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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
● CB
● 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 ) n1
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  ˆjˆ 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        m2 2 , S    d D x         m2  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
 m2 iab
(
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
► iab
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 , ic  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  φχχ


 1  exp   (  k ) / 2
1 h2
h2


log 
 1  exp   (  k ) / 2
32  16k




˚26˚
MIXING FERMIONS
EQUATION OF MOTION (homogeneous space):


 
ˆ
(  t    k  m)ˆ (k , t )  j (k , t ),
0
N


ˆj (k , t )   moiˆ 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'  ' kk ' ,  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 kh (t ), S kh (t )  (1  nkh ) ln( 1  nkh )  nkh ln( nkh ), nkh 

kh 
S


kh 

kh 
(t )  
1   kh
2
1   kh
2
 [0,1]
 1   kh  1   kh  1   kh 
 
,  kh [1,1]
ln 
ln 
2
 2 
 2 
(t )  2Fh (k ; t; t )  1  2n

kh 



 kh 
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  kh 
[ kh  ]

 kh  (t )
[ kh 
˚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 105 eV, m23
 m132  2.32 103 eV
sin 2 (213 )  0.10, sin 2 (2 23 )  0.97, sin 2 (212 )  0.861

P12 sin 2 (212 ) 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 ihˆ 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 (CB):
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: CB neutrinos do not oscillates (by assumption)
NB2: CB violates both lepton number and helicity
and CB 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 CB
(that is neither diagonal in helicity nor in lepton number)
APPLICATIONS: Need to understand better how neutrinos affect CB
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 ħ0, 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
2in 
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 RATEdec
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)


ic (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  m2 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 ' )  ZF (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)
► Zc,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 )iZab (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
  ikt 

 ikt

 
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+m² )
˚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:
ic (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 xp=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