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Wick calculus
Alexander Wurma兲
Western New England College, Springfield, Massachusetts 01119
Marcus Berg
Max Planck Institute for Gravitational Physics, Albert Einstein Institute, D-14476 Potsdam, Germany
共Received 19 February 2007; accepted 7 October 2007兲
In quantum field theory the “normal ordering” of operators is routinely used, which amounts to
shuffling all creation operators to the left. Potentially confusing is the occurrence in the literature of
normal-ordered functions, sometimes called “Wick transforms.” In this paper, we introduce the
reader to some basic results, ideas, and mathematical subtleties about normal ordering of operators
and functions. The intended audience are instructors of quantum field theory and researchers who
are interested but not specialists in mathematical physics. © 2008 American Association of Physics Teachers.
关DOI: 10.1119/1.2805232兴
I. INTRODUCTION
polynomials. Finally, we provide a brief example of how the
Wick transform can be utilized in a physical application.
Normal ordering was introduced in quantum field theory
by Wick in 1950 to avoid some infinities in the vacuum
expectation values of field operators expressed in terms of
creation and annihilation operators.1 The simplest example
of such infinities can be discussed starting from only nonrelativistic quantum mechanics and the simple harmonic oscillator; an infinite number of harmonic oscillators make up a
free quantum field. 共The reader who wishes a quick reminder
of some basic quantum field-theoretical concepts may find
comfort in Appendix A兲.
In addition to the usual concept of quantizing by promoting fields to operators, modern quantum field theory uses
functional integrals as basic objects. In the functional integral formalism we calculate physical quantities such as scattering cross sections and decay constants by integrating over
some polynomials in the fields and their derivatives 共see, for
example, Refs. 2 and 3兲. In the examples considered here, the
fields we want to integrate are equivalent to functions on
spacetime. As mentioned, a first step toward removing infinities and giving mathematical meaning to calculations in the
operator formalism is normal ordering. The analog of this
ordering for functional integrals is called the “Wick transform”.
In this paper we introduce the reader to an interesting connection, which is usually not discussed in introductory treatments of quantum field theory, among normal ordering, Wick
transforms, and Hermite polynomials. We will also be concerned with some basic questions that are usually glossed
over, i.e., what does the functional integral really mean? Although a complete answer is not known and beyond the
scope of this paper, we intend to give some flavor of the first
steps toward addressing this question and how the Wick
transform has been put to work in this regard.
The paper is organized as follows: We will use the harmonic oscillator to exhibit the connection between Wickordered polynomials and the familiar Hermite polynomials.
Then we turn to Wick transforms in the functional integral
formalism of field theory, where we show that there is again
a connection with Hermite polynomials. Several approaches
to Wick transforms are compared and shown to be equivalent. In passing, we observe how the standard quantum field
theory result known as “Wick’s theorem” follows directly in
this framework from well-known properties of the Hermite
65
Am. J. Phys. 76 共1兲, January 2008
http://aapt.org/ajp
II. WICK OPERATOR ORDERING
A. Simple harmonic oscillator
The Hamiltonian operator for the simple harmonic oscillator in nonrelativistic quantum mechanics has the form:
H = 21 共P2 + Q2兲,
共1兲
where we have, as usual, hidden Planck’s constant ប, the
mass m, and the angular frequency ␻ in the definitions of the
dimensionless momentum and position operators,
P⬅
1
冑បm␻
p̂,
Q⬅
冑
m␻
q̂,
ប
共2兲
so that
关P,Q兴 = − i.
共3兲
†
If we define the creation and annihilation operators a and a
by
1
冑2 共Q − iP兲,
共4兲
1
共5兲
关a,a†兴 = 1,
共6兲
a† =
a=
冑2 共Q + iP兲,
so that
we find that
H = 21 共P2 + Q2兲 = 21 共a†a + aa†兲 = a†a + 21 .
共7兲
The eigenvalues of this Hamiltonian operator 共see for example, Ref. 4兲 satisfy the sequence:
En = n + 21 ,
n = 0,1,2, . . . .
共8兲
In particular, the ground state energy 共or zero-point energy兲,
which is the lowest eigenvalue of the Hamiltonian, is nonzero:
H兩0典 = 21 兩0典.
© 2008 American Association of Physics Teachers
共9兲
65
This zero-point energy has observable physical consequences. As an illustration, it is possible to measure the zeropoint motion of the atoms in a crystal by studying the dispersion of light in the crystal. Classical theory predicts that
the oscillations of the atoms in the crystal, and therefore also
dispersion effects, cease to exist when the temperature is
near absolute zero. However, experiments demonstrate that
the dispersion of light reaches a finite nonzero value at very
low temperatures.
In quantum field theory a free scalar field can be viewed as
an infinite collection of harmonic oscillators as described in
Appendix A. If we proceed as before, each oscillator will
give a contribution to the zero-point energy, resulting in an
infinite energy, which seems like it could be a problem.
One way to remedy the situation is to define the ground
state as a state of zero energy, which can be achieved by
redefining the Hamiltonian. We subtract the contribution of
the ground state and define a Wick-ordered 共or normalordered兲 Hamiltonian, denoted by putting a colon on each
side, by
:H: = : 21 共a†a + aa†兲: ⬅ 21 共a†a + aa†兲 − 21 具0兩a†a + aa†兩0典
共10兲
⬅ a†a.
In this example the definition of Wick ordering can be
thought of as a redefinition of the ground state of the harmonic oscillator.
In the last equality in Eq. 共10兲 we see that all creation
operators end up on the left. A general prescription for Wick
ordering in quantum field theory in a creation/annihilation
operator formalism is “Permute all the a† and a, treating
them as if they commute, so that in the end all a† are to the
left of all a.” The resulting expression is the same:
共11兲
:H: = a†a.
B. Wick ordering and Hermite polynomials
共12a兲
共a† + a兲2 = a†2 + a†a + aa† + a2 ,
=a†2 + 2a†a + a2 + 关a,a†兴,
共12b兲
=:共a† + a兲2: + 关a,a†兴,
共12c兲
关by Eq. 共10兲兴.
共12d兲
If we arrange terms in a similar way for higher powers of
共a† + a兲, we find:
共13a兲
共a† + a兲3 = :共a† + a兲3: + 3共a† + a兲,
共13b兲
共a† + a兲4 = :共a† + a兲4: + 6:共a† + a兲2: + 3.
†
We can summarize the results using the notation a + a = q,
66
q2 = :q2: + 1,
共14a兲
q3 = :q3: + 3:q:,
共14b兲
Am. J. Phys., Vol. 76, No. 1, January 2008
共14c兲
Because we can recursively replace normal-ordered terms on
the right by expressions on the left that are not normalordered 共for example :q2: can be replaced by q2 − 1兲, we can
also invert these relations:
:q2: = q2 − 1 = He2共q兲,
共15a兲
:q3: = q3 − 3q = He3共q兲,
共15b兲
:q4: = q4 − 6q2 + 3 = He4共q兲,
共15c兲
where the polynomials Hen共q兲 are a scaled version of the
more familiar form of the Hermite polynomials Hn:
Hen共x兲 = 2−n/2Hn共x/冑2兲.
共16兲
In some of the mathematical physics literature, the Hen are
often just called Hn. Some of the many useful properties are
collected in Appendix B for easy reference 共a more complete
collection is given in Ref. 5, for example兲.
Because of the relation between operator Wick ordering
and Hermite polynomials, the literature sometimes defines
“Wick ordering” in terms of Hermite polynomials:
:qn: ⬅ Hen共q兲.
共17兲
Although q is an operator composed of noncommuting operators a and a†, this alternative definition naturally generalizes to Wick ordering of functions. We will explore this idea
in the next section.
One reason that the connection to Hermite polynomials is
not mentioned in the standard quantum field theory literature
is the fact 共which was also Wick’s motivation兲 that the
normal-ordered part is precisely the part that will vanish
when we take the vacuum expectation value. The traditional
way to define normal-ordering, the one given at the end of
Sec. II A 共“put a† to the left of a”兲, yields for powers of q:
n
The first connection between Wick ordering and Hermite
polynomials arises when we study powers of the 共dimensionless兲 position operator Q. For the harmonic oscillator the
eigenvalue of Q2 gives the variance of the oscillator from
rest. We have Q = 共a† + a兲 / 冑2, but to avoid cluttering the
equations with factors of 冑2, we will study powers of just
共a† + a兲:
=:共a† + a兲2: + 1
q4 = :q4: + 6:q2: + 3.
:qn: = 兺
i=1
冉冊
n
共a†兲n−iai ,
i
共18兲
which vanishes for any nonzero power n when applied to the
vacuum state 兩0典.
In other words, because we know that normal-ordered
terms vanish upon taking the vacuum expectation value, we
may not be interested in their precise form. However, if the
expectation value is not taken in vacuum 共for example, in a
particle-scattering experiment兲, this part does not vanish in
general, and there are many instances where the actual
normal-ordered expression itself is the one of interest.
III. FUNCTIONAL INTEGRALS AND THE WICK
TRANSFORM
Most contemporary courses on quantum field theory discuss functional integrals 共sometimes called path integrals;
however, only in nonrelativistic quantum mechanics do we
really integrate over paths兲. In a functional-integral setting,
the counterpart of the Wick ordering in the operator formalism is the Wick transform. This transform applies to functions and functionals. It can, like its quantum-mechanics
counterpart 关Eq. 共17兲兴, be defined by means of Hermite polynomials. But first, we briefly skip ahead and explain why
such a transform will prove to be useful.
Alexander Wurm and Marcus Berg
66
A. Integration over products of fields
In the functional integral formalism, physical quantities
such as scattering cross sections and decay constants are calculated by integrating over some polynomial in the fields and
their derivatives. The algorithmic craft of such calculations is
described in textbooks such as those of Peskin and
Schroeder2 and Ryder.3 Although there are examples of
physical effects that can be studied with functional integral
methods but not with ordinary canonical quantization,6 in the
scope of this paper we can only give examples of some results that can be derived more quickly or transparently using
functional integrals.
The polynomials we will consider in this section are those
of Euclidean fields 共fields defined on four-dimensional Euclidean space R4兲. Similar formalisms exist for Minkowski
fields 共fields defined on spacetime兲 with minor changes in the
equations 共see for example, Ref. 2兲. Because functional integrals over Minkowski fields are less mathematically developed than integrals over Euclidean fields, we shall restrict
our attention to Wick transforms of functions and functionals
of Euclidean fields—primarily polynomials and exponentials. The Wick transform, like the Hermite polynomials, has
orthogonality properties that turn out to be useful in quantum
field theory. First, we have to introduce a few mathematical
concepts.
In general, the covariance is a positive continuous nondegenerate bilinear form on the space of test functions. A
simple example is the covariance for a free Klein–Gordon
field,
C共f,g兲 =
冕
d4xd4yf共x兲DF共x − y兲g共y兲,
共22兲
where DF is the textbook Feynman propagator 共see Ref. 2 for
example兲. In the following we will often encounter the covariance at coincident test functions, here denoted as C共f , f兲.
To get to the point, a Gaussian measure d␮C is defined by
its covariance C as
冕
d␮C共⌽兲exp共− i具⌽, f典兲 = exp关− 21 C共f, f兲兴,
over a space Y of distributions ⌽.
For comparison, the usual Gaussian measure on Rd is defined by,9
1
共2␲兲d/2
冕
Rd
d dx
e−共1/2兲Q共x兲e−i具x⬘,x典 = e−共1/2兲W共x⬘兲 , 共24兲
共det D兲1/2
where 具x⬘ , x典Rd = x␮⬘ x␮ is the duality in Rd with x 苸 Rd and
x⬘ 苸 Rd, Q共x兲 is a quadratic form on Rd,
Q共x兲 = D␮␯x␮x␯ = 具Dx,x典Rd ,
1. Gaussian measures
共19兲
where the bracket 具,典 denotes duality, that is, ⌽ is such that it
yields a number when applied to a smooth test function f. In
many situations the distribution ⌽ is equivalent to a function
␾, which means this number is the ordinary integral:
具⌽, f典 =
冕
R4
d4x␾共x兲f共x兲.
共20兲
A familiar example of a distribution is the Dirac distribution
⌽ = ␦, for which we have
⌽共f兲 = 具␦, f典 = f共0兲.
共21兲
Just like a function, ⌽ in general belongs to an infinitedimensional space. To be able to integrate over this space
关not to be confused with the integral in Eq. 共20兲, which is
an ordinary integral over spacetime兴 we need a measure,
some generalization of the familiar d4x in the ordinary
integral in Eq. 共20兲. To this end, we need to introduce the
covariance. Given the action of the theory, we readily
calculate the classical equations of motion, extract the differential operator D, and invert it to find the covariance C.
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Am. J. Phys., Vol. 76, No. 1, January 2008
共25兲
and W共x⬘兲 is a quadratic form on Rd
Here, our aim is to specify the notation and to briefly
remind the reader how to integrate over Euclidean fields,
without going into too much detail. The standard mathematical framework to perform such integration is the theory of
Gaussian measures in Euclidean field theory.7,8
As a first try, we might define a field in the functional
integral as a function ␾ on R4. Fields in the functional integral might seem like functions at first glance, but can produce divergences that cannot, for instance, be multiplied in
the way that functions can. A more useful way to regard a
quantum field in the functional integral formalism is as a
distribution ⌽ acting on a space of test functions f:
⌽共f兲 ⬅ 具⌽, f典,
共23兲
Y
W共x⬘兲 = x␮⬘ C␮␯x␯⬘ = 具x⬘,Cx⬘典Rd ,
共26兲
such that
共27兲
DC = CD = 1.
In the more familiar R case 关Eq. 共24兲兴, the combination of
the standard measure and kinetic term corresponds to the
measure d␮C we introduced earlier. That is,
ddx / 共det D兲1/2e−1/2Q共x兲 is analogous to d␮C共⌽兲. The explicit
separation of d␮C into ddx / 共det D兲1/2 and e−1/2Q共x兲 will turn
out to be unnecessary for our discussion. By defining the
measure d␮C through Eq. 共23兲, we have not even specified
what such a separation would mean.
The covariance at incident points can be expressed as the
following integral, obtained by expanding Eq. 共23兲:
d
C共f, f兲 =
冕
d␮C共⌽兲具⌽, f典2 .
共28兲
The integral on the left-hand side of Eq. 共23兲 is the generating function of the Gaussian measure; let us denote this integral by Z共f兲. By successively expanding Eq. 共23兲, the nth
moment of the Gaussian measure can be compactly written
as
冏冕
冉 冊 冏
d␮C共⌽兲具⌽, f典n = − i
=
再
d
d␭
n
Z共␭f兲
␭=0
共n − 1兲 ! ! C共f, f兲n/2
n even
0
n odd,
共29兲
where n ! ! = n共n − 2兲共n − 4兲¯ is the semifactorial. For convenience, we introduce the following notation for the average
with respect to the Gaussian measure ␮C:
Alexander Wurm and Marcus Berg
67
具F关⌽共f兲兴典␮C ⬅
冕
d␮C共⌽兲F关⌽共f兲兴.
共30兲
Note the difference between the brackets 具 典␮C used for the
average and the brackets 具,典 used for duality.
:exp共␣⌽共f兲兲:C ⬅ 1 + ␣:⌽共f兲:C + 21 ␣2:⌽共f兲2:C + . . . ,
共35兲
where normal ordering is defined by Eq. 共33兲.
Problem 1. Show that
:exp共␣⌽共f兲兲:C =
2. Wick transforms: Definitions
We can use this set of definitions to define a Wick transform of functionals of fields. Our goal is to provide some
idea of how to address the difficult problem of making sense
out of products of distributions and integrals of such products. These products are ubiquitous in quantum field theory,
although their exact meaning is not usually discussed in introductory textbooks. To simplify the calculations, we define
the Wick transform of a power ⌽共f兲n ⬅ 具⌽n , f典 so as to satisfy an orthogonality property with respect to Gaussian integration. We recall the orthogonality properties of Hermite
polynomials 共Appendix B兲 and the definition 关Eq. 共17兲兴 and
define the Wick transform in terms of Hermite polynomials:
:⌽共f兲n:C = C共f, f兲n/2Hen
冉冑 冊
⌽共f兲
C共f, f兲
共31兲
.
兺k=0 bkxk 1 ⬁ k
= 兺 c kx ,
⬁
兺k=0 akxk a0 k=0
⬁
具exp关␣⌽共f兲兴典␮C =
=
共33a兲
␦
:⌽共f兲n:C = n:⌽共f兲n−1:C,
␦⌽
冕
d␮C共⌽兲:⌽共f兲n:C = 0,
n = 1,2, . . . ,
共33b兲
共33c兲
n = 1,2, . . . ,
where the functional derivative with respect to a distribution
is
␦
⌽共f兲 = f .
␦⌽
共34兲
n
Let us check that the Wick transform :⌽共f兲 :C defined by Eq.
共33兲 is the same as the Wick transform given in terms of
Hermite polynomials in Eq. 共31兲. This equivalence is to be
expected, because the Hermite polynomials satisfy similar
recursion relations, but it is a useful problem to check that it
works. To begin, we establish a property of Wick exponentials.
Let :exp共␣⌽共f兲兲:C be the formal series:
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Am. J. Phys., Vol. 76, No. 1, January 2008
冕
冕
=兺
n
d␮C共⌽兲exp共␣⌽共f兲兲
d␮C共⌽兲 兺
n
冕
␣2n
n!
␣n
⌽共f兲n
n!
冋
=exp
共38a兲
d␮C共⌽兲⌽共f兲2n ,
关by Eq. 共29兲兴
共32兲
:⌽共f兲0:C = 1,
共37兲
n
cn−kak − bn = 0. A comparison of the resulting
where cn + a10 兺k=1
series, term by term, to the power series expansion of the
left-hand side proves Eq. 共36兲.
Problem 2. Show the equivalence of Eqs. 共33兲 and 共31兲.
Solution. We can explicitly calculate the denominator in
Eq. 共36兲:
d␮C共⌽兲:⌽共f兲n:C:⌽共g兲m:C = ␦m,nn ! 共具⌽共f兲⌽共g兲典␮C兲n .
An entertaining problem is to prove Eq. 共32兲, which we will
do in Sec. III A 3 共paragraph 2兲.
The Wick transform can also be defined recursively by the
following equations:10
共36兲
Solution. We can evaluate the right-hand side of Eq. 共36兲 by
expanding the numerator and denominator in a power series
and dividing one power series by the other.
Note that this definition depends on the covariance C, and
that there is no analogous dependence in the analogous
harmonic-oscillator definition 关Eq. 共17兲兴.
The orthogonality of two Wick-transformed polynomials
is expressed by
冕
exp共␣⌽共f兲兲
.
具exp共␣⌽共f兲兲典␮C
册
1 2
␣ C共f, f兲 .
2
共38b兲
关by Eq. 共28兲兴
共38c兲
Thus, from Eq. 共36兲 we find
:exp关␣⌽共f兲兴:C = exp共␣⌽共f兲 − 21 ␣2C共f, f兲兲 .
共39兲
If we multiply the power series expansions of exp关␣⌽共f兲兴
and exp关1 / 2␣2C共f , f兲兴 and compare the result term by term
to the series expansion of the left-hand side of Eq. 共39兲, we
obtain
11
关 2n 兴
:⌽共f兲n:C =
n!
⌽共f兲n−2m
兺
m
!
共n
−
2m兲!
m=0
冉
1
− C共f, f兲
2
冊
m
.
共40兲
We rewrite Eq. 共40兲 as
关 2n 兴
:⌽共f兲n:C = C共f, f兲n/2 兺 共− 1兲m
m=0
⫻
冉冑 冊
⌽共f兲
C共f, f兲
n!
2 m ! 共n − 2m兲!
m
n−2m
,
共41兲
use the expression 共B1兲 for the defining series of the Hermite
polynomials given in Appendix B and recover Eq. 共31兲.
Alexander Wurm and Marcus Berg
68
3. Wick transforms: Properties
Many properties of Wick-ordered polynomials can be conveniently derived using the formal exponential series. For
simplicity, we assume that all physical quantities that we
wish to calculate 共for example, scattering cross sections兲 are
written with normalization factors of 1 / C共f , g兲, which in effect lets us set the coincident-point covariance to unity:
C共f , f兲 = 1. The covariance can be restored by comparison
with Eq. 共41兲. The properties in which we are interested are
useful problems:
Problem
3.
Show
that
:exp共⌽共f兲 + ⌽共g兲兲:C
= exp共−具⌽共f兲⌽共g兲典␮C兲 : exp共⌽共f兲兲:C : exp共⌽共g兲兲:C.
Solution.
:exp共⌽共f兲兲:C:exp共⌽共g兲兲:C ,
共42a兲
=exp共⌽共f兲 + ⌽共g兲兲exp共− 21 关具⌽共f兲2典␮C + 具⌽共g兲2典␮C兴兲,
B. Wick transforms and functional Laplacians
We can also define Wick transforms of functions by the
following exponential operator expression, which is convenient in many cases 共for example in two-dimensional quantum field theory settings, such as in Refs. 12 and 13兲:
1
: ␾n共x兲:C ⬅ e− 2 ⌬C␾n共x兲,
共42c兲
where we have used Eq. 共39兲 in the first equality, and, after
completing the square in the second factor of Eq. 共42b兲,
again in the second equality. Dividing both sides of Eq. 共42兲
by the second factor in Eq. 共42c兲 completes the proof.
Problem
4.
Show
that
具:⌽共f兲n:C : ⌽共g兲m:C : 典␮C
n
= ␦nmn ! 具⌽共f兲⌽共g兲典␮
C
Solution. If we take the expectation value of both sides of
Eq. 共42兲, we find
⌬C =
共43兲
using Eq. 共33兲. Expanding the exponentials on both sides and
comparing term by term completes the proof.
Problem 5. Show that :⌽共f兲n+1:C = n : ⌽共f兲n−1:C
− ⌽共f兲 : ⌽共f兲n:C.
Solution. This recursive relationship is a consequence of
the equivalence of Wick-ordered functions and Hermite
polynomials. The expression follows from the recursion relation for Hermite polynomials given in Appendix B.
Problem 6. The definition of Wick transforms given in Eq.
共33兲 can be generalized to several fields in a straightforward
manner. We quote here some results without proof 共details
can be found in Ref. 10兲. The reader may find it interesting to
check that it works:
:⌽共f 1兲 . . . ⌽共f n+1兲: = :⌽共f 1兲 . . . ⌽共f n兲:⌽共f n+1兲
冉 冕
: ␾4共y兲:C = exp −
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Am. J. Phys., Vol. 76, No. 1, January 2008
d4xd4x⬘C共x,x⬘兲
冉 冊冉冕
共48兲
冊
␦
␦
␾4共y兲.
␦␾共x兲 ␦␾共x⬘兲
冕
共44兲
共45兲
d4xd4x⬘C共x,x⬘兲
冊
⫻
␦
␦
␾4共y兲
␦␾共x兲 ␦␾共x⬘兲
+
1
1
−
2!
2
⫻
2
␦
␦
␾4共y兲.
␦␾共x兲 ␦␾共x⬘兲
冉 冊 冉冕
2
d4xd4x⬘C共x,x⬘兲
=
d4xd4x⬘C共x,x⬘兲
冊
共50兲
冕
冕
␦
␦
␾4共y兲
␦␾共x兲 ␦␾共x⬘兲
␦
4␾3共y兲␦共x⬘ − y兲,
␦␾共x兲
共51a兲
d4x,C共x,y兲12␾2共y兲␦共x − y兲 = 12␾2共y兲,
共51b兲
d4xd4x⬘C共x,x⬘兲
if the covariance is normalized to unity. We use this result in
the term:
冕
共46兲
1
2
All higher terms in the expansion are zero. We can now
evaluate each term separately. The second term is the integral
=
d␮C共⌽兲:⌽共f 1兲 . . . ⌽共f n兲::⌽共g1兲 . . . ⌽共gm兲: = 0
for n ⫽ m.
1
2
: ␾4共y兲:C = ␾4 + −
k=1
冕
␦
␦
,
␦␾共x兲 ␦␾共x⬘兲
共49兲
− 兺 C共f k, f n+1兲:⌽共f 1兲 . . . ⌽共f k−1兲
d␮C共⌽兲:⌽共f 1兲 . . . ⌽共f n兲: = 0,
d4xd4x⬘C共x,x⬘兲
which, again, depends on the covariance C.
Instead of proving Eq. 共47兲, which is straightforward, we
just illustrate the equivalence of definition 共47兲 and definition
共31兲 for the common example of a ␾4 power for which the
definition becomes
n
冕
冕
We expand the exponential and find
具:exp共⌽共f兲兲:C:exp共⌽共g兲兲:C典␮C = exp关具⌽共f兲⌽共g兲典␮C兴
⫻⌽共f k+1兲 . . . ⌽共f n兲:
共47兲
where the functional Laplacian is defined by
共42b兲
=:exp共⌽共f兲 + ⌽共g兲兲:C exp共具⌽共f兲⌽共g兲典␮C兲,
These latter multifield expressions reproduce, within this
functional framework, what is usually referred to as Wick’s
theorem in the creation/annihilation-operator formalism. In
this formalism it takes some effort to show this theorem;
here we find it somewhat easier by relying on familiar properties of the Hermite polynomials.
d4xd4x⬘,C共x,x⬘兲
=
冕
␦
␦
12␾2共y兲
␦␾共x兲 ␦␾共x⬘兲
d4xd4x⬘C共x,x⬘兲
␦
24␾共y兲␦共x⬘ − y兲,
␦␾共x兲
Alexander Wurm and Marcus Berg
共52a兲
69
=24
冕
d4xC共x,y兲␦共x − y兲 = 24,
共52b兲
with the same normalization of the covariance. We collect
these results with the appropriate coefficients from the expansion:
1
1 1
· 24,
: ␾ 共y兲:C = ␾ 共y兲 − · 12␾2共y兲 +
2
2! 22
4
4
= ␾4共y兲 − 6␾2共y兲 + 3 = He4关␾共y兲兴,
共53a兲
共53b兲
which completes the example.
C. Further reading
Although we hope to have given some flavor of some of
the techniques and ideas of quantum field theory
mathematical-physics style, we have given only a few examples and demonstrated some simple identities. For more
on the mathematical connection between Wick transforms on
function spaces and Wick-ordering of annihilation and creation operators, we recommend Refs. 7 and 8. For readers
more interested in fermions than the scalar fields we have
discussed, we recommend Ref. 14 as an introduction to Wick
ordering.
D. An application: Specific heat
In this section we discuss an example of a physical application of some of the results we have discussed. By using the
connection Eq. 共31兲 between the Wick transform and Hermite polynomials, we show how standard properties of those
polynomials can be exploited to simplify certain calculations.
Consider the familiar generating function of Hermite polynomials 关but for the scaled polynomials Eq. 共16兲兴:
e
x␣−␣2/2
␣n
= 兺 Hen共x兲 .
n!
共54兲
This generating function gives a shortcut to calculations in
two-dimensional quantum field theory, where normal ordering is often the only form of renormalization necessary. An
important issue is the scaling dimension of the normalordered exponential :eip␾:, where p is a momentum. 共The
real part of this operator can represent the energy of a system
where ␾ is the quantum field.兲 In other words, the question is
if we rescale the momentum p → ⌳p, equivalent to a rescaling x → ⌳−1x in coordinate space, how does the operator
:eip␾: scale? Because an exponential is dimensionless, we
might guess that the answer is that it does not scale at all,
that is, the scaling dimension is zero. This answer is not so
due to quantum effects induced by the normal ordering. In
terms of the functional integral we can easily calculate the
effect of the normal-ordering 共here, the Wick transform兲:
⬁
:eip␾:C = 兺
n=0
⬁
=兺
n=0
1
共ip兲n:␾n:C ,
n!
1 2
1
共ip兲nCn/2Hen共␾/冑C兲 = e 2 p Ceip␾ ,
n!
共55a兲
共55b兲
where we used the definition Eq. 共31兲 and the generating
function Eq. 共54兲. The covariance C is a logarithm in two
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Am. J. Phys., Vol. 76, No. 1, January 2008
dimensions; that is, the solution of the two-dimensional
Laplace equation is a logarithm. To regulate divergences
when p → ⬁, we introduce a cutoff ⌳ for the momentum,
which makes C = ln ⌳. If we substitute this result into Eq.
共55b兲, we obtain
2
:eip␾: = ⌳ p /2eip␾ .
共56兲
Thus the anomalous scaling dimension, usually denoted by
␥, is ␥ = p2 / 2 for the exponential operator. This result is important in conformal field theory 共see for example Ref. 15兲.
Here the p2 / 2 comes from the ␣2 / 2 in the generating function Eq. 共54兲.
How could such a quantum effect be measured? Consider
the two-dimensional Ising model with random bonds 共Ref.
13, p. 719兲. This model is the familiar Ising model, but the
coupling between the spins is allowed to fluctuate, that is, the
coupling becomes a space-dependent Euclidean field. The
energy of the system is described by 共the real part of兲 the
exponential operator :eip␾:. The anomalous dimension in Eq.
共56兲 leads to an expression for the specific heat 关see Ref. 13,
Eq. 共356兲兴, where the derivation is given using renormalization group methods. The specific heat is in principle directly
measurable as a function of the temperature, or more conveniently, as a function of ␪ = 共T − Tc兲 / Tc, a dimensionless deviation from the critical temperature. The renormalization
group description predicts a double logarithm dependence on
␪ that could not have been found by simple perturbation
theory; it uses as input the result from Eq. 共56兲.
Admittedly, the telegraphic description in the previous
paragraph does not do justice to the full calculation of the
specific heat in the two-dimensional Ising model with random bonds. Our purpose here is only to show how the Wick
transform reproduces the quantum effect in Eq. 共56兲, and
then to give some flavor of how this effect is measurable.
IV. CONCLUSION
We have shown that the scope of normal ordering has
expanded to settings beyond the original one of ordering
operators and we have discussed an interesting connection to
Hermite polynomials, which is usually not mentioned in
courses on quantum field theory. Several different definitions
of Wick ordering of functions have been discussed and their
equivalence established.
From a physics point of view, Wick ordering is part of the
process of renormalization, which systematically removes infinities from a theory 共that is, removes infinite terms from the
perturbation expansion of the functional integral兲. In general,
Wick ordering needs to be supplemented by additional rules
for renormalization 共see for example, Ref. 2兲 From a mathematics point of view, Wick ordering is helpful to make
sense of the polynomials of fields that are integrated over in
the functional integral.
For a deeper understanding and further applications of
these ideas, the interested reader is invited to consult the
cited literature, which is a selection of texts we have found
particularly useful. In particular, for the physics of functional
integrals we recommend Ref. 7. For a more mathematically
oriented treatment we have found Ref. 8 to be quite useful.
ACKNOWLEDGMENTS
We would like to thank C. DeWitt–Morette, P. Cartier, and
D. Brydges for many helpful discussions.
Alexander Wurm and Marcus Berg
70
APPENDIX A: WICK-ORDERING OF OPERATORS
IN QUANTUM FIELD THEORY
We briefly remind the reader how the need for Wick ordering arises in the operator formulation of quantum field
theory. All of this material is standard and can be found in
introductory books on quantum field theory 共for example,
Ref. 2兲, albeit in lengthier and more thorough form. We set
ប = 1 throughout.
Consider a real scalar field ␾共t , x兲 of mass m defined at all
points of four-dimensional Minkowski spacetime and satisfying the Klein-Gordon equation:
冉
冊
⳵2
− ⵜ2 + m2 ␾共t,x兲 = 0.
⳵ t2
共A1兲
The differential operator in parenthesis is one instance of
what we call D in the text. The classical Hamiltonian of this
scalar field is:
H=
1
兺 关共␲共t,x兲兲2 + 共ⵜ␾共t,x兲兲2 + m2␾2共t,x兲兴,
2 x
共A2兲
where ␲ is the variable canonically conjugate to ␾, ␲
= ⳵␾ / ⳵t. We can think of the first term as the kinetic energy
and the second as the shear energy. This classical system is
quantized in the canonical quantization scheme by treating
the field ␾ as an operator, and imposing equal-time commutation relations:
关␾共t,x兲, ␾共t,x⬘兲兴 = 0,
共A3a兲
关␲共t,x兲, ␲共t,x⬘兲兴 = 0,
共A3b兲
关␾共t,x兲, ␲共t,x⬘兲兴 = i␦3共x − x⬘兲.
共A3c兲
The plane-wave solutions of the Klein–Gordon equation are
known as the field modes, uk共t , x兲. Together with their respective complex conjugates uk*共t , x兲 they form a complete
orthonormal basis. Hence, the field ␾ can be expanded as
␾共t,x兲 = 兺 关akuk共t,x兲 + ak† uk*共t,x兲兴.
共A4兲
k
The equal time commutation relations for ␾ and ␲ are then
equivalent to
Ĥ =
冉
冊
1
1
共ak† ak + akak† 兲␻k = 兺 ak† ak + ␻k ,
兺
2 k
2
k
共A7兲
where in the last step we used the commutation relations
from Eq. 共A5c兲. A calculation of the vacuum energy reveals
a potential problem:
1
1
具0兩Ĥ兩0典 = 具0兩0典 兺 ␻k = 兺 ␻k → ⬁ ,
k 2
k 2
共A8兲
where we have used the normalization condition 具0 兩 0典 = 1.
This infinite constant can be removed as described in the
text.
Propagation amplitudes in quantum field theory 共and
hence scattering cross sections and decay constants兲 are
given in terms of expectation values of time-ordered products of field operators. These time-ordered products arise in
the interaction Hamiltonian of an interacting quantum field
theory. The goal is to calculate propagation amplitudes for
these interactions using time-dependent perturbation theory
familiar from quantum mechanics. To leading order in the
coupling constant, these products can be simplified, and the
zero-point constant energy removed by using Wick’s theorem.
The way Wick ordering is applied in practice to calculations in quantum field theory is through Wick’s theorem,
which gives a decomposition of time-ordered products of
field operators into sums of normal-ordered products of field
operators 共again, we refer to Ref. 2兲. Wick’s theorem appears
in the functional-integral formulation of the theory in Sec.
III A 3.
APPENDIX B: PROPERTIES OF Hen„x…
Here we list a few useful properties of the scaled Hermite
polynomials. More properties can be found in Ref. 5.
Defining series:
Hen共x兲 =
关 2n 兴
n!
兺 共− 1兲m m ! 2m共n − 2m兲! xn−2m ,
m=0
共B1兲
where 关n / 2兴 is the integer part of n / 2.
Orthogonality:
冕
⬁
dxe−x /2Hen共x兲Hem共x兲 = ␦mn冑2␲n ! .
2
共B2兲
−⬁
关ak,ak⬘兴 = 0,
共A5a兲
†
共A5b兲
关ak† ,ak⬘兴 = 0,
Generating function:
⬁
exp共x␣ −
1 2
2␣ 兲
= 兺 Hen共x兲
n=0
关ak,ak⬘兴 = ␦kk⬘ .
†
共A5c兲
These operators are defined on a Fock space, which is a
Hilbert space made of n-particle states 共n = 0 , 1 , . . . 兲. The
normalized basis ket vectors, denoted by 兩典, can be constructed starting from the vector 兩0典. The vacuum state 兩0典
has the property that it is annihilated by all the ak operators:
ak兩0典 = 0,
∀ k.
共A6兲
In terms of the frequency ␻k = c冑兩k兩 + m , the Hamiltonian
operator obtained from Eq. 共A2兲 is
2
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Am. J. Phys., Vol. 76, No. 1, January 2008
␣n
.
n!
共B3兲
Recursion relation:
Hen+1共x兲 = xHen共x兲 − nHen−1共x兲.
共B4兲
The first five polynomials are
He0共x兲 = 1,
共B5a兲
He1共x兲 = x,
共B5b兲
He2共x兲 = x2 − 1,
共B5c兲
He3共x兲 = x3 − 3x,
共B5d兲
Alexander Wurm and Marcus Berg
71
He4共x兲 = x4 − 6x2 + 3.
a兲
共B5e兲
Electronic address: [email protected]
G. C. Wick, “The evaluation of the collision matrix,” Phys. Rev. 80,
268–272 共1950兲.
2
Michael Peskin and Daniel V. Schroeder, An Introduction to Quantum
Field Theory 共Addison-Wesley, Reading, MA, 1995兲. See in particular
the chapter on functional integrals in quantum field theory.
3
Lewis H. Ryder, Quantum Field Theory 共Cambridge University Press,
Cambridge, 1985兲.
4
Jun J. Sakurai, Modern Quantum Mechanics, rev. ed. 共Addison-Wesley,
Reading, MA, 1994兲.
5
Milton Abramovitz and Irene A. Stegun, Handbook of Mathematical
Functions 共Dover, New York, 1965兲.
6
In particular, contributions to the S-matrix that have an essential singularity at zero coupling constant cannot be found by standard perturbative
expansion around zero coupling. Yet these “nonperturbative” contributions can be studied using functional integrals. Rates for decays that
would be strictly forbidden without these effects can be calculated; see
for example, Ref. 3, Sec. 10.5.
7
James Glimme and Arthur Jaffe, Quantum Physics, 2nd ed. 共Springer1
Verlag, New York, 1981兲.
Svante Janson, Gaussian Hilbert Spaces 共Cambridge University Press,
Cambridge, 1997兲.
9
To avoid dimension-dependent numerical terms 共powers of ␲, powers of
2兲 in the definition 共24兲 of the Gaussian measure we can, alternatively,
define it by 兰Rdddx / 共det D兲1/2e−␲Q共x兲e−2␲i具x⬘,x典 = e−␲W共x⬘兲. This definition
can be convenient in the simplest Fourier transforms for those who forget
where the 2␲ goes: f̂共p兲 = 兰dxe−2␲ipx f共x兲 yields an inverse f共x兲
= 兰dpe2␲ipx f̂共p兲, without the prefactor.
10
Barry Simon, The P共␾兲2 Euclidean (Quantum) Field Theory 共Princeton
University Press, Princeton, NJ, 1974兲.
11
Similar to the division of power series, the multiplication of a power
⬁
⬁
⬁
n
series is 共兺k=0
akxk兲共兺k=0
bkxk兲 = 兺n=0
dnxn, where dn = 兺m=0
ambn−m.
12
Joseph Polchinski, String Theory: An Introduction to the Bosonic String,
Vol. 1 共Cambridge University Press, Cambridge, 1998兲.
13
Claude Itzykson and Jean-Michel Drouffe, Statistical Field Theory, Vols.
1 and 2 共Cambridge University Press, Cambridge, 1989兲.
14
Manfred Salmhofer, Renormalization: An Introduction 共Springer-Verlag,
Berlin, 1999兲.
15
Edward Witten, “Perturbative quantum field theory,” in Quantum Fields
and Strings: A Course for Mathematicians, edited by P. Deligne et al.
共American Mathematical Society, Providence, RI, 1999兲, Vol. 1, p. 451.
8
Inertia Demonstration. We still do this demonstration today. A stiff card is placed on a pillar and a marble is
balanced atop the card. A quick blow with the hand or a snap with the fingertip on the edge of the card knocks it out
of the way, and the marble drops into a small depression in the top of the tower. For reliable demonstrations, the digit
is replaced by a strip of spring steel. This example is at Grinnell College in Iowa. Suggestion: make a half dozen of
these 共they can be made with blocks of wood and a section of an old hacksaw blade兲 and put them in the introductory
lab for students to play with before lab starts. 共Photograph and Notes by Thomas B. Greenslade, Jr., Kenyon College兲
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Am. J. Phys., Vol. 76, No. 1, January 2008
Alexander Wurm and Marcus Berg
72