Download Manifestly Covariant Functional Measures for Quantum Field Theory

Manifestly Covariant Functional Measures for Quantum Field Theory in
Curved Spacetime via Canonical Quantization∗
C. F. Richardson†
(Dated: January 12, 2017)
It is shown that manifestly generally covariant functional measures can be derived from
canonical quantization for quantum field theory in curved spacetime provided that the time
slices used in constructing the functional integrals are equally spaced in physical time rather
than in coordinate time and, similarly, discretization within time slices are chosen such that
points are distributed uniformly in physical distances rather than in coordinate distances.
Scalar, spinor and vector fields and quantum gravity are considered. In each case, manifestly
covariant functional measures are obtained.
I.
INTRODUCTION
In the conventional approach to deriving the functional integral representation of the partition
function in quantum field theory, one discretizes time into slices spaced a constant distance apart
in coordinate time. When quantum field theory is formulated in a curved spacetime background,
the measure resulting from this uniform coordinate discretization contains non-covariant factors
of g 00 = 1/N 2 where N is the lapse[1–3]. Toms[4] has argued that the factor of the lapse is a
reflection of a naive generalization of finite-dimensional quantum mechanics to field theory. Toms
showed that a covariant measure for the scalar field can be obtained by including an additional
factor in the phase space measure. Toms’ approach was applied to spinor and vector fields by
Basler[5]. However, Toms’ approach loses contact with canonical quantization where unitarity is
manifest and thus leaves open the question of whether the factors of lapse are needed to preserve
unitarity as suggested by Fradkin and Vilkovisky[1] (We will see, however, that such factors are
not needed for unitarity). Other authors[6–8] have obtained covariant functional measures but the
techniques employed leave the connection with canonical quantization obscure. We will show that
manifestly covariant measures can be obtained directly from canonical quantization.
In deriving the functional integral, it is typically assumed that one can take a continuum limit
and that the details of the discretization used to derive the results will not matter in this limit. We
show that this is not the case by demonstrating that using a different discretization results in an
inequivalent measure. In particular, we show that if time (and similarly, distances) is discretized
into slices spaced a constant physical time apart (i.e., δτ = N δt is constant) that no non-covariant
factors appear in the measure. The conclusion that using a different discretizations results in
inequivalent measures is consistent with Toms’ argument that field theory is not a straightforward limit of the finite dimensional case. By expanding fields in a covariant mode expansion,
we compare the measures obtained from uniform discretization in coordinate distances and from
uniform discretization in physical distances and conclude that the measure obtained from uniform
discretization in coordinate distances is off by a factor of the lapse for every bosonic degree of
freedom minus every fermionic degree of freedom.
We consider scalar, spinor, vector and tensor fields. In all cases, the measure can be written as
products of dai terms and ghost terms in the case of gauge fields, where ai are mode coefficients
in the expansion of the fields. No factors of the determinant of the metric or of the lapse appear
in the measure when written in this form.
∗
†
http://archive.org/details/CovariantFunctionalMeasure
[email protected]
2
II.
FUNCTIONAL MEASURE FOR SCALAR FIELD IN CURVED SPACETIME
We start by reviewing the standard functional integral approach to quantizing a scalar field in a
curved spacetime background. This is typically done by dividing spacetime into time slices having
uniform spacing in coordinate time. Because this is done in coordinate time, the selection of points
in the measure is not generally covariant and the resulting measure has odd factors of g 00 . The
resulting measure is therefore not manifestly generally covariant. After reviewing the standard
approach, we show that a manifestly covariant result can be obtained by discretizing uniformly in
physical distances rather than uniformly in coordinate lengths.
Starting from a Lagrangian
L=
√
g
1 αβ
g ∂α Φ∂β Φ − V (Φ) ,
2
(1)
we find that the momentum conjugate to Φ is
π=
δL
√
= gg 0α ∂α Φ.
δ∂0 Φ
(2)
We will work in Lorentzian spacetime and, in a minor departure from more conventional notation,
we define g to be the absolute value of the determinant of the metric tensor,
g ≡ | det gαβ |.
(3)
√
g
1 π2
πg 0i ∂i Φ
√ 1
+ 00 (g 0i ∂i Φ)2 − g g ij ∂i Φ∂j Φ + V (Φ) .
H = √ 00 −
00
2 gg
g
2g
2
(4)
The Hamiltonian can be written as
In order to define the functional integral, we define fields on a discrete set of points in spacetime
and insert a complete set of states at each point. The canonical commutation relation can be
written
[Φ(~x), π(~x0 )] = iδ (3) (~x − ~x0 ).
(3)
Integrating ~x and ~x0 over a coordinate 3-volume vc
(5)
= δ 3 , we have
[Φj , πk ] = iδjk /vc(3)
(6)
where j and k label points in a time slice. This implies, up to a constant normalization factor,
that
(3)
hΦj |πj i = eiπj Φj vc .
(7)
From this one obtains[9]
Z=
Y
Z
4
dπ(x)dΦ(x) exp i d x π(x)∂0 Φ(x) − H
(8)
xc
Q
where the notation xc indicates that the points are selected uniformly in coordinate space.
Before proceeding with
Qevaluating the partition function, consider how the functional measure
is defined. The quantity xc dΦ(x) is defined so that
Z
Y
√
4 √
dΦ(x) exp − d x gΦ(x)A(x)Φ(x) = det−1/2 ( gA)
(9)
xc
3
up to irrelevant constants. We can also define a measure for uniform discretization in physical
distances so that
Z
Y
4 √
dΦ(x) exp − d x gΦ(x)A(x)Φ(x) = det−1/2 (A)
(10)
xp
Q
where xp indicates that the points are selected uniformly in physical distance. We will return to
this second measure shortly.
Integrating out the momentum variables in equation (8), we have
Z
Y
g 1/4 (g 00 )1/2 dΦ(x) exp i d4 xL .
(11)
Z=
xc
So the functional measure is not manifestly generally covariant - neither the selection of points nor
the g 00 factor in the measure is covariant. In the literature it is often implicitly assumed that since
we envision taking the coordinate grid spacing to zero that the dependence on the selection of points
is removed in this limit. The appearance of g 00 in the measure then seems mysterious. However
the assumption that the selection of points is irrelevant is incorrect. We will now demonstrate this
by selecting a different set of points and obtaining a different measure as the grid spacing becomes
small.
Instead of discretizing uniformly in coordinates, we now discretize uniformly in physical distances. It is convenient to define
π
π
π̂ = p
=√
(12)
h
gg 00
where h ≡ | det hij | and hij is the metric induced on the spacelike time slice. The canonical
commutation relation can be written
p
[Φ(~x), π̂(~x0 )] = iδ (3) (~x − ~x0 )/ h(~x).
(13)
p
p
Multiplying both sides by h(~x) h(~x0 ) and integrating ~x and ~x0 over a proper 3-volume
p
p
(3)
h(~x0 )∆x0
(14)
vp(3) = h(~x)∆(3)
x =
(3)
(3)
where ∆x is the coordinate 3-volume at ~x needed for a proper 3-volume vp , and similarly for
(3)
∆x0 , we have
[Φj , πk ] = iδjk /vp(3)
(15)
where j and k label points in a time slice. This implies, up to a constant normalization factor,
that
(3)
hΦj |π̂j i = eiπ̂j Φj vp .
The partition function is then
Z
Z Y
1
4 √
Z=
dπ̂(x)dΦ(x) exp i d x gπ̂(x) ∂0 Φ(x) − H .
N
x
(16)
(17)
p
Note than canonical quantization dictates that the measure is either
Y
Y
dπ̂(x)dΦ(x) or
dπ(x)dΦ(x)
xp
xc
(18)
4
depending on how the discretization is done and, for example, it is not correct to simply replace
π with π̂ in the right hand side of equation (18). Integrating out the momentum in equation (17)
we find
Z
Z Y
4
Z=
dΦ(x) exp i d xL .
(19)
xp
So in contrast to the usual approach,Pthe canonical measure is manifestly
R 4 √ covariant.
Expanding Φ in a basis as Φ = k ak Φk with Φk satisfying d x gΦ∗k Φk0 = δk,k0 , equation
(19) becomes
Z
Z Y
Z=
dak exp i d4 xL .
(20)
k
This expression is also follows from
Z
Z Y
1/4
4
Z=
g dΦ(x) exp i d xL
(21)
xc
where the additional factor of g 1/4 arises from the difference between equations (9) and (10).
Equation (21) is equation (11) without the factor of (g 00 )1/2 = 1/N. Since physical states must
be distributed in physical distances not coordinate distances equation (19) which is equivalent to
equation (21) is the physically correct result. That equation (21) is not equivalent to equation (11)
implies that discretizing uniformly in coordinate distances does not give the correct result. This is
a reasonable conclusion since there is no reason to expect that the functional measure should not
contain information about how the underlying states are distributed.
The spurious factor of the lapse is a general feature of the measure derived by discretizing
uniformly in coordinate distances where we always obtain a spurious factor of 1/N for each bosonic
degree of freedom and a spurious factor of N for each fermionic degree of freedom. This can be
seen as follows. For actions quadratic in momenta, Unz[2, 3] showed that after integrating over
momenta (for discretization uniform in coordinate distances) the measure picks up a factor of
"
δ2L
sdet
δ(∂0 QA )δ(∂0 QB )
#1/2
(22)
where sdet is the superdeterminant defined by
sdetM =
det Mbb − Mbf Mf−1
f Mf b
det Mf f
(23)
where Mbb , Mf f , Mf b and Mbf are the boson-boson, fermion-fermion, fermion-boson, and bosonfermion submatrices of the supermatrix of equation (22). Following Unz’s argument for a system
with Nb bosonic degrees of freedom and Nf fermionic degrees of freedom in the case where we
discretize uniformly in physical distances, we find an additional factor of
N/g 1/4
(Nb −Nf )
(24)
compared to the case where we discretize uniformly in coordinate distances. This explicitly demonstrates the factors of the lapse that are missing when the measure is constructed by uniform coordinate discretization. In each of the examples that we will consider here, the measure determined
by discretizing uniformly in coordinates is missing the factor of N (Nb −Nf ) , however the factor of
5
g −(Nb −Nf )/4 does not represent a missing factor but rather is needed to cancel various factors of
g that arise when the field variables are decomposed into modes as in equation (20). The conclusion regarding the factors of the lapse holds for gauge theories, but there are differing factors of g
between the case of uniform coordinate discretization and uniform physical distance discretization
because the gauge fixing delta function and Fadeev determinant introduce differing g dependence.
III.
CONSTRAINED SYSTEM QUANTIZATION OF QED
In this section we quantize electrodynamics using the constrained system approach[10, 11].
The measure that we obtain will hold for Yang-Mills theory as well, since the additional nonabealian interactions in Yang-Mills theory do not involve the momenta. In this approach, the
functional integral is written as an integral over the canonical coordinates and canonical momenta
that includes delta functions for the gauge conditions, delta functions for the constraints and the
determinant of the Poisson brackets of the gauge conditions and constraints. In QED, A0 is not a
canonical variable, but is introduced to implement the delta function for the Gauss’ law constraint.
The canonical momentum in QED is
√
π̂ i = π i / h = N F i0 .
(25)
The part of the Hamiltonian quadratic in momenta can be written
Hquad =
√ 1 i
g π̂ hij π̂ j .
2
Integrating over the momenta then gives a factor of h−1/2 in the measure.
The Gauss’ law constraint can be written
√
√
0 = ∂i π i = ∂i ( hπ̂ i ) = (Di π̂ i ) h
(26)
(27)
where Di is the covariant derivative compatible with hij . We may take the delta function for the
constraint to be
Z
Z Y
i
i
4 √
(28)
δ(Di π̂ ) =
dλ exp i d x gλDi π̂
xp
and we can write
Z
4
√
i
d x gλDi π̂ =
Z
d4 xN λ∂i π i .
(29)
R
In order to produce a gauge and Lorentz covariant action, we need this to equal d4 xA0 ∂i π i . We
must therefore equate λ with A0 /N and so dλ = dA0 /N . The Poisson brackets implement the
gauge transformation
Z
√
(30)
δF Aj (~x) = d3 ~x0 hF (~x0 ){Aj (~x), Di (π̂ i (~x0 ))} = −∂j F (~x)
and therefore the determinant of the Poisson brackets can be replaced with the usual determinant
of the derivative of the gauge condition with respect to a gauge transformation. This gives
Z
Z Y
−1/2
4
Z=
dAµ g
δ(Φ) det (δα Φ) exp i d xL
(31)
xp
6
which is manifestly covariant.
We will now write this in a form that naturally arises in a theory we will present elsewhere[12].
For the present work, the form that we will derive has the property that all factors of the metric
are eliminated from the measure. Expand Aµ in modes as
X
i
Aµ (x) =
aik fµk
(x)
(32)
i
where fµk(x)
for i = 1...4 are the mode functions and aik are the expansion coefficients. Let us take
4 = ∂ λ (x) where λ (x) are orthonormal and take our gauge condition to be
fµk
µ k
k
0=Φ=
X
a4k λk .
(33)
k
P 4
The quantity
k ak λk is the gauge parameter of an infinitesimal gauge transformations about
Φ = 0 and so we have det (δα Φ) = 1. Since λk (x) is orthonormal,
Z
√
1/2
detk,xp (λk (x)) = detk0 k ( d4 x gλ∗k0 (x)λk (x)) = 1
(34)
where the notation detk,xp indicates that the determinant is of a matrix in k and x with x discretized
√
in physical distances and we have taken the physical 4-volume per point to be equal to d4 x g.[13]
This formula for a determinant is a special case of the general form
det M = det1/2 M † N M /det1/2 N
(35)
where † denotes Hermitian conjugate. Although we are dealing with real fields, it is convenient to
allow the modes to enter in complex conjugate pairs. In this case det M is real and so we may include
complex conjugation in equation (35). Alternatively, we could just use a transpose and instead of
expressions
encountering quantities like a∗k ak we would encounter quantities
Q like a−k ak . We will findQ
like equation (35) to be useful below. It follows that δ(Φ) = δ(a4k )/detk,xp (λk (x)) = δ(a4k ) and
so the partition function can be written as
!
Z
Z Y
i=3 Y
i
1/2
4
Z=
Kik0 ,jk exp i d xL
(36)
dak det
i=1 k
where
Z
Kik0 ,jk =
dAc
√ dAb
d4 x g ∗i g bc j .
dak0
dak
(37)
Note that the g −1/2 in equation (31) is precisely what is needed to combine with the determinant
of g bc from equation (37) to obtain equation (36) with no factors of g in the measure. If we chose
fki for i = 1, 2, 3 to be transverse, the determinant can be written as
det1/2 Kik0 ,jk = det1/2 (−).
(38)
The square root of the determinant can be written in terms of fermionic and real bosonic ghost
fields (which may be expanded in modes occurring in complex conjugate pairs) as
X
Z Y
j
j
1/2
i
i
i
i
i
det
Kik0 ,jk =
df¯k dfk dbk exp i
(f¯k0 fk + bk0 bk )Kik0 ,jk
(39)
k,k0
7
so that equation (36) can be written as
i=3 Y
Y
Z
Z=
!
daik
Y
df¯ki dfki dbik eiSef f
(40)
i=1 k
where
Z
Sef f =
d4 xL +
X
(f¯ki 0 fkj + bik0 bjk )Kik0 ,jk .
(41)
k,k0
We could alternatively write the bik0 bjk term as bi−k0 bjk which is the form we would have obtained if
we had used the transpose rather than the Hermitian conjugate in equation (35).
Equation (36) can be obtained from
Z
Z Y
1/4
4
dAµ g δ(Φ) det (δα Φ) exp i d xL
(42)
Z=
xc
and therefore equation (42) is equivalent to equation (31). This follows because of how various
formulas change by various factors of g when uniform coordinate discretization is used rather than
discretization uniform in physical distances. In particular, we have
Y
Y
δ(a4k )
(43)
δ(Φ) =
δ(a4k )/detk,xc (λk (x)) = g 1/4
giving a g 1/4 factor which combines with the g 1/4 term in the measure to give a g 1/2 term which
√
cancels with a factor of det1/2 ( gg ab ) = g 1/2 arising from using equation (35) to write the Jacobian
in terms of det1/2 Kik0 ,jk . However, if we were to derive the partition function from canonical
quantization using a uniform coordinate discretization, we would not obtain equation (42), but
would instead obtain the partition function
Z
Z Y
g 1/4
dAµ 2 δ(Φ) det (δα Φ) exp i d4 xL
(44)
N
x
c
which is off by two factors of N , one factor for each physical bosonic degree of freedom, which,
as we have noted, is a general feature of partition functions derived using uniform coordinate
discretization.
IV.
CANONICAL QUANTIZATION OF GAUGE FIXED LAGRANGIAN FOR QED
The previous section established that the measure for QED from canonical quantization in
physical coordinates is manifestly covariant. However, applying this procedure to quantum gravity
is problematic since integrating over the lapse times the Hamiltonian constraint does not result in
a delta function if N δt is held fixed. In this section we re-derive the results of the previous section
in a way that will readily extend to the quantum gravity case.
Following Unz[2, 3], we start with a gauge-fixed Lagrangian where we consider ghost fields
as well as the gauge field as the canonical variables. We then construct the partition function
by canonical quantization in the usual way and obtain equation (31). We take the gauge fixing
Lagrangian to be
√ 1
Lgf = − g (∇µ Aν g µν )2 .
2α
(45)
8
The ghost contribution is then
√
Lgh = − gg µν η̄∇µ ∂ν η
(46)
which is equivalent to
Lgh =
√
gg µν ∂µ η̄∂ν η.
(47)
Considering first QED in a fixed curved spacetime background, we see that the η-momenta integrals
can be done without introducing any factors of the metric in the measure. However, the gauge
fixing term contains a π̂ 0 contribution and this does contribute to the measure.
To determine this contribution, note that the gauge fixing terms produces contribution to the
momentum of
a
π̂gf
=−
N
(∂µ Ab )g µb g 0a
α
(48)
and a contribution to the part of the Hamiltonian density quadratic in momentum of
Hgf =
√ α a
b
.
g π̂gf gab π̂gf
2
(49)
a
to be equal to that given in equation (25) for a spatial and 0 for a = 0 and define
Define π̂EM
a . The total Hamiltonian density, which is defined as the sum of equations (26)
a
a
π̂tot = π̂EM + π̂gf
and (49), can be written as
Htot =
√ 1 a
b
g π̂T ot Mab π̂tot
2
(50)
where
(51)
Mab = gab + (α − 1)g 0a g 0b /g 00 = gab + (α − 1)na nb
p
and na is a unit timelike vector given by na = g 0a / g 00 . This form follows from using g 0c g 0a /g 00
a .
to project π̂Ta ot onto π̂gf
The determinant of Mab can be determined as follows. Denote spacelike unit vectors orthogonal
to na as n(j)a with j = 1, 2, 3 and define n(0)a ≡ na . (We will drop the parenthesis around the
vector label when discussing the corresponding 1-forms.) Write
Z Y
Z
√
(det M )−1/2 =
dv a exp − d4 x gv a Mab v b .
(52)
xp
P
i (i)a (x)Φ (x) where Φ is a set of basis functions normalized so that
Expand v a (x) =
k
k
k,i dk n
R 4 √ ∗
d x gΦk0 Φk = δk0 ,k . The action in equation (52) is then diagonal in the expansion coefficients
dik . If we change variables from v a (x) to dik , we have
Z Y
X
(det M )−1/2 =
d(dik )J exp −
na(i) Mab nb(j) dik dj−k
(53)
i,k
ijk
where J is the Jacobian associated with the variable change from v a (x) to dik . Since J is independent
of the field dik , we can perform the functional integral and we have up to irrelevant constants that
(det M )−1/2 = J. We can evaluate this Jacobian using equation (35). This gives
(det M )−1 = J 2 = det n(i)a gab n(j)b / det gab .
(54)
9
The numerator is independent of the metric and so (det M )−1/2 ∝ g −1/2 . Therefore the integrals
a results in a factor of g −1/2 in the measure and we obtain equation (31) in the limit that
over π̂tot
the gauge parameter α is taken to zero in order to recover the delta function.
Now consider the case where we wish to simultaneously quantize the EM field and the metric.
Equation (45) contains a term from the Christoffel symbols in the covariant derivative which couple
the momenta conjugate to the metric with momenta conjugate to the vector potential. Dealing
with such cross terms led Unz[2, 3] to make a low energy approximation; this is, however, not
√
necessary. The momenta cross terms can be eliminated by using õ = gAµ as the canonical
variable since
1
∇α Aα = √ (∂α Ãα ).
(55)
g
We can then independently treat the metric and vector potential momenta. For QED, we integrate
out the momenta conjugate to õ and after changing variables back to Aµ , we recover equation
(31).
If we had discretized uniformly in coordinates, we would modify equation (47) by dropping the
√
g so that we obtain the correct Fadeev determinant and we then obtain a factor of (g 00 )−1 in the
measure from performing the ghost momenta integral, a factor of (g 00 )2 g 1/2 from the πab integrals,
and a factor of g −1/4 from taking the limit that the gauge parameter goes to zero to obtain a delta
function. If we include an additional factor of N 2 - one for each physical degree of freedom, we
recover equation (42).
V.
SPINOR FIELDS
Consider Dirac fermions in a curved spacetime. The part of the Lagrangian involving time
derivatives of the spinor field can be written as (see, for example, [14])
√
giψ̄γ a eαa ∂α ψ
(56)
where eαa is the vierbein. The canonical momentum is therefore
π̂ = iψ̄γ a e0a N.
(57)
The Jacobian between π̂ and ψ̄ is proportional to det(iγ a e0a N ) which is unity as follows immediately
from writing the vierbein in upper tridiagonal form and from the Dirac representation of γ 0 . The
measure can therefore be written
Y
dψdψ̄.
(58)
xp
We can expand in a basis as ψ =
P
bik ψki and similarly for ψ̄. If we choose an orthonormal basis
Z
√
d4 x g ψ̄ki 0 ψkj = δij δk0 ,k ,
(59)
we can write the measure as
Y
dbik db̄ik .
(60)
ik
We now compare this with the measure resulting from discretizing uniformly in coordinate
distances. The momenta is
√
π = iψ̄γ a e0a g.
(61)
10
The Jacobian for changing variables from π to dψ̄ gives a factor of h−2 in the measure (recall that
Grassmann variables transform oppositely to ordinary variables). If we multiply the measure by a
factor of N −4 - one factor of 1/N for each physical fermionic degree of freedom, the measure can
be written
Y
1
dψdψ̄ 2 .
(62)
g
x
c
For coordinate discretization, the transformations to dik and d¯ik each give a factor of g (g 1/4 for
each of four independent spinor basis functions) and equation (62) then reduces to equation (60)
showing that equation (62) is equivalent to equation (58). This confirms that it is necessary to
include a factor of N −1 in the measure for each physical fermionic degree of freedom.
VI.
CANONICAL QUANTIZATION OF GAUGE FIXED LAGRANGIAN FOR
GENERAL RELATIVITY
We now quantize general relativity using the techniques of Section IV. We discretize uniformly
in physical distances which means that the coordinate positions of the discrete points vary with
the metric. This does not present any problems for the approach taken here. We take the gauge
fixing Lagrangian to be
√ 1
Lgf = − g (∂µ g µν )gνσ (∂λ g λσ ).
α
(63)
√
Lgh = − g η̄ν ∂µ ∇(µ η ν)
(64)
The ghost contribution is then
which is equivalent to
Lgh =
√
g∂µ η̃¯ν ∇(µ η̃ ν)
(65)
√
where η̃¯ν = g η̄ν and η̃ ν = √1g η ν .
Through the covariant derivative, Lgh contains a term coupling the time derivative of the metric
with the time derivative of the ghosts. Lgh contains a term contributing to Mf b , however Mbf is
zero and therefore the superdeterminant is simply det Mbb / det Mf f . We can therefore integrate out
the η̃ µ and η̃¯ν -momenta independently of the metric variables. This momenta integral produces
a factor proportional to det(gµν ) det(g µν − nµ nν ) which using the technique of Section IV is
independent of the metric and so does not give a non-constant contribution to the measure.
Dropping the total derivative in the “ΓΓ” version of the Einstein Hilbert action and combining
with the gauge fixing term of equation (63), we find that the part of the gauge fixed action quadratic
in time derivatives of the metric is proportional to
√
gg 00 M abcd gab,0 gcd,0
(66)
where
M abcd = Gabcd + λ1 S(na nc g bd ) + λ2 S(na nb g cd ).
(67)
Here Gabcd is the DeWitt supermetric[15], λ1 and λ2 are constants and na is the unit timelike vector
defined in Section IV. The notation S(..) indicate symmetrization in a, b; in c, d; and in (ab), (cd).
Integrating the momenta conjugate to the metric will give a factor of (det M )1/2 in the measure.
11
This determinant can be evaluated using the technique of Section IV. Label the tensors nj(a nkb)
as iab where i labels the 10 independent (in 4-d) possibilities. n(j)a are the vectors introduced
in Section IV and we have dropped the parenthesis around the vector label when discussing the
corresponding 1-forms. Write
Z Y
Z
√
−1/2
(det M )
=
dπ exp − d4 x gπab M abcd πcd .
(68)
xp
i i
Expand πab (x) =
k,i dk ab (x)Φk (x) where Φk is a set of basis functions normalized so that
R 4 √ ∗
d x gΦk0 Φk = δk0 ,k . The action in equation (68) is then diagonal in dik and we see that
(det M )−1/2 , up to irrelevant constants, is simply the Jacobian, J, associated with the variable
change from πab (x) to dik . Using Gabcd as a metric on the space spanned by πab (x), we see that
(det M )−1 = J 2 = det iab Gabcd jcd δk0 ,k / det G.
(69)
P
The numerator is independent of the metric and so (det M )1/2 ∝ (det G)1/2 ∝ g −5/2 . Taking the
limit that the gauge parameter goes to zero in order to obtain a delta function gives an additional
√
factor of g and so the measure becomes
Y 1
dgµν δ(Φα ) det(Lξβ Φα )
3
g
x
(70)
p
which is manifestly covariant.
In order to compare this measure with the coordinate-discretized measure that we will derive
below, let us expand the metric on a given gauge-slice in a basis of the six functions Ψik
ab . Then for
metrics infinitesimally close to this gauge slice, we can write
gab =
i=6
XX
k
aik Ψik
ab +
i=1
k
P P4
Take our gauge condition to be Φα = k
(70) is unity. The delta function becomes
YY
xp
i=4
XX
δ(Φα ) =
α
(71)
i=1
i+6 iα
i=1 ak ξk
i=4
YY
ik
ai+6 ∇(a ξb)
.
= 0 so that the determinant in equation
iα
δ(ai+6
k )/ det(ξk (x))
(72)
k i=1
where the determinant is between i, k variables and α, x variables. Choose ξkiα so that
Z
√
jβ
d4 x gξk∗iα
0 gαβ ξk = δij δk 0 ,k .
(73)
1/2
Then the determinant can be evaluated using det J = det(J † gJ)/ det g
= g −1/2 . The change
of variables between the metric and aik variables results in a Jacobian which can be evaluated using
1/2
the similar identity det J = det(J † GJ)/ det G
= g 5/2 . The measure is then simply
i=6
YY
daik det1/2 K
(74)
k i=1
where
Z
Kik0 ,jk =
dgcd
√ dgab
d4 x g ∗i Gabcd j .
dak0
dak
(75)
12
The square root of the determinant can be exponentiated with a combination of Grassmann and
real scalar ghosts as in equation (39), but this form is sufficient for facilitating comparison with
the measure for uniformly discretizing in coordinate distances, which we now derive.
If we had discretized uniformly in coordinates, we would obtain a factor of (g 00 )−4 in the measure
√
from performing the ghost momenta integral (after dropping the factor of g in equation (65) so
that the correct Fadeev determinant is generated), a factor of (g 00 )5 from the πab integrals, and
a factor of g −3/2 from taking the limit that the gauge parameter goes to zero to obtain a delta
function. If we include an additional factor of N 2 - one for each physical degree of freedom, we
find
Y 1
dgµν δ(Φα ) det(Lξβ Φα )
(76)
3/2
g
x
c
which is the standard result[1] except for the factors of the lapse. We now decompose the metric
as we did above. Since we are discretizing in coordinate time now, various expressions take on a
different form. We have
√
det(ξkiα (x)) = 1/(det( ggαβ ))1/2 = 1/g 3/2 .
(77)
This factor cancels the g 3/2 appearing in equation (76) The Jacobian associated with the change
√ 1/2
√
√
. In 4-d, det( gG) is a
of variables to aik is now given as det J = det(J t gGJ)/ det gG
constant. Thus we obtain equation (74) and therefore equation (76) is equivalent to equation (70).
VII.
CANONICAL QUANTIZATION OF GENERAL RELATIVITY VIA
CONSTRAINTS
It is not straightforward to quantize GR as aRconstrained system by discretizing uniformly in
√
physical distances. Consider the exponential of i d4 x gf (x). If we perform a functional integral
over the lapse by discretizing in coordinate distances we obtain products of δ(f (x)). If we perform
√
a functional integral over the lapse by discretizing in physical distances, d4 x g is the 4-volume
element which is held fixed and so we do not obtain delta functions. This means that if we naively
implement the procedure that we used for QED in section III, we would not correctly implement
the constraints in general relativity.
We have learned, however, that discretizing in coordinate distances rather than physical distances introduces a factor of N (Nf −Nb ) into the measure. To obtain the correct measure we have to
compensate for this. We will introduce the lapse factors by starting with a modified Hamiltonian:
H = HGR + HG (M )
(78)
where HGR is the Hamiltonian for general relativity and HG (M ) is the Hamiltonian for two
fermionic ghost fields having mass M that are to be integrated out. We will take the large M
limit so that the ghost field has no physical consequence. The only effect of HG will be to modify
the measure so that the spurious factors of N that occur from discretizing in coordinate time are
canceled. General relativity has 2 physical bosonic degrees of freedom and HG has two fermionic
degrees of freedom so there are no net spurious factors of N when equation (78) is quantized. This
is a Hermitian Hamiltonian and will therefore result in unitary evolution. The ability to derive
a partition function from such a modified Hamiltonian implies that factors of N do not affect
unitarity.
We can now proceed to quantize by discretizing in coordinate time in the usual way. The ghost
term enters as modification to H0 and the normal derivations of the measure for quantum gravity
go though otherwise unchanged. After integrating out the ghost fields we recover equation (76).
13
VIII.
CONCLUSION
We have considered scalar, spinor, vector, and tensor fields in curved spacetime. In all cases
manifestly covariant functional measures can be obtained from canonical quantization. We have
shown that taking the degrees of freedom to be uniformly distributed in physical distances yields
covariant measures directly from the standard canonical quantization procedure, while taking the
degrees of freedom to be uniformly distributed in coordinate distances results in a missing factor
of the lapse for each bosonic degree of freedom minus each fermionic degree of freedom. For the
case of constrained systems, we may quantize the gauge fixed Lagrangian instead of working with
constraints and obtain the same measure. Our results imply that the limit of small spacing depends
on how the limit is taken.
In all cases that we have considered, expanding in a covariant basis results in a measure with
√
√
no extra factors of N or g in the measure - all factors of g that appear in the measure are
precisely what is required to cancel the Jacobian factors resulting from changing variables to the
basis components. If any additional factors had appeared, they could be exponentiated giving
δ(0) terms in the effective action. So in all cases considered, the measure can be determined by
requiring covariance and by requiring that no δ(0) terms appear in the effective action.
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
E. S. Fradkin and G. A. Vilkovisky, Phys. Rev. D 8, 4241 (1973).
R. K. Unz, Nuovo Cimento 92A, 397 (1986).
R. K. Unz, Phys. Rev. D 32, 2539 (1985).
D. J. Toms, Phys. Rev. D 35, 3796 (1987).
M. Basler, Fortschr. Phys. 41, 1 (1993).
L. C. Botelho, Phys. Rev. D 38, 2464 (1988).
P. O. Mazur and E. Mottola, Nuclear Physics B 341, 187 (1990).
Z. Bern, S. K. Blau, and E. Mottola, Nuclear Physics B 341, 187 (1990).
S. Weinberg, The Quantum Theory of Fields, Vol. I (Cambridge University Press, 1995).
R. Marnelius, Acta Physica Polnica B13, 669 (1982).
H. J. Rothe and K. D. Rothe, Classical and Quantum Dynamics of Constrained Hamiltonian Systems
(World Scientific, 2010).
C. F. Richardson, “UV-Finite Gauge Theory and Quantum Gravity,” (2016), archive.org/details/DiscreteEmergentSpacetime.
In order to arrive at an integral from a discrete sum, we must assume that the spacetime spacing is
small compared to the length scale on which λk varies. Expressions such as equation (34) are only
formal since λk for large k varies arbitrarily rapidly and one must regularize such quantities before
sums can be converted into integrals.
M. Nakahara, Geometry, Topology, and Physics, 2nd Edition (Taylor and Francis, 2003).
B. S. DeWitt, Phys. Rev. 160, 1113 (1967).