Mode regularization, time slicing, Weyl ordering and phase space path integrals for quantum mechanical nonlinear sigma models

15 Pages·0.22 MB·English

by  Fiorenzo Bastianelli

ITP-SB-98-04
Mode regularization, time slicing, Weyl ordering
and phase space path integrals for quantum
mechanical nonlinear sigma models.
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n Fiorenzo Bastianelli1
a
J

5 Dipartimento di Fisica, Universit`a di Modena
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via Campi 213/A, I-41100, Modena, Italy

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v
5 Koenraad Schalm2 and Peter van Nieuwenhuizen3
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1
1
Institute for Theoretical Physics
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State University of New York at Stony Brook
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/ Stony Brook, New York, 11794-3840, USA
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Abstract
h
:
v
i
X A simple, often invoked, regularization scheme of quantum mechanical path in-
r tegrals in curved space is mode regularization: one expands fields into a Fourier
a
series, performs calculations with only the first M modes, and at the end takes the
limit M . This simple scheme does not manifestly preserve reparametrization
→ ∞
invariance of the target manifold: particular noncovariant terms of order h¯2 must be
added to the action in order to maintain general coordinate invariance. Regulariza-
tion by time slicing requires a different set of terms of order h¯2 which can be derived
from Weyl ordering of the Hamiltonian. With these counterterms both schemes give
the same answers to all orders of loops. As a check we perform the three-loop calcu-
lation of the trace anomaly in four dimensions in both schemes. We also present a
diagrammatic proof of Matthews’ theorem that phase space and configuration space
path integrals are equal.
1Also INFN, Sezione di Bologna, Italy. E-mail: [email protected]
2E-mail: [email protected]
3Research supported by National Science Foundation Grant Phy 9722101.
E-mail: [email protected]
1
I Introduction
Quantum mechanical pathintegrals incurved target spaces have variousapplica-
tions. Two decades ago they were used to quantize collective coordinates of solitons
[1], and one decade ago to calculate chiral [2] and trace [3, 4] anomalies in quantum
field theories. In the former case, the path integrals were regularized by expanding
the fluctuations about the solitons into normal modes, and cutting the sum over
modes off at a maximum energy or at a maximum number of modes [5]. In the
trivial vacuum, on the other hand, two methods have been widely used: the time
discretization method1 which has been pioneered by Dirac and Feynman, and the
mode truncation method already described in [6]. In the literature, both methods
have been used on a par.
In this article we point out the relation and the differences between the two
methods. Both the time discretization method and the mode truncation method
are particular regularization schemes which define the path integral. One expects
therefore that the results for physical objects, for example the transition element
T = xµ,t xµ,t , can be obtained from path integrals whose actions differ at most
h f f| i ii
by finite local counterterms. In quantum gauge field theories, one can fix such am-
biguities by requiring that Ward identities are satisfied. Similarly, for theories in
curved target space which are general coordinate invariant one can require that the
transition element be a “bi-scalar” (a scalar in xµ and xµ), but this fixes the ambigu-
f i
ities only up to covariant terms, namely a term proportional to the scalar curvature
R. Only an experiment, for example the measurement of the trace anomaly, can fix
the coefficient of this term. One can also view the Hamiltonian operator Hˆ as an
observable, and by writing T = xµ,t xµ,t = xµ exp( βHˆ) xµ where β = t t
h f f| i ii h f| −¯h | ii f− i
in Euclidean space, one can fix the ambiguities in T by requiring that it satisfies
the Schroedinger equation with Hˆ as Hamiltonian. If Hˆ itself contains no term
proportional to R, then the action will contain a term 1R as we will show, and
−8
T will have an R term in the exponent with coefficient 1 . If Hˆ contains a term
−12
αR, the coefficient of R in T will be ( 1 +α). In general different regularization
− 12
schemes can lead to different finite local counterterms of any order in h¯, but finite
local counterterms of order h¯ and h¯2 only are generated in the path integral action if
one considers different operator orderings in the Hamiltonian. A Hamiltonian which
is a target space scalar2 is
1
Hˆ = g−41(xˆ)pˆµg21(xˆ)gµν(xˆ)pˆνg−41(xˆ) (1)
2
but changing the order of the operators will in general destroy the invariance under
targetspacediffeomorphisms. Iftheambiguities duetousing different regularization
1Time discretization in the sector with solitons is complicated because the canonical mo-
menta (of the non-zero-modes) satisfy the equal-time commutation relation [π(x,t),ϕ(y,t)] =
′ ′
i¯h[δ(x y) φ (x)φ (y)/M ] where φ (x) is the classicalsoliton solution with mass M .
− 2The−gene−ratosrolfor tasorlget spascoel diffeomorspolhisms xˆµ xˆµ +ξµ(xˆ) is a sum of an orbital psaorlt
Gˆorb(ξ)= 1 (pˆ ξµ(xˆ)+ξµ(xˆ)pˆ ) and a spin part Gˆspin→(ξ)=(2∂ ξλg ξλ∂ g ) ∂ . Closure
2ih¯ µ µ µ λν − λ µν ∂gµν
of the commutator algebra fixes Gˆspin(ξ). An example of a target space diffeomorphism scalar
is (1), namely [Gˆorb(ξ)+Gˆspin(ξ),Hˆ]=0, as one may check by an explicit (tedious) calculation.
2
schemes correspond to the ambiguities due to different operator orderings, the terms
of order h¯3 and higher in the Hamiltonian should be the same in all cases. This
conclusion is corroborated by the observation that in these non-linear sigma models,
N-loop graphs are convergent by power counting for N 3.
In the time discretization method, a very clear conn≥ection between Hˆ and the
transitionelementexistsifoneusesphasespacepathintegrals(Feynman’sapproach)
instead of configuration space path integrals (Dirac’s approach), and rewrites the
HamiltonianinWeylorderedform. ThenonemayreplacetheWeylorderedoperator
Hˆ (pˆ,xˆ) in the kernel by the corresponding function at the midpoint (Berezin’s
W
theorem)
ǫ ǫ x +x
ˆ 1 2
dp x exp( H ) p p x exp H(p, ) dp x p p x . (2)
1 W 2 1 2
h | −h¯ | ih | i → −h¯ 2 h | ih | i
Z (cid:18) (cid:19)Z
The Weyl ordering of Hˆ in (1) leads to the following local finite correction to the
naive Hamiltonian H = 1gµνp p to be used in the path integral
2 µ ν
h¯2
∆V = (R+gµνΓ σΓ ρ), (3)
W µρ νσ
8
where Γ σ are the Christoffel symbols.3 This counterterm has been extensively
µν
discussed and used in the literature [7].
Both in the time discretization and in the mode truncation method, one writes
the paths as xµ(τ) = xµ (τ)+qµ(τ) for 1 τ 0, where xµ (τ) is a solution of the
free field equations witbhgthe correct bou−nda≤ry c≤onditions (xµbg(τ) = xµ+τ(xµ xµ))
bg f f − i
andoneexpandsthequantumfluctuationsqµ(τ)intoacompletesetofeigenfunctions
ofthe freefield equations with vanishing boundaryvalues andintegrates over afinite
number of coefficients qµ. The difference between the time discretization method
m
and mode regularization is that in the latter one uses the continuum action and
naive continuum Feynman rules for a finite number of modes and one chooses a
measure for the integrals over these modes which is usually normalized such that
it reproduces the standard result for the free particle if the interaction part of the
action, S , is set to zero. In the time discretization method, on the other hand, one
int
uses a discretized action whose discretized Feynman rules are derived (in explicit
form [8]) from the Hamiltonian starting point.
Up to this point we have made fairly obvious statements. Both regularization
schemes are well-defined, and this is how they are used in the literature. However,
it is already clear that this cannot be the whole truth, because with time slicing one
has the nontrivial ∆V in (3), whereas with mode truncation such a ∆V seems at
W
first to be absent. The precise way in which also mode truncation leads to a ∆V
will be presented in section II. We will see that one needs the following counterterm
for mode regularization
h¯2 1
∆V = R gµνgαβg Γ γΓ δ . (4)
MR γδ µα νβ
8 − 3
(cid:18) (cid:19)
3OurconventionsfortheRiemannandRiccitensorare: R σ =∂ Γ σ+Γ σΓ τ (ρ µ);
R = R µ. Hence, at the linearized level, R = 1(∂ρµν∂ h ∂ρ hµν ∂ ρhτ +µ2νh− ) w↔here
ρν ρµν µν 2 µ ν − µ ν − ν µ µν
h=ηµνg and h =ηνρ∂ h .
µν µ ρ µν
3
Asacheck weshallobtainthecorrecttraceanomalyinfourdimensions fromathree-
loop calculation. In Riemann normal coordinates, the Christoffel symbols Γ ρ in
µν
∆V vanish at the origin of the coordinate system, but at the three-loop level one
finds a contribution from ∆V by expanding each Christoffel symbol into qτ∂ Γ ρ
τ µν
and contracting the two quantum fields. Since ∂ Γ ρ contains a part proportional
τ µν
totheRiemanntensor, itisclearthat∆V yields anonvanishing contribution. Hence
dropping ∆V in Riemann normal coordinates is incorrect.
If one eliminates the momenta by integrating over them in the time discretized
path integral, one obtains N factors (detgµν)−1 at the N midpoint coordinates
2
1(qµ +qµ). Exponentiating these in the familiar Faddeev-Popov way one obtains
2 k+1 k
what we have called “Lee-Yang ghosts”, namely commuting aµ and anticommuting
bµ and cµ ghost fields [4]. Phase space path integrals are free from ambiguities
because they are finite. Clearly, upon transition from phase space to configuration
space, ambiguities are created; technically this is due to the fact that the momenta
are replaced by the q˙’s and ghosts, each of which introduces divergences and hence
ambiguities into the theory.
In the phase space approach, the vertices are different (for example 1p gµν(x)p
2 µ ν
instead of 1x˙µg (x)x˙ν), as well as the propagators ( p p is not proportional to
2 µν h µ νi
g g q˙αq˙β ), but the transition element should be the same (Matthews’ theorem).
µα νβ
h i
In section III we present a graphical proof.
In section IV we draw conclusions and we show that the coefficient of the R term
in ∆V is scheme independent and equals 1.
8
II Mode regularization
We now describe how one can define mode regularized path integrals in curved
space. Ideally, one would like to derive mode regularization from first principles, i.e.,
starting from the transition amplitude defined as a matrix element of the evolution
operatoracting intheHilbert space ofphysical states: xµ exp( βHˆ) xµ . However,
h f| −¯h | ii
thisderivationlooksquitedifficult, sothatweprefertotakeamorepragmaticcourse
ofaction, andattempt a direct definition of themoderegularized pathintegral. This
definitionwillbesupplemented bycertainconsistency requirements whichwespecify
later on.
The transition amplitude can formally be written as follows
x(0)=xf 1
xµ,t xµ,t = ˜x exp S ; (5)
h f f| i ii D −βh¯
Zx(−1)=xi (cid:20) (cid:21)
0 1
S = dτ g (x)(x˙µx˙ν +aµaν +bµcν),
µν
Z−1 2
˜x = dDx(τ)dDa(τ)dDb(τ)dDc(τ).
D
−1<τ<0
Y
We have shifted and rescaled the time parameter t t = βτ, and since all terms in
f
−
the action only depend on βh¯, we set h¯ = 1 from now on. Note that β then counts
the number of loops. We will evaluate the path integral in a perturbative expansion
4
in β and in the coordinate displacements ξµ about the final point: ξµ = xµ xµ.
i − f
Formally the path integral is a scalar since the action is a scalar and the ghost fields
make up a scalar measure on the space of paths.4 Mode regularization will destroy
this formal covariance. However, we will see that covariance can be recovered by
adding a suitable noncovariant counterterm ∆V to the action S.
We start parametrizing
xµ(τ) = xµ (τ)+qµ(τ), (6)
bg
where xµ (τ) is a background trajectory and qµ(τ) the quantum fluctuations. The
bg
background trajectory is taken to satisfy the free field equations of motion and is
a function linear in τ connecting xµ to xµ in the chosen coordinate system, thus
i f
enforcing the boundary conditions
xµ (τ) = xµ ξµτ; ξµ = xµ xµ. (7)
bg f − i − f
Then the quantum fields qµ(τ) should vanish at the time boundaries and can be
expanded into sines. For the Lee-Yang ghosts we use the same Fourier expansion;
this may be considered as part of our definition of mode regularization.5 Hence
∞ ∞
qµ(τ) = qµ sin(πmτ); aµ(τ) = aµ sin(πmτ);
m m
m=1 m=1
X X
∞ ∞
bµ(τ) = bµ sin(πmτ); cµ(τ) = cµ sin(πmτ). (8)
m m
m=1 m=1
X X
At this point the formal measure ˜x can be defined in terms of integration over
D
the Fourier coefficients
∞ D
˜x = q a b c = A dqµdaµdbµdcµ, (9)
D D D D D m m m m
m=1µ=1
Y Y
which fixes the path integral for a free particle up to the constant A
1 0 1
˜x exp Q = A; Q = dτ δ (q˙µq˙ν +aµaν +bµcν). (10)
µν
Z D (cid:20)−β (cid:21) Z−1 2
The constant A will be fixed later on from a consistency requirement. Note that
limiting the integration over the number of modes to a finite number M gives a nat-
ural regularization of the path integral. This regularization resolves the ambiguities
that show up in the continuum limit.
4The factors (detg(x ))1/2dxµ in the discretized path integral are target space scalars and
i i
exponentiating them leads to ghosts.
5AnotherarguQmenttojustify thattheghostsshouldbeexpandedintosinesisthatthe classical
solutions of their field equations are aµ =bµ =cµ =0, and that the quantum fluctuations do not
modify the boundary conditions of the classical solutions. In [8] it was shown that the results for
the transition amplitude do not change if one uses cosines for the ghosts.
5
Now we expand the action about the final point xµ and obtain
f
S = S +S = S +S +S +... (11)
2 int 2 3 4
where
0 1
S = dτ g (ξµξν +q˙µq˙ν +aµaν +bµcν),
2 µν
Z−1 2
0 1
S = dτ ∂ g (qα ξατ)(ξµξν +q˙µq˙ν +aµaν +bµcν 2q˙µξν), (12)
3 α µν
Z−1 2 − −
0 1
S = dτ ∂ ∂ g (qαqβ +ξαξβτ2 2qαξβτ)(ξµξν +q˙µq˙ν +aµaν +bµcν 2q˙µξν).
4 α β µν
Z−1 4 − −
All geometrical quantities, like g or ∂ g , are evaluated at the final point xµ, but
µν α µν f
for notational simplicity we do not exhibit this dependence. S is taken as the free
2
part and defines the propagators while S gives the vertices as usual. Therefore,
int
the quantum perturbative expansion reads:
1
hxµf,tf|xµi,tii = (D˜x) exp −βS = Ae−21βgµνξµξνhe−β1Sinti
Z (cid:20) (cid:21)
1 1 1
= Ae−21βgµνξµξν h1− βS3 − βS4 + 2β2S32i+O(β23) . (13)
(cid:18) (cid:19)
Aiming at a two-loop computation, we have kept only those terms contributing up
to O(β), taking into account that ξµ O(√β), as follows from the exponential
∼
appearing in the last line of (13). The propagators that follow from S are given by
2
qµ(τ)qν(σ) = βgµν(x )∆(τ,σ)
f
h i −
aµ(τ)aν(σ) = βgµν(x )••∆(τ,σ) (14)
f
h i
bµ(τ)cν(σ) = 2βgµν(x )••∆(τ,σ)
f
h i −
where ∆ is regulated by the mode cut-off introduced below (10):
M 2
∆(τ,σ) = sin(πmτ)sin(πmσ) , (15)
−π2m2
m=1(cid:20) (cid:21)
X
and has as continuum value
∆(τ,σ) = τ(σ +1)θ(τ σ)+σ(τ +1)θ(σ τ). (16)
− −
(A dot on the left(right) of ∆(τ,σ) indicates differentiation with respect to τ(σ).)
Using standard Wick contractions, we computed:
1 1 1
S = ∂ g ξαξµξν, (17)
3 α µν
h−β i −β 4
1 β 1
S = ∂ ∂ g (gµνgαβ gµαgνβ) (gµνξαξβ +gαβξµξν 2gµαξνξβ)
4 α β µν
h−β i 24 − − 24 −
(cid:20)
1 1
ξµξνξαξβ , (18)
−β 12
(cid:21)
6
1 β
S2 = ∂ g ∂ g (gαβgµνgλρ 4gαρgµνgβλ 6gαβgµλgνρ
h2β2 3i α µν β λρ 96 − −
(cid:20)
+4gαρgβµgνλ+4gαµgβλgνρ)
1
+ gµλgνρξαξβ +2(gαβgµλ gαλgµβ)ξνξρ+(2gαλgβρ gαβgλρ)ξµξν
48 − −
(cid:18)
+(2gµβgλρ 4gµλgβρ)ξαξν
−
(cid:19)
1 1
+ (gαβξµξνξλξρ 4gαλξµξνξβξρ +4gµλξαξνξβξρ)
β96 −
1 1
+ ξαξµξνξβξλξρ . (19)
β232
(cid:21)
This completes the calculation of the transition amplitude in the two-loop approxi-
mation using the mode regularized path integral.
Atthispointweshouldmakecontactwithotherschemes andtesttheconsistency
ofourrules. Todothat,weuseourtransitionamplitudetoobtainthetimeevolution
of an arbitrary wave function
Ψ(x ,t ) = dDx g(x ) xµ,t xµ,t Ψ(x ,t ). (20)
f f i i h f f| i ii i i
Z q
We need the factor g(x ) because the transition element is formally a biscalar as
i
we explained before,qbut then Ψ(x ,t ) is a scalar and hence also Ψ(x ,t ), which in
f f i i
turn implies that the measure must be a scalar as well.
Since the transition amplitude (13) is given in terms of an expansion around
the final point (x ,t ), we Taylor expand also the wave function Ψ(x ,t ) and the
f f i i
measure g(x ) in eq. (20) about this point
i
q
1
Ψ(xi,ti) = Ψ(xf,tf) β∂tΨ(xf,tf)+ξµ∂µΨ(xf,tf)+ ξµξν∂µ∂νΨ(xf,tf)+O(β23)
− 2
1
g(x ) = g(x ) 1+ξµΓ α + ξµξν(∂ Γ α +Γ αΓ β)+O(ξ3) . (21)
i f µα µ να µα νβ
2
q q (cid:18) (cid:19)
We then perform the integration over dDxµ = dDξµ in (20) and match the various
terms. The leading term fixes A
D −D
Ψ = A(2πβ)2Ψ A = (2πβ) 2, (22)
→
and the terms of order β give
1 1 1 1
β ∂ Ψ+ 2Ψ+ RΨ gµνgαβgγδ∂ g ∂ g Ψ+ gµνgαβgγδ∂ g ∂ g Ψ = 0.
t µ αγ ν βδ µ αγ β νδ
− 2∇ 8 −32 48
(cid:20) (cid:21)
(23)
This last equation means that the wave function Ψ satisfies the following Schroe-
dinger equation at the final point (xµ,t ):
f f
1
∂ Ψ = (H +∆V )Ψ = 2Ψ+∆V Ψ, (24)
t 0 eff eff
− −2∇
7
where the effective potential is given by
1 1 1
∆V = R+ gµνgαβgγδ∂ g ∂ g gµνgαβgγδ∂ g ∂ g
eff µ αγ ν βδ µ αγ β νδ
−8 32 − 48
1 1 1
= R gµνΓ βΓ α + gµνgαβgγδ∂ g ∂ g (25)
µα νβ µ αγ β νδ
−8 − 8 24
1 1
= R+ gµνgαβg Γ γΓ δ.
γδ µα νβ
−8 24
Clearly, to obtain the “free” Hamiltonian H from a path integral, one should sub-
0
tract the potential ∆V , and thus use in (5) the following classical action
eff
1 0 1
S = dτ g (x)x˙µx˙ν +β∆V (x) . (26)
µν eff
− β Z−1 (cid:18)−2β (cid:19)
With this counterterm mode regularization gives the same results as the time dis-
cretization scheme described in the introduction. Note that (25) is different from
the counterterm for time discretization. The difference is the last term in the second
line of (25).
Finally, as a non-trivial test, which was one of our motivations to carry out the
present research, we have computed the trace anomaly for a real conformal scalar
field in four dimensions using mode regularization andthe newly found counterterm.
Anomalies in Fujikawa’s approach are given by the regulated trace of the Jacobian
J of the symmetry transformation.
−βR
An = limTr(J e h¯ ). (27)
β→0
Atheoryofconsistentregulators exists[9]anditgivesusforconformalscalarfields
the Hamiltonian (1) minus an imRprovement potential h¯2 D−2 R. The Jacobian for
8(D−1)
trace anomalies is proportional to unity. In mode regularization (27) translates into
the evaluation of the expectation value of the Jacobian with respect to the path
integral based on the formal transition element (5) including the just calculated
counterterms andthe improvement term. Taking the trace of (5) means first putting
x = x which gives that x = x and then integrating over x with dDx g(x ).
f i bg f f f f
As the measure of (5) includes the constant A = (2πβ)−D, the β-independeqnt term
2
of the r.h.s. of (27) is given by the (D/2 + 1)-loop term in the path integral.
Expanding the metric in Riemann normal coordinates we find the first nonvanishing
contributions at the two-loop level: graphs with the topology of the number 8 and
the background values of the various potential terms. From (18) with ξµ = 0 one
obtains6
1 1 (1 (D 2)/(D 1))
+ = ( βh¯) R+( βh¯) − − − R. (28)
• − 6 −4 − 8
(cid:18) (cid:19)
The result is the same as obtained from time slicing and yields the correct trace
anomaly in D = 2 dimensions.
6In equations (28), (29), (30), (31) only the topology of the graphs is indicated, e.g., the figure
8 in (28) stands for all possible graphs of that shape, including ghost loops.
8
At the three-loop level we find three regular graphs
1 1
= ( βh¯)2 R2 , (29)
72 − −6 µν
(cid:18) (cid:19)
1 1
= ( βh¯)2 R2 , (30)
72 − −6 µνλρ
(cid:18) (cid:19)
1 1 1
= ( βh¯)2 2R+ R2 + R2 ,(31)
− 480∇ 720 µνλρ 1080 µν
(cid:18) (cid:19)
as well as a graph coming from the potential terms
1 1 (D 2)/(D 1) 1
= (βh¯)2 − − − 2R R2 . (32)
−6 16 ∇ − 72 µνλρ!
There are only two differences with respect to time slicing: with time slicing, the
factor 1 in (30) becomes 1, and the factor 1 in (32) becomes 1 . These
−6 −4 −72 −48
two modifications lead to the same final expression. Adding all contributions of
connected and disconnected graphs we find the correct result
d4x 1
An(Weyl)(spin0,D = 4) = g(x)σ(x) R2 R2 2R . (33)
(2π)2 720 µνλρ − µν −∇
Z q (cid:16) (cid:17)
where σ(x) is the arbitrary function appearing in the Jacobian: J = σ(x)δD(x y).
−
This shows that mode regularization works after all.7 The fact that the right answer
isobtainedsuggeststhatnonewcountertermsareneededinthisscheme. Acomplete
three-loop calculation in arbitrary coordinates and with a non-vanishing ξµ could
be used to test eq. (20) at order β2. As we noted in the introduction, however, all
three- and higher-loop graphs are power-counting convergent. This means that at
these orders in h¯ any consistent regularization scheme will yield the same answer.
We therefore claim that with the presently found counterterm, the mode regularized
path integral is consistent to all orders. This suggests that the difference is indeed
due to operator-orderings. It would be interesting to justify mode regularization
from first principles.
Before closing this section, we comment on how loop integrations in Feynman
diagrams are done. Mode cut-off allows one to disentangle ambiguities that appear
in the continuum limit of certain integrals over the ∆’s. Resorting to the mode
regulated expression for ∆, one can use partial integration and take boundary terms
into account if they are non-vanishing. Using partialintegration repeatedly, one gets
• •
expressions containing only ∆’s, ∆ ’s and ∆’s which are unambiguous in the con-
tinuum limit, and computes them there. Useful identities obtained at the regulated
7 In [4] the noncovariantpart of the counterterm (25) was missed. In Riemann normal coordi-
natesthistermaffectsonlythecoefficientoftheR2 terminthefourdimensionaltraceanomaly,
µνλρ
which was erroneously calculated.
9
level from (15) are
• • •• •
∆ (τ,τ)+ ∆(τ,τ) = ∂ (∆(τ,τ)),
τ
•• ••
∆(τ,σ) = ∆ (τ,σ). (34)
The first identity was used to compute (28). Here is a list of integrals that are easily
computed using partial integration8 and whose values are different from the time
discretization method:
• • ••
I = dτ τ (∆ + ∆) = 0,
1 σ=τ
|
Z
1
• • • •
I = dτdσ (∆) (∆ ) (∆ ) = , (35)
2
−12
ZZ
1
• • • •
I = dτdσ ∆ (∆) (∆ ) (∆ ) = .
3
180
ZZ
Time discretization would give I = 1, I = 1 and I = 7 .9
1 −2 2 −6 3 360
The non-vanishing value of I in time discretization is needed to compensate the
1
explicit factors of g−1/4(x ) appearing in the measure, see (36), and together with
i
I it is responsible for the different counterterms required in the two regularization
2
schemes. Finally, I leads to the different values for the coefficient of R2 in
3 µνλρ
eq. (30) in the two schemes. The counterterms with two Christoffel symbols are also
different in both schemes but the final result for the transition element (and hence
the trace anomaly) is the same. This confirms our approach to mode regularization.
III Phase space path integrals in curved space
and Matthews’ theorem
In the phase space approach to path integrals in curved space the vertices and
propagators are different from those used in the configuration space approach. For
example, the leading terms in the action are 1p gµν(x)p and 1x˙µg (x)x˙ν, respec-
2 µ ν 2 µν
tively, and the q˙µ(τ)q˙ν(σ) propagator contains a Dirac delta function but the
h i
p (τ)p (σ) propagator is completely regular. Although the vertices and propa-
µ ν
h i
gators differ, the result for the transition element should be the same (Matthews’
theorem10) [10], and the way this comes about is due to a new kind of ghosts [3, 4].
8To obtain I2, use (∆•)(•∆•) = 12∂τ(∆•)2, partially integrate, use ••∆ = ∆••, and then use
(∆••)(∆•)2 = 31∂σ(∆•)3. ToobtainI3,writetheintegrandas∆(•∆)12∂τ(∆•)2,partiallyintegrate,
and use then that ∆(••∆)(∆•)2 = ∆(∆••)(∆•)2 = ∆1∂ (∆•)3. Then partially integrate once
3 τ
more.
9In the time discretization scheme δ(σ τ) is a Kronecker delta function and θ(0) = 1. This
2
−
leadstoafullyconsistentschemeasshownin[8]. Forexample, dσdτδ(σ τ)θ(σ τ)θ(σ τ)=
1 − 1 − 1 −1 1
dσdτδ(σ τ)θ(σ τ)θ(τ σ) = whereas mode regularization would give and =
− •− − 4 • • RR •• 3 2 − 3 6
respectively. Using ∆(τ,σ) = σ+θ(τ σ), ∆ (τ,σ) = 1 δ(τ σ) and ∆ = δ(τ σ), the
RR − − − −
results below (35) follow.
10Matthews’ theorem holds for connected graphs and originally only applied to nonderivative
interactions.
10