Fundamental Commutation Relations Research Articles

Canonical momenta in digitized Su(2) lattice gauge theory: definition and free theory

Hamiltonian simulations of quantum systems require a finite-dimensional representation of the operators acting on the Hilbert space mathcal {H}. Here we give a prescription for gauge links and canonical momenta of an SU(2) gauge theory, such that the matrix representation of the former is diagonal in mathcal {H}. This is achieved by discretising the sphere S_3 isomorphic to SU(2) and the corresponding directional derivatives. We show that the fundamental commutation relations are fulfilled up to discretisation artefacts. Moreover, we directly construct the Casimir operator corresponding to the Laplace–Beltrami operator on S_3 and show that the spectrum of the free theory is reproduced again up to discretisation effects. Qualitatively, these results do not depend on the specific discretisation of SU(2), but the actual convergence rates do.

Algebraic Bethe ansatz for Q-operators of the open XXX Heisenberg chain with arbitrary spin

In this note we construct Q-operators for the spin s open Heisenberg XXX chain with diagonal boundaries in the framework of the quantum inverse scattering method. Following the algebraic Bethe ansatz we diagonalise the introduced Q-operators using the fundamental commutation relations. By acting on Bethe off-shell states and explicitly evaluating the trace in the auxiliary space we compute the eigenvalues of the Q-operators in terms of Bethe roots and show that the unwanted terms vanish if the Bethe equations are satisfied.

Fluctuation-dissipation theorem and fundamental photon commutation relations in lossy nanostructures using quasinormal modes

We provide theory and formal insight on the Green function quantization method for absorptive and dispersive spatial-inhomogeneous media in the context of dielectric media. We show that a fundamental Green function identity, which appears, e.g., in the fundamental commutation relation of the electromagnetic fields, is also valid in the limit of non-absorbing media. We also demonstrate how the zero-point field fluctuations yields a non-vanishing surface term in configurations without absorption, when using a more formal procedure of the Green function quantization method. We then apply the presented method to a recently developed theory of photon quantization using quasinormal modes [Franke et al., Phys. Rev. Lett. 122, 213901 (2019)] for finite nanostructures embedded in a lossless background medium. We discuss the strict dielectric limit of the commutation relations of the quasinormal mode operators and present different methods to obtain them, connected to the radiative loss for non-absorptive but open resonators. We show exemplary calculations of a fully three-dimensional photonic crystal beam cavity, including the lossless limit, which supports a single quasinormal mode and discuss the limits of the commutation relation for vanishing damping (no material loss and no radiative loss).

Quantum features in the orthogonality of optical modes for structured and plane-wave light.

A fundamental photon creation-annihilation commutation relation underpins the familiar quantum formulation of optics. However, an internal inconsistency becomes apparent in the pursuit of structured light applications. This requires the relationship between operator commutation and mode orthogonality to be recast in a form ensuring full consistency with the precepts of quantum theory. A suitable reformulation, shown to register correctly an intrinsic quantum uncertainty in the associated interactions, has special relevance to optical vortex physics-particularly with regard to information content-through its connection to the degrees of freedom in the associated radiation modes.

The Fourier Transform and Principles of Quantum Mechanics

From the hypotheses that the position-representation of a physical state is the Fourier transform of its momentum-representation and that the timerepresentation is the inverse Fourier transform of its energy-representation, we are able to obtain the Planck relation E = hν , the de Broglie relation p = h /λ , the Dirac fundamental commutation relation, the Schr?dinger equations, the Heisenberg uncertainty principle in quantum mechanics, and the annihilation/creation of a photon from excitation/de-excitation of an atom following Bohr.

Symplectic Structure of Intrinsic Time Gravity

The Poisson structure of intrinsic time gravity is analysed. With the starting point comprising a unimodular three-metric with traceless momentum, a trace-induced anomaly results upon quantization. This leads to a revision of the choice of momentum variable to the (mixed index) traceless momentric. This latter choice unitarily implements the fundamental commutation relations, which now take on the form of an affine algebra with SU(3) Lie algebra amongst the momentric variables. The resulting relations unitarily implement tracelessness upon quantization. The associated Poisson brackets and Hamiltonian dynamics are studied.

Intrinsic time quantum geometrodynamics

Quantum Geometrodynamics with intrinsic time development and momentric variables is presented. An underlying SU(3) group structure at each spatial point regulates the theory. The intrinsic time behavior of the theory is analyzed, together with its ground state and primordial quantum fluctuations. Cotton-York potential dominates at early times when the universe was small; the ground state naturally resolves Penrose's Weyl Curvature Hypothesis, and thermodynamic and gravitational `arrows of time' point in the same direction. Ricci scalar potential corresponding to Einstein's General Relativity emerges as a zero-point energy contribution. A new set of fundamental commutation relations without Planck's constant emerges from the unification of Gravitation and Quantum Mechanics.

Algebraic Bethe ansatz for Q-operators: the Heisenberg spin chain

We diagonalize Q-operators for rational homogeneous -invariant Heisenberg spin chains using the algebraic Bethe ansatz. After deriving the fundamental commutation relations relevant for this case from the Yang–Baxter equation we demonstrate that the Q-operators act diagonally on the Bethe vectors if the Bethe equations are satisfied. In this way we provide a direct proof that the eigenvalues of the Q-operators studied here are given by Baxter's Q-functions.

An Enlarged Canonical Quantization Scheme and Quantization of a Free Particle on Two-Dimensional Sphere

For a non-relativistic particle that freely moves on a curved surface, the fundamental commutation relations between positions and momenta are insufficient to uniquely determine the operator form of the momenta. With introduction of more commutation relations between positions and Hamiltonian and those between momenta and Hamiltonian, our recent sequential studies imply that the Cartesian system of coordinates is physically preferable, consistent with Dirac' observation. In present paper, we study quantization problem of the motion constrained on the two-dimensional sphere and develop a discriminant that can be used to show how the quantization within the intrinsic geometry is improper. Two kinds of parameterization of the spherical surface are explicitly invoked to investigate the quantization problem within the intrinsic geometry.

Effects of quantum gravity on black holes

In this review, we discuss the effects of quantum gravity on black hole physics. After a brief review of the origin of the minimal observable length from various quantum gravity theories, we present the tunneling method. To incorporate quantum gravity effects, we modify the Klein–Gordon equation and Dirac equation by the modified fundamental commutation relations. Then we use the modified equations to discuss the tunneling radiation of scalar particles and fermions. The corrected Hawking temperatures are related to the quantum numbers of the emitted particles. Quantum gravity corrections slow down the increase of the temperatures. The remnants are observed as [Formula: see text]. The mass is quantized by the modified Wheeler–DeWitt equation and is proportional to n in quantum gravity regime. The thermodynamical property of the black hole is studied by the influence of quantum gravity effects.

Four-dimensional Rotations in Coherent Optical Communications

To model electromagnetic wave propagation for coherent communications without polarization dependent losses, the unitary 2 ×2 Jones transfer matrix formalism is typically used. In this study, we propose an alternative formalism to describe such transformations based on rotations in four-dimensional (4d) Euclidean space. This formalism is usually more attractive from a communication theoretical perspective, since decisions and symbol errors can be related to geometric concepts such as Euclidean distances between points and decision boundaries. Since 4d rotations is a richer description than the conventional Jones calculus, having six rather than four degrees of freedom (DOF), we propose an extension of the Jones calculus to handle all six DOF. In addition, we show that the two extra DOF in the 4d description represents transformations that are nonphysical for propagating photons, since they does not obey the fundamental quantum mechanical boson commutation relations. Finally, we exemplify on how the nonphysical rotations can change the polarization-phase degeneracy of well-known constellations such as single-polarization QPSK, polarization-multiplexed (PM-)QPSK and polarization-switched (PS-) QPSK. For example, we show how PM-QPSK, which is well known to consist of four polarization states each having four-fold phase degeneracy, can be represented as eight states of polarizations, each with binary phase degeneracy.

Geometric momentum for a particle constrained on a curved hypersurface

The canonical quantization is a procedure for quantizing a classical theory while preserving the formal algebraic structure among observables in the classical theory to the extent possible. For a system without constraint, we have the so-called fundamental commutation relations (CRs) among positions and momenta, whose algebraic relations are the same as those given by the Poisson brackets in classical mechanics. For the constrained motion on a curved hypersurface, we need more fundamental CRs otherwise neither momentum nor kinetic energy can be properly quantized, and we propose an enlarged canonical quantization scheme with introduction of the second category of fundamental CRs between Hamiltonian and positions, and those between Hamiltonian and momenta, whereas the original ones are categorized into the first. As an N − 1 (N ⩾ 2) dimensional hypersurface is embedded in an N dimensional Euclidean space, we obtain the proper momentum that depends on the mean curvature. For the spherical surface, a long-standing problem in the form of the geometric potential is resolved in a lucid and unambiguous manner, which turns out to be identical to that given by the so-called confining potential technique. In addition, a new dynamical group SO(N, 1) symmetry for the motion on the sphere is demonstrated.

Quantum motion on a torus as a submanifold problem in a generalized Dirac’s theory of second-class constraints

A generalization of Dirac’s canonical quantization scheme for a system with second-class constraints is proposed, in which the fundamental commutation relations are constituted by all commutators between positions, momenta and Hamiltonian, so they are simultaneously quantized in a self-consistent manner, rather than by those between merely positions and momenta which leads to ambiguous forms of the Hamiltonian and the momenta. The application of the generalized scheme to the quantum motion on a torus leads to a remarkable result: the quantum theory is inconsistent if built up in an intrinsic geometric manner, whereas it becomes consistent within an extrinsic examination of the torus as a submanifold in three dimensional flat space with the use of the Cartesian coordinate system. The geometric momentum and potential are then reasonably reproduced.

A charged particle interacting with a stationary magnetic monopole: quantum mechanics based on the kinetic momentum operators

The standard quantum mechanical description of the motion of a charged particle in the field of a stationary magnetic monopole is notorious for the presence of unnatural singularities in the Hamiltonian operator originating in the vector potential A(r) used to describe the magnetic field of the monopole. In this paper, an elementary quantum mechanical formulation of the problem which involves only the physically observable field B(r) is presented. This is achieved by treating as a fundamental observable of the charged particle its kinetic momentum instead of the linear momentum p. An irreducible representation of the fundamental commutation relations involving the operators and is explicitly constructed. It is shown that the existence of an irreducible representation requires that Dirac’s charge quantization condition is satisfied. Also, it is demonstrated that, from the quantum mechanical perspective, the singularities (appearing when the vector potential is introduced) are in fact properties of coordinate representations of the fundamental commutation relations.

Remarks on Exactly Solvable Noncommutative Quantum Field

We study exactly the solvable noncommutative scalar quantum fieldmodels of (2n) or (2n+1) dimensions. By writing out anequivalent action of the noncommutative field, it is shown that thespecial condition B.θ = ±I in field theoretic context meansthe full restoration of the maximal U(∞) gauge symmetries brokendue to kinetic term. It is further shown that the model can be obtainedby dimensional reduction of a 2n-dimensional exactly solvablenoncommutative ϕ4 quantum field model closely related to the 1+1-dimensional Moyal/matrix-valued nonlinear Schrödinger (MNLS)equation. The corresponding quantum fundamental commutation relation ofthe MNLS model is also given explicitly.

QUANTUM MECHANICS AS A MEASUREMENT THEORY ON BICONFORMAL SPACE

Biconformal spaces contain the essential elements of quantum mechanics, making the independent imposition of quantization unnecessary. Based on three postulates characterizing motion and measurement in biconformal geometry, we derive standard quantum mechanics, and show how the need for probability amplitudes arises from the use of a standard of measurement. Additionally, we show that a postulate for unique, classical motion yields Hamiltonian dynamics with no measurable size changes, while a postulate for probabilistic evolution leads to physical dilatations manifested as measurable phase changes. Our results lead to the Feynman path integral formulation, from which follows the Schrödinger equation. We discuss the Heisenberg uncertainty relation and fundamental canonical commutation relations.

SCALE TRANSFORMATIONS ON THE NONCOMMUTATIVE PLANE AND THE SEIBERG–WITTEN MAP

We write down three kinds of scale transformations (i)–(iii) on the noncommutative plane. Transformation (i) is the analogue of standard dilations on the plane, transformation (ii) is a rescaling of the noncommutative parameter θ, and transformation (iii) is a combination of the previous two, whereby the defining relations for the noncommutative plane are preserved. The action of the three transformations is defined on gauge fields evaluated at fixed coordinates and θ. The transformations are obtained only up to terms which transform covariantly under gauge transformations. We give possible constraints on these terms. We show how the transformations (i) and (ii) depend on the choice of star product, and show the relation of (ii) to Seiberg–Witten transformations. Because transformation (iii) preserves the fundamental commutation relations it is a symmetry of the algebra. One has the possibility of implementing it as a symmetry of the dynamics, as well, in noncommutative field theories where θ is not fixed.

Polaritons of dispersive dielectric media

We quantize the electromagnetic field in a polar medium starting with the fundamental equations of motion. In our model the medium is described by a Lorenz-type dielectric function ε ( r , ω ) appropriate e.g. for ionic crystals, metals and inert dielectrics. There are no restrictions on the spatial behavior of the dielectric function, i.e. there can be many different polar media with arbitrary shapes. We assume no losses in our system so the dielectric function for the whole space is assumed as real. The quantization procedure is based on an expansion of the total field (transverse and longitudinal) in terms of the coupled (polariton) eigenmodes, so our theory gives the Hamiltonian of polaritons of dispersive dielectric media. We pay particular attention to the derivation of the fundamental (equal-time) commutation relations between the conjugate field operators. As an example, we apply our theory to the quasi-two-dimensional Wigner crystal, formed by electrons at very low temperature. We discuss the influence of the quantized electromagnetic field on the dynamics of Wigner electrons, i.e., on the dispersion relation of Wigner phonons. We expect a significant influence in the case when frequencies of (surface) polaritons and Wigner phonons coincide, and we discuss the energy spectrum of such a system.

Quantum optics of dispersive dielectric media

We quantize the electromagnetic field in a polar medium starting with the fundamental equations of motion. In our model the medium is described by a Lorenz--type dielectric function $\ensuremath{\epsilon}(\mathbf{r},\ensuremath{\omega})$ appropriate, e.g., for ionic crystals, metals, and inert dielectrics. There are no restrictions on the spatial behavior of the dielectric function, i.e., there can be many different polar media with arbitrary shapes. We assume no losses in our system so the dielectric function for the whole space is assumed as real. The quantization procedure is based on an expansion of the total field (transverse and longitudinal) in terms of the coupled (polariton) eigenmodes, and this approach incorporates all previous results derived for similar but restricted systems (e.g., without spatial or frequency dependence of coupled modes). Within the same model, we also quantize the Hamiltonian of a nonretarded electromagnetic field in polar media. Particular attention is paid to the derivation of the orthogonality and closure relations, which are used in a discussion of the fundamental (equal-time) commutation relations between the conjugate field operators.

Classical and quantum oscillators of sextic and octic anharmonicities

Classical oscillators of sextic and octic anharmonicities are solved analytically up to the linear power of λ ( anharmonic constant) by using Taylor series method. These solutions exhibit the presence of secular terms which are summed up for all orders. The frequency shifts of the oscillators for small anharmonic constants are obtained. It is found that the calculated shifts agree nicely with the available results to-date. The solutions for classical anharmonic oscillators are used to obtain the solutions corresponding to quantum anharmonic oscillators by imposing fundamental commutation relations between position and momentum operators.