξ, ξ ′ are real elements of G, then
θθ′ = −θ′ θ.
(14.75) {@}
A trivial consequence of this condition is that the square of each Grassmann element
vanishes, i.e., θ2 = 0. If θ = θ1 + iθ2 is a complex element of G, then θ2 = −θ∗ θ =
−2iθ1 θ2 is nonzero, but (θ∗ θ)2 = (θθ)2 = 0.
All properties of operator quantum field theory for fermions can be derived from
functional integrals if we find an appropriate extension of the integral formulas
in the previous sections to Grassmann variables. Integrals are linear functionals.
For Grassmann variables, these are completely determined from the following basic
integration rules, which for real θ are
dθ
√ ≡ 0,
2π
dθ
dθ
√ θ ≡ 1,
√ θn ≡ 0, n > 1.
2π
2π
√
For complex variable θ = (θ1 + iθ2 )/ 2, these lead to
Z
Z
dθ
√ ≡ 0,
2π
Z
Z
dθ∗
√ θ ≡ 1,
2π
Z
Z
dθ
√ θ∗ ≡ 1,
2π
Z
(14.76) {[email protected]}
dθ dθ∗
√ √ (θ∗ θ)n ≡ −δn1 , (14.77) {[email protected]}
2π 2π
with the definition dθdθ∗ ≡ −idθ1 dθ2 .
Note that these integration rules make the linear operation of integration in
(14.76) coincide with the linear operation of differentiation. A function F (θ) of a
real Grassmann variable θ, is determined by only two parameters: the zeroth- and
the first-order Taylor coefficients. Indeed, due to the property θ2 = 0, the Taylor
938
14 Functional-Integral Representation of Quantum Field Theory
series has only two terms F (θ) = F0 + F ′ θ, where F0 = F (0) and F ′ ≡ dF (θ)/dθ.
But according to (14.76), also the integral gives F ′ :
dθ
√ F (θ) = F ′ .
2π
Z
(14.78) {@}
The coincidence of integration and differentiation has the important consequence
that any changes in integration variables will not transform with the Jacobian, but
rather with the inverse Jacobian:
Z
dθ
√ =a
2π
Z
d(aθ)
√ ,
2π
(14.79) {12@foinvjac
We shall use this transformation property below in Eq. (14.85).
As far as perturbation theory is concerned, it is sufficient to define only Gaussian
functional integrals such as (14.40) and (14.89). In the discretized form, we may
derive the formula
"
Y Z
n
∞
−∞
#
!
i 2D X
dθ
√ n
exp
ǫ
θm Mmn θn = det 1/2 (ǫD M).
D
2
2πiǫ
m,n
(14.80) {[email protected]}
The right-hand side is the inverse of the bosonic result (14.40). In addition, there
is an important difference: only the antisymmetric part of the functional matrix
contributes.
If the matrix Mmn is Hermitian, complex Grassmann variables are necessary to
produce a nonzero Gaussian integral. For complex variables we have
"
Y Z
n
X
dθn dθn∗
∗
√
√
exp iǫ2D
θm
Mmn θn = det (ǫD M),
D
D
2πiǫ 2πiǫ
m,n
#
!
(14.81) {[email protected]}
the right-hand side being again the inverse of the corresponding bosonic result
(14.89):
We first prove the latter formula. After bringing the matrix Mmn to a diagonal
form via a unitary transformation, we obtain the product of integrals
"
Y Z
n
X
dθn dθn∗
√
√
exp iǫ2D
θn∗ Mn θn .
D
D
2πiǫ 2πiǫ
n
#
!
(14.82) {[email protected]}
Expanding the exponentials into a power series leaves only the first two terms, since
(θn∗ θn )2 = 0, so that the integral reduces to
YZ
m
√
dθn dθn∗
√
(1 + iǫ2D θn∗ Mn θn ).
D
D
2πiǫ 2πiǫ
(14.83) {[email protected]}
Each of these integrals is performed via the formulas (14.77), and we obtain the
product of eigenvalues Mn , which is the determinant:
Y
m
Mn = det M.
(14.84) {[email protected]}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
939
14.6 Functional Integral Representation for Fermions
For real fermion fields, we observe that an arbitrary real antisymmetric matrix
Mmn can always be brought to a canonical form C, that is zero except for 2 × 2
matrices c = iσ 2 along the diagonal, by a real orthogonal transformation T . Thus
M = T T CT . The matrix C has a unit determinant so that det T = det 1/2 (M).
′
Let θm
≡ Tmn θn , then the measure of integration in (14.80) changes according to
(14.79) as follows:
Y
Y
dθn = det T
dθn′ .
(14.85) {[email protected]}
n
n
Applying now the formulas (14.76), the Grassmannian functional integral (14.80)
can be evaluated as follows:
"
Y Z
n


#
#
!
"
′
X
X
Y Z dθn
dθn
′
√
√
exp iǫ2D
exp
θm
Cmn θn′
θm Mmn θn = det T
D
2πiǫD
2πiǫ
m,n
n
k,l
= det 1/2 (iǫD M).
(14.86) {[email protected]}
The integrals over θn in one dimension decompose into a product of two-dimensional
Grassmannian integrals involving the antisymmetric unit matrix c = iσ2 . They have
the generic form
"
Y Z
n
#
′
dθ2n
dθ′
′
′
√
√ 2n+1 1 + iǫ2D θ2n
= ǫ2D .
θ2n+1
D
D
2πiǫ
2πiǫ
(14.87) {@}
There is one such factor for every second lattice site, which changes det T = det 1/2 M
into det 1/2 (ǫD M), thus proving (14.80). See [4]. .
In the continuum limit, the result of this discussion can be summarized in an
extension of the Gaussian functional integral formulas (14.40) and (14.26) to
Z
i
Dφ(x)e 2
R
dD x dD x′ φ(x)M (x,x′ )φ(x′ )
1
= Det ∓ 2 M,
(14.88) {12@fo64new
where M(x, x′ ) is real symmetric or antisymmetric for bosons or fermions, respectively, and
Z
Dφ∗ (x)Dφ(x)ei
R
dD x dD x′ φ∗ (x)M (x,x′ )φ(x′ )
= Det ∓1 M,
(14.89) {12@fogaussia
where the matrix is Hermitian.
The functional integral formulation of fermions follows now closely that of bosons.
For N relativistic real fermion fields χa , we can obtain an amplitude Z̃[χ] from the
Fourier transform
Z̃[χ] =
Z
Dj(x)Z[j]e−i
R
dD x ja (x)χa (x)
,
(14.90) {12@fo85}
and find
Z̃[χ] = hR
ei
R
dD x L(χ,∂χ)
Dχei
R
dD x L0 (χ,∂χ)
i.
(14.91) {12@fo86}
940
14 Functional-Integral Representation of Quantum Field Theory
The functional
Y Z
Z[j] = N
a
Dχa ei
R
dD x L(χ,∂χ)
R
dD x L0 (χ,∂χ)
(14.92) {12@fo86}
,
with a normalization factor
N
−1
=
Y Z
a
i
Dχa e
(14.93) {@}
,
provides us with a functional integral representation of the generating functional of
all fermionic Green functions.
An obvious extension of this holds for complex fermion fields. In the case of a
Dirac field, for example, where the sources j are commonly denoted by η, we obtain
Z[η, η̄] = N
Z
∗ i
DψDψ e
R
dD x [ψ̄(i ∂/ −m)ψ+Lint (ψ)+η̄ψ+ψ̄η]
(14.94) {12@fo89}
with
N
−1
=
Z
′
DψDψ ∗eiψ̄(i ∂/ −m)ψ = Det iG−1
∂ − m).
0 (x, x ) = Det (i/
(14.95) {12@fo90}
As in the boson case, we shall from now on work with the unnormalized functional
without the factor N ,
Z[η, η̄] =
Z
i
Dψe
R
dD x [ψ̄(i ∂/ −m)ψ+Lint (ψ)+η̄ψ+ψ̄η]
,
(14.96) {12@fo89u}
which again has the advantage that the euclidean version of Z[0, 0] becomes directly
the thermodynamic partition function of the system.
The functional representations of the generating functionals can of course be
continued to a euclidean form, as in Section 14.5, thereby replacing operator quantum physics by statistical physics. The corresponding Gaussian formulas for boson
and fermion fields are the obvious generalization of Eqs. (14.88) and (14.89):
Z
1
Dφ(x)e− 2
R
dD x dD x′ φ(x)M (x,x′ )φ(x′ )
1
= Det ∓ 2 M,
(14.97) {12@fo64new
and
Z
Dφ(x)Dφ∗(x)e−
R
dD x dD x′ φ∗ (x)M (x,x′ )φ(x′ )
= Det ∓1 M.
(14.98) {12@fogaussi
Here we have defined the measure of the euclidean functional integration in the same
way as before in Eqs. (14.88) and (14.89), except without the factors i under the
square roots. The euclidean version of the generating functional (14.94) can be used
to obtain all Wick-rotated Green functions from functional derivatives Z[η, η̄].
H. Kleinert, PARTICLES AND QUANTUM FIELDS
14.7 Relation Between Z [j ] and the Partition Function
941
As a side result of the above development we can state the following functional
integral formulas known under the name Hubbard-Stratonovich transformations:
Z
i
Dϕ(x, t)e 2
R
d3 xdtd3 x′ dt′ [ϕ(x,t)A(x,t;x′ ,t′ )ϕ(x′ ,t′ )+2j(x,t)ϕ(x,t)δ3 (x−x′ ,t)δ(t−t′ )]
=e
Z
n o
i(± 2i Trlog 1i A)− 2i
Dψ ∗ (x, t)Dψ(x, t)ei
R
{
d3 xdtd3 x′ dt′
= ei(±iTrlogA)−i
R
R
d3 xdtd3 x′ dt′ j(x,t)A−1 (x,t;x′ ,t′ )j(x′ ,t′ )
,
(14.99) {te-2.24a}
}
ψ∗ (x,t)A(x,t;x′ ,t′ )ψ(x,t′ )+[η∗ (x,t)ψ(x)δ3 (x−x′ )δ(t−t′ )+c.c.]
d3 xdtd3 x′ dtη∗ (x,t)A−1 (x,t;x′ ,t′ )η(x′ ,t′ )
(14.100) {te-2.24b}
.
These formulas will be needed repeatedly in the remainder of this text. They are
the basis for the reformulation of many interacting quantum field theories in terms
of collective quantum fields.
14.7
Relation Between Z[j] and the Partition Function
{12@exam
The introduction of the unnormalized functional integral representation (14.58) for
Z[j] was motivated by the fact that, in the euclidean version (14.74), Z[0] is equal
to the thermodynamic partition function of the system, except for a trivial overall
factor. Let us verify this for a free field theory in D = 1 dimension. Then ZE [0] of
Eq. (14.74) becomes
Zω =
Z
(
Dx exp −
Z
β
0
1
ω2
dτ ẋ2 (τ ) + x2 (τ )
2
2
"
#)
.
(14.101) {[email protected]}
For D = 1, the fields φ(τ ) may be interpreted as paths x(τ ) in imaginary time
τ = −it, and we have changed the notation accordingly. In the exponent, we
recognize the euclidean version
Z
AE =
β
0
ω2
1
dτ ẋ2 (τ ) + x2 (τ )
2
2
"
#
(14.102) {@}
of the action of the harmonic oscillator:
A=
Z
tb
ta
ω2
1
dt ẋ2 (t) − x2 (t) ,
2
2
#
"
(14.103) {@}
for tb − ta = −ih̄β = −ih̄/kB T . Thus Zω in expression (14.114) is a quantumstatistical path integral for a harmonic oscillator. The measure of path integration is
defined as a product of integrals on a lattice of points τn = nǫ with n = 0, . . . , N + 1
on the τ -axis:
Z
Dx(τ ) =
N Z
Y
n=0
dx
√ n ,
2πǫ
(14.104) {12@fo54phix
942
14 Functional-Integral Representation of Quantum Field Theory
where xn ≡ x(τn ) and N + 1 = h̄β/ǫ. For a finite τ -interval h̄β, the paths have to
satisfy periodic boundary conditions
(14.105) {12@fuxperb
x(h̄β) = x(0),
as a reflection of the quantum-mechanical trace. On the τ -lattice, this implies
xN +1 = x0 , and the action becomes
+1
(xn − xn−1 )2
1 NX
+ ω 2 x2n .
= ǫ
2 n=1
ǫ2
"
AN
E
#
(14.106) {[email protected]}
This can be rewritten as
AN
E =
+1
1 NX
xn (−ǫ2 ∇∇ + ǫ2 ω 2 )xn ,
2ǫ n=1
(14.107) {[email protected]}
where ∇∇x denotes the lattice version of ẍ. It may be represented as an (N + 1) ×
(N + 1)-matrix

−ǫ2 ∇∇ =








2 −1 0
−1 2 −1
..
.
0
−1
0
0
0
...
...
0
0
0
0
−1
0
..
.
0 . . . −1 2 −1
. . . 0 −1 2





.



(14.108) {[email protected]}
The Gaussian functional integral can now be evaluated using formula (14.40), and
we obtain
Zω = detN +1 [−ǫ2 ∇∇ + ǫω 2 ]−1/2 .
(14.109) {[email protected]}
The determinant is calculated recursively,2 and yields
1
,
2 sinh(h̄ω̃β/2)
(14.110) {[email protected]}
2
ωǫ
ω̃ ≡ arsinh .
ǫ
2
(14.111) {[email protected]}
Zω =
where ω̃ is the auxiliary frequency
In the continuum limit ǫ → 0, the frequency ω̃ goes against ω, and Zω becomes
Zω =
1
.
2 sinh(h̄ωβ/2)
(14.112) {[email protected]}
This can be expanded as
Zω = e−h̄ω/2kB T + e−3h̄ω/2kB T + e−5h̄ω/2kB T + . . . ,
2
(14.113) {[email protected]
See the textbook [1] and Section 2.12 of the textbook [2].
H. Kleinert, PARTICLES AND QUANTUM FIELDS
14.7 Relation Between Z [j ] and the Partition Function
943
which is the quantum statistical partition function of the harmonic oscillator, as we
wanted to prove. The ground state has a nonzero energy, as observed in the operator
discussion in Chapter 7.32.
For the quantum-mechanical version of the functional integral (14.114)
Zω =
Z
( Z
Dx exp i
1
ω2
dt ẋ2 (t) − x2 (t)
2
2
"
tb
ta
#)
(14.114) {[email protected]}
,
we obtain, with the measure of functional integration analog to (14.104)
Z
Dx(t) =
N Z
Y
dx
√ n ,
2πiǫ
n=0
(14.115) {12@fo54phix
and the use of the Gaussian integral formula (14.40), the result
Zω = detN +1 [−ǫ2 ∇∇ − ǫω 2 ]−1/2 .
(14.116) {[email protected]
Thus we only have to replace ω → iω in (14.111)–(14.112). In the continuum limit,
we therefore obtain
1
.
2 sin(h̄ωβ/2)
Zω =
(14.117) {[email protected]
We end this section by mentioning that the path-integral representation of the
partition function (14.114) with the integration measure (14.104) can be obtained
from a euclidean phase space path integral3
Zω =
Z Z
Dp(τ )
exp
Dx(τ )
2πh̄
(Z
β
0
1
ω2
dτ ipẋ − p2 − x2
2
2
"
#)
(14.118) {[email protected]
by going to a τ -lattice and√integrating out the momentum variables. The momentum
integrals give the factors 2πǫ in the denominators of the measure (14.104). In the
quantum-mechanical version of (14.118)
Zω =
Z Z
Z tb
Dp(t)
ω2
1
Dx(t)
exp i
dt pẋ − p2 − x2
2πh̄
2
2
ta
(
"
the p-integrals produce the denominators containing the factors
(14.104).
The measure of functional integration on the τ -lattice
N Z
Y
n=0
∞
−∞
dxn
"
NY
+1 Z
n=1
∞
−∞
dpn
2πh̄
#)
√
,
(14.119) {[email protected]
i in the measure
#
(14.120) {@}
is the obvious generalization of the classical statistical weight in phase space
Z
∞
−∞
3
dx
Z
∞
−∞
dp
2πh̄
For a detailed discussion of the measure see Chapters 2 and 7 in the textbook [2].
(14.121) {@}
944
14 Functional-Integral Representation of Quantum Field Theory
to fluctuating paths with many variables xn = x(τn ).
For completeness, we write down the free energy Fω = −β log Zω associated with
the partition function (14.112):
Fω =
h̄ω
1
1
log[2 sinh(h̄ωβ/2)] =
+ log(1 − e−βh̄ω ).
β
2
β
(14.122) {12@freeenb}
At zero temperatures, only the ground-state oscillations contribute.
The detailed time-sliced calculation of the partition functions has to be compared
with the formal evaluation of the partition function (14.114) according to formula
(14.69), which would yield
Zω =
Z
(
Dx(τ ) exp −
Z
β
0
1
ω2
dτ ẋ2 (τ ) + x2 (τ )
2
2
"
#)
= Det −1/2 (−∂τ2 + ω 2 ).
(14.123) {[email protected]
In this continuum formulation, the right-hand side is at first meaningless. It differs
from the results obtained by proper time-sliced calculations (14.109)–(14.112) by a
temperature-dependent infinite overall factor. In order to see what this factor is, we
note that for a periodic boundary condition (14.105), the eigenvectors of the matrix
2
+ ω 2 . Since ωm grows with m like m2 ,
−∂τ2 + ω 2 are e−iωm τ with eigenvalues ωm
−1/2
the functional determinant Det
(−∂τ2 + ω 2 ) is strongly divergent. Indeed, the
bosonic lattice result (14.112) can only be obtained in the limit ǫ → 0 after dividing
out of the the lattice result (14.109) an equally divergent ω-independent functional
determinant and calculating the ratio
Det −1/2 (−∂τ2 + ω 2 )
detN +1 [−ǫ2 ∇∇ − ǫω 2 ]−1/2
−
−
−→
.
h̄βdet′N +1 [−ǫ2 ∇∇ − ǫω 2 ]−1/2 ǫ→0 h̄βDet ′ −1/2 (−∂τ2 )
(14.124) {12@furatiox
The prime in the denominator indicates that the zero-frequency ω0 must be omitted
to obtain a finite result. The factor 1/h̄β is the regularized contribution of the zero
frequency. This follows from a simple integral consideration. An integral
Z
b/2
−b/2
dx
2 2
√ e−ω x /2
2π
(14.125) {@}
√
gives 1/ω only for finite ω(≫ 1/b). In the limit ω → 0, it gives b/ 2π. In the path
integral, the zero mode is associated with the fluctuations of the average of the path
x(τ ). To indicate this origin of the factor h̄β in the ratio (14.124), we may write
(14.124) as
Det −1/2 (−∂τ2 + ω 2 )
,
(14.126) {12@furatioy
−1/2
Det R (−∂τ2 )
where the subscript R indicates the above regularization.
Explicitly, the ratio (14.124) is calculated from the product of eigenvalues:
2
Det −1/2 (−∂τ2 + ω 2 )
ωm
1 Y
.
=
2 + ω2
h̄βDet ′ −1/2 (−∂τ2 )
h̄βω m>0 ωm
(14.127) {12@furatiox
H. Kleinert, PARTICLES AND QUANTUM FIELDS
14.7 Relation Between Z [j ] and the Partition Function
945
The product can be found in the tables4 :
2
h̄βω/2
ωm
=
,
2
2
sinh(h̄βω/2)
m6=0 ωm + ω
Y
(14.128) {@}
so that (14.127) is equal to 1/2 sinh(h̄βω/2), and the properly renormalized partition
function (14.123) yields the lattice result (14.112). In a lattice calculation, the
−1/2
determinant Det R (−∂τ2 ) for β → ∞ is equal to unity.
It is curious to see that a formal evaluation of the functional determinant via the
analytic regularization procedure of Sections 7.12 and 11.5 is indifferent to this denominator. It produces precisely the same result from the formal expression (14.123)
as from the proper lattice calculation. Recalling formula (11.134) and setting D = 1,
we find
1
Det −1/2 (−∂τ2 + ω 2 ) = exp − Tr log(−∂τ2 + ω 2 )
2
)
(
Z
h̄ω
dω ′
1
′2
2
log(ω + ω ) = e−β 2 . (14.129) {12@fuzeroot
= exp − βh̄
2
2π
The exponent gives precisely the free energy at zero temperatures in Eq. (14.122).
We now admit finite temperatures. For this purpose, we have to replace the
integral over ω ′ by a sum over Matsubara frequencies (2.415), and evaluate
Z
Z
1 X
1 X
dω ′
dωm
2
2
log(ωm
+ ω 2 ).
log(ω ′2 + ω 2) +
log(ωm
+ ω2) =
−
h̄β m
2π
h̄β m
2π
(14.130) {@}
!
In contrast to the first term whose evaluation required the analytic regularization,
the second term is finite. It can be rewritten as
!Z
Z
∞
1
1 X
dωm
dω ′2 2
−
.
(14.131) {12@frewrite}
−
2
h̄β m
2π
ωm + ω ′2
ω
Recalling the summation formula (2.424), this becomes
−
Z
∞
ω2
dω ′2
2ω ′
"(
coth(h̄βω ′/2)
tanh(h̄βω ′/2)
)
−1
#
= ±
Z
∞
ω2
dω ′2
2
′
βh̄ω
2ω e ′ ∓ 1
= kB T 2 log(1 ∓ e−βh̄ω ).
(14.132) {12@dummye
For later applications, we have treated also the case of fermionic Matsubara frequencies (the lower case). Together with the zero-temperature result (14.129), we find
at any temperature
1
Zω = e−βFω = Det −1/2 (−∂τ2 + ω 2 ) = exp − Tr log(−∂τ2 + ω 2)
2
#)
( "
h̄ω
+ log(1 − e−βh̄ω ) ,
(14.133) {12@funzerot
= exp − β
2
in agreement with (14.122).
4
I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 1.431.2: sinh x =
Q∞
m=1 (1
+ x2 /m2 π 2 ).
946
14.8
14 Functional-Integral Representation of Quantum Field Theory
Bosons and Fermions in a Single State
The discussion in the last section cannot be taken over to fermionic variables x(τ ).
For fermions the action (14.114) vanishes identically, as a consequence of the symmetry of the functional matrix D(τ, τ ′ ) = (−∂τ2 + ω 2 )δ(τ − τ ′ ). A fermionic version
of the above path integral can only be introduced within the canonical formulation
(14.119) of the harmonic path integral. With the help of a canonical transformation
a† =
q
q
(14.134) {[email protected]
dτ (a∗ i∂t a − ωa∗ a).
(14.135) {[email protected]}
1/2h̄ω(ωx − ip), a =
this may be transformed into the action
Aω QM =
Z
tb
ta
1/2h̄ω(ωx + ip),
A canonical quantization with commutation and anticommutation rules
[â(t), ↠(t)]∓ = 1,
[↠(t), ↠(t)]∓ = 0,
[â(t), â(t)]∓ = 0
(14.136) {[email protected]}
produces a second-quantized Hilbert space of the type discussed in Chapter 2. Since
a† (t) and a(t) carry no space variables. They describe Bose and Fermi particles at
a single point.
The path-integral representation of the quantum-mechanical partition function
of this system is
Z
Da(t)Da∗ (t) iAω QM
e
.
(14.137) {[email protected]}
Zω QM ≡
2π
The measure of integration is
Da∗ Da
≡
2π
D a1 D a2
,
(14.138) {[email protected]}
2π
−∞ −∞
√
√
where a = (a1 + ia2 )/ 2 and a† = (a1 − ia2 )/ 2 are directly obtained from the
canonical measure of path integration in (14.118).
At a euclidean time τ = −it, the action of the free nonrelativistic field becomes
Z Z
Aω =
Z
h̄β
0
Z
∞
Z
∞
dτ (a∗ ∂τ a + ωa∗ a),
(14.139) {[email protected]}
and the thermodynamic partition function has the path-integral representation
Zω ≡
Z
Da(τ )Da∗ (τ ) −Aω
e
.
2π
(14.140) {[email protected]}
In this formulation, there is no problem of treating both bosons or fermions at
the same time. We simply have to assume the fields a(τ ), a∗ (τ ) to be periodic or
antiperiodic, respectively, in the imaginary-time interval h̄β:
a(h̄β) = ±a(0),
(14.141) {[email protected]}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
947
14.9 Free Energy of Free Scalar Fields
or in the sliced form
aN +1 = ±a0 .
(14.142) {[email protected]}
Using formula (14.98), the partition function can be written as
Zω =
Z
Da∗ Da
exp −
2π
"
Z
h̄β
0
∗
∗
dτ (a ∂τ a + ωa a)
#
= Det ∓1/2 (∂τ + ω).
(14.143) {[email protected]}
Contact with the previous oscillator calculation is established by observing that
in the determinant, the operator ∂τ + ω can be replaced by the conjugate operator
−∂τ +ω, since all eigenvalues come in complex-conjugate pairs, except for the m = 0 value, which is real. Hence the determinant of ∂τ + ω can be substituted everywhere
by
q
(14.144) {[email protected]}
det (∂τ + ω) = det (−∂τ + ω) = det (−∂τ2 + ω 2 ).
In the boson case, we thus reobtain the result (14.123). In both cases, we may
therefore write
Zω = Det ∓1/2 (−∂τ2 + ω 2 ).
(14.145) {@}
The right-hand side was evaluated for bosons in Eqs. (14.129)–(14.133). In anticipation of the present result, we have calculated the Matsubara sums up to Eq. (14.132)
for bosons and fermions, with the results
1
Zω = e−βFω = Det ∓1/2 (−∂τ2 + ω 2 ) = exp ∓ Tr log(−∂τ2 + ω 2)
2
( "
#)
h̄ω
= exp ∓ β
+ log(1 ∓ e−βh̄ω ) .
(14.146) {12@funzerot
2
This can be written as
Zω =
(
[sinh(h̄βω/2)]−1
cosh(h̄βω/2)
)
(
for
bosons,
fermions.
)
.
(14.147) {[email protected]
For bosons, the physical interpretation of this expression was given after
Eq. (14.110). The analog sum of Boltzmann factors for fermions is
Zω = eh̄ω/2kB T + e−h̄ω/2kB T .
(14.148) {[email protected]}
This shows that the fermionic system at a point has two states, one with no particle
and one with a single particle, where the no-particle state has a negative vacuum
energy, as observed in the operator discussion in Chapter 2.
14.9
Free Energy of Free Scalar Fields
The results of the last section are easily applied to fluctuating scalar fields. Consider
the free-field partition function
Z0 =
Z
Dφ(x)e−
R
dD x
1
2
[(∂φ)2 +m2 φ2 ] ,
{12@fescft}
(14.149) {12@fo80xx}
948
14 Functional-Integral Representation of Quantum Field Theory
which is of the general form (14.74) for j = 0. If the fields are decomposed into their
spatial Fourier components in a finite box of volume V ,
Z
i
1 X h ikx
φ(x) = √
e φk (τ ) + c.c. ,
2V k
(14.150) {12@fumoms
the partition function becomes
Z0 =
YZ
k
−
Dφk (τ )e
R h̄β
0
dt
1
2
{|φ̇k (τ )|2 +k2 |φk (τ )|2 +m2 |φk (τ )|2 } .
(14.151) {12@fo80xxy
For each k, the functional integral is obviously the same as in (14.114), so that
Z0 =
Y
(14.152) {@}
Zω(k) ,
k
where ω(k) ≡
√
k2 + m2 , and by (14.112)
Zω(k) =
1
.
2 sinh[h̄ω(k)β/2]
(14.153) {[email protected]
For a real field φ(x), the anticommuting alternative cannot be accommodated
into the functional integral (14.149). We must first go to the field-theoretic analog
of the path integral in phase space
Z0 =
Z Z
× exp
Dφ(x)
Dπ(x)
2πh̄
1
1
d4 x iπ(x)∂µ φ(x) − π 2 (x) − [(∇φ(x))2 + m2 φ2 (x)]
2
2
Z
, (14.154) {[email protected]
where π(x) are the canonical field momenta (7.1). After the Fourier decomposition
(14.150) and a similar one for π(x), we perform again a canonical transformation
corresponding to (14.134),
a†k (τ ) = √
1
[ω(k)φk (τ )−iπk (τ )],
2h̄ω
ak (τ ) = √
1
[ω(k)φk (τ )+iπk (τ )], (14.155) {[email protected]
2h̄ω
and arrive at the analog of (14.143):
Zω =
YZ
k
=
Y
Z h̄β
Da∗k Dak
dτ (a∗k ∂τ ak + ωa∗k ak )
exp −
2π
0
"
#
Det ∓1/2 (∂τ + ω(k)).
(14.156) {[email protected]}
k
This can be evaluated as in (14.147) to yield
Zω =
Y
k
(
{2 sinh[h̄βω(k)/2]}−1
2 cosh[h̄βω(k)/2]
)
for
(
bosons,
fermions.
)
.
(14.157) {[email protected]
H. Kleinert, PARTICLES AND QUANTUM FIELDS
949
14.10 Interacting Nonrelativistic Fields
The associated free energies are
F0 = −β log Z0 = ±
X
k
(
)
i
h
ω(k)
+ log 1 ∓ e−βh̄ω(k) .
2
(14.158) {12@freenscfn
For a complex field, the canonical transformation is superfluous. The field in the
partition function
Z
Z0 =
Dφ(x)Dφ∗(x)e−
R
dD x
1
2
(∂φ∗ ∂φ+m2 φ∗ φ)
(14.159) {12@fo80xxco
can directly be assumed to be of the bosonic or of the fermionic type. A direct
application of the Gaussian formula (14.98) leads to
Zω =
Y
Det [−∂τ2 + ω 2 (k)]∓1
k
=
Y
k
(
{2 sinh[h̄βω(k)/2]}−2
{2 cosh[h̄βω(k)/2]}2
)
for
(
bosons,
fermions.
)
,
(14.160) {[email protected]
with free energies twice as large as (14.158).
14.10
Interacting Nonrelativistic Fields
The quantization of nonrelativistic particles was amply discussed in Chapter 2 and
applied to many-body Bose and Fermi systems in Chapter 3. Her we shall demonstrate that a completely equivalent formulation of the second-quantized nonrelativistic field theory is possible with the help of functional integrals.
Consider a many-fermion system described by an action
A ≡ A0 + Aint =
Z
−
d3 xdtψ ∗ (x, t) [i∂t − ǫ(−i∇)] ψ(x, t)
1
2
Z
(14.161) {te-2.1}
d3 xdtd3 x′ dt′ ψ ∗ (x′ , t′ )ψ ∗ (x, t)V (x, t; x′ t′ )ψ(x, t)ψ(x′ , t′ )
with a translationally invariant two-body potential
V (x, t; x′ , t′ ) = V (x − x′ , t − t′ ).
(14.162) {te-2.2}
In the systems to be treated in this text we shall be concerned with a potential that
is, in addition, instantaneous in time
V (x, t; x′ , t′ ) = δ(t − t′ )V (x − x′ ).
(14.163) {te-2.2’}
This property will greatly simplify the discussion.
The fundamental field ψ(x) may describe bosons or fermions. The complete
information on the physical properties of the system resides in the Green functions.
950
14 Functional-Integral Representation of Quantum Field Theory
In the operator Heisenberg picture, these are given by the expectation values of the
time-ordered products of the field operators
G (x1 , t1 , . . . , xn , tn ; xn′ , tn′ , . . . , x1′ , t1′ )
= h0|T̂
†
†
(x1′ , t1′ )
ψ̂H (x1 , t1 ) · · · ψ̂H (xn , tn )ψ̂H
(xn′ , tn′ ) · · · ψ̂H
|0i.
(14.164) {te-2.3}
The time-ordering operator T̂ changes the position of the operators behind it in
such a way that earlier times stand to the right of later times. To achieve the final
ordering, a number of field transmutations are necessary. If F denotes the number
of transmutations of Fermi fields, the final product receives a sign factor (−1)F .
It is convenient to view all Green functions (14.164) as derivatives of the generating functional
Z
Z[η ∗ , η] = h0|T̂ exp i
h
†
d3 xdt ψ̂H
(x, t)η(x, t) + η ∗ (x, t)ψ̂H (x, t)
i
|0i, (14.165) {te-2.4}
namely
(14.166) {te-2.5}
G (x1 , t1 , . . . , xn , tn ; xn′ , tn′ , . . . , x1′ , t1′ )
n+n′
∗
δ
Z[η , η]
′
= (−i)n+n ∗
.
∗
δη (x1 , t1 ) · · · δη (xn , tn )δη(xn′ , tn′ ) · · · δη(x1′ , tn′ ) η=η∗ ≡0
Physically, the generating functional (14.165) describes the probability amplitude
for the vacuum to remain a vacuum in the presence of external sources η ∗ (x, t) and
η ∗ (x, t).
The calculation of these Green functionals is usually performed in the interaction
picture which can be summarized by the operator expression for Z:
Z[η ∗ , η] = Nh0|T exp iAint [ψ † , ψ] + i
Z
h
d3 xdt ψ † (x, t)η(x, t) + h.c.
i
|0i. (14.167) {te-2.6}
In the interaction picture, the fields ψ(x, t) possess free-field propagators and the
normalization constant N is determined by the condition [which is trivially true for
(14.165)]:
Z[0, 0] = 1.
(14.168) {te-2.7}
The standard perturbation theory is obtained by expanding exp{iAint } in (14.167)
in a power series and bringing the resulting expression to normal order via Wick’s
expansion technique. The perturbation expansion of (14.167) often serves conveniently to define an interacting theory. Every term can be pictured graphically and
has a physical interpretation as a virtual process.
Unfortunately, the perturbation series up to a certain order in the coupling constant is unable to describe several important physical phenomena. Examples are the
formation of bound states in the vacuum, or the existence and properties of collective
excitations in many-body systems. Those require the summation of infinite subsets
H. Kleinert, PARTICLES AND QUANTUM FIELDS
951
14.10 Interacting Nonrelativistic Fields
of diagrams to all orders. In many situations it is well-known which subsets have to
be taken in order to account approximately for specific effects. What is not so clear
is how such lowest approximations can be improved in a systematic manner. The
point is that, as soon as a selective summation is performed, the original coupling
constant has lost its meaning as an organizer of the expansion and there is need for
a new systematics of diagrams. This will be presented in the sequel.
As soon as bound states or collective excitations are formed, it is very suggestive
to use them as new quantum fields rather than the original fundamental particles ψ.
The goal would then be to rewrite the expression (14.167) for Z[η ∗ , η] in terms of
new fields whose unperturbed propagator has the free energy spectrum of the bound
states or collective excitations and whose Aint describes their mutual interactions.
In the operator form (14.167), however, such changes of fields are hard to conceive.
14.10.1
Functional Formulation
In the functional integral approach, the generating functional (14.165) is given by
[5]:
∗
Z[η , η] = N
Z
Dψ ∗ (x, t)Dψ(x, t)
∗
× exp iA[ψ , ψ] + i
Z
3
∗
d xdt [ψ (x, t)η(x, t) + c.c.] . (14.169) {te-2.8}
It is worth emphasizing that the field ψ(x, t) in the path-integral formulation is a
complex number and not an operator. All quantum effects are accounted for by
fluctuations; the path integral includes not only the classical field configurations but
also all classically forbidden ones, i.e., all those which do not run through the valley
of extremal action in the exponent.
By analogy with the development in Section 14.4 we take the interactions outside
the integral and write the functional integral (14.169) as
(
Z[η ∗ , η] = exp iAint
"
1 δ 1 δ
,
i δη i δη ∗
#)
Z0 [η ∗ , η],
(14.170) {te-2.25}
where Z0 is the generating functional of the free-field correlation functions, whose
functional integral looks like (14.169), but with the action being only the free-particle
expression
A0 [ψ ∗ , ψ] =
Z
dxdt ψ ∗ (x, t) [i∂t − ǫ(−i∇)] ψ(x, t),
(14.171) {te-2.26}
rather than the full A[ψ ∗ , ψ]= A0 [ψ ∗ , ψ]+Aint [ψ ∗ , ψ] of Eq. (14.161) in the exponent.
The functional integral is of the Gaussian type (14.89) with a matrix
A(x, t; x′ , t′ ) = [i∂t − ǫ(−i∇)] δ (3) (x − x′ )δ(t − t′ ).
(14.172) {te-2.27}
This matrix is the inverse of the free propagator
′ ′
A(x, t; x′ , t′ ) = iG−1
0 (x, t; x , t )
(14.173) {te-2.28}
952
14 Functional-Integral Representation of Quantum Field Theory
where
G0 (x, t; x′ , t′ ) =
dE Z d3 p −i[E(t−t′ )−p(x−x′ )]
i
e
.
4
2π
(2π)
E − ǫ(p) + iη
Z
(14.174) {te-2.29}
Inserting this into (14.100), we see that
Z0 [η ∗ , η] = N exp i ±iTr log iG−1
−
0
Z
d3 xdtd3 x′ dt′ η ∗ (x, t)G0 (x′ , t′ )η(x′ , t′ ) .
(14.175) {te-2.30a}
We now fix N in accordance with the normalization (14.168) to
(14.176) {te-}
N = exp [i (±iTr log iG0 )]
and arrive at
Z0 [η ∗ , η] = exp −
Z
d3 xdtd3 x′ dt′ η ∗ (x, t)G0 (x, t; x′ , t′ )η(x′ , t′ ) . (14.177) {te-2.30}
This coincides exactly with what would have been obtained from the operator expression (14.167) for Z0 [η ∗ , η] (i.e., without Aint ).
Indeed, according to Wick’s theorem [2,5,6], any time ordered product can be
expanded as a sum of normal products with all possible contractions taken via
Feynman propagators. The formula for an arbitrary functional of free fields ψ, ψ ∗ is
Z
T F [ψ ∗ , ψ] = exp d3 xdtd3 x′ dt′
δ
δ
G0 (x, t; x′ , t′ ) ∗
: F [ψ ∗ , ψ] : . (14.178) {te-2.31}
δψ(x, t)
δψ (x, t′ )
Applying this to
Z
∗
h0|T F [ψ , ψ]|0i = h0|T exp i
∗
∗
dxdt(ψ η + η ψ) |0i
(14.179) {te-}
one finds:
Z0 [η ∗ , η] = exp −
Z
dxdtdx′ dt′ η ∗ (x, t)G0 (x, t; x′ , t′ )η(x′ , t′ )
Z
× h0| : exp i
dxdt(ψ ∗ η + η ∗ ψ) : |0i.
(14.180) {te-2.32}
The second factor is equal to unity, thus proving the equality of this operatorially
defined Z0 [η ∗ , η] with the path-integral expression (14.177). Because of (14.170),
this equality holds also for the interacting geerating functional Z[η ∗ , η].
14.10.2
Grand-Canonical Ensembles at Zero Temperature
All these results are easily generalized from vacuum expectation values to thermodynamic averages at fixed temperatures T and chemical potential µ. The change
H. Kleinert, PARTICLES AND QUANTUM FIELDS
953
14.10 Interacting Nonrelativistic Fields
at T = 0 is trivial: The single particle energies in the action (14.161) have to be
replaced by
ξ(−i∇) = ǫ(−i∇) − µ
(14.181) {te-2.33}
and new boundary conditions have to be imposed upon all Green functions via an
appropriate iǫ prescription in G0 (x, t; x′ , t′ ) of (14.174) [see [2,7]]:
T =0
G0 (x, t; x′ , t′ ) =
Z
i
dEd3 p −iE(t−t′ )+ip(x−x′ )
e
. (14.182) {te-2.34}
(2π)4
E − ξ(p) + iη sgn ξ(p)
Note that, as a consequence of the chemical potential, fermions with ξ < 0 inside
the Fermi sea propagate backwards in time. Bosons, on the other hand, have in
general ξ > 0 and, hence, always propagate forward in time.
In order to simplify the notation we shall often use four-vectors p = (p0 , p) and
write the measure of integration in (14.182) as
dEd3 p
=
(2π)4
Z
Z
d4 p
.
(2π)4
(14.183) {te-}
Note that in a solid, the momentum integration is really restricted to a Brillouin
zone. If the solid has a finite volume V , the integral over spacial momenta becomes
a sum over momentum vectors,
Z
1 X
d3 p
=
,
(2π)3
V p
(14.184) {te-finitev}
and the Green function (14.182) reads
T =0
G0 (x, t; x′ , t′ ) ≡
Z
dE 1 X −ip(x−x′ )
i
.
e
0
2π V p
p − ξ(p) + iη sgn ξ(p)
(14.185) {te-}
The resulting power series expansions for the generating functional at zerotemperature T =0 Z[η ∗ , η] and nonzero coupling can be written down as before after
performing a Wick rotation in the complex energy plane in all energy integrals occurring in the expansions of formulas (14.180) and (14.167) in powers of the sources
η(x, t) and η ∗ (x, t). For this, one sets E = p0 ≡ iω and replaces
Z
∞
−∞
dE
→i
2π
Z
∞
−∞
dω
.
2π
(14.186) {te-2.35}
Then the Green function (14.182) becomes
T =0
G0 (x, t; x′ , t′ ) = −
Z
dω d3 p ω(t−t′ )+ip(x−x′ )
1
e
.
2π (2π)3
iω − ξ(p)
(14.187) {te-2.36}
Note that with formulas (14.170) and (14.177) the generating functional T =0 Z[η ∗ , η]
is the grand-canonical partition function in the presence of sources [7].
954
14 Functional-Integral Representation of Quantum Field Theory
Finally, we have to introduce arbitrary temperatures T . According to the standard rules of quantum field theory (for an elementary introduction see Chapter 2 in
Ref. [2]), we must continue all times to imaginary values t = iτ , restrict the imaginary time interval to the inverse temperature5 β ≡ 1/T , and impose periodic or
antiperiodic boundary conditions upon the fields ψ(x, −iτ ) of bosons and fermions,
respectively [2,7]:
ψ(x, −iτ ) = ±ψ(x, −i(τ + 1/T )).
(14.188) {te-}
When there is no danger of confusion, we shall drop the factor −i in front of the
imaginary time in field arguments, for brevity. The same thing will be done with
Green functions.
By virtue of (14.177), these boundary conditions wind up in all free Green functions, i.e., they have the property
T
T
G0 (x, τ + 1/T ; x′ , τ ′ ) ≡ ± G0 (x, −iτ ; x′ , −iτ ′ ).
(14.189) {te-2.38}
This property is enforced automatically by replacing the energy integrations
−∞ dω/2π in (14.187) by a summation over the discrete Matsubara frequencies [by
analogy with the momentum sum (14.184), the temporal “volume” being β = 1/T ]
R∞
Z
∞
−∞
X
dω
,
→T
2π
ωn
(14.190) {te-2.39}
which are even or odd multiples of πT
ωn =
(
2n
2n + 1
)
πT for
(
bosons
fermions
)
.
(14.191) {te-2.40}
The prefactor T of the sum over the discrete Matsubara frequencies accounts for the
density of these frequencies yielding the correct T → 0-limit.
Thus we obtain the following expression for the imaginary-time Green function
of a free nonrelativistic field at finite temperature (the so-called free thermal Green
function)
T
G0 (x, τ, x′ , τ ′ ) = − T
XZ
ωn
d3 p −iωn (τ −τ ′ )+ip(x−x′ )
1
e
.
3
(2π)
iωn − ξ(p)
(14.192) {te-2.41}
Incorporating the Wick rotation in the sum notation we may write
T
X
p0
= −iT
X
ωn
= −iT
X
,
p4
(14.193) {te-}
where p4 = −ip0 = ω. If temperature and volume are both finite, the Green function
is written as
T
G0 (x, τ, x′ , τ ′ ) = −
5
1
T X X −iωn (τ −τ ′ )+ip(x−x′ )
.
e
V p0 p
iωn − ξ(p)
(14.194) {te-2.41x}
We use natural units in this chapter, so that kB = 1, h̄ = 1.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
955
14.10 Interacting Nonrelativistic Fields
At equal space points and equal imaginary times, the sum can easily be evaluated.
One must, however, specify the order in which τ → τ ′ . Let η denote an infinitesimal
positive number and consider the case τ ′ = τ + η, where the Green function is
T
G0 (x, τ, x, τ + η) = − T
d3 p iωn η
1
e
.
3
(2π)
iωn − ξ(p)
XZ
ωn
(14.195) {te-2.41cc}
Then the sum is found after converting it into a contour integral
T
X
iωn η
e
ωn
T
1
=
iωn − ξ(p)
2πi
Z
C
dz
1
eηz
.
z/T
e ∓1z−ξ
(14.196) {te-3.18a}
The upper sign holds for bosons, the lower for fermions. The contour of integration
C encircles the imaginary z-axis in a positive sense, thereby enclosing all integer or
half-integer valued poles of the integrand at the Matsubara frequencies z = iωm (see
Fig. 2.8). The factor eηz ensures that the contour in the left half-plane does not
contribute.
By deforming the contour C into C ′ and contracting C ′ to zero we pick up the
pole at z = ξ and find
T
X
eiωn η
ωn
1
1
1
= ∓ ξ(p)/T
= ∓ ξ(p)/T
= ∓n(ξ(p)).
iωn − ξ(p)
e
∓1
e
∓1
(14.197) {te-3.18}
The phase eηz ensures that the contour in the left half-plane does not contribute.
The function on the right is known as the Bose or Fermi distribution function.
By subtracting from (14.197) the sum with ξ replaced by −ξ, we obtain the
important sum formula
T
X
ωn
ωn2
1
1
ξ(p)
=
coth±1
.
2
+ ξ (p)
2ξ(p)
T
(14.198) {te-3.18b}
In the opposite limit with τ ′ = τ − η, the phase factor in the sum would be
leading to a contour integral
−iωm η
e
−kB T
X
ωm
eiωm η
1
kB T
=±
iωm − ξ(p)
2πi
Z
C
dz
e−ηz
e−z/kB T
1
,
∓1z −ξ
(14.199) {te-3.18an}
and we would find 1 ± nξ(p) .
In the operator language, these limits correspond to the expectation values of
free non-relativistic field operators
T
†
†
G0 (x, τ ; x, τ + η) = h0|T̂ ψ̂H (x, τ )ψ̂H
(x, τ + η) |0i = ±h0|ψ̂H
(x, τ )ψ̂H (x, τ )|0i
T
†
†
G0 (x, τ ; x, τ − η) = h0|T̂ ψ̂H (x, τ )ψ̂H
(x, τ − η) |0i = h0|ψ̂H (x, τ )ψ̂H
(x, τ )|0i
†
= 1 ± h0|ψ̂H
(x, τ )ψ̂H (x, τ ∓ η)|0i .
The function n(ξ(p)) is the thermal expectation value of the number operator
956
14 Functional-Integral Representation of Quantum Field Theory
†
N̂ = ψ̂H
(x, τ )ψ̂H (x, τ ).
(14.200) {te-}
In the case of T 6= 0 ensembles, it is also useful to employ a four-vector notation.
The four-vector
pE ≡ (p4 , p) = (ω, p)
(14.201) {te-2.42p}
is called the euclidean four-momentum. Correspondingly, we define the euclidean
spacetime coordinate
xE ≡ (−τ, x).
(14.202) {te-2.42}
The exponential in (14.192) can be written as
pE xE = −ωτ + px.
(14.203) {te-}
Collecting integral and sum in a single four-dimensional summation symbol, we shall
write (14.192) as
T
G0 (xE − x′ ) ≡ −
1
T X
exp [−ipE (xE − x′E )]
.
V pE
ip4 − ξ(p)
(14.204) {te-2.43}
It is quite straightforward to derive the general T 6= 0 Green function from a
path-integral formulation analogous to (14.169). For this we consider classical fields
ψ(x, τ ) with the periodicity or anti-periodicity
ψ(x, τ ) = ±ψ (x, τ + 1/T ) .
(14.205) {te-2.44}
They can be Fourier-decomposed as
ψ(x, τ ) =
T X X −iωn τ +ipx
T X −ipE xE
e
a(pE )
e
a(ωn , p) ≡
V ωn p
V pE
(14.206) {te-2.45}
with a sum over even or odd Matsubara frequencies ωn . If a free action is now
defined as
A0 [ψ ∗ , ψ] = −i
Z
1/2T
−1/2T
dτ
Z
d3 x ψ ∗ (x, τ ) [−∂τ − ξ (−i∇)] ψ(x, τ ), (14.207) {te-2.47}
formula (14.100) renders [1,8]
T
∗
∓Tr log A+
Z0 [η , η] = e
R R 1/2T
−1/2T
dτ dτ ′
R
d3 xd3 x′ η∗ (x,τ )A−1 (x,τ,x′ ,τ ′ )η(x′ ,τ ′ )
,
(14.208) {te-2.48}
with the functional matrix
A(x, τ ; x′ , τ ′ ) = [∂τ + ξ (−i∇)] δ (3) (x − x′ )δ(τ − τ ′ ).
(14.209) {te-2.49}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
957
14.10 Interacting Nonrelativistic Fields
Its inverse A−1 is equal to the propagator (14.192), the Matsubara frequencies arising
due to the finite-τ interval of euclidean space together with the periodic boundary
condition (14.205).
Again, interactions are taken care of by multiplying TZ0 [η ∗ η] with the factor
(14.170). In terms of the fields ψ(x, τ ), the exponent has the form:
Aint =
1
2
×
Z
Z Z
1/2T
−1/2T
dτ dτ ′
d3 xd3 x′ ψ ∗ (x, τ )ψ ∗ (x′ , τ ′ )ψ(x′ , τ ′ )ψ(x, τ )V (x, −iτ ; x′ , −iτ ′ ). (14.210) {te-2.50}
In the case of a potential (14.163) that is instantaneous in time t, the potential of
the euclidean formulation becomes instantaneous in τ :
V (x, −iτ ; x′ , −iτ ′ ) = V (x − x′ ) iδ(τ − τ ′ ).
(14.211) {te-2.51}
In this case Aint can be written in terms of the interaction Hamiltonian as
Aint = i
Z
1/2T
−1/2T
(14.212) {te-2.52}
dτ Hint (τ ).
Thus the grand-canonical partition function in the presence of external sources may
be calculated from the path integral [8]:
T
∗
Z[η , η] =
Z
T
i A+
∗
Dψ (x, τ )Dψ(x, τ )e
R 1/2T
−1/2T
dτ
R
d3 x[ψ∗ (x,τ )η(x,τ )+c.c.]
(14.213) {te-2.53}
,
where the grand-canonical action is
T
∗
i A[ψ , ψ] = −
i
+
2
Z
1/2T
−1/2T
Z
1/2T
−1/2T
dτ
Z
d3 xψ ∗ (x, τ ) [∂τ + ξ(−i∇)] ψ(x, τ )
(14.214) {te-2.54}
Z
dτ dτ ′ d3 xd3 x′ ψ ∗ (x, τ )ψ ∗ (x′ , τ ′ )ψ(x, τ ′ )ψ(x, τ )V (x, −iτ ; x, −iτ ′ ).
The Green functions of the fully interacting theory are obtained from the functional derivatives
(14.215) {te-2.5}
G (x1 , τ1 , . . . , xn , τn ; xn′ , τn′ , . . . , x1′ , τ1′ )
′
δ n+n Z[η ∗ , η]
′
.
= (−i)n+n ∗
∗
δη (x1 , τ1 ) · · · δη (xn , τn )δη(xn′ , τn′ ) · · · δη(x1′ , τn′ ) η=η∗ ≡0
Explicitly, the right-hand side represent the functional integrals
Z
N Dψ ∗ (x, t)Dψ(x, t)ψ̂(x1 , τ1 )· · · ψ̂(xn , τn )ψ̂ ∗ (xn′ , τn′ )· · · ψ̂ ∗ (x1′ , τ1′ )ei
T
A[ψ∗ ,ψ]
.(14.216) {te-2.8x}
In the sequel, we shall always assume the normalization factor to be chosen in such
a way that Z[0, 0] is normalized to unity. Then the functional integrals (14.216) are
obviously the correlation function of the fields:
hψ̂(x1 , τ1 ) · · · ψ̂(xn , τn )ψ̂ ∗ (xn′ , τn′ ) · · · ψ̂ ∗ (x1′ , τ1′ )i.
(14.217) {te-2.3eu}
958
14 Functional-Integral Representation of Quantum Field Theory
In contrast to Section 1.2, the bra and ket symbols denote now a thermal average
of the classical fields.
If the generating functional of the interacting theory is evaluated in a perturbation expansion using formula (14.170), the periodic boundary conditions for the
free Green functions (14.189) will go over to the fully interacting Green functions
(14.215).
The functional integral expression (14.213) for the generating functional have
a great advantage in comparison to the equivalent operator formulation based on
(14.180) and (14.170). They share with ordinary integrals the extreme flexibility
with respect to changes in the field variables.
Summarizing we have seen that the functional (14.213) defines the most general
type of theory involving two-body forces. It contains all information on a physical
system in the vacuum as well as in thermodynamic ensembles. The vacuum theory
is obtained by setting T = 0 and µ = 0, and by continuing the result back from
T to physical times. Conversely, the functional (14.169) in the vacuum can be
generalized to ensembles in a straight-forward manner by first continuing the times
t to imaginary values −iτ via a Wick rotation in all energy integrals and then going
to periodic functions in τ .
There is a complete correspondence between the real-time generating functional
(14.169) and the thermodynamic imaginary-time expression (14.213). For this reason it will be sufficient to exhibit all techniques only in one version, for which we
shall choose (14.169). Note, however, that due to the singular nature of the propagators (14.174) in real energy-momentum, the thermodynamic formulation specifies
the way how to avoid singularities.
14.11
Interacting Relativistic Fields
Let us see how this formalism works for relativistic boson and fermion systems.
Consider a Lagrangian of Klein-Gordon and Dirac particles consisting of a sum
L ψ, ψ̄, ϕ = L0 + Lint .
(14.218) {@}
As in the case of nonrelativistic fields, all time ordered Green’s functions can be
obtained from the derivatives with respect to the external sources of the generating
functional
Z [η, η̄, j] = const × h0|T ei
R
dx(Lint +η̄ψ+ψ̄η+jϕ)
|0i.
(14.219) {2.10}
The fields in the exponent follow free equations of motion and |0i is the free-field
vacuum. The constant is conventionally chosen to make Z [0, 0, 0] = 1, i. e.
const = h0|T ei
R
dxLint (ψ,ψ̄,ϕ)
|0i
−1
.
(14.220) {2.11}
This normalization may always be enforced at the very end of any calculation such
that Z [η, η̄, j] is only interesting as far as its functional dependence is concerned.
Any constant prefactor is irrelevant.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
959
14.11 Interacting Relativistic Fields
It is then straight-forward to show that Z [η, η̄, j] can alternatively be computed
via the Feynman path-integral formula
Z [η, η̄, j] = const ×
Z
i
DψD ψ̄Dϕe
R
dx[L0 (ψ,ψ̄,ϕ)+Lint +η̄ψ+ψ̄η+jϕ]
.
(14.221) {2.12}
Here the fields are no more operators but classical functions (with the mental reservation that classical Fermi fields are anticommuting objects). Notice that contrary
to the operator formula (14.219) the full action appears in the exponent.
For simplicity, we demonstrate the equivalence only for one real scalar field ϕ(x).
The extension to other fields is immediate [9,10,51]. Note that it is sufficient to give
the proof for free fields, where
Z0 [j] = h0|T ei
R
dxj(x)ϕ(x)
= const ×
Z
i
Dϕe
|0i
R
dx[ 12 ϕ(x)(−✷x −µ2 )ϕ(x)+j(x)ϕ(x)]
.
(14.222) {2.13}
For if it holds there, a simple multiplication on both sides of (14.222) by the differential operator
i
e
R
δ
dxLint ( 1i δj(x)
)
(14.223) {2.14}
would extend it to the interacting functionals (14.219) or (14.221). But (14.222)
follows directly from Wick’s theorem, according to which any time-ordered product
of a free field can be expanded into a sum of normal products with all possible time
ordered contractions. This statement can be summarized in operator form valid for
any functional F [ϕ] of a free field ϕ(x):
1
T F [ϕ] = e 2
R
δ
δ
dxdy δϕ(x)
D(x−y) δϕ(y)
(14.224) {2.15}
: F [ϕ] : ,
where D(x − y) is the free-field propagator
Z
i
i
d4 q −iq(x−y)
D(x − y) =
δ(x
−
y)
=
e
. (14.225) {2.16}
−✷x − µ2 + iǫ
(2π)4
q 2 − µ2 + iǫ
Applying this to (14.224) gives
1
Z0 = e 2
R
δ
δ
dxdy δϕ̂(x)
D(x−y) δϕ̂(y)
R
1
= e− 2
R
1
= e− 2
dxdyj(x)D(x−y)j(y)
dxdyj(x)D(x−y)j(y)
h0| : ei
h0| : ei
.
R
R
dxj(x)ϕ̂(x)
dxj(x)ϕ̂(x)
: |0i
: |0i
(14.226) {2.17}
The last part of the equation follows from the vanishing of all normal products of
ϕ(x) between vacuum states.
Exactly the same result is obtained by performing the functional integral in
(14.222) and by using the functional integral formula (14.99). The matrix A is equal
to A(x, y) = (−✷x − µ2 ) δ(x − y), and its inverse yields the propagator D(x − y):
A−1 (x, y) =
1
δ(x − y) = −iD(x − y),
−✷x − µ2 + iǫ
(14.227) {te-2.18}
960
14 Functional-Integral Representation of Quantum Field Theory
yielding again (14.226).
The generating functional of a free Dirac field theory reads
Z0 [η, η̄] = h0|T ei
R
ˆ
(η̄ ψ̂+ψ̄η)dx
= const ×
Z
|0i
DψD ψ̄ei
where L0 (x) is the free-field Lagrangian
R
dx[L0 (ψ,ψ̄ )+η̄ψ+ψ̄η]
(14.228) {2.12b}
.
L0 (x) = ψ̄(x) (iγ µ ∂µ − M) ψ(x) = ψ̄(x)A(x.y)ψ(x),
(14.229) {@}
By analogy with the bosonic expression (14.226) we obtain for Dirac particles
1
Z0 [η̄, η] = e 2
R
− 21
= e
δ
δ
dxdy δψ(x)
G0 (x−y) δψ̄(y)
R
dxdy η̄(x)G0 (x−y)η(y)
R
1
= e− 2
dxdy η̄(x)G0 (x−y)η(y)
h0| : ei
h0| : ei
,
R
R
ˆ
dx(η̄ ψ̂+ψ̄)η
ˆ
dx(η̄ ψ̂+ψ̄)η
: |0i
: |0i
(14.230) {2.17b}
where G0 (x − y) is the free fermion propagator, related to the functional inverse of
the matrix A(x, y) by
A−1 (x, y) =
1
δ(x − y) = −iG0 (x − y).
− M + iǫ
iγ µ ∂µ
(14.231) {te-2.18d}
Note that it is Wick’s expansion which supplies the free part of the Lagrangian
when going from the operator form (14.224) to the functional version (14.221).
14.12
Plasma Oscillation
The functional formulation of second-quantized many-body systems allows us to
treat efficiently various collective phenomena. As a first example we shall consider
a many-electron system that interacts only via long-range Coulomb forces. The
Coulomb forces give rise to collective modes called plasmons.
The other extremely important example caused by attractive short-range interactions will be treated in the next chapter.
14.12.1
{PLOSCI}
Plasmon Fields
Let us give a first application of the functional method by transforming the grand
partition function (14.213) to plasmon coordinates.
For this, we make use of the Hubbard-Stratonovich transformation (14.99) and
observe that a two-body interaction (14.161) in the generating functional (14.213)
can always be produced (following Maxwell’s original ideas in electromagnetism) by
an auxiliary field ϕ(x) as follows:
i
exp −
2
Z
′
∗
∗
′
′
′
dxdx ψ (x)ψ (x )ψ(x)ψ(x )V (x, x )
= const ×
Z
i
Dϕ
2
Z
′
h
dxdx ϕ(x)V
−1
′
(14.232) {3.1}
′
∗
′
(x, x )ϕ(x )−2ϕ(x)ψ (x)ψ(x)δ(x − x )
i
.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
961
14.12 Plasma Oscillations
To abbreviate the notation, we have used a four-vector notation in which
dx ≡ d3 xdt,
x ≡ (x, t),
δ(x) ≡ δ 3 (x)δ(t).
The symbol V −1 (x, x′ ) denotes the functional inverse of the matrix V (x, x′ ), i.e., the
solution of the equation
Z
dx′ V −1 (x, x′ )V (x′ , x′′ ) = δ(x − x′′ ).
(14.233) {@}
The constant prefactor in (14.232) is [det V ]−1/2 . Absorbing this into the always omitted normalization factor N of the functional integral, the grand-canonical
partition function Ω = Z becomes
Z[η ∗ , η] =
Z
Dψ ∗ DψDϕ exp iA + i
Z
dx (η ∗ (x)ψ(x) + ψ ∗ (x)η(x)) , (14.234) {13-3.2}
where the new action is
∗
A[ψ , ψ, ϕ] =
Z
dxdx ψ ∗ (x) [i∂t − ξ(−i∇) − ϕ(x)] δ(x − x′ )ψ(x′ ) (14.235) {3.3}
′
1
+ ϕ(x)V −1 (x, x′ )ϕ(x′ ) .
2
Note that the effect of using formula (14.99) in the generating functional amounts
to the addition of the complete square in ϕ in the exponent:
Z
1Z
′
dxdx ϕ(x) − dyV (x, y)ψ ∗(y)ψ(y) V −1 (x, x′ )
2
Z
× ϕ(x′ ) −
dy ′V (x′ , y ′)ψ ∗ (y ′ )ψ(y ′) ,
(14.236) {@}
together with the additional integration over Dϕ. This procedure of going from
(14.161) to (14.235) is probably simpler mnemonically than via the formula (14.99).
The fact that the functional Z remains unchanged by this addition is obvious, since
the integral Dϕ produces only the irrelevant constant [det V ]−1/2 .
The physical significance of the new field ϕ(x) is easy to understand: ϕ(x) is
directly related to the particle density. At the classical level this is seen immediately
by extremizing the action (14.235) with respect to variations δϕ(x):
δA
= ϕ(x) −
∂ϕ(x)
Z
dyV (x, y)ψ ∗ (y)ψ(y) = 0.
(14.237) {3.4}
Quantum mechanically, there will be fluctuations around the field configuration
ϕ(x) determined by Eq. (14.237), causing a difference between the Green functions
involving
the fields ϕ(x) versus those involving the associated composite field operR
ators dyV (x, y)ψ ∗ (y)ψ(y). Due to the Gaussian nature of the Dϕ integration, the
962
14 Functional-Integral Representation of Quantum Field Theory
difference between the two is quite simple. for example, one can easily see that the
propagators of the two fields differ merely by the direct interaction:
hT (ϕ(x)ϕ(x′ ))i
= V (x − x′ ) + T
Z
dyV (x, y)ψ(y)
Z
dy ′V (x′ , y ′)ψ ∗ (y ′ )ψ(y ′)
(14.238) {3.5}
.
For the proof, the reader is referred to Appendix 14A. Note that for a potential
V which is dominantly caused by a single fundamental-particle exchange, the field
ϕ(x) coincides with the field of this particle: If, for example, V (x, y) represents the
Coulomb interaction
e2
δ(t − t′ ),
′
|x − x |
(14.239) {@}
4πe2 ∗
ψ (x, t)ψ(x, t),
∇2
(14.240) {3.6}
V (x, x′ ) =
then Eq. (14.237) amounts to
ϕ(x, t) = −
revealing the auxiliary field as the electric potential.
If the particles ψ(x) have spin indices, the potential will, in this example, be
thought of as spin conserving at every vertex, and Eq. (14.237) must be read as spin
R
contracted: ϕ(x) ≡ d4 yV (x, y)ψ ∗α (y)ψα (y). This restriction is initially applied
only for convenience, and can easily be dropped later. Nothing in our procedure
depends on this particular property of V and ϕ. In fact, V could arise from the
exchange of many different fundamental particles and their multiparticle configurations (for example π, ππ, σ, ϕ, etc. in nuclei) so that the spin dependence is the rule
rather than the exception.
The important point is now that the entire theory can be rewritten as a field
theory of only the auxiliary field ϕ(x). For this we integrate out ψ ∗ and ψ in
Eq. (14.234), and make use of formula (14.100) to obtain
Z[η ∗ , η] ≡ Ω[η ∗ , η] = NeiA ,
(14.241) {3.7}
where the new action is
A[ϕ] = ±Tr log
iG−1
ϕ
1Z
dxdx′ η ∗ (x)Gϕ (x, x′ )η(x′ ),
+
2
(14.242) {3.8}
with Gϕ being the Green function of the fundamental particles in an external classical field ϕ(x):
[i∂t − χ(−i∇) − ϕ(x)] Gϕ (x, x′ ) = iδ(x − x′ ).
(14.243) {3.9}
The field ϕ(x) is called a plasmon field. The new plasmon action can easily be
interpreted graphically. For this, one expands Gϕ in powers of ϕ:
Gϕ (x, x′ ) = G0 (x − x′ ) − i
Z
dx1 G0 (x − x1 )ϕ(x1 − x′ ) + . . .
(14.244) {3.10}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
963
14.12 Plasma Oscillations
i
E−ζ(p)
ϕ
ϕ ϕ
Figure 14.1 The last pure-current term of collective action (14.242). The original fundamental particle (straight line) can enter and leave the diagrams only via external currents,
emitting an arbitrary number of plasmons (wiggly lines) on its way.
{Fig. Ia}
Hence the couplings to the external currents η ∗ , η in (14.242) amount to radiating
one, two, etc. ϕ fields from every external line of fundamental particles (see Fig.
14.1).
A functional expansion of the Tr log expression in powers of ϕ gives
−1
±iTr log(iG−1
ϕ ) = ±iTr log(iG0 ) ± iTr log(1 + iG0 ϕ)
∞
X
1
(−iG0 ϕ)n .
= ±iTr log(iG−1
)
∓
iTr
0
n
n=1
(14.245) {3.11}
The first term leads to an irrelevant multiplicative factor in (14.241). The nth term
corresponds to a loop of the original fundamental particle emitting nϕ lines (see Fig.
14.2).
Figure 14.2 Non-polynomial self-interaction terms of plasmons arising from the Tr log
in (14.242). The nth term presents a single-loop diagram emitting n plasmons.
{Fig. Ib}
Let us now use the action (14.242) to construct a quantum field theory of plasmons. For this we may include the quadratic term
±iTr(G0 ϕ)2
1
2
(14.246) {3.12}
into the free part of ϕ in (14.242) and treat the remainder perturbatively. The free
propagator of the plasmon becomes
{0|T ϕ(x)ϕ(x′ )|0} ≡ (2s + 1)G0 (x′ , x).
(14.247) {3.13}
This corresponds to an inclusion into the V propagator of all ring graphs (see Fig.
14.3). It is worth pointing out that the propagator in momentum space Gpl (k)
contains actually two important physical informations. From the derivation at fixed
temperature it appears in the transformed action (14.242) as a function of discrete
964
14 Functional-Integral Representation of Quantum Field Theory
Figure 14.3 Free plasmon propagator containing an infinite sequence of single-loop corrections (“bubblewise summation”).
{Fig. II}
euclidean frequencies νn = 2πnT only. In this way it serves to set up a timeindependent fixed-T description of the system. The calculation (14.247), however,
renders it as a function in the whole complex energy plane. It is this function which
determines by analytic continuation the time-dependent collective phenomena for
real times6 .
With the propagator (14.247) and the interactions given by (14.245), the original theory of fundamental fields ψ ∗ , ψ has been transformed into a theory of ϕ
fields whose bare propagator accounts for the original potential which has absorbed
ringwise an infinite sequence of fundamental loops.
This transformation is exact. Nothing in our procedure depends on the statistics
of the fundamental particles nor on the shape of the potential. Such properties
are important when it comes to solving the theory perturbatively. Only under
appropriate physical circumstances will the field ϕ represent important collective
excitations with weak residual interactions. Then the new formulation is of great
use in understanding the dynamics of the system. As an illustration consider a dilute
fermion gas of very low temperature. Then the function ξ(−i∇) is ǫ(−i∇) − µ with
ǫ(−i∇) = −∇2 /2m.
14.12.2
Physical Consequences
Let the potential be translationally invariant and instantaneous:
V (x, x′ ) = δ(t − t′ )V (x − x′ ).
(14.248) {3.14}
Then the plasmon propagator (14.247) reads in momentum space
Gpl (ν, k) = V (k)
1
1 − V (k)π(ν, k)
(14.249) {3.15P}
where the single electron loop symbolizes the analytic expression7
π(ν, k) = 2
1
1
T X
.
2
V p iω − p /2m + µ i(ω + ν) − (p + k)2 /2m + µ
(14.250) {3.16}
6
See the discussion in Chapter 9 of the last of Ref. [7] and G. Baym and N.D. Mermin, J. Math.
Phys. 2, 232 (1961).
7
The factor 2 stems from the trace over the electron spin.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
965
14.12 Plasma Oscillations
The frequencies ω and ν are odd and even multiples of πT . In order to calculate
the sum we introduce a convergence factor eiωη , and rewrite (14.250) as
d3 p
1
π(ν, k) = 2
3
(2π) ξ(p + k) − ξ(p) − iν
"
#
X
1
1
eiωn η
×T
.
−
i(ωn + ν) − ξ(p + k) iωn − ξ(p)
ωn
Z
(14.251) {nolabel}
Using the summation formula (14.197), this becomes
π(ν, k) = 2
d3 p n(p + k) − n(p)
,
(2π)3 ǫ(p + k) − ǫ(p) − iν
Z
(14.252) {3.19}
or, after some rearrangement,
π(ν, k) = −2
Z
1
1
d3 p
.(14.253) {3.19r1}
n(p)
+
3
(2π)
ǫ(p + k) − ǫ(p) − iν ǫ(p − k) − ǫ(p) + iν
#
"
Let us study this function for real physical frequencies ω = iν where we rewrite it
as
d3 p
1
1
π(ω, k) = −2
, (14.254) {3.19r3}
n(p)
+
3
(2π)
ǫ(p+k)−ǫ(p) − ω ǫ(p−k)−ǫ(p) + ω
#
"
Z
which can be brought to the form
π(ω, k) = 2
k2
mω 2
1
d3 p
n(p)
.
(2π)3
(ω − p · k/m + iη)2 − (k 2 /2M)2
Z
(14.255) {3.19r11}
For |ω| > pF k/m + k 2 /2m, the integrand is real and we can expand
k2
π(ω, k) = 2
mω 2
Z

2p · k
p·k
d3 p
n(p) 1 +
+3
3
(2π)
mω
mω
p·k
+
mω
!3
!2

80(p · k)4 + m2 ω 2k 4
+ . . .. (14.256) {3.19r3}
+
16m2 ω 4
Zero Temperature
For zero temperature, the chemical potential coincides with the radius of the Fermi
sphere µ = pF , and all levels below the Fermi momentum are occupied, to that the
Fermi distribuion function is n(p) = Θ(p − pF ). Then the integral in (14.256) can
be performed trivially using the integral
2
Z
N
p3F
d3 p
n
(p)
=
=
n
=
,
T =0
(2π)3
V
3π 2
(14.257) {@3.19r4a}
966
14 Functional-Integral Representation of Quantum Field Theory
and we obtain

k2 n
3
π(ω, k) = 2 1 +
ω m
5
pF k
mω
!2
1
+
5
pF k
mω
!4

1 k4
+ . . . .
+
16 m2 ω 2
(14.258) {3.19r4}
Inserting this into (14.249) we find, for long wavelengths, the Green function
#−1
"
V (k) n
+ ...
Gpl (ν, k) ≈ V (k) 1 −
ω2 m
(14.259) {3.15bcd}
.
Thus the original propagator is modified by a factor
ǫ(ω, k) = 1 −
4πe2 n
+ ... .
ω2 m
(14.260) {sc@diele}
The dielectric constant vanishes at the frequency
ω = ωpl =
s
4πe2
,
m
(14.261) {@}
which is the famous plasma frequency of the electron gas. At this frequency, the
plasma propagator (14.249) has a pole on the real-ω axis, implying the existence of
an undamped excitation of the system.
For an electron gas we insert the Coulomb interaction (14.240), and obtain
#−1
4πe2
4πe2
G (ν, k) ≈ 2 1 −
n+ ...
k
mω 2
"
pl
(14.262) {3.15bc}
.
Thus the original Coulomb propagator is modified by a factor
ǫ(ω, k) = 1 −
4πe2
n + ...,
mω 2
(14.263) {@}
which is simply the dielectric constant.
The zero temperature limit can also be calculated exactly starting from the
expression (14.256), written in the form
d3 p
1
π(ω, k) = −2
Θ(p − pF )
+ (ω → −ω) . (14.264) {3.19r31}
3
(2π)
p · k + k 2 /2m − ω
"
Z
#
Performing the integral yields
π(ω, k) = −



1  2
mpF
k mω
1−
pF −
+
2

2π
2kpF
2
k
+ (ω → −ω).
!2

+ p2F  log


2
k + 2mω − 2kpF
k 2 + 2mω + 2kpF 
(14.265) {3.19r31}
The lowest terms of a Taylor expansion in powers of k agree with (14.258).
H. Kleinert, PARTICLES AND QUANTUM FIELDS
967
14.12 Plasma Oscillations
Short-Range Potential
Let us also find the real poles of Gpl (ν, k) for a short-rang potential where the
singularity at k = 0 is absent. Then a rotationally invariant [V (k)]−1 has the longwavelength expansion
[V (k)]−1 = [V (0)]−1 + ak2 + . . . ,
(14.266) {3.22}
−1
as long as [V (0)] is finite and positive, i.e., for a well behaved overall repulsive
R
potential satisfying V (0) = d3 xV (x) > 0. Then the Green function (14.249)
becomes


Gpl (ω, k) = ω 2 ω 2 [V (0)]

−1

n
3
+ aω 2 k 2 − k 2 1 +
m
5
This has a pole at ω = ±c0 k where
pF k
mω
!2
−1

+ . . .
.

(14.267) {3.23w}
n
.
(14.268) {@}
m
This is the spectrum of zero sound with the velocity c0 .
In the neighbourhood of the positive-energy pole, the propagator has the form
c0 = V (0)
Gpl (k0 , k) ≈ V (0) ×
|k|
.
ω − c0 |k|
(14.269) {3.25}
Nonzero Temperature
In order to discuss the case of nonzero temperature it is convenient to add and
subtract a term −n(p + k)n(k) in the numerator of (14.252), and rewrite it as
π(ν, k) = 2
d3 p n(p + k) [1 − n(p)] − n(p) [1 − n(p + k)]
,
(2π)3
ǫ(p + k) − ǫ(p) − iν
Z
(14.270) {3.19c}
which can be rearranged to
π(ν, k) = −4
Z
ǫ(p + k) − ǫ(p)
d3 p
n(p) [1 − n(p + k)]
.
3
(2π)
[ǫ(p + k) − ǫ(p]2 + ν 2
(14.271) {3.19d}
In the high-temperature limit the Fermi distribution becomes Boltzmannian,
2
n(p) ≈ e−β(p /2−µ) , and we evaluate again most easily expression (14.256) as follows:
π(ω, k) = −2
Z
0
∞
dσ
Z
d3 p −β(p2 /2−µ)−σ[ǫ(p+k)−ǫ(p)−ω]
e
+ (ω → −ω) .
(2π)3
(14.272) {3.19r34}
The right-hand side is equal to
Z
−
∞
0
d cos θ d3 p −β(p2 /2−µ)−σ[(pk cos θ/m+k2 /2m)−ω]
dσ
e
+ (ω → −ω). (14.273) {3.19r34}
2
(2π)3
−1
Z
1
Z
Performing the angular integral yields
−
m2
k2
Z
0
∞
d3 p −β(p2 /2−µ)+σω 1
dσ
e
(pk cosh pk − sinh pk) + (ω → −ω). (14.274) {3.19r34}
2π 2 (2π)3
p2
Z
968
14 Functional-Integral Representation of Quantum Field Theory
14.13
Pair Fields
The introduction of a scalar field ϕ(x) was historically the first way, invented by
Maxwell, to convert the Coulomb interaction in a theory (14.232) into a local field
theory. The resulting plasmon action depends only on the local field ϕ. There exists
an alternative way of converting the interaction between four fermions in (14.232)
into a new field theory. That is based on introducing a bilocal scalar field which has
been very successful to understand the properties of electrons in of superconductors.
It is a collective field complementary to the plasmon field. Generically it will be
called a pair field . It describes the dominant low-energy collective excitations in
systems such as type II superconductors, superfluid 3 He, excitonic insulators, etc.
The pair field is originally a bilocal field and will be denoted by ∆(x t; x′ t′ ), with
two space arguments and two time arguments. It is introduced into the generating
functional by rewriting the exponential of the interaction term in (14.232) in the
partition function (14.169) as a functional integral
exp −
i
2
Z
×e
Z
{PLOSCI2}
{PAFI}
dxdx′ ψ ∗ (x)ψ ∗ (x′ )ψ(x′ )ψ(x)V (x, x′ ) = const × D∆(x, x′ )D∆∗ (x, x′ )
R
i
2
n
o
1
∗
′
′
∗
∗ ′
′
dxdx′ |∆(x,x′ )|2 V (x,x
′ ) −∆ (x,x )ψ(x)ψ(x )−ψ (x)ψ (x )∆(x,x )
(14.275) {4.1}
.
In contrast to the similar-looking plasmon expression (14.232), the inverse 1/V (x, x′ )
in (14.275) is understood as a numeric division for each x, y, not as a functional
inverse. Hence the grand-canonical potential becomes
∗
Z[η, η ] =
Z
Dψ DψD∆ D∆ e
Z
dxdx′ {ψ ∗ (x) [i∂t − ξ(−i∇)] δ(x − x′ )ψ(x′ )
with the action
∗
∗
A[ψ , ψ, ∆ , ∆] =
∗
∗
iA[ψ∗ ,ψ,∆∗ ,∆]+i
R
dx(ψ∗ (x)η(x)+c.c.)
,
(14.276) {N4.2}
(14.277) {4.3X}
)
1
1
1
1
− ∆∗ (x, x′ )ψ(x)ψ(x′ ) − ψ ∗ (x)ψ ∗ (x′ )∆(x, x′ ) + |∆(x, x′ )|2
,
2
2
2
V (x, x′ )
where ξp ≡ εp − µ is the grand-canonical single particle energy (2.256). This new
action arises from the original one in (14.169) by adding to it the complete square
i
2
Z
dxdx′ |∆(x, x) − V (x′ , x)ψ(x′ )ψ(x)|2
1
,
V (x, x′ )
(14.278)4.3 {nolabel}
which removes
the fourth-order interaction term and gives, upon functional integraR
tion over D∆∗ D∆, merely an irrelevant constant factor to the generating functional.
At the classical level, the field ∆(x, x′ ) is nothing but a convenient abbreviation
for the composite field V (x, x′ )ψ(x)ψ(x′ ). This follows from the equation of motion
obtained by extremizing the new action with respect to δ∆∗ (x, x′ ). This yields
δA
1
=
[∆(x, x′ ) − V (x, x′ )ψ(x)ψ(x′ )] ≡ 0.
δ∆∗ (x, x′ )
2V (x, x′ )
(14.279) {4.4}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
969
14.13 Pair Fields
Quantum mechanically, there are Gaussian fluctuations around this solution which
are discussed in Appendix 14B.
Taking care of the spin components of the Fermi field, we can rewrite the first
line in the expression (14.278), which are quadratic in the fundamental fields ψ(x),
in a matrix form as
1 ∗
f (x)A(x, x′ )f (x′ )
2
!
1 †
[i∂t − ξ(−i∇)] δ(x − x′ )
−∆(x, x′ )
= f (x)
f (x′ ),(14.280) {4.5}
∗
′
′
−
∆
(x,
x
)
∓
[i∂
+
ξ(i∇)]
δ(x
−
x
)
2
t
where f (x) denotes the fundamental field doublet
f (x) =
!
ψ(x)
ψ ∗ (x)
(14.281) {4.5D}
and f † ≡ f ∗T , as usual. Here the field f ∗ (x) is not independent of f (x). Indeed,
there is an identity
†
f Af = f
T
0 1
1 0
!
(14.282) {4.5E}
Af.
Therefore, the real-field formula (14.99) must be used to evaluate the functional
integral for the generating functional
Z[η ∗ , η] =
Z
D∆∗ D∆ eiA[∆
∗ ,∆]− 1
2
R
dx
R
dx′ j † (x)G∆ (x,x′ )j(x′ )
,
(14.283) {4.6}
where j(x)
! collects the external source η(x) and its complex conjugate, j(x) ≡
η(x)
. Then the collective action (14.278) reads
η ∗ (x)
h
i
i
1
′
A[∆ , ∆] = ± Tr log iG−1
∆ (x, x ) +
2
2
∗
Z
dxdx′ |∆(x, x′ )|2
1
.
V (x, x′ )
(14.284) {4.7}
The 2 × 2 matrix G∆ denotes the propagator iA−1 which satisfies the functional
equation
Z
dx′′
!
[i∂t − ξ(−i∇)] δ(x−x′′ )
−∆(x, x′′ )
G∆ (x′′ , x′ ) = iδ(x−x′ ).
∗
′′
− ∆ (x, x )
∓ [i∂t + ξ(i∇)] δ(x−x′′ )
(14.285) {4.8}
!
G G∆
Writing G∆ as a matrix
, the mean-field equations associated with this
G†∆ G̃
action are precisely the equations used by Gorkov to study the behavior of type II
superconductors [12]. With Z[η ∗ , η] being the full partition function of the system,
the fluctuations of the collective field ∆(x, x′ ) can now be incorporated, at least in
principle, thereby yielding corrections to these equations.
970
14 Functional-Integral Representation of Quantum Field Theory
Let us set the sources in the generating functional Z[η ∗ , η] equal to zero and
investigate the behavior of the collective quantum field ∆. In particular, we want to
develop Feynman rules for a perturbative treatment of the fluctuations of ∆(x, x′ ).
As a first step we expand the Green function G∆ in powers of ∆ as
0 ∆
∆∗ 0
G∆ = G0 − iG0
!
G0 − G0
0 ∆
∆∗ 0
!
0 ∆
∆∗ 0
G0
!
G0 + . . . (14.286) {4.9}
with
i
δ(x − x′ )

i∂t − ξ(−i∇)

G0 (x, x′ ) = 

0
∓

0
i
δ(x − x′ )
i∂t + ξ(i∇)



.

(14.287) {@4.9}
We shall see later that this expansion is applicable only close to the critical temperature Tc . Inserting this expansion into (14.283), the source term can be interpreted graphically by the absorption and emission of lines ∆(k) and ∆∗ (k), respectively, from virtual zig-zag configurations of the underlying particles ψ(k), ψ ∗ (k) (see
Fig. 14.4)
i
ω−ξ(p)
∆∗ (ν ′ , q′ )
i
ν ′ −ω−ξ(q′ −p)
i
ν−ν ′ +ω−ξ(q−q′ +p)
+
+ . . . (14.286)
∆(ν, q)
Figure 14.4 Fundamental particles (fat lines) entering any diagram only via the external
currents in the last term of (14.283), absorbing n pairs from the right (the past) and
emitting the same number from the left (the future).
{SC-Fig. III}
The functional submatrices in G0 have the Fourier representation
T X
0
i
e−i(p t−px) ,
0
V p p − ξp
i
T X
−i(p0 t−px)
,
e
G̃0 (x, x′ ) = ±
V p −p0 − ξ−p
G0 (x, x′ ) =
(14.289) {nolabel}
(14.290) {4.10}
where we have used the notation ξp for the Fourier components ξ(p) of ξ(−i∇).
The first matrix coincides with the operator Green function
G0 (x − x′ ) = h0|T ψ(x)ψ † (x′ )|0i.
(14.291) {4.11}
The second one corresponds to
G̃0 (x − x′ ) = h0|T ψ †(x)ψ(x′ )|0i = ±h0|T ψ(x′ )ψ † (x) |0i
= ±G0 (x′ − x) ≡ ±[G0 (x, x′ )]T ,
(14.292) {4.11a}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
971
14.13 Pair Fields
where T denotes the transposition in the functional sense (i.e., x and x′ are interchanged). After a Wick rotation of the energy integration contour, the Fourier
components of the Green functions at fixed energy read
1
′
eip(x−x )
(14.293) {nolabel}
p iω − ξp
X
1
′
G̃0 (x − x′ , ω) = ∓
eip(x−x ) = ∓G0 (x′ − x, −ω). (14.294) {4.12}
−iω
−
ξ
−p
p
G0 (x − x′ , ω) = −
X
The Tr log term in Eq. (14.284) can be interpreted graphically just as easily by
expanding as in (14.286):
"
i
i
i
−1
± Tr log iG∆
= ± Tr log iG−1
∓ Tr −iG0
0
2
2
2
0
∆
∗
∆ 0
!
∆∗
#n
1
.(14.295) {4.13}
n
The first term only changes the irrelevant normalization N of Z. To the remaining
sum only even powers can contribute so that we can rewrite
∞
X
#n
i
∓i
(−)n
Tr
δ ∆
δ ∆∗
A[∆ , ∆] = ∓i
i∂t − ξ(−i∇)
i∂t + ξ(i∇)
n=1 2n
Z
1
1
+
dxdx′ |∆(x, x′ )|2
2
V (x, x′ )
Z
∞
X
1
1
An [∆∗ , ∆] +
=
dxdx′ |∆(x, x′ )|2
.
(14.296) {4.14}
2
V (x, x′ )
n=1
∗
"
!
!
This form of the action allows an immediate quantization of the collective field ∆.
The graphical rules are slightly more involved technically than in the plasmon case
since the pair field is bilocal. Consider at first the free collective fields which can be
obtained from the quadratic part of the action:
"
i
A2 [∆ , ∆] = − Tr
2
∗
!
#
!
i
i
δ ∆
δ ∆∗ . (14.297) {4.15}
i∂t − ξ(−i∇)
i∂t + ξ(i∇)
Variation with respect to ∆ displays the equations of motion
′
′
∆(x, x ) = iV (x, x )
"
!
i
i
δ ∆
δ
i∂t − ξ(−i∇)
i∂t + ξ(i∇)
!#
.
(14.298) {N4.16}
This equation coincides exactly with the Bethe-Salpeter equation [13] in the ladder approximation. Originally, this was set up for two-body bound-state vertex
functions, usually denoted in momentum space by
Γ(p, p′ ) =
Z
dxdx′ exp[i(px + p′ x′ )]∆(x, x′ ).
(14.299) {N4.16X}
Thus the free excitations of the field ∆(x, x′ ) consist of bound pairs of the original
fundamental particles. The field ∆(x, x′ ) will consequently be called pair field. If
972
14 Functional-Integral Representation of Quantum Field Theory
we introduce total and relative momenta q and P = (p − p′ )/2, then (14.298) can
be written as8
Γ(P |q) = −i
Z
i
d4 P ′
V (P − P ′ )
′
4
(2π)
q0 /2 + P0 − ξq/2+P′ + iη sgn ξ
i
.
× Γ(P ′ |q)
′
q0 /2 − P0 − ξq/2−P′ + iη sgn ξ
(14.300) {N4.17}
Graphically this formula can be represented as follows: The Bethe-Salpeter wave
Figure 14.5 Free pair field following the Bethe-Salpeter equation as pictured in this
diagram.
{Fig. IV}
function is related to the vertex Γ(P |q) by
Φ(P |q) = N
i
q0 /2 + P0 − ξq/2+P + iη sgn ξ
i
×
Γ(P |q).
q0 /2 + P0 − ξq/2+P + iη sgn ξ
(14.301) {4.18}
It satisfies
G0 (q/2 + P ) G0 (q/2 − P ) Φ(P |q) = −i
Z
dP ′
V (P, P ′)Φ(P ′ |q),
(2π)4
(14.302) {nolabel}
thus coinciding, up to a normalization, with the Fourier transform of the two-body
state wave functions
ψ(x, t; x′ , t′ ) = h0|T (ψ(x, t)ψ(x′ , t′ )) |B(q)i.
(14.303) {4.19}
If the potential is instantaneous, then (14.298) shows ∆(x, x′ ) to be factorizable
according to
∆(x, x′ ) = δ(t − t′ )∆(x, x′ ; t)
(14.304) {4.20}
so that Γ(P |q) becomes independent of P0 .
8
Here q abbreviates the four-vector q µ = (q 0 , q) with q0 = E.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
973
14.13 Pair Fields
Consider now the system at T = 0 in the vacuum. Then µ = 0 and ξp = εp > 0.
One can perform the P0 integral in (14.300) with the result
Γ(P|q) =
1
d3 P ′
V (P − P′ )
Γ(P′ |q).
4
(2π)
q0 − εq/2+P′ − εq/2−P′ + iη
Z
(14.305) {4.21}
Now the equal-time Bethe-Salpeter wave function
′
ψ(x, x ; t) ≡ N
Z
d3 Pdq0 d3 q
x + x′
exp
−i
q
t
−
q
·
− P · (x − x′ )
0
(2π)7
2
1
Γ(P|q)
×
q0 − εq/2+P − εq/2−P + iη
"
!#
(14.306) {4.22}
satisfies
h
i
i∂t − ǫ(−i∇) − ǫ(−i∇′ ) ψ(x, x′ ; t) = V (x − x)ψ(x, x′ ; t),
(14.307) {4.23}
which is simply the Schrödinger equation of the two-body system. Thus, in the
instantaneous case, the free collective excitations in ∆(x, x′ ) are the bound states
derived from the Schrödinger equation.
In a thermal ensemble, the energies in (14.300) have to be summed over the
Matsubara frequencies only. First, we write the Schrödinger equation as
Γ(P|q) = −
Z
d3 P′
V (P − P′ )l(P′ |q)Γ(P′|q)
(2π)3
(14.308) {4.24}
with
l(P|q) = −i
X
G0 (q/2 + P ) G̃0 (P − q/2)
= −i
X
q0 /2+P0 −ξq/2+P +iη sgn ξ q0 /2 − P0 −ξq/2−P +iη sgn ξ
P0
P0
i
(14.309) {nolabel}
i
.
After a Wick rotation and setting q0 ≡ iν, the replacement of the energy integration
by a Matsubara sum leads to
1
1
ωn i (ωn + ν/2) − ξq/2+P i (ωn − ν/2) + ξq/2−P
X
1
= T
ωn iν − ξq/2+P − ξq/2−P
l(P|q) = −T
X
"
1
1
×
−
i(ωn + ν/2) − ξq/2+P i(ωn − ν/2) + ξq/2−P
= −
h
± nq/2+P + nq/2−P
i
iν − ξq/2+P − ξq/2−P
.
#
(14.310) {4.25}
974
14 Functional-Integral Representation of Quantum Field Theory
Here we have used the frequency sum [see (14.197)]
T
X
ωn
1
1
= ∓ ξp /T
≡ ∓np .
iωn − ξp
e
∓1
(14.311) {te-3.18B}
with np being the occupation number of the state of energy ξp . Using the identity
np → ∓1 − np , the expression in brackets can be rewritten as −N(P, q) where
N(P|q) ≡ 1 ± nq/2+P + nq/2−P
!
ξq/2+P
ξq/2−P
1
=
tanh ∓1
,
+ tanh ∓1
2
2T
2T
(14.312) {nolabel}
so that
l(P|q) = −
N(P|q)
.
iν − ξq/2+P − ξ(q/2−P
(14.313) {nolabel}
Defining again a Schrödinger type wave function for T 6= 0 as in (14.306), the
bound-state problem can be brought to the form (14.305) but with a momentum
dependent potential V (P − P′ ) × N(P′ |q). Thus the Bethe-Salpeter equation at any
temperature reads
Γ(P|q) =
Z
1
d3 P ′
V (P − P′ )N(P′ |q)
Γ(P′|q). (14.314) {4.21X}
4
(2π)
q0 − εq/2+P′ − εq/2−P′ + iη
We are now ready to construct the propagator of the pair field ∆(x, x′ ) for T = 0.
In many cases, this is most simply done by considering Eq. (14.300) with a potential
λV (P, P ′) rather than V , and asking for all eigenvalues λn at fixed q. Let Γn (P |q)
be a complete set of vertex functions for this q. Then one can write the propagator
as
†
′
′
∆(P |q)∆ (P |q ) = −i
X
n
Γn (P |q)Γ∗n (P ′|q) (2π)4 δ (4) (q − q ′ )
λ − λn (q)
λ=1
(14.315) {4.26}
where a hook denotes, as usual, the Wick contraction of the fields. Obviously the vertex functions have to be normalized in a specific way, as discussed in Appendix 14A.
An expansion of (14.315) in powers of [λ/λn (q)]n exhibits the propagator of ∆
as a ladder sum of exchanges as shown in Fig. 14.6.
+
+
+ . . . (14.316)
Figure 14.6 Free pair propagator, amounting to a sum of all ladders of fundamental
potential exchanges. This is revealed explicitly by the expansion of (14.315) in powers of
(λ/λn (q)).
H. Kleinert, PARTICLES AND QUANTUM FIELDS
{Figu. V}
975
14.14 Competition of Plasmon and Pair Fields
For an instantaneous interaction, either side is independent of P0 , P0′ . Then the
propagator can be shown to coincide directly with the scattering matrix T of the
Schrödinger equation (14.307) and the associated integral equation in momentum
space (14.305) [see Eq. (14A.13)].
∆∆† = iT ≡ iV + iV
1
V.
E−H
(14.317) {4.27}
Consider now the higher interactions An , n ≥ 3 of Eq. (14.296). They correspond
to zig-zag loops shown in Fig. 14.7. These have to be calculated with every possible
Γn (P |q), Γ∗m(P |q) entering or leaving, respectively. Due to the P dependence at
+
+ . . . (14.318)
Figure 14.7 Self-interaction terms of the non-polynomial pair action (14.296) amounting
to the calculation of all single zig-zag loop diagrams absorbing and emitting n pair fields.
every vertex, the loop integrals become very involved. A slight simplification arises
for instantaneous potentials. Then the frequency sums can be performed. Only in
the special case of a completely local action, the full P -dependence disappears and
the integrals can be calculated. See Section IV.2 in Ref. [4].
14.14
Competition of Plasmon and Pair Fields
The Hubbard-Stratonovich transformation has a well-established place in manybody theory [8,9,4]. After it had been successfully applied in 1957 by Bardeen,
Cooper, and Schrieffer to explain the phenomenon of superconductivity by with
the so-called BCS theory [50], Nambu and Jona-Lasinio [14] discovered that the
same mechanism which explains the formation of an energy gap and Cooper pairs
of electrons in a metal can be used to understand the surprising properties of quark
masses in the physics of strongly interacting particles. This aspect of particle physics
will be explained in Chapter 26.
In many-body theory, the use of the Hubbard-Stratonovich transformation has
led to a good understanding of important collective physical phenomena such as
plasma oscillations and other charge-density type of waves, for example paramagnons
in superfluid He3 . It has put heuristic calculations such as the Gorkov’s derivation
[12] of the Ginzburg-Landau equations [15] on a reliable theoretical ground [4]. In
{te-vih}
976
14 Functional-Integral Representation of Quantum Field Theory
addition, it is in spirit close to the density functional theory [16] via the HohenbergKohn and Kohn-Sham theorems [17].
In Sections 14.12 and 14.13 we have used the Hubbard-Stratonovich transformation to rewrite the many-body action in two ways. One was theory of a local
plasmon field ϕ(x), the other a thery of a bilocal scalar pair field ∆(x, x′ ). In the
theory of collective excitation, either of the two transformations has been helpful to
understand either the behavior of electron gases or that of superconductors. In the
first case, one is able to deal efficiently with the oscillations of charge distributions in
the gas. In the second case one is able to see how the electrons in a superconductor
become bound to Cooper pairs, and how the binding gives rise to a frictionless flow
of the doubly charged bosonic pairs through the metal. In general, however, there
exists a competition between the two mechanisms.
The transformation was cherished by theoreticians since it allows them to reexpress a four-particle interaction exactly in terms of a collective field variable whose
fluctations can in principle be described by higher loop diagrams. The only bitter pill
is that any approximative treatment of a many-body system can describe interesting
physics only if calculations can be restricted to a few low-order diagrams. If this is
not the case, the transformation fails.
The trouble arises in all those many-body systems in which different collective
effects compete with similar relevance. Historically, an important example is the
fermionic superfluid He3 . While BCS superconductivity is described easily via the
Hubbard-Stratonovich transformation which turns the four-electron interaction into
a field theory of Cooper pairs, the same approach did initially not succeed in a liquid of He3 -atoms. Due to the strongly repulsive core of an atom, the forces in the
attractive p-wave are not sufficient to bind the Cooper pairs. Only by taking into
account the existence of another collective field that arises in the competing paramagnon channel was it possible to explain the formation of weakly-bound Cooper
pairs [18].
It is important to learn how to deal with this kind of mixture. The answer
is found with the help of Variational Perturbation Theory (VPT), which has been
discussed in Chapter 3 (see the pages 177 and 216). The key is to abandon the fluctuating collective quantum fields introduced by means of the Hubbard-Stratonovich
transformation. Instead one must turn to a variety of collective classical fields. After a perturbative calculation of the effective action, one obtains a functional that
depends on these classical fields. The dependence can be optimized, usually by extremizing their influence upon the effective action. In this way one is able to obtain
exponentially fast converging results.
It is the purpose of this section to point out how to circumvent the fatal focussing
of the Hubbard-Stratonovich transformation on a specific channel and to take into
account the competition between several competing channels.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
977
14.15 Ambiguity in the Selection of Important Channels
14.15
Ambiguity in the Selection of Important Channels
The basic weakness of the Hubbard-Stratonovich transformation lies in the different
possibilities of decomposing the fourth order interaction by a quadratic completion
with the help of an auxiliary field. The first is based on the introduction of a
scalar plasmon field ϕ(x) and the use of the quadratic-completion formula (14.232).
The other uses the introduction of a bilocal pair field ∆(x, x′ ) in combination with
the quadratic-completion formula (14.275). The trouble with both approaches is
that, when introducing an auxiliary field ϕ(x) or ∆(x, x′ ) and summing over all
fluctuations of one of the fields, the effects of the other is automatically included.
At first sight, this may appear as an advantage. Unfortunately, this is an illusion.
In either case, even the lowest-order fluctuation effect of the other is extremely hard
to calculate. That can be seen most simply in the simplest models of quantum
field theory such as the Gross-Neveu model (to be discussed in Chapter 23). There
the propagator of the quantum field ∆(x, x′ ) is a very complicated object. So it
is practically impossible to recover the effects OF ϕ(x) from the loop calculations
with these propagators. As a consequence, the use of a specific quantum field theory
must be abandoned whenever collective effects of different channels are important.
To be specific let us assume the fundamental interaction to be of the local form
Aloc
int
g
=
2
Z
x
ψα∗ ψβ∗ ψβ ψα
=g
Z
x
ψ↑∗ ψ↓∗ ψ↓ ψ↑ ,
(14.319) {4.28}
where the subscripts ↑, ↓ denote the spin directions of the fermion fields. For brevity,
we have absorbed the spacetime arguments x in the spin subscripts and written the
R
symbol x for an integral over spacetime and a sum over the spin indices of the
fermion field.
We now introduce auxiliary classical collective fields which are no longer assumed
to undergo functional fluctuations, and we replace the exponential in the interacting
version of the generating functional (14.276),
Z[η, η ∗ ] =
Z
Dψ ∗ Dψ eiA0 [ψ
∗ ,ψ]+iAloc +i
int
R
dx(ψ∗ (x)η(x)+c.c.)
(14.320) {nolabel}
,
identically by [34]
loc
eiAint = exp{i g
Z
x
i
= exp −
2
∗
∗
ψ↑,x
ψ↓,x
ψ↓,x ψ↑,x } = exp −
Z x
i
2
Z
x
fxT Mx fx × exp{iAnew
int } (14.321) {nolabel}
ψβ ∆∗βα ψα + ψα∗ ∆αβ ψβ∗ + ψβ∗ ρβα ψα + ψα∗ ραβ ψβ
×exp{iAnew
int }.
Here f (x) is here the doubled spinor field (14.281) with spin index:
f (x) =
ψα (x)
ψα∗ (x)
!
,
(14.322) {4.5F}
978
14 Functional-Integral Representation of Quantum Field Theory
and fxT denotes the transposed fundamental field doublet fxT = (ψα , ψα∗ ). The new
interaction reads
loc
Anew
int = Aint
Z
1Z T
g ∗ ∗
ψα ψβ ψβ ψα
(14.323) {nolabel}
fx Mx fx =
+
2 x
x 2
1
+ ψβ ∆∗βα ψα + ψα∗ ∆αβ ψβ∗ + ψα∗ ραβ ψβ .
2
We now define a further free action by the quadratic form
≡ A0 −
Anew
0
1
2
Z
1
fxT Mx fx = fx† A∆,ρ
x,x′ fx′ .
2
x
(14.324) {4.29}
with the functional matrix A∆,ρ
x,x′ being now equal to
!
[i∂t −ξ(−i∇)]δαβ +ραβ
∆αβ
.
∗
∆αβ
[i∂t +ξ(i∇)]δαβ −ραβ
(14.325) {4.5X}
The physical properties of the theory associated with the action A0 +Aloc
int can now be
derived as follows: first we calculate the generating functional of the new quadratic
action Anew
via the functional integral
0
Z0new [η, η ∗ ] =
Z
new +i
Dψ ∗ Dψ eiA0
R
dx(ψ∗ (x)η(x)+c.c.)
.
(14.326) {nolabel}
From its functional derivatives with respect to the sources ηα and ηα† we find the
new free propagators G∆ and Gρ . To higher orders, we expand the exponential
new
n new
with
eiAint in a power series and evaluate all expectation values (in /n!)h[Anew
int ] i0
the help of Wick’s theorem. They are expanded into sums of products of the free
particle propagators G∆ and Gρ . The sum of all diagrams up to a certain order g N
defines an effective collective action AN
eff as a function of the collective classical fields
∗
∆αβ , ∆βα , ραβ ,
Obviously, if the expansion is carried to infinite order, the result must be independent of the auxiliary collective fields since they were introduced and removed in
(14.322) without changing the theory. However, any calculation can only be carried
up to a finite order, and that will depend on these fields. We therefore expect the
best approximation to arise from the extremum of the effective action [6,21,54].
The lowest-order effective collective action is obtained from the trace of the
logarithm of the matrix (14.325):
h
i
i
A0∆,ρ = − Tr log iG−1
∆,ρ .
2
(14.327) {4.7x}
−1
The 2 × 2 matrix G∆,ρ denotes the propagator i[A∆,ρ
x,x′ ] .
To first order in perturbation theory we must calculate the expectation value
hAint i of the interaction (14.324). This is done with the help of Wick contractions
in the three channels, Hartree, Fock, and Bogoliubov:
hψ↑∗ ψ↓∗ ψ↓ ψ↑ i = hψ↑∗ ψ↑ ihψ↓∗ ψ↓ i − hψ↑∗ ψ↓ ihψ↓∗ ψ↑ i + hψ↑∗ ψ↓∗ ihψ↓ ψ↑ i.
(14.328) {@}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
979
14.15 Ambiguity in the Selection of Important Channels
For this purpose we now introduce the expectation values
˜ ∗ ≡ ghψ ∗ ψ ∗ i,
∆
αβ
α β
ρ̃αβ ≡
ghψα∗ ψβ i,
˜ βα ≡ ghψβ ψα i = [∆
˜ ∗ ]∗ ,
∆
αβ
ρ̃†αβ
∗
≡ [ρ̃βα ] ,
(14.329) {nolabel}
(14.330) {@FEXPv}
and rewrite hAnew
int i as
hAnew
int i =
1 Z ˜∗ ˜
1 Z ˜
˜ ∗ ∆βα + 2ρ̃αβ ραβ ).
(∆↓↑ ∆↓↑ − ρ̃↑↓ ρ̃↓↑ + ρ̃↑↑ ρ̃↓↓ ) −
(∆βα ∆∗βα + ∆
αβ
g x
2g x
˜ αβ , the diagonal matrix elements vanish, and ∆
˜ αβ has the
Due to the locality of ∆
˜ where cαβ is i times the Pauli matrix σ 2 . In the absence of a magnetic
form cαβ ∆,
αβ
field, the expectation values ρ̃αβ may have certain symmetries:
ρ̃↑↑ ≡ ρ̃ = ρ↓↓ ,
ρ̃↑↓ = ρ↓↑ ≡ 0,
(14.331) {@ASS}
so that (14.331) simplifies to
hAnew
int i =
1
g
Z h
x
i
˜ 2 + ρ̃2 ) − (∆∆
˜ ∗+∆
˜ ∗ ∆ + 2ρ̃ρ) .
(|∆|
(14.332) {@EXPE1}
The total first-order collective classical action A1∆,ρ is given by the sum
A1∆,ρ= A0∆,ρ +hAnew
int i.
(14.333) {@ExTRAc}
Now we observe that the functional derivatives of the zeroth-order action A0∆,ρ are
the free-field propagators G∆ and Gρ
δ
A0∆,ρ = [G∆ ]αβ ,
δ∆αβ
δ
A0∆,ρ = [Gρ ]αβ .
δραβ
(14.334) {@exep2}
Then we can extremize A1∆,ρ with respect to the collective fields ∆ and ρ, and find
that to this order these fields satisfy the gap-like equations
˜ x = g[G∆ ]x,x ,
∆
ρ̃x = g[Gρ ]x,x .
(14.335) {@GAPR}
If the fields satisfy (14.335), the extremal action has the value
A1∆,ρ = A0∆,ρ −
1
g
Z
x
(|∆|2 + ρ2 ).
(14.336) {@ExTRAc1}
Note how the theory differs, at this level, from the collective quantum field theory
derived via the Hubbard-Stratonovich transformation. If we assume that ρ vanishes
identically, the extremum of the one-loop action A1∆,ρ gives the same result as of
the mean-field collective quantum field action (14.284), which reads for the present
attractive δ-function in (14.319):
A1∆,0
=
A0∆,0
1
−
g
Z
x
|∆|2 .
(14.337) {@}
980
14 Functional-Integral Representation of Quantum Field Theory
On the other hand, if we extremize the action A1∆,ρ at ∆ = 0, we find the extremum
from the expression
A10,ρ = A00,ρ −
1
g
Z
x
ρ2 .
(14.338) {@ExTRAc3
The extremum of the first-order combined collective classical action (14.336) agrees
with the good-old Hartree-Fock-Bogolioubov theory.
The essential difference between this and the new approach arises in two issues:
• First, by being able to carry the expansion to higher orders: If the collective
quantum field theory is based on the Hubbard-Stratonovich transformation,
the higher-order diagrams must be calculated with the help of the propagators of the collective field such as h∆x ∆x′ i. These are extremely complicated
functions. For this reason, any loop diagram formed with them is practically
impossible to integrate. In contrast to that, the higher-order diagrams in the
present theory need to be calulated using only ordinary particle propagators
G∆ and Gρ of Eq. (14.334) and the interaction (14.324). Even that becomes,
of course, tedious for higher orders in g. At least,
there is a simple rule to
R
find the contributions of the quadratic terms 21 x fxT Mx fx in (14.322), given
the diagrams without these terms. One calculates the diagrams from only the
four-particle interaction, and collects the contributions up to order g N in an
N
N
effective action ÃN
∆,ρ . Then one replaces Ã∆,ρ by Ã∆−ǫg∆,ρ−ǫgρ and re-expands
P
i i
everything in powers of g up to the order g N , forming a new series N
i=0 g Ã∆,ρ .
Finally one sets ǫ equal to 1/g [23] and obtains the desired collective classic
action AN
∆,ρ as an expansion extending (14.336) to:
AN
∆,ρ
=
N
X
i=0
Ãi∆,ρ
Z
− (1/g) (|∆|2 + ρ2 ).
x
(14.339) {@ExTRAcN
Note that this action must merely be extremized. There are no more quantum
fluctuations in the classical collective fields ∆, ρ. Thus, at the extremum, the
action (14.339) provides us directly with the desired grand-canonical potential.
• The second essential difference with respect to the Hubbard-Stratonovich
transformation approach is the following: It becomes possible to study a rich
variety of competing collective fields without the danger of double-counting
Feynman diagrams. One simply generalizes the matrix Mx subtracted from
new
Aloc
int to define Aint in different ways. For instance, we may subtract and add a
vector field ψ † σ a ψS a containing the Pauli matrices σ a and study paramagnon
fluctuations, thus generalizing the assumption (14.331) and allowing a spontaneous magnetization in the ground state. Or one may do the same thing with
a term ψ † σ a ∇i ψAia + c.c. added to the previous term. In this way we derive
the Ginzburg-Landau theory of superfluid He3 as in [4,24].
H. Kleinert, PARTICLES AND QUANTUM FIELDS
981
14.16 Gauge Fields and Gauge Fixing
An important property of the proposed procedure is that it yields good results
even in the limit of infinitely strong coupling. It was precisely this property which
led to the successful calculation of critical exponents of all φ4 -theories in the textbook [21] since critical phenomena arise in the limit in which the unrenormalized
coupling constant goes to infinity [57]. This is in contrast to another possibility.
For example that of carrying the variational approach to highers order via the socalled higher effective actions [26]. These where discussed in Chapter 13. There one
extremizes the Legendre transforms of the generating functionals of bilocal correlation functions, which sums up all two-particle irreducible diagrams. That does not
give physically meaningful results [27] in the strong-coupling limit, even for simple
quantum-mechanical models, as we have shown in Section 13.12.
The mother of this approach is Variational Perturbation Theory (VPT). Its
origin was a variational approach developed for quantum mechanics some years
ago by Feynman and the author [53]. in the textbook [21] to quantum field theory.
It converts divergent perturbation expansions of quantum mechanical systems into
exponentially fast converging expansions for any coupling strength [54].
What we have shown in this section is that this powerful theory can easily
be transferred to many-body theory, if we identify a variety of relevant collective
classical fields, rather than a fluctuating collective quantum field suggested by the
Hubbard-Stratonovich Transformation. To lowerst order in the coupling constant
this starts out with the standard Hartree-Fock-Bogoliubov approximation, and allows to go to higher oders with arbitrarily high accuracy.
14.16
Gauge Fields and Gauge Fixing
The functional integral formalism developed in the previous sections does not immediately apply to electromagnetism and any other gauge fields. There are subtleties
which we are now going to discuss. These will lead to an explanation of the mistake
in the vacuum energy observed in Eq. (7.506) when quantizing the electromagnetic
field via Gupta-Bleuler formalism. Consider a set of external electromagnetic currents described by the current density j µ (x). Since charge is conserved, these satisfy
∂µ j µ (x) = 0.
{gffpg}
{GaUGEF}
(14.340) {[email protected]}
The currents are sources of electromagnetic fields Fµν determined from the field
equation (12.49),
∂ν F νµ (x) = j µ (x),
(14.341) {[email protected]}
if we employ natural units with c = 1. The action reads
1 2
A =
d x − Fµν
(x) − j µ (x)Aµ (x)
4
h
Z
i 1
1
4
2
2
=
dx
E (x) − B (x) − ρφ(x) − j · A(x) .
2
c
Z
4
(14.342) {[email protected]}
982
14 Functional-Integral Representation of Quantum Field Theory
The field strengths are the four-dimensional curls of the vector potential Aµ (x):
Fµν = ∂µ Aν − ∂ν Aµ ,
(14.343) {[email protected]}
that satisfy, for single-valued fields Aµ , the Bianchi identity
(14.344) {@}
ǫµνλκ ∂ν Fλκ = 0.
The decomposition (14.343) is not unique. If we add to Aµ (x) the gradient of an
arbitrary function Λ(x),
Aµ (x) → AΛµ (x) = Aµ (x) + ∂µ Λ(x),
(14.345) {[email protected]}
then Λ does not appear in the field strengths, assuming that it satisfies
(∂µ ∂ν − ∂ν ∂µ )Λ(x) = 0,
(14.346) {12@foxnonc
i.e., the derivatives in front of Λ(x) commute. In the theory of partial differential
equations, this is referred to as the Schwarz integrability condition for the function
Λ(x). In general, a function Λ(x) which satisfies (14.346) in a simply-connected
domain can be defined uniquely in this domain. Only if Λ(x) fulfills this condition,
the transformation (14.345) is called a local gauge transformation.
If the domain is multiply connected, there is more than one path along which
to continue the function Λ(x) from one spatial point to another and Λ(x) becomes
multi-valued. This happens, for example, if (14.346) is nonzero on a closed line in
three-dimensional space, in which case the set of paths between two given points
decomposes into equivalence classes, depending on how often the closed line is encircled. Each of these paths allows another continuation of Λ(x). By (14.346), such
functions are not allowed in gauge transformations (14.345).
Let us now see how we can construct the generating functional of fluctuating
free electromagnetic fields in the presence of external currents jµ (x). As in (14.58),
we would like to calculate a functional integral, now in the unnormalized version
corresponding to (14.58):
Z0 [j] =
Z
DAµ (x)ei
R
2 /4−j Aµ )
d3 x(−Fµν
µ
(14.347) {[email protected]}
.
2
The Fµν
-terms in the exponents can be partially integrated and rewritten as
−
1Z 4 2
1Z 4 µ 2
d xFµν =
d xA (∂ gµν − ∂µ ∂ν )Aν .
4
2
(14.348) {[email protected]}
Hence we can identify the functional matrix D(x, x′ ) in (14.34) as
Dµν (x, x′ ) = (∂ 2 gµν − ∂µ ∂ν )δ (4) (x − x′ ).
(14.349) {[email protected]}
Recalling (14.88) it appears, at first, as though the generating functional (14.347)
should simply be equal to
1
Z0 [j] = (Det G0 µν )1/2 e− 2
R
d4 xd4 x′ jµ (x)G0 µν (x,x′ )jµ (x′ )
,
(14.350) {[email protected]}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
983
14.16 Gauge Fields and Gauge Fixing
where, by analogy with (14.35), we get
−1
G0 µν (x, x′ ) = iDµν
(x, x′ ).
(14.351) {@}
We have divided out a normalization factor N = (Det G0 µν )−1/2 , assuming that
we are dealing with an unnormalized version of Z0 [j]. Unfortunately, however, the
expression (14.350) is meaningless, since the inverse of the functional matrix (14.349)
does not exist. In order to see this explicitly we diagonalize the functional part (i.e.,
the x, y -part) of G0 µν (x, x′ ) by considering the Fourier transform
Dµν (q) = −q 2 gµν + qµ qν .
(14.352) {[email protected]}
For every momentum q, this matrix has obviously an eigenvector with zero eigenvalue, namely qµ . This prevents us from inverting the matrix Dµν (q). Correspondingly, when trying to form the inverse determinant of the functional matrix Dµν in
(14.350), we encounter an infinite product of infinities, one for every momentum q.
The difficulty can be resolved using the fact that the action (14.342) is gauge
invariant. For Fµν this is trivially true; for the source term j µ (x)Aµ (x) this is a consequence of current conservation. Indeed, if we change Aµ according to (14.345), the
R 4 µ
source term is changed
by
d x j (x)∂µ Λ(x). With the help of a partial integration,
R 4
this is equal to − d x ∂µ jµ (x)(x)Λ(x), and this expression vanishes due to current
conservation (14.340).
Because of this invariance, not all degrees of freedom, which are integrated over
in the functional integral (14.347), are associated with a Gaussian integral. The
fluctuations corresponding to pure gauge transformations leave the exponent invariant. Since a path integral is a product of infinitely many integrals from minus to
plus infinity, the gauge invariance of the integral produces an infinite product of infinite factors. This is precisely the origin of the infinity that occurs in the functional
determinant.
This infinity must be controlled by restricting the functional integrals to field fluctuations via some specific gauge condition. For example, the restriction is achieved
by inserting into the integral gauge-fixing functionals. Several examples have been
used:
F1 [A] = δ[∂µ Aµ (x)],
F2 [A] = δ[∇ · A(x)],
F3 [A] = δ[A0 (x)],
F3 [A] = δ[A3n(x)],
o
i R 4
F4 [A] = exp − 2α
d x [∂µ Aµ (x)]2 ,
F5 [A] = Dζe−i
R
R
d4 x ζHζ/2
(Lorenz gauge)
(Coulomb gauge)
(Hamiltonian gauge)
(14.353) {12@fngauco
(axial gauge)
(generalized Lorenz gauge)
δ[∂µ Aµ −ζ]×Det −1/2H. (’t Hooft gauge)
The first is a δ-functional enforcing the Lorenz gauge at each spacetime point:
∂µ Aµ (x) = 0. The second enforces the Coulomb gauge, the third corresponds to
the axial gauge, and the fourth is a generalized form of the Lorenz gauge used
984
14 Functional-Integral Representation of Quantum Field Theory
by Feynman and which also serves to derive the Feynman diagrams of the GuptaBleuler quantization formalism, thereby correcting the mistake in the vacuum energy. The fifth, finally, is a generalization of the fourth used
R by ’t Hooft that arises
R
−i d4 x ζ 2 /2α
by rewriting the fourth as a path integral F4 [A] = Dζe
δ[∂µ Aµ − ζ] and
generalizing the constant 1/α to a functional matrix A(x, x′ ). If we insert any of
these gauge-fixing functionals Fi [A] into the path integral, then gauge-transform the
vector potential Aµ (x) à la (14.345), and integrate functionally over all Λ(x), the
integral receives a finite contribution from that gauge function Λ(x) which enforces
the desired gauge. The result is a gauge-invariant functional of Aµ (x):
Φ[A] =
Z
Λ
DΛ F [A ] =
Z
DΛ F [A + ∂Λ] .
(14.354) {12@fngauco
Explicitly, we find for the above cases (14.353), the normalization functionals:
Φ1 [A] =
Φ2 [A] =
Φ3 [A] =
Φ4 [A] =
Φ5 [A] =
Z
Z
Z
Z
Z
DΛF1 [AΛ ] =
DΛF2 [AΛ ] =
DΛF3 [AΛ ] =
DΛF4 [AΛ ] =
DΛF5 [AΛ ] =
Z
Z
Z
Z
Z
DΛδ[∂µ Aµ (x) + ∂ 2 Λ],
(14.355) {12@fngauco
DΛδ[∇ · A(x) + ∇2 Λ],
(14.356) {nolabel}
Dδ[∇A0 (x) + ∂ 0 Λ],
(14.357) {nolabel}
DΛ exp −
DΛ
Z
i
2α
Z
Dζ exp −
d4 x [∂µ Aµ + ∂ 2 Λ]2 ,
i
2
Z
(14.358) {nolabel}
d4 x ζHζ δ[∂µ Aµ + ∂ 2 Λ−ζ]
×Det −1/2 H × Det (∂ 2 ). (14.359) {12@fngauco
If we form the ratios Fi [A]/Φi [A], we obtain gauge fixing functionals which all yield
unity when integrated over all gauge transformations. If any of these are inserted into
the functional integral (14.347), they will all remove the gauge degree of freedom,
and lead to a finite functional integral which is the same for each choice of Fi [A].
Let us calculate the functionals Φi [A] explicitly. For Φ1,2,3 [A] we simply observe
a trivial identity for δ-functions
δ(ax) = a−1 δ(x).
(14.360) {@}
This is proved by multiplying both sides with a smooth function f (x) and integrating
over x. The functional generalization of this is
δ[OA] = Det −1 O δ[A],
(14.361) {@}
where O is an arbitrary differential operator acting on the field Aµ (x). From this
we find immediately the normalization functionals:
Φ1 [A] = Det −1 (∂ 2 ),
Φ2 [A] = Det −1 (∇2 ),
Φ3 [A] = Det −1 (∂ 0 ).
(14.362) {12@fngauco
(14.363) {12@fngauco
(14.364) {12@fngauco
H. Kleinert, PARTICLES AND QUANTUM FIELDS
985
14.16 Gauge Fields and Gauge Fixing
The fourth functional Φ4 [A] is simply a Gaussian functional integral. The additive term ∂µ Aµ (x) can be removed by a trivial shift of the integration variable
Λ(x) → Λ′ (x) = Λ(x) − ∂µ Aµ (x)/∂ 2 , under which the measure of integration remains invariant, DΛ = DΛ′ . Using formula (14.25) we obtain
√
(14.365) {12@fngauco5
Φ4 [A] = Det −1 (∂ 2 / α).
The functional determinants Φ−1
are called Faddeev-Popov determinants [49].
i
The Faddeev-Popov determinants in the four examples happen to be independent
of Aµ (x), so that we shall write them as Φi without arguments. This independence
is a very useful property. Complications arising for A-dependent functionals Φ[A]
will be illustrated below in an example.
We now study the consequences of inserting the gauge-fixing factors Fi [A]/Φi
into the functional integrands (14.347). For F4 [A]/Φ4 , the generating functional
becomes
Z0 [j] = (Det G0 µν )
1/2
Det
1/2
4
(∂ /α)
Z
DAµ (x)ei
R
2 − 1 (∂ µ A )2 −j µ A
d4 x[− 14 Fµν
µ
µ]
α
. (14.366) {[email protected]}
The free-field action in the exponent can be written in the form
A=
Z
1
1
(∂ µ Aµ )2 .
d x − (∂µ Aν )2 + 1 −
2
α
4
(14.367) {[email protected]}
The associated Euler-Lagrange equation is
1
∂ µ (∂A) = 0,
∂ A − 1−
α
2
µ
(14.368) {[email protected]}
which is precisely the field equation (7.381) of the covariant quantization scheme.
With the additional term in the action, the matrix (14.352) becomes
"
Dµν (q) = −q 2 gµν
#
1 qµ qν
− 1−
.
α q2
(14.369) {[email protected]}
This can be decomposed into projection matrices with respect to the subspaces
transverse and longitudinal to the four-vector q µ ,
l
Pµν
(q) =
qµ qν
,
q2
qµ qν
,
q2
(14.370) {[email protected]}
1 l
+ Pµν
(q) .
α
(14.371) {[email protected]}
l
Pµν
(q) = gµν −
as
Dµν (q) = −q
2
t
Pµν
(q)
It is easy to verify that the matrices are really projections, since they satisfy
l
t
l
l
t
Pµν
P lν λ = Pµν
, Pµν
P lν λ = Pµλ
, Pµν
P lν λ = 0.
(14.372) {12@foproject
986
14 Functional-Integral Representation of Quantum Field Theory
Similar projections appeared before in the three-dimensional subspace [see (4.334)
and (4.336)], where the projections were indicated by capital subscripts T , L.
Due to the relations (14.372), there is no problem in inverting Dµν (q), and we
find the free photon propagator in momentum space
i t
l
[P (q) + αPµν
(q)]
q 2 µν
#
qµ qν
− (1 − α) 2 .
q
−1
G0 µν (q) = iDµν
(q) = −
"
i
= − 2 gµν
q
(14.373) {[email protected]}
For α = 0, this is the propagator Gµν derived in the Gupta-Bleuler canonical field
quantization in Eq. (7.510), there obtained from canonical quantization rules with
a certain filling of the vacuum with unphysical states.
Taking into account the Faddeev-Popov determinant Φ−1
4 of (14.365), we obtain
for the generating functional (14.366):
1
Z[j] = [Det (−iG0 µν )]1/2 Det 1/2 (∂ 4 /α)e− 2
R
d4 xd4 x′ j µ (x)G0 µν (x,x′ )j ν (x′ )
.
(14.374) {[email protected]}
Let us calculate the functional determinant Det G0 µν . For this we take q µ in
the momentum representation (14.373) along the 0-direction, and see that there
are three spacelike eigenvectors of eigenvalues iq 2 , and one timelike eigenvector of
eigenvalue iα/q 2 . The total determinant is therefore
Det (−iG0 µν ) =
1
Det (−∂ 2 )2 Det (∂ 4 /α)
.
(14.375) {@}
The two prefactors of (14.374) together yield a factor
[Det (−iG0 µν )]1/2 Det 1/2 (∂ 4 /α) ∝
1
.
Det (−∂ 2 )
(14.376) {12@fsameas
Recalling the discussion of Eqs. (14.129)–(14.133), we see that the associated free
energy is
F0 = −β log Z0 = 2
X
k
(
)
i
h
ω(k)
+ log 1 − e−βh̄ω(k) .
2
(14.377) {12@freenscb
It contains precisely the energy of the two physical transverse photons. The unphysical polarizations have been eliminated by the Faddeev-Popov determinant in
(14.374).
We now understand why the Gupta-Bleuler formalism failed to get the correct
vacuum energy in Eq. (7.506). It has no knowledge of the Faddeev-Popov determinant.
Taking into account current conservation, the exponent in (14.374) reduces to
j µ (x)G0 µν (x, x′ )j ν (x′ ) = j µ (x)
i
jµ (x),
∂2
(14.378) {@}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
987
14.16 Gauge Fields and Gauge Fixing
so that the generating functional becomes
1
Z[j] = const × Det −1 (−∂ 2 )e− 2
R
d4 xd4 x′ j µ (x)(i/∂ 2 )jµ (x′ )
(14.379) {[email protected]}
,
where const is an infinite product of identical constant factors.
Let us see what happens in the other gauges (14.353). The results in the Lorenz
gauge are immediately obtained by going to the limit α → 0 in the previous calculation. In this limit, the functional Φ4 coincides with Φ1 , up to a trivial factor.
The Hamiltonian and the axial gauges are quite similar, so we may only discuss
one of them. In the Hamiltonian gauge, where the Faddeev-Popov determinant is
given by (14.364), the generating functional (14.374) becomes
1
Z[j] = (Det G0 µν )1/2 Det (∂0 )e− 2
R
d4 xd4 x′ j i (x)G0 ij (x,x′ )j j (x′ )
,
(14.380) {[email protected]}
where the matrix G0 ij (q) has only spatial entries, and is equal to
G0 ij (q) = iDij−1 (q)
(14.381) {@}
with
Dij (q) = q 2 δij + qi qj
(14.382) {[email protected]
being the spatial part of the 4×4 -matrix (14.352). With the help of 3×3 -projection
matrices
PijT (q) = δij − qi qj /q2
PijL (q)
(14.383) {[email protected]}
2
(14.384) {nolabel}
= qi qj /q ,
this can be decomposed as follows:
Dij (q) = q 2 PijT (q) + q 0 2 PijL (q).
(14.385) {[email protected]}
The inverse of this is
Dij−1 (q)
qi qj
1
1
1
= 2 PijT (q) + 0 2 PijL (q) = 2 δij − 0 2
q
q
q
q
!
(14.386) {[email protected]}
.
In the exponent of (14.380) we have to evaluate iDij−1 (q) between two conserved
currents, and find
"
#
i
1
j i∗ (q)G0 ij (q)j j (q) = − 2 j∗ (q) · j(q) − 0 2 q · j∗ (q) q · j(q) .
q
q
(14.387) {[email protected]}
Inserting the momentum space version of the local current conservation law
∂µ j µ (x) = 0:
q · j(q) = q 0 j 0 (q),
(14.388) {@}
we obtain
j i∗ (q)G0 ij (q)j j (q) =
i µ∗
j (q)jµ (q).
q2
(14.389) {[email protected]
988
14 Functional-Integral Representation of Quantum Field Theory
Let us calculate the functional determinant Det G0 µν in this gauge. From
(14.386) we see that G0 µν has two eigenvectors of eigenvalue i/q 2 , and one eigenvector of eigenvalue −i/q 0 2 . Hence:
Det G0 µν =
1
Det (−i∂ 2 )2 Det (i∂02 )
(14.390) {@}
.
The two prefactors in (14.380) together are therefore proportional to
(Det G0 µν )1/2 Det (∂0 ) ∝
1
,
Det (−i∂ 2 )
(14.391) {@}
which is the same as (14.376). Thus the generating functional (14.380) agrees with
the previous one in (14.379).
Let us also show that the Coulomb gauge leads to the same result. We rewrite
the exponent in (14.379) in momentum space as
Z
1
d xj (x) 2 jµ (x) =
−∂
4
µ
d4 q
1
1
c2 ρ(q) 2 ρ(q) − c2 j(q) 2 j(q) .
4
(2π)
q
q
"
Z
#
(14.392) {@WWIR}
Here we keep explicitly the light velocity c in all formulas since we want to rederive
the interaction equivalent to Eq. (12.85) where c is not set equal to unity. Writing
the denominator as q 2 = q02 − q2 and j 2 (q) = j2L (q) + j2T (q) with jL (q) = q · j(q)/|q|
and jT (q) · jL (q) = 0, we can bring (14.392) to the form
Z
1
1
1
d4 q
c2 ρ(q) 2
ρ(q) − jL (q) 2
jL (q) − jT (q) 2
jT (q) . (14.393) {@WWIR1}
4
2
2
(2π)
q0 − q
q0 − q
q0 − q2
"
#
Now we use the current conservation law cq0 ρ(q) = q · j(q) to rewrite (14.393) as
Z
2
d4 q
1
1
1
2
2 q0
c
ρ(q)
ρ(q)
−
c
ρ(q) 2
ρ(q) − jT (q) 2
jT (q) . (14.394) {@WWIR2}
4
2
2
2
2
(2π)
q −q
q
q0 − q
q0 − q2
"
#
The first two terms can be combined to
Aint
L = −
Z
1
d4 q 2
c
ρ(q)
ρ(q).
(2π)4
q2
(14.395) {10.68XX}
Undoing the Fourier transformation and multiplying this by e2 /c2 we find as a first
term in the interaction (12.84) the Coulomb term which is is the longitudinal part
of the interaction (12.83):
Aint
L
e2
1
1
d4 x E2L (x) =
d4 x ρ(x) 2 ρ(x)
= −
2
2
∇
Z
Z
1
e2
dt d3 xd3 x′ ρ(x, t)
ρ(x′ , t).
=−
8π
|x − x′ |
Z
Z
(14.396) {10.68X}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
989
14.17 Nontrivial Gauge and Faddeev-Popov Ghosts
The third term in (14.394) involves only the transverse current
Aint
T
=−
Z
1
d4 q ∗
jT (q) 2 jT (q).
4
(2π)
q
(14.397) {@WWIR2T}
It is the result of the transverse fields in the Lagrangian (12.54):
Aint
T =
1
4π
1
d4 x{[E2T (x) − B2 (x)] + j(x)AT (x)}.
c
Z
(14.398) {@}
It is found by integrating out the vector potential. Using Eq. (5.56), the transverse
interaction (14.397) can be rewritten as
Aint
T
=
Z
1 µ
1
d4 q
∗
int
j
(q)
j
(q)
+
|j0 (q)|2 = Aint
µ
tot − AL .
4
2
2
(2π)
q
q
"
#
(14.399) {@WWIR2TT
The transverse part of the electromagnetic action of a four-dimensional current
j µ (q) is the difference between the total covariant Biot-Savart interaction plus the
instantaneous Coulomb interaction.
{ENERGYRE
14.17
Nontrivial Gauge and Faddeev-Popov Ghosts
The Faddeev-Popov determinants in the above examples were all independent of the
fields. As such they were irrelevant for the calculation of any Green function. This,
however, is not always true.
As an example, consider the following nontrivial gauge-fixing functional (see also
[55])
i
h
F [A] = δ (∂A)2 + gA2 .
(14.400) {[email protected]}
As in Eqs. (14.355)–(14.359), we calculate
Φ[A] =
Z
Λ
DΛF [A ] =
Z
DΛ δ[∂A + gA2 + ∂ 2 Λ + 2gA∂Λ + g(∂Λ)2 ]. (14.401) {12@fngauco2
This path integral can trivially be performed by analogy with the ordinary integral
Z
1
dx δ(ax + bx2 ) = ,
a
(14.402) {[email protected]}
and yields the result
Φ[A] = Det (∂ 2 + 2gAµ ∂ µ )−1 .
(14.403) {[email protected]}
The generating functional is therefore
Z[j] =
Z
µ
2
µ
µ
2
Z
DA Det (∂ + 2gA ∂µ ) δ[∂µ A + gA ] exp i
1 2
− jµ Aµ .
d x − Fµν
4
(14.404) {[email protected]}
4
990
14 Functional-Integral Representation of Quantum Field Theory
Contrary to the previous gauges, the functional determinant is no longer a trivial
overall factor, but it depends now functionally on the field Aµ . It can therefore no
longer be brought outside the functional integral.
There is a simple way of including its effect within the usual field-theoretic
formalism. One introduces an auxiliary Faddeev-Popov ghost field. We may consider
the determinant as the result of a fluctuating complex fermion field c with a complexconjugate c∗ , and write
Det (∂ 2 + 2gAµ ∂µ ) =
Z
Dc∗ Dce−i
R
d4 x(∂c∗ ∂c−2gAµ c∗ ∂µ c)
(14.405) {[email protected]}
.
Note that the Fermi fields are necessary to produce the determinant in the numerator; a Bose field would have put it into the denominator. A complex field is taken
to make the determinant appear directly rather than the square-root of it.
The ghost fields interact with the photon fields. This interaction is necessary in
order to compensate the interactions induced by the constraint δ[∂Aµ + gA2 ] in the
functional integral.
It is possible to exhibit the associated cancellations order by order in perturbation
theory. For this we have to bring the integrand to a form in which all fields appear
in the exponent. This can be achieved for the δ-functional by observing that the
same representation (14.404) would be true with any other choice of gauge, say
δ[∂µ Aµ (x) + gA2 (x) − λ(x)],
(14.406) {[email protected]}
since this would lead to the same Faddeev-Popov ghost term (14.405). Therefore
we can average over all possible functions λ(x) with a Gaussian weight and replace
(14.406) just as well by
Z
i
Dλe− 2
R
d4 xλ2 (x)
δ[∂µ Aµ (x) + gA2 (x) − λ(x)].
(14.407) {[email protected]}
Now the generating functional has the form
Z[j] =
Z
DAµ
Z
Dc∗ Dc ei
R
d4 x(L−j µ Aµ )
(14.408) {[email protected]}
with a Lagrangian
2
1 2
1 L = − Fµν
∂A + gA2 − ∂µ c∗ ∂ µ c + 2gAµ c∗ ∂µ c.
−
4
2α
(14.409) {[email protected]}
The photon and ghost propagators are
Aµ (x)Aν (0) = −
Z
kµ kν
d4 k −ikx i
e
gµν − (1 − α) 2 ,
4
2
(2π)
k
k
(14.410) {408}
c∗ (x)c(0) = −
Z
d4 q −iqx i
e
.
(2π)4
q2
(14.411) {@}
"
#
H. Kleinert, PARTICLES AND QUANTUM FIELDS
991
14.17 Nontrivial Gauge and Faddeev-Popov Ghosts
Contrary to the previous gauges, there are now photon-ghost and photon-photon
interaction terms
g
g 2 2 2
− ∂µ Aµ A2 −
Aµ + 2gAµ c∗ ∂µ c
α
2α
(14.412) {[email protected]}
with the corresponding vertices
(14.413)
(14.414)
(14.415)
.
It can be shown that the Faddeev-Popov ghost Lagrangian has the property of
canceling all these unphysical contributions order by order in perturbation theory.
We leave it as an exercise to show, for example, that there is no contribution of
the ghosts to photon-photon scattering up to, say, second order in g, and that selfenergy corrections to the photon propagator due to photon and ghost loops cancel
exactly.
In the context of quantum electrodynamics, there is little sense in using a gauge
fixing term (14.400). The present discussion is, however, a useful warm-up exercise
to gauge-fixing procedures in nonabelian gauge theories, where the Faddeev-Popov
determinant will always be field dependent.
Of course, also the previous field-independent Faddeev-Popov determinants can
be generated from fermionic ghost fields. The determinant Φ−1
in (14.365), for
4
example, can be generated from a complex ghost field c(x) and its complex conjugate
c∗ by a functional integral
Φ−1
4
=
Z
∗
i
Dc Dc e
R
√
d4 x ∂c∗ ∂c/ α
.
(14.416) {[email protected]}
It should be pointed out that the signs of the kinetic term of these c-field Lagrangians are opposite to those of a normal field. If these fields were associated with
particles, their anticommutation rules would carry the wrong sign and the states
would have a negative norm. Such states are commonly referred to as ghosts, and
992
14 Functional-Integral Representation of Quantum Field Theory
this is the reason for the name of the fields c, c∗ . Note that the determinant cannot
be generated by a real fermion field via a functional integral
?
Φ−1
4 =
Z
Dc e−i
R
d4 x ∂ 2 c∂ 2 c/α
(14.417) {@}
.
The reason is that the differential operator ∂ 4 is a symmetric functional matrix, so
that the exponent vanishes after diagonalization by an orthogonal transformation.
In the language of ghost fields, the mistake in calculating the vacuum energy
in Eq. (7.506), that arose when quantizing the electromagnetic field via the GuptaBleuler formalism, can be phrased as follows. When fixing the gauge in the action
(7.376) by adding an Lagrangian density LGF (10.86), we must also add a ghost
Lagrangian density
√
Lghost = ∂c∗ ∂c/ α.
(14.418) {@GHOSTD
The ghost fields have to be quantized canonically, and the physical states must
satisfy, beside the Gupta-Bleuler subsidiary condition in Eq. (7.502), the condition
of being a vacuum to the ghost fields:
c|ψ“phys” i = 0.
(14.419) {12@fughostl
The Faddeev-Popov formalism is extremely useful offering many other possibilities of fixing a gauge and performing the functional integral for the generating
functional over all Aµ -components.
14.18
Functional Formulation of Quantum Electrodynamics
For quantum electrodynamics, the functional integral from which we can derive all
time-ordered vacuum expectation values reads [1,3,5]:
Z[j, η, η̄] =
Z
DψD ψ̄DAµ DD eiA−i
R
d4 x [ψ̄(x)η(x)+η̄(x)ψ(x)]−i
R
d4 x jµ (x)Aµ (x)
,
(14.420) {12@fintrepQ
where A is the sum of the free-field action (12.86) and the minimal interaction
(12.87), which we shall write here as
A=
Z
e
1
d4 x ψ̄ i/
∂ − A
/ − m ψ − F µν Fµν − D∂ µ Aµ + D 2 /2 .
c
4
(14.421) {[email protected]}
The Dirac field appears only quadratically in the action. It is therefore possible to
integrate it out using the Gaussian integral formula (14.95), and we obtain
Z[j, η, η̄] = Det (i/
∂ − M)
Z
′
DAµ DD eiA −
R
d4 x d4 x′ η̄(x)G0 (x,x′ )η(x′ )ψ(x)−i
′
where A is the action
A=
Z
1
d x − F µν Fµν − D∂ µ Aµ + D 2 /2 + Aeff [A].
4
4
R
d4 x jµ (x)Aµ (x)
,
(14.422) {12@fintrepQ
(14.423) {@}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
14.18 Functional Formulation of Quantum Electrodynamics
993
The effects of the electron are collected in the effective action
eff
ei∆A
= exp [Tr log (i/
∂ − e/
A − M) − Tr log (i/
∂ − M)] ,
(14.424) {12@@widthE
where TrO combines the functional trace of the operator O and the matrix trace
in the 4 × 4 space of Dirac matrices. We have performed a subtraction of the
infinite vacuum energy caused by the filled-negative energy states. The subtraction
is compensated by the determinant in the prefactor of Eq. (14.422).
14.18.1
Decay Rate of Dirac Vacuum in Electromagnetic Fields
An important application of the functional formulation (14.422) of quantum electrodynamics was made by Heisenberg and Euler [32,33]. They observed that in
a constant external electric field the vacuum becomes unstable. There exists a finite probability of creating an electron-positron pair. This process has to overcome
a large energy barrier 2Mc2 , but if the pair is separated sufficiently far, the total
energy of the pair can be made arbitrarily low, so that the process will occur with
nonzero probability. The rate can be derived from the result (6.254). For a large
total time ∆t, the time dependence of an unstable vacuum state will have the form
e−i(E0 −iΓ/2)∆t , where E0 is the vacuum energy, Γ the desired decay rate, and ∆t the
total time over which the amplitude is calculated. If Leff is the effective Lagrangian
density causing the decay, and Aeff the associated action, we identify
Γ
Aeff
= 2 Im
= 2 ImLeff .
V
V ∆t
(14.425) {12@@identde
We now make use of the fact that, due to the invariance under charge conjugation,
the right-hand side of (14.424) can depend only on M 2 . Thus we also have
exp [Tr log (i/
∂ −e/
A −M)] = exp [Tr log (i/
∂ −e/
A +M)]
1
[Tr log (i/
∂ − e/
A −M) (i/
∂ − e/
A +M)] , (14.426) {@}
= exp
2
and may use the product relation (6.107) to calculate (14.424) from half the trace
log of the Pauli operator in Eq. (6.241) to find the effective action
o
n
1
1
e
i∆A = Tr log [i∂ − eA(x)]2 − σ µ ν Fµ ν − M 2 − Tr log −∂ 2 − M 2 .
2
2
2
(14.427) {12@@Trextf}
We now use the integral identity
eff
Z ∞
i
dτ h iaτ
a
e − eibτ
log = −
b
τ
0
(14.428) {@}
and relation (6.206) to rewrite (14.427) as
i∆Aeff = −
1
2
Z
0
∞
2 e µν e
dτ −iτ (M 2 −iǫ)
2
e
trhx|eiτ {[i∂−eA(x)] + 2 σ Fµν } − e−iτ ∂ |xi. (14.429) {12@probabil
τ
994
14 Functional-Integral Representation of Quantum Field Theory
Recalling (6.189) and (6.206), the first, unsubtracted, term can be re-expressed as
i∆Aeff
1
1 Z ∞ dτ
1 Z ∞ dτ
−iτ Ĥ
=−
trhx|e
|xi = −
trhx, τ |x 0i.
2 0 τ
2 0 τ
(14.430) {12@proba01
Inserting (6.254), and subtracting the field-free second term in (14.429), we obtain
the contribution to the effective Lagrangian density
∆L
eff
1
=
2(2π)2
Z
0
∞
dτ
τ3
eEτ
2
− 1 e−iτ (M −iη) .
tanh eEτ
(14.431) {4@1steq5aa}
The integral over τ is logarithmically divergent at τ = 0. We can separate the
divergent term by a further subtraction, splitting
eff
∆Leff = ∆Leff
div + ∆LR
(14.432) {4@1steq5a0}
into a convergent integral
∆Leff
R
1
=
2(2π)2
Z
∞
0
dτ
τ3
e2 E 2 τ 2 −iτ (M 2 −iη)
eEτ
e
,
−1−
tanh eEτ
3
!
(14.433) {4@1steq5a}
and a divergent one
∆Leff
div
e2
1
= E2
2
3(2π)2
Z
∞
0
dτ −iτ (M 2 −iη)
e
.
τ
(14.434) {12@2@LogD
The latter is proportional to the electric part of the original Maxwell Lagrangian
density in (4.237). It can therefore be removed by renormalization. We add (14.434)
to the Maxwell Lagrangian density, and define a renormalized charge eR by the
equation
Z ∞
1
1
dτ −iτ (M 2 −iη)
1
1
=
+
e
≡
,
(14.435) {12@@Z3DE
2
eR
e2 12π 2 0 τ
Z3 e2
to obtain the modified electric Lagrangian density
LE =
e2 2
E .
2e2R
(14.436) {12@2@LogD
Now we redefine the electric fields by introducing renormalized fields
ER ≡
e
E,
eR
(14.437) {12@@RFIEL
and identify these with the physical fields. In terms of these, (14.436) takes again
the usual Maxwell form
1
LE = ER2 .
2
(14.438) {12@2@LogD
H. Kleinert, PARTICLES AND QUANTUM FIELDS
995
14.18 Functional Formulation of Quantum Electrodynamics
The finite effective Lagrangian density (14.433) possesses an imaginary part
which by Eq. (14.425) determines the decay rate of the vacuum per unit volume
Γ
V
1
= Im
(2π)2
Z
∞
0
dτ
τ3
e2 E 2 τ 2
eEτ
−1−
tanh eEτ
3
!
e−iτ (M
2 −iη)
(14.439) {4@1steq5}
.
For comparison we mention that, for a charged boson field, the expression
(14.424) is replaced by
eff
ei∆A
h
o
n
= exp −Tr log [i∂ − eA(x)]2 − M 2 + Tr log −∂ 2 − M 2
i
.
(14.440) {12@@widthE
Hence the last factor 4 cosh eEτ in (6.254) is simply replaced by −2, and the unsubtracted effective action (14.439) becomes
i∆Aeff
u
=
Z
0
∞
dτ
i
hx, τ |x 0i = −
τ
4(2π)2
Z
0
∞
dτ eEτ
2
e−iτ (M −iη) ,
3
τ sinh eEτ
(14.441) {4@1steq5B}
implying a twice subtracted effective Lagrangian density
∆Leff
R
1 Z ∞ dτ
=−
4(2π)2 0 τ 3
e2 E 2 τ 2 −iτ (M 2 −iη)
eEτ
e
,
−1+
sinh eEτ
6
(14.442) {4@1steq5aB}
dτ −iτ (M 2 −iη)
1
e
≡
.
τ
Z3 e2
(14.443) {12@@Z3DEF
!
and a charge renormalization
1
1
1
= 2− 2
2
eR
e
6π
Z
0
∞
The decay rate per unit volume is
ΓKG
1
= Im
V
4(2π)2
Z
0
∞
dτ
τ3
e2 E 2 τ 2
eEτ
−1+
sinh eEτ
6
!
e−iτ (M
2 −iη)
.
(14.444) {4@1steq5b}
The integrands in (14.439) and (14.444) are even in τ , so that the integrals for
the decay rate can be extended symmetrically to run over the entire τ -axis. After
this, the contour of integration can be closed in the lower half-plane and the integral
can be evaluated by Cauchy’s residue theorem. To find the pole terms we expand
the integrand for fermions in Eq. (14.439) as
∞
∞
X
X
(eετ )2
τ2
eετ
,
−1=
=
2
2
2 2
2
2
tanh eετ
n=−∞ (eετ ) + n π
n=1 τ + τn
τn ≡
nπ
.
eε
(14.445) {12@@expCO
The relevant poles lie at τ = −iτn and yield the result
∞
Γ
1 −nπM 2 /eE
e2 E 2 X
=
e
.
3
V
4π n=1 n2
(14.446) {12@@resuFE
A technical remark is necessary at this place concerning the integral (14.444).
At first sight it may appear as if the second subtraction term in the integrand
996
14 Functional-Integral Representation of Quantum Field Theory
e2 E 2 τ 2 /3 can be omitted [29]. First, it is unnecessary to arrive at a finite integral in
the imaginary part, and second, it seems to contribute only to the real part, since
for all even powers α > 2, the integral
Z
0
∞
1
dτ α −iτ (M 2 −iη)
τ e
=
Γ(α − 2)
3
τ
(iM 2 )α−2
(14.447) {12@@}
is real. The limit at hand α → 2, however, is an exception since for α ≈ 2, the
integral possesses an imaginary part due to the divergence at small τ
1
π
2
Γ(α
−
2)
≈
−γ
−
log
M
−
i
+ O(α − 1).
(iM 2 )α−2
2
(14.448) {12@@}
The right-hand side of Eq. (14.446) is a polylogarithmic function (2.274), so that
we may write
Γ
e2 E 2
2
=
ζ2 (e−πM /eE ).
(14.449) {12@@}
3
V
4π
For large fields, this has the so-called Robinson expansion [30]
ζν (e−α ) = Γ(1 − ν)αν−1 + ζ(ν) +
∞
X
1
(−α)k ζ(ν − k).
k!
k=1
(14.450) {4@dualsump
This expansion plays an important role in the discussion of Bose-Einstein condensation [31]. For ν → 2, the Robinson expansion becomes
π2
α2 α3
ζ2 (e ) =
+ (−1 + log α) α −
+
+ O(α5 ).
6
4
72
Hence we find the strong-field expansion
−α

(14.451) {4@dualsump

#
!
!2
!3
"

Γ
1 πM 2
e2 E 2  π 2
πM 2
πM 2
1 πM 2
−
=
+
−1
+
log
+
+
.
.
.
.

V
4π 3  6
eE
eE
4 eE
72 eE
(14.452) {12@@4decra
For bosons, we expand
∞
∞
X
X
(eEτ )2
τ2
eετ
nπ
n
(−1)
,
τ
≡
−1 = 2 (−1)n
=
2
.
n
sinh eετ
(eEτ )2 + n2 π 2
τ 2 + τn2
eE
n=1
n=1
(14.453) {12@@}
Comparison with (14.445) shows that the bosonic result for the decay rate differs
from the fermionic (14.446) by an alternating sign, accounting for the different
statistics. There is also a factor 1/2, since there is no spin. Thus we find the decay
rate per volume
∞
ΓKG
1
e2 E 2 1 X
2
(−1)n−1 2 e−nπM /eE .
=
(14.454) {12@@resuBO
3
V
4π 2 n=1
n
The sum can again be expressed in terms of the polylogarithmic function (14.450)
as follows:
∞
X
k
∞
∞
X
X
(−1)k−1 z k
zk
z2
ζ̃ν (z) ≡
=
−
2
= ζν (z) − 21−ν ζν (z 2 ).
ν
ν
ν
k
k
(2k)
k=1
k=1
k=1
(14.455) {12@@}
H. Kleinert, PARTICLES AND QUANTUM FIELDS
997
14.18 Functional Formulation of Quantum Electrodynamics
For z = e−α ≈ 1, the Robinson expansion (14.451) yields
ζ̃2 (e−α ) =
α2 α3
α5
π2
− α log 2 +
−
+
+ O(α8 ).
12
4
24 960
(14.456) {12@@}
The expansion ζ̃2 (e−α ) replaces the curly bracket in (14.452), if we set α = πM 2 /eE.
14.18.2
Constant Electric and Magnetic Background Fields
In the presence of both E and B fields, Eq. (6.253) reads
e
exp −i σ µ ν Fµ ν τ
2
=
0
e−ie ·(B−iE)τ
−ie ·(B+iE)τ
0
e
!
.
(14.457) {@which ise2}
The trace of this can be found by adding the traces of the 2 × 2 block matrices ee ·(−iB∓E)τ separately. These are equal to e−eλ1 τ + e−eλ2 τ and their complex
conjugates, respectively, where λ1 , λ2 are the eigenvalues of the matrix ·(−iB−E):
q
√
λ1,2 = ± (E + iB)2 = ± E2 − B2 + 2iE B.
(14.458) {12@@lambda
Thus we find
e
tr exp −i σ µ ν Fµ ν τ
2
= 2(cos eλ1 + cos eλ∗1 ).
(14.459) {@which ise3}
The eigenvalues are, of course, Lorentz-invariant quantities. They depend only on
the two quadratic Lorentz invariants of the electromagnetic field: the scalar S and
the pseudoscalar P defined by
1
P ≡ − Fµν F̃ µν = E B.
4
1 2
1
E − B2 ,
S ≡ − Fµν F µν =
4
2
(14.460) {12@@SPINV
In terms of these, Eq. (14.458) reads
√ √
λ1,2 = ± 2 S + iP ,
(14.461) {12@@}
which can be rewritten as
1/4
√ λ1,2 = ± 2 S 2 + P 2
(cos ϕ/2 + i sin ϕ/2) ,
(14.462) {12@@4stequ
where
tan ϕ =
P
,
S
(14.463) {12@@}
so that
cos ϕ = √
S
,
S2 + P 2
sin ϕ = √
P
,
S2 + P 2
(14.464) {12@@}
998
14 Functional-Integral Representation of Quantum Field Theory
q
implying that cos ϕ/2 =
(1 + cos ϕ)/2 and sin ϕ/2 =
(
cos ϕ/2
sin ϕ/2
)
q√
= √
S2 + P 2 ± S
2 (S 2 + P 2)1/4
q
(1 − cos ϕ)/2, or
(14.465) {12@@}
.
We shall abbreviate the result (14.462) by
λ1,2 = ± (ε + iβ) ,
(14.466) {12@@}
where
( )
ε
β
≡
q√
S2
+
P2
r
1 q 2
±S = √
(E − B2 )2 + 4(E B)2 ± (E2 − B2 ). (14.467) {12@@4stequ
2
In terms of ε and β, the invariants S and P in (14.460) become
1 2
1 2
1
E − B2 =
ε − β2 ,
S ≡ − Fµν F µν =
4
2
2
and the trace (14.459) is simply
e
tr exp −i σ µ ν Fµ ν τ
2
1
P ≡ − Fµν F̃ µν = E B = εβ,
4
(14.468) {12@@SPEP
= 2 cosh (ε + iβ) + 2 cosh (ε − iβ) = 4 cosh ε cos β. (14.469) {12@@4trace
Some special cases will simplify the upcoming formulas:
1. If B = 0, then ε reduces to |E|, whereas for E = 0, β reduces to |B|.
2. If E 6= 0 and B 6= 0 are orthogonal to each other, then we have either β = 0
for E > B, or ε = 0 for B > E. The formulas are then the same as for pure
electric or magnetic fields.
3. If E 6= 0 and B 6= 0 are parallel to each other, then ǫ = |E| =
6 0, β = |B| =
6 0.
In all these cases, the calculation of the exponential (14.457) can be done very
simply. Take the third case. Due to rotational symmetry, we can assume the fields
to point in the z-direction, B = Bẑ, E = Eẑ, and the exponential (14.457) has the
matrix form
e−ie σ3 (B−iE)τ
0
−ie σ3 (B+iE)τ
0
e
e
exp −i σ µ ν Fµ ν τ =
2




=
!
e−ie(B−iE)τ
0
0
0
ie(B−iE)τ
0
e
0
0
−ie(B+iE)τ
0
0
e
0
0
0
0
eie(B+iE)τ
(14.470) {@which ise4



.

This has the trace
e
tr exp −i σ µ ν Fµ ν τ = 2 cosh (E + iB) τ + 2 cosh (E − iB) τ = 4 cosh Eτ cos Bτ ,
2
(14.471) {12@@4trace
H. Kleinert, PARTICLES AND QUANTUM FIELDS
999
14.18 Functional Formulation of Quantum Electrodynamics
in agreement with (14.469). In fact, given an arbitrary constant field configuration
B and E, it is always possible to perform a Lorentz transformation to a coordinate
frame in which the transformed fields, call them BCF and ECF , are parallel. This
frame is called center-of-fields frame. The transformation has the form (4.285) and
(4.286) with a velocity of the transformation determined by
E×B
v/c
=
.
2
1 + (|v|/c)
|E|2 + |B|2
(14.472) {12@eblandau
By Lorentz invariance, we see that
E2 − B2 = E2CF − B2CF ,
E · B = ECF · BCF,
(14.473) {12@@}
which shows that |ECF| and |BCF | in Eq. (14.471) coincide with ε and β in
Eq. (14.468). This is the reason why Eq. (14.471) gives the general result for arbitrary constant fields, if E and B are replaced by |ECF | = ε and |BCF | = β.
These considerations permit us to present a simple alternative calculation of a
determinantal prefactor that occurred much earlier in Chapter 6, in particular in
Eqs. (6.214) and (6.225). For a general constant field strength Fµ ν , the basic matrix
that had to be diagonalized was eeF τ in Eq. (6.245). For an electric field pointing
in the z-direction, this has the form e−iM3 eEτ . In contrast to (6.253), this is a boost
matrix with rapidity ζ = eEτ in the defining 4 × 4 -representation [compare (4.63)].
The fact that the rapidity in the Dirac representation (6.252) was twice as large, has
its origin in the value of the gyromagnetic ratio 2 of the Dirac particle in Eq. (6.119).
If the field points in any direction, the obvious generalization is e−iMEeτ . In the
presence of a magnetic field, the generators of the rotation group (4.57) enter and
(6.245) can be written as eeF τ = e−ie(ME+LB)τ . This is the defining four-dimensional
representation of the complex Lorentz transformation, whose chiral Dirac representation was written down in (14.457), apart from the factor 2 multiplying the rapidity
and the rotation vectors.
As before in the Dirac representation, much labor is saved by working in the
center-of-fields frame where electric and magnetic fields are parallel and point in
the z-direction. Their lengths have the invariant values ε and β, respectively. The
associated transformation eeF τ has then the simple form




eeF τ = e−i(M3 ε+L3 β)eτ = 
cosh εeτ
0
0
− sinh εeτ
0
0
− sinh εeτ
cos βeτ − sin βeτ
0
sin βeτ
cos βeτ
0
0
cosh εeτ



.

(14.474) {12@@}
From this we find a matrix for sin eF τ = [e−i(M3 ε+L3 β)eτ − ei(M3 ε+L3 β)eτ ]/2:




sin eF τ = 
0
0
0
− sinh εeτ
0
0
− sinh εeτ
0
− sin βeτ
0
sin βeτ
0
0
0
1



,

(14.475) {12@@}
1000
14 Functional-Integral Representation of Quantum Field Theory
and from this




eF τ = 
0
0
−εeτ
0 −βeτ
0 

.
βeτ
0
0 
0
1
0
0
0
−εeτ

(14.476) {12@@}
This leads to the desired prefactor in the amplitude (6.225)
det
−1/2
sinh eF τ
eF τ
!
= eεβτ 2
eεβτ 2
.
sinh ετ sin βeτ
(14.477) {12@@}
Thus we can obtain the imaginary part of the vacuum energy in a constant
electromagnetic field if we simply replace the term eEτ coth eEτ in the integrand of
the rate formula (14.439) as follows:
eEτ
eετ
eβτ
→
.
tanh eEτ
tanh eετ tan eβτ
14.18.3
(14.478) {12@@expore
Decay Rate in a Constant Electromagnetic Field
magnetic field the effective Lagrangian density (14.433) becomes, after the replacement (14.478)
∆Leff
R
1
=
2(2π)2
Z
∞
0
e2 (ε2 − β 2 ) −iτ (M 2 −iη)
eετ
eβτ
dτ
e
. (14.479) {12@@EULH
−
1
−
τ 3 tanh eετ tan eβτ
3
#
"
The subtracted small-τ divergence lies in the integral [recall the equality ε2 − β 2 =
E2 − B2 from Eq. (14.468)]
∆Leff
div =
e2 2
1
(E − B2 )
2
3(2π)2
Z
∞
0
dτ −iτ (M 2 −iη)
e
,
τ
(14.480) {12@2@LogD
which is proportional to the full original Maxwell Lagrangian density in (4.237), and
can therefore be absorbed into it by a renormalization of the charge as in (14.435)
and of the fields
e
e
BR ≡
B, ER ≡
E.
(14.481) {12@@RFIEL
eR
eR
From twice the imaginary part of (14.489) we obtain the decay rate per unit volume extending Eq. (14.439) in an obvious manner. Inserting the expansion (14.445),
extending the integral over the entire τ -axis and rotating the contour of integration
as we did in evaluating (14.439), we find the generalization of Eq. (14.446) to constant electromagnetic fields
∞
1
e2 E 2 X
nπβ/ε
ΓKG
2
=
e−nπM /eE .
3
2
V
4π n=1 n tanh nπβ/ε
(14.482) {12@@resuFE
For bosons obeying the Klein-Gordon equation, we obtain, by analogy, the extension of (14.454) to constant electromagnetic fields:
∞
Γ
e2 E 2 1 X
(−1)n−1 nπβ/ε
2
=
e−nπM /eε .
3
2
V
4π 2 n=1 n
sinh nπβ/ε
(14.483) {12@@resuFE
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1001
14.18 Functional Formulation of Quantum Electrodynamics
14.18.4
Effective Action in a Purely Magnetic Field
If there is only a magnetic field, magnetic field the effective Lagrangian density
(14.433) becomes, after the replacement (14.478)
∆Leff
R
1
=
2(2π)2
Z
∞
0
e2 (ε2 − β 2 ) −iτ (M 2 −iη)
eετ
eβτ
dτ
e
. (14.484) {12@@EULH
−
1
−
τ 3 tanh eετ tan eβτ
3
#
"
The subtracted small-τ divergence lies in the integral [recall the equality ε2 − β 2 =
E2 − B2 from Eq. (14.468)]
∆Leff
div =
1 Z ∞ dτ −iτ (M 2 −iη)
e2 2
(E − B2 )
e
,
2
3(2π)2 0 τ
(14.485) {12@2@LogD
which is proportional to the full original Maxwell Lagrangian density in (4.237), and
can therefore be absorbed into it by a renormalization of the charge as in (14.435)
and of the fields
e
e
B, ER ≡
E.
(14.486) {12@@RFIEL
BR ≡
eR
eR
From twice the imaginary part of (14.489) we obtain the decay rate per unit volume extending Eq. (14.439) in an obvious manner. Inserting the expansion (14.445),
extending the integral over the entire τ -axis and rotating the contour of integration
as we did in evaluating (14.439), we find the generalization of Eq. (14.446) to constant electromagnetic fields
∞
ΓKG
e2 E 2 X
1
nπβ/ε
−nπM 2 /eE
=
e
.
V
4π 3 n=1 n2 tanh nπβ/ε
(14.487) {12@@resuFE
For bosons obeying the Klein-Gordon equation, we obtain, by analogy, the extension of (14.454) to constant electromagnetic fields:
∞
Γ
e2 E 2 1 X
(−1)n−1 nπβ/ε
2
=
e−nπM /eε .
3
2
V
4π 2 n=1 n
sinh nπβ/ε
14.18.5
(14.488) {12@@resuFE
Effective Action in a Purely Magnetic Field
If there is only a magnetic field, For a constant electromagnetic field the effective
Lagrangian density (14.433) becomes, after the replacement (14.478)
∆Leff
R
1
=
2(2π)2
Z
∞
0
eβτ
e2 (ε2 − β 2 ) −iτ (M 2 −iη)
eετ
dτ
e
. (14.489) {12@@EULH
−
1
−
τ 3 tanh eετ tan eβτ
3
#
"
The subtracted small-τ divergence lies in the integral [recall the equality ε2 − β 2 =
E2 − B2 from Eq. (14.468)]
∆Leff
div =
1
e2 2
(E − B2 )
2
3(2π)2
Z
0
∞
dτ −iτ (M 2 −iη)
e
,
τ
(14.490) {12@2@LogD
1002
14 Functional-Integral Representation of Quantum Field Theory
which is proportional to the full original Maxwell Lagrangian density in (4.237), and
can therefore be absorbed into it by a renormalization of the charge as in (14.435)
and of the fields
e
e
BR ≡
B, ER ≡
E.
(14.491) {12@@RFIEL
eR
eR
From twice the imaginary part of (14.489) we obtain the decay rate per unit volume extending Eq. (14.439) in an obvious manner. Inserting the expansion (14.445),
extending the integral over the entire τ -axis and rotating the contour of integration
as we did in evaluating (14.439), we find the generalization of Eq. (14.446) to constant electromagnetic fields
∞
e2 E 2 X
ΓKG
1
nπβ/ε
2
=
e−nπM /eE .
3
2
V
4π n=1 n tanh nπβ/ε
(14.492) {12@@resuFE
For bosons obeying the Klein-Gordon equation, we obtain, by analogy, the extension of (14.454) to constant electromagnetic fields:
∞
Γ
e2 E 2 1 X
(−1)n−1 nπβ/ε
−nπM 2 /eε
=
e
.
V
4π 3 2 n=1 n2 sinh nπβ/ε
14.18.6
(14.493) {12@@resuFE
Effective Action in a Purely Magnetic Field
If there is only a magnetic field, the integral representation (14.489) reduces to
[32,33]
∆Leff
R
1
=
2(2π)2
Z
0
∞
e2 B 2 τ 2 −iτ (M 2 −iη)
eBτ
e
.
−1+
tan eBτ
3
!
dτ
τ3
(14.494) {12@@EULH
The integral still contains a divergence at small τ . The associated divergent integral
is precisely a magnetic version of Eq. (14.434). It can be removed by the same type
of subtraction as before, with E 2 replaced by B 2 . Going to renormalized quantities
as in (14.435) and (14.491) and rotating the contour of integration clockwise to move
it away from the poles, a trivial change of the integration variable leads to
∆Leff
R
e2 B 2
=−
2(2π)2
∞
Z
0
1 s −sM 2 /eB
ds
e
.
coth s − −
2
s
s 3
(14.495) {12@@EULH
This is a typical Borel transformation of the expression in parentheses. It implies
that its power series expansion leads to coefficients of B 2n which grow like (2n)!.
The expansion has therefore a vanishing radius of convergence. It is an asymptotic
series, and we shall understand later the physical origin of this.
For Klein-Gordon particles, the result becomes
∆LR KG =
e2 B 2
(2π)2
Z
0
∞
1 s −sM 2 /eB
1
ds
e
.
−
+
s2 sin s s 6
(14.496) {12@@EULH
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1003
14.18 Functional Formulation of Quantum Electrodynamics
14.18.7
Heisenberg-Euler Lagrangian
By expanding (14.478) in powers of e, we obtain
eετ
1
e2 2
1 eβτ
4 τ
2
4
2 2
4
−
e
ε
−
β
ε
+
5ε
β
+
β
=
+
τ 3 tan eβτ tanh eετ τ 3 3τ
45
3 τ
+ e6
2ε6 + 7ε4 β 2 − 7ε2 β 4 − 2β 6 + . . . .
945
(14.497) {12@@}
Inserting this into (14.489) and performing the integral over τ leads to an expansion
in powers of the fields, whose lowest terms are, with e2 = 4πα:
o 16πα3 n
o
2α2 n 2
2 2
2
2
2 3
2
2
2
+
.
(E
−B
)
+
7(EB)
2(E
−B
)
+
13(E
−B
)(EB)
45M 4
315M 8
(14.498) {12@@smallfs
In each term we can replace α by αR = α(1 + O(α)), and the fields by the renormalized fields via (14.491). Then we obtain the same series as in (14.498) but
for the physical renormalized quantities, plus higher-order corrections in α for each
coefficient, which we ignore in this lowest-order calculation.
Each coefficient is exact to leading order in α. To illustrate the form of the
higher-order corrections we include, without derivation, the leading correction into
the first term, which becomes (see Appendix 14C for details)
∆Leff
R =
∆Leff
R =
2α2
45m4e
1+
40α
1315α
(E2 −B2 )2 + 7 1 +
(E · B)2 + . . . .(14.499) {12@Kleinert
9π
252π
Electrons in a Constant Magnetic Field
For Dirac particles in arbitrary constant fields we expand
eετ coth eετ =
eβτ cot eβτ =
∞
X
∞
X
(eετ )2
τ2
,
=
2
2 2
2
2
n=−∞ (eετ ) + n π
n=−∞ τ + τn,ε
∞
X
(eβτ )2
τ2
,
=
2
2
2 2
2
m=−∞ (eβτ ) − m π
m=−∞ τ − τm,β
∞
X
τn,ε ≡
nπ
,
eε
τm,β ≡
mπ
.
eβ
(14.500) {nolabel}
(14.501) {12@@}
The iη accompanying the mass term in the τ -integral (14.439) is equivalent to re2
2
placing e−iτ (M −iη) by e−iτ (1−iη)M , implying that the integral over all τ has to be
performed slightly below the real axis. Equivalently we may shift the τm,β slightly
upwards in the complex plane to τm,β +iǫ. This leads to an additional contribution to
eff
the action corresponding to a constant electromagnetic field ∆Leff = ∆Leff
div + ∆LR .
It contains a logarithmically divergent part
∆Leff
div
1
=
2(2π)2
Z
0
∞
∞
∞
X
1
1
dτ −iτ (1−iη)M 2 X
−
e
2
2
2
τ
m=1 τm,β
n=1 τn,ε
!
,
(14.502) {12@@LogDiV
1004
14 Functional-Integral Representation of Quantum Field Theory
and the finite part
1
=
2(2π)2
∆Leff
R
Z
!

∞
∞
∞
X
X
1 
1
τ2
τ2
dτ  X
2
−
−
1−2τ
2
2
3
2
2
2
2
τ n,m=−∞ τ + τn,ε τ − τm,β
m=1 τm,β
n=0 τn,ε
∞
0
2
× e−iτ (1−iη)M .
(14.503) {12@@4orL
Performing the sums
∞
X
1
n=1
k
τn,ε
eε
=
π
k
ζ(k),
∞
X
m=1
1
k
τm,β
=
eβ
π
!k
(14.504) {12@@4zeatf
ζ(k),
we see that (14.502) coincides with (14.490), as it should. The remaining sum is
finite:
∆Leff
R
∞
1 X
1
′
= 2
2
2
8π n,m=−∞ τn,ε + τm,β
Z
0
2
2
τ e−τ M
τ e−τ M
− . . . , (14.505) {12@@sumisfi
dτ 2
−
2
2 +iη
τ − τn,ε
τ 2 + τm,β
−iη
∞
"
#
where the dots indicate the subtractions. Now we decompose à la Sochocki [recall
Footnote 9 in Chapter 1]:
τ2
τ
=
2 + iη
− τn,ε
i
π
π
P
,
δ(τ + τn,ε ) − i δ(τ − τn,ε ) + τ 2
2
2
2
τ − τn,ε
(14.506) {12@@}
where P indicates the principal value under the integral. The integrals over the
δ-functions contribute
∞
X
i
e2 ε2 β 2 X
1
1
2
−τn,ε M 2
∆δ L =
e
=i
e−nπM /eε .
2
2
3
2
2
2
2
8π n>0,m6=0 τn,ε + τm,β
8π n>0,m≥0 n β + m ε
(14.507) {12@@}
This leads to a decay rate per volume which agrees with (14.493) if we expand in
that expression [recall (14.445) and (14.425)]:
eff
∞
X
X
(nπβ/ε)2
n2 β 2
nπβ/ε
= 1+
=
1
+
.
2
2 2
2 2
2 2
tanh nπβ/ε
m6=0 (nπβ/ε) + m π
m6=0 n β + m ε
(14.508) {12@@expCO
It remains to do the principal-value integrals. Here we use the formula9
J(z) ≡ P
Z
0
∞
dτ
i
1 h −z
τ e−τ
z
e
Ei(z)
+
e
Ei(−z)
,
=
−
τ 2 − z2
2
(14.509) {12@@Eiexp}
where
Ei(z) ≡
Z
z
−∞
dt
∞
X
P t
zk
e = log(−z) +
t
k=1 k k!
(14.510) {12@@}
9
I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press,
New York, 1980, Formulas 3.354.4 and 8.211.
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1005
14.18 Functional Formulation of Quantum Electrodynamics
is the exponential integral, γ being the Euler-Mascheroni constant
(14.511) {12@@EulerM
γ = 0.577216 . . . .
The function (14.509) has the small-z expansion
i
1h z
e log(z) + e−z log(−z) − γ cosh z
2
∞
∞
X
X
z 2l+1
z 2l
cosh z +
sinh z,
−
l=1 (2l + 1)(2l + 1)!
l=1 (2l)(2l)!
J(z) = −
(14.512) {12@@4SMal
and behaves for large z like
J(z) = −
1
6
120 5040
− 4 − 6 − 8 + ... .
2
z
z
z
z
(14.513) {12@@4LARG
Using these formulas, we find from the principal-value integrals in (14.505) [34]:
∆P Leff = −
∞
1 2
e2 X
(β an + ε2 bn ),
4π 4 n=1 n2
(14.514) {12@@findfr4
with
nπm2
nπε/β
Ci
an =
tanh nπε/β
eβ
!
nπm2
nπm2
cos
+ si
eβ
eβ
1 nπβ/ε
nπm2
bn = −
exp
2 tanh nπβ/ε
eε
nπm2
Ei −
eε
!
nπm2
sin
,
eβ
nπm2
+ exp −
eε
(14.515) {12@@find1}
nπm2
Ei
,
eε
(14.516) {12@@find2}
1
1
where Ci(z) ≡ 2i [Ei(iz) + Ei(−iz)] and si(z) ≡ 2i [Ei(iz) − Ei(−iz)] = Si(z) − π/2
are the cosine and sine integrals with the integral representations [35]:
Ci(z) ≡ −
Z
∞
z
!
dt
cos t = γ + log z +
t
Z
0
z
!
dt
(cos t − 1) , si(z) ≡ −
t
Z
∞
z
!
!
dt
sin t . (14.517) {@}
t
The prefactors in (14.515) and (14.516) come once more from sums of the type
(14.508).
For large arguments of J(z), the expansion (14.513) becomes
∞
1 X
1
′
∆P L = 2
2
2
8π n,m=−∞τn,ε + τm,β
eff
"
#
1
6
120
+ 4
+ 6
+ . . . . (14.518) {12@@findfr4
2
4
8
τn,ε M
τn,ε M
τn,ε M 12
Applying the summation formulas (14.504), this is seen to agree with the small-field
expansion (14.498).
For E = 0, the sum (14.514) contains only the n = 0 -terms. If B is large, the
n-sum is dominated by the first term in the expansion (14.512), and yields
∆P Leff =
∞
X
1
1
log(τm,β M 2 ).
2
2
2
8π m=1 τm,β
(14.519) {12@@findfr4
1006
14 Functional-Integral Representation of Quantum Field Theory
The leading part of this is
∆P Leff =
∞
e2 B 2
1 X
M2
1 2 2
eB
log
+ ... = −
e B log 2 ,
2
2
2
2
4π m=1 π m
eB
24π
M
(14.520) {12@@WEIS
as derived first by Weisskopf [36].
Note that this logarithmic behavior can be understood as the result of a lowest
expansion in e2 of an anomalous power behavior of the effective action
1
2
2
∆Leff = (E 2 − B 2 )1+e /12π .
2
(14.521) {12@@WEIS
Due to the smallness of e2 , this power can be observed only at very large field
strengths. In self-focussing materials, however, it is visible in present-day laser
beams [37].
14.18.8
Alternative Derivation for a Constant Magnetic Field
The case of a constant magnetic field can also be treated in a different way [38].
The eigenvalues of the euclidean Klein-Gordon operator of a free scalar field
OKG = −h̄2 ∂ 2 + M 2 c2
(14.522) {12@@}
λ p = p 2 + M 2 c2 .
(14.523) {12@@}
are
If a constant magnetic field is present that points in the z-direction, the kinetic
energy p21 + p22 in the xy-plane changes to Landau energies:
p21 + p22 → 2M ×
p21 + p22
1
, n = 0, 1, 2, . . . ,
= 2M × h̄ωL n +
2M
2
(14.524) {12@landaule
where ωL = eB/Mc is the Landau frequency. Thus the eigenvalues become
λn,p⊥ = p2⊥ + 2
h̄e
B(n + 1/2) + M 2 c2 ,
c
(14.525) {12@@}
where p⊥ is the two-dimensional momentum in the x3 − x4 -plane.
For a Dirac electron satisfying the Pauli equation (6.110), this changes to
λn,p⊥ = p2⊥ + 2
h̄e
B(n + 1/2 + s3 ) + M 2 c2 ,
c
s3 = ±1/2.
(14.526) {12@@SPDI}
In this expression, the famous g-factor which accounts for the anomalous magnetic
moment is approximated by the Dirac value 2. Radiative corrections would insert a
factor g/2 in front of σ3 , with g = 2 + α/π + . . . .
The contribution of electrons to the effective action in Eq. (14.427) can therefore
be written as
i∆Aeff
Z
Z
1 X
d 2 p⊥
d4 p
=
log
λ
−
log λp .
n,p
⊥
2 n,σ (2πh̄)2
(2πh̄)4
!
(14.527) {12@@Trextf
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1007
14.18 Functional Formulation of Quantum Electrodynamics
These are divergent expressions. In order to deal with them efficiently it is useful to
introduce the Hurwitz ζ-function employed in number theory [39,40]:
ζH (s, z) =
∞
X
1
,
s
n=0 (n + z)
v 6= 0, −1, −2, . . . .
Re(s) > 1 ,
(14.528) {12@hurwitz}
This Hurwitz ζ-function can be analytically continued in the s-plane to define an
analytic function with a single simple pole at s = 1. In quantum field theory, one
may extend this definition to an arbitrary eigenvalue spectrum of an operator O as
ζO (s) =
X
1
,
(λ/µd )s
(14.529) {12@@}
where µ is some mass parameter to make the eigenvalues of mass dimension d dimensionless. For large enough s, this is always convergent. By analytic continuation
in s, one can derive a finite value for the functional determinant of O:
Det O = eTr log O ,
Tr log O = −ζO′ (0).
(14.530) {12@@}
For the Dirac spectrum (14.526), the ζ-function becomes, in natural units,
∞ XZ
2
eB X
d 2 k⊥ M 2 + k⊥
+ eB(2n + 1 ± 1)
ζD (s) =
2
2π n=0 ±
(2π)
µ2
"
#−s
,
(14.531) {12@spzeta}
where the prefactor of eB/2π is the degeneracy of the Landau levels which ensures
P
that theR sum (eB/2π) n Rconverges in the limit B → 0 against the momentum
integral dk 1 dk 2 /(2π)2 = dk 2 /4π. Performing the integral over the momenta p⊥
yields
ζD (s) =
o
B 2s 1 n 2
1−s
2
1−s
. (14.532) {12@spzeta20
[M
+
2eBn]
+
[M
+
2eB(n
+
1)]
µ
8π 2
s−1
This can be expressed in terms of the Hurwitz ζ-function (14.528) as
M4
ζD (s) =
4π 2
eB
M2
2 X
∞
n=0
µ2
2eB
!s
From this we obtain

1 
M2
2ζH s − 1,
s−1
2eB
M2
(eB)2
′
−1,
−2ζ
=
H
4π 2
2eB
(
ζD′ (0)

2
1
M
+  +
6
2eB
!
!2  "
 1 + log
!
M2
−
2eB
M2
M2
−
log
2eB
2eB
2
µ
2eB
!#

!1−s 
. (14.533)
{12@spzeta2}
!
(14.534) {12@@zetapr5
.

Choosing µ = M and subtracting the zero-field contribution, which is simply
−3M 4 /32π 2 , we find
∆Leff
(eB)2 ′
M2
=
ζ
−1,
H
2π 2
2eB
(
!
+ ζH
M2
M2
−1,
log
2eB
2eB
!
!
1
1
− +
12 4
M2
2eB
!2 


,
(14.535) {12@spzetala
1008
14 Functional-Integral Representation of Quantum Field Theory
where we have used the property [41,39]
1
z z2
+ − .
(14.536) {12@@}
12 2
2
Contact with the previous result in (14.495) is established with the help of the
integral representation of the Hurwitz ζ-function [41]:
ζH (−1, z) = −
1
ζH (s, z) =
Γ(s)
Z
∞
0
e−z t ts−1
dt ,
1 − e−t
Re(s) > 1 ,
(14.537) {12@hurwitzi
Re(z) > 0.
The integral can be rewritten as
z −s sz −1−s 2s−1
z 1−s
+
+
+
ζH (s, z) =
s−1
2
12
Γ(s)
dt
∞
Z
t1−s
0
−2z t
e
1 t
coth t − −
, (14.538) {12@hurwitzi
t 3
this expression being valid for Re(s) > −2, where the integral converges [42]. From
this we evaluate the s-derivative at s = −1 as follows:
Z
1 t
z2
1 ∞ dt −2z t
1
′
coth t − −
. (14.539) {12@hurwitzd
−
− ζH (−1, z) log z −
e
ζH (−1, z) =
12
4
4 0 t2
t 3
Inserting this into (14.535) we recover exactly the previous Eq. (14.495).
Let us now derive the strong-field limit. For this we use the following relation
between the Hurwitz ζ-function and the Γ-function [41,39]:
Z z
z
z
′
ζH
(−1, z) = ζ ′(−1) − log(2π) − (1 − z) +
log Γ(x)dx .
(14.540) {12@zetalog}
2
2
0
This identity follows from an integration of Binet’s integral representation [43] of
log Γ(z). Thus we can write


M2
2eB
1
M2
3
(eB)
′
−
+
ζ
(−1)
−
+
∆L =
2

2π
12
4eB 4
2
eff

2
+ −
1
M
1
+
−
12 4eB 2
2
M
2eB
!2 
 log
2
M
2eB
!
!2
+
−
Z
M2
log(2π)
4eB
M 2 /2eB
0


dx log Γ(x) . (14.541) {nolabel}

In the strong-field limit, the range of integration in the last term vanishes, so we
can use the Taylor expansion [41,44] of log Γ(x):
log Γ(x) = − log x − γx +
∞
X
(−1)n
ζ(n) xn ,
n
n=2
(14.542) {12@loggamm
where ζ(n) is the usual Riemann ζ-function. This leads to :
∆L
eff

(eB)2  1
M2
3
′
−
=
+
ζ
(−1)
−
+
2π 2  12
4eB 4

1
M2
1
+ − +
−
12 4eB 2
M2
M2
1 − log
+
2eB
2eB
"
M2
2eB
!#
M2
2eB
!2
!2 
∞
X
M2
 log
2eB
(−1)n ζ(n)
+
n=2 n(n + 1)
!
M2
log(2π)
−
4eB
!2
γ
−
2
M2
2eB
M2
2eB
!n+1 


.
(14.543) {12@spmagst
H. Kleinert, PARTICLES AND QUANTUM FIELDS
1009
14.18 Functional Formulation of Quantum Electrodynamics
The leading behavior in the strong-field limit is
∆Leff =
2eB
(eB)2
log 2 + . . . ,
2
24π
M
(14.544) {12@1lspmag
in agreement with Weisskopf’s result (14.520).
Charged Scalar Field in a Constant Magnetic Field
For a charged scalar field obeying the Klein-Gordon equation there is no spin sum
and the ζ-function (14.531) becomes
∞ Z
2
eB X
+ eB(2n + 1)
d 2 k⊥ M 2 + k⊥
ζKG (s) =
2
2π n=0 (2π)
µ2
"
(eB)2
=
4π 2
"
µ2
2eB
!s
{12@strongsc
#−s
1
M2
1
ζH s − 1, +
(s − 1)
2 2eB
!#
.
(14.545) {12@sczeta}
Setting again µ = m, and subtracting the zero field contribution 3m4 /64π 2 , we
obtain
∆Leff
KG
2


(eB)
1
M2
′
= −
−1,
ζ
+
H
4π 2 
2 2eB
M2
+ 1 + log
2eB
"
!#
ζH
!
3
+
4
M2
2eB
1
M2
−1, +
2 2eB
!2
!)
.
(14.546) {12@sczetalag
In the Bose case, the equivalence with the previous result is shown with the help of
the integral representations of the Hurwitz ζ-function [compare (14.538)]
∞
1
1
e−t/2
ζH (s, 1/2 + z) =
dt , Re(s) > 1, Re(z) > −
e−z t ts−1
−t
Γ(s) 0
1−e
2
−1−s
s−1 Z ∞
1−s
sz
2
1
1 t
dt −2z t
z
.(14.547) {12@schurwit
−
+
e
− +
=
1−s
s−1
24
Γ(s) 0 t
sinh t t 6
Z
The second expression is valid for Re(s) > −2, where the subtracted integral
converges. We can therefore use it to find the derivative at s = −1 required in
Eq. (14.546):
1
z2 z2
(1 + log z) −
+ log z
24
4
2
Z
1
1 t
1 ∞ dt −2z t
.
e
−
+
−
4 0 t2
sinh t t 6
′
ζH
(−1, 1/2 + z) = −
(14.548) {12@schurwit
Inserting this into (14.546), we recover exactly the previous result (14.496).
In order to find a strong-field expansion of ∆Leff
KG we use the following relation
between the Hurwitz ζ-function and the log of the Γ-function [41,43]:
1
z
log 2 z 2
′
ζH
(−1, 1/2 + z) = − ζ ′ (−1) − log 2π −
+
+
2
2
24
2
Z
0
z
dz log Γ(x+ 12 ), (14.549) {12@sczetalog
1010
14 Functional-Integral Representation of Quantum Field Theory
where we have used the formula [42,45,46]
Z
1
2
0
log Γ(x) dx =
1
1 3
5
log 2 + log π + − ζ ′(−1) .
24
4
8 2
(14.550) {12@hint}
Then we expand
log Γ x + 1/2 =
∞
X
(−1)n−1 (1 − 2n )
1
log π − (γ + 2 log 2) x +
ζ(n) xn , (14.551) {12@scloggam
2
n
n=2
and obtain the strong-field expansion
∆Leff
KG

(eB)2  5
= −
4π 2  4
M2
2eB
!2

M2
2eB
1
1
+ −
24 2
log 2
M2
1
1 ′
−
log 2 −
− ζ (−1) −
2
24
4eB
2
∞
X
(−1)n−1 (1 − 2n )ζ(n)
+
n(n + 1)
n=2
M2
2eB
M2
2eB
!n+1 

with the leading behavior
∆Leff
KG =
Appendix 14A
!2  "
 1 + log
!2
M2
2eB
!#
(γ + 2 log 2)
,

2eB
(eB)2
log 2 + . . . .
2
96π
M
(14.552) {12@scmagst
(14.553) {12@1lscmag
Propagator of the Bilocal Pair Field
{@proppf}
Consider the Bethe-Salpeter equation (14.300) with a potential λV instead of V
Γ = −iλV G0 G0 Γ.
(14A.1) {NA.1}
Take this as an eigenvalue problem in λ at fixed energy-momentum q = (q 0 , q)= (E, q) of the bound
states. Let Γn (P |q) be all solutions, with eigenvalues λn (q). Then the convenient normalization of
Γn is:
Z
q
q
d4 P †
−i
Γn (P |q) G0
(14A.2) {A.2}
+ P G0
− P Γn′ (P |q) = δnn′ .
4
(2π)
2
2
If all solutions are known, there is a corresponding completeness relation (the sum may comprise
an integral over a continuous part of the spectrum)
q
q
X
+ P G0
− P Γn (P |q)Γ†n (P ′ |q) = (2π)4 δ (4) (P − P ′ ).
(14A.3) {A.3}
−i
G0
2
2
n
This completeness relation makes the object given in (14.315) the correct propagator of ∆. In
order to see this, write the free ∆-action A2 [∆† ∆] as
1 †
1
A2 = ∆
(14A.4) {A.4}
+ iG0 × G0 ∆
2
λV
H. Kleinert, PARTICLES AND QUANTUM FIELDS
Appendix 14A
Propagator of the Bilocal Pair Field
where we have used λV instead of V . The propagator of ∆ would have to satisfy
1
+ iG0 × G0 ∆∆† = i.
λV
1011
(14A.5) {A..5}
Indeed, by performing a short calculation, we can verify that this equation is fulfilled by the
spectral expansion (14.315). We merely have to use the fact that Γn and λn are eigenfunctions
and eigenvalues of Eq. (14A.1), and find that that
)
(
X Γn Γ†
1
n
+ iG0 × G0 × −iλ
λV
λ − λn (q)
n
= −iλ
X
1
λV
n
Γn Γ†n + iG0 × G0 Γn Γ†n
λ − λn (q)
X − λn (q) + 1
λ
(−iG0 × G0 Γn Γ†n )
= iλ
λ
−
λ
(q)
n
n
!
X
†
G0 × G0 Γn Γn = i.
= i −i
i
Note that the expansion of the spectral representation of the propagator in powers of λ
!
X X λ k
X Γn Γ†
n
†
†
Γn Γn
=i
∆∆ = −iλ
λ − λn (q)
λn (q)
n
n
k
(14A.6) {A.6}
(14A.7) {14A.7}
corresponds to the graphical sum over one, two, three, etc. exchanges of the potential λV . For
n = 1 this is immediately obvious since (14A.1) implies that
i
X
n
X λ
λ
Γn Γ†n =
λn (q)V G0 × G0 Γn Γ†n = iλV.
λn (q)
λn (q)
For n = 2 one can rewrite, using the orthogonality relation,
X λ
X λ 2
λ
Γn Γ†n =
= λV G0 × G0 λV .
Γn Γ†n G0 × G0 Γn′ Γ†n′
i
λ
(q)
λ
(q)
λ
n
n
n′ (q)
′
n
nn
(14A.8) {A.8}
(14A.9) {A.9}
This displays the exchange of two λV terms with particles propagating in between. The same
procedure applies at any order in λ. Thus the propagator has the expansion
∆∆† = iλV − iλV G0 × G0 iλV + . . . .
(14A.10) {A.10}
R
If the potential is instantaneous, the intermediate dP0 /2π can be performed replacing
G0 × G0 → i
1
E − E0 (P|q)
(14A.11) {A.11}
where
E0 (P|q) = ξ
q
q
+P +ξ
−P
2
2
is the free particle energy which may be considered as the eigenvalue of an operator H0 . In this
case the expansion (14A.10) reads
1
E − H0
†
∆∆ = i λV + λV
λV + . . . = iλV
.
(14A.12) {A.12}
E − H0
E − H0 − λV
1012
14 Functional-Integral Representation of Quantum Field Theory
We see it related to the resolvent of the complete Hamiltonian as
∆∆† = iλV (RλV + 1)
(14A.13) {A.13}
X ψn ψ †
1
n
=
E − H0 − λV
E − En
n
(14A.14) {A.14}
where
R≡
with ψn being the Schrödinger amplitudes in standard normalization. We can now easily determine
the normalization factor N in the connection between Γn and the Schrödinger amplitude ψn .
Eq. (29A.3) gives in the instantaneous case
Z
1
d3 P †
Γ (P|q)
Γn′ (P|q) = δnn′ .
(2π)3 n
E − H0
(14A.15) {A.15}
Inserting ψ from (14.306) renders
1
N2
Z
d3 P †
ψ (P|q)(E − H0 )ψn′ (P|q) = δnn′ .
(2π) n
(14A.16) {A.16}
Using finally the Schrödinger equation
(E − H0 )ψ = λV ψ,
(14A.17) {A.17}
we find
1
N2
Z
d3 P †
ψ (P|q) λV ψn′ (P|q) = δnn′ .
(2π)3 n
(14A.18) {A.18}
For the wave functions ψn (P|q) in standard normalization, the integral is equal to the energy
differential
λ
dE
.
dλ
For a typical calculation of a resolvent, the reader is referred to Schwinger’s treatment [47] of the
Coulomb problem. His result may directly be used for a propagator of electron hole pairs bound
to excitons.
Appendix 14B
Fluctuations around the Composite Field
{@flcompf}
Here we show that the quantum mechanical fluctuations around the classical equations of motion
(14.237) are quite simple to calculate. The exponent of (14.232) is extremized by the field
Z
ϕ(x) = dyV (x, y)ψ † (y)ψ(y).
(14B.1) {YB.1}
Similarly, the extremum of the exponent of (14.278) yields
∆(x, y) = V (x − y)ψ(x)ψ(y).
(14B.2) {YB.2}
For this let us compare the Green functions of ϕ(x) or ∆(x, y) with those of the composite operators
on the right-hand side of Eqs. (14B.1) or (14B.2). The Green functions of ϕ(x)R or ∆(x, y) are
generated by adding to the final actions (14.242) or (14.284) external currents dxϕ(x)I(x) or
H. Kleinert, PARTICLES AND QUANTUM FIELDS
Appendix 14B
Fluctuations around the Composite Field
1013
R
1/2 dxdy(∆(y, x)I † (x, y) + h.c.), respectively, and by forming functional derivatives δ/δI. The
Green functions of the composite operators, on the other hand, are obtained by adding
Z
Z
†
dx
dyV (x, y)ψ (y)ψ(y) K(x)
(14B.3) {nolabel}
1
2
Z
dx
Z
dyV (x − y)ψ(x)ψ(y)K † (x, y) + h.c.
(14B.4) {nolabel}
to the original actions (14.235) or (14.278), respectively, and by forming functional derivatives
δ/δK. It is obvious that the sources K(x) nad K(x, y) can be included in the final actions (14.242)
and (14.284) by simply replacing
Z
ϕ(x) → ϕ′ (x) = ϕ(x) − dx′ K(x′ )V (x′ , x),
(14B.5) {nolabel}
or
∆(x, y) → ∆′ (x, y) = ∆(x, y) − K(x, y).
(14B.6) {nolabel}
If one now shifts the functional integrations to these new translated variables and drops the irrelevant superscript “prime”, the actions can be rewritten as
Z
Z
1
′
−1
′
′
dxdx
ϕ(x)V
(x,
x
)ϕ(x
)+i
dxdx′ η † (x)Gϕ (x, x′ )η(x)
A[ϕ] = ±iTr log(iG−1
)+
ϕ
2
Z
Z
1
+
dxϕ(x) [I(x) + K(x)] +
dxdx′ K(x)V (x, x′ )K(x′ ),
(14B.7) {B.3}
2
or
Z
1
i
1
A[∆] = ± Tr log iG−1
+
dxdx′ |∆(x, x′ )|2
∆
2
2
V (x, x′ )
Z
i
1
dxdx′ j † (x)G∆ (x, x′ )
+
2
V (x, x′ )
Z
1
dxdx′ ∆(y, x) I † (x, y) + K † (x, y) + h.c.
+
2
Z
1
+
dxdx′ |K(x, x′ )|2 V (x, x′ ).
(14B.8) {B.4}
2
In this form the actions display clearly the fact that derivatives with respect to the sources K
or I coincide exactly, except for all possible insertions of the direct interaction
V . For example,
R
the propagators of the plasmon field ϕ(x) and of the composite operator dyV (x, y)ψ † (y)ψ(y) are
related by
δ (2) Z
δ (2) Z
−1
′
=
V
(x,
x
)
−
(14B.9) {nolabel}
δI(x)δI(x′ )
δK(x)δK(x′ )
Z
Z
= V −1 (x, x′ ) + h0|( dyV (x, y)ψ † (y)ψ(ϕ))( dy ′ V (x′ y ′ )ψ † (y ′ )ψ † (y ′ )ψ(y ′ ))|0i,
ϕ(x)ϕ(x′ ) = −
in agreement with (14.238). Similarly, one finds for the pair fields:
∆(x, x′ )∆† (y, y ′ ) = δ(x − y)δ(x′ − y ′ )iV (x − x′ )
+ h0|(V (x′ , x)ψ(x′ )ψ(x))(V (y ′ , y)ψ † (y)ψ † (y ′ ))|0i.
(14B.10) {B.6}
Note that the latter relation is manifestly displayed in the representation (14A.10) of the propagator
∆. Since
∆∆† = iV G(4) V,
(14B.11) {nolabel}
1014
14 Functional-Integral Representation of Quantum Field Theory
one has from (14B.10)
h0|V (ψψ)(ψ † ψ † V )|0i = V G(4) V,
(14B.12) {B.7}
which is correct remembering that G(4) is the full four-point Green function. In the equal-time
situation relevant for an instantaneous potential, G(4) is replaced by the resolvent R.
Appendix 14C
Two-Loop Heisenberg-Euler
Effective Action
{RITUSA}
The next correction to the Heisenberg-Euler Lagrangian density (14.489) is [38,48]
Z ∞ Z ∞
2
′
ie2
e4 β 2 ε2
(2) eff
′
∆ L = −
e−i(M −iη)(τ +τ )
dτ
dτ
′ sinh eετ sinh eετ ′
128π 4 0
sin
eβτ
sin
eβτ
0
n
o
× 4M 2 [S(τ )S(τ ′ ) + P (τ )P (τ ′ )] I0 − i I ,
(14C.1) {2lspbare}
where
S(τ ) ≡ cos eβτ cosh eετ
P (τ ) ≡ sin eβτ sinh eετ ,
,
and
1
b
log ,
b−a
a
a ≡ eβ (cot eβτ + cot eβτ ′ ) ,
2e2 β 2 cosh eε(τ − τ ′ )
p≡
,
sin eβτ sin(eβτ ′ )
I0 ≡
b
(q − p)
aq − bp
log −
,
(b − a)2
a ba(b − a)
b ≡ eε (coth eετ + coth eετ ′ ) ,
2e2 ε2 cos eβ(τ − τ ′ )
.
q≡
sinh eετ sinh eετ ′
I≡
(14C.2) {spii}
This expression contains divergences which require renormalization. First, there is a subtraction
of an infinity to make ∆(2) Leff vanish for zero fields. Then there are both charge and wave
function renormalizations, just as for the one-loop effective Lagrangian, which involves identifying
a divergent term in ∆(2) Leff of the form of the zero-loop Maxwell Lagrangian. This is done simply
by expanding the integrand to quadratic order in the fields β and ε. This divergence can be
absorbed by redefining the electric charge and the fields as
1/2
eR = e Z3
,
−1/2
BR = B Z 3
,
−1/2
ER = E Z3
(14C.3) {2lcharge}
where Z3 is some divergent normalization constant, which was given by (14.435) in the previous
−1/2
−1/2
result (14.491). The invariants β and ε are renormalized accordingly: βR = βZ3
, εR = εZ3
.
(2) eff
Then we re-express ∆ L in terms of the renormalized charges and fields. Finally, we have
to renormalize the mass:
m2R
=
2
∆(1) Leff
R (mR ) =
m20 + δM 2 ,
(1)
(1)
LR (m20 ) + δM 2
∂LR (m20 )
.
∂m20
(14C.4) {2lmass}
The second term in (14C.4) is of the order α2 , since δM 2 and ∆(1) Leff
R are both of order α. For
details of removing the divergencies, see the original papers in Refs. [48]. The final answer for the
renormalized two loop effective Lagrangian is
Z ∞ Z τ
K0 (τ )
ie2
′
′
(2) eff
dτ
dτ K(τ, τ ) −
∆ LR = −
64π 4 0
τ′
0
Z
∞
2
ie
5
2
(14C.5) {2lspr}
−
dτ K0 (τ ) log(iM τ ) + γ −
64π 4 0
6
H. Kleinert, PARTICLES AND QUANTUM FIELDS
Notes and References
1015
where γ ≈ 0.577... is Euler’s constant, and the functions K(τ, τ ′ ) and K0 (τ ) are
2
′
(eβ)2 (eε)2 4M 2 (S(τ ) S(τ ′ ) + P (τ ) P (τ ′ ))I0 − iI
K(τ, τ ′ ) = e−iM (τ +τ )
′
P (τ ) P (τ )
1
2i
5iτ τ ′
e2 (β 2 − ε2 )
2
2
′
2
′2
− ′
4M
−
2M
(τ
τ
−2τ
−2τ
)−
+
τ τ (τ +τ ′ )
τ + τ′
3
τ +τ ′
∂
eετ
1
eβτ
e2 (β 2 − ε2 )τ 2
2
−iM 2 τ
4M + i
. (14C.6) {spkss}
K0 (τ ) = e
−1+
∂τ τ 2 tan eβτ tanh eετ
3
The fields in this expression can be replaced by the renormalized fields, and everything is finite.
The lowest contribution is of the fourth power in the fields and reads
16 2
263
e6
2 2
2
+ ... ,
(14C.7) {2lspweak}
(β
−
ε
)
+
(β
ε)
∆(2) Leff =
64π 4 m4 81
162
which has been added to the one-loop result in Eq. (14.499). In the limit of strong magnetic fields
it yields
e4 β 2
eβ
(2) eff
∆ L =
log
+ constant + . . . .
(14C.8) {2lspstrong}
128π 4
πM 2
Notes and References
[1] R.P. Feynman and A.R. Hibbs, Path Integrals and Quantum Mechanics, McGraw-Hill, New
York (1968);
[2] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, World
Scientific Publishing Co., Singapore 1995, pp. 1–890.
[3] R.P. Feynman, Rev. Mod. Phys. 20, 367 (1948);
[4] H. Kleinert, Collective Quantum Fields, Fortschr. Physik 26, 565 (1978) (http://
klnrt.de/55). See also (http://klnrt.de/b7/psfiles/sc.pdf).
[5] J. Rzewuski, Quantum Field Theory II, Hefner, New York (1968).
[6] S. Coleman, Erice Lectures 1974, in Laws of Hadronic Matter, ed. by A. Zichichi, p. 172.
[7] See for example:
A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Methods of Quantum Field Theory in
Statistical Physics, Dover, New York (1975);
L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics, Benjamin, New York (1962);
A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New
York (1971).
[8] R.L. Stratonovich, Sov. Phys. Dokl. 2, 416 (1958);
J. Hubbard, Phys. Rev. Letters 3, 77 (1959);
B. Mühlschlegel, J. Math. Phys. 3, 522 (1962);
J. Langer, Phys. Rev. 134, A 553 (1964);
T.M. Rice, Phys. Rev. 140 A 1889 (1965); J. Math. Phys. 8, 1581 (1967);
A.V. Svidzinskij, Teor. Mat. Fiz. 9, 273 (1971);
D. Sherrington, J. Phys. C4 401 (1971).
[9] The first authors to employ such identities were
P.T. Mathews, A. Salam, Nuovo Cimento 12, 563 (1954), 2, 120 (1955).
[10] H.E. Stanley, Phase Transitions and Critical Phenomena, Clarendon Press, Oxford, 1971.
1016
14 Functional-Integral Representation of Quantum Field Theory
[11] For the introduction of collective bilocal fields in particle physics and applications see
H. Kleinert, On the Hadronization of Quark Theories, Erice Lectures 1976 on Particle
Physics, publ. in Understanding the Fundamental Constituents of Matter , Plenum Press
1078, (ed. by A. Zichichi). See also
H. Kleinert, Phys. Letters B 62, 429 (1976), B 59, 163 (1975).
[12] The mean-field equations associated with the pair fields of the electrons in a metal are
precisely the equations used by Gorkov to study the behavior of type II superconductors.
See, for example, p. 444 in the third of Refs. [52].
[13] H.A. Bethe and E.E. Salpeter in Encyclopedia of Physics (Handbuch der Physik), Springer,
Berlin, 1957, p. 405.
[14] Y. Nambu and G. Jona Lasinio, Phys. Rev. 122, 345 (1961); 124, 246 (1961).
[15] V.L. Ginzburg and L.D. Landau, Eksp. Teor. Fiz. 20, 1064 (1950).
[16] R.G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, Oxford, Oxford, 1989; K.U. Gross and R.M. Dreizler, Density Functional Theory, NATO Science
Series B, 1995.
[17] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)
W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
[18] A. Leggett, Rev. Mod. Phys. 47, 331 (1975).
[19] Note that the hermitian adjoint ∆∗↑↓ comprises transposition in the spin indices, i.e., ∆∗↑↓ =
∗
[∆↓↑ ] .
[20] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific, Singapore 2004 (http://klnrt.de/b5).
[21] H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4 -Theories, World Scientific,
2001 (klnrt.de/b8).
[22] H. Kleinert, Converting Divergent Weak-Coupling into Exponentially Fast Convergent
Strong-Coupling Expansions, Lecture presented at the Summer School on ”Approximation
and extrapolation of convergent and divergent sequences and series” in Luminy bei Marseille
in 2009 (arXiv:1006.2910).
[23] The alert reader will recognize her the so-called square-root trick of Chapter 5 in the textbook
Ref. [6].
[24] See the www page (http://klnrt.de/b7/psfiles/hel.pdf).
[25] H. Kleinert, Phys. Rev. D 57, 2264 (1998); Phys.Rev. D 60, 085001 (1999). (See also
klnrt.de/critical).
[26] C. De Dominicis, J. Math. Phys. 3, 938 (1962); C. De Dominicis and P.C. Martin, J.
Math. Phys. 5, 16, 31 (1964); J.M. Cornwall, R. Jackiw, and E.T. Tomboulis, Phys. Rev. D
10, 2428 (1974); H. Kleinert, Fortschr. Phys. 30, 187 (1982) (klnrt.de/82); Lett. Nuovo
Cimento 31, 521 (1981) (klnrt.de/77).
[27] H. Kleinert, Annals of Physics 266, 135 (1998) (klnrt.de/255).
[28] R.P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (klnrt.de/159).
[29] Such an omission was done in Eq. (4.117) of the textbook [55].
[30] J.E. Robinson, Phys. Rev. 83, 678 (1951).
[31] See Chapter 7 in the textbook [6].
H. Kleinert, PARTICLES AND QUANTUM FIELDS
Notes and References
1017
[32] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936). English translation available at
http://klnrt.de/files/heisenbergeuler.pdf.
[33] J. Schwinger, Phys. Rev. 84, 664 (1936); 93, 615; 94, 1362 (1954).
[34] U.D. Jentschura, H. Gies, S.R. Valluri, D.R. Lamm, E.J. Weniger, Canadian J. Phys. 80,
267 (2002) (hep-th/0107135).
[35] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York,
1972. See Section 5.2.
[36] V. Weisskopf, The electrodynamics of the vacuum based on the quantum theory of the electron, Kong. Dans. Vid. Selsk. Math.-Fys. Medd. XIV No. 6 (1936); English translation in:
Early Quantum Electrodynamics: A Source Book, A.I. Miller, Cambridge University Press,
1994.
[37] G. Mourou, T. Tajima, and S.V. Bulanov, Reviews of Modern Physics, 78, 309, (2006); and
references therein.
[38] G.V. Dunne, Heisenberg-Euler Effective Actions: 75 years on, Int.J. Mod. Phys. A 27 (2012)
1260004.
[39] E. Whittaker and G. Watson, A Course in Modern Analysis, 4th ed., Cambridge, 1950, pp.
268–269.
[40] http://mathworld.wolfram.com/HurwitzZetaFunction.html.
[41] A. Erdelyi (ed.), Higher Transcendental Functions, Vol. I, Kreiger, Florida, 1981.
[42] V. Adamchik, Symbolic and Numeric Computation of the Barnes Function, Conference on
applications of Computer Algebra, Albuquerque, June 2001; Contributions to the Theory of
the Barnes Function, (math.CA/0308086).
[43] E. Whittaker and G. Watson, A Course in Modern Analysis, 4th ed., Cambridge, 1950.
[44] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York,
1972.
[45] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Academic Press,
New York, 1972; Formula 6.441.1.
[46] E.W. Barnes, The theory of the G-function, Quart. J. Pure Appl. Math XXXI (1900) 264.
[47] J. Schwinger, J. Math. Phys. 5, 1606 (1964).
[48] V.I. Ritus, Lagrangian of an intense electromagnetic field and quantum electrodynamics at
short distances, Zh. Eksp. Teor. Fiz 69, 1517 (1975) [Sov. Phys. JETP 42, 774 (1975)];
Connection between strong-field quantum electrodynamics with short-distance quantum electrodynamics, Zh. Eksp. Teor. Fiz 73, 807 (1977) [Sov. Phys. JETP 46, 423 (1977)]; The
Lagrangian Function of an Intense Electromagnetic Field , in Proc. Lebedev Phys. Inst. Vol.
168, Issues in Intense-field Quantum Electrodynamics, V.L. Ginzburg, ed., (NovaScience
Pub., NY 1987); Effective Lagrange function of intense electromagnetic field in QED, (hepth/9812124).
[49] L.D. Faddeev and V.N. Popov Phys. Lett. B 25 29 (1967); See also
M. Ornigotti and A. Aiello, (arXiv:1407.7256).
[50] J. Bardeen, L. N. Cooper, and J.R. Schrieffer: Phys. Rev. 108, 1175 (1957).
See also the little textbook from the russian school:
N.N. Bogoliubov, E. A. Tolkachev, and D.V. Shirkov A New Method in the Theory o Superconductivity, Consultnts Bureau, New York, 1959.
1018
14 Functional-Integral Representation of Quantum Field Theory
[51] For the introduction of collective bilocal fields in particle physics and applications see
H. Kleinert, On the Hadronization of Quark Theories, Erice Lectures 1976 on Particle
Physics, publ. in Understanding the Fundamental Constituents of Matter , Plenum Press
1078, (ed. by A. Zichichi). See also
H. Kleinert, Phys. Letters B 62, 429 (1976), B 59, 163 (1975).
[52] See for example:
A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Methods of Quantum Field Theory in
Statistical Physics, Dover, New York (1975);
L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics, Benjamin, New York (1962);
A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Paricle Systems, McGraw-Hill, New
York (1971).
[53] R.P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (klnrt.de/159).
[54] H. Kleinert, Converting Divergent Weak-Coupling into Exponentially Fast Convergent
Strong-Coupling Expansions, Lecture presented at the Summer School on ”Approximation
and extrapolation of convergent and divergent sequences and series” in Luminy bei Marseille
in 2009 (arXiv:1006.2910).
[55] C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill (1985). See Section 12.2.
[56] For more details see (http://klnrt.de/b8/crit.htm).
[57] H. Kleinert, Phys. Rev. D 57, 2264 (1998); Phys.Rev. D 60, 085001 (1999). (See also
klnrt.de/critical).
H. Kleinert, PARTICLES AND QUANTUM FIELDS