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The Effective Action and Its Applications

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The Effective Action and Its Applications The Effective Action and its Applications Tom Charnock based on the lectures of Pete Millington Contents 1 Introduction and Motivation 2 2 Prerequisites 7 2.1 Convex Sets and Convex Functions and Functionals . 9 2.2 Legendre and Fenchel Transforms . 10 2.3 Schwinger Function . 10 3 n-Particle Irreducible (nPI) Effective Action 12 3.1 1PI Effective Action . 12 3.2 Two-point Effective Action . 14 3.2.1 2PI Effective Action . 14 4 (Non)-Equilibrium Field Theory 17 4.1 Imaginary-time Formalism . 17 4.2 Schwinger-Keldysh Closed Time Path Formalism . 17 4.3 Transport Equation . 19 5 Symmetries 21 5.1 Thermal Corrections . 21 5.2 Ward-Takahashi Identities . 23 5.3 Symmetry Preservation . 24 5.4 First-order Phase-transition . 24 6 Functional Renormalisation Group 25 6.1 Wetterich Equation . 25 6.2 First-order Phase Transitions . 27 1 Chapter 1 Introduction and Motivation y # + ∆# L # x x + ∆x L x − 2 2 Figure 1.1: Classically vibrating string which is transversly displaced. Start with a classically vibrating string with length L, tension T , and mass per unit length µ. If a segment of the string ∆x, between x and x + ∆x is displaced transversly from equilibrium then the net force of the displacement (for small displacements) is Fx = T cos # + ∆# − T cos # ≈ 0; (1.1) Fy = T sin # + ∆# − T sin # ≈ T ∆#: @2y From the variation in the angle ∆#, it can be seen that F ≈ T ∆x. y @x2 As the mass of this segment is µ∆x and the acceleration in the y direction is the second derivative of y with respect to time t, then it can be seen that @2y @2y 0 = µ − T ∆x: (1.2) @t2 @x2 This is the wave equation 1 @2y @2y 0 = − ; (1.3) v2 @t2 @x2 where v = pT/µ. This equation is not only invariant under translations in time and space 0 t ! t = t + a; a 2 R 0 (1.4) x ! x = x + b; b 2 R; but also under boosts (Lorentz transformations) ux t ! t0 = γ t − v2 (1.5) x ! x0 = γ x − ut ; 2 1 where γ = . p1 − u2=v2 " 1;1 These two sets of symmetries are the isometries of the 1+1 dimensional Poincar´egroup P+(1; 1) = R × SO(1; 1). µ A metric ηµν = diag(1; −1), can be introduced, along with contravariant and covariant vectors, x ≡ (ut; x) and ν xµ ≡ ηµν x = (ut; −x). As well as Poincar´einvariance, this classical wave equation is also invariant under discrete spacetime symmetries T : t ! −t (1.6) P : x ! −x; and charge conjugation C. Thus the equation is also invariant under CPT transformation. Due to the linearity and homogeneity of the wave equation it is further invariant under rescalings of y which can be seen by p φ(t; x) ≡ µy(t; x); (1.7) where the dimensionality of φ(t; x) is [φ] = M 1=2L1=2. This allows the wave equation to be written as φ(t; x) = 0; (1.8) µ where ≡ @µ@ is the 1+1 d'Alembertian. The Hamiltonian for this system is Z L=2 H(t) = dxH −L=2 (1.9) Z L=2 1 = dx Π2(t; x) + v2rφ(t; x)rφ(t; x); −L=2 2 where the Lagrangian L(t) and the Lagrangian density L (t; x) are Z L=2 1 µ L(t) = dx @µφ(t; x)@ φ(t; x) −L=2 2 Z L=2 (1.10) = dxL (t; x): −L=2 The action is Z tf S[φ] = dtL(t) ti (1.11) Z tf Z L=2 = dt L (t; x); ti −L=2 with the first functional variation Z tf Z L=2 @L @L δS[φ] = dt dx δφ + δ@µφ : (1.12) ti −L=2 @φ @@µφ It is assumed that the function and partial variations commute and that the surface terms can be neglected after integrating by parts. Setting this functional variation to zero gives the Euler-Lagrange equations Z tf Z L=2 @L @L 0 = dt dx − @µ δφ. (1.13) ti −L=2 @φ @@µφ Boundary conditions can be chosen to preserve as many symmetries as possible. Such conditions include • φ(t; x) is periodic on [−L=2; L=2]. • φ(t; x) is constant on the line x = vt. 3 The wave equation is a separable second-order partial differential equation 2 1 @ τ(t) 2 2 2 = k v τ(t) @t (1.14) 1 @2χ(x) = k2; χ(x) @x2 with φ(t; x) = τ(t)χ(x). Here k is discrete, k 2 fkn = 2πn=Ljn 2 Zg. φ(t; x) can be a function of only !n − knx, with !n = knv. It is written as +1 1 X 1 φ(t; x) = A e−i!nteiknx + A∗ ei!nte−iknx; (1.15) L j2! j n n n=−∞ n ∗ 1=2 3=2 −1 where An = A−n which have dimension [An] = M L T . Unlike in quantum field theory, the normalisation can be chosen such that +1 Z L=2 1 X dxφ2(t; x) = jA j2 L n (1.16) −L=2 n=−∞ < 1; 2 for any !njAnj which is a decreasing function of n. Substituting this into the Hamiltonian gives +1 1 X E = jA j2: (1.17) L n n=−∞ φ(t; x) can be split into components +1 1 X φ(t; x) = φ (t; x); (1.18) L n n=−∞ where the coefficients are 1 i _ An = φn(0; 0) + φn(0; 0) : (1.19) 2 !n The bandwidth theorem states that ∆t∆! ≥ 2π as seen with substitution using Fourier transforms. Supposing that 2 2 An have strong support in some region (jAnj ≈ jA¯j for n 2 [nmin; nmax]) then 1 ∆E ≈ jA¯j2∆n L (1.20) jA¯j2 = ∆!: 2πv Since !n = 2πnv=L then ∆! = 2π∆nv=L. From the ∆E relation it is seen that jA¯j2 ∆E∆t ≥ ; (1.21) v which is some number which we can name ~=2. Here ~ does not need to be related to the reduced Planck constant. The coefficients An, can be promoted to operatorsa ^n, which act on a Hilbert space spanned by some complete set y of states fjni; n 2 N0g.a ^n and its Hermitian conjugatea ^n, satisfy y [^an; a^m] = 2~!nLδnm; (1.22) (y) where n; m > 0. This commutator means that acting on states with thea ^n operators give y p yn a^nj0i = 2!nL~ a^ j0i = jni p n (1.23) a^njni = 2!nL~ a^ jmi = 2~!nL j0i 4 p p This meansa ^yjni = n + 1jn + 1i anda ^jni = njn − 1i. A conjugate operator T , can be introduced with the property that (y) (y) −1 a^−n ≡ T a^n T (y) (1.24) =a ^n : If the vacuum is T -symmetric (j0i = j0i∗ = j0iT ) then j − ni = jni∗. A field operator can also be introduced +1 1 X 1 φ^(t; x) = a^ e−ij!njteiknx +a ^y eij!njte−iknx; (1.25) L j2! j n n n=−∞ n along with the conjugate momentum operator +1 1 X 1 Π(^ t; x) = a^ e−ij!njteiknx − a^y eij!njte−iknx; (1.26) iL 2 n n n=−∞ where it can be seen that [φ^(t; x); φ^(t; x0)] = 0 i (1.27) [φ^(t; x); Π(^ t; x0)] = ~δ(x − x0): 2 The Hamiltonian operator for this system is +1 1 X H^ = a^y a^ (1.28) L n n n=−∞ which is normal ordered. The state vector is +1 1 X 1 An j (t)i = e−ij!njtjni: (1.29) L j2! j n=−∞ n ~ The expectation value for the energy is E = h (t)jH^ j (t)i +1 1 X (1.30) = jA j2: L n n=−∞ The state evolves with the Heisenberg equation of motion d i j (t)i = − H^ j (t)i (1.31) dt ~ 0 0 0 −iH^ (t0−t)= so there is an evolution operator j (t )i = U(t ; t)j (t)i where U(t; t ) = e ~. In the limit that L ! 1, ~kn becomes a momentum p, and 1 Z ~ X dp ! : (1.32) L 2π n=−∞ (y) (y) The operatorsa ^n become true creation and annihilation operators a (p) following [^a ; a^y ] ! [a(p); ay(p)] n n (1.33) = 2π2Eδ(p − p0) where E = jpjv and Z dp 1 φ^(t; x) ! a(p)e−iE(p)taipx + ay(p)eiE(p)te−ipx: (1.34) 2π 2E(p) 5 This is the description of a 1+1 dimensional field theory. Small anharmonic terms, such as λφ4 V (φ) = ; (1.35) 4! can be added. These terms will begin to dominate the evolution operator U(t0; t) and loop integrals need to be calculated. These depend on the two-point correlation function of the fluctuations 0 ^ ^ 0 0 i∆F (x; x ) = h0jT (φ(x0; x)φ(x0; x )j0i +1 ~ X 1 0 −ij! j(x −x0 ) ik (x−x0) 0 (1.36) = Θ(x − x )e n 0 0 e n + (x ! x ): L j2! j 0 0 n=−∞ n Whenever there is a system with fluctuations the \classical" equations of motion (those in the absence of fluctuations) are not enough to understand the full dynamics. A technology is needed to take into account the backreation from the fluctuations. This technology is the effective action which systematically organises pertubation theory in the scale of the fluctuations. As an example, for −βH stochastic fluctuations of scale β ≡ kBT , the Gibbs distribution is e which is expanded in orders of β.A well-defined variational procedure can be derived from which the full evolution equations from the system can be obtained.
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