Abstract:
We examine the longstanding problem of introducing a time observable in Quantum Mechanics; using the formalism of positive-operator-valued measures we show how to define such an observable in a natural way and we discuss some consequences.

Abstract:
The kinematic time operator can be naturally defined in relativistic and nonrelativistic quantum mechanics (QM) by treating time on an equal footing with space. The spacetime-position operator acts in the Hilbert space of functions of space and time. Dynamics, however, makes eigenstates of the time operator unphysical. This poses a problem for the standard interpretation of QM and reinforces the role of alternative interpretations such as the Bohmian one. The Bohmian interpretation, despite of being nonlocal in accordance with the Bell theorem, is shown to be relativistic covariant.

Abstract:
Possible theoretical frameworks for measurement of (arrival) time in the nonrelativistic quantum mechanics are reviewed. It is argued that the ambiguity between indirect measurements by a suitably introduced time operator and direct measurements by a physical clock particle has a counterpart in the corresponding classical framework of measurement of the Newtonian time based on the Hamiltonian mechanics.

Abstract:
Recent developments are reviewed and some new results are presented in the study of time in quantum mechanics and quantum electrodynamics as an observable, canonically conjugate to energy. This paper deals with the maximal Hermitian (but nonself-adjoint) operator for time which appears in nonrelativistic quantum mechanics and in quantum electrodynamics for systems with continuous energy spectra and also, briefly, with the four-momentum and four-position operators, for relativistic spin-zero particles. Two measures of averaging over time and connection between them are analyzed. The results of the study of time as a quantum observable in the cases of the discrete energy spectra are also presented, and in this case the quasi-self-adjoint time operator appears. Then, the general foundations of time analysis of quantum processes (collisions and decays) are developed on the base of time operator with the proper measures of averaging over time. Finally, some applications of time analysis of quantum processes (concretely, tunneling phenomena and nuclear processes) are reviewed. 1. General Introduction During almost ninety years (e.g., [1, 2]), it is known that time cannot be represented by a self-adjoint operator, with the possible exception of special abstract systems (such as an electrically charged particle in an infinite uniform electric field) and a system with the limited from both below and above energy spectrum (to see later)). (Namely that fact that time cannot be represented by a self-adjoint operator is known to follow from the semiboundedness of the continuous energy spectra, which are bounded from below (usually by the value zero). Only for an electrically charged particle in an infinite uniform electric field, and for other very rare special systems, the continuous energy spectrum is not bounded and extends over the whole energy axis from ？∞ to ∞.) This fact results to be in contrast with the known sircumstance that time, as well as space, in some cases plays the role just of a parameter, while in some other cases is a physical observable which ought to be represented by an operator. The list of papers devoted to the problem of time in quantum mechanics is extremely large (e.g., [3–51], and references therein). The same situation had to be faced also in quantum electrodynamics and, more in general, in relativistic quantum field theory (e.g., [12–14, 47, 50, 51]). As to quantum mechanics, the first set of known and cited articles is [3–21]. The second set of papers on time as an observable in quantum physics [22–51] appeared from the end of the

Abstract:
We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one but it is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it a pure state of some quantum system is described by a state section (along paths) of a (Hilbert) fibre bundle. Its evolution is determined through the bundle (analogue of the) Schr\"odinger equation. Now the dynamical variables and the density operator are described via bundle morphisms (along paths). The mentioned quantities are connected by a number of relations derived in this work. In the second part of this investigation we derive several forms of the bundle (analogue of the) Schr\"odinger equation governing the time evolution of state sections. We prove that up to a constant the matrix-bundle Hamiltonian, entering in the bundle analogue of the matrix form of the conventional Schr\"odinger equation, coincides with the matrix of coefficients of the evolution transport. This allows to interpret the Hamiltonian as a gauge field. Here we also apply the bundle approach to the description of observables. It is shown that to any observable there corresponds a unique Hermitian bundle morphism (along paths) and vice versa.

Abstract:
We propose a general construction of an observable measuring the time of occurence of an effect in quantum theory. Time delay in potential scattering is computed as a straightforward application.

We can not only bring time operator in quantum mechanics (non-relativistic) but also determine its Eigen value, commutation relation of its square with energy and some of the properties of time operator like either it is Hermitian or not, either its expectation value is real or complex for a wave packet etc. Exactly these are what I have done.

Abstract:
Some results are reviewed and developments are presented on the study of Time in quantum mechanics as an observable, canonically conjugate to energy. Operators for the observable Time are investigated in particle and photon quantum theory. In particular, this paper deals with the hermitian (more precisely, maximal hermitian, but non-selfadjoint) operator for Time which appears: (i) for particles, in ordinary non-relativistic quantum mechanics; and (ii) for photons, in first-quantization quantum electrodynamics.

Abstract:
We introduce a self-adjoint operator that indicates the direction of time within the framework of standard quantum mechanics. That is, as a function of time its expectation value decreases monotonically for any initial state. This operator can be defined for any system governed by a Hamiltonian with a uniformly finitely degenerate, absolutely continuous and semibounded spectrum. We study some of the operator's properties and illustrate them for a large equivalence class of scattering problems. We also discuss some previous attempts to construct such an operator, and show that the no-go theorems developed in this context are not applicable to our construction.

Abstract:
It is demonstrated that a nonrelativistic quantum scale anomaly manifests itself in the appearance of composite operators with complex scaling dimensions. In particular, we study nonrelativistic quantum mechanics with an inverse square potential and consider a composite s-wave operator O=\psi\psi. We analytically compute the scaling dimension of this operator and determine the propagator <0|T O O^{\dagger}|0>. The operator O represents an infinite tower of bound states with a geometric energy spectrum. Operators with higher angular momenta are briefly discussed.