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]

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]

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.

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.

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.

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 in spacetimes without CTCs can achieve results similar to Deutsch's criterion with any prescribed accuracy.^[18][19] This finding challenges claims that quantum simulations of CTCs are related to closed timelike curves as understood in general relativity. Their research also shows 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 of quantum physics.

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:

• is the density matrix of the system after interacting with the CTC.
• is the initial density matrix of the system before the time loop.
• is a transformation operator derived from the trace operation over the CTC, applied to the unitary evolution operator .

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.

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.

• Novikov self-consistency principle
• Grandfather paradox
• Causal loop
• Chronology protection conjecture
• Retrocausality