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 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 a quantum state extension problem

The problem is considered of finding, for a given pair of states on C ^*-algebras A ₁ A ₂ and A ₂ A ₃, a joint extension to A ₁ A ₂ A ₃. The fact that, in contrast to classical probability, such an extension may fail to exist, is related to the fact that different convex decompositions of the same quantum state need not have a common refinement. Improved necessary criteria for extensibility in terms of Bell's inequalities are derived, and are compared to the necessary and sufficient criteria, as well as to entropic bounds in the simplest case.     

Remarks on charm electroproduction in quantum chromodynamics

Predictions are made for the charm contribution to electroproduction structure functions both for small and large Q².     

Time symmetry and quantum measurements

Although the quantum laws are time-reversal invariant, a contradiction appears if two measurements performed by a single observer, and described according to these laws, are performed in two opposite directions of time. This contradiction leads to bringing forth the concept of an observer's private time, and then to building up a temporal parameter common to several observers from their private times. Time asymmetry turns out to be a consequence of the latter construction.     

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.     

Harnessing symmetry to control quantum transport

Controlling transport in quantum systems holds the key to many promising quantum technologies. Here we review the power of symmetry as a resource to manipulate quantum transport and apply these ideas to engineer novel quantum devices. Using tools from open quantum systems and large deviation theory, we show that symmetry-mediated control of transport is enabled by a pair of twin dynamic phase transitions in current statistics, accompanied by a coexistence of different transport channels. By playing with the symmetry decomposition of the initial state, one can modulate the importance of the different transport channels and hence control the flowing current. Motivated by the problem of energy harvesting, we illustrate these ideas in open quantum networks, an analysis that leads to the design of a symmetry-controlled quantum thermal switch. We review an experimental setup recently proposed for symmetry-mediated quantum control in the lab based on a linear array of atom-doped optical cavities, and the possibility of using transport as a probe to uncover hidden symmetries, as recently demonstrated in molecular junctions, is also discussed. Other symmetry-mediated control mechanisms are also described. Overall, these results demonstrate the importance of symmetry not only as an organizing principle in physics but also as a tool to control quantum systems.     

Yangian symmetry in integrable quantum gravity

Dimensional reduction of various gravity and supergravity models leads to effectively two-dimensional field theories described by gravity coupled G/H coset space σ-models. The transition matrices of the associated linear system provide a complete set of conserved charges. Their Poisson algebra is a semi-classical Yangian double modified by a twist which is a remnant of the underlying coset structure. The classical Geroch group is generated by the Lie-Poisson action of these charges. Canonical quantization of the structure leads to a twisted Yangian double with fixed central extension at a critical level.     

A short introduction to quantum symmetry

In quantum theory, symmetries more general than groups are possible. We give a general definition of a quantum symmetry, such that symmetry operations act on the Hilbert space of physical states and notions of unitarity, invariance and covariance are defined. Within this frame, weak quasi quantum groups are described as a natural generalization of group algebras. Consistency with locality distinguishes them from more general quantum symmetries. To find the new kinds of symmetry one should investigate low dimensional quantum systems such as two-dimensional layers.     

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.     

Diffeomorphism symmetry in simplicial quantum gravity

A construction of simplicial quantum gravity with an extremely strong form of diffeomorphism symmetry of the functional integral measure (Π d ℓ e^−S_pg) is proposed using the method of mathematical induction. By means of the formalism of introducing a continuous four-space approximating a lattice model in our previous paper this symmetry results in a “free gas” behavior of the vertices of the lattice on a C^∞ manifold. Furthermore, we argue that this kind of a simplicial quantum gravity model leads to a gravitational theory described by a local action in the long wavelength (continuum) limit.     

Antiunitary symmetry operators in quantum mechanics

A criterion to decide that some symmetries of a quantum system must be realized as antiunitary operators is given. It is based on some mathematical theorems about the second cohomology group of the symmetry group when expressed in terms of those of a normal subgroup and the corresponding factor group. It is also shown that this criterion implies that the only possibility for the unitary subgroup in the Galilean case is that generated by the space reflection and the connected component containing the identity; otherwise only massless systems would arise.     

Internal symmetry groups of quantum geons

In canonical quantum gravity asymptotically trivial diffeomorphisms not deformable to the identity can act nontrivially on the quantum state space. We show that for many 3-manifolds, the inequivalent diffeomorphisms comprise coverings in SU(2) of crystallographic groups. When the diffeomorphism R_2π associated with 2π-rotation is nontrivial, state vectors can have half-integral angular momentum; we list all 3-manifolds with R_2π trivial.     

Remarks on duality of 1 dimensional quantum spin models

We present some results on duality maps and ground states of 1 dimensional quantum spin models. We also give some applications to Kramers Wannier duality and the nonlocal transformation that Kennedy and Tasaki discovered in their study of Haldane phase of quantum antiferromagnetic spin models.