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Introduction Paths through time Interference Direction of time Early universe

Unidirectionality of time induced by T violation

Joan VaccaroCentre for Quantum Dynamics

Griffith UniversityAustralia

BU FUBU FU

BU FU

BU FU BU FU

BU FUBU FU BU FU

BU FU BU FU

( )N

11

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Introduction Paths through time Interference Direction of time Early universe

22

Direction of time arrows of time (Eddington)

thermodynamic (entropy doesn’t decrease)

cosmological (expanding universe)

electromagnetic (spontaneous emission not absorption)

psychological (remember past not future)

matter-antimatter, … (CP violation favours matter)

description of dynamical systems is time-symmetric

Introduction

[The Nature Of The Physical World, 1928]

unidirectionality of time problem

2

2

d x F

dt m ˆd i

Hdt

BUT why do we experience only one direction of time?

This is the

?

time

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Introduction Paths through time Interference Direction of time Early universe

Time reversal operator

)(t )(ˆ tT

p

p

L

L

[Wigner, Group theory (1959]

ˆ ˆ ˆT U K

unitary operator

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

Typical Schrodinger equation Backwards evolution is simply backtracking the forwards evolution

ˆi Ht

ˆ ˆHT

1ˆ ˆ ˆ ˆ ˆi T T HT Tt

1ˆ ˆ ˆ ˆT HT H

33

t

y

forwards

“backwards”H H

mirror symmetry

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Introduction Paths through time Interference Direction of time Early universe

But kaons don’t behave this way

1ˆ ˆ ˆ ˆT HT H

Violation of time reversal invariance- 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

gives time asymmetric dynamics

What effect does this have on the direction of time?

u s_

boson,neutral, ½ mp

lifetime 108s

44

Conventional answer: nothing!!!

1ˆ ˆ ˆT HT

t

y

forwards

“backwards”

H

broken mirror

a fundamental time asymmetry

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Introduction Paths through time Interference Direction of time Early universe

where

and = Hamiltonian for forward time evolution.

Model of the universe:▀ it is closed in the sense

that it does not interact with any other physical system

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

▀ both versions of the Hamiltonian shouldappear in the dynamical equation of motion

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

0

0ˆ( ) ( )F FU

ˆ ˆ( ) expF FU iH ˆ

FH

( 1)

Preprint arXiv:0911.4528

55

Paths through time

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Introduction Paths through time Interference Direction of time Early universe

66

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)

▀ 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 assign an equal statistical weighting to each

using Feynman’s sum over histories

[Feynman Rev. Mod. Phys. 20, 367 (1948)]

0ˆ ( )FU 0

ˆ ( )BU

0 0

BU FU

Constructing paths:

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Introduction Paths through time Interference Direction of time Early universe

which we call time-symmetric

evolution.

0 0

0

ˆ ˆ( ) ( )

ˆ ˆ ( ) ( )

F B

F B

U U

U U

0

0ˆ ˆ( ) ( ) ( )F BU + U

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

0

BU FU

0

77

c.f. double slit:The total amplitude for is proportional to

This is true for all states , so

Time-symmetric evolution over an additional time interval of is given by

0

BU FU

( )t

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Introduction Paths through time Interference Direction of time Early universe

Repeating this N times yields

0ˆ ˆ( )N ( ) ( )F B

NU + U

00

ˆ( )N

m

N

( )S N,m

▀ is a sum containing different terms

▀ is a sum over a set of paths each

comprising

forwards steps and backwards steps

Let

ˆ( )S N,m )(Nm

0ˆ ( )S N,m

m N m

88

0

BU FU

BU FU BU FU

BU FUBU FU BU FU

BU FU BU FU BU FUBU FU

( )Nt

ˆ(4,1)S

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Introduction Paths through time Interference Direction of time Early universe

The limit 0 ?

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 conventional clock device no time in conventional sense▀ Set to be a small physical time interval,

Planck time

445 10 s

99

0

BU FU

BU FU BU FU

BU FUBU FU BU FU

BU FU BU FU BU FUBU FU

( )N t

▀ fix total time and set . Take limit as .

▀ we find

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Introduction Paths through time Interference Direction of time Early universe

Interference

Multiple paths

ˆ(4,1)S

4 terms

interfere

ˆ(4,0)S ˆ(4,4)S

Example:

1010

0

BU FU

BU FU BU FU

BU FUBU FU BU FU

BU FU BU FU BU FUBU FU

( )N

t

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Introduction Paths through time Interference Direction of time Early universe

1111

The Zassenhaus (Baker-Campbell-Hausdorff ) formula

gives

Simplifying ˆ( )S N,m

ˆˆ ˆB FU N m U m ( )S N,m

Eigenvalue equation for

commutatorˆ ˆi [ , ]F BH H

degeneracytrace 1 projection op.

ˆ ( ) 1d ( )

eigenvalue

i

ˆ ˆ iA iB e e

2 3ˆ ˆˆ ˆ [ , ] ( )iB iA A B O e e e

2 3

0 0 0 0

exp ( ) (ˆ ˆ )N m v k

v u k j

v u k j O

[ , ]

F BH H

0 0ˆ

0

ˆˆN

n

( ) ( ) ( )F B

NU + U S N,m

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Introduction Paths through time Interference Direction of time Early universe

1 2

0

2

1

{exp[ ( ) ] 1}

[exp( ) 1]

m

km

k

i N k

ik

( , , )I N m where

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

degeneracy

eigenvalue

trace 1 projection op.

1212

Eigenvalues for j th kaon 17 210 sj

Eigenvalues for M kaons

Estimating eigenvalues

0

0( ) ,

a Kdi i

bdt K

ψM Γ ψ ψ

ˆ ˆi [ , ]F BH Hphenomenological model [Lee, PR 138, B1490 (1965)].

17 2SD 10 sM

0

8010M f

57 2SD 10 sf

Let

fraction

total # of particles

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Introduction Paths through time Interference Direction of time Early universe

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

( , , )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 d (( ) ), ,I N m

destructive interference

constructive interference

2 24

( )m N m N

1313

( )

( , , )I N m

width

( , , )I N m

8000, 400N m 500, 100N m

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Introduction Paths through time Interference Direction of time Early universe

1/2 13 10 sN f

total time

0ˆ ˆ( ) ( ) ( )F BN U N U N

Bi-evolution equation of motion

Only two paths survive if

1414

8010M f

fraction

total # of particles

#

kaons

0

BU FU

BU FU

BU FU

FUBU

( )N

t

Destructive interference

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Introduction Paths through time Interference Direction of time Early universe

0

BU FU

BU FU

BU FU

FUBU

( )N

t

0ˆ ˆ( ) ( ) ( )F BN U N U N

ˆBH

only observe evidence of in this branch

we observe only one of these terms

phenomenological unidirectionality of time

ˆFH

only observe evidence of in this branch

1515

Direction of time

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Introduction Paths through time Interference Direction of time Early universe

Shortest path through time

1616

BU FU

BU FU BU FU

BU FUBU FU BU FU

BU FU BU FU BU FUBU FU

BU FU

FU

shortest path

same time

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Introduction Paths through time Interference Direction of time Early universe

1717

Early universeT violation would be relatively rare, so no interference:

0

0

0

ˆ ˆ( ) ( ) ( ) ,

ˆ ˆexp( ) exp( )

ˆco

ˆ

s ( )

ˆcos( )

ˆF BN U U

iH iH

H

N

N

N

H

=F BH H

eigenstate of

with largest eigenvalue (i.e. 1)

= zero eigenvalue of

H

max max maxˆ NNA

Power method

ˆ ( ) 0H N ~ Hamiltonian constraint of the Wheeler-DeWitt eqn.

Hence

BU FU

BU FU BU FU

BU FUBU FU BU FU

BU FU BU FU BU FUBU FU

( )N t

largest eigenvalue

= history

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Introduction Paths through time Interference Direction of time Early universe

Summary

▀ must use Feynman’s sum over histories to account for both directions

▀ destructive interference leaves only 2 paths▀ physical evidence shows which path we experience▀ quantum algorithm for the shortest path to the

“future”...

1818

the unidirectionality of time

BU FU

BU FU BU FU

BU FUBU FU BU FU

BU FU BU FU BU FUBU FU

BU FU

FU

T invarianceT violation

0

BU FU

BU FU BU FU

BU FUBU FU BU FU

BU FU BU FU BU FUBU FU

( )N t

Universe has no reference for direction of time