In general relativity, theories about time travel have traditionally been explored based on its laws. However, incorporating quantum mechanics into the picture requires physicists to solve equations describing how probabilities (represented by density matrices) behave along closed timelike curves (CTCs). CTCs are loops in spacetime that could theoretically enable time travel.
In the 1980s, Igor Novikov proposed the self-consistency principle. This principle suggests that regardless of a time traveler's attempts to alter the past, calculations would always lead to a consistent history. However, Novikov's self-consistency principle challenges determinism, which holds that every event has a cause, some standard interpretations of quantum mechanics (particularly unitarity), which maintains a total probability of 1, and linearity, which calculates combined probabilities.[1]
There are two primary approaches to applying Novikov's self-consistency concept to quantum time travel. The first approach, the Deutsch prescription, utilizes a specific mathematical tool called a density matrix. The second approach employs a different concept, the state vector, and leads to theories deviating from standard interpretations of quantum mechanics.
Deutsch's prescription for closed timelike curves (CTCs)
In 1991, David Deutsch proposed a method using time evolution equations to describe quantum systems interacting with closed timelike curves (CTCs). This approach tackles paradoxes like the grandfather paradox,[2] where a time traveler could create a contradiction by going back in time and preventing their own birth. However, a criticism is that it suggests the time traveler might arrive in a parallel universe rather than their own past.
Method overview
To analyze the system, Deutsch divided it into two parts: a subsystem outside the CTC and the CTC itself. He then used a unitary operator (U) to capture the combined evolution of both parts over time. This approach relies on a specific mathematical description of quantum systems. The overall state is represented by combining the density matrices (ρ) for both the subsystem and the CTC using a tensor product (⊗). Notably, Deutsch assumed no initial connection between these two parts. While this assumption breaks time symmetry, he justifies it through arguments from measurement theory and the second law of thermodynamics.
The key idea of Deutsch's proposal is captured by this equation, which describes the fixed-point density matrix (ρCTC) for the CTC:
.
![{\displaystyle \rho _{\text{CTC}}={\text{Tr}}_{A}\left[U\left(\rho _{A}\otimes \rho _{\text{CTC}}\right)U^{\dagger }\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/899893034677f871b619e995c8b8b4139f5e0169)
Deutsch's proposal offers solutions that always return the CTC to a consistent state after a loop. This means any measurable property of the CTC will return to its initial value. However, this raises concerns. If a system retains memories after traveling through the CTC, it could create inconsistencies where the system has multiple possible pasts.
Furthermore, Deutsch's approach might conflict with standard probability calculations in quantum mechanics (path integrals) unless we allow for a system to have multiple histories. There can also be multiple solutions (fixed points) for the system's state after the loop, introducing randomness (nondeterminism). Deutsch suggested using the solution with the highest entropy, reflecting the tendency of systems to become more disordered over time (second law of thermodynamics).
To calculate the final state outside the CTC, a specific mathematical operation (trace) considers only the external system's state after the combined evolution of both. This combined evolution is described by a tensor product (⊗) of the density matrices for both systems, followed by applying a combined time evolution operator (U) to the entire system.
Implications and criticisms
This approach has intriguing implications for paradoxes like the grandfather paradox. Consider a scenario where everything except a single quantum bit (qubit) travels through a time machine and flips its value according to a specific operator:
- .
The most common solution for a qubit's state (represented by ρCTC) after interacting with a closed timelike curve (CTC) is described by a density matrix. This solution depends on a value (a) that blends the qubit's state between two possibilities. Notably, there isn't a single solution, but a range of possibilities.
Deutsch argues that the solution maximizing von Neumann entropy (a measure of how mixed the state is) is the most relevant. In this case, the qubit becomes a mix of starting at 0 and ending at 1, or vice versa. According to Deutsch's interpretation, which aligns with the many-worlds view of quantum mechanics, this doesn't create paradoxes because the qubit travels to a different parallel universe after interacting with the CTC.
Researchers have explored the potential of Deutsch's ideas. If his CTC time travel were possible, computers near a time machine might solve problems far exceeding the capabilities of classical computers.[3]
Deutsch proposed a specific criterion for simulating closed timelike curves (CTCs) using quantum mechanics. However, Tolksdorf and Verch showed that this criterion can be achieved with high accuracy even in quantum systems that lack CTCs.[4] This finding casts doubt on the idea that Deutsch's criterion is unique to quantum simulations of CTCs as theorized in general relativity. Their later work extended this concern, demonstrating that the criterion can also be fulfilled in classical systems governed by statistical mechanics.[5] These results suggest that Deutsch's criterion might not be specific to the peculiarities of quantum mechanics and may not be suitable for inferring possibilities of real time travel or its potential realization through quantum mechanics. Consequently, Tolksdorf and Verch argue that their findings raise questions about the validity of Deutsch's explanation of his time travel scenario using the many-worlds interpretation.
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, mathematical tools used in quantum mechanics to analyze probabilities.[6][7] Unlike classical approaches, path integrals allow for consistent histories even with CTCs. Lloyd argues that focusing on the state of the system outside the CTC is more relevant.
His proposal involves an equation describing how the density matrix (representing the system's state) outside the CTC is transformed after a time loop:
- , where .
![{\displaystyle \rho _{f}={\frac {C\rho _{i}C^{\dagger }}{{\text{Tr}}\left[C\rho _{i}C^{\dagger }\right]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66a8d66d08bd0e3ea453d3dae97759fc6cae15f0)
![{\displaystyle C={\text{Tr}}_{\text{CTC}}\left[U\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a14fd5f6200d9a3e61ef2ccc5cde74dae00af5)
This transformation depends on the trace (a specific sum) of another mathematical object calculated within the CTC. If this trace term is zero (), the equation has no solution, indicating an inconsistency like the grandfather paradox. Conversely, a non-zero trace leads to a unique solution for the external system's state.
![{\displaystyle {\text{Tr}}\left[C\rho _{i}C^{\dagger }\right]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f6bef6c7060324d7bc028c4a2809d8ff7acf90)
Entropy and computation
In 2001, Michael Devin proposed a model incorporating closed timelike curves (CTCs) into thermodynamics[8], suggesting it as a solution to the grandfather paradox.[9][10] This model introduces a "noise" factor to account for the imperfections of time travel, allowing it to explain how time travel might function without paradoxes.
Devin's model suggests that each time travel cycle involving a quantum bit (qubit) carries a usable form of energy (negentropy) proportional to the noise level. This implies a time machine could extract work from a heat source (thermal bath) in proportion to the negentropy gained. Similarly, the model suggests time machines could drastically reduce the effort needed to crack complex codes through trial and error. However, the model also predicts that usable energy and computing power become infinitely large as noise approaches zero. This implies that traditional classifications of computational problems (complexity classes) might not be applicable to time machines with very low noise levels. However, this concept remains theoretical.
See also
References
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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". Physical Review Letters. 106 (4): 040403. arXiv:1005.2219. Bibcode:2011PhRvL.106d0403L. doi:10.1103/PhysRevLett.106.040403. PMID 21405310. S2CID 18442086.
- ^ Lloyd, Seth; Maccone, Lorenzo; Garcia-Patron, Raul; Giovannetti, Vittorio; Shikano, Yutaka (2011). "The quantum mechanics of time travel through post-selected teleportation". Physical Review D. 84 (2): 025007. arXiv:1007.2615. Bibcode:2011PhRvD..84b5007L. doi:10.1103/PhysRevD.84.025007. S2CID 15972766.
- ^ Devin, Michael (2013-02-08), Thermodynamics of Time Machines, doi:10.48550/arXiv.1302.3298, retrieved 2024-06-29
- ^ Devin, Michael (2001). Thermodynamics of Time Machines(unpublished) (Thesis). University of Arkansas.
- ^ Devin, Michael (2013). "Thermodynamics of Time Machines". arXiv:1302.3298 [gr-qc].
Recent Comments