Gauge invariant Lagrangian construction for massive higher spin fermionic fields

We formulate a general gauge invariant Lagrangian construction describing the dynamics of massive higher spin fermionic fields in arbitrary dimensions. Treating the conditions determining the irreducible representations of Poincaré group with given spin as the operator constraints in auxiliary Fock space, we built the BRST charge for the model under consideration and find the gauge invariant equations of motion in terms of vectors and operators in the Fock space. It is shown that like in massless case [I.L. Buchbinder, V.A. Krykhtin, A. Pashnev, Nucl. Phys. B 711 (2005) 367, hep-th/0410215], the massive fermionic higher spin field models are the reducible gauge theories and the order of reducibility grows with the value of spin. In compare with all previous approaches, no off-shell constraints on the fields and the gauge parameters are imposed from the very beginning, all correct constraints emerge automatically as the consequences of the equations of motion. As an example, we derive a gauge invariant Lagrangian for massive spin 3/2 field.     

Equation for a particle with spin of 3/2 having anomolous magnetic moment

It is shown that it is possible to describe a particle with spin of 3/2 and anomolous magnetic moment within the theory of first-order relativistic wave equations in matrix form by introducing a minimum interaction with the external electromagnetic field. This is done by expanding the space of Lorentz group representations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 23–27, January, 1984.     

Spinor Structure and Matter Spectrum

Classification of relativistic wave equations is given on the ground of interlocking representations of the Lorentz group. A system of interlocking representations is associated with a system of eigenvector subspaces of the energy operator. Such a correspondence allows one to define matter spectrum, where the each level of this spectrum presents a some state of elementary particle. An elementary particle is understood as a superposition of state vectors in nonseparable Hilbert space. Classification of indecomposable systems of relativistic wave equations is produced for bosonic and fermionic fields on an equal footing (including Dirac and Maxwell equations). All these fields are equivalent levels of matter spectrum, which differ from each other by the value of mass and spin. It is shown that a spectrum of the energy operator, corresponding to a given matter level, is non-degenerate for the fields of type (l, 0) ⊕ (0, l), where l is a spin value, whereas for arbitrary spin chains we have degenerate spectrum. Energy spectra of the stability levels (electron and proton states) of the matter spectrum are studied in detail. It is shown that these stability levels have a nature of threshold scales of the fractal structure associated with the system of interlocking representations of the Lorentz group.     

A New Approach to Spinors and Some Representations of the Lorentz Group on Them

We give a geometric realization of space-time spinors and associated representations, using the Jordan triple structure associated with the Cartan factors of type 4, the so-called spin factors. We construct certain representations of the Lorentz group, which at the same time realize bosonic spin-1 and fermionic spin- wave equations of relativistic field theory, showing some unexpected relations between various low-dimensional Lorentz representations. We include a geometrically and physically motivated introduction to Jordan triples and spin factors.     

Wigner Rotations and Precession of Polarization

An easy and direct derivation of Thomas precession is obtained from infinitesimal Wigner rotations arising in unitary representations of the Poincaré group. For spin > 1/2, multipole parameters are studied from this point of view. The canonical 3-component definition of polarization arising naturally in this context is compared with formalisms which start form a pseudo 4-vector and an antisymmetric tensor respectively. The full Thomas equations, including Larmor precession, is derived using time derivatives of finite Wigner rotations. Exact solutions, with arbitrary initial conditions, are presented for constant magnetic fields and for orthogonal constant electric and magnetic fields. For a class of plane wave external fields exact solutions are obtained for the Dirac equation generalized by the inclusion of anomalous magnetic moment (Pauli) and electric dipole moment terms. Using the front form of dynamics, well-adapted to this context and coinciding with proper time dynamics, expectation values are calculated. The polarization pseudo 4-vector thus obtained is shown to satisfy the BMT equation, which is equivalent to the Thomas equation. This shows that the validity of the classical precession equations is not necessarily restricted to slowly varying external fields. These solutions can also be of interest in the study of spin 1/2 particles in laser fields and in the study of electric dipole moments.     

Nonrelativistic particles and wave equations

This paper is devoted to a detailed study of nonrelativistic particles and their properties, as described by Galilei invariant wave equations, in order to obtain a precise distinction between the specifically relativistic properties of elementary quantum mechanical systems and those which are also shared by nonrelativistic systems. After having emphasized that spin, for instance, is not such a specifically relativistic effect, we construct wave equations for nonrelativistic particles with any spin. Our derivation is based upon the theory of representations of the Galilei group, which define nonrelativistic particles. We particularly study the spin 1/2 case where we introduce a four-component wave equation, the nonrelativistic analogue of the Dirac equation. It leads to the conclusion that the spin magnetic moment, with its Landé factorg=2, is not a relativistic property. More generally, nonrelativistic particles seem to possess intrinsic moments with the same values as their relativistic counterparts, but are found to possess no higher electromagnetic multipole moments. Studying galilean electromagnetism (i.e. the theory of spin 1 massless particles), we show that only the displacement current is responsible for the breakdown of galilean invariance in Maxwell equations, and we make some comments about such a nonrelativistic electromagnetism. Comparing the connection between wave equations and the invariance group in both the relativistic and the nonrelativistic case, we are finally led to some vexing questions about the very concept of wave equations.     

Weyl Spinor and Solution of Massless Free Field Equations

The massless field equations for arbitrary spin in curved space-time arereconsidered. The general solution of the field equation in Robertson-Walkerspace-time that was previously determined is briefly discussed after explicitlyshowing that the Weyl spinor vanishes. The case of the Lemaître-Tolman-Bondispace-time is studied in detail. The general expression of the corresponding Weylspinor is obtained and some particular situations exploited. The spin-3/2 andspin-2 massless field equations are solved explicitly. The solutions are simplifiedby the existence of nontrivial algebraic constraints. The angular part of theequations is separated by the usual separation method and integrated directly.The other equations that are not separated in the radial and time dependence arereduced to a simple form. The results obtained are extended, as a consequenceof previous results, to the case of arbitrary spin. The solution of the general caseessentially reduces to the treatment of spin 3/2 and spin 2.     

Evolution Equation for a Joint Tomographic Probability Distribution of Spin-1 Particles

The nine-component positive vector optical tomographic probability portrait of quantum state of spin-1 particles containing full spatial and spin information about the state without redundancy is constructed. Also the suggested approach is expanded to symplectic tomography representation and to representations with quasidistributions like Wigner function, Husimi Q?function, and Glauber-Sudarshan P?function. The evolution equations for constructed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian are found. The evolution equations are also obtained in special case of the quantum system of charged spin-1 particle in arbitrary electro-magnetic field, which are analogs of non-relativistic Proca equation in appropriate representations. The generalization of proposed approach to the cases of arbitrary spin is discussed. The possibility of formulation of quantum mechanics of the systems with spins in terms of joint probability distributions without the use of wave functions or density matrices is explicitly demonstrated.     

Construction and classification of indecomposable finite-dimensional representations of the homogeneous Galilei group

We discuss finite-dimensional representations of the homogeneous Galilei group which, when restricted to its subgroup SO(3), are decomposed to spin 0, 1/2 and 1 representations. In particular we explain how these representations were obtained in our paper (M. de Montigny et al.: J. Phys. A39 (2006) 9365) via reduction of the classification problem to a matrix one admitting exact solutions, and via contraction of the corresponding representations of the Lorentz group. Finally, for discussed representations we derive all functional invariants.     

Properties of basis functions in O(3, 1) invariant expansions

Functions are discussed that form the bases of unitary irreducible representations of theO(3, 1) Lorentz group; some properties of these functions are derived for the case of spin 1/2.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 29–35, February, 1977.     

Algebraic study of a class of relativistic wave equations

First-order relativistic wave equations are considered whose irreducible matrix coefficients satisfy the simplest (except for the Dirac algebra) unique mass condition, (β · p)³ = p²(β · p), which is also sufficient to guarantee causality in a minimally coupled external electromagnetic field. All of the associated representations of SL(2, ©) are classified and studied up to and including those which are the direct sum of four irreducible components, (n, m), with either n or m less than two. A large number of families of representations are found which permit the algebraic condition to be satisfied. These are tabulated according to whether a Hermitian choice for β⁰ is possible and their spin content is given. If a unique spin is described, then the only possible representations are

$\text{(1) (}\text{n}\text{,0) ⊕ (}\text{n}\text{? 1/2, 1/2)}$

$\text{(2) (}\text{n}\text{,0) ⊕ (}\text{n}\text{+ 1/2, 1/2)}$

$\text{(3) (}\text{n}\text{+ 1/2, 1/2) ⊕ (}\text{n}\text{,0) ⊕ (}\text{n}\text{? 1/2, 1/2)}$

$\text{(4) (1,0) ⊕ (1/2, 1/2) ⊕ (0,1)}$

and their conjugates. If, in addition, the representation is assumed to be self-conjugate, then only the Dirac and Petiau-Duffin-Kemmer equations survive.     

<Emphasis Type="Italic">CPT</Emphasis> Groups of Higher Spin Fields

CPT groups of higher spin fields are defined in the framework of automorphism groups of Clifford algebras associated with the complex representations of the proper orthochronous Lorentz group. Higher spin fields are understood as the fields on the Poincaré group which describe orientable (extended) objects. A general method for construction of CPT groups of the fields of any spin is given. CPT groups of the fields of spin-1/2, spin-1 and spin-3/2 are considered in detail. CPT groups of the fields of tensor type are discussed. It is shown that tensor fields correspond to particles of the same spin with different masses.     

Dynamics of mean-field spin models from basic results in abstract differential equations

The infinite-volume limit of the dynamics of (generalized) mean-field spin models is obtained through a direct analysis of the equations of motion, in a large class of representations of the spin algebra. The resulting dynamics fits into a general framework for systems with long-range interaction: variables at infinity appear in the time evolution of local variables and spontaneous symmetry breaking with an energy gap follows from this mechanism. The independence of the construction of the approximation scheme in finite volume is proven.     

Current-induced spin injection and surface torque in ferromagnetic metallic junctions

The joint influence of two current-induced effects, namely, longitudinal nonequilibrium spin injection and surface torque, on spin-valve-type ferromagnetic metallic junctions is considered theoretically. The current flows normally to layer boundaries. The analysis is based on solving a system of coupled equations of motion for mobile electron and lattice magnetizations. The boundary conditions for the equations of motion are derived from the continuity condition for the total magnetization flux in these subsystems. A dispersion relation is derived for spin wave fluctuations depending on the current through the junction. The fluctuations become unstable at currents exceeding some threshold value (usually, 10⁶?3 × 10⁷ A/cm²). The joint action of longitudinal spin injection and torque lowers the instability threshold. Current-induced spin injection decreases spin wave frequencies near the threshold and can strengthen magnetization pinning at the injecting contact.     

Superfields with higher spin fermionic coordinates

We consider superspaces with fermionic coordinates belonging to representations of the Lorentz group other than spin 1/2. A general diagrammatic method is introduced which facilitates the identification of the various component fields comprising the superfields. Explicit examples of superfields with fermionic coordinates belonging to (1, 1/2)+(1/2, 1) and (3/2, 0)+(0, 3/2) representations of the Lorentz group are worked out in detail.