[Paper Review] Wave Equations for Invariant Infeld-van der Waerden Wave Functions for Photons and Their Physical Significance

J. G. Cardoso|arXiv (Cornell University)|Jun 9, 2010

Quantum and Classical Electrodynamics1 references3 citations

TL;DR

This paper derives wave equations for invariant spin-tensor electromagnetic wave functions in the Infeld-van der Waerden formalisms, showing they transform covariantly under the generalized Weyl gauge group. The key result is a wave equation for photon fields that couples to the Weyl curvature and Ricci scalar, with solutions potentially offering new insights into cosmic microwave background physics in cosmological spacetimes.

ABSTRACT

The inner structure of the γε-formalisms of Infeld and van der Waerden admits the occurrence of spin-tensor electromagnetic fields which bear invariance under the action of the generalized Weyl gauge group. A concise derivation of the wave equations for such fields is carried out explicitly along with the construction of a set of torsionless covariant-derivative expressions. It is emphatically pointed out that the integration of the wave equations arising herein may under certain circumstances produce significant insights into the situation concerning the description of some physical properties of the cosmic microwave background.

Motivation & Objective

• To derive wave equations for invariant spin-tensor electromagnetic fields in the $γ\varepsilon$-formalisms of Infeld and van der Waerden.
• To establish the role of the generalized Weyl gauge group in preserving invariance of these wave functions.
• To explore the physical significance of these wave equations in the context of cosmic microwave background radiation.
• To connect the derived equations to cosmological models, particularly Friedmann-Robertson-Walker spacetimes.
• To provide a covariant-derivative framework for solving these equations using torsionless connections.

Proposed method

• Derives the wave equation for the invariant spin-tensor field $\phi_A{}^B$ using the massless free-field condition $\nabla^{AB'}\phi_A{}^B = 0$.
• Applies the splitting $\nabla_{A'}^C\nabla^{AA'}\phi_A{}^B = \Delta^{AC}\phi_A{}^B - \frac{1}{2}M^{AC}\square\phi_A{}^B$ to decompose the d'Alembertian operator.
• Uses the curvature decomposition $W_{AA'BBCD} = M_{AB}\omega_{ABCD} + M_{A'B'}\omega_{A'B'CD}$ to express the field in terms of spinor curvature components.
• Constructs the wave equation $\left(\square + \frac{R}{3}\right)\phi_A{}^B = -2\Psi_{AD}{}^{BC}\phi_C{}^D$ by combining curvature and Ricci scalar terms.
• Demonstrates gauge invariance of the spin-connection terms $\vartheta_{a(BC)}$ and $\vartheta_a{}^{(BC)}$ in both formalisms.
• Considers conformally flat spacetimes to simplify the equation to $\left(\square + \frac{R}{3}\right)\phi_A{}^B = 0$, enabling application to cosmological models.

Experimental results

Research questions

• RQ1 How do invariant spin-tensor wave functions for photons arise naturally in the Infeld-van der Waerden $\gamma\varepsilon$-formalisms?
• RQ2 What is the form of the wave equation governing these invariant photon fields under the generalized Weyl gauge group?
• RQ3 How does the coupling of the photon wave function to the Weyl curvature and Ricci scalar influence its dynamics?
• RQ4 Can solutions to the derived wave equation provide new insights into the physical description of the cosmic microwave background?
• RQ5 What is the role of the energy-momentum tensor $T_{AA'}{}^{BB'} = \frac{1}{2\pi}\phi_A{}^B\phi_{A'}{}^{B'}$ in characterizing radiation in cosmological models?

Key findings

• The wave equation for the invariant spin-tensor photon field is derived as $\left(\square + \frac{R}{3}\right)\phi_A{}^B = -2\Psi_{AD}{}^{BC}\phi_C{}^D$, where $\Psi_{ABCD}$ is the Weyl spinor.
• The field $\phi_A{}^B$ transforms as a gauge-invariant spin-tensor under the generalized Weyl group, ensuring its physical consistency.
• In conformally flat spacetimes, the equation simplifies to $\left(\square + \frac{R}{3}\right)\phi_A{}^B = 0$, enabling systematic solution using distributional methods.
• The energy-momentum tensor $T_{AA'}{}^{BB'} = \frac{1}{2\pi}\phi_A{}^B\phi_{A'}{}^{B'}$ is proposed as a tool for computing radiation energy and momentum in cosmology.
• The covariant derivative expression $\nabla_a\phi_A{}^B = \partial_a\phi_A{}^B - \vartheta_{a(AC)}M^{BD}\phi_D{}^C + \vartheta_a{}^{(BC)}\phi_A{}^D M_{DC}$ ensures gauge invariance and proper index transformation.
• The formalism allows a unified treatment of electromagnetic wave propagation in curved spacetime, with direct relevance to cosmic microwave background physics.