Nonbijective canonical transformations and their representation in quantum mechanics

Canonical transformations to action and angle variables and their representation in quantum mechanics IV. Periodic potentials

A general theory of the realizations of Lie groups by means of canonical transformations in classical mechanics is given. The problem is the analog to that of the characterization of the projective representations in quantum mechanics considered by Wigner, Bargmann, and others in the case of the Galilei and the Lorentz group. However, no application to particular groups is given in this paper. It turns out that the generators yτ of the infinitesimal transformations in a canonical realization of a Lie group 𝒢 satisfy relations of the form {yρ, yσ} = cρστyτ + dρσ, where cρστ are the structure constands of 𝒢 and dρσ are constants depending on the particular realization. It also turns out that any canonical realization of 𝒢 can be reduced to a fundamental typical form by means of a constant canonical transformation in the phase space of the system. This typical form allows one to obtain a complete characterization of all the possible canonical realizations of 𝒢. Once a suitable definition of ``irreducible'' canonical realization is given, a simple classification can be obtained in terms of the values of certain functions of the generators yτ (canonical invariants). Recalling the correspondence {} → [], formal analogy appears to be achieved with the quantum mechanical case. Even more, a parallel development of the outlined theory in quantum mechanics is of interest in the construction of the invariant operators (even in case of non-semisimple groups) and of complete systems of commuting observables acting within the irreducible representations.

Recent work by Moshinsky et al. on the role and applications of canonical transformations in quantum mechanics has focused attention on some complex extensions of linear transformations mapping the position and momentum operators x̂ and p̂ to a pair η̂ and ζ̂ of canonically conjugate, but not necessarily Hermitian, operators. In this paper we show that for a continuum of complex linear canonical transformations, a related Hilbert space of entire analytic functions exists with a scalar product over the complex plane such that the pair η̂, ζ̂ can be realized in the Schrödinger representation η and −id/dη. We provide a unitary mapping onto the ordinary Hilbert space of square-integrable functions over the real line through an integral transform. The transform kernels provide a representation of a subsemigroup of SL(2,C). The well-known Bargmann transform is the special case when η̂ and iζ̂ are the harmonic oscillator raising and lowering operators. The Moshinsky-Quesne transform is regained in the limit when the canonical transformation becomes real, a case which contains the ordinary Fourier transform. We present a realization of these transforms through hyperdifferential operators.

We show that the group of linear canonical transformations in a 2N-dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N coordinates are diagonal. We show that this Sp(2N) group is the well-known dynamical group of the N-dimensional harmonic oscillator. Finally, we study the case of n particles in a q-dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)×O(q). An application to the calculation of matrix elements is given in a following paper.

We develop an alternative approach to time independent perturbation theory in non-relativistic quantum mechanics. The method developed has the advantage to provide in one operation the correction to the energy and to the wave function, additionally we can analyze the time evolution of the system. To verify our results, we apply our method to the harmonic oscillator perturbed by a quadratic potential. An alternative form of the Dyson series, in matrix form instead of integral form, is also obtained.

Time-dependent perturbation theory in quantum mechanics is divergent at long times when the perturbation induces a resonance between two eigenstates of the unperturbed Hamiltonian. Divergences in perturbation theory are also common in quantum field theory and in critical phenomena. The renormalization group (RG) was designed to deal with these divergences. In the last two decades, this procedure has been extended to dynamical systems where the perturbation theory diverges in the long-time limit. In this article, we first review the connection between RG in the context of field theory and RG in the context of dynamical systems. We then show that the long-time divergence in the resonant situation in the time-dependent perturbation theory in quantum mechanics can be removed by using a RG-aided calculational scheme.

We observe that, within the effective generating function formalism for the implementation of canonical transformations within wave mechanics, non-trivial canonical transformations which leave invariant the form of the Hamilton function of the classical analogue of a quantum system manifest themselves in an integral equation for its stationary state eigenfunctions. We restrict ourselves to that subclass of these dynamical symmetries for which the corresponding effective generating functions are necessaarily free of quantum corrections. We demonstrate that infinite families of such transformations exist for a variety of familiar conservative systems of one degree of freedom. We show how the geometry of the canonical transformations and the symmetry of the effective generating function can be exploited to pin down the precise form of the integral equations for stationary state eigenfunctions. We recover several integral equations found in the literature on standard special functions of mathematical physics. We end with a brief discussion (relevant to string theory) of the generalization to scalar field theories in 1+1 dimensions.

We apply the techniques of canonical transforms to equations of the type \[\left[ {A( t )\mathbb{P}^2 + B( t )\{ \mathbb{P}\mathbb{Q} + \mathbb{Q}\mathbb{P} \} + C( t )\mathbb{Q}^2 + D( t )\mathbb{Q} + E( t )\mathbb{P} + F( t )\mathbb{1}} \right]\psi ( q,t ) = - i\partial _t \psi ( q,t ),\]where $\mathbb{Q}$ and $\mathbb{P}$ are the quantum position and momentum operators. The time-dependent parameters of the ${\text{W}} \wedge {\text{SL}}( {{\text{2,}}R} )$ evolution operator are found through linear differential equations. In terms of these we give explicitly the Green’s function, all separating coordinates and similarity solutions of the equation. We analyze the behavior of Gaussian and coherent-state initial conditions in closed form and present a new interpretation of all the Lewis-Riesenfeld constants of motion.

This document is the Accepted Manuscript version of the following article: Dimitris Kakofengitis, Maxime Oliva, and Ole Steuernagel, ‘Wigner's representation of quantum mechanics in integral form and its applications’, Physical Review A, Vol. 95, 022127, published 27 February 2017. DOI: https://doi.org/10.1103/PhysRevA.95.022127 ©2017 American Physical Society.

The irreversible process of ‘wave packet collapse’ in quantum measurement contradicts the reversible characteristics of the Schrödinger equation and the entire quantum mechanics. This contradiction is actually the origin of measurement problems in quantum mechanics. How to resolve this contradiction is related to the understanding of some philosophical concepts. The Copenhagen School embraces the reversible/irreversible contradictions in quantum measurement through the principle of complementarity. Both von Neumann's measurement hypothesis and Everett's many world explanation adhere to the ‘universal position of the Schrödinger equation’, but due to their detachment from the thermodynamic mechanism analysis of quantum measurement processes, they fall into the dilemma of subjectivism or possible state materialization in ontology. Decoherence theory is a natural extension of dissipative structure theory in quantum mechanics, which helps to solve quantum measurement problems by combining the study of classical conversion mechanisms between reversibility and irreversibility in statistical mechanics. Due to the common statistical roots of the time arrow problem in classical mechanics and quantum mechanics, decoherence theory cannot be seen as a measurement theory that fully implements the ‘universal Schrödinger position’. Many contradictions in quantum mechanics may need to be cleverly resolved through various contradictions in classical physics.

The concept of a Lie algebra under constraints is developed in connection with the theory of non-bijective canonical transformations. A finite-dimensional vector space M, carrying a faithful linear representation of a Lie algebra L, is mapped into a lower-dimensional space, M in such a manner that a subalgebra L0 of L is mapped into D(L0)=0. The Lie algebra L under the constraint D(L0)=0 is the largest subalgebra L1 of L that can be represented faithfully on M. If L0 is semisimple, then L1 is shown to be the centraliser cent LL0. If L is semisimple and L0 is a one-dimensional diagonal subalgebra of a Cartan subalgebra of L, then L1 is shown to be the factor algebra cent LL0/L0. The later two results are applied to non-bijective canonical transformations generalising the Kustaanheimo-Stiefel transformation.

In this paper, I review one of the most important and interesting parts of my new book “Fuzzy Systems to Quantum Mechanics” (see Ref. 1). Several conclusions in this part are worth introducing here. First of all, the motion of a mass point in classic mechanics has also waviness and the wave function of the motion of a mass point is composed of wave functions of countably infinite microscopic particles. Secondly, based on the waviness of the motion of a mass point we surely know the new conclusion described as the wave-mass-point dualism in classic mechanics. And thirdly, by using the closed relation between the wave-mass-point dualism in classic mechanics and the wave-particle dualism in quantum mechanics, unified theory of classic mechanics and quantum mechanics is naturally formed.

While groups of point transformations such as rotations, translations and permutations have been extensively used in quantum mechanics, we cannot say the same of the more general groups of canonical transformations. To begin with, for the latter groups we have to determine their unitary representation in appropriate spaces. But once this is achieved canonical transformations can play as important a role in quantum mechanics as point transformations. We illustrate this in relation with the harmonic oscillator in one and many dimensions and for one and many particles. We show how radial matrix elements with respect to harmonic oscillator states can be determined group theoretically both in the Schroedinger and Heisenberg pictures. We discuss the application of canonical transformations to the three body nuclear problem. We briefly outline in the concluding section applications to Coulomb problems, many body systems, accidental degeneracy, and generalized integral transforms.

Quantum mechanics is a very successful theory that has impacted on many areas of physics, from pure theory to applications. However, it is difficult to interpret, and philosophical contradictions and counterintuitive results are apparent at a fundamental level. In this book, Laloë presents our current understanding of the theory. The book explores the basic questions and difficulties that arise with the theory of quantum mechanics. It examines the various interpretations that have been proposed, describing and comparing them and discussing their success and difficulties. The book is ideal for researchers in physics and mathematics who want to know more about the problems faced in quantum mechanics but who do not have specialist knowledge in the subject. It will also interest philosophers of science, as well as all scientists who are curious about quantum physics and its peculiarities.

A general comparison is presented between random electrodynamics and quantum electrodynamics for the two systems which can be solved exactly in both theories, free electromagnetic fields and point dipole oscillators. The $N$-point correlation functions of the fields are computed in both theories and are found to differ in general because of the dependence upon the order of the quantum operators within products of operators. However, if all products of quantum operators are symmetrized by taking all permutations of the operator order, then the two theories give identical results for the correlation functions. Analogous results hold to all orders in the fine-structure constant for dipole oscllators in quantum and random electrodynamics. The theories agree only if the quantum operator products are symmetrized. In the limit that the oscillator couplings to the radiation fields vanish, th oscllators can be regarded as mechanical oscillators in quantum mechanics and in random mechanics. The theory of random mechanics is defined in terms of this limit which uncouples a mechanical oscillator from the radiation field. The average values of oscillator variables in random mechanics agree with those of symmetrized products in quantum mechanics. The question is then raised as to the physical significance of the many quantum operators which differ only in the order of their factors. It is pointed out that some operator products which are regarded as physically important, such as the square of the angular momentum, indeed involve unsymmetrized products of operators. On this account the average values of the angular momentum squared in the ground state of an isotropic three-dimensional harmonic oscillator differ between the random-mechanical and quantum-mechanical descriptions. However, there seems to be no case in which experiments have shown that the (unsymmetrized) quantum operator value is to be preferred to that provided by random mechanics. The presence of thermal radiation is next treated for free electromagnetic fields and for dipole-oscillator systems. Despite extraordinary differences in the points of view toward thermal radiation taken by the two theories, the conclusion is the same as that found for zero temperature; the two theories agree in their average values if all products of quantum operators are symmetrized. Finally, as a further example of the power of random electrodynamics to give an account of phenomena where Lorentz's classical electron theory failed, we investigate the diamagnetism of a charged three-dimensional isotropic oscillator. The mathematical descriptions at finite temperature are developed in full random electrodynamics and quantum electrodynamics and in second-order perturbation theory in quantum mechanics.