Baxter operators for arbitrary spin

We construct Baxter operators for the homogeneous closed XXX spin chain with the quantum space carrying infinite- or finite-dimensional s?₂ representations. All algebraic relations of Baxter operators and transfer matrices are deduced uniformly from Yang-Baxter relations of the local building blocks of these operators. This results in a systematic and very transparent approach where the cases of finite- and infinite-dimensional representations are treated in analogy. Simple relations between the Baxter operators of both cases are obtained. We represent the quantum spaces by polynomials and build the operators from elementary differentiation and multiplication operators. We present compact explicit formulae for the action of Baxter operators on polynomials.     

Baxter operators for arbitrary spin II

This paper presents the second part of our study devoted to the construction of Baxter operators for the homogeneous closed XXX spin chain with the quantum space carrying infinite or finite-dimensional s?₂ representations. We consider the Baxter operators used in Bazhanov et al. (1996, 1997, 1999, 2010) [1] and [2], formulate their construction uniformly with the construction of our previous paper. The building blocks of all global chain operators are derived from the general Yang-Baxter operators and all operator relations are derived from general Yang-Baxter relations. This leads naturally to the comparison of both constructions and allows to connect closely the treatment of the cases of infinite-dimensional representation of generic spin and finite-dimensional representations of integer or half-integer spin. We prove not only the relations between the operators but present also their explicit forms and expressions for their action on polynomials representing the quantum states.     

Feynman propagator for a particle with arbitrary spin

Based on the solution to the Rarita-Schwinger equations, a direct derivation of the projection operator and propagator for a particle with arbitrary spin is worked out. The projection operator constructed by Behrends and Fronsdal is re-deduced and confirmed, and simplified in the case of half-integral spin; the general commutation rules and Feynman propagator for a free particle of any spin are derived, and explicit expressions for the propagators for spins 3/2, 2, 5/2, 3, 7/2, 4 are provided.Received: 13 March 2003, Revised: 24 April 2005, Published online: 6 July 2005     

The Casimir effect for fields with arbitrary spin

The Casimir force arises when a quantum field is confined between objects that apply boundary conditions to it. In a recent paper we used the two-spinor calculus to derive boundary conditions applicable to fields with arbitrary spin in the presence of perfectly reflecting surfaces. Here we use these general boundary conditions to investigate the Casimir force between two parallel perfectly reflecting plates for fields up to spin-2. We use the two-spinor calculus formalism to present a unified calculation of well-known results for spin-1/2 (Dirac) and spin-1 (Maxwell) fields. We then use our unified framework to derive new results for the spin-3/2 and spin-2 fields, which turn out to be the same as those for spin-1/2 and spin-1. This is part of a broader conclusion that there are only two different Casimir forces for perfectly reflecting plates—one associated with fermions and the other with bosons.     

Off-diagonal Bethe ansatz solution of the XXX spin chain with arbitrary boundary conditions

Employing the off-diagonal Bethe ansatz method proposed recently by the present authors, we exactly diagonalize the XXX spin chain with arbitrary boundary fields. By constructing a functional relation between the eigenvalues of the transfer matrix and the quantum determinant, the associated T–Q$T\text{–}Q$ relation and the Bethe ansatz equations are derived.     

Interacting Bhabha field with arbitrary spin

The Bhabha equation which describes a particle having multi-mass and spin states with maximum spin S, integer or half an odd integer, is considered. In the presence of interactions e.g. the electromagnetic interaction introduced through minimal coupling, it is found to yield the same (anti-) commutators as in the case of a free field. Also the propagation of an interacting field is causal. This shows the absence of inconsistencies of the type, pointed out by Johnson and Sudarshan. Finally the applicability of the methods of quantisation is discussed.     

Zitterbewegung of particles with arbitrary spin

The existence of zitterbewegung of particles with higher spins (s≥1) is proved. The investigation is based on the idea of Curie, Jordan, and Sudarshan that there are two aspects of relativistic invariance and also on the determination of the dynamical variables that describe systems with arbitrary spin obtained by Jordan and Mukunda. A number of new paradoxical properties of zitterbewegung unknown in Dirac theory is revealed. For example, particles with high spins (s≥1) can have a velocity greater than light. It is shown in a general form that elimination of zitterbewegung in all directions of space, or even only in a plane, is impossible. There can be only partial liquidation of zitterbewegung, in one of the directions of space, and then only at the price of violation of relativistic invariance of the theory. Finally, it is suggested that the paradoxical properties of zitterbewegung can be understood by redefining the momentum and mass operators. In this way, a connection between zitterbewgung and tachyons is established.     

Spin squeezing inequalities for arbitrary spin

We determine the complete set of generalized spin squeezing inequalities, given in terms of the collective angular momentum components, for particles with an arbitrary spin. They can be used for the experimental detection of entanglement in an ensemble in which the particles cannot be individually addressed. We also present a large set of criteria involving collective observables different from the angular momentum coordinates. We show that some of the inequalities can be used to detect k-particle entanglement and bound entanglement.     

Conformal invariant two and three-point functions for fields with arbitrary spin

The conformal invariant two and three-point functions for any “fundamental” fields with an arbitrary spin and scale dimensions are found in the Minkowsky x-space. The two-point functions for Dirac, symmetric and antisymmetric tensor fields are given. The three-point functions for two Dirac fields and one symmetrical tensor field, as well as any other field for which this function is nonvanishing, are given. In the case of conserved currents the Ward identities are considered.     

Clustering of Fermi particles with arbitrary spin

A single l-shell model is investigated for a system of fermions of spin s and an attractive s-wave, spin channel independent, interaction. The spectra and eigenvectors are determined exactly for different l,s values and particle numbers N. As a generalization of Cooper pairing it is shown that when N=mu(2s+1), mu=1,2,..., 2l+1, the ground state consists of clusters of (2s+1) particles. The relevance of the results for more general situations including the homogeneous system is briefly discussed.     

Relativistic scalar equations for a particle with an arbitrary spin

Relativistic Poincaré-invariant wave equations for zero-mass and heavy particles with an arbitrary spin are constructed on the basis of special infinite-dimensional representation of the Lorentz group. The equations form a compatible system of linear differential equations for an unknown scalar function and contain spin s as a parameter (arbitrary complex number). It is also shown that the equations obtained in this way include the well-known finite-component wave equations as a special case of half-integral or integral spin.State University, Omsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 39–44, June, 1992.     

Feynman propagator for an arbitrary half—integral spin

Based on the solution to Bargmann－Wigner equation for a particle with arbitrary half-integral spin, a direct derivation of the projection operator and propagator for a particle with arbitrary half-integral spin is worked out. The projection operator constructed by Behrends and Fronsdal is re-deduced and confirmed and simplified, the general commutation rules and Feynman propagator with additional non-covariant terms for a free particle with arbitrary half-integral spin are derived, and explicit expressions for the propagators for spins 3/2, 5/2 and 7/2 are provided.     

自旋为任意整数的传播子

以自旋为任意整数的自由粒子的波函数(Bargmann-Wigner方程的解)为基础，进一步研究了 自旋为任意整数的投影算符和传播子.证明了Behrends和Fronsdal所构造的投影算符是正确 的.导出了自旋为任意整数的场的一般对易规则和费恩曼传播子的一般表达式. 关键词： 整数自旋 投影算符 对易规则 费恩曼传播子     

GKS inequalities for arbitrary spin ising ferromagnets

Elementary proofs of the first and second Griffiths-Kelly-Sherman (GKS) inequalities are given for higher-spin Ising systems with a Hamiltonian containing only a quadratic form in the spin variables and integer powers of single spin variables. These proofs are obtained using Gaussian random variables. A slight generalization of previous results has been obtained in that the coefficients of the even powers of the spin variables are allowed to be negative.Work supported by NSF Grant GP-36564X.     

Supermultiplicity and the relativistic Coulomb problem with arbitrary spin

The Hamiltonian for n relativistic electrons without interaction but in a Coulomb potential is well known. If in this Hamiltonian we take r _′ ^u =r′, P _′ ^u =P′ with u=1,2,..., n, we obtain a one-body problem in a Coulomb field, but the appearance of n of the α _u , u=1,..., n, each of which corresponds to spin $\tfrac{1}{2}$ , indicates that we may have spins up to (n/2). We analyze this last problem first by denoting the 4×4 matrices α, β as direct products of 2×2 matrices which correspond to the ordinary spin, and a new concept, also related to the SU(2) group, which we call sign spin. In this new notation our problem depends on the sixteen generators of a U(4) group reduced along the chain Û(2)??(2) sub-groups associated with the ordinary and sign spins. We now make a change of variables in our Hamiltonian so a term ε related to the frequency ω of an oscillator, which will be our variational parameter, appears in it, and later construct the full states of the problem with a harmonic oscillator of frequency 1 and ordinary and sign spin parts. Finally we obtain the matrix representation of our Hamiltonian with respect to the states mentioned and discuss the energy spectra of the problem where the partition {h} representing the irrep of U(4) and j the total angular momentum, take the values {h}=[1], j= $\tfrac{1}{2}$ ; {h}=[11], j=0; {h}=[2], j=0.