- Email: [email protected]
Coordinate independence of quantum-mechanical path integrals
1 May 2000
Physics Letters A 269 Ž2000. 63–69 www.elsevier.nlrlocaterphysleta
Coordinate independence of quantum-mechanical path integrals H. Kleinert ) , A. Chervyakov 1 Freie UniÕersitat ¨ Berlin, Institut fur ¨ Theoretische Physik, Arnimallee14, D-14195 Berlin, Germany Received 27 January 2000; accepted 3 March 2000 Communicated by P.R. Holland
Abstract We develop simple rules for performing integrals over products of distributions in coordinate space. Such products occur in perturbation expansions of path integrals in curvilinear coordinates, where the interactions contain terms of the form q˙ 2 q n, which give rise to highly singular Feynman integrals. The new rules ensure the invariance of perturbatively defined path integrals under coordinate transformations. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction In the previous Letters w1,2x, we have presented a diagrammatic proof of reparametrization invariance of perturbatively defined quantum-mechanical path integrals. The proper perturbative definition of path integrals was shown to require an extension to a functional integral in D spacetime, and a subsequent analytic continuation to D s 1. In Ref. w1x the perturbative calculations were performed in momentum space, where Feynman integrals in a continuous number of dimensions D are known from the prescriptions of ’t Hooft and M. Veltman w3x. In Ref. w2x we have found the same results directly from the Feynman integrals in the 1 y ´-dimensional time space with the help of the Bessel representation of Green functions. The coordinate space calculation is
)
Corresponding author. Tel.: q49 30 838 3034r3337; fax: q49 30 838 3034. E-mail addresses: [email protected] ŽH. Kleinert., [email protected] ŽA. Chervyakov.. 1 On leave from LCTA, JINR, Dubna, Russia.
interesting for many applications, for instance, if one wants to obtain the effective action of a field system in curvilinear coordinates, where the kinetic term depends on the dynamic variable. Then one needs rules for performing temporal integrals over Wick contractions of local fields. In this Letter we want to show that the reparametrization invariance of perturbatively defined quantum-mechanical path integrals can be obtained in the coordinate space with the help of a simple but quite general arguments based on the inhomogeneous field equation for the Green function, and rules of the partial integration. The prove does not require the calculation of the Feynman integrals separately and remains valid for the functional integrals in an arbitrary space-time dimension D.
2. Problem with coordinate transformations Recall the origin of the difficulties with coordinate transformations in path integrals. Let x Žt . be
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 1 6 9 - 9
H. Kleinert, A. CherÕyakoÕr Physics Letters A 269 (2000) 63–69
64
the euclidean coordinates of a quantum-mechanical point particle of unit mass in a harmonic potential v 2 x 2r2 as a function of the imaginary time t s yit. Under a coordinate transformation x Žt . ™ q Žt . defined by x Žt . s f Ž q Žt .. s q Žt . q Ý`ns 2 a n q n Žt ., the kinetic term x˙ 2 Žt .r2 goes over into q˙ 2 Ž t . f X 2 Ž q Ž t . . r2. If the path integral over q Žt . is performed perturbatively, the expansion terms contains temporal integrals over Wick contractions which, after suitable partial integrations, are products of the following basic correlation functions
DŽ t y t X . ' ² q Ž t . q Ž t X . : s — , Et D Ž t y t X . ' ² q˙ Ž t . q Ž t X . : s - - — , X
X
Et Et X D Ž t y t . ' ² q˙ Ž t . q˙ Ž t . : s - - - -.
Ž 1. Ž 2. Ž 3.
The right-hand sides define the line symbols to be used in Feynman diagrams for the interaction terms. Explicitly, the first correlation function reads
D Ž t ,t X . s
1
X
2v
ey v
Ž 4.
The second correlation function Ž2. has a discontinuity
3. Model system To be specific, we shall prove the coordinate independence of the exactly solvable path integral of a point particle of unit mass in a harmonic potential v 2 x 2r2, over a large imaginary-time interval b , 2
2
Zv s D x Ž t . eyAAv w x x s eyTr log ŽyE qv . s ey bv 2 .
H
Ž 8. The action is Av s 12
H dt
x˙ 2 Ž t . q v 2 x 2 Ž t . .
Ž 9.
A coordinate transformation turns Ž8. into a path integral with a singular perturbation expansion. For simplicity we assume the coordinate transformation to preserve the symmetry x l yx of the initial oscillator, such its power series expansion starts out like x Žt . s f Ž q Žt .. s q y gq 3r3 q g 2 aq 5r5 y PPP , where g is a smallness parameter, and a an extra parameter. We shall see that the perturbation expansion is independent of a, such that a will merely serve to check the calculations. The transformation changes the partition function Ž8. into
X
Et D Ž t ,t X . s y 12 e Ž t y t X . eyv
Ž 5.
where
e Žtyt X . '2
t
XX
Hy`dt d Ž t
XX
ytX .
Ž 6.
is a distribution which has a jump at t s t X . The third correlation function Ž3. contains a d-function: v X Et Et X D Ž t ,t X . s d Ž t y t X . y ey v
Z s Dq Ž t . eyAAJ w q x eyAA w q x ,
Ž 10 .
H
where the transformed action A w q x s Av w q x q Aint w q x is decomposed into a free part Av w q x s 12
H dt
q˙ 2 Ž t . q v 2 q 2 Ž t . ,
Ž 11 .
and an interacting part, which reads to second order in g: Aint w q x
½
s 12 dt yg 2 q˙ 2 Ž t . q 2 Ž t . q
H
2v2 3
q4 Žt .
qg 2 Ž 1 q 2 a . q˙ 2 Ž t . q 4 Ž t . qv 2
ž
1
2a q
9
5
/
q6 Žt .
5
.
Ž 12 .
H. Kleinert, A. CherÕyakoÕr Physics Letters A 269 (2000) 63–69
The exponent in Ž10. contains an additional effective action AJ w q x coming from the Jacobian of the coordinate transformation: AJ w q x s yd Ž 0 . dt log
H
d f Ž qŽt . . d qŽt .
.
65
from the interaction Ž12., and one two-loop local diagram from the Jacobian action Ž14.:
Ž 13 .
Ž17.
This has the power series expansion AJ w q x s yd Ž 0 . dt ygq 2 Ž t . q g 2 Ž a y 12 . q 4 Ž t . .
H
We call a diagram local if it involves no temporal time integral. The Jacobian action Ž14. contributes further the nonlocal diagrams:
Ž 14 . For g s 0, the transformed partition function Ž10. coincides with Ž8.. When expanding Z of Eq. Ž10. in powers of g, we obtain a sum of Wick contractions with associated Feynman diagrams contributing to each order g n. This sum must vanish to ensure coordinate invariance of the path integral. By considering only connected Feynman diagrams, we shall obtain an expansion for the free energy F s Fv q
Ý
g n Fn ,
Ž 15 .
ns1
where Fv is the free energy of the unperturbed harmonic oscillator Ž8.. The coordinate invariance is ensured by the vanishing of all expansion terms Fn .
Ž18.
In the perturbative calculations to follow, we shall use dimensional regularization where d Ž0. s 0, according to a basic rule of t’Hooft and Veltman w3x. As a consequence, the last terms in F1 , F2Ž1., and the entire F2Ž2. are zero. In fact, the term AJ w q x may be omitted completely from the path integral Ž10.. The remaining diagrams are either of the threebubble type, or of the watermelon type, each with all possible combinations of the three line types Ž1. – Ž3.. The former are Ž19.
4. Expansion terms of free energy density The graphical expansion for the ground state energy will be carried here only up to three loops. The diagrams are composed of the three types of lines in Ž1. – Ž3., and new interaction vertices for each power of g. The diagrams coming from the Jacobian action Ž14. are easily recognized by an accompanying power of d Ž0.. To lowest order in g, there exists only three diagrams, two originated from the interaction Ž12., one from the Jacobian action Ž14.: Ž16. To order g 2 , we distinguish several contributions. First there are two three-loop local diagrams coming
and the latter: Ž20.
Since the equal-time expectation value ² q˙Žt . q Žt .: vanishes by Eq. Ž5., diagrams with a local contraction of a mixed line Ž2. are trivially zero, and have been omitted. In our previous Letters w1,2x, all integrals were calculated individually in D s 1 y ´ dimensions, taking the limit ´ ™ 0 at the end. Here we set up simple rules for finding the same results, which make the sum of all Feynman diagrams contributing to each order g n vanish.
H. Kleinert, A. CherÕyakoÕr Physics Letters A 269 (2000) 63–69
66
5. Basic properties of dimensionally regularized distributions
The second derivative of DŽ x . has the Fourier representation
The path integral Ž10. is extended to an associated functional integral in a D-dimensional coordinate space x, with coordinates xm ' Žt , x 2 , x 3 , . . . ., by replacing q˙ 2 Žt . in the kinetic term by Ž Em q Ž x .. 2 , where Em s ErE xm . The Jacobian action term Ž13. is omitted in dimensional regularization because of Veltman’s rule w3x:
Dmn Ž x . s y d/ D k
d Ž D. Ž 0 . s
d Dk
H Ž 2p .
s 0.
D
s d/ D k Ž ik . m1 . . . Ž ik . m n e i k x ,
H
Ž 22 .
with Em 1 . . . m n ' Em 1 . . . Em n , and with d/ D k ' d D krŽ2p . D . In dimensional regularization, alle these vanish at the origin as well: D
Ž 0 . s Hd/ k Ž ik . m1 . . . Ž ik . m n s 0,
dDk
H Ž 2p .
eik x D
2
k qv
2
,
Ž 24 .
d/ D k
Hk
2
Ž 27 .
H
k2qv2
eik x
s yd Ž D. Ž x . q v 2D Ž x . ,
Ž 28 .
which follows from the definition of the correlation function by the inhomogeneous field equation
Ž yEm2 q v 2 . q Ž x . s d Ž D. Ž x . .
Ž 29 .
Hd
D
x Dmm Ž x . s y1 q v 2 d D x D Ž x . ,
H
Ž 30 .
Inserting Veltman’s rule Ž21. into Ž28., we obtain
v
Dmm Ž 0 . s v 2 D Ž 0 . s
2
.
Ž 31 .
This ensures the vanishing of the first-order contribution Ž16. to the free energy yF1 s yg yDmm Ž 0 . q v 2D Ž 0 . D Ž 0 . s 0.
Ž 32 .
The same Eq. Ž28. allows us to calculate immediately the second-order contribution Ž17. from the local diagrams yF2Ž1. s y3 g 2
Ž 12 q a . Dmm Ž 0.
qv2
v Dy 2 s
Ž 4p .
Dr 2
ž
G 1y
D 2
/
1 s Ds1
2v
.
Ž 25 .
y5
The extension of the time derivative Ž2., D
Dm Ž x . s d/ k
H
eik x.
Ž 23 .
At the origin, it has the value
D Ž 0. s
k2
Dmm Ž x . s y d/ D k
Ds1
which is a more general way of expressing Veltman’s rule. In the extended coordinate space, the correlation function Ž1. becomes
DŽ x . s
k qv2
From Ž28. we have the relation between integrals
dmŽ 1D.. . . m nŽ x . ' Em 1 . . . m nd Ž D. Ž x .
dmŽ 1D.. . . m n
2
Contracting the indices yields
Ž 21 .
In our calculations, we shall encounter generalized d-functions, which are multiple derivatives of the ordinary d-function:
km kn
H
ikm k2qv2
ž
2 sy
e
ik x
Ž 26 .
vanishes at the origin, DmŽ0. s 0. This follows directly from a Taylor series expansion of 1rŽ k 2 q v 2 . in powers of k 2 , together with Eq. Ž23..
3
1
a q
18
5
/
v 2D Ž 0 . D 2 Ž 0 .
v 2D 3 Ž 0 . s y D™1
1 12 v
.
Ž 33 .
The other contributions to the free energy in the expansion Ž15. require rules for calculating products of two and four distributions, which we are now going to develop.
H. Kleinert, A. CherÕyakoÕr Physics Letters A 269 (2000) 63–69
6. Integrals over products of two distributions
67
Ž k 2 . 2 s Ž k 2 q v 2 . 2 y 2 v 2 Ž k 2 q v 2 . q v 4 , and use Ž23. to evaluate further
The simplest integrals of this type are
H Hd
D
x D 2 Ž x . s d/ D p d/ D k
H
d
Ž k2 .
2 d D x Dmm Ž x . s d/ D k
H
Ž k2qv2 .
Ž D.
Ž k q p. Ž p qv .Ž k2qv2 . 2
2
s y2 v
2
H
Ž k2qv . v Dy 4
s
Ž 4p . s
qv4
2 2
Ž2 yD. 2v
ž
G 2y Dr 2
2
D 2
H Žk
2
qv2 .
d/ D k
H
Ž k2qv2 .
2
s y2 v 2D Ž 0 . q v 4 d D x D 2 Ž x . .
H
/
D Ž 0. ,
2
d/ D k
d/ D k s
2
Ž 37 . Ž 34 .
Together with Ž34., we obtain the finite integrals
Hd
D
2 2 x Dmn Ž x . s d D x Dmm Ž x.
H
and s y2 v 2D Ž 0 . q v 4 d D x D 2 Ž x .
H
Hd
D
x Dm2 Ž x .
s y Ž 1 qD 2 . v 2D Ž 0 . .
An alternative way of deriving the equality Ž36. is to use partial integrations and the identity
s y d D x D Ž x . ydŽ D . Ž x . q v 2D Ž x .
H
Em Dmn Ž x . s En Dmm Ž x . ,
s D Ž 0. y v 2 d D x D 2 Ž x . s
H
D 2
D Ž 0. .
Ž 35 .
To obtain the second result we have performed a partial integration and used Ž28.. In contrast to the integrals Ž34. and Ž35., the integral 2 Ž kp . d Ž D. Ž k q p . D 2 D D Hd x Dmn Ž x . sHd/ p d/ k Ž k 2 q v 2 . Ž p 2 q v 2 .
D
s d/ k
H
Ž k2 .
2 s d D x Dmm Ž x.
H
Ž 39 .
which follows directly from the Fourier representation Ž26.. Finally, from Eqs. Ž34., Ž35., and Ž38., we observe the useful identity
Hd
D
2 x Dmn Ž x . q 2 v 2 Dm2 Ž x . q v 4 D 2 Ž x . s 0 ,
Ž 40 . which together with the inhomogeneous field Eq. Ž28. reduces the calculation of the second-order contribution of all three-bubble diagrams Ž19. to zero: 2 yF2Ž3. s yg 2D 2 Ž 0 . d D x Dmn Ž x.
H
2
Ž k2qv2 .
Ž 38 .
2
q2 v 2 Dm2 Ž x . q v 4 D 2 Ž x . s 0 .
Ž 41 .
Ž 36 .
diverges formally in D s 1 dimension. In dimensional regularization, however, we may decompose
7. Integrals products of four distributions More delicate integrals arise from the watermelon diagrams in Ž20. which contain products of four
H. Kleinert, A. CherÕyakoÕr Physics Letters A 269 (2000) 63–69
68
distributions, a nontrivial tensorial structure, and overlapping divergences w1,2x. Consider the first three diagrams: Ž42.
Invoking once more the inhomogeneous field Eq. Ž28. and Veltman’s rule Ž21., we obtain the integrals
Hd
D
2 x Dmm Ž x . D2 Ž x .
s y2 v 2D 3 Ž 0 . q v 4 d D x D 4 Ž x . ,
H
Ž43.
Ž 50 .
and
Hd
D
x Dmm Ž x . Dm2 Ž x . D Ž x .
Ž44. s v 2 d D x Dm2 Ž x . D 2 Ž x . .
H
To exhibit the subtleties with the tensorial structure, we introduce the integral 2 2 ID s d D x D 2 Ž x . Dmn Ž x . y Dmm Ž x. .
Ž 45 .
H
In D s 1 dimension, the bracket vanishes formally, but the limit D ™ 1 of the integral is nevertheless finite. We now decompose the Feynman diagram Ž42., into the sum
Hd
D
2 2 x D 2 Ž x . Dmn Ž x . s d D x D 2 Ž x . Dmm Ž x . q ID .
H
Ž 51 .
Due to Eq. Ž49., the integral Ž51. takes the form
Hd
D
x Dmm Ž x . Dm2 Ž x . D Ž x .
s 13 v 2D 3 Ž 0 . y 13 v 4 d D x D 4 Ž x . .
H
Ž 52 .
Partial integration, together with Eqs. Ž50. and Ž52., leads to
Hd
D
x Em Dl l Ž x . Dm Ž x . D 2 Ž x .
Ž 46 . To obtain an analogous decompositions for the other two diagrams Ž43. and Ž44. we derive a few useful relations using the inhomogeneous field Eq. Ž28., partial integrations, and Veltman’s rule Ž23.. First there is the relation D
s D 3 Ž 0. y v 2 d D x D 4 Ž x . .
Ž 47 .
H
By a partial integration, the left-hand side becomes
Hd
D
3
D
x Dmm Ž x . D Ž x . s y3 d x
H
Dm2
H
y 31 v 4 d D x D 4 Ž x . ,
H
Ž 53 .
A further partial integration, and use of Eqs. Ž39., Ž51., and Ž53., produces the decompositions of the second and third Feynman diagrams Ž43. and Ž44.: 4 d D x D Ž x . Dm Ž x . Dn Ž x . Dmn Ž x .
H
Ž x. D Ž x. , s y2 ID q 4v 2 d D x D 2 Ž x . Dm2 Ž x . ,
H
Ž 54 .
and
leading to D
y 2 d D x Dll Ž x . Dm2 Ž x . D Ž x . s 43 v 2D 3 Ž 0 .
2
Ž 48 .
Hd
H
3
y d x Dmm Ž x . D Ž x .
H
2 s y d D x Dll Ž x . D2 Ž x .
x Dm2 Ž x . D 2 Ž x .
s 13 D 3 Ž 0 . y 13 v 2 d D x D 4 Ž x . .
H
Hd Ž 49 .
D
x Dm2 Ž x . Dn2 Ž x .
s ID y 3 v 2 d D x D 2 Ž x . Dm2 Ž x . .
H
Ž 55 .
H. Kleinert, A. CherÕyakoÕr Physics Letters A 269 (2000) 63–69
We now make the important observation that the subtle integral ID of Eq. Ž45. appears in Eqs. Ž46., Ž54. and Ž55. in such a way that it drops out from the sum of the watermelon diagrams in Ž20.:
69
This cancels the finite contribution Ž33., thus making also the second-order free energy in Ž15. vanish, and confirming the invariance of the perturbatively defined path integral under coordinate transformations up to this order.
Ž56. 8. Summary Using Ž49. and Ž50., the right-hand side becomes a sum of completely regular expressions. Moreover, adding to this sum the last two watermelon-like diagrams in Eq. Ž20.: Ž57. and Ž58. we obtain for the contribution of all watermelon-like diagrams Ž20. the simple expression yF2Ž4. s y2 g 2
Hd
D
2 x D 2 Ž x . Dmm Ž x.
q5 v 2 Dm2 Ž x . q 23 v 4 D 2 Ž x . s 23 v 2D 3 Ž 0 . s
D™1
1 12 v
.
Ž 59 .
In this Letter we have set up simple rules for calculating integrals over products of distributions in configuration space which produce the same results as dimensional regularization in momentum space. For a path integral of a quantum-mechanical point particle in a harmonic potential, we have shown that these rules lead to a reparametrization-invariant perturbation expansions of path integral. Let us end with the remark that in the time-sliced definition of path integrals, reparametrization invariance has been established as long time ago in the textbook w4x. References w1x H. Kleinert, A. Chervyakov, Phys. Lett. B 464 Ž1999. 257, hep-thr9906156. w2x H. Kleinert, A. Chervyakov, Phys. Lett. B Ž2000., in press, quant-phr9912056. w3x G. ’t Hooft, M. Veltman, Nucl. Phys. B 44 Ž1972. 189. w4x H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, World Scientific, Singapore, 1995 Žwww.physik.fu-berlin.der ; kleinertrre.htmlab3..
Physics Letters A 269 Ž2000. 63–69 www.elsevier.nlrlocaterphysleta
Coordinate independence of quantum-mechanical path integrals H. Kleinert ) , A. Chervyakov 1 Freie UniÕersitat ¨ Berlin, Institut fur ¨ Theoretische Physik, Arnimallee14, D-14195 Berlin, Germany Received 27 January 2000; accepted 3 March 2000 Communicated by P.R. Holland
Abstract We develop simple rules for performing integrals over products of distributions in coordinate space. Such products occur in perturbation expansions of path integrals in curvilinear coordinates, where the interactions contain terms of the form q˙ 2 q n, which give rise to highly singular Feynman integrals. The new rules ensure the invariance of perturbatively defined path integrals under coordinate transformations. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction In the previous Letters w1,2x, we have presented a diagrammatic proof of reparametrization invariance of perturbatively defined quantum-mechanical path integrals. The proper perturbative definition of path integrals was shown to require an extension to a functional integral in D spacetime, and a subsequent analytic continuation to D s 1. In Ref. w1x the perturbative calculations were performed in momentum space, where Feynman integrals in a continuous number of dimensions D are known from the prescriptions of ’t Hooft and M. Veltman w3x. In Ref. w2x we have found the same results directly from the Feynman integrals in the 1 y ´-dimensional time space with the help of the Bessel representation of Green functions. The coordinate space calculation is
)
Corresponding author. Tel.: q49 30 838 3034r3337; fax: q49 30 838 3034. E-mail addresses: [email protected] ŽH. Kleinert., [email protected] ŽA. Chervyakov.. 1 On leave from LCTA, JINR, Dubna, Russia.
interesting for many applications, for instance, if one wants to obtain the effective action of a field system in curvilinear coordinates, where the kinetic term depends on the dynamic variable. Then one needs rules for performing temporal integrals over Wick contractions of local fields. In this Letter we want to show that the reparametrization invariance of perturbatively defined quantum-mechanical path integrals can be obtained in the coordinate space with the help of a simple but quite general arguments based on the inhomogeneous field equation for the Green function, and rules of the partial integration. The prove does not require the calculation of the Feynman integrals separately and remains valid for the functional integrals in an arbitrary space-time dimension D.
2. Problem with coordinate transformations Recall the origin of the difficulties with coordinate transformations in path integrals. Let x Žt . be
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 1 6 9 - 9
H. Kleinert, A. CherÕyakoÕr Physics Letters A 269 (2000) 63–69
64
the euclidean coordinates of a quantum-mechanical point particle of unit mass in a harmonic potential v 2 x 2r2 as a function of the imaginary time t s yit. Under a coordinate transformation x Žt . ™ q Žt . defined by x Žt . s f Ž q Žt .. s q Žt . q Ý`ns 2 a n q n Žt ., the kinetic term x˙ 2 Žt .r2 goes over into q˙ 2 Ž t . f X 2 Ž q Ž t . . r2. If the path integral over q Žt . is performed perturbatively, the expansion terms contains temporal integrals over Wick contractions which, after suitable partial integrations, are products of the following basic correlation functions
DŽ t y t X . ' ² q Ž t . q Ž t X . : s — , Et D Ž t y t X . ' ² q˙ Ž t . q Ž t X . : s - - — , X
X
Et Et X D Ž t y t . ' ² q˙ Ž t . q˙ Ž t . : s - - - -.
Ž 1. Ž 2. Ž 3.
The right-hand sides define the line symbols to be used in Feynman diagrams for the interaction terms. Explicitly, the first correlation function reads
D Ž t ,t X . s
1
X
2v
ey v
Ž 4.
The second correlation function Ž2. has a discontinuity
3. Model system To be specific, we shall prove the coordinate independence of the exactly solvable path integral of a point particle of unit mass in a harmonic potential v 2 x 2r2, over a large imaginary-time interval b , 2
2
Zv s D x Ž t . eyAAv w x x s eyTr log ŽyE qv . s ey bv 2 .
H
Ž 8. The action is Av s 12
H dt
x˙ 2 Ž t . q v 2 x 2 Ž t . .
Ž 9.
A coordinate transformation turns Ž8. into a path integral with a singular perturbation expansion. For simplicity we assume the coordinate transformation to preserve the symmetry x l yx of the initial oscillator, such its power series expansion starts out like x Žt . s f Ž q Žt .. s q y gq 3r3 q g 2 aq 5r5 y PPP , where g is a smallness parameter, and a an extra parameter. We shall see that the perturbation expansion is independent of a, such that a will merely serve to check the calculations. The transformation changes the partition function Ž8. into
X
Et D Ž t ,t X . s y 12 e Ž t y t X . eyv
Ž 5.
where
e Žtyt X . '2
t
XX
Hy`dt d Ž t
XX
ytX .
Ž 6.
is a distribution which has a jump at t s t X . The third correlation function Ž3. contains a d-function: v X Et Et X D Ž t ,t X . s d Ž t y t X . y ey v
Z s Dq Ž t . eyAAJ w q x eyAA w q x ,
Ž 10 .
H
where the transformed action A w q x s Av w q x q Aint w q x is decomposed into a free part Av w q x s 12
H dt
q˙ 2 Ž t . q v 2 q 2 Ž t . ,
Ž 11 .
and an interacting part, which reads to second order in g: Aint w q x
½
s 12 dt yg 2 q˙ 2 Ž t . q 2 Ž t . q
H
2v2 3
q4 Žt .
qg 2 Ž 1 q 2 a . q˙ 2 Ž t . q 4 Ž t . qv 2
ž
1
2a q
9
5
/
q6 Žt .
5
.
Ž 12 .
H. Kleinert, A. CherÕyakoÕr Physics Letters A 269 (2000) 63–69
The exponent in Ž10. contains an additional effective action AJ w q x coming from the Jacobian of the coordinate transformation: AJ w q x s yd Ž 0 . dt log
H
d f Ž qŽt . . d qŽt .
.
65
from the interaction Ž12., and one two-loop local diagram from the Jacobian action Ž14.:
Ž 13 .
Ž17.
This has the power series expansion AJ w q x s yd Ž 0 . dt ygq 2 Ž t . q g 2 Ž a y 12 . q 4 Ž t . .
H
We call a diagram local if it involves no temporal time integral. The Jacobian action Ž14. contributes further the nonlocal diagrams:
Ž 14 . For g s 0, the transformed partition function Ž10. coincides with Ž8.. When expanding Z of Eq. Ž10. in powers of g, we obtain a sum of Wick contractions with associated Feynman diagrams contributing to each order g n. This sum must vanish to ensure coordinate invariance of the path integral. By considering only connected Feynman diagrams, we shall obtain an expansion for the free energy F s Fv q
Ý
g n Fn ,
Ž 15 .
ns1
where Fv is the free energy of the unperturbed harmonic oscillator Ž8.. The coordinate invariance is ensured by the vanishing of all expansion terms Fn .
Ž18.
In the perturbative calculations to follow, we shall use dimensional regularization where d Ž0. s 0, according to a basic rule of t’Hooft and Veltman w3x. As a consequence, the last terms in F1 , F2Ž1., and the entire F2Ž2. are zero. In fact, the term AJ w q x may be omitted completely from the path integral Ž10.. The remaining diagrams are either of the threebubble type, or of the watermelon type, each with all possible combinations of the three line types Ž1. – Ž3.. The former are Ž19.
4. Expansion terms of free energy density The graphical expansion for the ground state energy will be carried here only up to three loops. The diagrams are composed of the three types of lines in Ž1. – Ž3., and new interaction vertices for each power of g. The diagrams coming from the Jacobian action Ž14. are easily recognized by an accompanying power of d Ž0.. To lowest order in g, there exists only three diagrams, two originated from the interaction Ž12., one from the Jacobian action Ž14.: Ž16. To order g 2 , we distinguish several contributions. First there are two three-loop local diagrams coming
and the latter: Ž20.
Since the equal-time expectation value ² q˙Žt . q Žt .: vanishes by Eq. Ž5., diagrams with a local contraction of a mixed line Ž2. are trivially zero, and have been omitted. In our previous Letters w1,2x, all integrals were calculated individually in D s 1 y ´ dimensions, taking the limit ´ ™ 0 at the end. Here we set up simple rules for finding the same results, which make the sum of all Feynman diagrams contributing to each order g n vanish.
H. Kleinert, A. CherÕyakoÕr Physics Letters A 269 (2000) 63–69
66
5. Basic properties of dimensionally regularized distributions
The second derivative of DŽ x . has the Fourier representation
The path integral Ž10. is extended to an associated functional integral in a D-dimensional coordinate space x, with coordinates xm ' Žt , x 2 , x 3 , . . . ., by replacing q˙ 2 Žt . in the kinetic term by Ž Em q Ž x .. 2 , where Em s ErE xm . The Jacobian action term Ž13. is omitted in dimensional regularization because of Veltman’s rule w3x:
Dmn Ž x . s y d/ D k
d Ž D. Ž 0 . s
d Dk
H Ž 2p .
s 0.
D
s d/ D k Ž ik . m1 . . . Ž ik . m n e i k x ,
H
Ž 22 .
with Em 1 . . . m n ' Em 1 . . . Em n , and with d/ D k ' d D krŽ2p . D . In dimensional regularization, alle these vanish at the origin as well: D
Ž 0 . s Hd/ k Ž ik . m1 . . . Ž ik . m n s 0,
dDk
H Ž 2p .
eik x D
2
k qv
2
,
Ž 24 .
d/ D k
Hk
2
Ž 27 .
H
k2qv2
eik x
s yd Ž D. Ž x . q v 2D Ž x . ,
Ž 28 .
which follows from the definition of the correlation function by the inhomogeneous field equation
Ž yEm2 q v 2 . q Ž x . s d Ž D. Ž x . .
Ž 29 .
Hd
D
x Dmm Ž x . s y1 q v 2 d D x D Ž x . ,
H
Ž 30 .
Inserting Veltman’s rule Ž21. into Ž28., we obtain
v
Dmm Ž 0 . s v 2 D Ž 0 . s
2
.
Ž 31 .
This ensures the vanishing of the first-order contribution Ž16. to the free energy yF1 s yg yDmm Ž 0 . q v 2D Ž 0 . D Ž 0 . s 0.
Ž 32 .
The same Eq. Ž28. allows us to calculate immediately the second-order contribution Ž17. from the local diagrams yF2Ž1. s y3 g 2
Ž 12 q a . Dmm Ž 0.
qv2
v Dy 2 s
Ž 4p .
Dr 2
ž
G 1y
D 2
/
1 s Ds1
2v
.
Ž 25 .
y5
The extension of the time derivative Ž2., D
Dm Ž x . s d/ k
H
eik x.
Ž 23 .
At the origin, it has the value
D Ž 0. s
k2
Dmm Ž x . s y d/ D k
Ds1
which is a more general way of expressing Veltman’s rule. In the extended coordinate space, the correlation function Ž1. becomes
DŽ x . s
k qv2
From Ž28. we have the relation between integrals
dmŽ 1D.. . . m nŽ x . ' Em 1 . . . m nd Ž D. Ž x .
dmŽ 1D.. . . m n
2
Contracting the indices yields
Ž 21 .
In our calculations, we shall encounter generalized d-functions, which are multiple derivatives of the ordinary d-function:
km kn
H
ikm k2qv2
ž
2 sy
e
ik x
Ž 26 .
vanishes at the origin, DmŽ0. s 0. This follows directly from a Taylor series expansion of 1rŽ k 2 q v 2 . in powers of k 2 , together with Eq. Ž23..
3
1
a q
18
5
/
v 2D Ž 0 . D 2 Ž 0 .
v 2D 3 Ž 0 . s y D™1
1 12 v
.
Ž 33 .
The other contributions to the free energy in the expansion Ž15. require rules for calculating products of two and four distributions, which we are now going to develop.
H. Kleinert, A. CherÕyakoÕr Physics Letters A 269 (2000) 63–69
6. Integrals over products of two distributions
67
Ž k 2 . 2 s Ž k 2 q v 2 . 2 y 2 v 2 Ž k 2 q v 2 . q v 4 , and use Ž23. to evaluate further
The simplest integrals of this type are
H Hd
D
x D 2 Ž x . s d/ D p d/ D k
H
d
Ž k2 .
2 d D x Dmm Ž x . s d/ D k
H
Ž k2qv2 .
Ž D.
Ž k q p. Ž p qv .Ž k2qv2 . 2
2
s y2 v
2
H
Ž k2qv . v Dy 4
s
Ž 4p . s
qv4
2 2
Ž2 yD. 2v
ž
G 2y Dr 2
2
D 2
H Žk
2
qv2 .
d/ D k
H
Ž k2qv2 .
2
s y2 v 2D Ž 0 . q v 4 d D x D 2 Ž x . .
H
/
D Ž 0. ,
2
d/ D k
d/ D k s
2
Ž 37 . Ž 34 .
Together with Ž34., we obtain the finite integrals
Hd
D
2 2 x Dmn Ž x . s d D x Dmm Ž x.
H
and s y2 v 2D Ž 0 . q v 4 d D x D 2 Ž x .
H
Hd
D
x Dm2 Ž x .
s y Ž 1 qD 2 . v 2D Ž 0 . .
An alternative way of deriving the equality Ž36. is to use partial integrations and the identity
s y d D x D Ž x . ydŽ D . Ž x . q v 2D Ž x .
H
Em Dmn Ž x . s En Dmm Ž x . ,
s D Ž 0. y v 2 d D x D 2 Ž x . s
H
D 2
D Ž 0. .
Ž 35 .
To obtain the second result we have performed a partial integration and used Ž28.. In contrast to the integrals Ž34. and Ž35., the integral 2 Ž kp . d Ž D. Ž k q p . D 2 D D Hd x Dmn Ž x . sHd/ p d/ k Ž k 2 q v 2 . Ž p 2 q v 2 .
D
s d/ k
H
Ž k2 .
2 s d D x Dmm Ž x.
H
Ž 39 .
which follows directly from the Fourier representation Ž26.. Finally, from Eqs. Ž34., Ž35., and Ž38., we observe the useful identity
Hd
D
2 x Dmn Ž x . q 2 v 2 Dm2 Ž x . q v 4 D 2 Ž x . s 0 ,
Ž 40 . which together with the inhomogeneous field Eq. Ž28. reduces the calculation of the second-order contribution of all three-bubble diagrams Ž19. to zero: 2 yF2Ž3. s yg 2D 2 Ž 0 . d D x Dmn Ž x.
H
2
Ž k2qv2 .
Ž 38 .
2
q2 v 2 Dm2 Ž x . q v 4 D 2 Ž x . s 0 .
Ž 41 .
Ž 36 .
diverges formally in D s 1 dimension. In dimensional regularization, however, we may decompose
7. Integrals products of four distributions More delicate integrals arise from the watermelon diagrams in Ž20. which contain products of four
H. Kleinert, A. CherÕyakoÕr Physics Letters A 269 (2000) 63–69
68
distributions, a nontrivial tensorial structure, and overlapping divergences w1,2x. Consider the first three diagrams: Ž42.
Invoking once more the inhomogeneous field Eq. Ž28. and Veltman’s rule Ž21., we obtain the integrals
Hd
D
2 x Dmm Ž x . D2 Ž x .
s y2 v 2D 3 Ž 0 . q v 4 d D x D 4 Ž x . ,
H
Ž43.
Ž 50 .
and
Hd
D
x Dmm Ž x . Dm2 Ž x . D Ž x .
Ž44. s v 2 d D x Dm2 Ž x . D 2 Ž x . .
H
To exhibit the subtleties with the tensorial structure, we introduce the integral 2 2 ID s d D x D 2 Ž x . Dmn Ž x . y Dmm Ž x. .
Ž 45 .
H
In D s 1 dimension, the bracket vanishes formally, but the limit D ™ 1 of the integral is nevertheless finite. We now decompose the Feynman diagram Ž42., into the sum
Hd
D
2 2 x D 2 Ž x . Dmn Ž x . s d D x D 2 Ž x . Dmm Ž x . q ID .
H
Ž 51 .
Due to Eq. Ž49., the integral Ž51. takes the form
Hd
D
x Dmm Ž x . Dm2 Ž x . D Ž x .
s 13 v 2D 3 Ž 0 . y 13 v 4 d D x D 4 Ž x . .
H
Ž 52 .
Partial integration, together with Eqs. Ž50. and Ž52., leads to
Hd
D
x Em Dl l Ž x . Dm Ž x . D 2 Ž x .
Ž 46 . To obtain an analogous decompositions for the other two diagrams Ž43. and Ž44. we derive a few useful relations using the inhomogeneous field Eq. Ž28., partial integrations, and Veltman’s rule Ž23.. First there is the relation D
s D 3 Ž 0. y v 2 d D x D 4 Ž x . .
Ž 47 .
H
By a partial integration, the left-hand side becomes
Hd
D
3
D
x Dmm Ž x . D Ž x . s y3 d x
H
Dm2
H
y 31 v 4 d D x D 4 Ž x . ,
H
Ž 53 .
A further partial integration, and use of Eqs. Ž39., Ž51., and Ž53., produces the decompositions of the second and third Feynman diagrams Ž43. and Ž44.: 4 d D x D Ž x . Dm Ž x . Dn Ž x . Dmn Ž x .
H
Ž x. D Ž x. , s y2 ID q 4v 2 d D x D 2 Ž x . Dm2 Ž x . ,
H
Ž 54 .
and
leading to D
y 2 d D x Dll Ž x . Dm2 Ž x . D Ž x . s 43 v 2D 3 Ž 0 .
2
Ž 48 .
Hd
H
3
y d x Dmm Ž x . D Ž x .
H
2 s y d D x Dll Ž x . D2 Ž x .
x Dm2 Ž x . D 2 Ž x .
s 13 D 3 Ž 0 . y 13 v 2 d D x D 4 Ž x . .
H
Hd Ž 49 .
D
x Dm2 Ž x . Dn2 Ž x .
s ID y 3 v 2 d D x D 2 Ž x . Dm2 Ž x . .
H
Ž 55 .
H. Kleinert, A. CherÕyakoÕr Physics Letters A 269 (2000) 63–69
We now make the important observation that the subtle integral ID of Eq. Ž45. appears in Eqs. Ž46., Ž54. and Ž55. in such a way that it drops out from the sum of the watermelon diagrams in Ž20.:
69
This cancels the finite contribution Ž33., thus making also the second-order free energy in Ž15. vanish, and confirming the invariance of the perturbatively defined path integral under coordinate transformations up to this order.
Ž56. 8. Summary Using Ž49. and Ž50., the right-hand side becomes a sum of completely regular expressions. Moreover, adding to this sum the last two watermelon-like diagrams in Eq. Ž20.: Ž57. and Ž58. we obtain for the contribution of all watermelon-like diagrams Ž20. the simple expression yF2Ž4. s y2 g 2
Hd
D
2 x D 2 Ž x . Dmm Ž x.
q5 v 2 Dm2 Ž x . q 23 v 4 D 2 Ž x . s 23 v 2D 3 Ž 0 . s
D™1
1 12 v
.
Ž 59 .
In this Letter we have set up simple rules for calculating integrals over products of distributions in configuration space which produce the same results as dimensional regularization in momentum space. For a path integral of a quantum-mechanical point particle in a harmonic potential, we have shown that these rules lead to a reparametrization-invariant perturbation expansions of path integral. Let us end with the remark that in the time-sliced definition of path integrals, reparametrization invariance has been established as long time ago in the textbook w4x. References w1x H. Kleinert, A. Chervyakov, Phys. Lett. B 464 Ž1999. 257, hep-thr9906156. w2x H. Kleinert, A. Chervyakov, Phys. Lett. B Ž2000., in press, quant-phr9912056. w3x G. ’t Hooft, M. Veltman, Nucl. Phys. B 44 Ž1972. 189. w4x H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, World Scientific, Singapore, 1995 Žwww.physik.fu-berlin.der ; kleinertrre.htmlab3..