Dr Mark Hadley

A Gravitational Explanation for Quantum Theory

& non-time-orientable manifolds

Mark Hadley

• Particles as solutions of the field equations• An explanation for Quantum theory• A unification of the forces of Nature

• A realist interpretation

• An antidote to string theory

Einstein’s dream

explanation

Mark Hadley

But…Models must give particle properties

Charge, mass etc

AND

Interactions

AND

Particle behaviourQuantum Theory

Mark Hadley

Two problems (at least!)

1. Interactions• Topology change requires a non-trivial

causal structure – Geroch, R P (1967)

2. Quantum theory • is incompatible with local realism

Mark Hadley

Topology Change and GRA topology change cannot take place in GR

without either:• Singularities appearing.

– A breakdown of GR

• Closed timelike curves.– Which need negative energy sources for their

creation.

• A failure of time orientability– interesting!!

Mark Hadley

Topology change

• A Simple Model in 1+1D

t

1 1S S

1S

Mark Hadley

Consequences of non time-orientable manifolds

• Charge and the topology of spacetime

Diemer and Hadley Class. Quantum Grav. Vol. 16 (1999) 3567-3577

• Spin half and classical general relativity

Class. Quantum Grav. Vol. 17 (2000) 4187-4194

• The orientability of spacetime

Class. Quantum Grav. Vol. 19 (2002) 4565-4571

Mark Hadley

Definition of electric charge:

2 3

3

3

if V is orientable

0 if 0 (no sources of electric charge)

S V

V

Q dS

dV

d*F = *J

*F

d*F

d*F

2 3

3

3

.dS

. if V is co orientable

0 if . 0 (no sources of electric charge)

S V

V

Q

dV

E

Div E

Div E

If the spacetime is not time orientable then

V3 is not co-orientable

* Operator is not globally defined.

Is not globally defined even when F is well defined.

If V3 is not orientable then use divergence theorem.

ˆE = F.t

Mark Hadley

The Faraday Tensor F

0

0

0

0

x y z

x z y

y z x

z y x

E E E

E B BF

E B B

E B B

Mark Hadley

Examples of non-orientable surfaces• Mobius Strip

• Wormholes

• Monopoles

• Einstein Rosen Bridge

Mark Hadley

Mobius Strip

Mark Hadley

Einstein Rosen Bridgeis not time-orientable

Einstein Rosen bridge: Phys Rev 48, 73 (1935)

Mark Hadley

Spin half

• Intrinsic spin is about the transformation of an object under rotations.

• If a particle is a spacetime manifold with non-trivial topology, how does it transform under a rotation?

Mark Hadley

Rotations of a manifold Defining a rotation on an asymptotically flat manifold with non trivial

topology.( )

( ) ( ) ( )

(0)

( ) R( ) as | |

R M M

R R x R x x M

R x x x M

R x x x

4 3 3

R( )( ,0) ( ( ) ,1) plus conditions above

R( ) ( )

M M M

x R x

R

Physical rotation is defined on a causal spacetime. Model spacetime as a line bundle over a 3-manifold

Mark Hadley

( )

( ) { ( ) : , [0,1]}

x R x

R x x M

( ,0) ( ( ) ,1)

( ) {R( , ) : , [0,1]}

x R x

t t t x x M t

A rotation defines a path in a 3-manifold

A physical rotation defines a world line in a spacetime

Defines a time direction !!

Mark Hadley

A physical rotation of a non-time-orientable spacetime

R( )( ,0) ( ( ) , ( )( ))

( )( ) 1 as | |

x R x x

x x

Fixed point

R( )( ,0) ( , ) 0x x

The exempt points form a closed 2 dimensional surface.

Exempt point

R ( )(x,0) = (x,0)

Mark Hadley

R(0) ( ,0) ( ,0)

R(2 ) ( ,0) ( (2 ) , ( )) ( , ( ))

R(4 ) ( ,0) ( (4 ) , ( )) ( , ( ))

x x

x st x R x x x x

x st x R x x x x

The exempt points prevent a 360 degree rotation being an isometry,

but a 720 degree isometry can be always be constructed.

If time is not orientable then:

Mark Hadley

• An object that transforms in this way would need to be described by a spinor.

– Tethered rocks (Hartung)– Waiter with a tray (Feynman)– Cube within a cube (Weinberg)– Demo

Mark Hadley

Acausal Manifolds and Quantum theory• With time reversal as part of the measurement

process – due to absorption/topology change.

• The initial conditions may depend upon the measurement apparatus.

→A non-local hidden variable theory.→Resulting in the probability structure of quantum

theory.

Mark Hadley

The essence of quantum theory

• Propositions in Classical physics satisfy Boolean Logic

• Propositions in quantum theory do not satisfy the distributive law– They form an orthomodular lattice

Mark Hadley

Evolving 3-manifolds…

• Prepare a beam of electrons

Stern Gerlach

X

Y

Mark Hadley

Spin measurement• Venn diagram of all 3-manifolds

X↑

X↓

Y←Y→

X↑ Y→

X↓ Y→ X↓ Y←

X↑ Y←

All manifolds consistent with the state preparation

X↑ Y→

X↓ Y→ X↓ Y←

X↑ Y←

Mark Hadley

• Cannot be prepared experimentally

• Cannot be described by quantum theory

• Is a local hidden variable theory

• Would violate Bell’s inequalities in an EPR experiment.

• Is NOT context dependent

{M: X↑ and Y→}

Mark Hadley

Geometric models

• We cannot model particles as 3-D solutions that evolve in time.

• Need context dependence

• Non-locality

• Non-trivial causal structure as part of a particle: 4-geon

Mark Hadley

4-geon• Non-trivial causal structure as part of the

particle.• Particle and its evolution are inseparable.• Time reversal is part of a measurement

• Context dependent– Signals from the “future” experimental set up.– Measurement can set non-redundant boundary

conditions

Mark Hadley

Spin measurement

X↑∩ Y→ = ∅

X↑

X↓ Y←

Y→

State preparation

x-measurement y-measurement

Sets of 3 manifolds

Incompatible boundary conditions

Mark Hadley

• How do calculate probabilities if Boolean Logic does not apply?

• That is the question the Gerard ‘t Hooft is looking for !

Mark Hadley

From General Relativity to Quantum Mechanics

a) Jauch (1968) Beltrametti and Cassinelli (1981)

b) Ballentine(1989) Weinberg(1995)

General Relativity

Quantum Logic

Hilbert Space

Schrödinger’s equation

Planck’s constantetc.

a

a

b

Mark Hadley

A gravitational explanation for quantum theory

• Aims to explain– QM– Particle spectrum– Fundamental interactions

• Predictions– No graviton (Gravity waves are just classical waves)

– Spin-half

–Parity is conserved

Mark Hadley

See:• The Logic of Quantum Mechanics Derived From Classical General

RelativityFoundations of Physics Letters Vol. 10, No.1, (1997) 43-60.

• Topology change and context dependenceInternational Journal of Theoretical Physics Vol. 38 (1999) 1481-149

• Time machines and Quantum theory

MG11 July 2006 Berlin

• A gravitational explanation of quantum mechanics

FFP8 October 2006 Madrid

Mark Hadley

GR may be the unifying theory

after all