On the realization of symmetries in quantum mechanics

The aim of this paper is to give a simple, geometric proof of Wigner’s theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner’s theorem is not fully appreciated in general. It is Wigner’s theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigner’s theorem is nothing else but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics.     

Polynomial supersymmetry and dynamical symmetries in quantum mechanics

A polynomial generalization of supersymmetry in quantum mechanics in one and two dimensions is proposed. Polynomial superalgebras in one dimension are classified. In two dimensions, a detailed analysis is made for supercharges of second order with respect to derivatives and it is shown that in all cases the binomial superalgebra leads to hidden dynamical symmetry generated by the central charge.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 3, pp. 463–478, September, 1995.     

Extended supersymmetry and hidden symmetries in one-dimensional matrix quantum mechanics

We study properties of nonlinear supersymmetry algebras realized in the one-dimensional quantum mechanics of matrix systems. Supercharges of these algebras are differential operators of a finite order in derivatives. In special cases, there exist independent supercharges realizing an (extended) supersymmetry of the same super-Hamiltonian. The extended supersymmetry generates hidden symmetries of the super-Hamiltonian. Such symmetries have been found in models with (2×2)-matrix potentials.     

On the notion of coexistence in quantum mechanics

The notion of coexistence of quantum observables was introduced to describe the possibility of measuring two or more observables together. Here we survey the various different formalisations of this notion and their connections. We review examples illustrating the necessary degrees of unsharpness for two noncommuting observables to be jointly measurable (in one sense of the phrase). We demonstrate the possibility of measuring together (in another sense of the phrase) noncoexistent observables. This leads us to a reconsideration of the connection between joint measurability and noncommutativity of observables and of the statistical and individual aspects of quantum measurements.     

On point interaction in quantum mechanics

For the Schrödinger operator corresponding to the point interaction, a direct definition is given in terms of a singular perturbation.     

Discrete symmetries in quantum scattering

Using the discrete symmetries of the Klein—Gordon, Dirac, and Schrödinger wave equations, we obtain from one solution, considered as a function of the quantum numbers and the parameters of the potentials, three other solution. Taken together, these solutions form two complete sets of solutions of the wave equation. The coefficients of the linear relations between the functions of these sets — the connection coefficients — are related in a simple manner to the wave transmission and reflection amplitudes. By virtue of the discrete symmetries of the wave equation, the connection coefficients satisfy certain symmetry relations. We show that in a number of simple cases, the behavior of the wave function near the center of formation of an additional wave determines the amplitude of the wave that is formed at infinity.P. N. Lebedev Physics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98, No. 1, pp. 60–79, January, 1994.     

On quantum mechanics and generalized Clifford algebras

In this note we give elementary examples of the naturalness of generalized Clifford algebras appearance, in some particular quantum mechanical models. First Weyl’s program [1] for quantum kinematics for the case of simplest Galois fieldsZ _n is realized in terms of generalized Clifford algebras. Dynamics might then be introduced, following the ideas of Hanney and Berry [2], as shown in [3]. Second the coherent state picture of the finite dimensional “Z _n — Quantum Mechanics” is presented. In the last part the known coherent states ofq-deformed quantum oscillators (q≡ω) are explicitly shown in the generalized Grassman algebras and the generalized Clifford algebras settings. Presented atThe Polish-Mexican Seminar, Kazimierz Dolny, August 1998 — Poland. 176     

On the use of Lie-Bäcklund operators in quantum mechanics

This paper uses Lie-Bäcklund operators to study the connection between classical and quantum-mechanical invariants and their relations to symmetry and to separation of variables. The problem of an isomorphic correspondence between classical and quantum mechanics is studied concretely and constructively. For functions at most quadratic in the momenta an isomorphism is possible which agrees with Weyl's transform and which takes invariants into invariants. It is not possible to extend the isomorphism indefinitely. The requirement that an invariant goes into an invariant may necessitate variants of Weyl's transform. This is illustrated for the case of cubic invariants. Finally, the case of a specific value of energy is considered.     

Hyperbolic quantum mechanics

We start to develop the quantization formalism in a hyperbolic Hilbert space. Generalizing Born’s probability interpretation, we found that unitary transformations in such a Hilbert space represent a new class of transformations of probabilities which describe a kind of hyperbolic interference. The most interesting problem which prompted by our investigation is to find experimental evidence of hyperbolic interference. The hyperbolic quantum formalism can also be interesting as a new theory of probability waves that can be developed in parallel with the standard quantum theory. Comparative analysis of these two wave theories could be useful for understanding of the role of various structures of the standard quantum formalism. In particular, one of distinguishing feature of the hyperbolic quantum formalism is the restricted validity of the superposition principle.     

Perturbation of resonances in quantum mechanics

Schrödinger operators on L²($\text{R}$³) of the form ?Δ + V_λ with potentials V_λ real-analytic in λ are discussed. The analytic structure in V_λ and k (with k² the energy variable) of the resolvent kernel, the eigenvalues and resonances is exhibited and we obtain in particular convergent perturbation expansions for the resonances and the corresponding resonance functions. The lower order expansion coefficients are computed explicitly. The resonances and the corresponding functions are also computed for a particle moving under the action of n point interactions. This gives asymptotic low energy information about Schrödinger Hamiltonians with short range potentials. The perturbation theory of resonances, eigenvalues and of the corresponding functions for Hamiltonians describing n point interactions perturbed by a potential is also given.