The Everett Interpretation of Quantum Mechanics: 50 years on 19 – 21 July 2007

Time-symmetric quantum mechanics and the Many-Worlds Interpretation

Lev Vaidman

The two-state vector formalism of quantum mechanics

The standard (one-state vector) description of a quantum system at time t

t

1t P 1

0FREEH

We assume:

1y

1z

1x

1x

y

x

x

z

( )tt

The one-state vector description of a quantum system at all times:

1y

1z

1x

1x ( )tt

The time reversed description of a quantum system

x

x

z

y

y

x

z

x

Backward Evolving Quantum State

The Quantum State Evolving Backward

1y

1z

1x

1x ( )tt

x

x

z

y

y

x

z

x

The two-state vector description of a quantum system:

( )t

x x

x z

x y

t

2t P 1

1t P 1

Time symmetric description of a pre- and post-selected quantum system

The two-state vector

t

P 1

1t

2t

P 1

Strong measurement: The Aharonov-Bergmann-Lebowitz (ABL) formula:

?C

2

2

PProb( )

Pi

C c

C ci

C c

described by the two-state vector:

Measurements performed on a pre- and post-selected system

Weak measurement: The Aharonov-Albert-Vaidman effect:

w

CC

Weak value

The three box paradox

t

2t

1t

3t

Where is the ball?

?

1

3A B C

1

3A B C

A B C

1

3A B C 1

3A B C

The three box paradox

t

2t

1

3A B C

1t 1

3A B C

3t

It is in

A B C

Aalways !

2

2 2Prob( 1) 1A

A

A B C

A B C A B C

A B C A B C A B C A B C

PP

P P

1

3A B C 1

3A B C

The three box paradox

t

2t

1t

3t

It is always in

B

1

3A B C

1

3A B C

A B C

2

2 2Prob( 1) 1B

B

B A C

A B C A B C

A B C A B C A B C A B C

PP

P P

1

3A B C 1

3A B C

A single photon “sees” two balls

t

2t

1t

It scatters exactlyas if there weretwo balls

1

3A B C

1

3A B C

A B C

Y. Aharonov and L. Vaidman Phys. Rev. A 67, 042107 (2003) 

Weakly coupled (numerous) particles “see” two balls

t

2t

1t

1

3A B C

1

3A B C

A B C

The tree of worlds picture of the MWI

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.

  II. The earth or a region of it; the universe or a part of it.

I. Human existence; a period of this.

world, n

OED

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

What is “a world” in the many-worlds tree picture?

A world is a branch of the Universal Wave Function consistent with the classically described state of macroscopic objects.

The tree of worlds

A A BB

A BA 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 A BB

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 A BB

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

21

3

1 2 3

21

3

1 2 3

21

3

1 2 3

1

3A B C

1

3A B C

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

A A BB

Is this the two-state vector which describes the Universe?

No! The backward evolving quantum state is equal to the forward evolving state!

A A BB

Is this the two-state vector which describes the Universe?

No! The backward evolving quantum state is equal to the forward evolving state!

A A BB

Is this the two-state vector which describes the Universe?

No! The backward evolving quantum state is equal to the forward evolving state!

Is this the two-state vector which describes the Universe?No! The backward evolving quantum state is equal to the forward evolving state!

2

2

PProb( )

Pi

C c

C ci

C c

2Prob( ) PC cC c

( )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)

?

1y

1z

1x

1x

( )tt

x

x

z

y

y

x

z

x

The two-state vector description of a quantum system:

( )t

x x

x z

x y

in a particular world:

1y

1z

( )i tt

x

z

y

y

y

x

z

The two-state vector description of a quantum system

( )i tin the Universe:

z z

x x x x x x

1x

1z

1x 1x 1x

x xxx1x 1x 1x 1x

( )t ( )t

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.

The theory is not testable

A backward evolving equal-weight mixture can be added

But it might provide a framework for (possibly testable) cosmological theory.