# Quantum Probability from Temporal Structure

## Abstract

**:**

## 1. Introduction

- (a)
- An ontological postulate—The state of a physical system is represented by a wavefunction $\left|\Psi \right.\u232a$;
- (b)
- A dynamical postulate—The state evolves deterministically according to the time-dependent Schrödinger equation (TDSE);
- (c)
- A composition postulate—The state space of a composite system is the tensor product of the spaces of its subsystems;
- (d)
- A statistical postulate—The probability of each measurement outcome is given by the Born measure.

## 2. The Universal Wavefunction

#### 2.1. General Considerations

**Completeness**:The wavefunction is all that exists—it contains all physical properties of nature at all moments in time;**Measurement Physicality**: Measurements are physical processes occurring within temporal regions of the universal wavefunction;**Event Symmetry**: The local description of nature is independent of event location. There are no ontologically privileged spacetime points;**Self-Location**: Temporal boundary constraints provide the only information an observer can use to locate themselves within the wavefunction.

**Self-Location**.

**Definition**

**1.**

**Completeness**must then answer the question:

**Event Symmetry**—if, given two connected points in time, one is a source and the other a sink, and the dynamics are allowed to be time symmetric, it must follow that both points are sources and both are sinks for all time regions they are connected to.

#### 2.2. The Universal Wavefunction on the Keldysh Contour

**Ontological postulate**

**Dynamical postulate**

## 3. One Fixed Point

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Statistical postulate (Vaidman rule)**:

## 4. The Born Measure

**Measurement Physicality**and

**Self-Location**, the physical process of an experiment occurs within a region of $\left|{\Psi}_{U}\right.\u232a$ subject to the boundary constraints determined by the preparation. The measure of existence is now evaluated for the simplest type of quantum history—a two-time measurement—with ${N}_{t}=2$.

## 5. Three Fixed Points

**Event Symmetry**. For comparison, the standard Schrödinger dynamics used in the consistent histories framework is illustrated in Figure 5c. We also note that the TSVF formalism allows oppositely-oriented states to overlap at the intermediate measurement time (the backwards-travelling vector from the future is represented as a ‘bra’ state in the conjugate Hilbert space ${\mathcal{H}}_{{t}_{2}}^{\u2020}$) [21], which is prevented by branch-independence in the FPF.

## 6. Conclusions

- It is logically parsimonious. The statistical postulate supplies the meaning of probability. However, the mathematical form of probability is not postulated, but derived from ontic and dynamical structure.
- It describes deterministic unitary quantum mechanics with a multiple-event structure which may have implications for quantum gravity [14].
- It makes no theoretical distinction between past, present and future times. A fixed point is simply a crossing point for quantum histories.
- It contains no genuine randomness, only integrals over temporal regions of the wavefunction.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**The purple region represents the measure of existence connecting the prepared fixed point state ${\u27e6\psi \u27e7}_{{t}_{1}}$ to a measurement at ${t}_{2}$ described by the fixed point ${\u27e6\varphi \u27e7}_{{t}_{2}}$. The black lines represent processes connected to the prepared state. The blue lines represent regions of the universal wavefunction that are incompatible with the preparation.

**Figure 4.**The purple region represents the measure of existence corresponding to the ABL measure in an experiment connecting the pre- and postselection fixed points ${\u27e6\psi \u27e7}_{{t}_{1}}$ and ${\u27e6\varphi \u27e7}_{{t}_{2}}$ to a measurement at t corresponding to the fixed point ${\u27e6{a}_{i}\u27e7}_{t}$.

**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 Schrödinger dynamics.

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Ridley, M.
Quantum Probability from Temporal Structure. *Quantum Rep.* **2023**, *5*, 496-509.
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**AMA Style**

Ridley M.
Quantum Probability from Temporal Structure. *Quantum Reports*. 2023; 5(2):496-509.
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**Chicago/Turabian Style**

Ridley, Michael.
2023. "Quantum Probability from Temporal Structure" *Quantum Reports* 5, no. 2: 496-509.
https://doi.org/10.3390/quantum5020033