Quantum Dynamics without the Wave Function

Abstract

When suitably generalized and interpreted, the path-integral offers an alternative to the more familiar quantal formalism based on state-vectors, selfadjoint operators, and external observers. Mathematically one generalizes the path-integral-as-propagator to a {\it quantal measure} $\mu$ on the space $\Omega$ of all ``conceivable worlds'', and this generalized measure expresses the dynamics or law of motion of the theory, much as Wiener measure expresses the dynamics of Brownian motion. Within such ``histories-based'' schemes new, and more ``realistic'' possibilities open up for resolving the philosophical problems of the state-vector formalism. In particular, one can dispense with the need for external agents by locating the predictive content of $\mu$ in its sets of measure zero: such sets are to be ``precluded''. But unrestricted application of this rule engenders contradictions. One possible response would remove the contradictions by circumscribing the application of the preclusion concept. Another response, more in the tradition of ``quantum logic'', would accommodate the contradictions by dualizing $\Omega$ to a space of ``co-events'' and effectively identifying reality with an element of this dual space.

plainTeX, 24 pages, no figures. To appear in a special volume of {\it Journal of Physics A: Mathematical and General} entitled ``The Quantum Universe'' and dedicated to Giancarlo Ghirardi on the occasion of his 70th birthday. Most current version is available at http://www.physics.syr.edu/~sorkin/some.papers/ (or wherever my home-page may be)

References in corpus (7)

• Quantum Mechanics as Quantum Measure Theory
• "Observables" in causal set cosmology
• Quantum Measure Theory and its Interpretation
• The Random Walk in Generalized Quantum Theory
• Some Identities for the Quantum Measure and its Generalizations
• General Covariance and the "Problem of Time" in a Discrete Cosmology
• A Topos Perspective on State-Vector Reduction

Cited by in corpus (51)

• Is the quantum state real? An extended review of $\psi$-ontology theorems
• The causal set approach to quantum gravity
• Laws of Nature as Constraints
• Scalar Field Theory on a Causal Set in Histories Form
• An exercise in "anhomomorphic logic"
• Quantum States as Ordinary Information
• Stable Homology as an Indicator of Manifoldlikeness in Causal Set Theory
• Quantum Mechanics with Extended Probabilities
• Quantum measure and integration theory
• The Kochen-Specker Theorem Revisited in Quantum Measure Theory
• Causal Sets: Quantum gravity from a fundamentally discrete spacetime
• Understanding quantum mechanics: a review and synthesis in precise language
• A histories perspective on characterising quantum non-locality
• Hilbert Spaces from Path Integrals
• Do We Have Any Viable Solution to the Measurement Problem?
• Dynamical Wave Function Collapse Models in Quantum Measure Theory
• Quantum Measure Theory: A New Interpretation
• Dynamics & Predictions in the Co-Event Interpretation
• Quantum measures and the coevent interpretation
• The emergence of probabilities in anhomomorphic logic
• Quantum integrals and anhomomorphic logics
• An anhomomorphic logic for quantum mechanics
• Path Integrals and Reality
• The Universe as a Quantum Computer
• Is the spacetime metric Euclidean rather than Lorentzian?
• Distinguishing initial state-vectors from each other in histories formulations and the PBR argument
• Spacetime Path Integrals for Entangled States
• Two-Site Quantum Random Walk
• Algorithmic Causal Sets and the Wolfram Model
• Hilbert Space Representations of Decoherence Functionals and Quantum Measures
• Physical Logic
• Contrary Inferences in Consistent Histories and a Set Selection Criterion
• The coevent formulation of quantum theory
• Quantum Reality Filters
• Does a Quantum Particle Know its Own Energy?
• Recovering General Relativity from a Planck scale discrete theory of quantum gravity
• Composing decoherence functionals
• A "problem of time" in the multiplicative scheme for the $n$-site hopper
• Probability, Preclusion and Biological Evolution in Heisenberg-Picture Everett Quantum Mechanics
• Transition probability spaces in loop quantum gravity
• The Universe and The Quantum Computer
• Reality, No Matter How You Slice It
• Coevents as Beables
• If time had no beginning
• Continuous measurement on a causal set with and without a boundary
• Evolving Realities for Quantum Measure Theory
• Consistent Histories as Valuations
• Explanation, Evolution and Subjective Probability in Everett Quantum Mechanics with Positive Preclusion
• The relativistic reason for quantum probability amplitudes
• Phenomenological quantum mechanics: I. Phenomenology of quantum observables
• Examples of quantum integrals