Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 67
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For this reason, the transverse part has zero frequency. This feature,observed here in lowest order of the fluctuation expansion, is a very general one,and can be found in the effective action to any loop order. In quantum field theory,there exists a theorem asserting this called Nambu-Goldstone theorem. It statesthat if a quantum field theory without long-range interactions has a continuoussymmetry which is broken by a nonzero expectation value of the field correspondingH.
Kleinert, PATH INTEGRALS3113.23 Path Integral Calculation of Effective Action by Loop Expansionto the present X [recall (3.659)], then the fluctuations transverse to it have a zeromass. They are called Nambu-Goldstone modes or, because of their bosonic nature,Nambu-Goldstone bosons. The exclusion of long-range interactions is necessary,since these can mix with the zero-mass modes and make it massive. This happens, forexample, in a superconductor where they make the magnetic field massive, giving ita finite penetration depth, the famous Meissner effect. One expresses this pictoriallyby saying that the long-range mode can eat up the Nambu-Goldstone modes andbecome massive. The same mechanism is used in elementary particle physics toexplain the mass of the W ± and Z 0 vector bosons as a consequence of having eatenup a would be Nambu-Goldstone boson of an auxiliary Higgs-field theory.In quantum-mechanical systems, however, a nonzero expectation value with theassociated zero frequency mode in the transverse direction is found only as an artifactof perturbation theory.
If all fluctuation corrections are summed, the minimum ofthe effective potential lies always at the origin. For example, it is well known, thatthe ground state wave functions of a particle in a double-well potential is symmetric,implying a zero expectation value of the particle position. This symmetry is causedby quantum-mechanical tunneling, a phenomenon which will be discussed in detailin Chapter 17. This phenomenon is of a nonperturbative nature which cannotbe described by an effective potential calculated order by order in the fluctuationexpansion. Such a potential does, in general, posses a nonzero minimum at someX0 somewhere near the zero-order minimum (3.680).
Due to this shortcoming, itis possible to derive the Nambu-Goldstone theorem from the quantum-mechanicaleffective action in the loop expansion, even though the nonzero expectation valueX0 assumed in the derivation of the zero-frequency mode does not really exist inquantum mechanics. The derivation will be given in Section 3.24.The use of the initial action to approximate the effective action neglecting corrections caused by the fluctuations is referred to as mean-field approximation.3.23.3Corrections from Quadratic FluctuationsIn order to find the first h̄-correction to the mean-field approximation we expandthe action in powers of the fluctuations of the paths around the classical solutionδx(t) ≡ x(t) − xcl (t),(3.692)and perform a perturbation expansion.
The quadratic term in δx(t) is taken to bethe free-particle action, the higher powers in δx(t) are the interactions. Up to secondorder in the fluctuations δx(t), the action is expanded as follows:A[xcl + δx] +Zdt j(t) [xcl (t) + δx(t)]= A[xcl ] +1+2ZZdt j(t) xcl (t) +Zdt(δA δx(t)j(t) +δx(t) x=xcl δ2Aδx(t0 ) + O (δx)3 .dt dt δx(t)0 δx(t)δx(t ) x=xcl0(3.693)3123 External Sources, Correlations, and Perturbation TheoryThe curly bracket multiplying the linear terms in the variation δx(t) vanish dueto the extremality property of the classical path xcl expressed by the equation ofmotion (3.668). Inserting this expansion into (3.667), we obtain the approximateexpressionZ[j] ≈ e(i/h̄){A[xcl ]+Rdt j(t)xcl (t)}ZDδx expi Z h̄2dt dt0δ Aδx(t)0 δx(t)δx(t )δx(t0 ) .x=xcl(3.694)√We now observe that the fluctuations δx(t) will be of average size h̄ due to theh̄-denominatorin the Fresnel exponent.
Thus the fluctuations (δx)n are of average√ nsize h̄ . The approximate path integral (3.694) is of the Fresnel type and my beintegrated to yield(i/h̄){A[xcl ]+eRdt j(t)xcl (t)}δ2Adetδx(t)δx(t0 )"#−1/2(3.695)x=xcl(i/h̄){A[xcl ]+ dt j(t)xcl (t)+i(h̄/2)Tr log[δ2 A/δx(t)δx(t0 )|x=xcl }R=e.Comparing this with the left-hand side of (3.667), we find that to first order in h̄,the effective action may be recovered by equatingΓ[X] +Zdt j(t)X(t) = A[xcl [j]] +Zih̄δ 2 A [xcl [j]]dt j(t) xcl (t)[j] + Tr log.
(3.696)2δx(t)δx(t0 )In the limit h̄ → 0, the tracelog term disappears and (3.696) reduces to the classicalexpression (3.669).To include the h̄-correction into Γ[X], we expand W [j] asW [j] = W0 [j] + h̄W1 [j] + O(h̄2 ).(3.697)Correspondingly, the path X differs from Xcl by a correction term of order h̄:X = xcl + h̄ X1 + O(h̄2 ).(3.698)Inserting this into (3.696), we findZΓ[X] + dt jX = A [X −h̄X1 ] +Zdt jX − h̄δ2Ai+ h̄ Tr log2δxa δxbZx=X−h̄X1dt jX1+ O h̄2 .(3.699)Expanding the action up to the same order in h̄ givesΓ[X] = A[X] − h̄Z iδ 2 A δA[X]+ j X1 + h̄ Tr log+ O h̄2 .
(3.700)dtδX2δxa δxb x=X()H. Kleinert, PATH INTEGRALS3.23 Path Integral Calculation of Effective Action by Loop Expansion313Due to (3.668), the curly-bracket term is only of order h̄2 , so that we find the oneloop form of the effective action1 2 ω 2 2 g 2 2Γ[X] = Γ0 [X] + h̄Γ1 [X] =dtẊ − Xa −Xa224!ig222+δ X + 2Xa Xb . (3.701)h̄ Tr log −∂t − ω −26 ab cZ#"Using the decomposition (3.683), the tracelog term can be written as a sum oftransversal and longitudinal partsii(2)(2)h̄ Tr log ΓL (t1 , t2 )ab + (N − 1)h̄ Tr log ΓT (t1 , t2 )ab(3.702)22ii= h̄ Tr log −∂t2 − ωL2 (X) + (N − 1)h̄ Tr log −∂t2 − ωT2 (X) .22h̄Γ1 [X] =What is the graphical content in the Green functions at this level of approximation? Assuming ω 2 > 0, we find for j = 0 that the minimum lies at X̄ = 0, as inthe mean-field approximation.
Around this minimum, we may expand the tracelogin powers of X. For the simplest case of a single X-variable, we obtainigiX2iih̄ Tr log −∂t2 −ω 2 − X 2 = h̄ Tr log −∂t2 −ω 2 + h̄ Tr log 1+ 2ig2222−∂t −ω 2 2!n∞ ih̄ Xh̄g n1222TrX−i= i Tr log −∂t − ω − i.(3.703)22 n=12 n−∂t2 − ω 2!If we insertG0 =−∂t2i,− ω2(3.704)this can be written as∞h̄h̄ Xgi Tr log −∂t2 − ω 2 − i−i22 n=12nn1 Tr G0 X 2 .n(3.705)More explicitly, the terms with n = 1 and n = 2 read:h̄− g dt dt0 δ(t − t0 )G0 (t, t0 )X 2 (t0 )4Zg2+ih̄dt dt0 dt00 δ 4 (t − t00 )G0 (t, t0 )X 2 (t0 )G0 (t0 , t00 )X 2 (t00 ) + . . .
.16Z(3.706)The expansion terms of (3.705) for n ≥ 1 correspond obviously to the Feynmandiagrams (omitting multiplicity factors)A[xcl ] =(3.707)3143 External Sources, Correlations, and Perturbation TheoryThe series (3.705) is therefore a sum of all diagrams with one loop and any numberof fundamental X 4 -verticesTo systematize the entire expansion (3.705), the tracelog term is [compare(3.546)] pictured by a single-loop diagramh̄1i Tr log −∂t2 − ω 2 =22.(3.708)The first two diagrams in (3.707) contribute corrections to the vertices Γ(2) andΓ .
The remaining diagrams produce higher vertex functions and lead to moreinvolved tree diagrams. In Fourier space we find from (3.706)(4)ig dk(3.709)Γ (q) = q − ω − h̄22 2π k − ω 2 + iη#"Zg2iidk(4)Γ (qi ) = g − i+ 2 perm .22π k 2 − ω 2 + iη (q1 + q2 − k)2 − ω 2 + iη(3.710)(2)22ZWe may write (3.709) in Euclidean form asgΓ (q) = −q − ω − h̄2dk122π k + ω 2g 122,= − q + ω + h̄2 2ωg2Γ(4) (qi ) = g − h̄ [I (q1 + q2 ) + 2 perm] ,2(2)22Z(3.711)(3.712)with the Euclidean two-loop integralI(q1 + q2 ) =Zdk1i,222π k + ω (q1 + q2 − k)2 + ω 2(3.713)to be calculated explicitly in Chapter 10.
It is equal to J((q1 + q2 )2 )/2π with thefunctions J(z) of Eq. (10.258).For ω 2 < 0 where the minimum of the effective action lies at X̄ 6= 0, the expansionof the trace of the logarithm in (3.701) must distinguish longitudinal and transverseparts.3.23.4-2Effective Action to Order hLet us now find the next correction to the effective action.19 Instead of truncatingthe expansion (3.693), we keep all terms, reorganizing only the linear and quadraticterms as in (3.694). This yieldse(i/h̄){Γ[X]+jX} = ei(h̄/2)W [j] = e(i/h̄){(A[xcl ]+jxcl )+(ih̄/2)Tr log Axx [xcl ]} e(i/h̄)h̄192W2 [xcl ].(3.714)R.
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