arXiv subjects

Patrick L. Nash

Publications and source records attributed to Patrick L. Nash.

5 recordsLinked to original sources

arXiv2015

A Remarkable New Identity Satisfied by the Dirac Matrices of a Bilocal Field Theory

In 1925 Elie Cartan described `triality' \cite{CARTAN25}, \cite{CARTAN} as a symmetry between SO$(8; \mathbb{C})$ vectors and the two types of Spin$(8; \mathbb{C})$ spinor. It is known that the reduced generators of the Clifford algebra $\mathbb{C}_{8}$ defined on the real, eight-dimensional Euclidean space $\mathbb{E}_{8}$ satisfy an identity that guarantees the existence of matrix representations (acting on the vector and spinor bundles of $\mathbb{E}_{8}$) of triality. Analogously, let $\mathbb{E}_{4,4}$ denote a real eight-dimensional pseudo-Euclidean vector space that is endowed with an indefinite inner product with signature $(+,+,+,-\,;\,-,-,-,+)$. As a normed vector space, $\mathbb{E}_{4,4} \cong M_{3,1} \times {}^{*}\!M_{3,1}$, where $M_{3,1}$ and ${}^{*}\!M_{3,1}$ denote real four-dimensional Minkowski spacetimes, with opposite signatures. %Clearly, bilocal Minkowski field theories may be cast on the $\mathbb{E}_{4,4}$ spacetime. The reduced generators (i.e., the Dirac matrices) of the pseudo Clifford algebra $\mathbb{C}_{4,4}$ defined on $\mathbb{E}_{4,4}$ satisfy an identity $\,$ \cite{NASH86} $\,,\,$ \cite{NASH90} that guarantees the existence of invertible linear mappings between each of the two types of $\overline{S0(4,4; \mathbb{R})}$ spinor and the ${S0(4,4; \mathbb{R})}$ vector, thereby realizing matrix representations of triality that act on the vector and spinor bundles of the spacetime $\mathbb{E}_{4,4}$. In this note we generalize this identity (see Eq.[\ref{newIdentity}]).

gr-qc↗

arXiv2015

Pseudo-Euclidean Gravity

A new theory of a (flat) spacetime gravitational interaction is presented. This theory follows almost effortlessly from a new Lagrangian formulation of Maxwell's theory for photons and electrons (and positrons) whose associated Euler Lagrange equations imply the conventional Maxwell equations, but which possesses new \textbf{\emph{bosonic}} degrees of freedom that may be associated with a fundamental gravitational interaction. The precise character of this gravitational interaction with photons is explicitly defined in terms of a local U(1)-invariant Lagrangian in Eq.[\ref{Lagrangian3}]. The new formulation of Maxwell's theory is cast on the real, eight dimensional pseudo-Euclidean vector space defined by the split octonion algebra, regarded as a vector space over $\mathbb{R}$, and denoted $ \mathbb{R}^{4,4} \cong M_{3,1} \oplus {}^*M_{3,1}$. (Here $M_{3,1}$ denotes real four-dimensional Minkowski space-time and $ {}^*M_{3,1}$ denotes its dual; $\mathbb{R}^{4,4}$ resembles the phase space of a single relativistic particle.) This gravitational interaction is carried by a field that defines an algebraically distinguished element of the split octonion algebra, namely, the multiplicative unit element. We call this interaction the "unit" interaction, since any equivalence with Newton-Einstein gravity has yet to be established.

gr-qc↗

arXiv2015

Spinor-Unit Field Representation of Electromagnetism Applied to a Model Inflationary Cosmology

The new spinor-unit field representation of the electromagnetism \cite{Nash2010} (with quark and lepton sources) is integrated via minimal coupling with standard Einstein gravitation, to formulate a Lagrangian model of the very early universe. The solution of the coupled Euler-Lagrange field equations yields a scale factor $a(t)$ (comoving coordinates) that initially exponentially increases $N$ e-folds from $a(0) \approx 0$ to $a_{1} = a(0) {e}^{N} $ ($N$ = 60 is illustrated), then exponentially decreases, then exponentially increases to $a_{1}$, and so on almost periodically. (Oscillatory cosmological models are not knew, and have been derived from string theory and loop quantum gravity.) It is not known if the scale factor escapes this periodic trap. This model is noteworthy in several respects: $\{1\}$ All fundamental fields other than gravity are realized by spinor fields. $\{2\}$ A plausible connection between the \emph{unit} field $\mathbf{u}$ and the generalization of the photon wave function with a form of Dark Energy is described, and a simple natural scenario is outlined that allocates a fraction of the total energy of the Universe to this form of Dark Energy. $\{3\}$ A solution of an analog of the pure Einstein-Maxwell equations is found. This approach is in contrast with the method followed to obtain a solution of the well known Friedmann model of a radiation-dominated universe.

gr-qc↗

arXiv2014

Possible consistent extra time dimensions in the early universe

Gravity cannot be quantized unless the quantized theory is cast on a manifold whose concomitant number of physical space dimensions and number of physical time dimensions correspond to physical reality, and not simply to the perception of reality. At present, the accepted number of physical time dimensions is dictated more by folklore than by science. In this paper we discuss a model of the early universe in which the number of physical time dimensions is four, and formulate Theorem[\ref{tj}], which underlies an explanation of why the extra time dimensions do not source unphysical effects. In this paper we describe a new model of gravitational inflation that is driven by dark energy and "mediated" by a real massless scalar inflaton field $φ$ whose potential is identically equal to zero. The coupled Einstein gravitational and inflaton field equations are formulated on an eight-dimensional spacetime manifold of \textbf{four space} dimensions and \textbf{four time} dimensions. We find explicit solutions to these field equations that exhibit temporal exponential \textbf{deflation of three of the four time dimensions}, and then study the dynamics of a massive complex scalar field $ψ$ that propagates on the background ground state Einstein gravitational field to determine whether its quantum fluctuations $δψ$ are stable or unstable. We compute explicit approximate solutions to the $δψ$ field equations that are \textbf{stable}, meaning that the quantum fluctuations $δψ$ of the field $ψ$ do not grow exponentially with time. \textbf{Instabilities} driven by the momenta associated to the three extra time dimensions do not appear in the physical solutions of the field equations of this model.

physics.gen-ph↗

arXiv1994

Order $\hbar$ Corrections to the Classical Dynamics of a Particle with Intrinsic Spin Moving in a Constant Magnetic Field

$O(\hbar)$ effects that modify the classical orbit of a charged particle are described for the case of a classical spin-1/2 particle moving in a constant magnetic field, using a manifestly covariant formalism reported previously. It is found that the coupling between the momentum and spin gives rise to a shift in the cyclotron frequency, which is explicitly calculated. In addition the orbit is found to exhibit $O(\hbar)$ oscillations along the axis of the uniform static magnetic field whenever the gyromagnetic ratio $g$ of the particle is not 2. This oscillation is found to occur at a frequency $\propto$ $\frac{g}{2} - 1$, and is an observable source of electric dipole radiation.

acc-phys↗