Coordinate independence of quantum-mechanical q, qq. path integrals. H. Kleinert ), A. Chervyakov Introduction

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vvv Phyic Letter A 10045 000 xxx www.elevier.nlrlocaterpla Coordinate independence of quantum-mechanical q, qq path integral. Kleinert ), A.

1 vvv Phyic Letter A xxx Coordinate independence of quantum-mechanical q, qq path integral. Kleinert ), A. Chervyakov 1 Freie UniÕeritat Berlin, Intitut fur Theoretiche Phyik, Arnimallee14, Berlin, Germany Received 10 May 000 Abtract We develop imple rule for performing integral over product of ditribution in coordinate pace. Such product occur in perturbation expanion of path integral in curvilinear coordinate, where the interaction contain term of the form q q n, which give rie to highly ingular Feynman integral. The new rule enure the invariance of perturbatively defined path integral under coordinate tranformation. q 000 Publihed by Elevier Science B.V. 1. Introduction In the previou Letter w1; x, we have preented a diagrammatic proof of reparametrization invariance of perturbatively defined quantum-mechanical path integral. The proper perturbative definition of path integral wa hown to require an extenion to a functional integral in pacetime, and a ubequent analytic continuation to 1. In Ref. wx 1 the perturbative calculation were performed in momentum pace, where Feynman integral in a continuou q PII of original article S Ž qq Pleae note that thi article, which wa originally publihed on pp of Vol. 69, i now reprinted in it entirety and in it corrected verion. The Publiher apologie to the author and reader for any inconvenience and confuion caued a a reult of eriou production error in the original publication. ) Correponding author. Tel.: q r3337; fax: q addree: kleinert@phyik.fu-berlin.de Kleinert., chervyak@phyik.fu-berlin.de Ž A. Chervyakov.. 1 On leave from LCTA, JINR, ubna, Ruia. number of dimenion are known from the prewx 3. In Ref. wx cription of t ooft and M. Veltman we have found the ame reult directly from the Feynman integral in the 1 y -dimenional time pace with the help of the Beel repreentation of Green function. The coordinate pace calculation i intereting for many application, for intance, if one want to obtain the effective action of a field ytem in curvilinear coordinate, where the kinetic term depend on the dynamic variable. Then one need rule for performing temporal integral over Wick contraction of local field. In thi Letter we want to how that the reparametrization invariance of perturbatively defined quantum-mechanical path integral can be obtained in the coordinate pace with the help of a imple but quite general argument baed on the inhomogeneou field equation for the Green function, and rule of the partial integration. The prove doe not require the calculation of the Feynman integral eparately and remain valid for the functional integral in an arbitrary pace-time dimenion r00r$ - ee front matter q 000 Publihed by Elevier Science B.V. PII: S

2 . Kleinert, A. CherÕyakoÕrPhyic Letter A Problem with coordinate tranformation Recall the origin of the difficultie with coordinate tranformation in path integral. Let xž t. be the euclidean coordinate of a quantum-mechanical point particle of unit ma in a harmonic potential v x r a a function of the imaginary time tyit. Under a coordinate tranformation xž t. qž t. de- Ž. ` fined by x t f q t q t qý aq n Ž t. n n, the kinetic term x Ž t. r goe over into q Ž t. f X qž t. r. If the path integral over qž t. i performed perturbatively, the expanion term contain temporal integral over Wick contraction which, after uitable partial integration, are product of the following baic correlation function X X tyt ' ² qž t. qž t.:, Ž 1. X X E tyt ' ² q Ž t. qž t.: --, t X X EE X tyt ' ² q Ž t. q Ž t.: Ž 3. t t The right-hand ide define the line ymbol to be ued in Feynman diagram for the interaction term. Explicitly, the firt correlation function read 1 X X yv <tyt <

3 and an interacting part, which read to econd order in g: 1 Aintwqx dt yg q t q t ½ v 4 q q Ž t. 3. Kleinert, A. CherÕyakoÕrPhyic Letter A qg 1q a q t q t ž / 5 1 a 6 qv q q Ž t.. Ž The exponent in Ž 10. contain an additional effective action A wqx J coming from the Jacobian of the coordinate tranformation: d f qž t. AJ wqxydž 0. dt log. Ž 13. d qž t. Thi ha the power erie expanion w x AJ q yd 0 dt ygq t 1 4 qg ay q t. 14 For g 0, the tranformed partition function Ž 10. coincide with Ž. 8. When expanding Z of Eq. Ž 10. in power of g, we obtain a um of Wick contraction with aociated Feynman diagram contributing to each order g n. Thi um mut vanih to enure coordinate invariance of the path integral. By conidering only connected Feynman dia-

4 4. Kleinert, A. CherÕyakoÕrPhyic Letter A Ž.Ž. poible combination of the three line type 1 3. The former are In our calculation, we hall encounter generalized d-function, which are multiple derivative of the ordinary d-function: and the latter: 19 d Ž. Ž x.'e d Ž. Ž x. m 1...mn m 1...mn 1 n d/ kž ik. m... Ž ik. m e ikx, with E m...m ' E m...e m, and with d/ k ' 1 n 1 n d krž p.. In dimenional regularization, alle thee vanih at the origin a well: d Ž. Ž 0. d/ kž ik. m... Ž ik. m 0, Ž 3. m 1...mn 1 n 0 Since the equal-time expectation value ² q Ž t. qž t.: vanihe by Eq. Ž. 5, diagram with a local contraction of a mixed line Ž. are trivially zero, and have been omitted. In our previou Letter w1; x, all integral were calculated individually in 1 y dimenion, taking the limit 0 at the end. ere we et up imple rule for finding the ame reult, which make the um of all Feynman diagram contributing to each order g n vanih. 5. Baic propertie of dimenionally regularized ditribution The path integral Ž 10. i extended to an aociated functional integral in a -dimenional coordinate pace x, with coordinate x ' Ž t, x, x,... m 3, by Ž. replacing q t in the kinetic term by Emq x, where Em ErE x m. The Jacobian action term Ž 13. i omitted in dimenional regularization becaue of wx Veltman rule 3 : d k Ž. d Ž Ž 1. Ž p. which i a more general way of expreing Veltman rule. In the extended coordinate pace, the correlation function Ž. 1 become d k e ikx Ž x., Ž 4. Ž p. k qv At the origin, it ha the value / d/ k v y 1 Ž 0. r Gž 1y. k qv Ž 4p. 1 v Ž 5. Ž. The extenion of the time derivative, ik m ikx mž x. d/ k e Ž 6. k qv vanihe at the origin, Ž. m 0 0. Thi follow di- rectly from a Taylor erie expanion of 1rŽk qv. in power of k, together with Eq. Ž 3.. The econd derivative of Ž x. ha the Fourier repreentation km k n ikx mn Ž x. yd/ k e. Ž 7. k qv Contracting the indice yield k ikx mm x y d/ k e k qv yd Ž. Ž x. qv Ž x., Ž 8.

5 . Kleinert, A. CherÕyakoÕrPhyic Letter A which follow from the definition of the correlation function by the inhomogeneou field equation ye qv qž x. d Ž. Ž x.. Ž 9. Ž m. From 8 we have the relation between integral d x Ž x. y1qv d x Ž x., Ž 30. mm Inerting Veltman rule Ž 1. into Ž 8., we obtain v mmž 0. v Ž 0.. Ž Thi enure the vanihing of the firt-order contribu- tion 16 to the free energy yf1 yg ymmž 0. qv Ž 0. Ž Ž 3. The ame Eq. Ž 8. allow u to calculate immediately the econd-order contribution Ž 17. from the local diagram Ž1. 1 yf y3g qa mm 0 ž / 1 a y5 q vž 0. Ž y v Ž 0. y. Ž v The other contribution to the free energy in the expanion Ž 15. require rule for calculating product of two and four ditribution, which we are now going to develop. 6. Integral over product of two ditribution The implet integral of thi type are d Ž. Ž kqp. Ž p qv.ž k qv. d x x d/ pd/ k d/ k Ž k qv. v y 4 Ž 4p. r ž / G y Ž y. Ž 0., Ž 34. v and d x m Ž x. Ž. y d x x yd x qv x Ž 0. yv d x Ž x. Ž 0.. Ž 35. To obtain the econd reult we have performed a partial integration and ued Ž 8.. In contrat to the integral Ž 34. and Ž 35., the integral Ž. Ž kp. d Ž kqp. d x mn x d/ pd/ k Ž k. d/ k Ž k qv.ž p qv. Ž k qv. d x mmž x. Ž 36. diverge formally in 1 dimenion. In dimenional regularization, however, we may decompoe Ž. Ž. Ž. 4 k k qv yv k qv qv, and ue Ž 3. to evaluate further Ž k. d x mm Ž x. d/ k Ž k qv. d/ k yv Ž k qv. d/ k 4 qv Ž k qv. yv Ž 0. qv 4 d x Ž x.. Together with 34, we obtain the finite integral d x Ž x. d x Ž x. mn mm yv Ž 0. qv 4 d x Ž x. Ž 37. yž 1q. v Ž 0.. Ž 38.

6 6. Kleinert, A. CherÕyakoÕrPhyic Letter A An alternative way of deriving the equality Ž 36. i to ue partial integration and the identity Em mn Ž x. En mmž x., Ž 39. which follow directly from the Fourier repreentation Ž 6.. Finally, from Eq. Ž 34., Ž 35., and Ž 38., we oberve the ueful identity 4 d x mnž x. qv mž x. qv Ž x. 0, Ž 40. which together with the inhomogeneou field Eq. Ž 8. reduce the calculation of the econd-order contribution of all three-bubble diagram Ž 19. to zero: Ž3. yf yg 0 d x mn x qv m x 4 qv Ž x. 0. Ž Integral product of four ditribution More delicate integral arie from the watermelon diagram in Ž 0. which contain product of four ditribution, a nontrivial tenorial tructure, and overlapping divergence w1; x. Conider the firt three diagram: 4 To exhibit the ubtletie with the tenorial tructure, we introduce the integral I d x x mn x y mm x. 45 In 1 dimenion, the bracket vanihe formally, but the limit 1 of the integral i neverthele finite. We now decompoe the Feynman diagram Ž 4., into the um d x Ž x. Ž x. d x Ž x. Ž x. qi. mn mm Ž 46. To obtain an analogou decompoition for the other two diagram Ž 43. and Ž 44. we derive a few ueful relation uing the inhomogeneou field Eq. Ž 8., partial integration, and Veltman rule Ž 3.. Firt there i the relation y d x Ž x. 3 Ž x. 3 Ž 0. yv d x 4 Ž x.. mm Ž 47. By a partial integration, the left-hand ide become d x Ž x. 3 Ž x. y3 d x Ž x. Ž x., mm leading to m Ž d x m x x 3 0 y 3v d x x. Ž 49. Invoking once more the inhomogeneou field Eq. 8 and Veltman rule 1, we obtain the integral d x mmž x. Ž x. 43 yv 3 Ž 0. qv 4 d x 4 Ž x., Ž 50. and d x mmž x. mž x. Ž x. 44 v d x m Ž x. Ž x.. Ž 51.

7 . Kleinert, A. CherÕyakoÕrPhyic Letter A ue to Eq. Ž 49., the integral Ž 51. take the form d x mmž x. mž x. Ž x v 0 y 3v d x x. 5 Partial integration, together with Eq. Ž 50. and Ž 5., lead to d x Em llž x. mž x. Ž x. adding to thi um the lat two watermelon-like diagram in Eq. 0 : and 57 yd x llž x. Ž x. yd x llž x. mž x. Ž x v 0 y 3v d x x, 53 A further partial integration, and ue of Eq. Ž 39., Ž 51., and Ž 53., produce the decompoition of the econd and third Feynman diagram Ž 43. and Ž 44.: 4d x Ž x. mž x. nž x. mnž x. y I q4v d x Ž x. Ž x., Ž 54. and d x m Ž x. n Ž x. I y3v d x Ž x. Ž x.. Ž 55. We now make the important obervation that the ubtle integral I of Eq. Ž 45. appear in Eq. Ž 46., Ž 54. and Ž 55. in uch a way that it drop out from the um of the watermelon diagram in Ž 0.: m m 56 Uing 49 and 50, the right-hand ide become a um of completely regular expreion. Moreover, 58 we obtain for the contribution of all watermelon-like diagram 0 the imple expreion Ž4. yf y g d x x mm x 4 q5v m x q 3 v x v Ž 0.. Ž v Thi cancel the finite contribution Ž 33., thu making alo the econd-order free energy in Ž 15. vanih, and confirming the invariance of the perturbatively defined path integral under coordinate tranformation up to thi order. 8. Summary In thi Letter we have et up imple rule for calculating integral over product of ditribution in configuration pace which produce the ame reult a dimenional regularization in momentum pace. For a path integral of a quantum-mechanical point particle in a harmonic potential, we have hown that thee rule lead to a reparametrization-invariant perturbation expanion of path integral. Let u end with the remark that in the time-liced definition of path integral, reparametrization invariance ha been etablihed a long time ago in the textbook wx 4.

8 8. Kleinert, A. CherÕyakoÕrPhyic Letter A Reference wx 1. Kleinert, A. Chervyakov, Phy. Lett. B 464 Ž , hep-thr wx. Kleinert, A. Chervyakov, Phy. Lett. B Ž 000., in pre, quant-phr wx 3 G. t ooft, M. Veltman, Nucl. Phy. B 44 Ž wx 4. Kleinert, Path Integral in Quantum Mechanic, Statitic, and Polymer Phyic, World Scientific, Singapore, 1995 Ž ; kleinertrre.htmlab3..

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