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Functional-Integral Calculation of Effective Action. Loop Expansion

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Functional-Integral Calculation of Effective Action. Loop Expansion It is simplicity that makes the uneducated more effective than the educated. Aristoteles (384 BC–322 BC) 22 Functional-Integral Calculation of Effective Action. Loop Expansion The functional-integral formula for Z[j] can be employed for a rather direct evalua- tion of the effective action of a theory. Diagrammatically, the result will be the loop expansion organized by powers of the Planck quantumh ¯ [1]. 22.1 General Formalism Consider the generating functional of all Green functions Z[j]= eiW [j]/¯h, (22.1) where W [j] is the generating functional of all connected Green functions. The vac- uum expectation of the field, the average Φ(x) φ(x) , (22.2) ≡h i is given by the first functional derivative Φ(x)= δW [j]/δj(x). (22.3) This can be inverted to yield j(x) as an x-dependent functional of Φ(x): j(x)= j[Φ](x), (22.4) which may be used to form the Legendre transform of W [j]: Γ[Φ] W [j] d4xj(x)Φ(x), (22.5) ≡ − Z where j(x) on the right-hand side is replaced by (22.4). The result is the effective action of the theory. The first functional derivative of the effective action gives back the current δΓ[Φ] = j(x). (22.6) δΦ(x) − 1253 1254 22 Functional-Integral Calculation of Effective Action. Loop Expansion This equation justifies the name effective action for the functional Γ[Φ]. In the ab- sence of an external current j(x), the effective action Γ[Φ] is extremal on the physical field expectation value (22.2). Thus it plays exactly the same role for the fully in- teracting quantum theory as the action does for the classical field configurations. The Legendre transformation (22.5) can be inverted to recover the generating functional W [j] from the effective action: W [j] = Γ[Φ] + d4xj(x)Φ(x). (22.7) Z Let us compare this with the functional-integral formula for the generating functional Z[j]: 4 φ(x)e(i/¯h) A[φ]+ d xj(x)φ(x) Z[j]= D { } . (22.8) φ(x)e(i/¯hR)A0[φ] R D Using (22.1) and (22.5), this impliesR a functional-integral formula for the effective action Γ[Φ]: i Γ[Φ]+ d4xj(x)Φ(x) (i/¯h) A[φ]+ d4xj(x)φ(x) e h¯ = φ(x)e , (22.9) { } N D { } R Z R where is a normalization factor: N −1 = φ(x)e(i/¯h)A0[φ] . (22.10) N Z D We have explicitly displayed the fundamental quantum of actionh ¯, which is a mea- sure for the size of quantum fluctuations. There are many physical systems for which quantum fluctuations are rather small, except in the immediate vicinity of certain critical points. For these systems, it is desirable to develop a method of solving (22.9) for Γ[Φ] as a series expansion in powers ofh ¯, taking into account successively increasing quantum fluctuations. In the limith ¯ 0, the path integral over the field φ(x) in (22.8) is dominated → by the classical solution φcl(x) which extremizes the exponent δ A = j(x). (22.11) ∂φ − φ=φcl(x) At this level we therefore identify 4 4 W [j] = Γ[Φ] + d xj(x)Φ(x) = [φcl]+ d xj(x)φcl(x). (22.12) Z A Z Of course, φcl(x) is a functional of j(x), so that we may write it more explicitly as φcl[j](x). By differentiating W [j] with respect to j, we have from Eqs. (22.3) and (22.7): δW δΓ δΦ δΦ Φ(x)= = +Φ+ j . (22.13) δj δΦ δj δj 22.1 General Formalism 1255 Inserting the classical field equation (22.11), this becomes δ δφcl δφcl Φ(x)= A + φcl + j = φcl. (22.14) δφcl δj δj Thus, to this approximation, Φ(x) coincides with the classical field φcl(x). For the φ4-theory with O(N)-symmetry, the action is 2 4 1 2 m 2 g 2 2 [φ]= d x (∂φa) φa φa . (22.15) A " 2 − 2 − 4! # Z Replacing the φ φ + δφ = Φ + δφ on the right-hand side of Eq. (22.12), we a → cl a s a therefore obtain the lowest-order result for the effective action (which is of zeroth order inh ¯): 2 4 1 2 m 2 g 2 2 Γ[Φ] [Φ] = d x (∂Φa) Φa Φa . (22.16) ≈A "2 − 2 − 4! # Z It equals the fundamental action, and is also referred to as the mean field approxi- mation to the effective action. We have shown in Chapter 13 that all vertex functions can be obtained from the derivatives of Γ[Φ] with respect to Φ at the equilibrium value of Φ for j = 0. We 2 1 begin by assuming that m > 0. Then Γ[Φ] has an extremum at Φa 0, and there (n) ≡ are only two types of non-vanishing vertex functions Γ (x1,...,xn): For n = 2, we obtain the lowest-order two-point function 2 2 (2) δ Γ δ Γ (x1, x2)ab = A ≡ δΦa(x1)δΦb(x2) φa(x1)φb(x2) Φa=0 φa=Φa=0 = ( ∂2 m2)δ δ(4) (x x ). (22.17) − − ab 1 − 2 This determines the inverse of the propagator: (2) −1 Γab (x1, x2) = [ihG¯ ]ab(x1, x2). (22.18) Thus we find in this zeroth-order approximation that Gab(x1, x2) is equal to the free propagator: Gab(x1, x2)= G0ab(x1, x2). (22.19) For n = 4, we find the lowest-order four-point function 4 (4) δ Γ Γabcd(x1, x2, x3, x4) = gTabcd, (22.20) ≡ δΦa(x1)δΦb(x2)δΦc(x3)δΦd(x4) where 1 T = (δ δ + δ δ + δ δ ) (22.21) abcd 3 ab cd ac bd ad bc 1Also the case m2 < 0 corresponds to a physical state in a different phase and will be discussed separately. In fact, the phase transition is the reason for the nonexistence of tachyon particles, i.e., of particles that move faster than the speed of light. See the pages 1123, 1294 and 1311. 1256 22 Functional-Integral Calculation of Effective Action. Loop Expansion is the fundamental vertex [compare (22.16)] for the local interaction of strength g. The effective action has the virtue that all diagrams of the theory can be ob- tained from tree diagrams built from propagators Gab(x1, x2) and vertex functions. This was explained in Chapter 13. In lowest approximation, we recover precisely the subset of all original Feynman diagrams with a tree-like topology. These are the diagrams which do not involve any loop integration. Since the limith ¯ 0 corresponds to the classical equations of motion with no quantum fluctuations,→ we conclude: Classical field theory corresponds to tree diagrams. The use of the initial action as an approximation to the effective action neglecting fluctuations is often referred to as mean-field theory. 22.2 Quadratic Fluctuations In order to find theh ¯-correction to this approximation we expand the action in powers of the fluctuations of the field around the classical solution δφ(x) φ(x) φ (x), (22.22) ≡ − cl and perform a perturbation expansion. The quadratic term in δφ(x) is taken to be the free-field action, the higher powers in δφ(x) are the interactions. Up to second order in the fluctuations δφ(x), the action is expanded as follows: 4 [φcl + δφ]+ d xj(x)[φcl(x)+ δφ(x)] A Z 4 4 δ = [φcl]+ d xj(x)φcl(x)+ d x j(x)+ A δφ(x) A ( δφ(x) Z Z φ=φcl δ2 + d4xd4y δφ(x) A δφ(y)+ (δφ) 3 . (22.23) δφ(x)δφ(y) O Z φ=φcl The curly bracket term linear in the variation δφ vanishes due to the extremality property of the classical field φcl expressed by the field equation (22.11). Inserting this expansion into (22.9), we obtain the approximate expression 2 4 i δ Z[j] e(i/¯h) A[φcl]+ d xjφcl δφ exp d4xd4y δφ(x) A δφ(y) . ≈N { } D h¯ δφ(x)δφ(y) R Z Z φ=φcl (22.24) We now observe that the fluctuations in δφ will be of average size √h¯ due to theh ¯-denominator in the Fresnel integrals over δφ in (22.24). Thus the fluctuations (δφ)n are on the average of relative orderh ¯n/2. If we ignore corrections of the order h¯3/2, the fluctuations remain quadratic in δφ and we may calculate the right-hand side of (22.24) as 2 4 δ e(i/¯h) A[φcl]+ d xj(x)φcl(x) det A N { } " δφ(x)δφ(y)# R φ=φcl 4 2 1/2 (i/¯h) A[φd]+ d xj(x)φcl(x)+i(¯h/2)Tr log[δ A/δφ(x)δφ(y)|φ=φcl = (det iG0) e { }. (22.25) R 22.2 Quadratic Fluctuations 1257 Comparing this with the left-hand side of (22.9), we find that to first order inh ¯ the effective action may be recovered by equating 2 4 4 ih¯ δ Γ[Φ] + d xj(x)Φ(x)= [φcl]+ d xjφcl + Tr log A (φcl) . (22.26) Z A Z 2 δφ(x)δφ(y) In the limith ¯ 0, the trace log term disappears and (22.26) reduces to the classical action. → To include theh ¯-correction into Γ[Φ], we expand W [j] as W [j]= W [j]+¯hW [j]+ (¯h2). (22.27) 0 1 O 2 Correspondingly, the field Φ differs from Φcl by a correction of the orderh ¯ : Φ= φ +¯hφ + (¯h2).
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