[Paper Review] Intrinsic momentum in Poincare gauge theory

Michel Leclerc|ArXiv.org|Oct 22, 2005

Noncommutative and Quantum Gravity Theories3 citations

TL;DR

This paper argues that in Poincaré gauge theory, fermions do not possess intrinsic momentum because minimal coupling of translational gauge fields to spinors leads to inconsistencies. By analyzing the Poincaré group as a limit of de Sitter and conformal groups, the study shows that such coupling cannot arise from symmetry breaking, and the only consistent ground state lacks Poincaré invariance, invalidating the notion of intrinsic momentum for fermions.

ABSTRACT

While it is generally accepted, in the framework of Poincare gauge theory, that the Lorentz connection couples minimally to spinor fields, there is no general agreement on the coupling of the translational gauge field to fermions. We will show that the assumption that spinors carry a full Poincare representation leads to inconsistencies, whose origins will be traced back by considering the Poincare group both as the contraction of the de Sitter group, and as a subgroup of the conformal group. As a result, the translational fields do not minimally couple to fermions, and consequently, fermions do not possess an intrinsic momentum.

Motivation & Objective

• To investigate whether fermions can carry a full Poincaré representation and thus couple minimally to translational gauge fields in Poincaré gauge theory.
• To resolve the inconsistency arising when spinors are assumed to transform under the full Poincaré group, including translations.
• To determine whether the residual Lorentz theory can emerge from spontaneous breaking of translational gauge symmetry.
• To examine the role of the Higgs-like field $ y^a $ in constructing the tetrad and its implications for gauge invariance and consistency.
• To assess whether a Poincaré-invariant ground state exists in conformal or de Sitter gauge theories when translational fields are coupled to fermions.

Proposed method

• Analyzes the Poincaré gauge theory using the connection one-form $ ( ilde{\Gamma}^{ab}, \Gamma^a) $, with $ \Gamma^a $ transforming under translations as $ \delta \Gamma^a = -D\varepsilon^a + \varepsilon^a{}_b \Gamma^b $.
• Introduces the Higgs-like field $ y^a $, transforming as $ \delta y^a = \varepsilon^a{}_b y^b + \varepsilon^a $, to define the tetrad $ e^a = \Gamma^a + D y^a $, which is translationally invariant.
• Constructs a Dirac-like Lagrangian $ \mathcal{L} $ using two-component spinors $ \psi_1, \psi_2 $, with minimal coupling to the Lorentz connection and a field $ A $, but not to the translational gauge field.
• Applies the Wigner-Inönü contraction of the de Sitter group $ SO(4,1) $ to the Poincaré group, showing that the resulting theory does not allow minimal coupling of translational fields to fermions.
• Treats Poincaré gauge theory as a subcase of conformal gauge theory $ SO(4,2) $, demonstrating that no Poincaré-invariant ground state exists when translational fields are coupled to fermions.
• Uses the constraint $ y^A y^A = 0 $ and $ y^A z^A = 0 $ to show that only trivial or inconsistent ground states emerge, leading to a vanishing Lagrangian.

Experimental results

Research questions

• RQ1 Can fermions in Poincaré gauge theory carry a full Poincaré representation, including intrinsic momentum via minimal coupling to translational gauge fields?
• RQ2 Does the residual Lorentz theory emerge from spontaneous breaking of translational gauge symmetry in a consistent Poincaré gauge framework?
• RQ3 Is there a Poincaré-invariant ground state in the conformal gauge theory that would allow minimal coupling of translational fields to fermions?
• RQ4 Does the Wigner-Inönü contraction of the de Sitter group to the Poincaré group permit minimal coupling of translational gauge fields to spinors in the resulting theory?
• RQ5 Can a consistent, non-vanishing Lagrangian be constructed for fermions with minimal coupling to the translational gauge field while preserving Poincaré invariance?

Key findings

• Minimal coupling of translational gauge fields to spinors leads to inconsistencies in Poincaré gauge theory, invalidating the assumption that fermions possess intrinsic momentum.
• The Lagrangian constructed with minimal coupling to the translational field $ A $ is not derivable from a conformally invariant Dirac equation, as the required symmetry breaking fails.
• In the conformal gauge theory framework, no non-trivial Poincaré-invariant ground state exists that supports the minimal coupling of translational fields to fermions.
• The only consistent ground state in the conformal approach requires $ y^A = (0,0,0,0,a,a) $, but this leads to a vanishing Lagrangian, rendering the theory trivial.
• The Wigner-Inönü contraction of the de Sitter group to the Poincaré group does not yield minimal coupling of translational fields to fermions, confirming the absence of intrinsic momentum.
• The tetrad field $ e^a = \Gamma^a + D y^a $, constructed from the Higgs-like field $ y^a $, is the only field in the theory that couples minimally to the full Poincaré connection, highlighting its foundational role in spacetime geometry.