Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 77
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. . + a2q Ck−1,(3C.90)and perform otherwise the same steps as for the potential gr2q /4 alone.Appendix 3DFeynman Integrals for T =/0The calculation of the Feynman integrals (3.547) can be done straightforwardly with the help ofthe symbolic program Mathematica. The first integral in Eqs. (3.547) is trivial.
The second andforth integrals are simple, since one overall integration over, say, τ3 yields merely a factor h̄β, dueto translational invariance of the integrand along the τ -axis. The triple integrals can then be splitasZ h̄βZ h̄βZ h̄βdτ1 dτ2 dτ3 f (|τ1 − τ2 |, |τ2 − τ3 |, |τ3 − τ1 |)00= h̄β= h̄βZ0h̄βZ h̄β0Z00h̄βdτ1 dτ2 f (|τ1 − τ2 |, |τ2 |, |τ1 |)dτ2Z0τ2dτ1 f (τ2 − τ1 , τ2 , τ1 ) +(3D.1)Z0h̄βdτ2Zh̄βτ2!dτ1 f (τ1 − τ2 , τ2 , τ1 ) ,to ensure that the arguments of the Green function have the same sign in each term. The linesrepresent the thermal correlation functionG(2) (τ, τ 0 ) =h̄ cosh ω[|τ − τ 0 | − h̄β/2].2M ωsinh(ωh̄β/2)(3D.2)With the dimensionless variable x ≡ ωh̄β, the result for the quantities α2LV defined in (3.547) inthe Feynman diagrams with L lines and V vertices isx1coth ,2211=(x + sinh x) ,8 sinh2 x2a2 =(3D.3)α42(3D.4)3643 External Sources, Correlations, and Perturbation Theory11x3xxx2+2xcosh+3cosh+6xsinh−3cosh,64 sinh3 x2222211=(6 x + 8 sinh x + sinh 2x) ,256 sinh4 x21xx3x1=−40 cosh + 24 x2 cosh + 35 cosh4096 sinh5 x2222x3x5x+ 72 x sinh + 12 x sinh,+ 5 cosh22211=− 48 + 32 x2 − 3 cosh x + 8 x2 cosh x16384 sinh6 x2+ 48 cosh 2 x + 3 cosh 3 x + 108 x sinhx ,11=(5 +24 cosh x) ,24 sinh2 x2xx3x113xcosh,+9sinh+sinh=72 sinh3 x222211=(30 x + 104 sinhx + 5 sinh2x) .2304 sinh4 x2α63 =(3D.5)α82(3D.6)α103α123α62α83α1030(3D.7)(3D.8)(3D.9)(3D.10)(3D.11)For completeness, we have also listed the integrals α62 , α83 , and α1030 , corresponding to the threediagrams,,,(3D.12)respectively, which occur in perturbation expansions with a cubic interaction potential x3 .
Thesewill appear in a modified version in Chapter 5.In the low-temperature limit where x = ωh̄β → ∞, the x-dependent factors α2LV in Eqs. (3D.3)–(3D.11) converge towards the constants1/2, 1/4, 3/16, 1/32, 5/(8 · 25 ),3/(8 · 26 ),1/12, 1/18, 5/(9 · 25 ),(3D.13)respectively. From these numbers we deduce the relations (3.550) and, in addition,a62 →2 6a ,3a83 →8 8a ,9a1030 →5 10a .9(3D.14)In the high-temperature limit x → 0, the Feynman integrals h̄β(1/ω)V −1 a2LV with L lines and Vvertices diverge like β V (1/β)L . The first V factors are due to the V -integrals over τ , the secondare the consequence of the product of n/2 factors a2 . Thus, a2LV behaves for x → 0 likea2LV∝h̄MωLxV −1−L .(3D.15)Indeed, the x-dependent factors α2LV in (3D.3)–(3D.11) grow likeα2α42α63α82α103≈ 1/x + x/12 + .
. . ,≈ 1/x + x3 /720,≈ 1/x + x5 /30240 + . . . ,≈ 1/x3 + x/120 − x3 /3780 + x5 /80640 + . . . ,≈ 1/x3 + x/240 − x3 /15120 + x7 /6652800 + . . . ,H. Kleinert, PATH INTEGRALSAppendix 3D Feynman Integrals for T 6= 0α123α62α83α1030≈≈≈≈3651/x4 + 1/240 + x2 /15120 − x6 /4989600 + 701 x8 /34871316480 + . . .
,1/x2 + x2 /240 − x4 /6048 + . . . ,1/x2 + x2 /720 − x6 /518400 + . . . ,1/x3 + x/360 − x5 /1209600 + 629 x9 /261534873600 + . . . .(3D.16)For the temperature behavior of these Feynman integrals see Fig. 3.16. We have plotted thereduced Feynman integrals â2LV (x) in which the low-temperature behaviors (3.550) and (3D.14)have been divided out of a2L.V1.2a210.80.60.40.2â2LVL/x0.10.20.30.50.42LFigure 3.16 Plot of reduced Feynman integrals âV (x) as a function of L/x = LkB T /h̄ω.The integrals (3D.4)–(3D.11) are indicated by decreasing dash-lengths.The integrals (3D.4) and (3D.5) for a42 and a63 can be obtained from the integral (3D.3) for a2by the operationnnh̄n∂1 ∂h̄n− 2−=,(3D.17)n!M n∂ωn!M n2ω ∂ωwith n = 1 and n = 2, respectively.
This follows immediately from the fact that the Green function0G(2)ω (τ, τ ) =∞X0h̄1,e−iωm (τ −τ ) 2h̄M β m=−∞ωm + ω 2(3D.18)with ω 2 shifted to ω 2 + δω 2 can be expanded into a geometric series∞X0h̄1δω 2h̄2(2)−iω(τ−τ)0mpace−1.9cmG√ω2 +δω2 (τ, τ ) =e−2 + ω22 + ω 2 )2h̄M β m=−∞ωmh̄ (ωm# 2 2h̄3δω+.
. . ,(3D.19)pace1.0cm +2h̄(ωm + ω 2 )3which corresponds to a series of convoluted τ -integralsZM δω 2 h̄β(2)(2)00G√ω2 +δω2 (τ, τ 0 ) = G(2)dτ1 G(2)(τ,τ)−ω (τ, τ1 )Gω (τ1 , τ )ωh̄02 Z h̄β Z h̄βM δω 2(2)(2)0+dτ1 dτ2 G(2)ω (τ, τ1 )Gω (τ1 , τ2 )Gω (τ2 , τ ) + . . . .h̄00(3D.20)In the diagrammatic representation, the derivatives (3D.17) insert n points into a line. In quantumfield theory, this operation is called a mass insertion.
Similarly, the Feynman integral (3D.7) isobtained from (3D.6) via a differentiation (3D.17) with n = 1 [see the corresponding diagrams in(3.547)]. A factor 4 must be removed, since the differentiation inserts a point into each of the four3663 External Sources, Correlations, and Perturbation Theorylines which are indistinguishable. Note that from these rules, we obtain directly the relations 1, 2,and 4 of (3.550).Note that the same type of expansion allows us to derive the three integrals from the one-loopdiagram (3.546). After inserting (3D.20) into (3.546) and re-expanding the logarithm we find theseries of Feynman integralsω 2 +δω 2−−−→M δω 2+h̄−M δω 2h̄212+M δω 2h̄313−... ,from which the integrals (3D.3)–(3D.5) can be extracted. As an example, consider the Feynmanintegral1= h̄β a42 .ωIt is obtained from the second-order Taylor expansion term of the tracelog as follows:21h̄2∂1[−2βVω ].(3D.21)− h̄β a41 =2 ω2!M 2 ∂ω 2A straightforward calculation, on the other hand, yields once more a42 of Eq.
(3D.5).Notes and ReferencesThe theory of generating functionals in quantum field theory is elaborated byJ. Rzewuski, Field Theory, Hafner, New York, 1969.For the usual operator derivation of the Wick expansion, seeS.S. Schweber, An Introduction to Relativistic Quantum Field Theory, Harper and Row, New York,1962, p. 435.The derivation of the recursion relation in Fig. 3.10 was given inH. Kleinert, Fortschr. Physik. 30, 187 (1986) (http://www.physik.fu-berlin.de/~kleinert/82), Fortschr. Physik. 30, 351 (1986) (ibid.http/84).See in particular Eqs. (51)–(61).Its efficient graphical evaluation is given inH.
Kleinert, A. Pelster, B. Kastening, M. Bachmann, Recursive Graphical Construction of Feynman Diagrams and Their Multiplicities in x4 - and in x2 A-Theory, Phys. Rev. D 61, 085017(2000) (hep-th/9907044).This paper develops a Mathematica program for a fast calculation of diagrams beyond five loops,which can be downloaded from the internet at ibid.http/b3/programs.The Mathematica program solving the Bender-Wu-like recursion relations for the general anharmonic potential (3C.26) is found in the same directory. This program was written in collaborationwith W. Janke.The path integral calculation of the effective action in Section 3.23 can be found inR. Jackiw, Phys. Rev. D 9, 1686 (1974).See alsoC.
De Dominicis, J. Math. Phys. 3, 983 (1962),C. De Dominicis and P.C. Martin, ibid. 5, 16, 31 (1964),B.S. DeWitt, in Dynamical Theory of Groups and Fields, Gordon and Breach, N.Y., 1965,A.N. Vassiliev and A.K. Kazanskii, Teor. Math. Phys. 12, 875 (1972),J.M. Cornwall, R. Jackiw, and E. Tomboulis, Phys. Rev. D 10, 1428 (1974),and the above papers by the author in Fortschr.
Physik 30.H. Kleinert, PATH INTEGRALSNotes and References367The path integral of a particle in a dissipative medium is discussed in A.O. Caldeira and A.J.Leggett, Ann. Phys. 149, 374 (1983), 153; 445(E) (1984).See alsoA.J. Leggett, Phys.
Rev. B 30, 1208 (1984);A.I. Larkin, and Y.N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 86, 719 (1984) [Sov. Phys. JETP 59,420 (1984)]; J. Stat. Phys. 41, 425 (1985);H. Grabert and U. Weiss, Z. Phys. B 56, 171 (1984);L.-D. Chang and S. Chakravarty, Phys. Rev. B 29, 130 (1984);D. Waxman and A.J. Leggett, Phys. Rev. B 32, 4450 (1985);P. Hänggi, H. Grabert, G.-L. Ingold, and U. Weiss, Phys. Rev. Lett. 55, 761 (1985);D.
Esteve, M.H. Devoret, and J.M. Martinis, Phys. Rev. B 34, 158 (1986);E. Freidkin, P. Riseborough, and P. Hänggi, Phys. Rev. B 34, 1952 (1986);H. Grabert, P. Olschowski and U. Weiss, Phys. Rev. B 36, 1931 (1987),and in the textbookU.
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