Quantum time
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Quantum time
John
Ashmead
Table of Contents
1. Abstract
2. Time and quantum mechanics
The problem of time
Two views of the river
Relativity
Quantum mechanics
Laboratory time and quantum time
Requirements
Working assumptions
Plan of attack
3. Formal development
Overview
Feynman path integrals
Wave functions
Paths
Lagrangian
Sum over paths
Convergence
Normalization
Explicit expression
Schrödinger equation
Derivation
Unitarity
Gauge transformations
Operator formalism
Hamiltonian path integrals
Derivation
Closing the circle
Geodesic definition of laboratory time
Discussion
4. Comparison of temporal quantization to standard quantum theory
Non-relativistic limit
Semi-classical approximation
Overview
Derivation
Applications
Quantum fluctuations in time
Discussion
Stationary states
Scattering
Schrödinger equation in relative time
Free Schrödinger equation
Time dependent magnetic field
Time independent electric fields
Time dependent electric field
Bound states
Why do bound states exist?
Stationary states as the bound states
Estimate of uncertainty in time
Evolution of non-stationary states
Evolution of Gaussian test functions
Discussion
Discussion
5. Experimental tests
Overview
Slits in time
Free case
Single slit
Double slit
Lindner's experiments
Discussion
Electric and magnetic fields
Time dependent magnetic field
Time independent electric field
Time dependent electric field
Aharonov-Bohm experiments
Aharonov-Bohm experiment in space
Aharonov-Bohm experiment in time
Discussion
6. Discussion
Review
Requirements
Advantages
Open questions
A. Appendices
Morlet wavelets
Morlet wavelet decomposition
Resolution of unity
Calculation of admissibility constant
Discussion
Free particles
Time/space representation
Energy/momentum representation
Time/momentum representation
Estimates of uncertainty in time/energy
B. Acknowledgments
Bibliography
List of Figures
2.1.
Laboratory time
2.2.
Quantum time
2.3.
Evolution of wave function
3.1.
Four formalisms
3.2.
Path integrals
3.3.
First step of calculation
3.4.
Alice and Bob have rather different notions of time
3.5.
Geodesics
4.1.
Zeitbewegung
5.1.
Double slit
5.2.
Hard-edged case
5.3.
Gates as sources
5.4.
Electric field as source
5.5.
Single and double sources in time
5.6.
Time dependent magnetic field
5.7.
Time independent electric field
5.8.
Time dependent electric field
5.9.
Aharonov-Bohm experiment in space
5.10.
Aharonov-Bohm experiment in time
A.1.
Real and imaginary parts of the mother wavelet
A.2.
Admissibility constant as a function of
f