Hilbert Spaces from Path Integrals

Abstract

It is shown that a Hilbert space can be constructed for a quantum system starting from a framework in which histories are fundamental. The Decoherence Functional provides the inner product on this "History Hilbert space". It is also shown that the History Hilbert space is the standard Hilbert space in the case of non-relativistic quantum mechanics.

22 pages. Minor updates to match published version

References in corpus (7)

• Quantum Mechanics as Quantum Measure Theory
• Quantum Logic and the Histories Approach to Quantum Theory
• Quantum Dynamics without the Wave Function
• Feynman Propagator for a Free Scalar Field on a Causal Set
• An exercise in "anhomomorphic logic"
• The Random Walk in Generalized Quantum Theory
• A Bell Inequality Analog in Quantum Measure Theory

Cited by in corpus (15)

• The causal set approach to quantum gravity
• Schwinger's Picture of Quantum Mechanics III: The statistical interpretation
• On extending the Quantum Measure
• Pitfalls of Path Integrals: Amplitudes for Spacetime Regions and the Quantum Zeno Effect
• Renormalisation group flows connecting a $4-\epsilon$ dimensional Hermitian field theory to a $\mathcal{P}\mathcal{T}$-symmetric theory for a fermion coupled to an axion
• Two-Site Quantum Random Walk
• An Argument for Strong Positivity of the Decoherence Functional
• Ontological models for quantum theory as functors
• Contrary Inferences in Consistent Histories and a Set Selection Criterion
• The coevent formulation of quantum theory
• Amplitudes for Spacetime Regions and the Quantum Zeno Effect: Pitfalls of Standard Path Integral Constructions
• Composing decoherence functionals
• The Universe and The Quantum Computer
• (2+1) Lorentzian quantum cosmology from spin-foams: opportunities and obstacles for semi-classicality
• Self-normalizing Path Integrals