A mathematical formulation is proposed representing time as a dual matrix structure: physical timeis represented by a matrix T, while its structural or relational time is expressed by the adjoint matrix T†This representation allows for interpreting quantum coupling and decoupling as transformations betweensubspaces of minimal entropy and maximal organization. We prove the equivalence between temporalreversibility and the invertibility of the associated operators.
A mathematical formulation is proposed representing time as a dual matrix structure: physical timeis represented by a matrix T, while its structural or relational time is expressed by the adjoint matrix T†This representation allows for interpreting quantum coupling and decoupling as transformations betweensubspaces of minimal entropy and maximal organization. We prove the equivalence between temporalreversibility and the invertibility of the associated operators.
The standard formulation of quantum theory assumes a predefined notion of time. This is a major obstacle in the search for a quantum theory of gravity, where the causal structure of space-time is expected to be dynamical and fundamentally probabilistic in character. Here, we propose a generalized formulation of quantum theory without predefined time or causal structure, building upon a recently introduced operationally time-symmetric approach to quantum theory. The key idea is a novel isomorphism between transformations and states which depends on the symmetry transformation of time reversal. This allows us to express the time-symmetric formulation in a time-neutral form with a clear physical interpretation, and ultimately drop the assumption of time. In the resultant generalized formulation, operations are associated with regions that can be connected in networks with no directionality assumed for the connections, generalizing the standard circuit framework and the process matrix framework for operations without global causal order. The possible events in a given region are described by positive semidefinite operators on a Hilbert space at the boundary, while the connections between regions are described by entangled states that encode a nontrivial symmetry and could be tested in principle. We discuss how the causal structure of space-time could be understood as emergent from properties of the operators on the boundaries of compact space-time regions. The framework is compatible with indefinite causal order, timelike loops, and other acausal structures.
We propose an alternative notion of time reversal in open quantum systems as represented by linear quantum operations, and a related generalization of classical entropy production in the environment. This functional is the ratio of the probability to observe a transition between two states under the forward and the time reversed dynamics, and leads, as in the classical case, to fluctuation relations as tautological identities. As in classical dynamics in contact with a heat bath, time reversal is not unique, and we discuss several possibilities. For any bistochastic map its dual map preserves the trace and describes a legitimate dynamics reversed in time, in that case the entropy production in the environment vanishes. For a generic stochastic map we construct a simple quantum operation which can be interpreted as a time reversal. For instance, the decaying channel, which sends the excited state into the ground state with a certain probability, can be reversed into the channel transforming the ground state into the excited state with the same probability.
Abstract Pauli's theorem asserts that no self-adjoint time operator can exist that is canonically conjugate to a Hamiltonian with a spectrum bounded from below, preventing time and energy from both being operators in standard quantum mechanics. In this paper, we propose a new reformulation that circumvents this constraint by inverting the roles of time and energy: time becomes a self-adjoint operator, while energy is treated as an external parameter. This dual perspective yields a mathematically equivalent yet physically distinct formulation, where quantum systems evolve according to internal clocks. The model accounts for the measurement problem without invoking collapse, offering an observer-relative interpretation tied to temporal misalignment. The proposed model also suggests a fresh perspective on the relationship between quantum systems and spacetime geometry.
The aim of this work is the mathematical analysis of the physical time-reversal operator and its definition as a geometrical structure\QTR{bf}{, }in such a way that it could be generalized to the purely mathematical realm. Rigorously, only having such a ``time-reversal structure'' it can be decided whether a dynamical system is time-symmetric or not.\QTR{it}{\}The ``time-reversal structures'' of several important physical and mathematical examples are presented, showing that there are some mathematical categories whose objects are the (classical or abstract) ``time-reversal systems'' and whose morphisms generalize the Wigner transformation.
The dynamics of an open quantum system can be described by a quantum operation: A linear, complete positive map of operators. Here, I exhibit a compact expression for the time reversal of a quantum operation, which is closely analogous to the time reversal of a classical Markov transition matrix. Since open quantum dynamics are stochastic, and not, in general, deterministic, the time reversal is not, in general, an inversion of the dynamics. Rather, the system relaxes toward equilibrium in both the forward and reverse time directions. The probability of a quantum trajectory and the conjugate, time reversed trajectory are related by the heat exchanged with the environment.
We analyze the symmetries of open dissipative models for quantum Liouvillian dynamical systems. Our analysis yields two representation independent symmetry relations. One of the symmetry relations, star-conjugation symmetry, is shown to represent a necessary and sufficient condition for ensuring the reality of observables and the Hermiticity of the system density operator. The other symmetry relation represents a necessary and sufficient condition for the realization of microscopic reversibility. These symmetry relations and the structure of the global dynamics are shown to give rise to a number of generalized symmetry relations for quantities expressible as matrix elements of the retarded, advanced, and global propagators. It is shown that the generalized symmetry relations may be used to facilitate the solution of spectral and temporal problems. This is demonstrated in a concrete way in an application entailing the use of dual Lanczos transformation theory to obtain an exact solution of a prototype model for excited state dynamics. Apart from elucidating the symmetries of open dissipative systems, the reported analysis reveals that there are two different and equally legitimate schemes for performing time-reversal transformations in the matrix formulation of quantum Liouvillian dynamics. The properties and ramifications of these time-reversal transformation schemes are discussed in detail.