Representations Of Lorentz Group Research Articles

Positive-Energy Dirac Particles and Dark Matter

The relativistic positive-energy wave equation proposed by P. Dirac in 1971 is an old but largely forgotten subject. The purpose of this note is to speculate that particles described by this equation (called here Dirac particles) are natural candidates for the dark matter. The reasoning is based on a fact that the internal structure of such particles simply prohibits their interaction with electromagnetic fields (at least with the minimal coupling) which is exactly what is required for dark matter. Dirac particles have quite unusual properties. In particular, they are transformed by an infinite-dimensional representation of the homogeneous Lorentz group, which clearly distinguishes them from all known elementary particles described by finite-dimensional representations and hints to a physics beyond the Standard Model. To clarify the topic, a brief review of the main features of the above-mentioned Dirac equation is given.

Ideal gas thermodynamics with an invariant energy scale

A viable approach towards Quantum Gravity is the Doubly Special Relativity (DSR) framework in which an observer-independent finite energy upper bound (or a finite smallest length scale) appears quite naturally. In this work, we have studied the thermodynamic properties of an ideal gas in a specific DSR framework, known as the Magueijo-Smolin (MS) Model. We use the fact that DSR can be considered as nonlinear representation of Lorentz Group. Subsequently, various thermodynamic parameters of ideal gas have been derived in this modified framework to compare the corresponding deviations from the usual scenario due to the presence of the invariant energy (length) scale.

Low degree Lorentz invariant polynomials as potential entanglement invariants for multiple Dirac spinors

A system of multiple spacelike separated Dirac particles is considered and a method for constructing polynomial invariants under the spinor representations of the local proper orthochronous Lorentz groups is described. The method is a generalization of the method used in Johansson (2022) for the case of two Dirac particles. All polynomials constructed by this method are identically zero for product states. The behaviour of the polynomials under local unitary evolution that acts unitarily on any subspace defined by fixed particle momenta is described. By design all of the polynomials have invariant absolute values on this kind of subspaces if the evolution is locally generated by zero-mass Dirac Hamiltonians. Depending on construction some polynomials have invariant absolute values also for the case of nonzero-mass or additional couplings. Because of these properties the polynomials are considered potential candidates for describing the spinor entanglement of multiple Dirac particles, with either zero or arbitrary mass or additional couplings. Polynomials of degree 2 and 4 are derived for the cases of three and four Dirac spinors. For three spinors no non-zero degree 2 polynomials are found but 67 linearly independent polynomials of degree 4 are identified. For four spinors 16 linearly independent polynomials of degree 2 are constructed as well as 26 polynomials of degree 4 selected from a much larger number. The relations of these polynomials to the polynomial spin entanglement invariants of three and four non-relativistic spin-12 particles are described. Moreover, it is described how degree 4 polynomials for five spinors can be constructed and how degree 2 polynomials can be constructed for any even number of spinors.

On some relations between hyper Bessel–Clifford, Macdonald and Meijer functions and hyper Hankel–Clifford integral transforms

Using a representation of the unimodular Lorentz group, we derive some relations between hyper Bessel–Clifford, Macdonald and Meijer functions. We obtained them as two additional theorems (continual and countable) for a functional defined on the above group and a pair of basis functions belonging to representation spaces. Introducing a hyper analogue of the known first and second Hankel–Clifford integral transforms and writing the continual addition theorem for a particular case, we obtain a simple formula for the sum of these transforms of Macdonald function.

Mass dimension one fields with Wigner degeneracy: A theory of dark matter

Whatever dark matter is, it must be one irreducible unitary representation of the inhomogeneous Lorentz group or another. We here develop a formalism of mass dimension one fermions and bosons of spin one half, and show that they provide natural dark matter candidates. By construction, they are covariant under space-time translations and boosts. However, incorporating the rotational symmetry is non-trivial and requires introducing a two-fold Wigner degeneracy thus doubling the degrees of freedom for particles and anti particles from two to four. With Wigner degeneracy, we have a well-defined theory of mass dimension one fields of spin one half that are physically distinct from the Dirac field. They are local, Lorentz covariant and have positive definite Hamiltonians. The developed framework also has the potential to resolve the cosmological constant problem, and supply dark energy.

Candidate entanglement invariants for two Dirac spinors

We consider two spacelike separated Dirac particles and construct five invariants under the spinor representations of the local proper orthochronous Lorentz groups. All of the constructed Lorentz invariants are identically zero for product states. The behaviour of the Lorentz invariants under local unitary evolutions that act unitarily on any subspace with fixed particle momenta is studied. All of the Lorentz invariants have invariant absolute values on such subspaces if the evolutions are generated by local zero-mass Dirac Hamiltonians. Some of them also for the case of nonzero-mass. Therefore, they are considered potential candidates for describing spinor entanglement of two Dirac particles, with either zero or arbitrary mass. Furthermore, their relations to the Wootters concurrence is investigated and their representations in the Foldy-Wouthuysen picture is given.

Some Formulas for Ordinary and Hyper Bessel–Clifford Functions Related to the Proper Lorentz Group

In this paper, we show that the matrix elements of some Lorentz group representation operators and bases transform operators acting in the representation space may be expressed in terms of the modified Bessel–Clifford functions and their multi-index analogs introduced by the authors via Delerue hyper Bessel functions.

Lepton-Antineutrino Entanglement and Chiral Oscillations

Dirac bispinors belong to an irreducible representation of the complete Lorentz group, which includes parity as a symmetry yielding two intrinsic discrete degrees of freedom: chirality and spin. For massive particles, chirality is not dynamically conserved, which leads to chiral oscillations. In this contribution, we describe the effects of this intrinsic structure of Dirac bispinors on the quantum entanglement encoded in a lepton-antineutrino pair. We consider that the pair is generated through weak interactions, which are intrinsically chiral, such that in the initial state the lepton and the antineutrino have definite chirality but their spins are entangled. We show that chiral oscillations induce spin entanglement oscillations and redistribute the spin entanglement to chirality-spin correlations. Such a phenomenon is prominent if the momentum of the lepton is comparable with or smaller than its mass. We further show that a Bell-like spin observable exhibits the same behavior of the spin entanglement. Such correlations do not require the knowledge of the full density matrix. Our results show novel effects of the intrinsic bispinor structure and can be used as a basis for designing experiments to probe chiral oscillations via spin correlation measurements.

Lorentz invariance of basis tensor gauge theory

Basis tensor gauge theory (BTGT) is a vierbein analog reformulation of ordinary gauge theories in which the vierbein field describes the Wilson line. After a brief review of the BTGT, we clarify the Lorentz group representation properties associated with the variables used for its quantization. In particular, we show that starting from an SO(1,3) representation satisfying the Lorentz-invariant U(1,3) matrix constraints, BTGT introduces a Lorentz frame choice to pick the Abelian group manifold generated by the Cartan subalgebra of U(1,3) for the convenience of quantization even though the theory is frame independent. This freedom to choose a frame can be viewed as an additional symmetry of BTGT that was not emphasized before. We then show how an [Formula: see text] permutation symmetry and a parity symmetry of frame fields natural in BTGT can be used to construct renormalizable gauge theories that introduce frame-dependent fields but remain frame independent perturbatively without any explicit reference to the usual gauge field.

Spin 1 Spinor Construction with Clifford Algera and Dirac Spin 1/2 Spinors

A compatible spin 1 spinor representation with Clifford algebra (1,3) (or 1,3 Cl ) is derived for both (1 / 2,1 / 2) and (1, 0) (0,1)  Lorentz group representations with spin 1/2 particles Dirac spinors in 1,3 Cl . The relation between the two different representations of spin 1 spinors is analogous to the relation between the electromagnetic vector potential field A and the electromagnetic field strength tensor F  . From this relationship, the two representations are combined by the formula u( p, )  ( p, )  p / m. We also note that the Grassmann basis provides more convenient basis for spin 1 spinors especially in chiral representations of (1, 0) (0,1)  , even though the Clifford basis is more fitting for spin 1/2 and (1/ 2,1/ 2) spinor representations for both helicity and handedness.

Single particle entanglement of a massive relativistic particle: Dirac bispinors and spin 1/2 states

In this paper we study the single particle entanglement induced by a Lorentz boost. We described the particle as a solution of the free Dirac equation, a Dirac bispinor, and compare the induced momentum-spin entanglement with the result obtained in the widely considered framework of relativistic spin 1/2 states. The spin linear entropy for both approaches agree in the ultra relativistic limit. We also verify that the spin-momentum entanglement differers from the spin entropy for the bispinorial case, indicating a true multipartite entanglement involving the degrees of freedom of a Dirac bispinor state: momentum, spin and intrinsic parity. The fact that Dirac bispinors belongs to an irreducible representation of the complete Lorentz group, which also included parity as a symmetry, is the ultimate reason behind such non-trivial structure.

Renormalization of a model for spin-1 matter fields

In this work, the one-loop renormalization of a theory for fields transforming in the (1,0)oplus (0,1) representation of the Homogeneous Lorentz Group is studied. The model includes an arbitrary gyromagnetic factor and self-interactions of the spin 1 field, which has mass dimension one. The model is shown to be renormalizable for any value of the gyromagnetic factor.

Reflection positivity on spheres

In this article we specialize a construction of a reflection positive Hilbert space due to Dimock and Jaffe–Ritter to the sphere $${{\mathbb {S}}}^n$$. We determine the resulting Osterwalder–Schrader Hilbert space, a construction that can be viewed as the step from euclidean to relativistic quantum field theory. We show that this process gives rise to an irreducible unitary spherical representation of the orthochronous Lorentz group $$G^c = \mathop {{\mathrm{O}}{}}\nolimits _{1,n}({{\mathbb {R}}})^{\uparrow }$$ and that the representations thus obtained are the irreducible unitary spherical representations of this group. A key tool is a certain complex domain $$\Xi $$, known as the crown of the hyperboloid, containing a half-sphere $${{\mathbb {S}}}^n_+$$ and the hyperboloid $${{\mathbb {H}}}^n$$ as totally real submanifolds. This domain provides a bridge between those two manifolds when we study unitary representations of $$G^c$$ in spaces of holomorphic functions on $$\Xi $$. We connect this analysis with the boundary components which are the de Sitter space and a bundle over the space of future pointing lightlike vectors.

Fradkin Equation for a Spin-3/2 Particle in the Presence of External Electromagnetic and Gravitational Fields

Fradkin’s model for a spin-3/2 particle in the presence of external fields is investigated. Applying the general Gel’fand–Yaglom formalism, we develop this model on the base of a set of six irreducible representations of the proper Lorentz group, making up a 20-component wave function. Applying the standard requirements such as the relativistic invariance, single nonzero mass, spin S =3/2, P-symmetry, and existence of a Lagrangian for the model, we derive a set of spinor equations, firstly in the absence of external fields. The 20-component wave function consists of a bispinor and a vector-bispinor. In the absence of external fields, the Fradkin model reduces to the minimal Pauli–Fierz (or Rarita–Schwinger) theory. Details of this equivalence are given. Then we take the presence of external electromagnetic fields into account. It turns out that the Fradkin equation in the minimal form contains an additional interaction term governed by electromagnetic tensor Fab. In addition, we consider the external curved space-time background. In the generally covariant case, the Fradkin equation contains the additional gravitational interaction term governed by the Ricci tensor Rab. If the electric charge of a particle is zero, the Fradkin model remains correct and describes a neutral Majorana-type spin-3/2 particle interacting additionally with the geometric background through the Ricci tensor.

A note on higher spin symmetry in the IIB matrix model with the operator interpretation

We study the IIB matrix model in an interpretation where the matrices are differential operators defined on curved spacetimes. In this interpretation, coefficients of higher derivative operators formally appear to be massless higher spin fields. In this paper, we examine whether the unitary symmetry of the matrices includes appropriate higher spin gauge symmetries. We focus on fields that are bosonic and relatively simple in the viewpoint of the representation of Lorentz group. We find that the additional auxiliary fields need to be introduced in order to see the higher spin gauge symmetries explicitly. At the same time, we point out that a part of these extra fields are gauged-away, and the rest of part can be written in terms of a totally symmetric tensor field. The transformation to remove its longitudinal components exists as well. As a result, we observe that the independent physical DoF are the transverse components of that symmetric field, and that the theory describes the corresponding higher spin field. We also find that the field is not the Fronsdal field, rather the generalization of curvature.

Lorentz Group Projector Technique for Decomposing Reducible Representations and Applications to High Spins

The momentum-independent Casimir operators of the homogeneous spin-Lorentz group are employed in the construction of covariant projector operators, which can decompose anyone of its reducible finite-dimensional representation spaces into irreducible components. One of the benefits from such operators is that any one of the finite-dimensional carrier spaces of the Lorentz group representations can be equipped with Lorentz vector indices because any such space can be embedded in a Lorentz tensor of a properly-designed rank and then be unambiguously found by a projector. In particular, all the carrier spaces of the single-spin-valued Lorentz group representations, which so far have been described as 2 ( 2 j + 1 ) column vectors, can now be described in terms of Lorentz tensors for bosons or Lorentz tensors with the Dirac spinor component, for fermions. This approach facilitates the construct of covariant interactions of high spins with external fields in so far as they can be obtained by simple contractions of the relevant S O ( 1 , 3 ) indices. Examples of Lorentz group projector operators for spins varying from 1 / 2 –2 and belonging to distinct product spaces are explicitly worked out. The decomposition of multiple-spin-valued product spaces into irreducible sectors suggests that not only the highest spin, but all the spins contained in an irreducible carrier space could correspond to physical degrees of freedom.

Quaternion Electromagnetism and the Relation with Two-Spinor Formalism

By using complex quaternion, which is the system of quaternion representation extended to complex numbers, we show that the laws of electromagnetism can be expressed much more simply and concisely. We also derive the quaternion representation of rotations and boosts from the spinor representation of Lorentz group. It is suggested that the imaginary “i” should be attached to the spatial coordinates, and observe that the complex conjugate of quaternion representation is exactly equal to parity inversion of all physical quantities in the quaternion. We also show that using quaternion is directly linked to the two-spinor formalism. Finally, we discuss meanings of quaternion, octonion and sedenion in physics as n-fold rotation.

Quaternion generalization of super-Poincaré group

Super-Poincaré algebra in [Formula: see text] space–time dimensions has been studied in terms of quaternionic representation of Lorentz group. Starting the connection of quaternion Lorentz group with [Formula: see text] group, the [Formula: see text] spinors for Dirac and Weyl representations of Poincaré group are described consistently to extend the Poincaré algebra to super-Poincaré algebra for [Formula: see text] space–time.

Spin portal to dark matter

In this work we study the possibility that dark matter fields transform in the $(1,0)\oplus(0,1)$ representation of the Homogeneous Lorentz Group. In an effective theory approach, we study the lowest dimension interacting terms of dark matter with standard model fields, assuming that dark matter fields transform as singlets under the standard model gauge group. There are three dimension-four operators, two of them yielding a Higgs portal to dark matter. The third operator couple the photon and $Z^0$ fields to the higher multipoles of dark matter, yielding a \textit{spin portal} to dark matter. For dark matter ($D$) mass below a half of the $Z^0$ mass, the decays $Z^0\to \bar{D} D$ and $H\to \bar{D}D$ are kinematically allowed and contribute to the invisible widths of the $Z^0$ and $H$. We calculate these decays and use experimental results on these invisible widths to constrain the values of the low energy constants finding in general that effects of the spin portal can be more important that those of the Higgs portal. We calculate the dark matter relic density in our formalism, use the constraints on the low energy constants from the $Z^0$ and $H$ invisible widths and compare our results with the measured relic density, finding that dark matter with a $(1,0)\oplus(0,1)$ space-time structure must have a mass $M>43 ~ GeV$.

Revisiting EPRL: All Finite-Dimensional Solutions by Naimark’s Fundamental Theorem

In this paper, we research all possible finite-dimensional representations and corresponding values of the Barbero–Immirzi parameter contained in EPRL simplicity constraints by using Naimark’s fundamental theorem of the Lorentz group representation theory. It turns out that for each nonzero pure imaginary with rational modulus value of the Barbero–Immirzi parameter $$\gamma = i \frac{p}{q}, p, q \in Z, p, q \ne 0$$ , there is a solution of the simplicity constraints, such that the corresponding Lorentz representation is finite-dimensional. The converse is also true—for each finite-dimensional Lorentz representation solution of the simplicity constraints $$(n, \rho )$$ , the associated Barbero–Immirzi parameter is nonzero pure imaginary with rational modulus, $$\gamma = i \frac{p}{q}, p, q \in Z, p, q \ne 0$$ . We solve the simplicity constraints with respect to the Barbero–Immirzi parameter and then use Naimark’s fundamental theorem of the Lorentz group representations to find all finite-dimensional representations contained in the solutions.