Thermal Quantum Field Theory in Real and Imaginary Time. Daniele Teresi

Transcription

1 Thermal Quantum Field Theory in Real and Imaginary Time University of Manchester 42nd BUSSTEPP - Durham University

2 WHAT IS THERMAL QFT? ORDINARY VACUUM QFT few in and out particles: 0 φφ... φ 0 S fi = f, + i, : no temporal information about the evolution of the system EQUILIBRIUM THERMAL QFT huge amount of particles, stationary state expectation values, not asymptotic amplitudes NON-EQUILIBRIUM THERMAL QFT huge amount of particles expectation values time evolution of the system

3 WHAT IS THERMAL QFT? ORDINARY VACUUM QFT few in and out particles: 0 φφ... φ 0 S fi = f, + i, : no temporal information about the evolution of the system EQUILIBRIUM THERMAL QFT huge amount of particles, stationary state expectation values, not asymptotic amplitudes NON-EQUILIBRIUM THERMAL QFT huge amount of particles expectation values time evolution of the system

4 WHAT IS THERMAL QFT? ORDINARY VACUUM QFT few in and out particles: 0 φφ... φ 0 S fi = f, + i, : no temporal information about the evolution of the system EQUILIBRIUM THERMAL QFT huge amount of particles, stationary state expectation values, not asymptotic amplitudes NON-EQUILIBRIUM THERMAL QFT huge amount of particles expectation values time evolution of the system

5 WHY THERMAL QFT? QUANTUM FIELD THEORY AT FINITE TEMPERATURE Collider Physics: quark-gluon plasma, nuclear matter Early Universe: electroweak phase transition, nucleosynthesis NONEQUILIBRIUM QUANTUM FIELD THEORY Baryogenesis - third Sakharov s condition Quantum effects in Leptogenesis Reheating at the end of inflation Heavy-ion collisions

6 Imaginary time Real time IMAGINARY-TIME FORMALISM CANONICAL ENSEMBLE ρ = 1 Z e βh = 1 Z e i( iβ)h evolution operator in imaginary-time Tr ρ = 1 Z φ(x) e βh φ(x) = 1 Z φ(x) φ(x), t iβ φ(x), t φ(x)

7 Imaginary time Real time IMAGINARY-TIME FORMALISM CANONICAL ENSEMBLE ρ = 1 Z e βh = 1 Z e i( iβ)h evolution operator in imaginary-time Tr ρ = 1 Z φ(x) e βh φ(x) = 1 Z φ(x) φ(x), t iβ φ(x), t φ(x) Im(t) t Re(t) t - ιβ

8 Imaginary time Real time IMAGINARY-TIME FREE PROPAGATOR imaginary time euclidean theory periodicity discrete frequencies MATSUBARA PROPAGATOR 0 1 (iω n, k) = ωn 2 + ωk 2 ω n = 2πn/β ω k = k 2 + m 2 Same diagrams as in the T=0 theory Discrete summations to evaluate loop corrections Analytic continuation to obtain real-time physical functions

9 Imaginary time Real time IMAGINARY-TIME FREE PROPAGATOR imaginary time euclidean theory periodicity discrete frequencies MATSUBARA PROPAGATOR 0 1 (iω n, k) = ωn 2 + ωk 2 ω n = 2πn/β ω k = k 2 + m 2 Same diagrams as in the T=0 theory Discrete summations to evaluate loop corrections Analytic continuation to obtain real-time physical functions

10 Imaginary time Real time REAL-TIME FORMALISM UNITARITY Tr ρ = Tr{ρ S S} = 1 Z Tr{e i( iβ)h S S}

11 Imaginary time Real time REAL-TIME FORMALISM UNITARITY Tr ρ = Tr{ρ S S} = 1 Z Tr{e i( iβ)h S S} Im(t) -T T C 1 C 2 Re(t) -T - ιβ T = can be factorized

12 Imaginary time Real time REAL-TIME FREE PROPAGATOR PROPAGATOR contour-ordered propagator 2 2 matrix ( ) i (x y) = i F (x y) < (x y) > (x y) D (x y) VERTICES 1 = iλ 2 = +iλ More diagrams than in the T=0 theory Analytic continuation not required Generalizable to non-equilibrium

13 Imaginary time Real time REAL-TIME FREE PROPAGATOR PROPAGATOR contour-ordered propagator 2 2 matrix ( ) i (x y) = i F (x y) < (x y) > (x y) D (x y) VERTICES 1 = iλ 2 = +iλ More diagrams than in the T=0 theory Analytic continuation not required Generalizable to non-equilibrium

14 Imaginary time Real time REAL-TIME FREE PROPAGATOR FREE SCALAR FIELD 0 (k) = ( 1 k 2 m 2 i n(k)2πδ(k 2 m 2 ) i [n(k)+θ( k 0 )]2πδ(k 2 m 2 ) +iɛ i [n(k)+θ(k 0 )]2πδ(k 2 m 2 1 ) k 2 m 2 i n(k)2πδ(k 2 m 2 ) iɛ ) Equilibrium boundary condition = n(k) = 1 e β k 0 1

15 Dyson s equation Self-energy calculations SELF-ENERGY IMAGINARY-TIME RESUMMED PROPAGATOR (iω n, k) = 1 0 (iω n,k) 1 + Π(iω n,k) REAL-TIME RESUMMED PROPAGATORS F (k) = k 2 m 2 +Π(k) [k 2 m 2 +RΠ R (k)] 2 +[IΠ R (k)] 2 R/A (k) = 1 k 2 m 2 +Π R/A (k) ( ) Π = Π Π< Π > Π

16 Dyson s equation Self-energy calculations SELF-ENERGY IMAGINARY-TIME RESUMMED PROPAGATOR (iω n, k) = 1 0 (iω n,k) 1 + Π(iω n,k) REAL-TIME RESUMMED PROPAGATORS F (k) = k 2 m 2 +Π(k) [k 2 m 2 +RΠ R (k)] 2 +[IΠ R (k)] 2 R/A (k) = 1 k 2 m 2 +Π R/A (k) ( ) Π = Π Π< Π > Π

17 Dyson s equation Self-energy calculations SELF-ENERGY IMAGINARY-TIME RESUMMED PROPAGATOR (iω n, k) = 1 0 (iω n,k) 1 + Π(iω n,k) REAL-TIME RESUMMED PROPAGATORS F (k) = k 2 m 2 +Π(k) [k 2 m 2 +RΠ R (k)] 2 +[IΠ R (k)] 2 R/A (k) = 1 k 2 m 2 +Π R/A (k) ( ) Π = Π Π< Π > Π QUASIPARTICLES M 2 (k) = m 2 RΠ R (k) Γ = IΠ R /M

18 Dyson s equation Self-energy calculations φ 3 THEORY - IΠ R Im T m 2 k 1.5 m THERMAL CUTTING RULES k 2 4 m 2 IΠ R related to thermal disappearance of particles

19 Dyson s equation Self-energy calculations φ 3 THEORY

20 WHAT I COULDN T TALK ABOUT fermions, gauge interactions IR problems: Hard-Thermal-Loop techniques phenomenology: quark-gluon plasma, early universe,... CJT formalism (2PI effective action): resummed self-consistent equations for the propagator Goldstone theorem not satisfied 1... Non-equilibrium resummed formalism (based on CJT): Kadanoff-Baym equations Non-equilibrium applications: thermalization of quantum fields, resonant leptogenesis,... Non-equilibrium perturbation theory 1 D. Teresi and A. Pilaftsis, Goldstone-symmetry improved CJT formalism, in preparation