T violation, direction of time and general relativity

11/28/28RQIWRQIW 09 09

T violation,direction of time and

general relativityJoan VaccaroGriffith University

0

BU FU

BU FU BU FU

BU FUBU FU BU FU

BU FU BU FU BU FUBU FU

( )N

RQIWRQIW 09 09

Arrows Paths through time Interference Unidirectionality General Relativity

22/28/28

Arrows of time ▀ Emerge from phenomenological time asymmetric dynamics

Cannot be derived from first principlesMust be inserted into physical theories by hand

past future

cosmological arrowbig bang expanding universe

electromagnetic arrow

thermodynamic arrow

psychological arrow

increasing entropy

no memory of the future

spontaneous emission

memory of the past

low entropy

excited atom

due to asymmetrical boundary conditions

RQIWRQIW 09 09


33/28/28

matter-antimatter excess ofmatter

balance ofmatter & antimatter

The matter-antimatter arrow - due to a small (0.2%) violation of CP & T invariance in neutral Kaon decay- discovered in 1964 by Cronin & Fitch (Nobel Prize 1980)- partially accounts for observed dominance of matter over antimatter.- dismissed as not directly affecting the nature of time or everyday life.

due to time asymmetric dynamics Time reversal operator

)(t )(ˆ tT

p

p

L

L

Wigner, Group theory (1959), Messiah, Quantum Mechanics (1961) Ch XV

ˆ ˆ ˆT U K

unitary operator- depends on spin

anti-unitary operator- action is complex conjugation

* *ˆ ˆ ˆ0 1 0 1K a b a K b K † † ˆˆ ˆ ˆ ˆ 1K K K K

RQIWRQIW 09 09


44/28/28

Fundamental question T inversion symmetry violation implies

How should one incorporate the two Hamiltonians, and , in one equation of motion?

1ˆ ˆ ˆ ˆT HT H H 1ˆ ˆ ˆT HT

Schrodinger’s equation Backwards evolution is simply backtracking the forwards evolution

ˆi Ht

ˆ ˆHT

1ˆ ˆ ˆ ˆ ˆi T T HT Tt

y

t

forwards

backwards

ˆ ( ) ( )T t t

1ˆ ˆ ˆ ˆT HT H

RQIWRQIW 09 09


55/28/28

Physical system:▀ composed of matter and fields in a

manner consistent with the visible portion of the universe

▀ the system is closed in the sense that it does not interact with any other physical system

▀ no external clock and so analysis needs to be unbiased with respect to the direction of time

▀ convenient to differentiate the two directions of time as "forwards" and "backwards”

Possible paths through time

Forwards and Backwards evolutionEvolution of state over time interval in the forward direction

where and = Hamiltonian for forward time evolution.

0

0ˆ( ) ( )F FU

ˆ ˆ( ) expF FU iH ˆ

FH( 1)

arXiv:0911.4528

RQIWRQIW 09 09


66/28/28

Evolution of state over time interval in the backward direction

where

and = Hamiltonian for backward time evolution.

0

0ˆ( ) ( )B BU

ˆ ˆ( ) expB BU iH 1ˆ ˆ ˆ ˆ

B FH T H T ( 1)

Internal clocks:▀ assume Hamiltonian of internal clocks is time reversal

invariant during normal operation

▀ This gives an operational meaning of the parameter as a time interval.

(clock)H

(clock) 1 (clock)ˆ ˆ ˆ ˆT H T H

Constructing paths:▀ and are probability amplitudes

for the system to evolve from to via two paths in

time ▀ we have no basis for favouring one path over the other

so attribute an equal weighting to each [Feynman RMP 20, 367 (1948)]

0ˆ ( )FU 0

ˆ ( )BU 0

0

BU FU

RQIWRQIW 09 09


77/28/28

Principle: The total probability amplitude for the system to evolve from one given state to another is proportional to the sum of the probability amplitudes for all possible paths through time.

The total amplitude for is proportional to

This is true for all states , so

which we call the time-symmetric evolution of the system. Time-symmetric evolution over an additional time interval of is given by

0ˆ ˆ( ) ( )F BU U

0

0ˆ ˆ( ) ( ) ( )F BU U

2

0ˆ ˆ ˆ ˆ(2 ) ( ) ( ) ( ) ( ) ( )F B F BU U U U

0

BU FU

0

BU FU

( )

0

RQIWRQIW 09 09


88/28/28

Repeating this for N such time intervals yields

0ˆ ˆ( ) ( ) ( )

N

F BN U U

0

BU FU

BU FU BU FU

BU FUBU FU BU FU


( )N0

0

ˆ( ) ( , )N

n

N S N n n

▀ is a sum containing different terms

▀ each term has factors of and factors of

▀ is a sum over a set of paths each

comprising

forwards steps and backwards steps

Let

ˆ( , )S N n n )(Nn

n ˆ ( )FU ˆ ( )BU N n

0ˆ( , )S N n n

n N n

ˆ( , )S N n n

RQIWRQIW 09 09


99/28/28

Consider the limit 0▀ fix total time and set . Take limit as .

▀ we find

tott tott

N N

2

tot

1 1ˆ ˆ ˆ ˆ( ) ( ) exp[ ( ) ] ( )2 2

1 ˆ ˆexp[ ( ) ] as 2

NN

F B F BN

F B

U U i H H O

i H H t N

effective Hamiltonian=0 for clock device no time in conventional sense

▀ Set to be the smallest physical time interval, Planck time

445 10 s

0

BU FU

BU FU BU FU

BU FUBU FU BU FU


( )N

RQIWRQIW 09 09


1010/28/28

InterferenceMultiple paths

0

BU FU

BU FU BU FU

BU FUBU FU BU FU


( )Nˆ(3,1)S

4 terms

interfere

ˆ(4, 0)S ˆ(0, 4)S

Example:

RQIWRQIW 09 09


1111/28/28

Use the Zassenhaus (Baker-Campbell-Hausdorff ) formula

for arbitrary operators and and parameter to get

We eventually find that

2 3ˆ ˆˆ êxp( )exp( )exp( [ , ] ( ))iB iA A B Oˆ êxp( )exp( )iA iB

2 3ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) exp [ , ] ( )B F F B F BU m U n U n U m nm H H O

2 3

0 0 0 0

ˆ ˆ( , ) exp ( ) exp

ˆ êxp ( ) [ , ] ( )

B FN n v k

F Bv u k j

S N n n i N n H in H

v u k j H H O

BA

nested

sums

Simplifying the expression forˆ( , )S N n n

Using eigenvalue equation for commutator

we find

F B[ , ]i H H

RQIWRQIW 09 09


1212/28/28

1 2

0

2

1

{exp[ ( ) ] 1}( , , )

[exp( ) 1]

n

kn

k

i N kI N n n

ik

where

ˆ ˆ ˆ ˆ( , ) [( ) ] ( ) ( , , ) ( ) ( )B FS N n n U N n U n I N n n d degeneracy

eigenvalue

trace 1 projection op.

( , , )I N n n

4

24

( )n N n N

RQIWRQIW 09 09


1313/28/28

Eigenvalues for j th kaon17 2 17 210 s or +10 sj

Eigenvalues for M kaons17 2 17 2 ranges from 10 s to + 10 sj

j

M M

17 2SD

110 s

2M 8010M f

57 2SD

110 s

2f

SDSD 0

Let

fraction

Estimating eigenvalues 0

0( ) ,

a Kdi i

bdt K

ψM Γ ψ ψ

RQIWRQIW 09 09


1414/28/28

Comparison of with ( , , )I N n n ( )

( , , )I N n n ( , , )I N n n

1dˆ (0)d

ˆ ˆ ˆ ˆ( , ) [( ) ] ( ) ( , , ) ( ) ( )B FS N n n U N n U n I N n n d

destructive

interference

0ˆ (0) 0

Assume

constructive

interference

RQIWRQIW 09 09


1515/28/28

Destructive interference

( ) 1n N n

1/3 231/3 2/3

SD

1 10 sN f

( , , )I N n n is much narrower

than

( )

forward steps backward steps

total time

0

BU FU

BU FU BU FU

BU FUBU FU BU FU


if

Consider:

( )N ˆ( , )S N n n


RQIWRQIW 09 09


1616/28/28

1/2 13 10 sN f

total time

if 0

BU FU

BU FU BU FU

BU FUBU FU BU FU


( )N

0ˆ ˆ( ) ( ) ( )F BN U N U N

Bi-evolution equation of motion

RQIWRQIW 09 09


1717/28/28

0ˆ ˆ( ) ( ) ( )F BN U N U N

Smoking gun: evidence left in the state

Unidirectionality of time

1( )t Let

1 0ˆ ( )BU t 1 0

ˆ ( )FU t

1 1 1 0

1 1 0

ˆ ˆ( ) ( ) ( )

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )

ˆ ˆ( ) ( )

F B

F F B B

F B

F B

t a U t a U t a

U a U t U a U t

U a U a

ˆ ( ) FB

U a FB

ˆ ( ) FB

U a FB

Hamiltonians and leave distinguishable evidence in state

if 0F B F B

ˆBHˆ

FH

RQIWRQIW 09 09


1818/28/28

0

BU FU

BU FU BU FU

BU FUBU FU BU FU


( )N

▀ and representevolution in opposite directions of time

▀ in each case corroborating evidence of

Hamiltonian is left in the state

1 1 1 0

1 1 0

, ,

ˆ ˆ( 2 ) ( 2 ) ( 2 )

ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( )

ˆ ˆ( ) ( )

F B

F F F B B B

F F B F

F F B B

t a U t a U t a

U a U a U t U a U a U t

U a U a

Repeating...

...leaves corroborating evidence in the state

1 0ˆ ( )BU t 1 0

ˆ ( )FU t Interpretation

Our experience

▀ Experiments give evidence of exactly one of the Hamiltonians ˆ

BHˆFH or

RQIWRQIW 09 09


1919/28/28

Compare with universe obeying T invariance

ˆ ˆ ˆF BH H H In this case

00

( ) exp[ ( 2 ) ]N

n

NN i N n H

n

0

U U

U U U U

U UU U U U

U U U U U UU U

( )N

Most likely paths

for2

Nn

▀ clocks don’t tick (show t=0 on average)▀ no physical evidence of direction of

time

RQIWRQIW 09 09


2020/28/28

Recall

What about zero eigenvalues?

0 0 0 Let

0ˆ (0) 0

0 0ˆ (0)

0

00

ˆ ˆ( ) ( ) ( )

ˆ ˆ ( ) ( )

B F

N

B Fn

N U N U Nt

NU N n U n

n

Then


mixed Hamiltonians – not observed

we see Hamiltonian of one of these branches

RQIWRQIW 09 09


2121/28/28

Schrödinger’s equation for bi-evolution

0ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )B B F Ft t U t U t U t U t

0

( ) ( ) ˆ ˆ ˆ ˆ( ) ( ) ( )B B F F

t t tiH U t iH U t O t

t

t

0

( ) ˆ ˆ ˆ ˆ( ) ( ) ( )B B F F

d tiH U t iH U t O

dt

( ) ( ) ( )B Fd t d t d t

dt dt dt

( ) ˆ ( )BB B

d tiH t

dt

( ) ˆ ( )FF F

d tiH t

dt

Consider time increment

Rate of change

Take limit

i.e.

ignore

RQIWRQIW 09 09


2222/28/28

Consider Robertson-Walker-Friedman universeGeneral Relativity

22 2 2 2 2 2 2

2( ) ( sin )

1

drds dt a t r d d

kr

Metric:

Friedman equations:

2

2

4( 3 )

3

8

3

a Gp

a

a G k

a a

scale

parameter

closedflatopen

101

k

Square root of last equation:

2

8

3

a G k

a a

+ve root-ve root

( ) ( )

( ) ( )

( ) ( )

a t a t

p t p t

t t

RQIWRQIW 09 09


2323/28/28

▀ CP and T violation expected to occur

in latter part of inflation

▀ Before this period, direction of

time is uncertain

0

BU FU

BU FU BU FU

BU FUBU FU BU FU


( )N

consider a path with a changing direction of evolution

“backwards” evolution is in direction of

decreasing t▀ depends on

length of path

( )a

t

RQIWRQIW 09 09


2424/28/28

▀ CP and T violation expected to occur

in latter part of inflation

▀ Before this period, direction of

time is uncertain

0

BU FU

BU FU BU FU

BU FUBU FU BU FU


( )N

consider a path with a changing direction of evolution

“backwards” evolution is in direction of

decreasing t▀ depends on

length of path

( )a

t

Consider massless

balloon containing

a gas

normal component

of tension in

membrane

pressure

of gas

balloon expands in

both directions of

time

F

F

motion of

molecule

in both

directions

of time

evolution

RQIWRQIW 09 09


2525/28/28

tt

3t

2t

1t

3( )a t

2( )a t

1( )a t

3t

2t

1t

3( )a t

2( )a t

1( )a t

is length of path

0

BU FU

BU FU BU FU

BU FUBU FU BU FU


( )N

t

Unchanging direction of time(conventional GR) Following a

path through time

...same topology

Path length – GR time coordinate

Foliation of spacetime

space-like slices

RQIWRQIW 09 09


2626/28/28

t

3t

2t

1t

3( )a t

2( )a t

1( )a t

While is a “good” coordinate for GR, the net time traversed is what clocks measure and what quantum fields depend on.

tt

is length of path

0

BU FU

BU FU BU FU

BU FUBU FU BU FU


t

tnet time

0

t

t t

Two time coordinates

net time = cosmic time = time since big bang

net time

path length

RQIWRQIW 09 09


2727/28/28

t

( )a t

t

( )a t

values of net time (cosmic time)

t

path length

net time

(cosmic time)

CP and T violation from here onwards

CP and T violation from here onwards

inflation

exp( )N

t N

34 2

63

10 s

10N

1310 s

4310 (present day value)

RQIWRQIW 09 09


2828/28/28

Summary

▀ Feynman path integral method▀ T violation causes destructive interference of zigzagging

paths▀ empirical evidence determines which branch

▀ early universe – no T violation - direction of time is uncertain▀ Friedman equations: expansion in both directions –

coordinate for GR is path length▀ radiation and clocks “slow” – cosmic time

Q. Is inflation due to uncertain direction of time?

Unidirectionality of time

Implications for general relativity

tinflation

ˆFUˆ

BU

T violation, direction of time and general relativity

Documents

Transcript of T violation, direction of time and general relativity