Remarks on quaternion quantum mechanics

It is shown that quaternion quantum mechanics can be deduced from a general principle for constructing the mathematical apparatus of a physical theory of submicroscopic events. In the axiomatics of the theory, this principle may be formally characterized by postulating invariance (with respect to the probability function), in propositional calculus, of a certain set endowed with a prescribed algebraical structure.The author would like to thank Prof. D. J. Newman (London) for valuable critical comments and for his kind interest in this work.     

Remarks on supersymmetric quantum mechanics at finite temperature

We discuss the finite temperature behavior of supersymmetric quantum mechanics. We give some general results on energy eigenvalues and eigenstates and discuss their implications on susy breaking. For some models we derive exact solutions. In addition we remark on peculiarities occurring if the formalism of thermo field dynamics is used in supersymmetric theories.     

Remarks on the notion of state in quantum mechanics

In the present paper two kinds of quantum-theoretical states are considered: the physical state determined by a complete observation and the intrinsic state which comprises the values of the observed as well as the unobserved observables. It will be shown that the future values of all these observables are determined. Causality is therefore valid, though not verifiable.     

Supersymmetric quantum mechanics living on topologically non-trivial Riemann surfaces

Supersymmetric quantum mechanics is constructed in a new non-Hermitian representation. Firstly, the map between the partner operators H ^(±) is chosen antilinear. Secondly, both these components of a super-Hamiltonian $ \mathcal{H} $ \mathcal{H} are defined along certain topologically non-trivial complex curves r ^(±)(x) which spread over several Riemann sheets of the wave function. The non-uniqueness of our choice of the map $ \mathcal{T} $ \mathcal{T} between ‘tobogganic’ partner curves r ⁽⁺⁾(x) and r ⁽⁻⁾(x) is emphasized.     

Hydrogen atom in quantum mechanics and quantization on curved surfaces

New quantization rules for classical systems are obtained using the Titchmarsh expansion. These rules generalize the conventional ones and are reduced to them when a transition to Cartesian coordinates exists. An equation generalizing the Schrödinger equation to arbitrary natural systems is found. The principle of minimal constraint (strong equivalence principle) makes it possible to extend this equation to any curved spaces.     

Statistical mechanics of Coulomb gases as quantum theory on Riemann surfaces

Statistical mechanics of a 1D multivalent Coulomb gas can be mapped onto non-Hermitian quantum mechanics. We use this example to develop the instanton calculus on Riemann surfaces. Borrowing from the formalism developed in the context of the Seiberg-Witten duality, we treat momentum and coordinate as complex variables. Constant-energy manifolds are given by Riemann surfaces of genus g ≥ 1. The actions along principal cycles on these surfaces obey the ordinary differential equation in the moduli space of the Riemann surface known as the Picard-Fuchs equation. We derive and solve the Picard-Fuchs equations for Coulomb gases of various charge content. Analysis of monodromies of these solutions around their singular points yields semiclassical spectra as well as instanton effects such as the Bloch bandwidth. Both are shown to be in perfect agreement with numerical simulations.     

Remarks on quantum groups

We give a Poisson-bracket realization of SL _q (2) in the phase space ². We then discuss the physical meaning of such a realization in terms of a modified (regularized) toy model, the nonregularized version of which is due to Klauder.Some general remarks and suggestions are also presented in this Letter.     

Remarks on quantum symmetry

We discuss the concept of Quantum Symmetry in quantum field theory, and in particular the role of the gauge principle. We present a scheme how quantum symmetries can be realized in a Hilbert space, and sketch its construction from the theory of superselection sectors of the gauge invariant (observable) quantities. The approach is independent of (Drinfeld's) quantum groups.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June, 1992.     

N=2 supersymmetric quantum mechanics on Riemann surfaces with meromorphic superpotentials

We constructN=2 supersymmetric quantum Hamiltonians with meromorphic superpotentials on compact Riemann surfaces and investigate the topological properties of these Hamiltonians.L ₂-cohomology groups for supercharge (a deformed operator) are considered and the Witten index for the supersymmetric Hamiltonian with meromorphic superpotential is calculated in terms of Euler characteristic of the Riemann surface and the degree of a divisor of poles for the differential of the superpotential.This work was supported, in part, by a Soros Foundation Grant awarded by the American Physical Society     

From conformal quantum mechanics on bordered Riemann surfaces to covariant string field theory

The Faddeev-Popov determinant of the reparametrization ghosts of the closed bosonic string on bordered Riemann surfaces is investigated. It is found that the standard ghost action has to be supplemented by additional boundary terms. These additional terms lead to a nontrivial ghost transition amplitude. It is shown that boundary variations of ghost transition amplitudes are generated by the ghost Virasoro operators. The relevance of the boundary variation operator for string field theory is illustrated. A new interpretation of the BRST operator in terms of boundary variations is obtained and the BRST invariance of the transition amplitudes is demonstrated. A generalization of the Siegel gauge of second quantized closed bosonic string field theory is presented.     

Remarks on phase transitions of surfaces

Clean surfaces of pure crystals may exhibit superstructures at low temperature which disappear at a critical temperature. We discuss the order and characteristics of these crystallographic phase transitions of surfaces.     

Quaternionic quantum mechanics is consistent with complex quantum mechanics

Quaternionic quantum mechanics is investigated in the light of the great success of complex quantum mechanics. It is shown that to reproduce the results of complex quantum mechanics, quaternionic quantum mechanics must contain complex quantum mechanics.     

The transitions among classical mechanics,quantum mechanics,and stochastic quantum mechanics

Various formalisms for recasting quantum mechanics in the framework of classical mechanics on phase space are reviewed and compared. Recent results in stochastic quantum mechanics are shown to avoid the difficulties encountered by the earlier approach of Wigner, as well as to avoid the well-known incompatibilities of relativity and ordinary quantum theory. Specific mappings among the various formalisms are given.     

p-adic quantum mechanics

An extension of the formalism of quantum mechanics to the case where the canonical variables are valued in a field ofp-adic numbers is considered. In particular the free particle and the harmonic oscillator are considered. In classicalp-adic mechanics we consider time as ap-adic variable and coordinates and momentum orp-adic or real. For the case ofp-adic coordinates and momentum quantum mechanics with complex amplitudes is constructed. It is shown that the Weyl representation is an adequate formulation in this case. For harmonic oscillator the evolution operator is constructed in an explicit form. For primesp of the form 4l+1 generalized vacuum states are constructed. The spectra of the evolution operator have been investigated. Thep-adic quantum mechanics is also formulated by means of probability measures over the space of generalized functions. This theory obeys an unusual property: the propagator of a massive particle has power decay at infinity, but no exponential one.     

Generalized quantum mechanics

A convex scheme of quantum theory is outlined where the states are not necessarily the density matrices in a Hilbert space. The physical interpretation of the scheme is given in terms of generalized “impossibility principles”. The geometry of the convex set of all pure and mixed states (called a statistical figure) is conditioned by the dynamics of the system. This provides a method of constructing the statistical figures for non-linear variants of quantum mechanics where the superposition principle is no longer valid. Examples of that construction are given and its possible significance for the interrelation between quantum theory and general relativity is discussed.     

Copenhagen quantum mechanics

In our quantum mechanics courses, measurement is usually taught in passing, as an ad-hoc procedure involving the ugly collapse of the wave function. No wonder we search for more satisfying alternatives to the Copenhagen interpretation. But this overlooks the fact that the approach fits very well with modern measurement theory with its notions of the conditioned state and quantum trajectory. In addition, what we know of as the Copenhagen interpretation is a later 1950s development and some of the earlier pioneers like Bohr did not talk of wave function collapse. In fact, if one takes these earlier ideas and mixes them with later insights of decoherence, a much more satisfying version of Copenhagen quantum mechanics emerges, one for which the collapse of the wave function is seen to be a harmless book keeping device. Along the way, we explain why chaotic systems lead to wave functions that spread out quickly on macroscopic scales implying that Schrödinger cat states are the norm rather than curiosities generated in physicists’ laboratories. We then describe how the conditioned state of a quantum system depends crucially on how the system is monitored illustrating this with the example of a decaying atom monitored with a time of arrival photon detector, leading to Bohr’s quantum jumps. On the other hand, other kinds of detection lead to much smoother behaviour, providing yet another example of complementarity. Finally we explain how classical behaviour emerges, including classical mechanics but also thermodynamics.