Jakub Rembieliński

Coherent states for a quantum particle on a circle

The coherent states for a quantum particle on a circle are introduced. The Bargmann representation within the actual treatment provides the representation of the algebra , where U is unitary, which is a direct consequence of the Heisenberg algebra , but it is more adequate for the study of circular motion.

On the quantum differential calculus and the quantum holomorphicity

Under some natural assumptions [less restrictive than in the paper by Wess and Zumino (preprint CERN‐TH‐5697/90, LAPP‐TH‐284/90)] differential calculi on the quantum plane are found and investigated. Complex structure, complex derivatives, and holomorphic functions on the quantum plane are defined. Generalized Cauchy–Riemann equations are given and solved.

Lorentz-covariant reduced spin density matrix and Einstein-Podolsky-Rosen–Bohm correlations

We show that it is possible to define a Lorentz-covariant reduced spin density matrix for massive particles. Such a matrix allows one to calculate the mean values of observables connected with spin measurements (average polarizations). Moreover, it contains not only information about polarization of the particle but also information about its average kinematical state. We also use our formalism to calculate the correlation function in the Einstein-Podolsky-Rosen-Bohm type experiment with massive relativistic particles.

Einstein-Podolsky-Rosen correlations of spin measurements in two moving inertial frames

The formula for the correlation function of spin measurements of two particles in two moving inertial frames is derived within Lorentz-covariant quantum mechanics formulated in the absolute synchronization framework. These results are the first exact Einstein-Podolsky-Rosen correlation functions obtained for Lorentz-covariant quantum-mechanical systems in moving frames under physically acceptable conditions, i.e., taking into account the localization of the particles during the detection and using the spin operator with proper transformation properties under the action of the Lorentz group. Some special cases and approximations of the calculated correlation function are given. The resulting correlation function can be used as a basis for a proposal of a decisive experiment for a possible existence of a quantum-mechanical preferred frame.

Tritium decay and the hypothesis of tachyonic neutrinos

Abstract. Numerous recent measurements indicate an excess of counts near the endpoint of the electron energy spectrum in tritium decay. We show that this effect is expected if the neutrino is a tachyon. Results of calculations, based on a unitary (causal) theory of tachyons, are presented. The hypothesis of tachyonic neutrinos also offers a natural explanation of the vector-axial (V-A) structure of the weak leptonic current in neutrino interactions.

Coherent states of a charged particle in a uniform magnetic field

The coherent states are constructed for a charged particle in a uniform magnetic field based on coherent states for the circular motion which have recently been introduced by the authors.

On the uncertainty relations and squeezed states for the quantum mechanics on a circle

The uncertainty relations for the position and momentum of a quantum particle on a circle are identified as minimized by the corresponding coherent states. The squeezed states in the case of the circular motion are introduced and discussed in the context of the uncertainty relations.

Tachyons and Preferred Frames

Quantum field theory of spacelike particles is investigated in the framework of an absolute causality scheme preserving Lorentz symmetry. It is related to an appropriate choice of the synchronization procedure (definition of time). In this formulation existence of field excitations (tachyons) distinguishes an inertial frame (privileged frame of reference) via spontaneous breaking of the so-called synchronization group. In this scheme the relativity principle is broken but Lorentz symmetry is exactly preserved in agreement with local properties of the observed world. It is shown that tachyons are associated with unitary orbits of Poincare mappings induced from the SO(2) little group instead of the SO(2,1) one. Therefore the corresponding elementary states are labeled by helicity. The cases of the helicity λ = 0 and are investigated in detail and a corresponding consistent field theory is proposed. In particular, it is shown that the Dirac-like equation proposed by Chodos et al.,1 inconsistent in the standard formulation of QFT, can be consistently quantized in the presented framework. This allows us to treat more seriously the possibility that neutrinos might be fermionic tachyons, as is suggested by experimental data about neutrino masses.2–4

An open quantum system approach to EPR correlations in K0K¯0 system

Abstract We find the time evolution of the system of two non-interacting unstable particles, distinguishable as well as identical ones, in arbitrary reference frame having only the Kraus operators governing the evolution of its components in the rest frame. We than calculate in the rigorous way Einstein–Podolsky–Rosen quantum correlation functions for K 0 K ¯ 0 system in the singlet state taking into account CP -violation and decoherence and show that the results are exactly the same despite the fact we treat kaons as distinguishable or identical particles which means that the statistics of the particles plays no role, at least in considered cases.

Unstable particles as open quantum systems

We present the probability-preserving description of the decaying particle within the framework of quantum mechanics of open systems, taking into account the superselection rule prohibiting the superposition of the particle and vacuum. In our approach the evolution of the system is given by a family of completely positive trace-preserving maps forming a one-parameter dynamical semigroup. We give the Kraus representation for the general evolution of such systems, which allows one to write the evolution for systems with two or more particles. Moreover, we show that the decay of the particle can be regarded as a Markov process by finding explicitly the master equation in the Lindblad form. We also show that there are remarkable restrictions on the possible strength of decoherence.