Should There Be a Spin-Rotation Coupling for a Dirac Particle?

It was argued by Mashhoon that a spin-rotation coupling term should add to the Hamiltonian operator in a rotating frame, as compared with the one in an inertial frame. For a Dirac particle, the Hamiltonian and energy operators H and E in a given reference frame were recently proved to depend on the tetrad field. We argue that this non-uniqueness of H and E really is a physical problem. We show that a tetrad field contains two informations about local rotation, which usually do not coincide. We compute the energy operator in the inertial and the rotating frame, using three different tetrad fields. We find that Mashhoon’s term is there if the spatial triad rotates as does the reference frame—but then it is also there in the energy operator for the inertial frame. In fact, if one uses the same given tetrad field, the Dirac Hamiltonian operators in two reference frames in relative rotation differ only by the angular momentum term. If the Mashhoon effect is to exist for a Dirac particle, the tetrad field must be selected in a specific way for each reference frame.     

A non‐uniqueness problem of the Dirac theory in a curved spacetime

The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under which a change of the coefficient fields leads to an equivalent Hamiltonian operator H, or to an equivalent energy operator E. We do that for the standard version of the gravitational Dirac equation, and for two alternative equations based on the tensor representation of the Dirac fields. The latter equations may be defined when the spacetime is four‐dimensional, noncompact, and admits a spinor structure. We find that, for each among the three versions of the equation, the vast majority of the possible coefficient changes do not lead to an equivalent operator H, nor to an equivalent operator E, whence a lack of uniqueness. In particular, we prove that the Dirac energy spectrum is not unique. This non‐uniqueness of the energy spectrum comes from an effect of the choice of coefficients, and applies in any given coordinates.     

Locally inertial reference frames in Lorentzian and Riemann‐Cartan spacetimes

In this paper the concept of locally inertial reference frames ( LIRF s) in Lorentzian and Riemann‐Cartan spacetime structures is scrutinized. A rigorous mathematical definition of a LIRF in both structures is given, something that needs preliminary a clear mathematical distinction between the concepts of observers, reference frames, naturally adapted coordinate functions to a given reference frame and which properties may characterize an inertial reference frame (if any) in the Lorentzian and Riemann‐Cartan structures. Hopefully, the paper clarifies some obscure issues associated to the concept of a LIRF appearing in the literature, in particular the relationship between LIRF s in Lorentzian and Riemann‐Cartan spacetimes and Einstein's most happy thought, i.e., the equivalence principle.     

On Properties of Hamiltonian Structures for a Class of Evolutionary PDEs

In a recent paper we proved that for certain class of perturbations of the hyperbolic equation u _t = f (u)u _x , there exist changes of coordinate, called quasi-Miura transformations, that reduce the perturbed equations to the unperturbed one. We prove in the present paper that if in addition the perturbed equations possess Hamiltonian structures of certain type, the same quasi-Miura transformations also reduce the Hamiltonian structures to their leading terms. By applying this result, we obtain a criterion of the existence of Hamiltonian structures for a class of scalar evolutionary PDEs and an algorithm to find out the Hamiltonian structures.      

Spacetime Quantization, Elementary Particles, and Cosmology

Relativistic quantum mechanics is generalized to account for a universally constant quantum of length a. Its value depends on the total convertible energy content of our universe: E_u = hc/2a. The eigenvalues of all (x,y,z,ct) coordinates are integer or half-integer multiples of a in every particular inertial frame. There are thus several spacetime lattices of lattice-constant a: the normal lattice contains the origin of the chosen frame, while inserted lattices are displaced by a/2 along one or several reference axes. States of motion are defined by possible variations of -functions on any one of these lattices. Particle states are defined by their relative phases, specified by four new quantum numbers, u_x, u_y, u_z, u_ct = 0, ±1, ±2,.... They account for all known elementary particles and yield a natural extension of the standard model. Spacetime quantization solves also the EPR paradox and other difficulties that subsisted in the usual continuum theories. It defines inertial frames and is related to cosmology.     

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.     

Relativistic Landau‐He‐McKellar‐Wilkens quantization induced by noninertial effects

We show that the relativistic analogue of the Landau‐He‐McKellar‐Wilkens quantization can be achieved through the noninertial effects of the Fermi‐Walker reference frame without assuming the existence of a magnetic charge density and discuss the nonrelativistic limit of the energy levels. We also obtain the Dirac spinors for positive‐energy values parallel and antiparallel to the z axis of the spacetime and obtain the Gordon decomposition of the Dirac probability current J^μ.     

Review: Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge

We define the rest-frame instant form of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find a quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges. The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.     

Can Dirac quantization of constrained systems be fulfilled within the intrinsic geometry?

For particles constrained on a curved surface, how to perform quantization within Dirac’s canonical quantization scheme is a long-standing problem. On one hand, Dirac stressed that the Cartesian coordinate system has fundamental importance in passing from the classical Hamiltonian to its quantum mechanical form while preserving the classical algebraic structure between positions, momenta and Hamiltonian to the extent possible. On the other, on the curved surface, we have no exact Cartesian coordinate system within intrinsic geometry. These two facts imply that the three-dimensional Euclidean space in which the curved surface is embedded must be invoked otherwise no proper canonical quantization is attainable. In this paper, we take a minimum surface, helicoid, on which the motion is constrained, to explore whether the intrinsic geometry offers a proper framework in which the quantum theory can be established in a self-consistent way. Results show that not only an inconsistency within Dirac theory occurs, but also an incompatibility with Schrödinger theory happens. In contrast, in three-dimensional Euclidean space, the Dirac quantization turns out to be satisfactory all around, and the resultant geometric momentum and potential are then in agreement with those given by the Schrödinger theory.     

F‐theory and unpaired tensors in 6D SCFTs and LSTs

We investigate global symmetries for 6D SCFTs and LSTs having a single “unpaired” tensor, that is, a tensor with no associated gauge symmetry. We verify that for every such theory built from F‐theory whose tensor has Dirac self‐pairing equal to −1, the global symmetry algebra is a subalgebra of . This result is new if the F‐theory presentation of the theory involves a one‐parameter family of nodal or cuspidal rational curves (i.e., Kodaira types I₁ or II) rather than elliptic curves (Kodaira type I₀). For such theories, this condition on the global symmetry algebra appears to fully capture the constraints on coupling these theories to others in the context of multi‐tensor theories. We also study the analogous problem for theories whose tensor has Dirac self‐pairing equal to −2 and find that the global symmetry algebra is a subalgebra of . However, in this case there are additional constraints on F‐theory constructions for coupling these theories to others.     

The Analysis of Lagrangian and Hamiltonian Properties of the Classical Relativistic Electrodynamics Models and Their Quantization

The Lagrangian and Hamiltonian properties of classical electrodynamics models and their associated Dirac quantizations are studied. Using the vacuum field theory approach developed in (Prykarpatsky et al. Theor. Math. Phys. 160(2): 1079–1095, 2009 and The field structure of a vacuum, Maxwell equations and relativity theory aspects. Preprint ICTP) consistent canonical Hamiltonian reformulations of some alternative classical electrodynamics models are devised, and these formulations include the Lorentz condition in a natural way. The Dirac quantization procedure corresponding to the Hamiltonian formulations is developed. The crucial importance of the rest reference systems, with respect to which the dynamics of charged point particles is framed, is explained and emphasized. A concise expression for the Lorentz force is derived by suitably taking into account the duality of electromagnetic field and charged particle interactions. Finally, a physical explanation of the vacuum field medium and its relativistic properties fitting the mathematical framework developed is formulated and discussed.     

Treatment of the intrinsic Hamiltonian in particle-number nonconserving theories

We discuss the implications of using an intrinsic Hamiltonian in theories without particle-number conservation, e.g., the Hartree–Fock–Bogoliubov approximation, where the Hamiltonian's particle-number dependence leads to discrepancies if one naively replaces the particle-number operator by its expectation value. We develop a systematic expansion that fixes this problem and leads to an a posteriori justification of the widely-used one- plus two-body form of the intrinsic kinetic energy in nuclear self-consistent field methods. The expansion's convergence properties as well as its practical applications are discussed for several sample nuclei.     

The large-amplitude motions in quasi-symmetric top molecules with internal C3v rotors: The vibration-torsion-rotation Hamiltonian of a two-top molecule

An expression for the vibration-torsion-rotation Hamiltonian of a molecule with triatomic nonrigid frame and two internal C_3v rotors has been derived. Three large-amplitude motions, namely skeletal bending and two torsions, are removed from the vibrational problem and are considered together with the rotational problem. The Hamiltonian obtained is applicable to any two-C_3v-top molecule with triatomic linear or bent frame. The zeroth-order skeletal bending-torsion-rotation Hamiltonian is derived and the method of solving the corresponding Schrödinger equation is discussed. The Hamiltonian obtained with only slight modifications is shown to be applicable to any single-C_3v-top molecule with a quasi-linear tetratomic nonrigid frame or to the problem of the large-amplitude bending motion in a pentatomic quasi-linear molecule.     

On the Dimensions of the Chiral and Dilatation Symmetry Breaking Hamiltonian Densities

Denoting by l_u and l_δ the dimensions (if any) of the chiral and dilation symmetry breaking Hamiltonian densities u(x) and δ(x), respectively, the property (4 – l_u) (l_u – l_δ) > 0 (which is already known in special cases) is derived for a much larger class of models. We furthermore obtain a simple and general explicit expression for u(x) in terms of the current divergences, and review the physical applications of this (or an analogous, almost obvious) expression.     

Spin-rotation coupling and Fermi–Walker transport

It is demonstrated that a spin-rotation coupling term only appears in the Hamiltonian of a Dirac particle if the rotating frame obeys the rules of Fermi–Walker transport. This is illustrated by examples in Minkowski space and the paper concludes with some speculations about spin-rotation coupling in Kerr space-time.     

Landau electron in a rotating environment: A general factorization of time evolution

For the Landau problem with a rotating magnetic field and a confining potential in the (changing) direction of the field, we derive a general factorization of the time evolution operator that includes the adiabatic factorization as a special case. The confining potential is assumed to be of a general form and it can correspond to nonlinear Heisenberg equations of motion. The rotation operator associated with the solid angle Berry phase is used to transform the problem to a rotating reference frame. In the rotating reference frame, we derive a natural factorization of the time evolution operator by recognizing the crucial role played by a gauge transformation. The major complexity of the problem arises from the coupling between motion in the direction of the magnetic field and motion perpendicular to the field. In the factorization, this complexity is consolidated into a single operator which approaches the identity operator when the potential confines the particle sufficiently close to a rotating plane perpendicular to the magnetic field. The structure of this operator is clarified by deriving an expression for its generating Hamiltonian. The adiabatic limit and non-adiabatic effects follow as consequences of the general factorization which are clarified using the magnetic translation concept.     

The Invariant-Relat ed Unitary Transformation Method and the Evolution of a Dirac Field in Friedmann-Robertson-Walker Flat Spacetimes

On the basis of the generalized invariant formulation, the invariant-related unitary transformation method is used to study the evolution of a quantum Dirac field in Friedmann-Robertson-Walker spatially flat spacetime. We first solve the functional Schrodinger equation for a free Dirac field and obtain the exact solutions. We then investigate the way of extending the method to treat the case in which there is an interaction between the Dirac field and a scalar field.     

Many fields interaction: Beam splitters and waveguide arrays

We study the interaction of many fields. We obtain an effective Hamiltonian for this system by using a method recently introduced that produces a small rotation to the Hamiltonian that allows to neglect some terms in the rotated Hamiltonian. We show that coherent states remain coherent under the action of a quadratic Hamiltonian and by solving the eigenvalue and eigenvector problem for tridiagonal matrices we also show that a system of n interacting harmonic oscillators, initially in coherent states, remain coherent during the interaction.     

Coulomb-Like Interactions in the Dirac Equation

The Dirac equation for the Coulomb-like problem is modified by incorporating minimal interactions into the Dirac Hamiltonian, that keep the 1/r potential dependence. We determine the general energy eigenvalues and the corresponding eigenfunctions.     

Search for heavy sterile neutrinos in trileptons at the LHC

We present a search strategy for both Dirac and Majorana sterile neutrinos from the purely leptonic decays of W~±→e~±e~±μ~?ν and μ~±μ~± e~?ν at the 14 TeV LHC. The discovery and exclusion limits for sterile neutrinos are shown using both the Cut-and-Count(CC) and Multi-Variate Analysis(MVA) methods. We also discriminate between Dirac and Majorana sterile neutrinos by exploiting a set of kinematic observables which differ between the Dirac and Majorana cases. We find that the MVA method, compared to the more common CC method, can greatly enhance the discovery and discrimination limits. Two benchmark points with sterile neutrino mass m N =20 GeV and 50 GeV are tested. For an integrated luminosity of 3000 fb~(-1), sterile neutrinos can be found with 5σ significance if heavy-to-light neutrino mixings |U_(Ne)|~2~|U_(Nμ)|~2~10~(-6), while Majorana vs. Dirac discrimination can be reached if at least one of the mixings is of order 10~(-5).