Quantum mechanics of time travel

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The theoretical study of time travel generally follows the laws of general relativity. Quantum mechanics requires physicists to solve equations describing how probabilities behave along closed timelike curves (CTCs), which are theoretical loops in spacetime that might make it possible to travel through time. [1] [2] [3] [4]

Contents

In the 1980s, Igor Novikov proposed the self-consistency principle. [5] According to this principle, any changes made by a time traveler in the past must not create historical paradoxes. If a time traveler attempts to change the past, the laws of physics will ensure that events unfold in a way that avoids paradoxes. This means that while a time traveler can influence past events, those influences must ultimately lead to a consistent historical narrative.

However, Novikov's self-consistency principle has been debated in relation to certain interpretations of quantum mechanics. Specifically, it raises questions about how it interacts with fundamental principles such as unitarity and linearity. Unitarity ensures that the total probability of all possible outcomes in a quantum system always sums to 1, preserving the predictability of quantum events. Linearity ensures that quantum evolution preserves superpositions, allowing quantum systems to exist in multiple states simultaneously. [6]

There are two main approaches to explaining quantum time travel while incorporating Novikov's self-consistency principle. The first approach uses density matrices to describe the probabilities of different outcomes in quantum systems, providing a statistical framework that can accommodate the constraints of CTCs. The second approach involves state vectors, [7] which describe the quantum state of a system. Both approaches can lead to insights into how time travel might be reconciled with quantum mechanics, although they may introduce concepts that challenge conventional understandings of these theories. [8] [9]

Deutsch's prescription for closed timelike curves (CTCs)

In 1991, David Deutsch proposed a method to explain how quantum systems interact with closed timelike curves (CTCs) using time evolution equations. This method aims to address paradoxes like the grandfather paradox, [10] [11] which suggests that a time traveler who stops their own birth would create a contradiction. One interpretation of Deutsch's approach is that it allows for self-consistency without necessarily implying the existence of parallel universes.

Method overview

To analyze the system, Deutsch divided it into two parts: a subsystem outside the CTC and the CTC itself. To describe the combined evolution of both parts over time, he used a unitary operator (U). This approach relies on a specific mathematical framework to describe quantum systems. The overall state is represented by combining the density matrices (ρ) for both parts using a tensor product (⊗). [12] While Deutsch's approach does not assume initial correlation between these two parts, this does not inherently break time symmetry. [10]

Deutsch's proposal uses the following key equation to describe the fixed-point density matrix (ρCTC) for the CTC:

.

The unitary evolution involving both the CTC and the external subsystem determines the density matrix of the CTC as a fixed point, focusing on its state.

Ensuring Self-Consistency

Deutsch's proposal ensures that the CTC returns to a self-consistent state after each loop. However, if a system retains memories after traveling through a CTC, it could create scenarios where it appears to have experienced different possible pasts. [13]

Furthermore, Deutsch's method may not align with common probability calculations in quantum mechanics unless we consider multiple paths leading to the same outcome. There can also be multiple solutions (fixed points) for the system's state after the loop, introducing randomness (nondeterminism). Deutsch suggested using solutions that maximize entropy, aligning with systems' natural tendency to evolve toward higher entropy states.

To calculate the final state outside the CTC, trace operations consider only the external system's state after combining both systems' evolution.

Implications and criticisms

Deutsch's approach has intriguing implications for paradoxes like the grandfather paradox. For instance, if everything except a single qubit travels through a time machine and flips its value according to a specific operator:

.

Deutsch argues that maximizing von Neumann entropy is relevant in this context. In this scenario, outcomes may mix starting at 0 and ending at 1 or vice versa. While this interpretation can align with many-worlds views of quantum mechanics, it does not necessarily imply branching timelines after interacting with a CTC. [14]

Researchers have explored Deutsch's ideas further. If feasible, his model might allow computers near a time machine to solve problems beyond classical capabilities; however, debates about CTCs' feasibility continue. [15] [16]

Despite its theoretical nature, Deutsch's proposal has faced significant criticism. [17] For example, Tolksdorf and Verch demonstrated that quantum systems without CTCs can achieve results similar to Deutsch's criterion with high accuracy. [18] [19] This finding challenges claims about its uniqueness for quantum simulations of CTCs as theorized in general relativity. Their research suggests that classical systems governed by statistical mechanics could also meet these criteria [20] without invoking peculiarities attributed solely to quantum mechanics. Consequently, they argue that their findings raise doubts about Deutsch's explanation of his time travel scenario using many-worlds interpretations.

Lloyd's prescription: Post-selection and time travel with CTCs

Seth Lloyd proposed an alternative approach to time travel with closed timelike curves (CTCs), based on "post-selection" and path integrals. [21] Path integrals are a powerful tool in quantum mechanics that involve summing probabilities over all possible ways a system could evolve, including paths that do not strictly follow a single timeline. [22] Unlike classical approaches, path integrals can accommodate histories involving CTCs, although their application requires careful consideration of quantum mechanics' principles.

He proposes an equation that describes the transformation of the density matrix, which represents the system's state outside the CTC after a time loop:

, where .

In this equation:

The transformation relies on the trace operation, which summarizes aspects of the matrix. If this trace term is zero (), it indicates that the transformation is invalid in that context, but does not directly imply a paradox like the grandfather paradox. Conversely, a non-zero trace suggests a valid transformation leading to a unique solution for the external system's state.

Thus, Lloyd's approach aims to filter out histories that lead to inconsistencies by allowing only those consistent with both initial and final states. This aligns with post-selection, where specific outcomes are considered based on predetermined criteria; however, it does not guarantee that all paradoxical scenarios are eliminated.

Entropy and computation

Michael Devin (2001) proposed a model that incorporates closed timelike curves (CTCs) into thermodynamics, [23] suggesting it as a potential way to address the grandfather paradox. [24] [25] This model introduces a "noise" factor to account for imperfections in time travel, proposing a framework that could help mitigate paradoxes.

Devin's model posits that each cycle of time travel involving a quantum bit (qubit) carries a concept related to "negentropy" which represents a decrease in disorder. However, the relationship between negentropy and usable energy in this context requires further clarification and is not universally established. The model suggests that the amount of negentropy is influenced by the noise level introduced during time travel, implying that a time machine could extract work from a thermal bath based on this relationship.

Moreover, Devin's model indicates that a time machine could potentially reduce the computational effort required to solve complex problems, such as cracking codes through trial and error. CTCs could allow for more efficient computation because the system can effectively "reuse" information from different timelines, leading to faster problem-solving capabilities.

However, the model also suggests that as the noise level approaches zero, usable energy and computational power could increase significantly. This raises questions about physical limits, as infinite extraction of energy would contradict established thermodynamic principles. In addition, while Devin's model proposes intriguing ideas about computational complexity in relation to time machines, it does not fundamentally invalidate existing complexity classes. Devin's model remains theoretical and speculative, with no experimental evidence confirming its predictions.

See also

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References

  1. ↑ Smeenk, Christopher; Arntzenius, Frank; Maudlin, Tim (2023), "Time Travel and Modern Physics", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Spring 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-07-04
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  3. ↑ Ringbauer, Martin; Broome, Matthew A.; Myers, Casey R.; White, Andrew G.; Ralph, Timothy C. (2014-06-19). "Experimental simulation of closed timelike curves". Nature Communications. 5 (1): 4145. doi:10.1038/ncomms5145. ISSN   2041-1723. PMID   24942489. Archived from the original on 2024-07-01. Retrieved 2024-07-15.
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  5. ↑ "Time Travel Explained: The Novikov Self-Consistency Principle And Its Implications". Time Quiver. 2024-02-07. Archived from the original on 2024-07-16. Retrieved 2024-07-04.
  6. ↑ Friedman, John; Morris, Michael; Novikov, Igor; Echeverria, Fernando; Klinkhammer, Gunnar; Thorne, Kip; Yurtsever, Ulvi (15 September 1990). "Cauchy problem in spacetimes with closed timelike curves" (PDF). Physical Review. 42 (6): 1915–1930. Bibcode:1990PhRvD..42.1915F. doi:10.1103/PhysRevD.42.1915. PMID   10013039. Archived (PDF) from the original on 24 July 2018. Retrieved 11 August 2019.
  7. ↑ "4.2: States, State Vectors, and Linear Operators". Physics LibreTexts. 2022-01-13. Retrieved 2024-07-04.
  8. ↑ Allen, John-Mark A. (2014-10-10), Treating Time Travel Quantum Mechanically, doi:10.48550/arXiv.1401.4933 , retrieved 2024-11-27
  9. ↑ "Quantum Physics Time Travel - Consensus Academic Search Engine". consensus.app. Retrieved 2024-11-27.
  10. 1 2 Deutsch, David (15 Nov 1991). "Quantum mechanics near closed timelike lines". Physical Review. 44 (10): 3197–3217. Bibcode:1991PhRvD..44.3197D. doi:10.1103/PhysRevD.44.3197. PMID   10013776.
  11. ↑ Lindley, David (2011-02-04). "Time Travel without Regrets". Physics. 27 (4): 5. arXiv: 1005.2219 . Bibcode:2011PhRvL.106d0403L. doi:10.1103/PhysRevLett.106.040403. PMID   21405310. Archived from the original on 2024-07-16. Retrieved 2024-07-04.
  12. ↑ Michael A. Nielsen, Isaac L. Chuang. "Quantum Computation and Quantum Information" (PDF). Archived (PDF) from the original on 2024-04-20. Retrieved 2024-07-04.
  13. ↑ Lucas, Dunlap (2015). "The Metaphysics of D-CTCs: On the Underlying Assumptions of Deutsch's Quantum Solution to the Paradoxes of Time Travel". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 56: 39. arXiv: 1510.02742 . Bibcode:2016SHPMP..56...39D. doi:10.1016/j.shpsb.2016.09.001.
  14. ↑ Wallace, David (2003-09-01). "Everettian rationality: defending Deutsch's approach to probability in the Everett interpretation". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. Quantum Information and Computation. 34 (3): 415–439. arXiv: quant-ph/0303050 . Bibcode:2003SHPMP..34..415W. doi:10.1016/S1355-2198(03)00036-4. ISSN   1355-2198. Archived from the original on 2024-07-16. Retrieved 2024-07-04.
  15. ↑ Aaronson, Scott; Watrous, John (Feb 2009). "Closed Timelike Curves Make Quantum and Classical Computing Equivalent". Proceedings of the Royal Society. 465 (2102): 631–647. arXiv: 0808.2669 . Bibcode:2009RSPSA.465..631A. doi:10.1098/rspa.2008.0350. S2CID   745646.
  16. ↑ Billings, Lee. "Time Travel Simulation Resolves "Grandfather Paradox"". Scientific American. Archived from the original on 2024-06-23. Retrieved 2024-07-16.
  17. ↑ "A problem with David Deutsch's model of time travel". Conjectures and Refutations. 2015-09-02. Archived from the original on 2023-06-04. Retrieved 2024-07-16.
  18. ↑ Tolksdorf, Juergen; Verch, Rainer (2018). "Quantum physics, fields and closed timelike curves: The D-CTC condition in quantum field theory". Communications in Mathematical Physics. 357 (1): 319–351. arXiv: 1609.01496 . Bibcode:2018CMaPh.357..319T. doi:10.1007/s00220-017-2943-5. S2CID   55346710.
  19. ↑ Yuan, Xiao; Assad, Syed M.; Thompson, Jayne; Haw, Jing Yan; Vedral, Vlatko; Ralph, Timothy C.; Lam, Ping Koy; Weedbrook, Christian; Gu, Mile (2015). "Replicating the benefits of Deutschian closed timelike curves without breaking causality" (PDF). npj Quantum Information. 1: 15007. arXiv: 1412.5596 . Bibcode:2015npjQI...115007Y. doi:10.1038/npjqi.2015.7. Archived (PDF) from the original on 2024-07-16. Retrieved 2024-07-04.
  20. ↑ Tolksdorf, Juergen; Verch, Rainer (2021). "The D-CTC condition is generically fulfilled in classical (non-quantum) statistical systems". Foundations of Physics. 51 (93): 93. arXiv: 1912.02301 . Bibcode:2021FoPh...51...93T. doi:10.1007/s10701-021-00496-z. S2CID   208637445.
  21. ↑ Lloyd, Seth; Maccone, Lorenzo; Garcia-Patron, Raul; Giovannetti, Vittorio; Shikano, Yutaka; Pirandola, Stefano; Rozema, Lee A.; Darabi, Ardavan; Soudagar, Yasaman; Shalm, Lynden K.; Steinberg, Aephraim M. (27 January 2011). "Closed Timelike Curves via Postselection: Theory and Experimental Test of Consistency" (PDF). Physical Review Letters. 106 (4) 040403. doi:10.1103/PhysRevLett.106.040403. hdl:1721.1/63096. PMID   21405310. S2CID   18442086.
  22. ↑ Lloyd, Seth; Maccone, Lorenzo; Garcia-Patron, Raul; Giovannetti, Vittorio; Shikano, Yutaka (2011). "The quantum mechanics of time travel through post-selected teleportation" (PDF). Physical Review D. 84 (2) 025007. doi:10.1103/PhysRevD.84.025007. hdl:1721.1/66971. S2CID   15972766.
  23. ↑ Devin, Michael (2013-02-08), Thermodynamics of Time Machines, arXiv: 1302.3298
  24. ↑ Devin, Michael (2001). Thermodynamics of Time Machines(unpublished) (Thesis). University of Arkansas.
  25. ↑ Devin, Michael (2013). "Thermodynamics of Time Machines". arXiv: 1302.3298 [gr-qc].
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