Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 64
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3.9. The zeroth-order solutionG0 WG0 = 8 1−1 4λG0 WG0 G0 + 2G0 λG20 WG0 + WG0 G20 λG20 WG0 +4!2Figure 3.9 Diagrammatic representation of functional differential equation (3.603). Forthe purpose of finding the multiplicities of the diagrams, it is convenient to represent hereby a vertex the coupling strength −λ/4! rather than g/4 in Section 3.20.2963 External Sources, Correlations, and Perturbation Theoryto this equation is obtained by setting λ = 0:1(3.604)W (0) [G0 ] = Tr log(G0 ).2Explicitly, the right-hand side is equal to the one-loop contribution to the free energyin Eq.
(3.546), apart from a factor −β.The corrections are found by iteration. For systematic treatment, we write W [G0 ]as a sum of a free and an interacting part,W [G0 ] = W (0) [G0 ] + W int [G0 ],(3.605)insert this into Eq. (3.603), and find the differential equation for the interactingpart:−λ 2λG0 WGint0 + (G40 WGint0 G0 + 3G30 WGint0 + G40 WGint0 WGint0 ) = 6G.34! 0(3.606)This equation is solved iteratively.
Setting W int [G0 ] = 0 in all terms proportionalto λ, we obtain the first-order contribution to W int [G0 ]:−λ 2G.(3.607)4! 0This is precisely the contribution (3.542) of the two-loop Feynman diagram (apartfrom the different normalization of g).In order to see how the iteration of Eq. (3.606) may be solved systematically,let us ignore for the moment the functional nature of Eq. (3.606), and treat G0 asan ordinary real variable rather than a functional matrix. We expand W [G0 ] in aTaylor series:!p∞X1−λintW [G0 ] =Wp(G0 )2p ,(3.608)p!4!p=1W int [G0 ] = 3and find for the expansion coefficients the recursion relationWp+1 = 4 [2p (2p − 1) + 3(2p)] Wp +p−1Xq=1pq!2q Wq × 2(p − q)Wp−q .
(3.609)Solving this with the initial number W1 = 3, we obtain the multiplicities of theconnected vacuum diagrams of pth order:3, 96, 9504, 1880064, 616108032, 301093355520, 205062331760640,185587468924354560, 215430701800551874560, 312052349085504377978880.(3.610)To check these numbers, we go over to Z[G] = eW [G0 ] , and find the expansion:∞X1−λ1WpZ[G0 ] = exp Tr log G0 +24!p=1 p!∞X−λ1zp= Det1/2 [G0 ] 1 +4!p=1 p!!p!p(G0 )2p (G0 )2p .(3.611)H. Kleinert, PATH INTEGRALS2973.22 Generating Functional of Connected Correlation FunctionsThe expansion coefficients zp count the total number of vacuum diagrams of orderp.
The exponentiation (3.611) yields zp = (4p − 1)!!, which is the correct number ofWick contractions of p interactions x4 .In fact, by comparing coefficients in the two expansions in (3.611), we may deriveanother recursion relation for Wp :!!p−1p−1p−1Wp + 3Wp−1 + 7·5 ·3+ . . . + (4p −5)!!12p−1!= (4p −1)!!, (3.612)which is fulfilled by the solutions of (3.609).In order to find the associated Feynman diagrams, we must perform the differentiations in Eq. (3.606) functionally.
The numbers Wp become then a sum ofdiagrams, for which the recursion relation (3.609) readsWp+1=4G40p−1Xd2d3W+3·GW+p0pd∩d ∩2q=1pq!!!dd,Wq G20 · G20Wd∩d ∩ p−q(3.613)where the differentiation d/d∩ removes one line connecting two vertices in all possibleways. This equation is solved diagrammatically, as shown in Fig. 3.10.Wp+1 = 4"G40#p−1Xd2dddp223Wp + 3 · G0Wp +Wq G0 · G0Wp−qqd ∩2d∩d∩d∩q=1Figure 3.10 Diagrammatic representation of recursion relation (3.609).
A vertex represents the coupling strength −λ.Starting the iteration with W1 = 3, we have dWp /d ∩ = 6and2d Wp /d ∩ = 6. Proceeding to order five loops and going back to the usual vertex notation −λ, we find the vacuum diagrams with their weight factors as shownin Fig. 3.11. For more than five loops, the reader is referred to the paper quotedin Notes and References, and to the internet address from which Mathematica programs can be downloaded which solve the recursion relations and plot all diagramsof W [0] and the resulting two- and four-point functions.23.22.4Correlation Functions from Vacuum DiagramsThe vacuum diagrams contain information on all correlation functions of the theory.One may rightly say that the vacuum is the world.
The two- and four-point functionsare given by the functional derivatives (3.592) of the vacuum functional W [K].Diagrammatically, a derivative with respect to K corresponds to cutting one line ofa vacuum diagram in all possible ways. Thus, all diagrams of the two-point function2983 External Sources, Correlations, and Perturbation Theorydiagrams and multiplicitiesg1g12!2g3g3413!ggqq + 7224 q lgqq mq+ 3456 q l1728 Tqq + 2592gggq q!gqggggq q q+ 1728 gq gq g!i gqgq gq qlqqq qg1q qmqggqqqqqlq62208 q q + 66296 q gq + 248832 q + 497664 T+ 165888 + 248832 qlq 4!gg!qggqqqgqggggggqqqqgggqqgqg+ 248832165888 q lqg + 62208q + 124416qgFigure 3.11Vacuum diagrams up to five loops and their multiplicities.
Thetotal numbers to orders gn are 3, 96, 9504, 1880064, respectively. In contrast toFig. 3.10, and to the previous diagrammatic notation in Fig. 3.7, a vertex standshere for −λ/4! for brevity. For more than five loops see the tables on the internet(http://www.physik.fu-berlin/~kleinert/b3/programs).G(2) can be derived from such cuts, multiplied by a factor 2. As an example, considerthe first-order vacuum diagram of W [K] in Fig.
3.11. Cutting one line, which ispossible in two ways, and recalling that in Fig. 3.11 a vertex stands for −λ/4! ratherthan −λ, as in the other diagrams, we find11(2)W1 [0] =−−−→ G1 (τ1 , τ2 ) = 2 × 2.(3.614)88The second equation in (3.592) tells us that all connected contributions to thefour-point function G(4) may be obtained by cutting two lines in all combinations,and multiplying the result by a factor 4. As an example, take the second-ordervacuum diagrams of W [0] with the proper translation of vertices by a factor 4!,which are11W2 [0] =+.(3.615)1648Cutting two lines in all possible ways yields the following contributions to the connected diagrams of the two-point function:11+4·3·G(4) = 4 × 2 · 1 ·.(3.616)1648It is also possible to find all diagrams of the four-point function from the vacuumdiagrams by forming a derivative of W [0] with respect to the coupling constant −λ,H.
Kleinert, PATH INTEGRALS2993.22 Generating Functional of Connected Correlation Functionsand multiplying the result by a factor 4!. This follows directly fromthe fact thatRthis differentiation applied to Z[0] yields the correlation function dτ hx4 i. As anexample, take the first diagram of order g 3 in Table 3.11 (with the same vertexconvention as in Fig. 3.11):1W2 [0] =.(3.617)48Removing one vertex in the three possible ways and multiplying by a factor 4! yieldsG(4) = 4! ×3.22.51348.(3.618)Generating Functional for Vertex Functions.Effective ActionApart from the connectedness structure, the most important step in economizing thecalculation of Feynman diagrams consists in the decomposition of higher connectedcorrelation functions into one-particle irreducible vertex functions and one-particleirreducible two-particle correlation functions, from which the full amplitudes caneasily be reconstructed.
A diagram is called one-particle irreducible if it cannot bedecomposed into two disconnected pieces by cutting a single line.There is, in fact, a simple algorithm which supplies us in general with sucha decomposition. For this purpose let us introduce a new generating functionalΓ[X], to be called the effective action of the theory. It is defined via a Legendretransformation of W [j]:−Γ[X] ≡ W [j] − Wj j.(3.619)Here and in theRfollowing, we use a short-hand notation for the functional multiplication, Wj j = dτ Wj (τ )j(τ ), which considers fields as vectors with a continuousindex τ .
The new variable X is the functional derivative of W [j] with respect toj(τ ) [recall (3.569)]:X(τ ) ≡δW [j]≡ Wj(τ ) = hxij(τ ) ,δj(τ )(3.620)and thus gives the ground state expectation of the field operator in the presence ofthe current j. When rewriting (3.619) as−Γ[X] ≡ W [j] − X j,(3.621)and functionally differentiating this with respect to X, we obtain the equationΓX [X] = j.(3.622)This equation shows that the physical path expectation X(τ ) = hx(τ )i, where theexternal current is zero, extremizes the effective action:ΓX [X] = 0.(3.623)3003 External Sources, Correlations, and Perturbation TheoryWe shall study here only physical systems for which the path expectation value isa constant X(τ ) ≡ X0 .
Thus we shall not consider systems which possess a timedependent X0 (τ ), although such systems can also be described by x4 -theories byadmitting more general types of gradient terms, for instance x(∂ 2 − k02 )2 x. The ensuing τ -dependence of X0 (τ ) may be oscillatory.17 Thus we shall assume a constantX0 = hxi|j=0,(3.624)which may be zero or non-zero, depending on the phase of the system.Let us now demonstrate that the effective action contains all the informationon the proper vertex functions of the theory. These can be found directly from thefunctional derivatives:δδΓ(n) (τ1 , . .
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