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 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.
This thesis examines the quantum-mechanical geometric phase with a view toward time reversal symmetry considerations. The idea of time reversal in quantum mechanics is investigated, disagreements and inconsistencies in the literature are examined, and the action of the time reversal operator is extended to time-dependent Hamiltonians. With this background, and using a definition of time reversal symmetry based on the evolution operator, I demonstrate that the existence of a non-zero geometric phase can in all cases be attributed to a breakdown of time reversal symmetry in some form. This result holds for both adiabatic and nonadiabatic evolutions, and for arbitrary dimensional parameter spaces. I explore the role of the geometric phase in a two-level Kramers system described by a parameter-dependent Hamiltonian such that the two levels can become degenerate for some value of the parameters, and discuss, from a mathematical point of view, the monopole geometric potential that results. I then extend this analysis by considering a pair of Kramers doublets, each doublet degenerate due to time reversal symmetry, where the parameters can be chosen so that each of the pair of doublets becomes degenerate with the other. I find the explicit functional forms for the two resulting nonabelian geometric gauge potentials and show that they can be identified exactly with the only two gauge-inequivalent SU(2) monopole potentials of Yang. Furthermore, following a conformal transformation these potentials can be mapped to those of the SU(2) instanton/anti-instanton pair. Finally I examine the relevance of the geometric phase to the molecular physics of time-odd systems. Time-odd coupling in molecular physics is a much under-studied area, with many potentially interesting results. Specifically I study time-odd coupling in Jahn-Teller systems under the Born-Oppenheimer approximation, where the electronic position states are coupled to the lattice momentum rather than the usual time-even lattice position. As an example I solve the E ⊗ (b₁ ⊕ b₂ ⊕ a₂) Jahn-Teller system exactly, showing that once again monopole-like geometric potentials arise, and comment on how this affects the angular momentum of the lattice subsystem.
Time is used in non-relativistic quantum theory as a classical concept. On the one hand, time is the parameter entering Schrödinger's equation and measured by an external laboratory clock. But it could be considered that time also plays the role of an observable when a given event occurs (e.g., when a particle crosses through a potential barrier), and one would expect that like every observable, it should be represented in the theory by an operator whose properties are predictors of the outcome of time measurements made on physical systems. Several time operators have been proposed in different publications, but still, there is not a widely accepted consensus. That subject, among others concerned with time in quantum mechanics, is analyzed in the present chapter, where we focus on the analysis of time-reversal symmetry in non-relativistic quantum systems. The search for symmetries in Physics helps to study the structure of theories. Briefly speaking, a symmetry is a transformation that does not change the structure of the theory. We start with the analysis of Schrödinger’s Equation verifying that for obtaining a time-symmetric description of the dynamics of a particle, the operation of time-reversal must include conjugation represented by the action of an antiunitary operator acting over a wave function. In particular, we analyze quantum systems whose evolution is determined by a set of external parameters varying cyclically, discussing how even though these parameters return to their initial values, the state of the system acquires a geometric phase, which depends on the path followed by the external parameters in the parameter space and which in general can be associated with the lack of the time-reversal symmetry of the system.
In standard physics, time reversal (T) is a mathematical abstraction: an antiunitary operator that formally reverses the direction of evolution. But what does it mean physically? How do you "reverse time" when you can't reverse scrambled eggs, un-mix gases, or un-bake a cake? This paper provides the answer that physics has been missing for a century. Using nothing but the language of cardboard tubes, colored corks, and paper sheets—objects you could find in a child's workshop—we solve what the most sophisticated quantum field theories cannot: what pure time reversal actually means when every transformation must be physically realizable, not just mathematically definable. The discovery is both radical and simple: T isn't about going backward. It's about stopping. ḱ → 0⁺: the geometric frame rate drops to zero. Billions of intermediate quantum states—the "movie" of a process—vanish into geometric silence. What remains when the frames stop generating? Only the bookends of reality: the beginning and the end. The proton and its anti-proton. The initial configuration and its final counterpart. No rewinding, no backtracking—just the collapse of history into its boundary conditions. This is more than a reinterpretation; it's an ontological revolution. Time isn't a dimension you travel along—it's a process you can halt. The arrow isn't reversed; it's erased. This work completes the "A Tale of Deep Symmetry" trilogy, providing the missing geometric interpretation of T that preserves the sacred CPT theorem while respecting a rigorous mechanical ontology. For physicists tired of mathematical abstractions that obscure rather than illuminate... For philosophers of science wrestling with the nature of time and reality... For AI researchers watching machines ask better questions than humans... For anyone who believes that the deepest truths about our universe can—and should—emerge from models simple enough to build at your kitchen table. The tubes and corks were never just toys. They were the Rosetta Stone for time itself. **Keywords:** Time reversal paradox, operational physics, geometric ontology, AI-driven discovery, CPT symmetry, foundational crisis, simplicity in science
In this paper I consider the concept of time in a general way as daily human time andthen within physics with relation to mathematics. I focus the attention on quantum mechanics, with its particular peculiarities, examining peculiar important questions like the temporal asymmetry, the Prigogine’s position and the time-reversal operator of Wigner. I conclude considering the theme of the temporal asymmetry in relation to decoherence and irreversibility. Interesting imputs related to education science will be done.
Dynamical collapse models embody the idea of a physical collapse of the wave function in a mathematically well-defined way. They involve modifications to the standard rules of quantum theory in order to describe collapse as a physical process. This appears to introduce a time reversal asymmetry into the dynamics since the state at any given time depends on collapses in the past but not in the future. Here we challenge this conclusion by demonstrating that, subject to specified model constraints, collapse models can be given a structurally time symmetric formulation in which the collapse events are the primitive objects of the theory. Three different examples of time asymmetries associated with collapse models are then examined and in each case it is shown that the same dynamical rule determining the collapse events works in both the forward and backward in time directions. Any physically observed time asymmetries that arise in such models are due to the asymmetric imposition of initial or final time boundary conditions, rather than from an inherent asymmetry in the dynamical law. This is the standard explanation of time asymmetric behaviour resulting from time symmetric laws.
This is Part 1 of a four part paper, intended to redress some of the most fundamental confusions in the subject of physical time directionality, and represent the concepts accurately. There are widespread fallacies in the subject that need to be corrected in introductory courses for physics students and philosophers. Parts 1 and 2 are about quantum mechanics, Part 3 is about fundamental concepts, and Part 4 is about cosmology. We start in Part 1 by analysing the time reversal symmetry of quantum probability laws. Time reversal symmetry is defined as the property of invariance under the time reversal transformation, T: t -> -t. It is shown that quantum mechanics (classical or relativistic) is strongly time asymmetric in its probability laws. This contradicts the orthodox analysis, found throughout the conventional literature on physical time, which claims that quantum mechanics is time symmetric or reversible. This is widely claimed as settled scientific fact, and large philosophical and scientific conclusions are drawn from it. But it is an error. The fact is that while quantum mechanics is widely claimed to be reversible on the basis of two formal mathematical properties (that it does have), these properties do not represent invariance under the time reversal transformation. A recent experiment (Batalhao at alia, 2015) showing irreversibility of quantum thermodynamics is discussed as an illustration of this result. Most physicists remain unaware of the errors, decades after they were first demonstrated. Orthodox specialists in the philosophy of time who are aware of the error continue to refer to the ‘time symmetry’ or ‘reversibility’ of quantum mechanics anyway – and exploit the ambiguity to claim false implications about physical time reversal symmetry in nature. The excuse for perpetrating the confusion is that, since it is has now become customary to refer to the formal properties of quantum mechanics as ‘reversibility’ or ‘time reversal symmetry’, we should just keep referring to them by this name, even though they are not time reversal symmetry. This causes endless confusion, in attempts to explain the physical irreversibility of our universe, and in philosophical discussions of implications of physics for the nature of time. The failure of time reversal symmetry in quantum mechanics changes the interpretation of modern physics in a deep way. It changes the problem of explaining the real irreversibility found throughout nature.