Bispinor Field Research Articles

Biquaternion representation of the Dirack equations and bispinor fields

Biquaternion representation of the Dirack equations and bispinor fields

Supersymmetric relativistic quantum mechanics in time-domain

A supersymmetric relativistic quantum theory in the temporal domain is developed for bi-spinor fields satisfying the Dirac equation. The simplest time-domain supersymmetric theory can be postulated for fields with time-dependent mass, showing an equivalence with the bosonic supersymmetric theory in time-domain. Solutions are presented and they are used to produce probability oscillations between mass states. As an application of this idea, we study the two-neutrino oscillation problem, showing that flavour state oscillations may emerge from the supersymmetry originated by the time-dependence of the unique mass of the neutrino.

Static Casimir Condensate of the Bispinor Field in the Friedmann Universe

The quantum Casimir condensate of the massive bispinor in the compact Friedmann universe has been calculated in the static approximation. The Abel–Plana formula has been used to renormalize divergent series.

EoS of Casimir vacuum of massive fields in Friedmann universe

In this paper we study equations of state of Casimir vacuum of massive scalar field and massive bispinor field in compact Friedmann universe. With use of the Abel–Plana formula the renormalization of divergent series for calculation of the quantum means of operators is implemented.

On the quantum-mechanics of a single photon

It is shown that a Dirac(-type) equation for a rank-two bi-spinor field ψph on Minkowski (configuration) spacetime furnishes a Lorentz-covariant quantum-mechanical wave equation in position-space representation for a single free photon. This equation does not encounter any of the roadblocks that have obstructed previous attempts (by various authors) to formulate a quantum-mechanical photon wave equation. In particular, it implies that the photon wave function ψph yields conserved non-negative Born-rule-type quantum probabilities and that its probability current density four-vector transforms properly under Lorentz transformations. Moreover, the eigenvalues of the pertinent photon Dirac Hamiltonian and the vector eigenvalues of the photon momentum operator yield the familiar Einstein relations E = ℏω and p = ℏk, respectively. Furthermore, these spin-1 wave modes are automatically transversal without the need of an additional constraint on the initial data. Some comments on other proposals to set up a photon wave equation are supplied as well.

Auxiliary tensor fields for Sp(2, ℝ) self-duality

The coset Sp(2, ℝ)/U(1) is parametrized by two real scalar fields. We generalize the formalism of auxiliary tensor (bispinor) fields in U(1) self-dual nonlinear models of abelian gauge fields to the case of Sp(2, ℝ) self-duality. In this new formulation, Sp(2, ℝ) duality of the nonlinear scalar-gauge equations of motion is equivalent to an Sp(2, ℝ) invariance of the auxiliary interaction. We derive this result in two different ways, aiming at its further application to supersymmetric theories. We also consider an extension to interactions with higher derivatives.

Generalized Bargmann-Wigner equations in curved space-time

The Bargmann-Wigner equations are formulated to represent bosonic fields in terms of fermionic fields in curved space-time. The bispinor field is explicitly studied including lower spin states. The correspondence among fermions and bosons through bispinors is applied to the superradiance problem in Kerr black-hole spacetime. Superradiance phenomena of the type of positive frequency (0 < ω) and negative momentum near the horizon (pH < 0) are shown not to occur.

New approach to duality-invariant nonlinear electrodynamics

We survey a new approach to the duality-invariant systems of nonlinear electrodynamics, based on introducing auxiliary bi-spinor fields. In this approach, the entire information about the given self-dual system is encoded in the U(1) invariant interaction of the auxiliary fields, while the standard self-dual Lagrangians appear on shell as a result of eliminating auxiliary fields by their equations of motion. Starting from the simplest U(1) duality, we show how this approach can be generalized to the U(N) duality (with N independent Maxwell field strengths), as well as to self-dual systems of = 1 supersymmetric electrodynamics. Also, it works perfectly for self-dual systems with higher derivatives in the action.

Bispinor auxiliary fields in duality-invariant electrodynamics revisited: TheU(N)case

We update and detail the formulation of the duality-invariant systems of $N$ interacting Abelian gauge fields with $N$ auxiliary bispinor fields added. In this setting, the self-duality amounts to $U(N)$ invariance of the nonlinear interaction of the auxiliary fields. The $U(N)$ self-dual Lagrangians arise after solving the nonlinear equations of motion for the auxiliary fields. We also elaborate on a new extended version of the bispinor field formulation involving some additional scalar auxiliary fields and study $U(N)$ invariant interactions with derivatives of the auxiliary bispinor fields. Such interactions generate higher-derivative $U(N)$ self-dual theories.

Bispinor auxiliary fields in duality-invariant electrodynamics revisited

Motivated by recent progress in studying the duality-symmetric models of nonlinear electrodynamics, we revert to the auxiliary tensorial (bispinor) field formulation of the $O(2)$ duality proposed by us in Ivanov and Zupnik [Nucl. Phys. B618, 3 (2001); Yad. Fiz. 67, 2212 (2004)]. In this approach, the entire information about the given duality-symmetric system is encoded in the $O(2)$-invariant interaction Lagrangian which is a function of the auxiliary fields ${V}_{\ensuremath{\alpha}\ensuremath{\beta}}$, ${\overline{V}}_{\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\alpha}}\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\beta}}}$. We extend this setting to duality-symmetric systems with higher derivatives and show that the recently employed ``nonlinear twisted self-duality constraints'' amount to the equations of motion for the auxiliary tensorial fields in our approach. Some other related issues are briefly discussed and a few instructive examples are explicitly worked out.

Geometric Origin of Physical Constants in a Kaluza-Klein Tetrad Model

An important feature of Kaluza-Klein theories is their ability to relate fundamental physical constants to the radii of higher dimensions. In previous Kaluza-Klein theory, which unifies the electromagnetic field with gravity as dimensionless components of a Kaluza-Klein metric, i) all fields have the same physical dimensions, ii) the Lagrangian has no explicit dependence on any physical constants except mass, and hence iii) all physical constants in the field equations except for mass originate from geometry. While it seems natural in Kaluza-Klein theory to add fermion fields by defining higher-dimensional bispinor fields on the Kaluza-Klein manifold, these Kaluza-Klein theories do not satisfy conditions (i), (ii), and (iii). In this paper, we show how conditions (i), (ii), and (iii) can be satisfied by including bispinor fields in a tetrad formulation of the Kaluza-Klein model, as well as in an equivalent teleparallel model. This demonstrates an unexpected feature of Dirac's bispinor equation, since conditions (i), (ii), (iii) imply a special relation among the terms in the Kaluza-Klein or teleparallel Lagrangian that would not be satisfied in general.

Hestenes' Tetrad and Spin Connections

Defining a spin connection is necessary for formulating Dirac's bispinor equation in a curved space-time. Hestenes has shown that a bispinor field is equivalent to an orthonormal tetrad of vector fields together with a complex scalar field. In this paper, we show that using Hestenes' tetrad for the spin connection in a Riemannian space-time leads to a Yang-Mills formulation of the Dirac Lagrangian in which the bispinor field is mapped to a set of Yang-Mills gauge potentials and a complex scalar field. This result was previously proved for a Minkowski space-time using Fierz identities. As an application we derive several different non-Riemannian spin connections found in the literature directly from an arbitrary linear connection acting on Hestenes' tetrad and scalar fields. We also derive spin connections for which Dirac's bispinor equation is form invariant. Previous work has not considered form invariance of the Dirac equation as a criterion for defining a general spin connection.

Supersymmetric [formula omitted] models inspired by deconstruction

We consider 4-dimensional N = 1 supersymmetric SO ( 10 ) models inspired by deconstruction of 5-dimensional N = 1 supersymmetric orbifold SO ( 10 ) models and high-dimensional non-supersymmetric SO ( 10 ) models with Wilson line gauge symmetry breaking. We discuss the SO ( 10 ) × SO ( 10 ) models with bi-fundamental link fields where the gauge symmetry can be broken down to the Pati–Salam, SU ( 5 ) × U ( 1 ) , flipped SU ( 5 ) × U ( 1 ) ′ or the Standard Model like gauge symmetry. We also propose an SO ( 10 ) × SO ( 6 ) × SO ( 4 ) model with bi-fundamental link fields where the gauge symmetry is broken down to the Pati–Salam gauge symmetry, and an SO ( 10 ) × SO ( 10 ) model with bi-spinor link fields where the gauge symmetry is broken down to the flipped SU ( 5 ) × U ( 1 ) ′ gauge symmetry. In these two models, the Pati–Salam and flipped SU ( 5 ) × U ( 1 ) ′ gauge symmetry can be further broken down to the Standard Model gauge symmetry, the doublet–triplet splittings can be obtained by the missing partner mechanism, and the proton decay problem can be solved. We also study the gauge coupling unification. We briefly comment on the interesting variation models with gauge groups SO ( 10 ) × SO ( 6 ) and SO ( 10 ) × flipped SU ( 5 ) × U ( 1 ) ′ in which the proton decay problem can be solved.

Measuring a Kaluza-Klein radius smaller than the Planck length

Hestenes has shown that a bispinor field on a Minkowski space-time is equivalent to an orthonormal tetrad of one-forms together with a complex scalar field. More recently, the Dirac and Einstein equations were unified in a tetrad formulation of a Kaluza-Klein model which gives precisely the usual Dirac-Einstein Lagrangian. In this model, Dirac's bispinor equation is obtained in the limit for which the radius of higher compact dimensions of the Kaluza-Klein manifold becomes vanishingly small compared with the Planck length. For a small but finite radius, the Kaluza-Klein model predicts the velocity splitting of single fermion wave packets. That is, the model predicts that a single fermion wave packet will split into two wave packets with slightly different group velocities. The observation of such wave packet splits would determine the size of the Kaluza-Klein radius. If wave packet splits were not observed in experiments with currently achievable accuracies, the Kaluza-Klein radius would be bounded by at most ${10}^{\ensuremath{-}25}$ times the Planck length.

Bitetrad as geometrodynamical variable for GL(2, )-invariant models of mutually interacting gravitational and bispinor fields

Abstract We suggest a model of the mutually interacting gravitational and bispinor fields considered within the GL(2, C )-gauge framework.

Electromagnetic field in Finsler and associated spaces

The electromagnetic field and its interaction with the leptons is introduced in Finsler space. This space is also considered as the microlocal space-time of the extended hadrons. The field equations for the Finsler space have been obtained from the classical field equations by quantum generalization of this space-time below a fundamental length-scale. On the other hand, the classical field equations are derived from a property of the fields on the autoparallel curve of the Finsler space. The field equations for the associated spaces of the Finsler space, which are macroscopic spaces, such as the large-scale space-time of the universe and the usual Minkowski space-time, can also be obtained for the case of Finslerian bispinor fields separable as the direct products of fields depending on the position coordinates with those depending on the directional arguments. The equations for the coordinate-dependent fields are the usual field equations with the cosmic time-dependent masses of the leptons. The other equations of the directional variable-dependent fields are solved here. Also, the lepton current and the continuity equation are considered. The form-invariance of the field equations under the general coordinate transformations of the Finsler spaces has been discussed.

Higgs Field—Fermion Coupling in the Tensor Dirac Theory

In previous work, the Dirac and Einstein equations were unified in a tetradformulation of a Kaluza—Klein model which gives precisely the usualDirac—Einstein Lagrangian. In this model, the self-adjoint modes of the tetraddescribe gravity, whereas the isometric modes of the tetrad together with a scalarfield describe fermions. The tetrad Kaluza—Klein model is based on a constrainedYang—Mills formulation of the Dirac Lagrangian in which the bispinor field Ψis mapped to a set of SL(2, R) × U(1) gauge potentialsAKa and a complex scalarfield ρ. In this paper we generalize the map Ψ →(AKa, ρ) to multiplets of nbispinor fields representing a fermion multiplet as in standard electroweak theory.We show that the Lagrangian for bispinor multiplets used in the Standard Modelbecomes a constrained Yang—Mills Lagrangian, for which the Higgs fielddetermines a noninvariant gauge metric, thereby breaking the full gauge symmetry.

Q-Deformed grassmann fields and the two-dimensional Ising model

We construct an exact representation of the Ising partition function in the form of the SLq(2, R)-invariant functional integral for the lattice-free q-fermion field theory (q=−1). It is shown that the q-fermionization allows one to rewrite the partition function of the eight-vertex model in an external field through a functional integral with four-fermion interaction. To construct these representations, we define a lattice (l, q, s)-deformed Grassmann bispinor field and extend the Berezin integration rules to this field. At q=−1, l=s=1, we obtain the lattice q-fermion field which allows us to fermionize the two-dimensional Ising model. We show that the Gaussian integral over (q, s)-Grassmann variables is expressed through the (q, s)-deformed Pfaffian which is equal to square root of the determinant of some matrix at q=±1, s=±1.

Unobservability of bispinor two-valuedness in Minkowski space-time

Currently there is some interest in studying the tensor forms of the Dirac equation to elucidate the physical consequences, if any, of the two-valuedness of bispinors and the possibility of the constrained tensor fields admitting Fermi quantization. In this paper, we first argue that n-particle, square-integrable, positive energy bispinor fields in a Minkowski space-time cannot be physically distinguished from constrained tensor fields. That is, the two-valuedness of bispinors could be observed only by the unphysical superposition of positive and negative energy states, which is forbidden in quantum field theory. We also demonstrate that the bispinor and tensor Hamiltonian systems have equivalent Fermi quantizations. Although the tensor Hamiltonian system is noncanonical, representing the tensor Poisson brackets as commutators for the Heisenberg operators directly leads to Fermi quantization without the use of bispinors.

Gravitational anomalies: Antisymmetric tensor gauge field versus bispinor field

It is shown perturbatively that the gravitational anomaly of the chiral bispinor field is twice as large as that of the self-dual antisymmetric tensor gauge field; in the previous literature they were supposed to agree with each other. This implies that a naive application of Fujikawa's path integral method leads to a wrong result for the anomaly of the self-dual antisymmetric tensor gauge field.