Relativistic hamiltonian equations for any spin

We discuss 2(2J + 1)-component Poincaré-invariant Hamiltonian theories that describe free particles of definite mass and spin and that are subject to the conditions (a) every observable $\text{O}$ is either Hermitian or pseudo-Hermitian (i.e., O = ?₃O⁺?₃) and (b) the theory is invariant under the discrete symmetries. Our treatment is based on the Heisenberg equations of motion and on the Lie algebra of the Poincaré group. Explicit formulas are found for the generators of this algebra, including the Hamiltonian H, and all relations between the operators Γ and H that are both necessary and sufficient for $\text{K =}\text{1}\text{2}\text{[x}\text{, H]}{}_{\mathrm{+}}\text{+ Γ}$ to generate Lorentz boosts are found. To illustrate the utility of our results, we apply them to obtain explicit generalizations of the Dirac equation to any spin, by requiring that Γ = 0, and of the Sakata-Taketani spin-0 and spin-1 equations to any spin, by requiring that $\text{Γ = ??}{}_3\text{(}\text{1}\text{2m}\text{)}\text{S × p}$.     

Spacetime path formalism for massive particles of any spin

Earlier work presented spacetime path formalism for relativistic quantum mechanics arising naturally from the fundamental principles of the Born probability rule, superposition, and spacetime translation invariance. The resulting formalism can be seen as a foundation for a number of previous parametrized approaches to relativistic quantum mechanics in the literature. Because time is treated similarly to the three-space coordinates, rather than as an evolution parameter, such approaches have proved particularly useful in the study of quantum gravity and cosmology. The present paper extends the foundational spacetime path formalism to include massive, non-scalar particles of any (integer or half-integer) spin. This is done by generalizing the principle of translational invariance used in the scalar case to the principle of full Poincaré invariance, leading to a formulation for the non-scalar propagator in terms of a path integral over the Poincaré group. Once the difficulty of the non-compactness of the component Lorentz group is dealt with, the subsequent development is remarkably parallel to the scalar case. This allows the formalism to retain a clear probabilistic interpretation throughout, with a natural reduction to non-relativistic quantum mechanics closely related to the well-known generalized Foldy-Wouthuysen transformation.     

Higher spin gauge theories in any dimension

Some general properties of higher spin gauge theories are summarized, with the emphasize on the nonlinear theories in any dimension. To cite this article: M.A. Vasiliev, C. R. Physique 5 (2004).

Résumé

Des propriétés générales des théories de jauge de grands spins sont présentées, en insistant particulièrement sur les théories non-linéaires en dimensions diverses. Pour citer cet article : M.A. Vasiliev, C. R. Physique 5 (2004).     

Do Harrison transformations possess any physical meaning?

We propose to give a physical interpretation of the Harrison transformations, so far considered merely as generation techniques of exact solutions of the Einstein-Maxwell equations, as the mechanism which makes a certain class of electrovacuum test fields (Papapetrou fields) gravitate themselves.     

Operator oriented optimization of block spin transformations

We propose a generally applicable method to optimize block spin transformations. Working in the sector of even operators the renormalization transformation is adjusted such that the resulting renormalized trajectory is close to the trajectory of a simple few-parameter hamiltonian. The idea is tested within the d = 2 Ising model and leads to clear improvements in the determination of the critical exponents.     

Unitary transformations with spin angular momentum operators

Useful formulae for replacing an exponentiated spin scalar product by a finite sum of powers of this product, are presented in the context of Coulomb interactions between magnetic ions and conduction electrons in metals.     

Consistent real-time propagators for any spin,mass, temperature and density

The time-path method in finite temperature field theory is extended to arbitrary covariant fields. Explicit expressions for the free thermal propagators are obtained using the multi-mass Klein-Gordon divisor. The key formula which shows that the interacting theory is free of singularities is derived. Finally, a simple method for the determination of free massless propagators is given.     

A complete Hermitian operator basis set for any spin quantum number.

A new Hermitian operator basis set for spins of any quantum number is presented for use in simulations of NMR experiments. The advantage with a Hermitian operator basis is that the Liouville-von Neumann equation, including relaxation with dynamic frequency shifts, is real. Real algebra makes numerical calculations faster and simplifies physical interpretation of the equation system as compared to complex algebra. The unity operator is included in the Hermitian operator basis, which makes it easy to rewrite the inhomogeneous Liouville-von Neumann equation into a homogeneous form. The unity operator also simplifies physical interpretation of the equation system for coupled spin systems.     

Validation of the Eriksen method for the exact Foldy-Wouthuysen representation

The Eriksen method is proven to yield a correct and exact result when a sufficient condition of exact transformation to the Foldy-Wouthuysen (FW) representation is satisfied. Therefore, the Eriksen method is confirmed as valid. This makes it possible to establish the limits within which the approximate “step-by-step” methods are applicable. The latter is done by comparing the relativistic formulas for a Hamiltonian operator in FW representation (obtained using those methods) and the known expression for the first terms of a series, which defines the expansion of this operator in powers of v/c as found by applying the Eriksen method.     

Calculation of Higher-Order Foldy-Wouthuysen Transformation Hamiltonian

The Foldy-Wouthuysen Hamiltonian of a light atomic system that has an ma^8 contribution to energy levels is calculated. The case of a Dirac-Coulomb field is discussed. The results can be used for relativistic and radiative corrections to energy levels in the low-energy part. A divergent operator δ^2（r） emerges. This is probably due to the nature of the point-like charge source. The effective method of radiation calculation may be re-checked.     

Rigorous bounds for critical temperatures of O(N) spin models by transformations of block spin type

We obtain new upper bounds of critical temperatures of N-vector (Heisenberg) models. We apply a transformation of block spin type to random walk representations of the spin models, which was developed by Fröhlich et al. more than a decade ago. Though the transformation is applied just one time, the upper bounds are considerably improved.     

Resonances of any spin in pion photoproduction and nucleon Compton scattering

Describing the vertices by form factors and the propagators by their spectral representations, we calculate the contribution of the exchange diagramsγπ→M ^j→ ¯N N andγN→N ^j+1/2→γN to the invariant and helicity amplitudes, for meson (M _j) and baryon (N ^j+1/2) resonances of any spin (j integer ≧ 1) and both parities ((?)^j and (?)^j+1).     

The (BFKL) Pomeron-\gamma^*-\gamma vertex for any conformal spin

To study diffractive photon production at HERA, we compute the projection of the impact-factor on the BFKL leading-order eigenfunctions for non-zero transfer. This calculation supplements former ones performed for n = 0. We provide an expression for and check that all the other components are zero. Received: 6 September 1999 / Revised version: 21 November 1999 / Published online: 17 March 2000