Thermal Field Theory to All Orders in Gradient Expansion Peter Millington p.w.millington@sheffield.ac.uk Astro-Particle Theory and Cosmology Group, University of Sheffield, UK, Consortium for Fundamental Physics arXiv: 1211.3152 PM & Apostolos Pilaftsis (University of Manchester) Thursday, 6 th December, 2012 Discrete 2012 CFTP, IST, Universidade Tecnica de Lisboa Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 1 / 15
Outline 1. Introduction 2. Formalism 3. Master Time Evolution Equations 4. Simple Example 5. Conclusions Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 2 / 15
Introduction Motivation the density frontier: ultra-relativistic many-body dynamics early Universe: ◮ baryon asymmetry of the Universe ◮ electroweak phase transition ◮ reheating/preheating ◮ relic densities ‘terrestrial:’ ◮ quark gluon plasma/glasma/color glass condensates Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 3 / 15
Introduction Current Approaches (semi-classical) Boltzmann transport equations ◮ effective resummation of finite-width effects Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 4 / 15
Introduction Current Approaches (semi-classical) Boltzmann transport equations ◮ effective resummation of finite-width effects Kadanoff–Baym ⇒ quantum Boltzmann equations ◮ incorporation of off-shell effects ◮ truncated gradient expansion in time derivative ◮ separation of time scales and quasi-particle approximation ◮ varying definitions of physical observables, e.g. particle number density Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 4 / 15
Introduction Current Approaches (semi-classical) Boltzmann transport equations ◮ effective resummation of finite-width effects Kadanoff–Baym ⇒ quantum Boltzmann equations ◮ incorporation of off-shell effects ◮ truncated gradient expansion in time derivative ◮ separation of time scales and quasi-particle approximation ◮ varying definitions of physical observables, e.g. particle number density underlying perturbation series contain pinch singularities: δ 2 ( p 2 − m 2 ) Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 4 / 15
Canonical Quantisation Boundary Conditions No assumption of adiabatic hypothesis. QM pictures have a finite microscopic time of coincidence ˜ t i : Φ S ( x ;˜ t i ) = Φ I (˜ t i , x ;˜ t i ) = Φ H (˜ t i , x ;˜ t i ) Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 5 / 15
Canonical Quantisation Boundary Conditions No assumption of adiabatic hypothesis. QM pictures have a finite microscopic time of coincidence ˜ t i : Φ S ( x ;˜ t i ) = Φ I (˜ t i , x ;˜ t i ) = Φ H (˜ t i , x ;˜ t i ) ⇒ interactions switched on at ˜ t i ⇒ initial density matrix ρ (˜ t i ;˜ t i ) specified fully in on-shell Fock states ⇒ finite lower bound on time integrals in path-integral action Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 5 / 15
Canonical Quantisation Canonical Commutation Relations Interaction-picture creation and annihilation operators satisfy: = (2 π ) 3 2 E ( p ) δ (3) ( p − p ′ ) e − iE ( p )(˜ t − ˜ a ( p , ˜ t ;˜ t i ) , a † ( p ′ , ˜ t ′ ;˜ t ′ ) � � t i ) Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 6 / 15
Canonical Quantisation Canonical Commutation Relations Interaction-picture creation and annihilation operators satisfy: = (2 π ) 3 2 E ( p ) δ (3) ( p − p ′ ) e − iE ( p )(˜ t − ˜ a ( p , ˜ t ;˜ t i ) , a † ( p ′ , ˜ t ′ ;˜ t ′ ) � � t i ) Ensemble Expectation Value (EEV) at macroscopic time t = ˜ t f − ˜ t i : �•� t = tr ρ (˜ t f ;˜ t i ) • tr ρ (˜ t f ;˜ t i ) Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 6 / 15
Canonical Quantisation Canonical Commutation Relations Interaction-picture creation and annihilation operators satisfy: = (2 π ) 3 2 E ( p ) δ (3) ( p − p ′ ) e − iE ( p )(˜ t − ˜ a ( p , ˜ t ;˜ t i ) , a † ( p ′ , ˜ t ′ ;˜ t ′ ) � � t i ) Ensemble Expectation Value (EEV) at macroscopic time t = ˜ t f − ˜ t i : �•� t = tr ρ (˜ t f ;˜ t i ) • tr ρ (˜ t f ;˜ t i ) Most general EEVs permitted: � a ( p , ˜ t f ;˜ t i ) a † ( p ′ , ˜ t f ;˜ t i ) � t = (2 π ) 3 2 E ( p ) δ (3) ( p − p ′ ) + 2 E 1 / 2 ( p ) E 1 / 2 ( p ′ ) f ( p , p ′ , t ) � a † ( p ′ , ˜ t f ;˜ t i ) a ( p , ˜ t f ;˜ t i ) � t = 2 E 1 / 2 ( p ) E 1 / 2 ( p ′ ) f ( p , p ′ , t ) Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 6 / 15
b b b Schwinger–Keldysh CTP Formalism − i � i � Ω t d 4 x J − ( x )Φ H ( x ) � Ω t d 4 x J + ( x )Φ H ( x ) � � ¯ �� � ˜ t f ;˜ Z [ ρ, J ± , t ] = tr T e ρ H t i T e � ˜ 2 , ˜ t i = − t t f = + t x 0 ∈ � 2 Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 7 / 15
b b b Schwinger–Keldysh CTP Formalism − i � i � Ω t d 4 x J − ( x )Φ H ( x ) � Ω t d 4 x J + ( x )Φ H ( x ) � � ¯ �� � ˜ t f ;˜ Z [ ρ, J ± , t ] = tr T e ρ H t i T e � ˜ 2 , ˜ t i = − t t f = + t x 0 ∈ � 2 Im t z (0) = ˜ ˜ t i C + Re t z (1 / 2) = ˜ ˜ t f − iǫ/ 2 z (1) = ˜ ˜ t i − iǫ C − z ( u ) − ˜ macroscopic time t = Re ˜ t i initial conditions: observation: macroscopic time t = ˜ t f − ˜ macroscopic time t = 0 t i ⇒ finite upper and lower bounds on time integrals in path-integral action. Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 7 / 15
Non-Homogeneous Free Propagators Propagator Double-Momentum Representation ( − ) i i ∆ 0 F(D) ( p , p ′ , ˜ t f ;˜ p 2 − M 2 + ( − ) i ǫ (2 π ) 4 δ (4) ( p − p ′ ) t i ) = Feynman (Dyson) +2 π | 2 p 0 | 1 / 2 δ ( p 2 − M 2 )˜ 0 | 1 / 2 δ ( p ′ 2 − M 2 ) f ( p , p ′ , t ) e i ( p 0 − p ′ 0 )˜ t f 2 π | 2 p ′ t i ) = 2 πθ (+( − ) p 0 ) δ ( p 2 − M 2 )(2 π ) 4 δ (4) ( p − p ′ ) + ( − )ve- i ∆ 0 > ( < ) ( p , p ′ , ˜ t f ;˜ freq. +2 π | 2 p 0 | 1 / 2 δ ( p 2 − M 2 )˜ 0 | 1 / 2 δ ( p ′ 2 − M 2 ) f ( p , p ′ , t ) e i ( p 0 − p ′ 0 )˜ t f 2 π | 2 p ′ Wightman i Retarded ( p 0 + ( − ) i ǫ ) 2 − p 2 − M 2 (2 π ) 4 δ (4) ( p − p ′ ) i ∆ 0 R(A) ( p , p ′ ) = (Advanced) Pauli- i ∆ 0 ( p , p ′ ) = 2 πε ( p 0 ) δ ( p 2 − M 2 )(2 π ) 4 δ (4) ( p − p ′ ) Jordan t i ) = 2 πδ ( p 2 − M 2 )(2 π ) 4 δ (4) ( p − p ′ ) i ∆ 0 1 ( p , p ′ , ˜ t f ;˜ Hadamard +2 π | 2 p 0 | 1 / 2 δ ( p 2 − M 2 )2˜ 0 | 1 / 2 δ ( p ′ 2 − M 2 ) f ( p , p ′ , t ) e i ( p 0 − p ′ 0 )˜ t f 2 π | 2 p ′ i Principal- i ∆ 0 P ( p , p ′ ) = P p 2 − M 2 (2 π ) 4 δ (4) ( p − p ′ ) part Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 8 / 15
Diagrammatics 2 M 2 Φ 2 + ∂ µ χ † ∂ µ χ − m 2 χ † χ − g Φ χ † χ 1 2 ∂ µ Φ ∂ µ Φ − 1 L = χ k ′ k 1 1 a b Φ Φ q q ′ k 2 k ′ 2 χ Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 9 / 15
Diagrammatics 2 M 2 Φ 2 + ∂ µ χ † ∂ µ χ − m 2 χ † χ − g Φ χ † χ 1 2 ∂ µ Φ ∂ µ Φ − 1 L = χ k ′ k 1 1 a b Φ Φ q q ′ k 2 k ′ 2 χ 1. time-dependent, energy-non-conserving vertices: �� � � � � � � ∼ − ig t t p 0 δ (3) 2 π sinc p i i 2 i i Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 9 / 15
Diagrammatics 2 M 2 Φ 2 + ∂ µ χ † ∂ µ χ − m 2 χ † χ − g Φ χ † χ 1 2 ∂ µ Φ ∂ µ Φ − 1 L = χ k ′ k 1 1 a b Φ Φ q q ′ k 2 k ′ 2 χ 1. time-dependent, energy-non-conserving vertices: �� � � � � � � ∼ − ig t t p 0 δ (3) 2 π sinc p i i 2 i i 2. momentum-non-conserving, non-homogeneous free propagators Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 9 / 15
Physically Meaningful Observables Construct from EEVs of field operators: t = tr ρ (˜ t f ;˜ t i )Φ( x ;˜ t i )Φ( y ;˜ t i ) � Φ( x ;˜ t i )Φ( y ;˜ t i ) � tr ρ (˜ t f ;˜ t i ) Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 10 / 15
Physically Meaningful Observables Construct from EEVs of field operators: t = tr ρ (˜ t f ;˜ t i )Φ( x ;˜ t i )Φ( y ;˜ t i ) � Φ( x ;˜ t i )Φ( y ;˜ t i ) � tr ρ (˜ t f ;˜ t i ) Physically meaningful observables must be equal-time and picture-independent. Peter Millington (University of Sheffield) TFT to All Orders in Gradients Discrete 2012, IST, Lisboa 10 / 15
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