A Gravitational Explanation for
Quantum Theory
&
non-time-orientable manifolds
Dr Mark Hadley
Einstein’s dream
• Particles as solutions of the field equations
• An explanation for Quantum theory
• A unification of the forces of Nature
• A realist interpretation explanation
• An antidote to string theory
Mark Hadley
But…
Models must give particle properties
Charge, mass etc
AND
Interactions
AND
Particle behaviour
Quantum 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 GR
A 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
S
1
S S
1
t
1
Mark Hadley
Consequences of non timeorientable manifolds
•
Charge and the topology of spacetime
Diemer and Hadley Class. Quantum Grav. Vol. 16 (1999) 35673577
•
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:
d * F = *J

 
d * FdV if V3 is orientable
0
if d * F  0 (no sources of electric charge)
Q
S2 V3
V3
*FdS
If V3 is not orientable then use divergence theorem.

 
Q
S2 V3
V3
0
E.dS
Div.E dV if V3 is co orientable
if Div.E  0 (no sources of electric charge)
If the spacetime is not time orientable then
V3 is not co-orientable
*
Operator is not globally defined.
E = F.tˆ Is not globally defined even when F is well defined.
Mark Hadley
The Faraday Tensor F
 0

 Ex

F 
 Ey

  Ez
Ex
0
 Bz
By
Ey
Bz
0
 Bx
Ez 

 By 
Bx 

0 
Mark Hadley
Examples of non-orientable
surfaces
•
•
•
•
Mobius Strip
Wormholes
Monopoles
Einstein Rosen Bridge
Mark Hadley
Mobius Strip
Mark Hadley
Einstein Rosen Bridge
is 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.
R ( ) M  M
R ( ) R( ) x  R(   ) x x  M
R (0) x  x x  M
R ( ) x  R( ) x as | x | 
Physical rotation is defined on a causal spacetime.
Model spacetime as a line bundle over a 3-manifold
M4  M3 


 M3
R( )( x, 0)  ( R( ) x,1)
 R( )  R( )
plus conditions above
Mark Hadley
A rotation defines a path in a 3-manifold
x  R ( ) x
  ( )  {R( ) x : x  M ,   [0,1]}
A physical rotation defines a world line in a spacetime
( x, 0)  ( R( ) x,1)
 (t )  {R(t , t ) x : x  M , t  [0,1]}
Defines a time direction !!
Mark Hadley
A physical rotation of a non-time-orientable spacetime
R( )( x, 0)  ( R( ) x,  ( )( x))
 ( )( x)  1 as | x | 
Fixed point
R( )( x, 0)  ( x, )   0
Exempt point
R ( )(x,0) = (x,0)
The exempt points form a closed
2 dimensional surface.
Mark Hadley
If time is not orientable then:
 R(0)
( x, 0)  ( x, 0)
 R(2 )  x st ( x, 0)  ( R(2 ) x,  ( x))  ( x,  ( x))
x  R(4 ) st ( x, 0)  ( R(4 ) 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.
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
Y
X
Stern
Gerlach
Mark Hadley
Spin measurement
• Venn diagram of all 3-manifolds
X↑
X↓
X↑ Y→X↑ Y→
X↓ Y→X↓ Y→
Y→
All manifolds
consistent with the
state preparation
X↑ Y←X↑ Y←
X↓ Y←X↓ Y←
Y←
Mark Hadley
{M: X↑ and Y→}
•
•
•
•
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
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
Incompatible
boundary conditions
Sets of 3 manifolds
State preparation
x-measurement
y-measurement
X↑
Y→
X↓
Y←
X↑∩ Y→ = ∅
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
General
Relativity
Quantum
Logic
Hilbert
Space
Schrödinger’s
equation
Planck’s constant
etc.
a) Jauch (1968)
Beltrametti and Cassinelli (1981)
b
a
b) Ballentine(1989)
Weinberg(1995)
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
Relativity
Foundations of Physics Letters Vol. 10, No.1, (1997) 43-60.
•
Topology change and context dependence
International 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
Mark Hadley