Spin invariants and uniqueness of spin operators

The problem of uniqueness of spin operators for Dirac particles is examined. To this end, invariant spin relations based on the spin projection in the rest frame are constructed. Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 7–11, 1999.     

Dipole-dipole interactions of high-spin paramagnetic centers in disordered systems

Dipole-dipole interactions between distant paramagnetic centers (PCs) where at least one PC has spinS>1/2 are examined. The results provide a basis for the application of pulsed electron-electron double resonance method to the measurement of distances between PCs involving high-spin species. A projection operator technique based on spectral decomposition of the secular Hamiltonian is used to calculate electron paramagnetic resonance (EPR) line splitting caused by the dipole coupling. This allows calculation of operators projecting an arbitrary wave function onto high-spin PC eigenstates when the eigenvectors of the Hamiltonian are not known. The effective spin vectors — that is, the expectation values for vector spin operators in the PC eigenstates — are calculated. The dependence of these effective spin vectors on the external magnetic field is calculated. There is a qualitative difference between pairs having at least one integer spin (non-Kramers PC) and pairs of two half-integer (Kramers PC) spins. With the help of these effective spin vectors, the dipolar line shape of EPR lines is calculated. Analytical relations are obtained for PCs with spinS=1/2 and 1. The dependence of Pake patterns on variations of zero-field splitting, Zeeman energy, temperature and dipolar coupling are illustrated.     

A projection operator method for the analysis of magnetic neutron form factors

A set of projection operators in matrix form has been derived on the basis of decomposition of the spin density into a series of fully symmetrized cubic harmonics. This set of projection operators allows a formulation of the Fourier analysis of magnetic form factors in a convenient way. The presented method is capable of checking the validity of various theoretical models used for spin density analysis up to now. The general formalism is worked out in explicit form for the fcc and bcc structures and deals with that part of spin density which is contained within the sphere inscribed in the Wigner-Seitz cell. This projection operator method has been tested on the magnetic form factors of nickel and iron.     

The Spin Interaction of a Dirac Particle in an Aharonov-Bohm Potential in First Order Scattering

For a Dirac particle in an Aharonov-Bohm (AB) potential, it is shown that the spin interaction (SI) operator which governs the transitions in the spin sector of the first order S-matrix is related to one of the generators of rotation in the spin space of the particle. This operator, which is given by the projection of the spin operator Σ along the direction of the total momentum of the system, and the two operators constructed from the projections of the Σ operator along the momentum transfer and the z-directions close the SU(2) algebra. It is suggested, then, that these two directions of the total momentum and the momentum transfer form some sort of natural intrinsic directions in terms of which the spin dynamics of the scattering process at first order can be formulated conveniently. A formulation and an interpretation of the conservation of helicity at first order using the spin projection operators along these directions is presented.     

Eine phasenraumdarstellung für spinsysteme

We start with the definition of two mapping operators, one of them is the projection operator onto coherent spin states. With the help of these operators we derive a mapping theorem which defines a correspondence between the operators in spin space andc-number functions of a certain class. It is shown that this correspondence is one-to-one. The quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase space average of classical statistical mechanics. We also derive a product theorem which allows us to transcribe the equations of motion for operators into equivalent equations for thec-number functions. As an illustration of the theory, some examples are discussed.     

Controllability of spin 1 systems and realization of ternary SWAP gate in two spin 1 systems coupled with Ising interaction

In this paper,we investigate the controllability of spin 1 systems and the realization of ternary gates.Using dipole and quadrupole operators as the orthogonal basis of su(3) algebra,we discuss the controllability of one spin 1 systems and offer the concept of a complete set of control operators first.Then we present the controllability of two spin 1 systems coupled with Ising interaction and the transforming relations of the drift process of the system.Finally the specific realization of the ternary SWAP gate in these systems is discussed.It takes 9 drift processes and 25 basic control processes.     

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.     

Coherent spin states and energy-phase uncertainty relation

In analogy to what has been done for the quantum harmonic oscillator, two non-commuting phase operators cos Φ and spin Φ are here defined for a multi-spin system in terms of the angular momentum operators. These operators are used to introduce a satisfactory energy-phase uncertainty relation. In the classical limit it is possible to establish a correspondence between the phase operators cos Φ and sin Φ and the classical functions cos ? and sin ?, where ? is the azimuthal angle of the angular momentum. First results are reported indicating that the coherent spin states satisfy, in the classical limit, the energy-phase minimum-uncertainty relations here introduced.     

Projection operators and supplementary conditions for superfields with arbitrary spin

It is shown that a superfield with an external spin j and a non-vanishing mass contains four irreducible representations of the supersymmetry algebra. A general method is proposed for extracting these irreducible multiplets out of the superfield using projection operators constructed with the help of Casimir operators. The same decomposition is expressed in terms of supplementary differential conditions.     

Operators of observables for a neutral particle with anomalous magnetic moment in an electromagnetic field

The explicit form of operators of kinetic momenta and spin projection for a neutral particle with an anomalous magnetic moment in constant homogeneous electromagnetic field is found. The possible applications of the obtained results in neutrino physics are considered.     

Quantum Supergroups V. Braid Group Action

We construct a braid group action on quantum covering groups. We further use this action to construct a PBW basis for the positive half in finite type which is pairwise-orthogonal under the inner product. This braid group action is induced by operators on the integrable modules; however, these operators satisfy spin braid relations.     

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.     

Irrelevant operators and momentum-shell recursion relations ind=2 + ∈ dimensions

The momentum-shell recursion relations of Nelson and Pelcovits for ann-vector model near two dimensions are reexamined. The renormalization of the infinite set of relevant and marginal operators present in the system is studied. Ambiguities obtained in the ensuing recursion relations are shown to involve irrelevant operators only, thus justifying the procedure of Nelson and Pelcovits. The cases of finite external fieldh and finite spin anisotropyg are both considered.     

The magnetic properties of weakly doped layered copper oxides: Relaxation function in the model of strongly correlated charge carriers

The method of projection operators is applied to the two-dimensional model of strongly correlated charge carriers to explain the magnetic properties of weakly doped layered cuprates in the paramagnetic state. The theory explains the observed special features of the behavior of the imaginary part of the dynamic spin susceptibility averaged over the Brillouin zone over wide temperature and frequency ranges.     

Collective coordinate theory for light pulses in fibers: The reduced projection operators

We consider optical waveguides in which light pulses may execute complex dynamical behaviors including translational motions accompanied with strong internal vibrations. Such systems necessarily generate various types of collective motion, in which each collective mode is describable by a collective coordinate. We present a novel projection operator formalism for deriving the equations of motion of the collective coordinates and coupled fields. This formalism is built up by treating separately the dynamics of the pulse phase and that of its amplitude, that is, by using two distinct projection operators (one for the amplitude, and the other one for the phase). This new pair of operators, which we call reduced projection operators, has as main virtue of having dimension reduced by half as compared to that of the conventional operators that includes both pulse amplitude and phase together. The main interest of the reduced projection operators lies in the ease with which the equations of motion are derived when compared with the amount of algebra needed to obtain the same equations from the conventional projection operators. We provide examples of concrete situations that illustrate the effectiveness of the collective-coordinate approach based on the reduced projection operators.     

Bi-local baryon interpolating fields with two flavors

We construct bi-local interpolating field operators for baryons consisting of three quarks with two flavors, assuming good isospin symmetry. We use the restrictions following from the Pauli principle to derive relations/identities among the baryon operators with identical quantum numbers. Such relations that follow from the combined spatial, Dirac, color, and isospin Fierz transformations may be called the (total/complete) Fierz identities. These relations reduce the number of independent baryon operators with any given spin and isospin. We also study the Abelian and non-Abelian chiral transformation properties of these fields and place them into baryon chiral multiplets. Thus we derive the independent baryon interpolating fields with given values of spin (Lorentz group representation), chiral symmetry (U _L (2)×U _R (2) group representation) and isospin appropriate for the first angular excited states of the nucleon.     

Massless equation for an antisymmetric tensor field

Starting from the formalism of covariant spin projection operators we present a general derivation and analysis of massless equations with zero helicity for an antisymmetric tensor field. We show that the minimal electromagnetic interaction for gauge-invariant equations is inconsistent.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 95–99, September, 1987.     

Generalized nonadditive entropies and quantum entanglement

We examine the inference of quantum density operators from incomplete information by means of the maximization of general nonadditive entropic forms. Extended thermodynamic relations are given. When applied to a bipartite spin 1 / 2 system, the formalism allows one to avoid fake entanglement for data based on the Bell-Clauser-Horne-Shimony-Holt observable, and, in general, on any set of Bell constraints. Particular results obtained with the Tsallis entropy and with an introduced exponential entropic form are also discussed.     

A path integral to quantize spin

We present a model for a classical spinning particle, characterized by spin magnitude, arbitrary but fixed, and continuously varying direction. A gauge freedom of the model reflects the choice of canonical coordinates in the phase space, which is spherical. We formulate the path integral for the model and find, unexpectedly, that the phase space must be punctured at the poles. It then follows that both the total spin and spin projection along any axis are quantized. The model has rotational invariance and yields the usual quantum mechanics of spin, including commutation relations, in a simple way.     

Relativistic Spin Operators

A systematic theory on the appropriate spin operators for the relativistic states is developed.For a massive relativistic particle with arbitrary nonzero spin,the spin operator should be replaced with the relativistic one,which is called in this paper as moving spin.Further the concept of moving spin is discussed in the quantum field theory.A new operator,field quanta spin is defined and in terms of the generators of Poincare group the moving spin of field system is constructed.It is shown that,in virtue of the two operators,problems in quantum field concerned spin can be neatly settled.