arXiv subjects

Kaare Borchsenius

Publications and source records attributed to Kaare Borchsenius.

5 recordsLinked to original sources

arXiv2025

Canonical quantization in a spinor substructure of Minkowski space

We factorize the space-time coordinates of Minkowski space into Weyl spinors with components in a split Clifford algebra. Poisson brackets are defined for spinor-valued canonical variables and applied to the quantization of point particles and strings. In particular, we obtain the Lorentz algebra for the quantum string, and show that the string supports both integral and half-integral spin states. The Clifford algebra is augmented with the octonions through an R-algebra tensor product, and we apply the results of Manogue, Schray and Dray on octonionic Lorentz transformations to obtain a Lorentz invariant string action in ten dimensions.

math-ph↗

arXiv2017

Matrix mechanics of the relativistic point particle and string in Clifford space

We resolve the space-time canonical variables of the relativistic point particle into inner products of Weyl spinors with components in a Clifford algebra and find that these spinors themselves form a canonical system with generalized Poisson brackets. For N particles, the inner products of their Clifford coordinates and momenta form two NxN Hermitian matrices X and P which transform under a U(N) symmetry in the generating algebra. This is used as a starting point for defining matrix mechanics for a point particle in Clifford space. Next we consider the string. The Lorentz metric induces a metric and a scalar on the world sheet which we represent by a Jackiw-Teitelboim term in the action. The string is described by a polymomenta canonical system and we find the wave solutions to the classical equations of motion for a flat world sheet. Finally, we show that the SL(2.C) charge and space-time momentum of the quantized string satisfy the Poincare algebra.

math-ph↗

arXiv2013

Constructing Quantum Mechanics from a Clifford substructure of the relativistic point particle

We show that the quantized free relativistic point particle can be understood as a string in a Clifford space which generates the space-time coordinates through its inner product. The generating algebra is preserved by a unitary symmetry which becomes the symmetry of the quantum states. We start by resolving the space-time canonical variables of the point particle into inner products of Weyl spinors with components in a Clifford algebra. Next, we show that a system of N particles has a U(N) symmetry that mixes the Clifford coordinates and momenta belonging to different particles. The inner products of these variables are assembled into Hermitian matrices X and P which are employed in defining a general unitarily invariant dynamical system. When X and P commute, this system can be gauged back into the original system of independent particles. When they do not commute, the system becomes irreducible and infinite and generates a space-time canonical system formally identical to Matrix Mechanics. The continuum limit is identified as a particular parametrization of a relativistic string in Clifford space.

math-ph↗

arXiv2001

Clifford spinors and the relativistic point particle

We examine the structure of the Clifford algebra associated with a Hermitian bilinear form and apply the result to a dynamical model of the relativistic point particle. The dynamics of the particle is described by a Dirac spinor with components in a Clifford algebra. This spinor determines, through the Clifford algebra, both the space-time coordinates and their conjugate momenta and satisfies a first order equation of motion which leads to the usual space-time canonical equations of motion. The constraints appear as the equations of motion for the einbein and spin connection which are needed to ensure the reparametrization invariance and local Lorentz invariance of the action.

hep-th↗

arXiv2000

Degenerate space-time paths and the non-locality of quantum mechanics in a Clifford substructure of space-time

The quantized canonical space-time coordinates of a relativistic point particle are expressed in terms of the elements of a complex Clifford algebra which combines the complex properties of SL(2.C) and quantum mechanics. When the quantum measurement principle is adapted to the generating space of the Clifford algebra we find that the transition probabilities for twofold degenerate paths in space-time equals the transition amplitudes for the underlying paths in Clifford space. This property is used to show that the apparent non-locality of quantum mechanics in a double slit experiment and in an EPR type of measurement is resolved when analyzed in terms of the full paths in the underlying Clifford space. We comment on the relationship of this model to the time symmetric formulation of quantum mechanics and to the Wheeler-Feynman model.

quant-ph↗