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˚ 1˚. ENTROPY AND DECOHERENCE IN QUANTUM THEORIES. Based on : Jurjen F. Koksma, Tomislav Prokopec and Michael G. Schmidt, Phys. Rev. D (2011 ) [ arXiv:1102.4713 [hep-th]]; arXiv:1101.5323 [quant-ph]; Annals Phys. (2011), arXiv:1012.3701 [quant-ph];

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## ENTROPY AND DECOHERENCE IN QUANTUM THEORIES

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**˚ 1˚**ENTROPY AND DECOHERENCE IN QUANTUM THEORIES Based on: Jurjen F. Koksma, Tomislav Prokopec and Michael G. Schmidt, Phys. Rev. D (2011) [arXiv:1102.4713 [hep-th]]; arXiv:1101.5323 [quant-ph]; Annals Phys. (2011), arXiv:1012.3701 [quant-ph]; Phys. Rev. D 81 (2010) 065030 [arXiv:0910.5733 [hep-th]] Annals Phys. 325 (2010) 1277 [arXiv:1002.0749 [hep-th]] Tomislav Prokopec and Jan Weenink, [arXiv:1108.3994[gr-qc]]+ in preparation Tomislav Prokopec, ITP & Spinoza Institute, Utrecht University Nikhef, Mar 30 2012**˚ 2˚**PLAN ENTROPY as a physical quantity and decoherence ENTROPY of (harmonic) oscillators ● bosonic oscillator ● fermionic oscillator ENTROPY and DECOHERENCE in relativistic QFT’s APPLICATIONS ● CB ● neutrino oscillations and decoherence DISCUSSION**˚ 3˚**VON NEUMANN ENTROPY von Neumann entropy (of a closed system): ..OBEYS A HEISENBERG EQUATION: CLOSED SYSTEM as a result, von Neumann entropy is conserved: Consequently, von Neumann entropy is conserved, hence USELESS. However: vN entropy is constant if applied to closed systems, where all dof’s and their correlations are known. In practice: never the case!**˚ 4˚**OPEN SYSTEMS ◙ OPEN SYSTEMS (S) interact with an environment (E). If observer (O) does not perceive SE correlations (entanglement), (s)he will detect a changing (increasing?) vN entropy. Proposal: vN entropy (of S) is a quantitative measure for decoherence. OPEN SYSTEM • von Neumann entropy is • not any more conserved S E NB: entropy/decoherence is an observer dependent concept. Hence, arguably there is no unique way of defining it. Some argue: useless. In practice: has shown to be very useful.**˚ 5˚**ENTROPY, DECOHERENCE, ENTANGLEMENT • system (S) + environment (E) + observer (O) • E interacts very weakly with O: unobservable • O sees a reduced density matrix: • Tracing over E is not unitary: destroys entanglement; responsible for decoherence & entropy generation • DIVISION S-E can be in physical space: traditional entropy; black holes; CFTs Srednicki, 1992**˚ 6˚**CORRELATOR APPROACH TO DECOHERENCE BASED ON (UNITARY, PERTURBATIVE) EVOLUTION OF 2-pt FUNCTIONS (in field theory or quantum mechanics) Koksma, Prokopec, Schmidt (‘09, ‘10), Giraud, Serreau (‘09) ADVANTAGES: • NEW INSIGHT: decoherence/entropy increase is due to unobservable • higher order correlations (non-gaussianities) in the S-E sector: • realisation of COARSE GRAINING. • evolution is in principle unitary: reduction of does not affect the evolution, i.e. it happens in the channel: O-S, and not S-E • (almost) classical systems tend to behave stochastically, i.e. • there is a stochastic force, kicking particles in unpredictable ways. • Examples: Solar Planetary System; Large scalar structure of the Universe**˚ 7˚**DECOHERENCE AND CLASSICIZATION A theory that explains how quantum systems become (more) classical Zeh (1970), Joost, Zurek (1981) & others Phase space picture: EARLY TIME t LATE TIME t’>t p(t) p(t’) x(t) x(t’) EVOLUTION: IRREVERSIBLE! – in discord with quantum mechanics! Decoherence has gained in relevance: EPR paradox; quantum computational systems**˚ 8˚**HARMONIC OSCILLATORS**˚ 9˚**BOSONIC OSCILLATORS (bHO) ● HAMILTONIAN & HAMILTON EQUATIONS ● GAUSSIAN DENSITY OPERATOR • NB: knowing (t), (t), (t) is equivalent to solving the problem exactly! ● THE FOLLOWING TRANSFORMATION DIAGONALISES :**˚10˚**BOSONIC OSCILLATOR: GAUSSIAN ENTROPY ● DIAGONAL DENSITY OPERATOR ● INTRODUCE A FOCK BASIS: ● IN THIS BASIS: ● Can relate parameters in (, , ) to correlators: ● AN INVARIANT OF A GAUSSIAN DENSITY OPERATOR**˚11˚**GAUSSIAN ENTROPY ● in terms of and ● is an invariant measure (statistical particle number) of the phase space volume of the state in units of ħ/2. ENTROPY GROWTH IS THUS PARAMETRIZED BY THE GROWTH OF THE PHASE SPACE AREA (in units of ħ) (t) p q ● is the probability that there are n particles in the state.**˚12˚**ENTROPY FOR 1+1 bHO ►UNITARY EVOLUTION (black); REDUCED EVOLUTION (gray) ► LEFT: nonresonant regime; RIGHT: resonant regime (PERT. MASTER EQ) (ENTROPY) TIME TIME NB: relatively small Poincaré recurrence time. • NB: grey: UNPHYSICAL SECULAR • GROWTH AT LATE TIMES • NB: If initial conditions are Gaussian, the evolution is linear and will preserve Gaussianity. Scorr will be generated by <xq>0 correlators.**˚13˚**ENTROPY FOR 50+1 bHO ►UNITARY EVOLUTION (black); REDUCED EVOLUTION (gray) ► LEFT: nonresonant regime; RIGHT: resonant regime (PERT. MASTER EQ) (ENTROPY) TIME TIME • NB: gray: UNPHYSICAL SECULAR • GROWTH AT LATE TIMES • (PERT. MASTER EQ) NB: exponentially large Poincaré recurrence time.**˚14˚**FERMIONIC OSCILLATORS (fHO) Tomislav Prokopec and Jan Weenink, in preparation ● LAGRANGIAN & EQUATIONS OF MOTION FOR fHOs ● DENSITY OPERATOR FOR fHO ..or: ● DENSITY OPERATOR IN THE FOCK SPACE REPRESENTATION**˚15˚**ENTROPY OF FERMIONIC OSCILLATOR ● INVARIANT PHASE SPACE AREA: : (statistical) number of particles ● ENTROPY OF fHO ALSO FOR FERMIONS: ENTROPY IS PARAMETRIZED BY THE PHASE SPACE INVARIANT (in units of ħ) (can be >0 or <0)**˚16˚**ENTROPY FOR 1+1 fHO ► LEFT PANEL: WEAK COUPLING RIGHT: STRONG COUPLING ENTROPY ENTROPY TIME TIME NB1: MAX ENTROPY ln(2) approached, but never reached. • NB2: For 2 oscillators, small Poincare recurrence time: quick return to initial state.**˚17˚**ENTROPY FOR 50+1 fHO ► LEFT: LOW TEMPERATURE RIGHT: HIGH TEMPERATURE random frequencies i[0,5]0 ENTROPY TIME TIME evenly distributed frequencies i[0,5]0 NB: exponentially large Poincaré recurrence time. When i<<1, Smax=ln(2) reached**˚18˚**ENTROPY AND DECOHERENCE IN FIELD THEORIES**˚19˚**TWO INTERACTING SCALARS ACTION: Can solve pertubatively for the evolution of (S) and (E) • O only sensitive to (near) coincident Gaussian (2pt) correlators. Cubic interaction generates non-Gaussian S-E correlations: Sng,corr, e.g. 3pt fn: NB: Expressible in terms of (non-coincident!) Gaussian S-E (2pt) correlators**˚20˚**EVOLUTION EQUATIONS Kadanoff, Baym (1961); Hu (1987) In the in-in formalism: the keldysh propagator i is a 2x2 matrix: ► are the time ordered (Feynman) and anti-time ordered propagators ► are the Wightman functions ► is the self-energy (self-mass). At one loop: ► are the thermal correlators. Solve the above KB Eq.: spatially homogeneous limit; m=0 PROBLEM: scattering in presence of thermal bath**˚20˚**QUANTUM FIELD THEORY: 2 SCALARS 1 LOOP SCHWINGER-DYSON EQUATION FOR & : = + = + + NB: INITIALLY we put in a pure state at T=0 (vacuum) & in a thermal state at temp. T STATISTICAL & CAUSAL CORRELATORS: 1 LOOP KADANOFF-BAYM EQUATIONS (in Schwinger-Keldysh formalism): ► are the renormalised `wave function’ and self-masses**˚21˚**RESULTS FOR SCALARS**˚23˚**STATISTICAL CORRELATOR AT T>0 LOW TEMPERATURE HIGH TEMPERATURE ► t-t’: DECOHERENCE DIRECTION**˚24˚**PHASE SPACE AREA AND ENTROPY AT T>0 TIME TIME HIGH TEMPERATURE LOW TEMPERATURE ► Entropy reaches a value Smswe can (analytically) calculate.**˚25˚**DECOHERENCE RATE @ T>0 ► decoherence rate can be well approximated by perturbative one-particle decay rate:**˚26˚**MIXING FERMIONS EQUATION OF MOTION (homogeneous space): Helicity h is conserved: work with 2 spinors . Diagonalise: ENTROPY ..can be diagonalised a (diagonal) Fock representation:**˚27˚**ENTROPY OF FERMIONIC FIELDS ● FERMIONIC ENTROPY: FOR FERMIONIC FIELDS: ENTROPY PER DOF ALSO PARAMETRIZED BY THE PHASE SPACE INVARIANT**˚28˚**RESULTS FOR FERMIONS**˚29˚**ENTROPY OF TWO MIXING FERMIONS ● TOTAL ENTROPY OF THE SYSTEM FIELD ► LEFT PANEL: LOW TEMP. 0=1RIGHT: HI TEMP:0=1/2 HI TEMP:0=1/10 ● TERMALISATION RATE**˚30˚**APPLICATIONS TO NEUTRINOS**˚31˚**NEUTRINOS Mark Pinckers, Tomislav Prokopec, in preparation There are 3 active (Majorana) left-handed neutrino species, that mix and possibly violate CP symmetry. Majorana condition implies that each neutrino has 2 dofs (helicities): IN GAUSSIAN APPROXIMATION, ONE CAN DEFINE GENERAL INITIAL CONDITIONS FOR NEUTRINOS IN TERMS OF EQUAL TIME STATISTICAL CORRELATORS:**˚32˚**NEUTRINO OSCILLATIONS IF INITIALLY PRODUCED IN A DEFINITE FLAVOUR, NEUTRINOS DO OSCILLATE: INITIAL MUON • BLUE = MUON ; • RED = TAU ; • BLACK=ELECTRON OSCILLATIONOS ARE A MANIFESTATION OF QUANTUM COHERENCE, BUT ARE NOT GENERIC! INITIAL ELECTRON **˚34˚**NEUTRINOS NEED NOT OSCILLATE WE FOUND GENERAL CONDITIONS ON F’s UNDER WHICH NEUTRINOS DO NOT OSCILLATE. EXAMPLES (WHEN MAJORANA NEUTRINOS DO NOT OSCILLATE): EXAMPLE A: other (mixed) correllators vanish. Q: can one construct such a state in laboratory? NB: albeit neutrinos coming e.g. from the Sun are coherent and do oscillate, when averaged over the source localtion, oscillations tend to cancel, and one observes neutrino deficit, but no oscillations.**˚34˚**COSMIC NEUTRINO BACKGROUND EXAMPLE B: thermal cosmic neutrino background (CB): Current temperature: In flavour diagonal basis: NB1: CB neutrinos do not oscillates (by assumption) NB2: CB violates both lepton number and helicity and CB contains a calculable lepton neutrino condensate. NB3: A similar story holds for supernova neutrinos (they are believed to be approximately thermalised). NB4: Can construct a diagonal thermal density matrix for CB (that is neither diagonal in helicity nor in lepton number) APPLICATIONS: Need to understand better how neutrinos affect CB**˚35˚**CONCLUSIONS DECOHERENCE: the physical process by which quantum systems become (more) classical, i.e. they become classical stochastic systems. Von Neumann entropy (of a suitable reduced sub-system) is a good quantitative measure of decoherence, and can be applied to both bosonic and fermionic systems. Correlator approach to decoherence is based on perturbative evolution of 2 point functions & neglecting observationally inaccessible (non-Gaussian) correlators. Our methods permit us to study decoherence/classicization in realistic (quantum field theoretic) settings. There is no classical domain in the usual sense: phase space area – and therefore the `size’ of the system – never decreases in time. Particular realisations of a stochastic system (recall: large scale structure of our Universe) behave (very) classically.**APPLICATIONS**˚35b˚ Classicality of scalar & tensor cosmological perturbations (observable in CMB?) Baryogenesis: CP violation (requires coherence) Quantum information Thermal cosmic neutrino background: - relation to lepton number and baryogenesis via leptogenesis Lab experiments on neutrinos; neutrinos from supernovae**˚36˚**INTUITIVE PICTURE: WIGNER FUNCTION WIGNER FUNCTION: GAUSSIAN STATE (momentum space: per mode): p ENTROPY ~ effective phase space area of the state**˚37˚**WIGNER FUNCTION: SQUEEZED STATES PURE STATE (=1,Sg=0) MIXED STATE (>1,Sg>0) NB: ORIGIN OF ENTROPY GROWTH: neglected S-E (nongaussian) correlators STATISTICAL ENTROPY: GENERALISED UNCERTAINTY RELATION:**˚38˚**WIGNER FUNCTION AS PROBABILITY GAUSSIAN ENTROPY: WIGNER ENTROPY (Wigner function = quasi-probability) • THE AMOUNT OF QUANTUMNESS IN THE STATE: the difference of the two entropies:**˚39˚**WIGNER FUNCTION OF NONGAUSSIAN STATE POSITIVE KURTOSIS : NEGATIVE KURTOSIS : Q: can non-Gaussianity – e.g. a negative curtosis – break the Heiselberg uncertainty relation? Naïve Answer: YES(!?); but it is probably wrong.**˚40˚**CLASSICAL STOCHASTIC SYSTEMS BROWNIAN PARTICLE (3 dim) ● exhibits walk of a drunken man/woman ● distance traversed: d ~ t DISTRIBUTION OF GALAXIES IN OUR UNIVERSE (2dF): ● amplified vacuum fluctuations ● we observe one realisation (breaks homogeneity of the vacuum) NB: first order phase transitions also spont. break spatial homogeneity of a state. NB2: planetary systems are stochastic, and essentially unstable.**˚41˚**RESULTS: CHANGING MASS**˚42˚**CHANGING MASS CASE ► RELEVANCE: ELECTROWEAK SCALE BARYOGENESIS: axial vector current is generated by CP violating scatterings of fermions off bubble walls in presence of a plasma. ►Since the effect vanishes when ħ0, quantum coherence is important. ►ANALOGOUS EFFECT: double slit with electrons in presence of air PROBLEMS: ►non-equilibrium dynamics in a plasma at T>0; ►non-adiabatically changing mass term; BUBBLE WALL: m²(t) ►apply to Yukawa coupled fermions. TIME: t**˚43˚**CHANGING MASS: STATISTICAL PROPAGATOR ► NOTE: ADDITIONAL OSCILLATORY STRUCTURE**˚44˚**DELTA: FREE CASE, CHANGING MASS ►the state gets squeezed, but the phase space area is conserved EXACT SOLUTION: in terms of hypergeometric functions ► CONSTANT GAUSSIAN ENTROPY Pure + frequency mode at t- becomes a mixture of + & - frequency solutions at t+ Mixing amplitude: (t) TIME Particle production:**˚45˚**MASS CHANGE AT T>0 LOW T MASS INCREASE: T=/2, k=,h=4, m=2 LOW T MASS DECREASE: T=/2, k=,h=4,m=2 time time NB: ENTROPY CHANGES AT THE ONE PARTICLE DECAY RATEdec NB2: MASS CHANGES MUCH FASTER THAN ENTROPY:**˚46˚**MASS CHANGE AT T>0 HIGH T MASS INCREASE: T=2, k=, h=3, m=2 HIGH T MASS DECREASE: T=2, k=, h=3,m=2 time**˚47˚**EVOLUTION OF SQUEEZED STATES ► of relevance for baryogenesis: changing mass induces squeezing (coherent effect) HIGH T: 2r=ln(5), =/2 T=2m, h=3m, k=m LOW T: 2r=ln(5), =0 T=2m, h=3m, k=m time time NB: ADDITIONAL OSCILLATIONS DECAY AT THE RATE = dec. QUANTUM COHERENCE IS NOT DESTROYED BY THERMAL EFFECTS. CONJECTURE: THIN WALL BG UNAFFECTED BY THERMAL EFFECTS. ► related work: Herranen, Kainulainen, Rahkila (2007-10)**˚48˚**KADANOFF-BAYM EQUATIONS IMPORTANT STEPS: calculate 1 loop self-masses renormalise using dim reg solve for the causal and statistical correlators (must be done numerically, since it involves memory effects) calculate the (gaussian) entropy of (S) KB equations can be written in a manifestly causal and real form: Berges, Cox (1998); Koksma, TP, Schmidt (2009) ► here: m² is the renormalised mass term (the only renormalisation needed at 1loop) ► are the renormalised `wave function’ and self-masses**˚49˚**SELF-MASSES LOCAL VACUUM MASS COUNTERTERM RENORMALISED VACUUM SELF-MASSES ► CURIOUSLY: we could not find these expressions in literature or textbooks ► there are also thermal contributions to the self-masses (which are complicated) ► there is also the subtlety with KB eqs: in practice t0=- should be made finite. But then there is a boundary divergence at t=t0, which can be cured by (a) adiabatically turning on coupling h, or (b) by modifying the initial state.

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