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˚ 1˚ ENTROPY AND DECOHERENCE IN QUANTUM THEORIES Tomislav Prokopec, ITP & Spinoza Institute, Utrecht University 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 Nikhef, Mar 30 2012 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 ˚ 2˚ VON NEUMANN ENTROPY ˚ 3˚ von Neumann entropy (of a closed system): S vN (t ) Tr[ ˆ ln( ˆ )], ˆ (t ) density operator ..OBEYS A HEISENBERG EQUATION: t ˆ (t ) [ Hˆ , ˆ ] CLOSED SYSTEM as a result, von Neumann entropy is conserved: d S vN (t ) 0 S vN (t ) constant. dt S vN (t ) const. 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! OPEN SYSTEMS ˚ 4˚ ◙ 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 E S von Neumann entropy is not any more conserved d S vN (t ) 0 dt S vN (t ) increases (?) in time 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. ENTROPY, DECOHERENCE, ENTANGLEMENT ˚ 5˚ system (S) + environment (E) + observer (O) E interacts very weakly with O: unobservable O sees a reduced density matrix: ˆ red TrE [ ˆS E ] S vN (t ) Tr[ ˆ red ln( ˆ red )] Tracing over E is not unitary: destroys entanglement; responsible for decoherence & entropy generation 2 Tr[ ˆ red ] Tr[ ˆ red ] 1 DIVISION S-E can be in physical space: traditional entropy; black holes; CFTs Srednicki, 1992 CORRELATOR APPROACH TO DECOHERENCE ˚ 6˚ 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 p(t) LATE TIME t’>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 BOSONIC OSCILLATORS (bHO) ˚ 9˚ ● HAMILTONIAN & HAMILTON EQUATIONS 2 N ˆ p 1 2 2 Hˆ (t ) m qˆ Hˆ int (t ), Hˆ int (t ) i qˆqi 2m 2 i 1 d pˆ (t ) q Hˆ , dt d qˆ (t ) p Hˆ dt ● GAUSSIAN DENSITY OPERATOR 1 exp 12 (t ) pˆ 2 (t )qˆ 2 (t ){qˆ , pˆ } , [qˆ , pˆ ] , ( 1) Z NB: knowing (t), (t), (t) is equivalent to solving the problem exactly! ˆ g (t ) ● THE FOLLOWING TRANSFORMATION DIAGONALISES : bˆ(t ) ˆ 1 q 2 ˆp , bˆ (t ) ˆ 1 q 2 ˆp , [bˆ, bˆ ] 1 ˚10˚ BOSONIC OSCILLATOR: GAUSSIAN ENTROPY ● DIAGONAL DENSITY OPERATOR ˆ g (t ) 1 exp (t )( Nˆ 1 / 2) , 2 , Nˆ bˆ bˆ, Z 1 1 e σ Z ● INTRODUCE A FOCK BASIS: n , n 0,1,.., I n n n 0 ● IN THIS BASIS: ˆ g (t ) n nn ˆ 1 n n n , n (t ) , n ( t ) N (1 n ) n1 e 1 ● Can relate parameters in (, , ) to correlators: 1 p 2 2 ln( Z ) n , 2 1 qˆ 2 2 ln( Z ) n , 2 qˆ , pˆ 1 2 n 2 ● AN INVARIANT OF A GAUSSIAN DENSITY OPERATOR 2 4 qˆ 2 ˆ2 p 1 2 qˆ , pˆ 2 1 2n 2 coth 2 2 GAUSSIAN ENTROPY ● in terms of and ˚11˚ n (t ) Nˆ 1 1 1 1 ln ln 2 2 2 2 S (n 1) ln n 1 n ln n S Tr ˆ g (t ) ln( ˆ g (t )) 1 n (t ) Nˆ 2 ● is an invariant measure (statistical particle number) of the phase space volume of the state in units of ħ/2. p q ENTROPY GROWTH IS THUS PARAMETRIZED BY THE GROWTH OF THE PHASE SPACE AREA (in units of ħ) (t) ● n (t )n n n n /(1 n ) n 1 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; (ENTROPY) RIGHT: resonant regime (PERT. MASTER EQ) SE SS SS TIME SS TIME SS-E 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. ENTROPY FOR 50+1 bHO ˚13˚ ►UNITARY EVOLUTION (black); REDUCED EVOLUTION (gray) ► LEFT: nonresonant regime; RIGHT: resonant regime (PERT. MASTER EQ) S S (ENTROPY) SS SS TIME TIME NB: gray: UNPHYSICAL SECULAR GROWTH AT LATE TIMES (PERT. MASTER EQ) NB: exponentially large Poincaré recurrence time. FERMIONIC OSCILLATORS (fHO) ˚14˚ Tomislav Prokopec and Jan Weenink, in preparation ● LAGRANGIAN & EQUATIONS OF MOTION FOR fHOs 1 1 ˆ ˆ L (t ) t t (t ) ˆ ˆjˆ ˆ ˆj , 2 2 N ˆj (t ) i (t )ˆ i i 1 t (t )ˆ ˆj , t (t )ˆ ˆj ● DENSITY OPERATOR FOR fHO ˆ (t ) ..or: ˆ ,ˆ 1, 1 exp a(t ) Nˆ , Z ˆ (t ) (1 n ) (2n 1) Nˆ , n Nˆ ˆ ˆ , Nˆ 2 Nˆ , Z 1 e a 1 1 , n (a ) th a e 1 e 1 ● DENSITY OPERATOR IN THE FOCK SPACE REPRESENTATION ˆ (t ) 0 (1 n ) 0 1 n 1 , Fock space : 0 , 1 ˚15˚ ENTROPY OF FERMIONIC OSCILLATOR ● INVARIANT PHASE SPACE AREA: 1 a (t ) 2 F (t ; t ) 1 2n (t ) tanh , F (t ; t ' ) ˆ (t ),ˆ (t ' ) 2 2 n (t ) : (statistical) number of particles ● ENTROPY OF fHO S (t ) 1 2 1 ln 2 1 1 ln 2 2 S (t ) (1 n ) ln( 1 n ) n ln( n ), n , [1,1] 1 2 [0,1] ALSO FOR FERMIONS: ENTROPY IS PARAMETRIZED BY THE PHASE SPACE INVARIANT (in units of ħ) (can be >0 or <0) [ ] (t ) [ ] ˚16˚ ENTROPY FOR 1+1 fHO ► LEFT PANEL: WEAK COUPLING ENTROPY RIGHT: STRONG COUPLING 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 TWO INTERACTING SCALARS ACTION: ˚19˚ S S S Sint 1 1 1 1 S d D x m2 2 , S d D x m2 2 2 2 2 2 1 Sint d D x h 2 3 , 3! 2 Can solve pertubatively for the evolution of (S) and (E) Stot SS SE Scorr 0; SS Sg,S Sng,S 0, Scorr Sg,corr Sng,corr Icorr 0 O only sensitive to (near) coincident Gaussian (2pt) correlators. Cubic interaction generates non-Gaussian S-E correlations: Sng,corr, e.g. 3pt fn: ( x) ( x' ) ( x' ' ) ~ h d D y ( x) ( y ) ( x' ) ( y ) ( x' ) ( y ) NB: Expressible in terms of (non-coincident!) Gaussian S-E (2pt) correlators ˚20˚ EVOLUTION EQUATIONS Kadanoff, Baym (1961); Hu (1987) 2 ac cb 3 m2 iab ( x ; x ' ) d yM ( x ; y ) ci ( y ; x ' ) D ab i D ( x x' ), a, b c In the in-in formalism: the keldysh propagator i is a 2x2 matrix: i i i i i ► i , i are the time ordered (Feynman) and anti-time ordered propagators ► i , i are the Wightman functions ► M ab is the self-energy (self-mass). At one loop: 2 ih 2 ab iM ( x; y ) i ( x; y) 2 ► iab are the thermal correlators. ( x; y ) ab 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: F 1 1 i i , ic i i 2 2 1 LOOP KADANOFF-BAYM EQUATIONS (in Schwinger-Keldysh formalism): 2 t 2 t m F (k , t , t ' ) k2 2 2 c d Z (k , t, ) (k , , t ' ) d M (k , t, ) (k , , t ' ) 0 d Z (k , t, ) F (k , , t ' ) Z (k , t , ) (k , , t ' ) d M (k , t , ) F (k , , t ' ) M (k , t , ) (k , , t ' ) 0 t k m (k , t , t ' ) k 2 2 t 2 t c c c t t' 2 t k2 c , th t' c F c t c , th F , th c, F c, F ► Z , M are the renormalised `wave function’ and self-masses c ˚21˚ RESULTS FOR SCALARS STATISTICAL CORRELATOR AT T>0 F (k , t ; t ' ) HIGH TEMPERATURE LOW TEMPERATURE ► t-t’: DECOHERENCE DIRECTION ˚23˚ PHASE SPACE AREA AND ENTROPY AT T>0 TIME TIME LOW TEMPERATURE ► Entropy reaches a value ˚24˚ HIGH TEMPERATURE Sms we can (analytically) calculate. DECOHERENCE RATE @ T>0 ˚25˚ ► decoherence rate can be well approximated by perturbative one-particle decay rate: dec ( ms ) 0 dec φχχ 1 exp ( k ) / 2 1 h2 h2 log 1 exp ( k ) / 2 32 16k ˚26˚ MIXING FERMIONS EQUATION OF MOTION (homogeneous space): ˆ ( t k m)ˆ (k , t ) j (k , t ), 0 N ˆj (k , t ) moiˆ i (k , t ) i 1 Helicity h is conserved: work with 2 spinors ˆ h (k , t ) . Diagonalise: hk 1 m 3 ˆ ( t )ˆ h (k , t ) jh (k , t ), k 2 | m |2 0 3 ENTROPY ˆ h (k , t ),ˆ h ' ' (k ' , t ) hh' ' kk ' , h *11 12 12 22 1 ˆ (t ) exp ˆ h (k , t ) h (k , t )ˆ h (k , t ), Z k ,h ..can be diagonalised ˆ (t ) 1 d d d ˆ ˆ exp h (k , t ) h (k , t ) h (k , t ), Z k ,h dh 0 a (diagonal) Fock representation: ˆ (t ) ˆ kh kh (t ), ˆ kh (t ) (1 n kh ) (2n kh ˆ ˆ 1) N kh , N kh ˆ h (k , t )ˆ h (k , t ) 0 ENTROPY OF FERMIONIC FIELDS ˚27˚ ● FERMIONIC ENTROPY: S (t ) S kh (t ), S kh (t ) (1 nkh ) ln( 1 nkh ) nkh ln( nkh ), nkh kh S kh kh (t ) 1 kh 2 1 kh 2 [0,1] 1 kh 1 kh 1 kh , kh [1,1] ln ln 2 2 2 (t ) 2Fh (k ; t; t ) 1 2n kh kh 1 , Fh (k ; t; t ' ) ˆ h (k ; t ),ˆ h (k ; t ' ) (t ) tanh 2 2 FOR FERMIONIC FIELDS: ENTROPY PER DOF ALSO PARAMETRIZED BY THE PHASE SPACE INVARIANT kh [ kh ] kh (t ) [ kh ˚28˚ RESULTS FOR FERMIONS ˚29˚ ENTROPY OF TWO MIXING FERMIONS ● TOTAL ENTROPY OF THE SYSTEM FIELD ► LEFT PANEL: LOW TEMP. 0=1 ● TERMALISATION RATE T RIGHT: HI TEMP: 0=1/2 HI TEMP: 0=1/10 ˚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): c C ( ) T 2 * IN GAUSSIAN APPROXIMATION, ONE CAN DEFINE GENERAL INITIAL CONDITIONS FOR NEUTRINOS IN TERMS OF EQUAL TIME STATISTICAL CORRELATORS: 1 Fhij (k ; t ; t ) ˆ hi (k ; t ),ˆ hj (k ; t ' ) 2 t ' t , i, j 1,2 (flavour), t' t NEUTRINO OSCILLATIONS ˚32˚ IF INITIALLY PRODUCED IN A DEFINITE FLAVOUR, NEUTRINOS DO OSCILLATE: 2 m122 7.6 105 eV, m23 m132 2.32 103 eV sin 2 (213 ) 0.10, sin 2 (2 23 ) 0.97, sin 2 (212 ) 0.861 P12 sin 2 (212 ) sin 2 m2 x / 4E INITIAL MUON BLUE = MUON ; RED = TAU ; BLACK=ELECTRON INITIAL ELECTRON OSCILLATIONOS ARE A MANIFESTATION OF QUANTUM COHERENCE, BUT ARE NOT GENERIC! NEUTRINOS NEED NOT OSCILLATE ˚34˚ WE FOUND GENERAL CONDITIONS ON F’s UNDER WHICH NEUTRINOS DO NOT OSCILLATE. EXAMPLES (WHEN MAJORANA NEUTRINOS DO NOT OSCILLATE): EXAMPLE A: ni ni 1, (i 1,2) & 1 2 ˆ i ˆ i 0, nih (k , t ) ˆ ihˆ ih Fih (k , t ) 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: T (4 / 11)1/ 3 T 1.95K, T 2.73K In flavour diagonal basis: 1 1 ˆ i ,ˆ i ˆ i ,ˆ i Fi Fi 1 ni ni , (i 1,2) 2 2 kh kh 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 CONCLUSIONS ˚35˚ 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 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 ˚35b ˚ ˚36˚ INTUITIVE PICTURE: WIGNER FUNCTION WIGNER FUNCTION: W[ x , p, t ] D( x x' ) e ip( x x ') [ x, x' , t ], x ( x x' ) / 2 GAUSSIAN STATE (momentum space: per mode): p 2 4 2 2 F(t, t) t t'F(t, t') t't tF(t, t') t't x ENTROPY ~ effective phase space area of the state Sg 1 1 1 1 Log Log 2 2 2 2 ˚37˚ WIGNER FUNCTION: SQUEEZED STATES PURE STATE (=1,Sg=0) MIXED STATE (>1,Sg>0) EVOLUTION NB: ORIGIN OF ENTROPY GROWTH: neglected S-E (nongaussian) correlators STATISTICAL ENTROPY: Sg (n 1) Log n 1 n Log n , n GENERALISED UNCERTAINTY RELATION: 4 1 2 x2 p2 x, p 2 2 2 1 n 0, S 0 1 : uncorr.regions 2 ˚38˚ WIGNER FUNCTION AS PROBABILITY GAUSSIAN ENTROPY: Sg (n 1) ln( n 1) n ln( n) ln( n) 1 1 1 O(n 2 ), n 2n 2 WIGNER ENTROPY (Wigner function = quasi-probability) S W ln( n) 1 THE AMOUNT OF QUANTUMNESS IN THE STATE: the difference of the two entropies: S Sg SW 1 1 2 O(n 3 ), n 1 2n 6n ˚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. 4 1 2 x2 p2 x, p 2 2 2 1 n 0, S 0 ˚40˚ CLASSICAL STOCHASTIC SYSTEMS DISTRIBUTION OF GALAXIES IN OUR UNIVERSE (2dF): ● amplified vacuum fluctuations ● we observe one realisation (breaks homogeneity of the vacuum) BROWNIAN PARTICLE (3 dim) ● exhibits walk of a drunken man/woman ● distance traversed: d ~ t 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 CHANGING MASS CASE ˚42˚ ► 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; ► apply to Yukawa coupled fermions. BUBBLE WALL: m²(t) TIME: t CHANGING MASS: STATISTICAL PROPAGATOR ► NOTE: ADDITIONAL OSCILLATORY STRUCTURE ˚43˚ DELTA: FREE CASE, CHANGING MASS ˚44˚ ► the state gets squeezed, but the phase space area is conserved ► CONSTANT GAUSSIAN ENTROPY TIME EXACT SOLUTION: in terms of hypergeometric functions Pure + frequency mode at t- becomes a mixture of + & - frequency solutions at t+ Mixing amplitude: (t) Particle production: n k 2 sinh 2 / 1 , out in sinh in / sinh out / 2 2 in 1 , in , out n 2in 2 MASS CHANGE AT T>0 LOW T MASS INCREASE: T=/2, k=, h=4, m= 2 ˚45˚ 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: m / m dec 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 ˚46˚ EVOLUTION OF SQUEEZED STATES ˚47˚ ► 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) KADANOFF-BAYM EQUATIONS ˚48˚ 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) ic (k , t , t ' ) i i (k , t , t ' ), F (k , t , t ' ) 1 i i (k , t , t ' ) 2 calculate the (gaussian) entropy of (S) Sg 1 1 1 1 Log Log 2 2 2 2 2 4 2 F(t, t) F(t, t') t t' 2 tF(t, t') t't t't KB equations can be written in a manifestly causal and real form: Berges, Cox (1998); Koksma, TP, Schmidt (2009) 2 t 2 c d Z t k m (k , t , t ' ) k 2 2 t 2 t c t' 2 t k 2 m2 F (k , t , t ' ) t2 k 2 t' d Z t (k , t , ) (k , , t ' ) d M c,th (k , t , )c (k , , t ' ) 0 c t c (k , t , ) F (k , , t ' ) ZF (k , t , )c (k , , t ' ) d M c, th (k , t , ) F (k , , t ' ) M F, th (k , t , )c (k , , t ' ) 0 ► here: m² is the renormalised mass term (the only renormalisation needed at 1loop) ► Zc,F , M c,F are the renormalised `wave function’ and self-masses SELF-MASSES ˚49˚ LOCAL VACUUM MASS COUNTERTERM D ih 2 1 D 4 2 iM c,ct ( x; x' ) i D ( x x' ) D/2 16 ( D 3)( D 4) RENORMALISED VACUUM SELF-MASSES iM ab (k , t , t ' ) (t2 k 2 )iZab (k , t , t ' ) iZ iZ h2 64 2 ik|t| ik|t| k ci( 2k | t |) i si( 2k | t |) e E log 2 2 | t | i 2 e h2 64k 2 ikt ikt k e log i sign( t ) e ci( 2 k | t |) i sign( t ) si( 2 k | t |) 2 2 | t | E 2 ► 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. PHASE SPACE AREA AND ENTROPY AT T=0 h=4m, k= m ˚50˚ ms ENTROPY TIME TIME ► evolution towards the new (interacting) vacuum with stationary ms (calculated) ms ► initial conditions `forgotten’ ► ms reached at perturbative rate =decoherence/entropy growth rate: dec pert,tree 1 h2 32 ► wiggles (in part) due to imperfect memory kernel TIME ENTROPY AT T>0 ● ENTROPY ˚51˚ ● ms as a function of coupling h, T=2m, k=m LOW TEMPERATURE vs VACUUM CASE: T= m /10 (black) & T=0 (gray), h=4m, k=m NB: COUPLING h IS PERTURBATIVE UP TO h~3 (k²+m² ) ˚52˚ TWO POINT FUNCTION QUANTUM COMPUTATION ˚53˚ Feynman; Shore (factoring into primes) CLASSICAL LOGICAL GATES E.g. NAND GATE 00 10 01 11 10 10 11 01 NOT GATE 1 0 0 1 QUANTUM LOGICAL GATES 2 STATE SYSTEM WAVE FUNCTION: 2 2 0 1 , 1 {1,0} Bloch sphere: {{,} | ||²+||²=1} quantum NOT GATE {0,1} 0 1 NOT 1 1 0 * general q-gate: any `rotation’ on the Bloch sphere; e.g. Pauli matrices: rotation around x, y and z axes) MAIN PROBLEM of quantum computation: how to reduce decoherence of q-gates ˚54˚ A MEASURE OF DECOHERNECE: GAUSSIAN VON NEUMANN ENTROPY CAUSAL (SPECTRAL) FUNCTION (PAULI-JORDAN, SCHWINGER) 2-pt GREEN FUNCTION: ic (t ' ; t ) Tr( [x(t), x(t' )]) STATISTICAL (HADAMARD) 2-pt GREEN FUNCTION: 1 F (t ' ; t ) Tr {x(t), x(t' )} 2 PROGRAM: one solves the perturbative dynamical equations for c & F of S+E one calculates the Gaussian von Neumann entropy Sg of S: Sg 1 1 1 1 Log Log 2 2 2 2 2 2 F( t ; t ) F( t ; t ' ) F( t ; t ' ) t t' t t ' t t 't 2 Gaussian density matrix: gauss ( x, x' ; t ) N exp a(t ) x 2 b(t ) x'2 2c(t ) xx' 2 (t ) INTERMEDIATE SUMMARY ˚55˚ CONVENTIONAL APPROACH: S+E E weakly coupled SS Tr red log red 0 Evolve red TrE NEW FRAMEWORK: S+E E weakly coupled Stot SS SE Scorr 0; Evolve 2pt correlators for S & E: c , F perturbatively SS Sg,S Sng,S 0, Sg,S Tr S log S 0 Scorr Sg,corr Sng,corr Icorr 0 BROWNIAN PARTICLE ˚56˚ DYNAMICS: LANGEVIN EQUATION mv v V ' ( x) F (t ), F (t ) F (t ' ) 2 kBT (t t ' ) ► Describes motion of a Brownian particle (Einstein); of a drunken man/woman; also: inflaton fluctuations during inflation (Starobinsky; Woodard; Tsamis; TP) ► v=dx/dt; F(t)=Markovian (noise), V(x)= potential, = friction coefficient WHEN V(x)=0: t LATE TIME ENTROPY: S (1 / 2) log t / t0 1, t 0 2 /[ m(k BT ) 2 ] grows without limit Q: How can we understand this unlimited growth of phase space area? ˚57˚ BROWNIAN PARTICLE 2 Consider a free moving quantum particle (described by a wave packet) Quantum evolution: preserves the minimum phase space area xp=ħ/2 EARLY TIME t p(t’) LATE TIME t’>t p(t) x(t) x(t’) 2 p BROWNIAN PARTICLE gets thermal kicks: keeps p constant! 2m But x keeps growing!: explains the (unlimited) growth of phase space area. k BT 2