[Paper Review] The Free Particle in Deformed Special Relativity

Florian Girelli, Tomasz Konopka+2 more|HAL (Le Centre pour la Communication Scientifique Directe)|Dec 9, 2005

Relativity and Gravitational Theory4 citations

TL;DR

This paper proposes a 5-dimensional relativistic particle action with two constraints (mass and Planck-scale energy scale) that, upon gauge fixing, reduces to 4D Deformed Special Relativity (DSR) with de Sitter momentum space. By choosing different gauge conditions—such as 5D dilatation for Snyder basis or light-cone gauge for bicrossproduct basis—the standard DSR models emerge from a unified 5D action, providing a geometric origin for different DSR bases and clarifying their physical interpretation via coordinate choices tied to measurement protocols.

ABSTRACT

The phase space of a classical particle in DSR contains de Sitter space as the space of momenta. We start from the standard relativistic particle in five dimensions with an extra constraint and reduce it to four dimensional DSR by imposing appropriate gauge fixing. We analyze some physical properties of the resulting theories like the equations of motion, the form of Lorentz transformations and the issue of velocity. We also address the problem of the origin and interpretation of different bases in DSR.

Motivation & Objective

• To derive Deformed Special Relativity (DSR) from a fundamental 5D relativistic particle action with two constraints.
• To show how different DSR bases (Snyder and bicrossproduct) arise from distinct gauge fixing conditions in the 5D framework.
• To clarify the physical interpretation of momentum space curvature and coordinate choices in DSR by relating them to measurement protocols in quantum gravity.
• To establish that physical predictions in DSR are independent of gauge fixing, while the choice of coordinates reflects operational definitions of spacetime observables.

Proposed method

• Formulate a 5D relativistic particle action with a mass constraint and a Planck-scale constraint (κ), using a 10-dimensional phase space.
• Apply gauge fixing to reduce the 10D phase space to an 8D reduced phase space with de Sitter momentum space.
• Use the 5D dilatation generator as a gauge fixing condition to recover the Snyder basis; use the light-cone gauge to recover the bicrossproduct basis.
• Derive the equations of motion, Lorentz transformations, and velocity-momentum relations in both bases from the reduced 4D theory.
• Analyze the role of the fifth dimension as an effective degree of freedom from quantum gravity, possibly linked to renormalization scale or a dynamical cutoff.
• Demonstrate that physical observables must commute with both constraints, ensuring gauge invariance and independence from coordinate choice.

Experimental results

Research questions

• RQ1 How can Deformed Special Relativity (DSR) with de Sitter momentum space be derived from a single 5D relativistic particle action?
• RQ2 What is the geometric and physical origin of different DSR bases, such as the Snyder and bicrossproduct bases?
• RQ3 How do gauge fixing conditions in 5D correspond to physical measurement protocols in 4D spacetime?
• RQ4 Why are physical predictions in DSR independent of the choice of coordinates, even though different bases use different momentum and position variables?
• RQ5 What is the physical interpretation of the fifth dimension in the 5D formulation of DSR, and how might it relate to quantum gravity or renormalization?

Key findings

• The 5D relativistic particle with two constraints (mass and κ) reduces to 4D DSR upon gauge fixing, with the momentum space becoming de Sitter geometry.
• The Snyder basis arises from gauge fixing using the 5D dilatation generator, while the bicrossproduct basis emerges from a 5D light-cone gauge condition.
• The equations of motion, Lorentz transformations, and velocity-momentum relations in both bases are consistently derived from the 5D action.
• Physical predictions in DSR are independent of gauge fixing, confirming that only gauge-invariant quantities are observable.
• The fifth dimension is interpreted as an effective degree of freedom from quantum gravity, potentially linked to a dynamical cutoff or renormalization scale.
• The choice of coordinates in DSR corresponds to operational definitions of spacetime measurements, linking abstract phase space to physical observables.