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Authors
H. D. Zeh
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The concept of time as used in various applications and interpretations of quantum theory is briefly reviewed.
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Via the proper-time eigenstates (event states) instead of the proper-mass eigenstates (particle states), free-motion time-of-arrival theory for massive spin-1/2 particles is developed at the level of quantum field theory. The approach is based on a position-momentum dual formalism. Within the framework of field quantization, the total time-of-arrival is the sum of the single event-of-arrival contributions, and contains zero-point quantum fluctuations because the clocks under consideration follow the laws of quantum mechanics.
The symmetry of quantum theory under time reversal has long been a subject of controversy because the transition probabilities given by Borns rule do not apply backward in time. Here, we resolve this problem within a rigorous operational probabilistic framework. We argue that reconciling time reversal with the probabilistic rules of the theory requires a notion of operation that permits realizations via both pre- and post-selection. We develop the generalized formulation of quantum theory that stems from this approach and give a precise definition of time-reversal symmetry, emphasizing a previously overlooked distinction between states and effects. We prove an analogue of Wigners theorem, which characterizes all allowed symmetry transformations in this operationally time-symmetric quantum theory. Remarkably, we find larger classes of symmetry transformations than those assumed before. This suggests a possible direction for search of extensions of known physics.
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 apply the recently developed general theory of quantum time distributions arXiv:2010.07575 to find the distribution of arrival times at the detector. Even though the Hamiltonian in the absence of detector is hermitian, the time evolution of the system before detection involves dealing with a non-hermitian operator obtained from the projection of the hermitian Hamiltonian onto the region in front of the detector. Such a formalism eventually gives rise to a simple and physically sensible analytical expression for the arrival time distribution, for arbitrary wave packet moving in one spatial dimension with negligible distortion.
Quantum Theory, similar to Relativity Theory, requires a new concept of space-time, imposed by a universal constant. While velocity of light $c$ not being infinite calls for a redefinition of space-time on large and cosmological scales, quantization of action in terms of a finite, i.e. non vanishing, universal constant $h$ requires a redefinition of space-time on very small scales. Most importantly, the classical notion of time, as one common continuous time variable and nature evolving continuously in time, has to be replaced by an infinite manifold of transition rates for discontinuous quantum transitions. The fundamental laws of quantum physics, commutation relations and quantum equations of motion, resulted from Max Borns recognition of the basic principle of quantum physics: {bf To each change in nature corresponds an integer number of quanta of action}. Action variables may only change by integer values of $h$, requiring all other physical quantities to change by discrete steps, quantum jumps. The mathematical implementation of this principle led to commutation relations and quantum equations of motion. The notion of point in space-time looses its physical significance; quantum uncertainties of time, position, just as any other physical quantity, are necessary consequences of quantization of action.
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