Is quantum space-time infinite dimensional

Almost all presentations of Dirac theory in first or second quantization in Physics (and Mathematics) textbooks make use of covariant Dirac spinor fields. An exception is the presentation of that theory (first quantization) offered originally by Hestenes and now used by many authors. There, a new concept of spinor field (as a sum of non homogeneous even multivectors fields) is used. However, a carefully analysis (detailed below) shows that the original Hestenes definition cannot be correct since it conflicts with the meaning of the Fierz identities. In this paper we start a program dedicated to the examination of the mathematical and physical basis for a comprehensive definition of the objects used by Hestenes. In order to do that we give a preliminary definition of algebraic spinor fields (ASF ) and Dirac-Hestenes spinor fields (DHSF ) on Minkowski spacetime as some equivalence classes of pairs (Ξu, ψ Ξu ), where Ξu is a spinorial frame field and ψ Ξu is an appropriate sum of multivectors fields (to be specified below). The necessity of our definitions are shown by a carefull analysis of possible formulations of Dirac theory and the meaning of the set of Fierz identities associated with the 'bilinear covariants' (on Minkowski spacetime) made with ASF or DHSF. We believe that the present paper clarifies some misunderstandings (past and recent) appearing on the literature of the subject. It will be followed by a sequel paper where definitive definitions of ASF and DHSF are given as appropriate sections of a vector bundle called the left spin-Clifford bundle. The bundle formulation is essential in order to be possible to produce a coherent theory for the covariant derivatives of these fields on arbitrary RiemannCartan spacetimes. The present paper contains also Appendices (A-E) which exhibits a truly useful collection of results concerning the theory of

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“…The reversion operator is the anti-automorphism 28 We reserve the notation V for real vector spaces.…”

Section: Definition 15mentioning

confidence: 99%

Algebraic and Dirac–Hestenes spinors and spinor fields

Rodrigues

2004

Journal of Mathematical Physics

122

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“…In order to generalize Maxwell theory of electrodynamics we review briefly the extended relativity theory in C-spaces 7,9,11 by starting with a natural generalization of the notion of a spacetime interval in Minkowski space ds 2 to C-space. This is given by the scalar part of the Clifford geometric product of two infinitesimal displacements:…”

Section: Extending Relativity From Minkowski Spacetime To C-spacementioning

confidence: 99%

“…The minimal Planck scale then emerges as another universal invariant in constructing an extended scale relativity theory in C-spaces. 7,16 To continue along the same path of special relativity in Minkowski space, one considers the analog of Lorentz transformations in C-spaces which transform a polyvector X into another polyvector X given by X = RXR −1 with…”

Section: Extending Relativity From Minkowski Spacetime To C-spacementioning

confidence: 99%

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GENERALIZED p-FORMS ELECTRODYNAMICS IN CLIFFORD SPACES

Castro

2004

Mod. Phys. Lett. A

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Using Clifford algebraic methods we describe how to generalize Maxwell theory of Electrodynamics asociated with ordinary point-charges to a generalized Maxwell theory in Clifford spaces involving extended charges and p-forms of arbitrary rank. Clifford algebras contain the appropriate algebraic-geometric features to implement the principle of polydimensional transformations (branes of different dimensionality are rotated into each other) that could possibly lead to a proper formulation of string and M theory.

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“…A somewhat different approach has its roots in string and brane theories. It has led to the concept of the so called Clifford space [10]- [13] or C-space, a manifold whose tangent space is Clifford algebra. C-space is a generalization of spacetime, and describes a geometry of oriented r-loops.…”

Section: Introductionmentioning

confidence: 99%

On the Unification of Interactions by Clifford Algebra

Pavšič

2010

Adv. Appl. Clifford Algebras

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A theory based on the concept of 16-dimensional Clifford space C, a manifold whose tangent space is Clifford algebra, is investigated. The elements of the space C are oriented r-volumes, r = 0, 1, 2, 3, associated with extended objects such as strings and branes. Although the latter objects form an infinite dimensional configuration space, they can be sampled in terms of a finite dimensional subspace, namely, the Clifford space. The connection and the curvature of C describe what, from the point of view of 4-dimensional spacetime, appear as gravitation and Yang-Mills gauge fields and field strengths. This is demonstrated on the case of a Clifford algebra valued field Ψ(X) which depends on position X in C and satisfies the generalized Dirac equation. Mathematics Subject Classification (2000). Primary 15A66, 83E30; Secondary 11E88, 83E15, 81R99.

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Is quantum space-time infinite dimensional

Cited by 49 publications

References 14 publications

Algebraic and Dirac–Hestenes spinors and spinor fields

Algebraic and Dirac–Hestenes spinors and spinor fields

GENERALIZED p-FORMS ELECTRODYNAMICS IN CLIFFORD SPACES

On the Unification of Interactions by Clifford Algebra

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