Download The two-state vector description of a quantum system

Time-symmetric quantum mechanics and
the Many-Worlds Interpretation
Lev Vaidman
The Everett Interpretation of Quantum Mechanics: 50 years on 19 – 21 July 2007
The two-state vector formalism
of quantum mechanics
The standard (one-state vector) description of a quantum
system at time t
We assume:
t
t1

P  1
H FREE  0
The one-state vector description of a quantum system
at all times:
t
x
x 1
x
x 1
z
z 1
y
y 1
( t )
The time reversed description of a quantum system
x
t
x
x
x 1
 (t )
x 1
Backward Evolving Quantum State
x

z
The Quantum State Evolving Backward
z 1
z
y
y
y 1
The two-state vector description of a quantum system:
x
t
x
x
x
 (t )
x 1
x
x
x
z
x
y
x 1
z
z 1
z
y
y
y 1
( t )
Time symmetric description of a pre- and post-selected
quantum system
t2
t
t1
P  1

P  1
The two-state vector

Measurements performed on a pre- and post-selected system
described by the two-state vector:


Strong measurement: The Aharonov-Bergmann-Lebowitz (ABL) formula:
t2
P  1

Prob(C  c ) 
C?
t

t1
 PC c 

 PC ci 
i
P  1
Weak measurement: The Aharonov-Albert-Vaidman effect:
Weak value
2
C 
Cw 

2
The three box paradox


t3

1
3
 

A  B  C
1
3


A  B  C
1
3

A  B  C


t2

t
?
Where is the ball?

t1
 
A
1
3

A  B  C
B
C

The three box paradox


t3

1
3
 

A  B  C
1
3


A  B  C
1
3

A  B  C


t2

t
It is in
always !

t1
 
A
Prob(PA  1) 
1
3

A  B  C

C
B
 A
 A
A
B  C  PA  A  B  C
B  C  PA  A  B  C
  A
2

B  C P  A  B
2
B C
C

2
1
The three box paradox


t3

1
3
 

A  B  C
1
3


A  B  C
1
3

A  B  C


t2

t
B
It is always in

t1
 
A
Prob(PB  1) 
1
3

A  B  C
 A
C
B
 A

B  C  PB  A  B  C
B  C  PB  A  B  C
  A
2

B  C P  A  B
2
A C
C

2
1
A single photon “sees” two balls
 
1
3

A  B  C
Y. Aharonov and L. Vaidman
Phys. Rev. A 67, 042107 (2003)

t2
It scatters exactly
as if there were
two balls

t

t1
 
A
1
3

A  B  C
B
C

Weakly coupled (numerous) particles “see” two balls
 
1
3
 
1
3

A  B  C

t2

t

t1
A

A  B  C
B
C

The tree of worlds picture of the MWI
What is “a world” in the many-worlds tree picture?
world, n
I. Human existence; a period of this.
II. The earth or a region of it; the universe or a part of it.
OED
The World is a name for the planet Earth seen from a human point of view,
as a place inhabited by human beings. It is often used to mean the sum of
human experience and history, or the 'human condition' in general.
Wikipedia
A world is the totality of (macroscopic) objects: stars, cities, people,
grains of sand, etc. in a definite classically described state.
The MWI in SEP
A world is a branch of the Universal Wave Function consistent with the
classically described state of macroscopic objects.
The tree of worlds
A
B
A
B
A
B
A
B
A world consist of:
•"classical" macroscopic objects rapidly measured by the environment,
• quantum objects measured only occasionally (at world splitting events),
• weakly coupled quantum objects
A
B
A
B
A world consist of:
•"classical" macroscopic objects rapidly measured by the environment,
• quantum objects measured only occasionally (at world splitting events),
• weakly coupled quantum objects
A
B
A
B
A world consist of:
•"classical" macroscopic objects rapidly measured by the environment,
• quantum objects measured only occasionally (at world splitting events)
which described by the two-state vectors,
• weakly coupled quantum objects
1
1
2
3
2
3
1
1
2
3
2
3
1
1
2
2
3
3
A
B
C
 
1
3

A  B  C

 
1
3

A  B  C

Forward evolving branch of the universal wave function
does not describe all we should know about a world.
The (different) backward evolving state has to be added.
Is this the two-state vector which describes the Universe?
 ( t ) ( t )
Is this the two-state vector which describes the Universe?
No! The backward evolving quantum state is equal to the
forward evolving state!
A
B
A
B
Is this the two-state vector which describes the Universe?
No! The backward evolving quantum state is equal to the
forward evolving state!
A
B
A
B
Is this the two-state vector which describes the Universe?
No! The backward evolving quantum state is equal to the
forward evolving state!
A
B
A
B
Is this the two-state vector which describes the Universe?
No! The backward evolving quantum state is equal to the
forward evolving state!
Prob(C  c)   PC c 
2
Prob(C  c ) 
 PC c 

2
 PC ci 
2
i
 ( t ) ( t )
( t ) ( t )
Forward evolving branch of the universal wave function
does not describe all we should know about a world.
The (different) backward evolving state has to be added.
But, this backward evolving state has meaning only in this
world. It does not exist in the physical world (Universe)
?
The two-state vector description of a quantum system:
in a particular world:

(
t
)

(
t
)
x
t
x 1
x
x
x
x
x
x
z
x
y
x 1
z
z 1
z
y
y
y 1
The two-state vector description of a quantum system
in the Universe:
 ( t ) ( t )
 (t )
 i (t )
t
x
x
x 1
 x  1
x
x
x
x
x 1
 x  1
z
x
x
x 1
 x  1
x
x
x
 x  1
z
z
 z  1
y
y
y 1
x
x 1
z
z 1
y
i
Forward evolving branches of the universal wave function
do not describe all we should know about these worlds.
The (different) backward evolving states have to be added.
But, these backward evolving states have meaning only in
every world separately. They do not exist in the Universe
The multiverse: the tree of worlds
The Universe: the trivial two-state vector
( t )
( t )
Multiple Many-Worlds Interpretation
The Universe is an equal-weight mixture of
all quantum states of an orthonormal basis
Like one side of the teleportation machine
for universes
S
Multiple Many-Worlds Interpretation
The Universe is an equal-weight mixture of
all quantum states of an orthonormal basis
Like one side of the teleportation machine
for universes
It is very, very symmetric.
A backward evolving equal-weight mixture can be added
The theory is not testable
But it might provide a framework for (possibly testable)
cosmological theory.