[Paper Review] Maximal-acceleration phase space relativity from Clifford algebras

Carlos Castro|ArXiv.org|Aug 20, 2002

Algebraic and Geometric Analysis8 references3 citations

TL;DR

This paper proposes a maximal-acceleration relativity principle in phase space using Clifford algebras, unifying the speed of light and the Planck scale as fundamental invariants. By constructing a phase-space Clifford algebra, it derives the maximal proper acceleration bound $ a = c^2 / \Lambda $, consistent with Caianiello's and Finslerian models, and argues that Clifford spaces provide a more physically natural framework than kappa-deformed Poincaré algebras for quantum gravity unification.

ABSTRACT

We present a new physical model that links the maximum speed of light with the minimal Planck scale into a maximal-acceleration Relativity principle in the spacetime tangent bundle and in phase spaces (cotangent bundle). Crucial in order to establish this link is the use of Clifford algebras in phase spaces. The maximal proper-acceleration bound is a = c^2/ Λin full agreement with the old predictions of Caianiello, the Finslerian geometry point of view of Brandt and more recent results in the literature. We present the reasons why an Extended Scale Relativity based on Clifford spaces is physically more appealing than those based on kappa-deformed Poincare algebras and the inhomogeneous quantum groups operating in quantum Minkowski spacetimes. The main reason being that the Planck scale should not be taken as a deformation parameter to construct quantum algebras but should exist already as the minimum scale in Clifford spaces.

Motivation & Objective

• To establish a maximal-acceleration principle in spacetime tangent and phase spaces by unifying the speed of light and the Planck scale as universal invariants.
• To demonstrate that Clifford algebras in phase space naturally generate the maximal acceleration bound $ a = c^2 / \Lambda $, consistent with prior models.
• To argue that Clifford spaces (C-spaces) provide a more physically consistent foundation for quantum gravity than kappa-deformed Poincaré algebras or Moyal star-product deformations.
• To show that the Planck scale $ \Lambda $ is not a deformation parameter but a fundamental minimum scale inherent in the structure of Clifford algebras.
• To resolve issues in quantum gravity models such as non-abelian momentum addition and non-convergent group structures in kappa-deformed frameworks by using abelian, group-like C-space Lorentz transformations.

Proposed method

• Constructs a two-dimensional phase-space Clifford algebra using basis elements $ e_p, e_q $ satisfying $ e_p e_q = -e_q e_p $, with $ j = e_p e_q $ acting as the imaginary unit.
• Defines a Clifford vector $ Q = q e_q + p e_p $, whose left and right multiplications by $ e_q $ yield complex-like structures $ z = q + j p $ and $ z^* = q - j p $, linking to harmonic oscillator formalism.
• Generalizes the phase-space Clifford algebra to higher dimensions and constructs a C-phase-space generalization of the Nesterenko action for sub-maximally accelerated particles.
• Uses the cyclic trace property of gamma matrices to ensure invariance of the line element $ d\Sigma^2 = dX \cdot dX $ under C-space Lorentz transformations $ R = \exp[i(\theta I + \theta^\mu \gamma_\mu + \cdots)] $.
• Derives the maximal proper acceleration bound $ a = c^2 / \Lambda $ from the phase-space Clifford algebra structure, ensuring that $ dX \cdot dX > 0 $ only when generalized velocities exceed $ \Lambda $.
• Compares the C-space approach to kappa-deformed Poincaré algebras, showing that the latter rely on noncanonical, non-invariant basis changes and lack group structure in all bases, unlike the C-space framework.

Experimental results

Research questions

• RQ1 Can a maximal-acceleration principle be derived from a fundamental phase-space Clifford algebra structure?
• RQ2 How does the Planck scale $ \Lambda $ emerge as a minimum scale in a Clifford algebra framework, rather than as a deformation parameter?
• RQ3 Why is the C-space approach with intrinsic minimum scale more physically consistent than models based on kappa-deformed Poincaré algebras?
• RQ4 What is the role of noncanonical transformations in kappa-deformed models, and why do they undermine the invariance of quantum algebras?
• RQ5 Can the C-space Lorentz group structure resolve the non-abelian momentum addition problem present in kappa-deformed frameworks?

Key findings

• The maximal proper acceleration bound is derived as $ a = c^2 / \Lambda $, confirming earlier predictions by Caianiello and Finslerian geometry.
• The use of Clifford algebras in phase space naturally generates complex structures via $ j = e_p e_q $, providing a geometric origin for imaginary units without ad-hoc postulation.
• The line element $ d\Sigma^2 = dX \cdot dX $ is invariant under C-space Lorentz transformations due to the cyclic trace property of gamma matrices, ensuring consistency across polyvector components.
• C-space Lorentz transformations form a true group, unlike kappa-deformed models where group structure may fail to converge in certain bases.
• The Planck scale $ \Lambda $ is not a deformation parameter but a fundamental minimum scale required to unify p-branes of different dimensions via dimensional consistency.
• Kappa-deformed Poincaré algebras are shown to be isomorphic to Moyal star-product deformations of classical algebras, but this leads to non-unique quantum algebras under noncanonical transformations, undermining physical consistency.