Terms Of Spinors Research Articles

Bundle Theory of Improper Spin Transformations

We give a geometrical description of the action of the parity operator (\(\hat{P}\)) on non-relativistic spin 1/2 Pauli spinors in terms of bundle theory. The relevant bundle, \(SU(2)\odot Z_2\to O(3)\), is a non-trivial extension of the universal covering group \(\hbox{\it SU}(2)\to \hbox{\it SO}(3). \hat{P}\) is the non-relativistic limit of the corresponding Dirac matrix operator \({\cal P}=i\gamma_0\) and obeys \(\hat{P}^2=-1\). From the direct product of O(3)$ by \(Z_2\), naturally induced by the structure of the Galilean group, we identify, in its double cover, the time-reversal operator (\(\hat{T}\)) acting on spinors, and its product with \(\hat{P}. \hat{P}\) and \(\hat{T}\) generate the group \(Z_4 \times Z_2\). As in the case of parity, \(\hat{T}\) is the non-relativistic limit of the corresponding Dirac matrix operator \({\cal T}=\gamma^3 \gamma^1\), and obeys \(\hat{T}^2=-1\).

How to generate families of spinors

Using a technique [Mankoč Borštnik et al., J. Math. Phys. 43, 5782 (2002)] to construct a basis for spinors and “families” of spinors in terms of Clifford algebra objects, we define other Clifford algebra objects, which transform the state of one “family” of spinors into the state of another “family” of spinors, changing nothing but the “family” number. The proposed transformation works—as does the technique—for all dimensions and any signature and might open a path to understanding families of quarks and leptons [Mankoč Borštnik, Phys. Lett. B 292, 25 (1992); J. Math. Phys. 34, 3731 (1993); Int. J. Theor. Phys. 40, 315 (2001); Borštnik Bračič and Mankoč Borštnik, hep-ph/0301029].

Clifford algebras obtained by twisting of group algebras

We investigate the construction and properties of Clifford algebras by a similar manner as our previous construction of the octonions, namely as a twisting of group algebras of Z 2 n by a cocycle. Our approach is more general than the usual one based on generators and relations. We obtain, in particular, the periodicity properties and a new construction of spinors in terms of left and right multiplication in the Clifford algebra.

Generalized Weierstrass representation for surfaces in terms of Dirac-Hestenes spinor field

A representation of generalized Weierstrass formulae for an immersion of generic surfaces into a 4-dimensional complex space in terms of spinors treated as minimal left ideals of Clifford algebras is proposed. The relation between integrable deformations of surfaces via mVN-hierarchy and integrable deformations of spinor fields on the surface is also discussed.

Geometro-stochastically quantized fields with internal spin variables

The use of internal variables for the description of relativistic particles with arbitrary mass and spin in terms of scalar functions is reviewed and applied to the stochastic phase space formulation of quantum mechanics. Following Bacry and Kihlberg a four-dimensional internal spin space S̄ is chosen possessing an invariant measure and being able to represent integer as well as half integer spins. S̄ is a homogeneous space of the group SL(2,C) parametrized in terms of spinors α∈C2 and their complex conjugates ᾱ. The generalized scalar quantum mechanical wave functions may be reduced to yield irreducible components of definite physical mass and spin [m,s], with m⩾0 and s=0,12,1,32,… , with spin described in terms of the usual (2s+1)-component fields. Viewed from the internal space description of spin this reduction amounts to a restriction of the variable α to a compact subspace of S̄, i.e., a “spin shell” Sr=2s2 of radius r=2s in C2. This formulation of single particles or single antiparticles of type [m,s] is then used to study the geometro-stochastic (i.e., quantum) propagation of amplitudes for arbitrary spin on a curved background space–time possessing a metric and axial vector torsion treated as external fields. A Poincaré gauge covariant path integral-like representation for the probability amplitude (generalized wave function) of a particle with arbitrary spin is derived satisfying a second order wave equation on the Hilbert bundle constructed over curved space–time. The implications for the stochastic nature of polarization effects in the presence of gravitation are pointed out and the extension to Fock bundles of bosonic and fermionic type is briefly mentioned.

Quantization of magnetic top in the spinor representation

Classical magnetic top described in terms of spinors is quantized by applying Dirac's method of quantization for systems with constraints.

Spin densities in pseudoclassical kinetic theory

It is shown that pseudoclassical many-particle systems allow for observable consequences of the spin degrees of freedom, in contrast to the one-particle system, where a consistent formulation of spin space density is impossible. After quantization, this density can then be directly compared to the decomposition of the Wigner operator in terms of spin matrices, in both the nonrelativistic and the relativistic context. From an expansion of the Dirac spinors in terms of large and small components it is seen that in the nonrelativistic limit the pseudoscalar, the timelike component of the axial-vector current, and the spatial components of the vector current vanish. The spatial components of the axial-vector current vanish in the pseudoclassical limit. The pseudoclassical appearance of spin is due to the spin-tensor contribution.

Compact expression for a bilinear combination of Dirac spinors in terms of world tensors

A compact expression is obtained for the product of two Dirac spinors in the form of a linear combination of 16 Dirac matrices. A convenient form of expression of the results, permitting its use for analytic calculations of the amplitudes of many-particle processes, is found. It is shown that a bilinear combination of Dirac spinors can be expressed in terms of the 4-momentum vectors and spins of the corresponding particles together with a common phase factor.

Spinors over a Cone, Dirac Operator, and Representations of Spin(4, 4)

The representations of Spin(4, 2) e seem to be of particular physical interest since its quotient SO(4, 2) e is the conformal group of the spacetime. Kostant [7] has considered the Laplacian on a projective cone in R 8 and has shown that the kernel H of the Laplacian is an irreducible unitary representation of SO(4, 4) e . Moreover, it is known [9] that there is a Howe pair ( SO(2), SO(4, 2) e ) in SO(4, 4) e and that the weight spaces of SO(2) in H are irreducible representations of SO(4, 2) e . He conjectured that the analogous facts should hold for the Dirac operator acting on the spinor fields on S 3 × S 3 ⊆ R 8. We introduce a hitherto unknown embedding of the bundle of spinors on S 3 × S 3 in the bundle of spinors on R 8. This embedding makes it possible to obtain a particularly convenient expression for the Dirac operator on S 3 × S 3 in terms of spinors and vector fields on the underlying R 8 in particular the conformal invariance is manifest. We explicitly describe the kernel F of the restriction of the Dirac operator to even spinor fields and show that it is an irreducible representation of Spin(4, 4) e . We also show that it has an invariant sesquilinear scalar product (with is not, however, positive definite). The existence of this scalar product can also be established by showing that F ⊆ S + ⊗ H where S + the vector space of even spinors. We identify within Spin(4, 4) e a Howe pair consisting of SO(2) and Spin(4, 2) e and show that the weight spaces of SO(2) in F are irreducible representations of Spin(4, 2) e .

Density matrix of relativistic nuclear matter.

We investigate the structure of the density matrix of relativistic nuclear matter at large momentum (k\ensuremath{\gg}${\mathit{k}}_{\mathit{F}}$) making use of the \ensuremath{\sigma}+\ensuremath{\omega} model of Walecka. The density matrix may be written in terms of spinors describing nucleons of modified mass and also has exotic components that contain spinors describing negative-energy states (``antinucleons''). We calculate the modification of the mean-field density matrix due to the admixture of two-particle, two-hole states in the ground state of the relativistic theory and find that the excitation of the exotic components of the density matrix is so strong and coherent as to preclude the use of perturbation theory. We suggest that an expansion in terms of reaction matrices will not improve the situation and conclude that a consistent picture may be obtained only if one limits oneself to a space spanned by positive-energy spinors (describing nucleons of shifted mass). We also consider the exchange of pions in the calculation of high-momentum components of the density matrix. However, in the case of pion exchange, we find that sensible results may not be obtained (even for the nonexotic components) unless we include tensor and short-range correlations. (The latter calculations have not been performed as yet.) Further, we note that a theory without vertex cutoffs (meson-nucleon form factors) leads to totally unacceptable results, as the depletion of the Fermi sea is again much too large.

Geometrical unified theory of connection fields and frame fields.

A theory of connections on a principal bundle with structure group SL(4,R), the highest-dimensional simple subgroup of the Clifford algebra associated with a space-time of signature (1,-1,-1,-1), is used to interpret Dirac's equation geometrically. The proposed theory unifies Dirac's equation with the equations of gravitation and electrodynamics. The Lorentz equation of motion for charges is a consequence of the field equations when the connection reduces to a component which may be identified as the electromagnetic field. If this part of the generalized connection does not vanish, we obtain the correct expression for the electric current in terms of spinors and the correct minimal coupling in the first nonrelativistic approximation.

The Differential Approach to Spinors and Their Symmetries

A formulation of spinor analysis in space-time is given in terms of smooth sections of a real Clifford bundle. Its relation to the two-component complex calculus for spinor components is elucidated. Treating spinors in terms of inhomogeneous differential forms carryingPIN 3,1 andSPIN 3,1 representations enables the discrete covariances of the Maxwell-Dirac system to be induced naturally from smooth isometries of the space-time metric. Attention is drawn to the distinction between the Dirac and Kahler equations in curved space when expressed in this geometric formulation.

The differential approach to spinors and their symmetries

A formulation of spinor analysis in space-time is given in terms of smooth sections of a real Clifford bundle. Its relation to the two-component complex calculus for spinor components is elucidated. Treating spinors in terms of inhomogeneous differential forms carryingPIN 3,1 andSPIN 3,1 representations enables the discrete covariances of the Maxwell-Dirac system to be induced naturally from smooth isometries of the space-time metric. Attention is drawn to the distinction between the Dirac and Kähler equations in curved space when expressed in this geometric formulation.

Arbitrary spin fermions on the lattice

Lattice actions are constructed for free Dirac and Majorana fermions of arbitrary (halfinteger) spin using various extensions of the spin 1 2 Kogut-Susskind, Kähler and Wilson formalisms. In each case, the spectrum degeneracy and preservation of gauge invariance is analysed, and the equivalence or non-equivalence to previously constructed actions is determined. The Kogut-Susskind and lattice Kähler actions are then written explicitly in terms of spinors to demonstrate how the degenerate fermions couple on the lattice and how the original spinorial actions are recovered (or to recovered) in the continuum limit. Both degenerate and non-degenerate mass terms are dealt with and the various U(1) invariances of the lattice actions are pointed out.

Electromagnetic and gravitational Hertz potentials

With generality of complex relativity, the classical theory of the electromagnetic Hertz potentials is outlined in terms of spinors and forms. Particularly interesting are D (0,1) and D (1,0) null Hertz potentials. Then, a new spinorial approach to heavens (ℋ spaces) is proposed, which reveals in their structure the presence of the left null gravitational Hertz potential [of the type D (0,2)]. The relevant hints which follow from our results and concern the structure of the most general solutions of the Einstein vacuum equations (type G⊗G), are discussed, in particular on the level of the linearized theory.

Invariants and classification of gauge fields

A classification of gauge fields in terms of spinors and the rank of a matrix of Lorentz invariants is proposed. A peeling theorem for the Yang-Mills field in the asymptotic limit is shown to follow from an earlier work of Roskies. An analogous theorem is proved for all gauge fields in the geometric optics limit.

Some basic properties of Killing spinors

The concept of Killing spinor is analyzed in a general way by using the spinorial formalism. It is shown, among other things, that higher derivatives of Killing spinors can be expressed in terms of lower order derivatives. Conformal Killing vectors are studied in some detail in the light of spinorial analysis: Classical results are formulated in terms of spinors. A theorem on Lie derivatives of Debever–Penrose vectors is proved, and it is shown that conformal motion in vacuum with zero cosmological constant must be homothetic, unless the conformal tensor vanishes or is of type N. Our results are valid for either real or complex space–time manifolds.

A characterization of the Einstein tensor in terms of spinors

All tensors of contravariant rank two which are divergence-free on one index, concomitants of a spinor field σiAX′ together with its first two partial derivatives, and scalars under spin transformations are constructed. The Einstein and metric tensors are the only candidates.

Electroproduction of higher spin resonances within a generalized Joss-Weinberg formalism

Electromagnetic currents for transitions from spin s to another spin s' (s' is an arbitrary irteger for s = 0 and an arbitrary halfinteger for s = 1/2) are presented in terms of spinors satisfying Weinbergs equation and generalized 2(2s + 1) x 2(2s' + 1) JoosWeinberg matrices. When the currents are transformed to the Breit frame, they involve terms which are interpreted as form factors for different electromagnetic multipole transitions. Assuming current-current coupling with one -photon exchange, the unpolarized cross section for high energy electron scattering by spin-s target (s = 0, 1/2) at electroproduction of a spin- s' resonance is obtained. (LBS)

Flat-space metric in the quaternion formulation of general relativity

Using the quaternion representation of the Riemannian space-time of general relativity it is formulated the description of the gravitational field in flat space. The method presently used is an extension for the quaternion fields of a formalism introduced by Rosen. The theorem of quasi-inertia is established. The equations of the gravitational field are obtained in terms of spinors which are function of quaternions defined on the flat space. The equation of motion of a test particle is also obtained.