# 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

## References

- Feynman, R.P. The concept of probability in quantum mechanics. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability; University of California Press: Berkeley, CA, USA, 1951; Volume 533. [Google Scholar]
- Pusey, M.F.; Barrett, J.; Rudolph, T. On the reality of the quantum state. Nat. Phys.
**2012**, 8, 475–478. [Google Scholar] [CrossRef][Green Version] - Colbeck, R.; Renner, R. Is a system’s wave function in one-to-one correspondence with its elements of reality? Phys. Rev. Lett.
**2012**, 108, 150402. [Google Scholar] [CrossRef] [PubMed] - Ringbauer, M.; Duffus, B.; Branciard, C.; Cavalcanti, E.G.; White, A.G.; Fedrizzi, A. Measurements on the reality of the wavefunction. Nat. Phys.
**2015**, 11, 249–254. [Google Scholar] [CrossRef][Green Version] - Deutsch, D. Quantum theory of probability and decisions. Proc. R. Soc. Lond. A
**1999**, 455, 3129–3137. [Google Scholar] [CrossRef][Green Version] - Zurek, W.H. Environment-assisted invariance, entanglement, and probabilities in quantum physics. Phys. Rev. Lett.
**2003**, 90, 120404. [Google Scholar] [CrossRef][Green Version] - Zurek, W.H. Probabilities from entanglement, Born’s rule p
_{k}=∣ψ_{k}∣^{2}from envariance. Phys. Rev. A**2005**, 71, 052105. [Google Scholar] [CrossRef][Green Version] - Wallace, D. The Emergent Multiverse: Quantum Theory according to the Everett Interpretation; Oxford University Press: Oxford, UK, 2012. [Google Scholar]
- Sebens, C.T.; Carroll, S.M. Self-locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics. Br. J. Philos. Sci.
**2018**, 69, 25–74. [Google Scholar] [CrossRef][Green Version] - Vaidman, L. All is Ψ. J. Phys. Conf. Ser.
**2016**, 701, 012020. [Google Scholar] [CrossRef][Green Version] - Vaidman, L. Derivations of the Born rule. In Quantum, Probability, Logic: The Work and Influence of Itamar Pitowsky; Springer Nature: Cham, Switzerland, 2020; pp. 567–584. [Google Scholar]
- Vaidman, L. On schizophrenic experiences of the neutron or why we should believe in the many-worlds interpretation of quantum theory. Int. Stud. Philos. Sci.
**1998**, 12, 245–261. [Google Scholar] [CrossRef][Green Version] - Horwitz, L.; Arshansky, R.; Elitzur, A. On the two aspects of time: The distinction and its implications. Found. Phys.
**1988**, 18, 1159–1193. [Google Scholar] [CrossRef] - Maccone, L. A fundamental problem in quantizing general relativity. Found. Phys.
**2019**, 49, 1394–1403. [Google Scholar] [CrossRef][Green Version] - Page, D.N.; Wootters, W.K. Evolution without evolution: Dynamics described by stationary observables. Phys. Rev. D
**1983**, 27, 2885. [Google Scholar] [CrossRef] - Marletto, C.; Vedral, V. Evolution without evolution and without ambiguities. Phys. Rev. D
**2017**, 95, 043510. [Google Scholar] [CrossRef][Green Version] - Maccone, L.; Sacha, K. Quantum measurements of time. Phys. Rev. Lett.
**2020**, 124, 110402. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pauli, W. Die Allgemeinen Prinzipien der Wellenmechanik; Springer: Berlin/Heidelberg, Germany, 1933. [Google Scholar]
- Unruh, W.G.; Wald, R.M. Time and the interpretation of canonical quantum gravity. Phys. Rev. D
**1989**, 40, 2598. [Google Scholar] [CrossRef][Green Version] - Aharonov, Y.; Bergmann, P.G.; Lebowitz, J.L. Time symmetry in the quantum process of measurement. Phys. Rev.
**1964**, 134, B1410. [Google Scholar] [CrossRef] - Aharonov, Y.; Vaidman, L. The two-state vector formalism: An updated review. Time Quantum Mech.
**2008**, 734, 399–447. [Google Scholar] - Aharonov, Y.; Cohen, E.; Gruss, E.; Landsberger, T. Measurement and collapse within the two-state vector formalism. Quant. Stud. Math. Found.
**2014**, 1, 133–146. [Google Scholar] [CrossRef][Green Version] - Aharonov, Y.; Albert, D.Z. Is the usual notion of time evolution adequate for quantum-mechanical systems? I. Phys. Rev. D
**1984**, 29, 223. [Google Scholar] [CrossRef] - Aharonov, Y.; Popescu, S.; Tollaksen, J.; Vaidman, L. Multiple-time states and multiple-time measurements in quantum mechanics. Phys. Rev. A
**2009**, 79, 052110. [Google Scholar] [CrossRef][Green Version] - Heller, M. Temporal parts of four dimensional objects. Philos. Stud.
**1984**, 46, 323–334. [Google Scholar] [CrossRef] - Aharonov, Y.; Popescu, S.; Tollaksen, J. Each instant of time a new universe. In Quantum Theory: A Two-Time Success Story: Yakir Aharonov Festschrift; Springer: Milan, Italy, 2014; pp. 21–36. [Google Scholar]
- Lundeen, J.S.; Steinberg, A.M. Experimental joint weak measurement on a photon pair as a probe of Hardy’s paradox. Phys. Rev. Lett.
**2009**, 102, 020404. [Google Scholar] [CrossRef][Green Version] - Vaidman, L. Past of a quantum particle. Phys. Rev. A
**2013**, 87, 052104. [Google Scholar] [CrossRef][Green Version] - Curic, D.; Richardson, M.C.; Thekkadath, G.S.; Flórez, J.; Giner, L.; Lundeen, J.S. Experimental investigation of measurement-induced disturbance and time symmetry in quantum physics. Phys. Rev. A
**2018**, 97, 042128. [Google Scholar] [CrossRef][Green Version] - Watanabe, S. Symmetry of physical laws. Part III. Prediction and retrodiction. Rev. Mod. Phys.
**1955**, 27, 179. [Google Scholar] [CrossRef] - de Beauregard, C.O. Time symmetry and interpretation of quantum mechanics. Found. Phys.
**1976**, 6, 539–559. [Google Scholar] [CrossRef] - Cramer, J. The transactional interpretation of quantum mechanics. Rev. Mod. Phys.
**1986**, 58, 647. [Google Scholar] [CrossRef] - Wharton, K.B. Time-symmetric quantum mechanics. Found. Phys.
**2007**, 37, 159–168. [Google Scholar] [CrossRef] - Price, H. Toy models for retrocausality. Stud. Hist. Philos. Sci. Part B Stud. Hist. Philos. Mod. Phys.
**2008**, 39, 752–761. [Google Scholar] [CrossRef][Green Version] - Argaman, N. Bell’s theorem and the causal arrow of time. Am. J. Phys.
**2010**, 78, 1007–1013. [Google Scholar] [CrossRef][Green Version] - Leifer, M.S.; Pusey, M.F. Is a time symmetric interpretation of quantum theory possible without retrocausality? Proc. R. Soc. A Math. Phys. Eng. Sci.
**2017**, 473, 20160607. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zych, M.; Costa, F.; Pikovski, I.; Brukner, Č. Bell’s theorem for temporal order. Nat. Comm.
**2019**, 10, 3772. [Google Scholar] [CrossRef] [PubMed][Green Version] - Castro-Ruiz, E.; Giacomini, F.; Belenchia, A.; Brukner, Č. Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems. Nat. Comm.
**2020**, 11, 2672. [Google Scholar] [CrossRef] - Drummond, P.D.; Reid, M.D. Retrocausal model of reality for quantum fields. Phys. Rev. Res.
**2020**, 2, 033266. [Google Scholar] [CrossRef] - Keldysh, L.V. Diagram technique for nonequilibrium processes. Zh. Eksp. Teor. Fiz
**1964**, 47, 151–165. [Google Scholar] - Stefanucci, G.; van Leeuwen, R. Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction; Cambridge University Press: Cambridge, UK, 2013. [Google Scholar]
- Ridley, M.; Tuovinen, R. Formal equivalence between partitioned and partition-free quenches in quantum transport. J. Low Temp. Phys.
**2018**, 191, 380–392. [Google Scholar] [CrossRef][Green Version] - Tang, G.M. Full-counting statistics of charge and spin transport in the transient regime: A nonequilibrium Green’s function approach. Phys. Rev. B
**2014**, 90, 195422. [Google Scholar] [CrossRef][Green Version] - Esposito, M.; Ochoa, M.A.; Galperin, M. Quantum thermodynamics: A nonequilibrium Green’s function approach. Phys. Rev. Lett.
**2015**, 114, 080602. [Google Scholar] [CrossRef] [PubMed][Green Version] - Aeberhard, U.; Rau, U. Microscopic perspective on photovoltaic reciprocity in ultrathin solar cells. Phys. Rev. Lett.
**2017**, 118, 247702. [Google Scholar] [CrossRef] - Hořava, P.; Mogni, C.J. String perturbation theory on the Schwinger-Keldysh time contour. Phys. Rev. Lett.
**2020**, 125, 261602. [Google Scholar] [CrossRef] - Tuovinen, R.; Golež, D.; Eckstein, M.; Sentef, M.A. Comparing the generalized Kadanoff-Baym ansatz with the full Kadanoff-Baym equations for an excitonic insulator out of equilibrium. Phys. Rev. B
**2020**, 102, 115157. [Google Scholar] [CrossRef] - Atanasova, H.; Lichtenstein, A.I.; Cohen, G. Correlated nonequilibrium steady states without energy flux. Phys. Rev. B
**2020**, 101, 174316. [Google Scholar] [CrossRef] - Ridley, M.; Talarico, N.W.; Karlsson, D.; Gullo, N.L.; Tuovinen, R. A many-body approach to transport in quantum systems: From the transient regime to the stationary state. J. Phys. A Math. Theor.
**2022**, 55, 273001. [Google Scholar] [CrossRef] - Griffiths, R.B. What quantum measurements measure. Phys. Rev. A
**2017**, 96, 032110. [Google Scholar] [CrossRef][Green Version] - Hartle, J.; Hertog, T. One bubble to rule them all. Phys. Rev. D
**2017**, 95, 123502. [Google Scholar] [CrossRef][Green Version] - Isham, C.J.; Linden, N. Continuous histories and the history group in generalized quantum theory. J. Math. Phys.
**1995**, 36, 5392–5408. [Google Scholar] [CrossRef][Green Version] - Isham, C.J.; Linden, N.; Savvidou, K.; Schreckenberg, S. Continuous time and consistent histories. J. Math. Phys.
**1998**, 39, 1818–1834. [Google Scholar] [CrossRef][Green Version] - Oreshkov, O.; Cerf, N.J. Operational formulation of time reversal in quantum theory. Nat. Phys.
**2015**, 11, 853–858. [Google Scholar] [CrossRef][Green Version] - Gell-Mann, M.; Hartle, J.B. Quantum mechanics in the light of quantum cosmology. In Foundations of Quantum Mechanics in the Light of New Technology: Selected Papers from the Proceedings of the First through Fourth International Symposia on Foundations of Quantum Mechanics; World Scientific Publishing: Singapore, 1996; pp. 347–369. [Google Scholar]
- Dürr, D.; Goldstein, S.; Tumulka, R.; Zanghï, N. Bohmian mechanics and quantum field theory. Phys. Rev. Lett.
**2004**, 93, 090402. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ghirardi, G.C.; Rimini, A.; Weber, T. Unified dynamics for microscopic and macroscopic systems. Phys. Rev. B
**1986**, 34, 470. [Google Scholar] [CrossRef] [PubMed] - Vinante, A.; Mezzena, R.; Falferi, P.; Carlesso, M.; Bassi, A. Improved noninterferometric test of collapse models using ultracold cantilevers. Phys. Rev. Lett.
**2017**, 119, 110401. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bohm, D. A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Phys. Rev.
**1952**, 85, 166. [Google Scholar] [CrossRef] - Everett, H. “Relative state” formulation of quantum mechanics. Rev. Mod. Phys.
**1957**, 29, 454. [Google Scholar] [CrossRef][Green Version]

**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