Narrow Results

We prove a double coset formula for induced representations of compact Lie groups. We apply it to the representation rings of unitary and symplectic groups to obtain Hopf algebras. We also construct a Heisenberg algebra representation based on the restiction and induction of representations of unitary groups.

In the Penrose limit, AdS×S space turns into a Cahen–Wallach (CW) space whose Killing vectors satisfy a Heisenberg algebra. This algebra is mapped onto the holographic screen on the boundary of AdS. We show that the Heisenberg algebra on the boundary of AdS may be obtained directly from the CW space by appropriately constraining the states defined on it. The transformations generated by the constraint are similar to gauge transformations. The "holographic screen" on the CW space is thus obtained as a "gauge-fixing" condition.

Using the partition of the number p-1 into p-1 real parts which are not equal with each other necessarily, we develop the unitary parasupersymmetry algebra of arbitrary order p so that the well-known Rubakov–Spiridonov–Khare parasupersymmetry becomes a special case of the developed one. It is shown that the developed algebra is realized by simple harmonic oscillator and Landau problem on a flat surface with the symmetries of h₃ and h₄ Heisenberg–Lie algebras. For this new parasupersymmetry, the well-known unitary condition is violated, however, unitarity of the corresponding algebra is structurally conserved. Moreover, the components of the bosonic Hamiltonian operator are derived as functions from the mean value of the partition numbers with their label weight function.

We show that the non-Hermitian Hamiltonians of the simple harmonic oscillator with and symmetries involve a pseudo generalization of the Heisenberg algebra via two pairs of creation and annihilation operators which are -pseudo-Hermiticity and -anti-pseudo-Hermiticity of each other. The non-unitary Heisenberg algebra is represented by each of the pair of the operators in two different ways. Consequently, the coherent and the squeezed coherent states are calculated in two different approaches. Moreover, it is shown that the approach of Schwinger to construct the su(2), su(1, 1) and sp(4, ℝ) unitary algebras is promoted so that unitary algebras with more linearly dependent number of generators are made.

We herein construct a Heisenberg-like algebra for the one-dimensional quantum free Klein–Gordon equation defined on the interval of the real line of length L. Using the realization of the ladder operators of this Heisenberg-type algebra in terms of physical operators we build a (3+1)-dimensional free quantum field theory based on this algebra. We introduce fields written in terms of the ladder operators of this Heisenberg-type algebra and a free quantum Hamiltonian in terms of these fields. The mass spectrum of the physical excitations of this quantum field theory is given by , where n=1,2,… and m_q is the mass of a particle in a relativistic infinite square-well potential of width L.

We present the result concerning the perturbative computation of the scattering process 1 + 2 → 1′ + 2′ up to second order in the coupling constant. This was obtained in the context of a deformed quantum field theory which is interpreted as a phenomenological theory describing the scattering of spin-0 composite particles.

We consider the continuous symmetry group underlying Born's reciprocity principle, namely the so-called quaplectic group, the semidirect product of time-space-energy-momentum coordinate transformations with the Weyl-Heisenberg group. In two dimensional Minkowski space this group is Q(1, 1) ≅ U(1, 1) ⋉ H(2), or in Euclidean space Q(2) ≅ U(2) ⋉ H(2). For the 'scalar' system in the sense of induced representations, unitary irreducible representations are carried on a Fock space equivalent to that used by Schwinger as a model of the SU(2) angular momentum algebra, or by Holman and Biedenharn as a model of SU(1,1). Using this construction we consider the branching rules and state labelling problem for the reduction of Q(2) and Q(1, 1) to the 'physical' Euclidean and Poincaré subalgebras, respectively. The results serve to illustrate the difficulties of any consideration of Born reciprocity as an extended symmetry principle of nature.

In this paper, we show that a quadratic Lagrangian, with no constraints, containing ordinary time derivatives up to the order $m$ of $N$ dynamical variables, has $2mN$ symmetries consisting in the translation of the variables with solutions of the equations of motion. We construct explicitly the generators of these transformations and prove that they satisfy the Heisenberg algebra. We also analyze other specific cases which are not included in our previous statement: the Klein–Gordon Lagrangian, $N$ Fermi oscillators and the Dirac Lagrangian. In the first case, the system is described by an equation involving partial derivatives, the second case is described by Grassmann variables and the third shows both features. Furthermore, the Fermi oscillator and the Dirac field Lagrangians lead to second class constraints. We prove that also in these last two cases there are translational symmetries and we construct the algebra of the generators. For the Klein–Gordon case we find a continuum version of the Heisenberg algebra, whereas in the other cases, the Grassmann generators satisfy, after quantization, the algebra of the Fermi creation and annihilation operators.

We apply Lie algebra deformation theory to the problem of identifying the stable form of the quantum relativistic kinematical algebra. As a warm up, given Galileo's conception of spacetime as input, some modest computer code we wrote zeroes in on the Poincaré-plus-Heisenberg algebra in about a minute. Further ahead, along the same path, lies a three-dimensional deformation space, with an instability double cone through its origin. We give physical as well as geometrical arguments supporting our view that moment, rather than position operators, should enter as generators in the Lie algebra. With this identification, the deformation parameters give rise to invariant length and mass scales. Moreover, standard quantum relativistic kinematics of massive, spinless particles corresponds to non-commuting moment operators, a purely quantum effect that bears no relation to spacetime non-commutativity, in sharp contrast to earlier interpretations.

In this paper, we will first clarify the physical meaning of having a minimum measurable time. Then we will combine the deformation of the Dirac equation due to the existence of minimum measurable length and time scales with its deformation due to the doubly special relativity. We will also analyze this deformed Dirac equation in curved spacetime, and observe that this deformation of the Dirac equation also leads to a nontrivial modification of general relativity. Finally, we will analyze the stochastic quantization of this deformed Dirac equation on curved spacetime.

We give two alternate presentations of the Frobenius Heisenberg category, ${\mathscr{H}}ei{s}_{F,k}$, defined by Savage, when the Frobenius algebra $F=F_1\oplus \cdots \oplus F_n$ decomposes as a direct sum of Frobenius subalgebras. In these alternate presentations, the morphism spaces of ${\mathscr{H}}ei{s}_{F,k}$ are given in terms of planar diagrams consisting of strands “colored” by integers $i=1,\dots ,n$, where a strand of color $i$ carries tokens labelled by elements of $F_i.$ In addition, we prove that when $F$ decomposes this way, the tensor product of Frobenius Heisenberg categories, ${\mathscr{H}}ei{s}_{F_1,k}\otimes \cdots \otimes {\mathscr{H}}ei{s}_{F_n,k},$ is equivalent to a certain subcategory of the Karoubi envelope of ${\mathscr{H}}ei{s}_{F,k}$ that we call the partial Karoubi envelope of ${\mathscr{H}}ei{s}_{F,k}$.

In this paper we study the stationary scattering problem of the Aharonov–Bohm (AB) effect. To achieve this goal we construct a Hamiltonian from the most general representations of the Heisenberg algebra. Such representations are defined on a non-simply-connected manifold which we set as the flat circular annulus. By means of the von Neumann's self-adjoint extensions formalism, the scattering data are then provided. No solenoid is considered in this paper. The corresponding Hamiltonian is based on a topological quantum degree of freedom inherent in such representations. This variable stands for the magnetic vector gauge potential at quantum level. Our outcomes confirm the topological nature of this effect.

In this paper, we construct optical normal magnetic electromotive microscale by electroosmotic potential in quasi Heisenberg algebra. Also, we obtain optical quasi normal electric Heisenberg optimistic density in quasi Heisenberg algebra. Finally, we demonstrate quasi normal electric Heisenberg antiferromagnetic microscale with numerous potentials in quasi Heisenberg algebra.

We consider the continuous symmetry group underlying Born's reciprocity principle, namely the so-called quaplectic group, the semidirect product of time-space-energy-momentum coordinate transformations with the Weyl-Heisenberg group. In two dimensional Minkowski space this group is Q(1, 1) ≅ U(1, 1) ⋉ H(2), or in Euclidean space Q(2) ≅ U(2) ⋉ H(2). For the 'scalar' system in the sense of induced representations, unitary irreducible representations are carried on a Fock space equivalent to that used by Schwinger as a model of the SU(2) angular momentum algebra, or by Holman and Biedenharn as a model of SU(1,1). Using this construction we consider the branching rules and state labelling problem for the reduction of Q(2) and Q(1, 1) to the 'physical' Euclidean and Poincaré subalgebras, respectively. The results serve to illustrate the difficulties of any consideration of Born reciprocity as an extended symmetry principle of nature.

We describe the nontrivial central extensions CE(Heis) of the Heisenberg algebra and their representation as sub–algebras of the Scheoedinger algebra. We also present the characteristic and moment generating functions of the random variable corresponding to the self-adjoint sum of the generators of CE(Heis).

The following sections are included:

• Introduction and statement of the problem

• Random variables in CeHeis_ℂ(1)

• Infinite divisibility

• Kernels and matrices

• Positive definite kernels

• Functions of positive definite matrices and kernels

• Conditionally positive definite kernels

• Infinitely divisible kernels

• The Kolmogorov decomposition theorem for ℂ–valued PD kernels

• Boson Fock spaces

• Infinitely divisible kernels and Boson Fock spaces

• References

A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is presented. A natural approach to building coherent states associated to deformed algebras is then deduced.