Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 61
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The second-order result (3.515), for example, follows directly from (3.537) and(3.538), the latter giving(n)E2=Xk6=n(n)ak,1 hn|V̂ |ki =X hk|V̂ |nihn|V̂ |ki(k)k6=nE0(n)− E0.(3.539)If the potential V̂ = V (x̂) is a polynomial in x̂, its matrix elements hn|V̂ |ki are nonzero only for nin a finite neighborhood of k, and the recursion relations consist of finite sums which can be solvedexactly.3.20Calculation of Perturbation Series via FeynmanDiagramsThe expectation values in formula (3.484) can be evaluated also in another waywhich can be applied to all potentials which are simple polynomials pf x. Then the2823 External Sources, Correlations, and Perturbation Theorypartition function can be expanded into a sum of integrals associated with certainFeynman diagrams.The procedure is rooted in the Wick expansion of correlation functions in Section 3.10.
To be specific, we assume the anharmonic potential to have the formgV (x) = x4 .4(3.540)The graphical expansion terms to be found will be typical for all so-called ϕ4 theories of quantum field theory.To calculate the free energy shift (3.484) to first order in g, we have to evaluatethe harmonic expectation of Aint,e . This is written ashAint,e iω =g4h̄βZ0dτ hx4 (τ )iω .(3.541)The integrand contains the correlation function(4)hx(τ1 )x(τ2 )x(τ3 )x(τ4 )iω = Gω2 (τ1 , τ2 , τ3 , τ4 )at identical time arguments. According to the Wick rule (3.302), this can be expanded into the sum of three pair terms(2)(2)(2)(2)(2)(2)Gω2 (τ1 , τ2 )Gω2 (τ3 , τ4 ) + Gω2 (τ1 , τ3 )Gω2 (τ2 , τ4 ) + Gω2 (τ1 , τ4 )Gω2 (τ2 , τ3 ),(2)where Gω2 (τ, τ 0 ) are the periodic Euclidean Green functions of the harmonic oscillator [see (3.301) and (3.248)].
The expectation (3.541) is therefore equal to theintegralZg h̄β(2)hAint,e iω = 3dτ Gω2 (τ, τ )2 .(3.542)4 0The right-hand side is pictured by the Feynman diagram.3Because of its shape this is called a two-loop diagram. In general, a Feynman diagram consists of lines meeting at points called vertices. A line connecting two points(2)represents the Green function Gω2 (τ1 , τ2 ). A vertex indicates a factor g/4h̄ and avariable τ to be integrated over the interval (0, h̄β). The present simple diagram hasonly one point, and the τ -arguments of the Green functions coincide. The numberunderneath counts how often the integral occurs.
It is called the multiplicity of thediagram.To second order in V (x), the harmonic expectation to be evaluated ishA2int,e iω 2 Zg=40h̄βdτ2Zh̄β0dτ1 hx4 (τ2 )x4 (τ1 )iω .(3.543)H. Kleinert, PATH INTEGRALS2833.20 Calculation of Perturbation Series via Feynman Diagrams(8)The integral now contains the correlation function Gω2 (τ1 , . . . , τ8 ) with eight timearguments. According to the Wick rule, it decomposes into a sum of 7!! = 105(2)products of four Green functions Gω2 (τ, τ 0 ). Due to the coincidence of the time arguments, there are only three different types of contributions to the integral (3.543):hA2int,e iω 2 Zg=4h̄β0dτ2Zh̄β0(2)h(2)(2)dτ1 72Gω2 (τ2 , τ2 )Gω2 (τ2 , τ1 )2 Gω2 (τ1 , τ1 )(2)(2)(2)i+24Gω2 (τ2 , τ1 )4 + 9Gω2 (τ2 , τ2 )2 Gω2 (τ1 , τ1 )2 .(3.544)The integrals are pictured by the following Feynman diagrams composed of threeloops:.72249They contain two vertices indicating two integration variables τ1 , τ2 .
The first twodiagrams with the shape of three bubbles in a chain and of a watermelon, respectively, are connected diagrams, the third is disconnected . When going over to thecumulant hA2int,e iω,c , the disconnected diagram is eliminated.To higher orders, the counting becomes increasingly tedious and it is worthdeveloping computer-algebraic techniques for this purpose.
Figure 3.7 shows thediagrams for the free-energy shift up to four loops. The cumulants eliminate preciselyall disconnected diagrams. This diagram-rearranging property of the logarithm isvery general and happens to every order in g, as can be shown with the help offunctional differential equations.βF = βFω +31+3!1−2!!+7224+2592+1728!+3456+ ...1728Figure 3.7 Perturbation expansion of free energy up to order g3 (four loops).The lowest-order term βFω containing the free energy of the harmonic oscillator[recall Eqs. (3.242) and (2.519)]1βh̄ωFω = log 2 sinhβ2!(3.545)is often represented by the one-loop diagram11(2)βFω = − Tr log Gω2 = −22h̄βZ0h̄βh(2)idτ log Gω2 (τ, τ ) = −12.(3.546)2843 External Sources, Correlations, and Perturbation TheoryWith it, the graphical expansion in Fig. 3.7 starts more systematically with oneloop rather than two. The systematics is, however, not perfect since the line in theone-loop diagram does not show that integrand contains a logarithm.
In addition,the line is not connected to any vertex.All τ -variables in the diagrams are integrated out. The diagrams have no openlines and are called vacuum diagrams.The calculation of the diagrams in Fig. 3.7 is simplified with the help of a factorization property: If a diagram consists of two subdiagrams touching each otherat a single vertex, its Feynman integral factorizes into those of the subdiagrams.Thanks to this property, we only have to evaluate the following integrals (omittingthe factors g/4h̄ for each vertex)=Zh̄β(2)dτ Gω2 (τ, τ ) = h̄βa2 ,0=Zh̄βZ h̄β=Zh̄βZ0000≡ h̄β1(2)dτ1 dτ2 Gω2 (τ1 , τ2 )2 ≡ h̄β a42 ,ωh̄βZ h̄β1ω02=Zh̄βZ h̄βZ h̄β00≡ h̄β=Z01ω020(2)(2)(2)dτ1 dτ2 dτ3 Gω2 (τ1 , τ2 )Gω2 (τ2 , τ3 )Gω2 (τ3 , τ1 )3a103 ,h̄βZ h̄βZ h̄β0(2)1(2)dτ1 dτ2 Gω2 (τ1 , τ2 )4 ≡ h̄β a82 ,ω=h̄βZ h̄β0(2)a63 ,Z0(2)dτ1 dτ2 dτ3 Gω2 (τ1 , τ2 )Gω2 (τ2 , τ3 )Gω2 (τ3 , τ1 )(2)(2)(2)dτ1 dτ2 dτ3 Gω2 (τ1 , τ2 )2 Gω2 (τ2 , τ3 )2 Gω2 (τ3 , τ1 )221a12(3.547)3 .ωNote that in each expression, the last τ -integral yields an overall factor h̄β, due tothe translational invariance along the τ -axis.
The others give rise to a factor 1/ω,for dimensional reasons. The temperature-dependent quantities a2LV are labeled bythe number of vertices V and lines L of the associated diagrams. Their dimensionis length to the nth power [corresponding to the dimension of the n x(τ )-variablesin the diagram]. For more than four loops, there can be more than one diagram foreach V and L, such that one needs an additional label in a2LV to specify the diagramuniquely.
Each a2Lmaybewrittenasaproductofthebasiclength scale (h̄/Mω)LVmultiplied by a function of the dimensionless variable x ≡ βh̄ω:≡ h̄βa2LV=h̄Mω!LαV2L (x).(3.548)H. Kleinert, PATH INTEGRALS2853.20 Calculation of Perturbation Series via Feynman DiagramsThe functions αV2L (x) are listed in Appendix 3D.As an example for the application of the factorization property, take the Feynman integral of the second third-order diagram in Fig.
3.7 (called a “daisy” diagrambecause of its shape):=Z0h̄βZ h̄βZ h̄β00(2)(2)(2)dτ1 dτ2 dτ3 Gω2 (τ1 , τ2 )Gω2 (τ2 , τ3 )Gω2 (τ3 , τ1 )(2)(2)(2)× Gω2 (τ1 , τ1 )Gω2 (τ2 , τ2 )Gω2 (τ3 , τ3 ).It decomposes into a product between the third integral in (3.547) and three powersof the first integral:→×3.Thus we can immediately write1= h̄βω2a63 (a2 )3 .In terms of a2LV , the free energy becomes1g 2 2 4 2g 43a −72a a2 a + 24a82(3.549)42!h̄ω 4 3 hig12122592a2 (a42 )2 a2 + 1728a63 (a2 )3 + 3456a10a+1728a+ ...
.+ 2 2333!h̄ ω 4 F = Fω +In the limit T → 0, the integrals (3.547) behave likea42→ a4 ,a63a82→1 8a,23 12→a ,8a103a1233 6a,25 10→a ,8→(3.550)and the free energy reduces toF = 2gh̄ω g 4+ 3a −24442a8 3g1+h̄ω44 · 333a121h̄ω2+ ... .(3.551)In this limit, it is simpler to calculate the integrals (3.547) directly with the zero0(2)temperature limit of the Green function (3.301), which is Gω2 (τ, τ 0 ) = a2 e−ω|τ −τ |with a2 = h̄/2ωM [see (3.248)].
The limits of integration must, however, be shiftedR h̄β/2R∞dτby half a period to −h̄β/2 dτ before going to the limit, so that one evaluates −∞R∞rather than 0 dτ (the latter would give the wrong limit since it misses the left-handside of the peak at τ = 0). Before integration, the integrals are conveniently splitas in Eq. (3D.1).2863.213 External Sources, Correlations, and Perturbation TheoryPerturbative Definition of Interacting Path IntegralsIn Section 2.15 we have seen that it is possible to define a harmonic path integralwithout time slicing by dimensional regularization. With the techniques developedso far, this definition can trivially be extended to path integrals with interactions, ifthese can be treated perturbatively. We recall that in Eq. (3.480), the partition function of an interacting system can be expanded in a series of harmonic expectationvalues of powers of the interaction.
The procedure is formulated most convenientlyin terms of the generating functional (3.486) using formula (3.487) for the generating functional with interactions and Eq. (3.488) for the associated partition. Theharmonic generating functional on the right-hand side of (3.487),Zω [j] =I(1Dx exp −h̄Z0h̄β)M 2dτ(ẋ + ω 2 x2 ) − jx2,(3.552)can be evaluated with analytic regularization as described in Section (2.15) andyields, after a quadratic completion [recall (3.243), (3.244)]:)(1 Z h̄β Z h̄β 01dτ j(τ )Gpω2 ,e (τ − τ 0 )j(τ 0 ) , (3.553)dτZω [j] =exp2 sin(ωh̄β/2)2Mh̄ 00where Gpω2 ,e (τ ) is the periodic Green function (3.248)Gpω2 ,e (τ ) =1 cosh ω(τ − h̄β/2),2ω sinh(βh̄ω/2)τ ∈ [0, h̄β].(3.554)As a consequence, Formula (3.487) for the generating functional of an interactingtheory1Z[j] = e− h̄R h̄β0dτ V (h̄δ/δj(τ ))Zω [j] ,(3.555)is completely defined by analytic regularization.
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