Quantum probability from temporal structure

Abstract: The Born probability measure describes the statistics of measurements in which observers self-locate themselves in some region of reality. In $ψ$-ontic quantum theories, reality is directly represented by the wavefunction. We show that quantum probabilities may be identified with fractions of a universal multiple-time wavefunction containing both causal and retrocausal temporal parts. This wavefunction is defined in an appropriately generalized history space on the Keldysh time contour. Our deterministic formulation of quantum mechanics replaces the initial condition of standard Schrödinger dynamics with a network of `fixed points' defining quantum histories on the contour. The Born measure is derived by summing up the wavefunction along these histories. We then apply the same technique to the derivation of the statistics of measurements with pre- and post-selection.

• The paper presents a deterministic formulation of quantum mechanics that derives the Born measure from a universal multiple-time wavefunction over the Keldysh contour.
• It employs a Fixed Point Formulation integrating causal and retrocausal dynamics to reformulate quantum measurement without wavefunction collapse.
• The framework offers potential applications in quantum gravity and supports a time-symmetric, Everettian interpretation of quantum dynamics.

Quantum Probability from Temporal Structure

Introduction

The paper "Quantum Probability from Temporal Structure" (2112.10929) proposes a novel deterministic formulation of quantum mechanics to explain the appearance of probabilistic elements without resorting to wavefunction collapse or auxiliary assumptions. It introduces a universal multiple-time wavefunction over the Keldysh time contour, integrating causal and retrocausal temporal parts, to derive the Born measure of quantum mechanics.

Quantum Mechanics Reformation

Traditional quantum mechanics is governed by four central postulates: the ontological representation as a wavefunction, deterministic evolution via the TDSE, composite state spaces as tensor products, and the Born statistical measure. This paper challenges the statistical measure by deriving it within a deterministic framework based on history spaces, aiming to address the measurement problem without invoking inherent randomness or collapse.

The deterministic formulation recognizes quantum probabilities as fractions of a universal wavefunction mapped on a generalized history space with the Keldysh contour as a time domain (Figure 1).

Figure 1: The Keldysh time contour in the time interval $\left[t_{1},t_{2}\right]$.

Universal Wavefunction Model

The paper posits that the universal wavefunction consists of temporal sequences of macroscopic and microscopic properties without approximation or environmental simplifications (Figure 2). It suggests a complete $\psi$-ontic view where the wavefunction encompasses all physical properties eternally. Measurements are considered physical processes within defined temporal regions of the universal wavefunction.

Figure 2: A single fixed point on the Keldysh contour.

Fixed Point Formulation (FPF)

The fixed point formulation (FPF) represents quantum histories through fixed points on the Keldysh contour that act as both 'sources' and 'sinks' in all time directions. This formulation reduces independent postulates by deriving the Born measure from unitary dynamics and wavefunction structure, thus achieving a simpler physical basis for quantum probability (Figure 3).

Figure 3: The purple region represents the measure of existence connecting the prepared fixed point state $\left\llbracket\psi\right\rrbracket_{t_1}$.

Derivation of the Born Measure

The FPF employs temporally local constraints to derive the Born measure from the temporal structure of the wavefunction. This measure is quantified by regions of wavefunction associated with quantum measurement events (Figure 4), thereby grounding probability in physical ontology. It equates the measure of existence with quantum probability without stochastic collapse.

Figure 4: The purple region represents the measure of existence corresponding to the ABL measure in an experiment connecting the pre- and post-selection fixed points $\left\llbracket\psi\right\rrbracket_{t_1}$ and $\left\llbracket\phi\right\rrbracket_{t_2}$ to a measurement at $t$.

ABL Rule and Extended Histories

The ABL rule, interpreted within this framework, accounts for all temporal regions between fixed points, correcting limitations in time-symmetric approaches such as TSVF (Figure 5). This formulation applies to extended quantum histories with multiple measurements, reinforcing unitary evolution without randomness.

Figure 5: Schematic representation of the regions and direction of time propagation between three consecutive boundary conditions considered within (a) the FPF, (b) the TSVF, and (c) standard Schr{\"o}dinger dynamics.

Conclusion

This paper proposes a deterministic perspective on quantum mechanics, linking quantum probability to temporal wavefunction structures, free from stochastic elements. The FPF offers a coherent and simplified description compatible with quantum theory principles, with potential implications for fields like quantum gravity. The approach supports an Everettian interpretation, emphasizing time-symmetric branching. Future work may explore this foundational mechanism in complex quantum systems, potentially reshaping our understanding of quantum mechanics.