Skip to main content Accessibility help
×
Hostname: page-component-7dd5485656-wxk4p Total loading time: 0 Render date: 2025-10-30T17:31:50.026Z Has data issue: false hasContentIssue false

9 - Insights into Quantum Time Reversal from the Classical Schrödinger Equation

from Part III - The Arrow of Time and Time-Reversal Invariance

Published online by Cambridge University Press:  28 October 2025

Cristian López
Affiliation:
Université de Lausanne, Switzerland
Olimpia Lombardi
Affiliation:
Universidad de Buenos Aires, Argentina
Get access

Summary

The need to implement time reversal via complex conjugation in quantum theory has always been a bit of a puzzle. Why should i go to –i under temporal reflection when it has no spatiotemporal dimensions? I’ll provide a new insight into this question by showing how the little-appreciated “quantum-looking” classical Schrödinger equation of Schiller and Rosen faces the exact same problem. Since we know how to escape this problem classically, this observation teaches us one way to solve the problem quantum mechanically too. Big picture: if I’m right, the puzzle over quantum time reversal is connected to the interpretation of quantum theory.

Information

Type
Chapter
Information
The Arrow of Time
From Local Systems to the Whole Universe
, pp. 184 - 197
Publisher: Cambridge University Press
Print publication year: 2025

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Book purchase

Temporarily unavailable

References

Albert, D. Z. (2000). Time and Chance. Cambridge, MA: Harvard University Press.CrossRefGoogle Scholar
Allori, V. (2022). “On the Galilean invariance of the pilot-wave theory.” Foundations of Physics, 52 (5): 121.CrossRefGoogle Scholar
Benseny, A., Tena, D., and Oriols, X. (2016). “On the classical Schrodinger equation.” Fluctuation and Noise Letters, 15, 1640011.CrossRefGoogle Scholar
Bohm, D. (1952). “A suggested interpretation of the Quantum Theory in terms of ‘Hidden’ Variables, I and II.” Physical Review, 85(2): 166193.CrossRefGoogle Scholar
Callender, C. (2000). “Is time ‘handed’ in a quantum world?Proceedings of the Aristotelian Society, 100(3): 247269.Google Scholar
Callender, C. (2015). “One world, One beable.” Synthese 192: 31533177.CrossRefGoogle Scholar
Callender, C. (2023). “Quantum mechanics: Keeping it real?” British Journal for the Philosophy of Science, 74(4): 837–851.CrossRefGoogle Scholar
Callender, C. (2023). “The prodigy that time forgot: The incredible and untold story of John von Newton.” In Bassi, A., Goldstein, S., Tumulka, R., and Zanghì, N. (eds.), Physics and the Nature of Reality: Essays in Memory of Detlef Dürr, Springer, pp. 5161.Google Scholar
Chetrite, R., Muratore-Ginanneschi, P., and Schwieger, K. E. (2021). “Schrödinger’s 1931 paper ‘On the reversal of the laws of nature’” [“Über die Umkehrung der Naturge- setze”, Sitzungsberichte der preussischen Akademie der Wissenschaften, physikalisch-mathematische Klasse, 8 N9 144–153]. EPJ H 46, 28.CrossRefGoogle Scholar
Dürr, D., Goldstein, S., and Zanghì, N. (1997). “Bohmian mechanics and the meaning of the wave function.” In Cohen, R. S., Horne, M., and Stachel, J. (eds.), Experimental Metaphysics – Quantum Mechanical Studies for Abner Shimony, Volume One, (Boston Studies in the Philosophy of Science 193), Boston: Kluwer Academic Publishers.Google Scholar
Earman, J. (2002). “What time reversal is and why it matters.” International Studies in Philosophy of Science, 16(3): 245264.CrossRefGoogle Scholar
Gao, S. (2022). “Understanding time reversal in quantum mechanics: A new derivation.” Foundation of Physics, 52, 114. DOI: https://doi.org/10.1007/s10701-022-00634-1.CrossRefGoogle Scholar
Ghose, P. (2018). “The quantum-like face of classical mechanics.” DOI: https://doi.org/10.48550/arXiv.1801.02499.CrossRefGoogle Scholar
Hall, M. J. W., Deckert, D., & Wiseman, H. M. (2014). “Quantum phenomena modeled by interactions between many classical worlds.” Physical Review, X, 4, 041013.Google Scholar
Holland, P. (1993). The Quantum Theory of Motion. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Johns, O. (2016). Analytical Mechanics for Relativity and Quantum Mechanics (2nd ed.). Oxford: Oxford University Press.Google Scholar
Koopman, B. O. (1931). “Hamiltonian systems and transformations in Hilbert space.” Proceedings of the National Academy of Science USA, 17: 315318.CrossRefGoogle ScholarPubMed
Krylov, N. (1979). Work on the Foundations of Statistical Physics. Princeton: Princeton University Press.Google Scholar
López, C. (2021). “The physics and the philosophy of time reversal in standard quantum mechanics.” Synthese, 199 (5–6): 1426714292.CrossRefGoogle Scholar
Malament, D. B. (2004). “On the time reversal invariance of classical electromagnetic theory.” Studies in History and Philosophy of Modern Physics, 35: 295315.CrossRefGoogle Scholar
McCoy, C. (2020). “An alternative interpretation of statistical mechanics.” Erkenntnis, 85, 1: 121.CrossRefGoogle Scholar
Prigogine, I. (1996). The End of Certainty. New York: The Free Press.Google Scholar
Roberts, B. (2022). Reversing the Arrow of Time. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Roman, P. (1960). Theory of Elementary Particles. New York: Interscience.CrossRefGoogle Scholar
Romano, D. (2021). “On the alleged extra-structures of quantum mechanics.” Foundation of Physics, 51, 29. DOI: https://doi.org/10.1007/s10701-021-00426-z.CrossRefGoogle Scholar
Rosen, N. (1964). “The relation between classical and quantum mechanics.” American Journal of Physics: 597600.Google Scholar
Rosen, N. (1986). “Quantum particles and classical particles.” Foundations of Physics 16(8): 687700.CrossRefGoogle Scholar
Sakurai, J. J. (1994). Modern Quantum Mechanics. Boston, MA: Addison-Wesley.Google Scholar
Schiller, R. (1962). “Quasi-classical theory of the non-spinning electron.” Physical Review, 125: 11001108.CrossRefGoogle Scholar
Schleich, W. P., Greenberger, D. M, Kobe, D. H., and Scully, M. O. (2013). “Schrödinger equation revisited.” Proceedings of the National Academy of Science USA, 110 (14): 53745379.CrossRefGoogle ScholarPubMed
Sebens, C. (2015). “Quantum mechanics as classical physics.” Philosophy of Science 82(2): 266291.CrossRefGoogle Scholar
Shimbori, T., and Kobayashi, T. (2000). “‘Velocities’ in quantum mechanics,” Appendix A of Journal of Physics A33 7637 DOI: https://doi.org/10.48550/arXiv.quant-ph/0004086.CrossRefGoogle Scholar
Struyve, W. (2022). “Time-reversal invariance and ontology.” British Journal for the Philosophy of Science. DOI: https://doi.org/10.1086/721089.CrossRefGoogle Scholar
Sudarshan, E. C. G. (1976). “Interaction between classical and quantum systems and the measurement of quantum observables.” Pramana 6(3), 117126.CrossRefGoogle Scholar
Von Neumann, J. (1932). “Zur Operatorenmethode in der klassischen Mechanik.” Annals of Mathematics, 33, 587642.CrossRefGoogle Scholar
Wigner, E. P. (1931). Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra. New York: Academic Press.Google Scholar

Accessibility standard: WCAG 2.1 AA

Why this information is here

This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.

Accessibility Information

The PDF of this book complies with version 2.1 of the Web Content Accessibility Guidelines (WCAG), covering newer accessibility requirements and improved user experiences and achieves the intermediate (AA) level of WCAG compliance, covering a wider range of accessibility requirements.

Content Navigation

Table of contents navigation
Allows you to navigate directly to chapters, sections, or non‐text items through a linked table of contents, reducing the need for extensive scrolling.
Index navigation
Provides an interactive index, letting you go straight to where a term or subject appears in the text without manual searching.

Reading Order & Textual Equivalents

Single logical reading order
You will encounter all content (including footnotes, captions, etc.) in a clear, sequential flow, making it easier to follow with assistive tools like screen readers.
Short alternative textual descriptions
You get concise descriptions (for images, charts, or media clips), ensuring you do not miss crucial information when visual or audio elements are not accessible.

Visual Accessibility

Use of colour is not sole means of conveying information
You will still understand key ideas or prompts without relying solely on colour, which is especially helpful if you have colour vision deficiencies.

Structural and Technical Features

ARIA roles provided
You gain clarity from ARIA (Accessible Rich Internet Applications) roles and attributes, as they help assistive technologies interpret how each part of the content functions.

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×