Principle Of Minimal Coupling Research Articles

Bulk plasmaritons: Wave-mechanical and second-quantized theories

Inspired by a recently established quantum theory for bulk and surface plasmons [Jung and Keller, Phys. Rev. A 103, 063501 (2021).], wave-mechanical and second-quantized quantum electrodynamic theories for bulk plasmaritons in a homogeneous jellium are presented. Starting from the ``inner'' structure of the transversely polarized classical plasmariton mode, it is argued that a plasmariton quasiparticle may be formed by an always attached pair consisting of a transverse (gauge) photon and a never observable transverse plasmon. A first-quantized plasmariton theory is established from a plasmariton Klein-Gordon equation. It is shown that the plasmariton can be perceived as a diamagnetically driven spin-1 boson quasiparticle. A Lagrangian-Hamiltonian formalism is used to extend the first-quantized plasmariton theory to the second-quantized level. The minimal coupling principle is used to change the free QED theory into a theory coupling the plasmariton to an electromagnetic gauge field. Using the microscopic transverse Lindhard (random phase approximation) dielectric function, it is shown that for small wave numbers, the classical Boltzmann equation part of the Lindhard dielectric function results in a hydrodynamic Brewster branch, the quantization of which follows that of a vectorial Klein-Gordon equation. For plasmariton wave numbers larger than the Fermi wave number, a pure electrostatic model (with no magnetic-field component) results in a harmonic-oscillator description of the plasmariton's polarization field. With inspiration and elements from the Power-Zinau-Wolley description, the Pauli-Fierz theory for particle-tied transverse photons, and the propagator theory of Keller for the spatial localization of transverse photons, a formalism in which the plasmariton quasiparticle is dressed by a cloud of transverse (long-wavelength) photons is developed. The quantized radiated part of the electromagnetic field is described by a free-field quantization of the transverse part of the displacement field (${\mathbf{D}}_{T}$). The total Hamiltonian is diagonalized, and the resulting dispersion relation obtained. The interaction between the transverse radiated and attached photons implies that a strongly localized and charged plasmariton quasiparticle with its tied photons vibrates as a simple harmonic oscillator in the ``external'' radiative transverse photon field.

Anderson Localization for Electric Quantum Walks and Skew-Shift CMV Matrices

We consider the spectral and dynamical properties of one-dimensional quantum walks placed into homogenous electric fields according to a discrete version of the minimal coupling principle. We show that for all irrational fields the absolutely continuous spectrum of these systems is empty, and prove Anderson localization for almost all (irrational) fields. This result closes a gap which was left open in the original study of electric quantum walks: a spectral and dynamical characterization of these systems for typical fields. Additionally, we derive an analytic and explicit expression for the Lyapunov exponent of this model. Making use of a connection between quantum walks and CMV matrices our result implies Anderson localization for CMV matrices with a particular choice of skew-shift Verblunsky coefficients as well as for quasi-periodic unitary band matrices.

Bulk and surface plasmons: Wave-mechanical and second-quantized theories

In this paper we formulate the quantum theory for longitudinal (rotational free) collective electron modes, the so-called plasmons. Starting from the jellium dispersion relation, a first-quantized wave-mechanical theory is established. We show that plasmon quantum (quasi)particles are charged bosons described by complex Klein-Gordon fields satisfying a ``relativistic'' scalar Klein-Gordon equation in which the speed of light $c$ is replaced by a velocity $a$ on the order of the Fermi velocity. Based on a formally ($c\ensuremath{\rightarrow}a$) covariant description, the first-quantized theory is extended to the second-quantized level via a Lagrangian formalism. We show how the second-quantized theory enables the study of the plasmon quantum particles interaction with an electromagnetic gauge field via a modification of the free plasmon Lagrangian density by the minimal coupling principle. Utilizing the Weyl expansion, first- and second-quantized theories for surface-plasmon quantum particles are established in close analogy to those of the bulk plasmon quantum particle. This work opens up for the study of, e.g., squeezed, entangled, and coherent plasmon states in line with what has been studied theoretically and experimentally for photons for many years.

The coupling of matter and spacetime geometry

The geometrical formulation of gravity is not unique and can be set up in a variety of spacetimes. Even though the gravitational sector enjoys this freedom of different geometrical interpretations, consistent matter couplings have to be assured for a steady foundation of gravity. In generalised geometries, further ambiguities arise in the matter couplings unless the minimal coupling principle (MCP) is adopted that is compatible with the principles of relativity, universality and inertia. In this work, MCP is applied to all standard model gauge fields and matter fields in a completely general (linear) affine geometry. This is also discussed from an effective field theory perspective. It is found that the presence of torsion generically leads to theoretical problems. However, symmetric teleparallelism, wherein the affine geometry is integrable and torsion-free, is consistent with MCP. The generalised Bianchi identity is derived and shown to determine the dynamics of the connection in a unified fashion. Also, the parallel transport with respect to a teleparallel connection is shown to be free of second clock effects.

Generalized electromagnetic fields associated with the Coulomb-like atom problem

It is well known that the principle of minimal coupling in quantum mechanics establishes a unique interaction form for a charged particle. By properly redefining the canonical commutation relations between (canonical) conjugate components of position and momentum of the particle, e.g., a $$\pi ^{-}$$ meson, we restate the Klein–Gordon equation for the Coulomb-like problem incorporating a generalized minimal electromagnetic replacement. The corresponding interaction keeps the $$1/\vert \mathbf {q}\vert $$ dependence in both the scalar potential $$V(\vert \mathbf {q} \vert )$$ and the vector potential $$\mathbf {A}(\mathbf {q})(\vert \mathbf {A}(\mathbf {q})\vert \sim 1/\vert \mathbf {q}\vert )$$. This equation can be exactly solved in closed form. Thus, we present a novel relativistic quantum-mechanical model which can be further explored.

Generalized Electromagnetic Fields Associated with the Hydrogen-Like Atom Problem

Abstract It is known that the principle of minimal coupling in quantum mechanics determines a unique interaction form for a charged particle. By properly redefining the canonical commutation relation between (canonical) conjugate components of position and momentum of the particle, e.g. an electron, we restate the Dirac equation for the hydrogen-like atom problem incorporating a generalized minimal electromagnetic coupling. The corresponding interaction keeps the 1 / | q | $1/\left|\mathbf{q}\right|$ dependence in both the scalar potential V ( | q | ) $V\left({\left|\mathbf{q}\right|}\right)$ and the vector potential A ( q ) $\mathbf{A}\left(\mathbf{q}\right)$ ( | A ( q ) | ∼ 1 / | q | $\left|{\mathbf{A}\left(\mathbf{q}\right)}\right|\sim 1/\left|\mathbf{q}\right|$ ). This problem turns out to be exactly solvable; moreover, the eigenstates and eigenvalues can be obtained in an elementary fashion. Some feasible models within this approach are discussed. Then we make a few remarks about the breaking of supersymmetry. Finally, we briefly comment on the possible Lie algebra (dynamical symmetry algebra) of these relativistic quantum systems.

The [formula omitted]-matrix for surface boundary states: An application to photoemission for Weyl semimetals

We present a new theory of photoemission for Weyl semimetals. We derive this theory using a model with a boundary surface at z=0. Due to the boundary, the self adjoint condition needs to be verified in order to ensure physical solutions. The solutions are given by two chiral zero modes which propagate on the boundary. Due to the Coulomb interaction, the chiral boundary model is in the same universality class as interacting graphene. The interactions cause a temperature dependence of the velocity and life time. Using the principle of minimal coupling, we identify the electron–photon Hamiltonian. The photoemission intensity is computed using the S-matrix formalism. The S-matrix is derived using the initial photon state, the final state of a photoelectron and a hole in the valence band. The photoemission reveals the final valence band dispersion ħv(±ky−k0)+ħΩ after absorbing a photon of frequency Ω (k0 represents the shift in the momentum due to the crystal potential). The momentum in the z direction is not conserved, and is integrated out. As a result, the scattering matrix is a function of the parallel momentum . We observe two dimensional contours, representing the “Fermi arcs”, which for opposite spin polarization have opposite curvature. This theory is in agreement with previous experimental observations.

Group theoretical derivation of the minimal coupling principle.

The group theoretical methods worked out by Bargmann, Mackey and Wigner, which deductively establish the Quantum Theory of a free particle for which Galileian transformations form a symmetry group, are extended to the case of an interacting particle. In doing so, the obstacles caused by loss of symmetry are overcome. In this approach, specific forms of the wave equation of an interacting particle, including the equation derived from the minimal coupling principle, are implied by particular first-order invariance properties that characterize the interaction with respect to specific subgroups of Galileian transformations; moreover, the possibility of yet unknown forms of the wave equation is left open.

Eikonal approximation, Finsler structures, and implications for Lorentz-violating photons in weak gravitational fields

The current article shall contribute to understanding the classical analogue of the minimal photon sector in the Lorentz-violating Standard-Model Extension (SME). It is supposed to complement all studies performed on classical point-particle equivalents of SME fermions. The classical analogue of a photon is not a massive particle being described by a usual equation of motion, but a geometric ray underlying the eikonal equation. The first part of the paper will set up the necessary tools to understand this correspondence for interesting cases of the minimal SME photon sector. In conventional optics the eikonal equation follows from an action principle, which is demonstrated to work in most (but not all) Lorentz-violating cases as well. The integrands of the action functional correspond to Finsler structures, which establishes the connection to Finsler geometry. The second part of the article treats Lorentz-violating light rays in a weak gravitational background by implementing the principle of minimal coupling. Thereby it is shown how Lorentz violation in the photon sector can be constrained by measurements of light bending at massive bodies such as the Sun. The phenomenological studies are based on the currently running ESA mission GAIA and the planned NASA/ESA mission LATOR. The final part of the paper discusses certain aspects of explicit Lorentz violation in gravity based on the setting of Finsler geometry.

On gauge invariance and minimal coupling

Abstract The principle of minimal coupling has been used in the study of Higgs boson interactions to argue that certain higher dimensional operators in the low-energy effective theory generalization of the Standard Model are suppressed by loop factors, and thus smaller than others. It also has been extensively used to analyze beyond-the-Standard-Model theories. We show that in field theory, and even in quantum mechanics, the concept of minimal coupling is ill-defined and inapplicable as a general principle, and give many pedagogical examples which illustrate this fact. We also clarify some related misconceptions about the dynamics of strongly coupled gauge theories. Many arguments in the literature on Higgs boson interactions that use minimal coupling, particularly in pseudo-Goldstone Higgs theories, are inherently flawed.

Electrodynamics onκ-Minkowski space-time

In this paper, we derive Lorentz force and Maxwell's equations on kappa-Minkowski space-time up to the first order in the deformation parameter. This is done by elevating the principle of minimal coupling to non-commutative space-time. We also show the equivalence of minimal coupling prescription and Feynman's approach. It is shown that the motion in kappa space-time can be interpreted as motion in a background gravitational field, which is induced by this non-commutativity. In the static limit, the effect of kappa deformation is to scale the electric charge. We also show that the laws of electrodynamics depend on the mass of the charged particle, in kappa space-time.

The magnetic Weyl calculus

In the presence of a variable magnetic field, the Weyl pseudodifferential calculus must be modified. The usual modification, based on “the minimal coupling principle” at the level of the classical symbols, does not lead to gauge invariant formulas if the magnetic field is not constant. We present a gauge covariant quantization, relying on the magnetic canonical commutation relations. The underlying symbolic calculus is a deformation, defined in terms of the magnetic flux through triangles, of the classical Moyal product.

Gauge principle revisited: Towards a unification of space–time and internal gauge interactions

The minimal coupling principle is revisited under the quantum perspectives of the space–time symmetry. This revision is better realized on a group approach to quantization (GAQ) where group cohomology and extensions of groups play a preponderant role. We first consider the case of the electromagnetic potential; the Galilei and/or Poincaré group is (noncentrally) extended by the “local” U(1) group. The resulting group can also be seen as a central extension, parametrized by both the mass and the electric charge, of an infinite-dimensional group, on which GAQ leads to the dynamics of a particle moving in the presence of an electromagnetic field. Then we try the gravitational interaction of a particle by making the space–time translations “local.” However, promoting to “local” the space–time subgroup of the true symmetry of the quantum free relativistic particle, i.e., the centrally extended by U(1) Poincaré group, results in a new electromagneticlike force of pure gravitational origin. This is a consequence of the space–time translations not being an invariant subgroup of the extended Poincaré group and constitutes a preliminary attempt to a nontrivial mixing of space–time and internal gauge interactions.

ON A QUANTUM EQUIVALENCE PRINCIPLE

The logical consistency of a description of quantum theory in the context of general relativity, which includes minimal coupling principle, is analyzed from the point of view of Feynman's formulation in terms of path integrals. We will argue from this standpoint and use an argument that claims the incompleteness of the general relativistic description of gravitation, which emerges as a consequence of the gravitationally induced phases of the so-called flavor-oscillation clocks, that the postulates of quantum theory are logically incompatible with the usual minimal coupling principle. It will be shown that this inconsistency could emerge from the fact that the required geometrical information to calculate the probability of finding a particle at any point of the respective manifold does not lie in a region with finite volume. Then we put forth a new quantum minimal coupling principle in terms of a restricted path integral, and along the ideas of this model not only the propagator of a free particle is calculated but also the conditions under which we recover Feynman's case for a free particle are deduced. The effect on diatomic interstellar molecules is also calculated. The already existing relation between restricted path integral formalism and decoherence model will enable us to connect the issue of a quantum minimal coupling principle with the collapse of the wave function. From this last remark we will claim that the geometrical structure of the involved manifold acts as, always present, a measuring device on a quantum particle. In other words, in this proposal we connect the issue of a quantum minimal coupling principle with a claim which states that gravity could be one of the physical entities which results in the collapse of the wave function.

Squeezing and Dynamical Symmetries

A systematic formalism for dealing withnonrelativistic time-dependent quantum Hamiltonians ispresented. The starting point is the Lewis andRiesenfeld invariant isospectral operator I(x, t). Wediscuss three examples: the generalized harmonicoscillator, the conformal oscillator, and the infinitesquare well with a moving boundary. We obtain the sameresults for the generalized harmonic oscillator as other approaches. In the case of the squarewell with a moving boundary, an effective interactionappears which seems to be due to the time dependence ofthe boundary. Consistency with the principle of minimal coupling and gauge invariance isobtained. Some interesting physical applications aresuggested.

The generalized harmonic oscillator and the infinite square well with a moving boundary

For the one-dimensional generalized harmonic oscillator we obtain in this paper its wavefunctions in closed form by means of two independent methods which are based respectively in (i) the algebraic properties of the dynamical symmetry of the system and (ii) the construction of an invariant operator. The total equivalence of these two formulations is shown and quantal properties are described in terms of a classical solution of the equations of motion. Two possible reductions for the system exist: the static harmonic oscillator and the free particle. In the latter case the quantum system becomes a Fermi oscillator or equivalently it can describe a free particle in a well with one moving boundary which in turn follows certain classical rules. The time-dependent boundary conditions in the well play the role of an effective interaction acting on the particle. The formalism is shown to be compatible with the gauge principle of minimal coupling and several different gauges are constructed and analysed.

The intrinsic magnetic moment of elementary particles

In a purely nonrelativistic formulation, a Hamiltonian is obtained for the description of states of a charged particle with spin, interacting with an external electromagnetic field. For the specific cases of spin-1/2 and spin-1, the simplest choices for the Hamiltonian functions lead to a value for the (lowest order) intrinsic magnetic moment equal to the Bohr magneton in both cases. This is the same as is obtained in treatments that start from a relativistic formulation, as in the Dirac theory. The nonrelativistic and relativistic formulations both employ the ‘‘minimal coupling principle,’’ but the resulting Hamiltonians with couplings are, for very general considerations, not completely determined. However, the corresponding ambiguity that occurs in the nonrelativistic theory is also present in the relativistic formulation. The simpler nonrelativistic theory is more transparent in exhibiting these features of the problem and dispels the notion that spin (and the associated magnetic moment) require relativistic formulations for a proper understanding.

Quantum cosmology and Eddington's large cosmic numbers

A theory that contains three fundamental constants from which one can build length, time, and mass (or force) etalons satisfies with that a necessary criterion of a “universal unified field theory.” In order to interpret such a theory physically, one has to translate it into the Galilei-Newtonian language. This leads to classical “pictures” whose compatibility is ensured by introducing appropriate measurement-theoretical principles which imply corresponding uncertainty relations. In this paper we compare different (mainly gravitational) theories from the point of view of the fundamental constants underlying each case, and of the respective uncertainty relations. Assuming Eddington's hypothesis of large cosmic numbers, it is argued in particular that in quantized general relativity one arrives at less stringent limitations on cosmology than in other conceivable approaches satisfying the principle of minimal coupling. These limitations, however, are in all cases strong enough to bar the way to quantum cosmology. Instead one is led to a Diophantean concept.

Kalusa-Klein approach in higher-dimensional theories of gravity with torsion

The Kaluza-Klein approach is investigated in the Einstein-Cartan-Sciama-Kibble (ECSK) theory and in a Brans-Dicke-type model with torsion. In particular, the authors focus their attention on the possible coupling of gauge fields and torsion, the principle of minimal coupling and the possible physical meaning of the fields which originate from the higher-dimensional torsion fields. Also, it is known that Brans-Dicke-type models with torsion have an extended conformal symmetry. They investigate the fate of this symmetry when going (via dimensional reduction) to four dimensions. They find that the vectorial part of the torsion acquires a nonvanishing value, thus breaking the symmetry of the original five-dimensional theory.

Infinite-dimensional analogs of the minimal coupling principle and of the Poincaré lemma for differential two-forms

In classical mechanics, the minimal interaction principle for a charged particle in an external electromagnetic field describes in symplectic language the orbit of the Hamiltonian formalism on the cotangent bundle T * E under the affine action on the canonical two-form d p ∧ d q of the additive group ⋀ 1( E). Local description of this orbit amounts to the Poincaré Lemma for differential two-forms. We present an infinite-dimensional analog of these where the manifold E is replaced by a bundle π : E → M. Nonabelian case, corresponding to the external field being of Yang-Mills rather than electromagnetic type, is also discussed. In this case the orbit turns out to be much simpler and “smaller” than the abelian one.