Hilbert space representation of an algebra of observables forq-deformed relativistic quantum mechanics

Using a representation of theq-deformed Lorentz algebra as differential operators on quantum Minkowski space, we define an algebra of observables for a q-deformed relativistic quantum mechanics with spin zero. We construct a Hilbert space representation of this algebra in which the square of the massp ² is diagonal.     

Realizations ofU q(su(1, 1)) by deformed quantum harmonic oscillator algebras and associated deformed Heisenberg algebras

Forsu(1, 1)-symmetric Hamiltonians of quantum mechanical systems (e.g. single-mode quantum harmonic oscillator, radial Schrödinger equation for Coulomb problem or isotropic quantum harmonic oscillator, etc.), the Heisenberg algebra of phase-space variables in two dimensions satisfy the bilinear commutation relation [ip,x]=1 (in normal units). Also there are different realizations ofsu(1, 1) by the generators of quantum harmonic oscillator algebra. We seek here the forms of deformed Heisenberg algebras (bilinear in deformedx and ip) associated with deformedsu(1, 1)-symmetric Hamiltonians. These forms are not unique in contrast to the undeformed case; and these forms are obtained here by considering different realizations of the deformedsu(1, 1) algebra by deformed oscillator algebras (satisfying different bilinear relations in deformed creation and annihilation operators), and then imposing different conditions (e.g. the deformed Heisenberg algebra of the form of the undeformed one, the form of realizations of the deformedsu(1, 1) algebra by deformed phase-space variables being the same as that ofsu(1, 1) algebra by undeformed phase-space variables, etc.), assuming linear relations between deformed phase-space variables and deformed creation-annihilation operators (as it is done in the undeformed case), we get different Heisenberg algebras. These facts are revealed in the case of a two-body Calogero model in its centre of mass frame (and for no other integrable systems in one-dimension having potential of the formV(x _i ? x_j).     

Relativistic non-Hamiltonian mechanics

Relativistic particle subjected to a general four-force is considered as a nonholonomic system. The nonholonomic constraint in four-dimensional space-time represents the relativistic invariance by the equation for four-velocity u_μu^μ + c² = 0, where c is the speed of light in vacuum. In the general case, four-forces are non-potential, and the relativistic particle is a non-Hamiltonian system in four-dimensional pseudo-Euclidean space-time. We consider non-Hamiltonian and dissipative systems in relativistic mechanics. Covariant forms of the principle of stationary action and the Hamilton’s principle for relativistic mechanics of non-Hamiltonian systems are discussed. The equivalence of these principles is considered for relativistic particles subjected to potential and non-potential forces. We note that the equations of motion which follow from the Hamilton’s principle are not equivalent to the equations which follow from the variational principle of stationary action. The Hamilton’s principle and the principle of stationary action are not compatible in the case of systems with nonholonomic constraint and the potential forces. The principle of stationary action for relativistic particle subjected to non-potential forces can be used if the Helmholtz conditions are satisfied. The Hamilton’s principle and the principle of stationary action are equivalent only for a special class of relativistic non-Hamiltonian systems.     

Second-order superintegrable quantum systems

A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n ? 1 functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, then the system is second-order superintegrable, the most tractable case and the one we study here. Such systems have remarkable properties: multi-integrability and separability, a quadratic algebra of symmetries whose representation theory yields spectral information about the Schrödinger operator, and deep connections with expansion formulas relating classes of special functions. For n = 2 and for conformally flat spaces when n = 3, we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here, we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension.     

Q operators for the simple quantum relativistic Toda Chain

We investigate the simple quantum relativistic Toda chain. The ultralocal simple Weyl algebra pair is associated with each site of the chain. Weyl’s q is considered to be inside a unit circle. Both independent Baxter operators Q are constructed explicitly as series in local Weyl generators. The operator-valued Wronskian of Q-s is also calculated.     

Star products and quantum groups in quantum mechanics and field theory

Hopf algebras and quantum groups have recently been applied to the analysis of the combinatorics of Feynman graphs in relativistic quantum field theory. On the other hand, in accordance with the program of deformation quantization, the relation between star products and the perturbative expansion in field theory has also been the subject of intensive study. In the present work we clarify the relation between these two approaches. We show how these techniques can be applied in a unified way to quantum systems with a finite number of degrees of freedom and to quantum field theories. In particular, we find that the time-ordered product of quantum fields is the Weyl transform of a certain twisted product. We also show that one can pass from systems involving bosons to systems with fermions, essentially just by replacing the symmetric algebra of the relevant vector space by its exterior algebra.     

Collision theory for massless Fermions

Starting from the basic postulates of local relativistic quantum theory, the asymptotic incoming and outgoing collision states of massless Fermions are constructed. The corresponding Hilbert spaces have Fock structure and thus allow the usual definition of anS-matrix. In contrast to the massive case, there are geometric relations between the local nets of the underlying field algebra and the asymptotic fields.     

Geometric Quantization of Relativistic Hamiltonian Mechanics

A relativistic Hamiltonian mechanical system is seen as a conservative Dirac constraint system on the cotangent bundle of a pseudo-Riemannian manifold. We provide geometric quantization of this cotangent bundle where the quantum constraint serves as a relativistic quantum equation.     

Smearing of Observables and Spectral Measures on Quantum Structures

An observable on a quantum structure is any σ-homomorphism of quantum structures from the Borel σ-algebra of the real line into the quantum structure which is in our case a monotone σ-complete effect algebra with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean σ-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there corresponds a spectral measure.     

Group manifold analysis of the structure of relativistic quantum dynamics

We present a group law, derived as a contraction of the conformal group, from which we obtain by using a canonical procedure a relativistic quantum system with an invariant evolution parameter (the proper time) and where the position operator belongs to the Lie algebra of the group. The restriction of the theory to the mass shell breaks part of the symmetry; of the previous 15 generators, only 10 remain which generate an action of the Poincaré group defining an orbit in the former group manifold. Some comments on the relativistic position operator are also made.     

Supersymmetric Quantum Mechanics and SUSY Dependent SU(2) Symmetry for a Spin-1/2 Charged Particle in a Magnetic Field

We find that in a supersymmetric quantum mechanics (SUSY QM) system, in addition to supersymmetric algebra, an associated SU(2) algebra can be obtained by using semiunitary (SUT) operator and projection operator, and the relevant constants of motion can be constructed. Two typical quantum systems are investigated as examples to demonstrate the above finding. The first example is the quantum system of a nonrelativistic charged particle moving in x-y plane and coupled to a magnetic field along z axis. The second example is provided with the Dirac particle in a magnetic field. Similarly there exists an SU_τ(2) \otimes SU_σ(2) symmetry in the context of the relativistic Pauli Hamiltonian squared. We show that there exists also an SU(2) symmetry associated with the supersymmetry of the Dirac particle.     

Quantum constraint algebra for a closed bosonic string in a gravitational and dilaton background

We derive the quantum constraint algebra for a closed bosonic string moving in a gravitational and dilaton background to first order in '. The hamiltonian approach is used to directly compute the quantum constraint commutators and calculate the c-and q-number anomalies that arise at the quantum level. The requirement that the algebra preserves the conformal invariance leads to the known background field equations.     

Stochastic physical origin of the quantum operator algebra and phase space interpretation of the Hilbert space formalism: The relativistic spin zero case

Based on a recent association of quantum observable algebra with stochastic processes in the frame of the causal stochastic interpretation of quantum mechanics, a relativistic Hilbert space is defined for the Klein-Gordon case. It is demonstrated that unitary transformations in Hilbert space reflect canonical transformations in the associated phase space, manifesting thus an underlying symplectic structure.     

Quantum Einstein’s equations and constraints algebra

In this paper we shall address this problem: Is quantum gravity constraints algebra closed and what are the quantum Einstein’s equations. We shall investigate this problem in the de-Broglie-Bohm quantum theory framework. It is shown that the constraint algebra is weakly closed and the quantum Einstein’s equations are derived.     

Classification of the quantum two-dimensional superintegrable systems with quadratic integrals and the Stäckel transforms

The two-dimensional quantum superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion. There are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones. Analytic formulas for the involved integrals are calculated in all the cases. All the known quantum superintegrable systems with quadratic integrals are classified as special cases of these six general classes. The coefficients of the quadratic associative algebra of integrals are calculated and they are compared to the coefficients of the corresponding coefficients of the Poisson quadratic algebra of the classical systems. The quantum coefficients are similar to the classical ones multiplied by a quantum coefficient -?² plus a quantum deformation of order ?⁴ and ?⁶. The systems inside the classes are transformed using Stäckel transforms in the quantum case as in the classical case. The general form of the Stäckel transform between superintegrable systems is discussed.     

Phase-space Lagrangians for null spinning strings

The striking fact that normal-ordered null strings have the same critical dimension as their usual non-zero tension siblings can be understood from the observation that one must, in the tensionless case, keep all the conjugate momenta as independent dynamical variables, thus doubling the number of physical degrees of freedom. The fermionic momenta give rise to a second-class constraint which cannot be solved covariantly, but can be successfully incorporated into the first-class constraint algebra after gauge-fixing. The ghost contributions to the anomaly consist oftwo b?c (and alsotwo β?γ systems in the supersymmetric case), of the single Virasoro sub(super)algebra for the closed null (spinning) string. In the appropriate gauge, the null (super)string is (super)chiral.     

Algebraic approach to the geometry of the (super-) string model

We discuss the extension of constraint algebras to include subsidiary constraints within a larger algebra. The interplay between various mathematical aspects of this procedure is described. Tools from Lie algebra cohomology and differential geometry are used to gain new insights into BRS techniques for nonabelian constrained systems. We show that cohomology considerations restrict our formalism to non-semisimple constraint algebras, such as the (super-) string model; we illustrate the ideas by presenting concrete results for this case.     

Relativistic quantum effects of Dirac particles simulated by ultracold atoms

Quantum simulation is a powerful tool to study a variety of problems in physics, ranging from high-energy physics to condensed-matter physics. In this article, we review the recent theoretical and experimental progress in quantum simulation of Dirac equation with tunable parameters by using ultracold neutral atoms trapped in optical lattices or subject to light-induced synthetic gauge fields. The effective theories for the quasiparticles become relativistic under certain conditions in these systems, making them ideal platforms for studying the exotic relativistic effects. We focus on the realization of one, two, and three dimensional Dirac equations as well as the detection of some relativistic effects, including particularly the well-known Zitterbewegung effect and Klein tunneling. The realization of quantum anomalous Hall effects is also briefly discussed.     

QCD on an Infinite Lattice

We construct a mathematically well–defined framework for the kinematics of Hamiltonian QCD on an infinite lattice in ${\mathbb{R}^3}$ , and it is done in a C*-algebraic context. This is based on the finite lattice model for Hamiltonian QCD developed by Kijowski, Rudolph e.a.. To extend this model to an infinite lattice, we need to take an infinite tensor product of nonunital C*-algebras, which is a nonstandard situation. We use a recent construction for such situations, developed by Grundling and Neeb. Once the field C*-algebra is constructed for the fermions and gauge bosons, we define local and global gauge transformations, and identify the Gauss law constraint. The full field algebra is the crossed product of the previous one with the local gauge transformations. The rest of the paper is concerned with enforcing the Gauss law constraint to obtain the C*-algebra of quantum observables. For this, we use the method of enforcing quantum constraints developed by Grundling and Hurst. In particular, the natural inductive limit structure of the field algebra is a central component of the analysis, and the constraint system defined by the Gauss law constraint is a system of local constraints in the sense of Grundling and Lledo. Using the techniques developed in that area, we solve the full constraint system by first solving the finite (local) systems and then combining the results appropriately. We do not consider dynamics.