Algebra Of Quantum Operators Research Articles

Mutual averaged non-commutativity of quantum operator algebras

We introduce an elementary measure of non-commutativity between two algebras of quantum operators acting on the same Hilbert space. This quantity, which we call Mutual Averaged Non-commutativity (MAN), is a simple generalization of a type of averaged Out-of-Time-Order-Correlators used in the study of quantum scrambling and chaos. MAN is defined by a Haar averaged squared norm of a commutator and for some types of algebras is manifestly of entropic nature. In particular, when the two algebras coincide the corresponding self-MAN can be fully computed in terms of the structural data of the associated Hilbert space decomposition. Properties and bounds of MAN are established in general and several concrete examples are discussed. Remarkably, for an important class of algebras, —which includes factors and maximal Abelian ones—MAN can be expressed in the terms of the algebras projections CP-maps. Assuming that the latter can be enacted as physical processes, one can devise operational protocols to directly estimate the MAN of a pair of algebras.

Statistical tensor analysis of the MQ MR signals generated by weak quadrupole interactions

Multiple quantum NMR signals that appear in the presence of weak quadrupole interactions were formulated using statistical tensors (Fano, 1957). The approach aimed to present a concise and a computer-based tool for a detailed analysis and modification of the MQ pulse sequences. The calculation avoids a lengthy procedure of utilizing exponential operators and, moreover, the same formulae are applicable for any interval in the TQ pulse sequence, as well as any spin value. The quantum operator algebra was implemented using “Mathematica” software (Wolfram Inc.). The results of tensor's evolutions in the TQ pulse sequence were graphically illustrated using corresponding spherical harmonics. The visualization takes into consideration the parity properties of irreducible tensors and the corresponding spherical harmonics.

On Clifford groups in quantum computing

The term Clifford group was introduced in 1998 by D. Gottesmann in his investigation of quantum error-correcting codes. The simplest Clifford group in multiqubit quantum computation is generated by a restricted set of unitary Clifford gates - the Hadamard, π/4-phase and controlled-X gates. Because of this restriction the Clifford model of quantum computation can be efficiently simulated on a classical computer (the Gottesmann-Knill theorem). However, this fact does not diminish the importance of the Clifford model, since it may serve as a suitable starting point for a full-fledged quantum computation.In the general case of a single or composite quantum system with finite-dimensional Hilbert space the finite Weyl-Heisenberg group of unitary operators defines the quantum kinematics and the states of the quantum register. Then the corresponding Clifford group is defined as the group of unitary operators leaving the Weyl-Heisenberg group invariant. The aim of this contribution is to show that our comprehensive results on symmetries of the Pauli gradings of quantum operator algebras - covering any single as well as composite finite quantum systems - directly correspond to Clifford groups defined as quotients with respect to U(1).

Quantum Physical Origin of Lorentz Transformations

Our aim is to derive the symmetries of the space-time, i.e. the Lorentz transformations, from discrete symmetries of the interactions between the most fundamental constituents of matter, in particular quarks and leptons. The role of Pauli’s exclusion principle in the derivation of the SL(2,C), symmetry is put forward as the source of the macroscopically observed Lorentz symmetry. Then Pauli’s principle is generalized for the case of the Z3 grading replacing the usual Z2 grading, leading to ternary commutation relations for quantum operator algebras. In the case of lowest dimension, with two generators only, it is shown how the cubic combinations Z3-graded elements behave like Lorentz spinors, and the binary product of elements of this algebra with an element of the conjugate algebra behave like Lorentz vectors.

On the relationship between the Wigner–Moyal approach and the quantum operator algebra of von Neumann

In this paper we discuss the close relationship between the Wigner---Moyal algebra and the original non-commutative quantum algebra introduced by von Neumann in 1931. We show that the "distribution function", F(P, X, t) is simply the quantum mechanical density matrix for a single particle where the coordinates, X and P, are not the coordinates of a point particle, but the mean co-ordinate of a cell structure (a `blob') in phase space. This provides an intrinsically non-local and non-commutative description of an individual, which only becomes a point particle in the commutative limit. In this general structure, the Wigner function appears as a transition probability amplitude which accounts for the appearance of negative values. The Moyal and Baker brackets play a significant role in the time evolution, producing the quantum Hamilton---Jacobi equation used in the Bohm approach. It is the non-commutative structure based on a symplectic geometry that generates a generalised phase space for quantum processes.

The quantum Arnold transformation

Using a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations, including systems with friction linear in velocity, can be related to the quantum free-particle dynamical system. This transformation provides a basic (Heisenberg–Weyl) algebra of quantum operators, along with well-defined Hermitian operators which can be chosen as evolution-like observables and complete the entire Schrödinger algebra. It also proves to be very helpful in performing certain computations quickly, to obtain, for example, wavefunctions and closed analytic expressions for time-evolution operators.

Observation of lasing modes with exotic localized wave patterns from astigmatic large-Fresnel-number cavities

We investigate the lasing modes in large-Fresnel-number laser systems with astigmatism effects. Experimental results reveal that numerous lasing modes are concentrated on exotic patterns corresponding to intriguing geometries. We theoretically use the quantum operator algebra to construct the wave representation for manifesting the origin of the localized wave patterns.

Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review

A novel algebraic topology approach to supersymmetry (SUSY) and symmet- ry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconduc- tors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes ex- tended symmetries and quantum groupoid, algebroid and functorial representations of non- Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier- Stieltjes transforms, and duality relations link, respectively, the quantum groups and quan- tum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasi-triangular, quasi-Hopf algebras, bialgebroids, Grassmann-Hopf algebras and higher dimensional algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang-Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant general relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin) models with the extended quantum symmetry of entangled, 'string-net condensed' (ground) states.

Noncommutative field theory from twisted Fock space

We construct a quantum field theory in noncommutative spacetime by twisting the algebra of quantum operators (especially, creation and annihilation operators) of the corresponding quantum field theory in commutative spacetime. The twisted Fock space and S-matrix consistent with this algebra have been constructed. The resultant S-matrix is consistent with that of Filk\cite{Filk}. We find from this formulation that the spin-statistics relation is not violated in the canonical noncommutative field theories.

A Non-Associative Quantum Mechanics

A non-associative quantum mechanics is proposed in which the product of three and more operators can be non-associative one. The multiplication rules of the octonions define the multiplication rules of the corresponding operators with quantum corrections. The self-consistency of the operator algebra is proved for the product of three operators. Some properties of the non-associative quantum mechanics are considered. It is proposed that some generalization of the non-associative algebra of quantum operators can be helpful for understanding of the algebra of field operators with a strong interaction.

Involutions on the algebra of physical observables from reality conditions

Some aspects of the algebraic quantization program proposed by Ashtekar are revisited in this article. It is proven that, for systems with first-class constraints, the involution introduced on the algebra of quantum operators via reality conditions can never be projected unambiguously to the algebra of physical observables, i.e., of quantum observables modulo constraints. It is nevertheless shown that, under sufficiently general assumptions, one can still induce an involution on the algebra of physical observables from reality conditions, though the involution obtained depends on the choice of particular representatives for the equivalence classes of quantum observables.

Commutative quantum operator algebras

A key notion bridging the gap between quantum operator algebras [26] and vertex operator algebras [4, 9] is the definition of the commutativity of a pair of quantum operators (see Section 2). This is not commutativity in any ordinary sense, but it is clearly the correct generalization to the quantum context. In [26] we give a definition of a commutative quantum operator algebra. We show in [26] that a vertex operator algebra gives rise to a special case of a CQOA. The main purpose of the current paper is to further develop the foundations for a complete mathematical theory of CQOAs. We give proofs of most of the relevant results announced in [26], and we carry out some calculations with sufficient detail to enable the interested reader to become proficient with the algebra of commuting quantum operators.

The quantum relativistic harmonic oscillator: generalized Hermite polynomials

A relativistic generalisation of the algebra of quantum operators for the harmonic oscillator is proposed. The wave functions are worked out explicitly in configuration space. Both the operator algebra and the wave functions have the appropriate c→∞ limit. This quantum dynamics involves an extra quantization condition mc 2/ωℏ = 1, 3 2 , 2 , … of a topological character.

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.

Realistic physical origin of the quantum observable operator algebra in the frame of the causal stochastic interpretation of quantum mechanics: The relativistic spin-zero case.

The deduction by Guerra and Marra of the usual quantum operator algebra from a canonical variable Hamiltonian treatment of Nelson's hydrodynamical stochastic description of real nonrelativistic Schr\"odinger waves is extended to the causal stochastic interpretation given by Guerra and Ruggiero and by Vigier of relativistic Klein-Gordon waves. A specific representation shows that the Poisson brackets for canonical hydrodynamical observables become ``averages'' of quantum observables in the given state. Stochastic quantization thus justifies the standard procedure of replacing the classical particle (or field) observables with operators according to the scheme ${p}_{\ensuremath{\mu}}$\ensuremath{\rightarrow}-i\ensuremath{\Elzxh}${\ensuremath{\partial}}_{\ensuremath{\mu}}$ and ${L}_{\ensuremath{\mu}\ensuremath{\nu}}$\ensuremath{\rightarrow}-i\ensuremath{\Elzxh}(${x}_{\ensuremath{\mu}}$${\ensuremath{\partial}}_{\ensuremath{\nu}}$-${x}_{\ensuremath{\nu}}$${\ensuremath{\partial}}_{\ensuremath{\mu}}$).

Canonical quantization of solitons: The breather sector

The sine-Gordon system is quantized in the breather sector. The algebra of quantum operators is obtained and their occurrence in the Heisenberg field operator is displayed explicitly. The quantum Hamiltonian, including the interaction between the breather and the particle quanta, is determined.

Quantum theory of spin-3/2 field in Einstein spaces

A quantum theory of a free massless spin-3/2 field on Einstein spaces (Rab=Λgab) is formulated in an algebraic framework. Attention is confined to the structure of the quantum operator algebra. In particular, the issue of positivity of the anticommutator is investigated and found to depend on whether or not the space-time admits “rero-frequency” neutrino solutions. Using methods developed for the purpose, a class of space-time that does not admit neutrino “zero modes” is characterized. An appendix introduces a useful technique for obtaining an initial value formulation of spinor field equations.

The origin of the algebra of quantum operators in the stochastic formulation of quantum mechanics

The origin of the algebra of the non-commuting operators of quantum mechanics is explained in the general Fényes-Nelson stochastic models in which the diffusion constant is a free parameter. This is achieved by continuing the diffusion constant to imaginary values, a continuation which destroys the physical interpretation, but does not affect experimental predictions. This continuation leads to great mathematical simplification in the stochastic theory, and to an understanding of the entire mathematical formalism of quantum mechanics. It is more than a formal construction because the diffusion parameter is not an observable in these theories.

Quantum fields in curved space-times

The problem of obtaining a quantum description of the (real) KleinGordon system in a given curved space-time is discussed. An algebraic approach is used. The *-algebra of quantum operators is constructed explicitly and the problem of finding its *-representation is reduced to th at of selecting a suitable complex structure on the real vector space of the solutions of the (classical) Klein-Gordon equation. Since, in a static space-time, there already exists, a satisfactory quantum field theory, in this case one already knows what the ‘ correct ’ complex structure is. A physical characterization of this * correct ’ complex structure is obtained. This characterization is used to extend quantum field theory to nonstatic space-times. Stationary space-times are considered first. In this case, the issue of extension is completely straightforward and the resulting theory is the natural generalization of the one in static space-times. General, non-stationary space-times are then considered. In this case the issue of extension is quite complicated and we only present a plausible extension. Although the resulting framework is well-defined mathematically, the physical interpretation associated with it is rather unconventional. Merits and weaknesses of this framework are discussed.