Diploma thesis - Fachbereich Physik

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24 CHAPTER 2. EFFECTIVE ACTION Accordingly, a functional Taylor expansion of the action A[x] has to be performed around an arbitrarily chosen background X(τ): A[X + δx] = A[X] + + 1 2 + 1 6 + 1 24 ∫ ¯hβ 0 ∫ ¯hβ 0 ∫ ¯hβ 0 ∫ ¯hβ 0 dτ 1 δA[X] δX(τ 1 ) δx(τ 1) (2.115) ∫ ¯hβ dτ 1 dτ 2 0 ∫ ¯hβ ∫ ¯hβ dτ 1 dτ 2 dτ 3 0 δ 2 A[X] δX(τ 1 )δX(τ 2 ) δx(τ 1)δx(τ 2 ) 0 ∫ ¯hβ ∫ ¯hβ ∫ ¯hβ dτ 1 dτ 2 dτ 3 dτ 4 × 0 0 δ 3 A[X] δX(τ 1 )δX(τ 2 )δX(τ 3 ) δx(τ 1)δx(τ 2 )δx(τ 3 ) 0 δ 4 A[X] δX(τ 1 )δX(τ 2 )δX(τ 3 )δX(τ 4 ) δx(τ 1)δx(τ 2 )δx(τ 3 )δx(τ 4 ) + . . . . Again, δx(τ) := x(τ) − X(τ) denotes the deviation of the path x(τ) from the background X(τ). Similar to the procedure above, the first order term, ∫ ¯hβ 0 dτ 1 δA[X] δX(τ 1 ) δx(τ 1) , (2.116) will be neglected, and terms being of higher than second order in δx define the interaction A (int) [δx] := 1 6 ∫ ¯hβ 0 + 1 24 ∫ ¯hβ ∫ ¯hβ dτ 1 dτ 2 dτ 3 ∫ ¯hβ 0 0 0 ∫ ¯hβ ∫ ¯hβ ∫ ¯hβ dτ 1 dτ 2 dτ 3 dτ 4 × The integral kernel is defined as 0 0 δ 3 A[X] δX(τ 1 )δX(τ 2 )δX(τ 3 ) δx(τ 1)δx(τ 2 )δx(τ 3 ) (2.117) 0 δ 4 A[X] δX(τ 1 )δX(τ 2 )δX(τ 3 )δX(τ 4 ) δx(τ 1)δx(τ 2 )δx(τ 3 )δx(τ 4 ) + . . . . G −1 (τ 1 , τ 2 ) := and the partition function (2.114) can be written in the form δ 2 A[X] δX(τ 1 )δX(τ 2 ) , (2.118) Z = exp {− 1¯h } A[X] (2.119) ∮ { × Dδx exp − 1 ∫ ¯hβ ∫ ¯hβ dτ 1 dτ 2 G −1 (τ 1 , τ 2 )δx(τ 1 )δx(τ 2 ) − 1¯h } 2¯h A(int) [δx] . 0 0 In order to determine the second-order contribution to the partition function, it is helpful to remember the result of the one-dimensional Gaußian integral ∫ ∞ dx exp (− 1 ) √ 2π 2 Ax2 = [A > 0] . (2.120) A −∞
2.7. BACKGROUND METHOD: PATH INTEGRAL 25 In D dimensions, A can be taken as a matrix, and the Gaußian integral evaluates to ∫ ∞ ( d D x exp − 1 ) 2 xT Ax = √ (2π)D/2 [A = A † , det(A) > 0] . (2.121) det(A) −∞ The path integral in (2.119) constitutes an infinite-dimensional integration. Therefore, one would expect that the second-order contribution to the partition function is the generalization of the result (2.121) to infinite dimension: ∮ Dδx exp [ − 1 2¯h ∫ ¯hβ 0 ∫ ¯hβ ] dτ 1 dτ 2 G −1 (τ 1 , τ 2 )δx(τ 1 )δx(τ 2 ) = 0 1 √ det(G −1 ) . (2.122) The determinant of the integral kernel is defined as the product of its eigenvalues λ k , which are obtained by solving the eigenvalue problem ∫ ¯hβ 0 with δx k (τ) being periodic in the imaginary time: dτ 2 G −1 (τ 1 , τ 2 ) δx k (τ 2 ) = Mλ k δx k (τ 1 ) , (2.123) δx k (0) = δx k (¯hβ) . (2.124) Accordingly, the right-hand side of (2.122) can be rewritten in the form ( 1 √ det(G −1 ) = exp − 1 ) 2 Tr ln G−1 , (2.125) where the trace-log of the integral kernel G −1 (τ 1 , τ 2 ) is defined by Tr lnG −1 := ∑ k lnλ k . (2.126) The preceding argument lacks, of course, mathematical rigor. In Appendix A, the partition function for the harmonic oscillator is calculated, and an analogous procedure could be applied to demonstrate that the result (2.122) – (2.126) is indeed valid. Since the application of the background method implies a vanishing external current j, one obtains from (2.24) Z[X] = exp {− 1¯h } Γ[X] . (2.127) Thus, when neglecting the interactions, i.e. when setting A (int) [δx] = 0, from the last identity, (2.119), and (2.122) – (2.126), one obtains the first order approximation of the effective action: Γ (0) [X] + Γ (1) [X] = A[X] + ¯h 2 Tr ln G−1 ; (2.128) and the complete effective action is given by [4, Ch. 3] Γ[X] = A[X] + ¯h 2 Tr ln G−1 + Γ (int) [X] . (2.129) The zeroth-order term A[X] in (2.129) is referred to as ‘tree-level’. As before, the remaining interactions Γ (int) [X] can be characterized in terms of Feynman diagrams. The respective Feynman rules are now:
Page 1: Beyond Effective Potential via Vari
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Page 6 and 7: 6 CHAPTER 1. INTRODUCTION ries. For
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5.2. BACKGROUND METHOD FOR EFFECTIV
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88 CHAPTER 6. OUTLOOK where the ima
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Appendix A Harmonic Generating Func
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93 Evaluating the x 0 -, Re x m -,
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95 and that the eigenvalues of G
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98 APPENDIX B. STANDARD INTEGRALS R
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100 APPENDIX B. STANDARD INTEGRALS
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Appendix C Shooting Method In this
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x → −∞, or vice-versa. Thus,
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List of Figures 4.1 Schematic compa
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Bibliography [1] Physik Journal 3 (
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BIBLIOGRAPHY 115 [29] W. Janke and
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Diploma thesis - Fachbereich Physik

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