Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 62
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By expanding the exponential prefactor as in Eq. (3.490), the full generating functional is obtained from the harmonicone without any further path integration. Only functional differentiations are required to find the generating functional of all interacting interacting correlationfunctions Z[j] from the harmonic one Zω [j].This procedure yields the perturbative definition of arbitrary path integrals.It is widely used in the quantum field theory of particle physics15 and criticalphenomena16 It is also the basis for an important extension of the theory of distributions to be discussed in detail in Sections 10.6–10.11.It must be realized, however, that the perturbative definition is not a completedefinition. Important contributions to the path integral may be missing: all those15C.
Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill (1985).H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4 -Theories, World Scientific,Singapore 2001, pp. 1–487 (www.physik.fu-berlin.de/~kleinert/re.html#b8).16H. Kleinert, PATH INTEGRALS2873.22 Generating Functional of Connected Correlation Functionswhich are not expandable in powers of the interaction strength g.
Such contributionsare essential in understanding many physical phenomena, for example, tunneling, tobe discussed in Chapter 17. Interestingly, however, information on such phenomenacan, with appropriate resummation techniques to be developed in Chapter 5, alsobe extracted from the large-order behavior of the perturbation expansions, as willbe shown in Subsection 17.10.4.3.22Generating Functional of Connected CorrelationFunctionsIn Section 3.10 we have seen that the correlation functions obtained from the functional derivatives of Z[j] via relation (3.295) contain many disconnected parts. Thephysically relevant free energy F [j] = −kB T log Z[j], on the other hand, containsonly in the connected parts of Z[j].
In fact, from statistical mechanics we know thatmeaningful description of a very large thermodynamic system can only be given interms of the free energy which is directly proportional to the total volume V . Thepartition function Z = e−F/kB T has no meaningful infinite-volume limit, also calledthe thermodynamic limit, since it contains a power series in V .
Only the free energy density f ≡ F/V has an infinite-volume limit. The expansion of Z[j] divergestherefore for V → ∞. This is why in thermodynamics we always go over to the freeenergy density by taking the logarithm of the partition function. This is calculatedentirely from the connected diagrams.Due to this thermodynamic experience we expect the logarithm of Z[j] to provideus with a generating functional for all connected correlation functions. To avoidfactors kB T we define this functional asW [j] = log Z[j],(3.556)and shall now prove that the functional derivatives of W [j] produce precisely theconnected parts of the Feynman diagrams for each correlation function.Consider the connected correlation functions Gc(n) (τ1 , .
. . , τn ) defined by the functional derivativesδδG(n)···W [j] .(3.557)c (τ1 , . . . , τn ) =δj(τ1 )δj(τn )Ultimately, we shall be interested only in these functions with zero external current,where they reduce to the physically relevant connected correlation functions. For thegeneral development in this section, however, we shall consider them as functionalsof j(τ ), and set j = 0 only at the end.Of course, given all connected correlation functions G(n)c (τ1 , . . .
, τn ), the full(n)correlation functions G (τ1 , . . . , τn ) in Eq. (3.295) can be recovered via simplecomposition laws from the connected ones. In order to see this clearly, we shallderive the general relationship between the two types of correlation functions inSection 3.22.2. First, we shall prove the connectedness property of the derivatives(3.557).2883.22.13 External Sources, Correlations, and Perturbation TheoryConnectedness Structure of Correlation FunctionsWe first prove that the generating functional W [j] collects only connected diagramsin its Taylor coefficients δ n W/δj(τ1 ) .
. . δj(τn ). Later, after Eq. (3.585), we shall seethat these functional derivatives comprise all connected diagrams in G(n) (τ1 , . . . , τn ).Let us write the path integral for the generating functional Z[j] as follows (herewe use natural units with h̄ = 1):Z[j] =ZDx e−Ae [x,j]/h̄,(3.558)with the actionAe [x, j] =Zh̄β0M 22 2dτẋ + ω x + V (x) − j(τ )x(τ ) .2(3.559)In the following structural considerations we shall use natural physical units in whichh̄ = 1, for simplicity of the formulas.
By analogy with the integral identityZdxd −F (x)e= 0,dxwhich holds by partial integration for any function F (x) which goes to infinity forx → ±∞, the functional integral satisfies the identityZDxδe−Ae [x,j] = 0,δx(τ )(3.560)since the action Ae [x, j] goes to infinity for x → ±∞. Performing the functionalderivative, we obtainZDxδAe [x, j] −Ae [x,j]e= 0.δx(τ )(3.561)To be specific, let us consider the anharmonic oscillator with potential V (x) =λx4 /4!. We have chosen a coupling constant λ/4! instead of the previous g in (3.540)since this will lead to more systematic numeric factors. The functional derivative ofthe action yields the classical equation of motionλδAe [x, j]= M(−ẍ + ω 2x) + x3 − j = 0,δx(τ )3!(3.562)which we shall write asλ 3δAe [x, j]= G−1x − j = 0,0 x+δx(τ )3!(3.563)where we have set G0 (τ, τ 0 ) ≡ G(2) to get free space for upper indices.
With thisnotation, Eq. (3.561) becomesZDx(G−10 x(τ ))λ+ x3 (τ ) − j(τ ) e−Ae [x,j] = 0.3!(3.564)H. Kleinert, PATH INTEGRALS3.22 Generating Functional of Connected Correlation Functions289We now express the paths x(τ ) as functional derivatives with respect to the sourcecurrent j(τ ), such that we can pull the curly brackets in front of the integral. Thisleads to the functional differential equation for the generating functional Z[j]:"(3.565)δδδ···Z[j],δj(τ1 ) δj(τ2 )δj(τn )(3.566)δλδG−1+ 0 δj(τ )3! δj(τ )#3− j(τ ) Z[j] = 0.With the short-hand notationZj(τ1 )j(τ2 )...j(τn ) [j] ≡where the arguments of the currents will eventually be suppressed, this can bewritten asG−10 Zj(τ ) +λZ− j(τ ) = 0.3! j(τ )j(τ )j(τ )(3.567)Inserting here (3.556), we obtain a functional differential equation for W [j]:G−10 Wj +λWjjj + 3Wjj Wj + Wj3 − j = 0.3!(3.568)We have employed the same short-hand notation for the functional derivatives ofW [j] as in (3.566) for Z[j],Wj(τ1 )j(τ2 )...j(τn ) [j] ≡δδδ···W [j],δj(τ1 ) δj(τ2 )δj(τn )(3.569)suppressing the arguments τ1 , .
. . , τn of the currents, for brevity. Multiplying (3.568)functionally by G0 givesλWj = − G0 Wjjj + 3Wjj Wj + Wj3 + G0 j.3!(3.570)We have omitted the integralover the intermediate τ ’s, for brevity. More specifically,R0we have written G0 j for dτ G0 (τ, τ 0 )j(τ 0 ). Similar expressions abbreviate all functional products. This corresponds to a functional version of Einstein’s summationconvention.Equation (3.570) may now be expressed in terms of the one-point correlationfunctionG(1)c = Wj ,(3.571)defined in (3.557), asG(1)chiλ(1)(1)(1) 3+G+ G0 j.= − G0 Gc jj + 3Gc j G(1)cc3!(3.572)2903 External Sources, Correlations, and Perturbation TheoryThe solution to this equation is conveniently found by a diagrammatic proceduredisplayed in Fig.
3.8. To lowest, zeroth, order in λ we haveG(1)c = G0 j.(3.573)From this we find by functional integration the zeroth order generating functionalW0 [j]W0 [j] =Z1Dj G(1)c = jG0 j,2(3.574)up to a j-independent constant. Subscripts of W [j] indicate the order in the interaction strength λ.Reinserting (3.573) on the right-hand side of (3.572) gives the first-order expressionG(1)= −G0ciλh3G0 G0 j + (G0 j)3 + G0 j,3!(3.575)represented diagrammatically in the second line of Fig. 3.8. Equation (3.575) can beintegrated functionally in j to obtain W [j] up to first order in λ.
Diagrammatically,this process amounts to multiplying each open lines in a diagram by a current j,and dividing the arising j n s by n. Thus we arrive atW0 [j] + W1 [j] =1λλjG0 j − G0 (G0 j)2 −(G0 j)4 ,2424(3.576)Figure 3.8 Diagrammatic solution of recursion relation (3.570) for the generating functional W [j] of all connected correlation functions. First line represents Eq. (3.572), second(3.575), third (3.576).
The remaining lines define the diagrammatic symbols.H. Kleinert, PATH INTEGRALS2913.22 Generating Functional of Connected Correlation Functionsas illustrated in the third line of Fig. 3.8. This procedure can be continued to anyorder in λ.The same procedure allows us to prove that the generating functional W [j] collects only connected diagrams in its Taylor coefficients δ n W/δj(x1 ) . .
. δj(xn ). Forthe lowest two orders we can verify the connectedness by inspecting the third line inFig. 3.8. The diagrammatic form of the recursion relation shows that this topologicalproperty remains true for all orders in λ, by induction. Indeed, if we suppose it tobe true for some n, then all G(1)c inserted on the right-hand side are connected, andso are the diagrams constructed from these when forming G(1)c to the next, (n + 1)st,order.Note that this calculation is unable to recover the value of W [j] at j = 0 whichis an unknown integration constant of the functional differential equation.
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