Tomonaga's Conjecture on Photon Self-Energy

TL;DR

This paper supports Tomonaga's conjecture that the photon self-energy should be zero, demonstrating that its non-zero contribution violates Lorentz invariance and that the photon wave function does not undergo renormalization.

Contribution

It provides a theoretical argument and mathematical clarification confirming that the photon self-energy vanishes, challenging conventional renormalization assumptions.

Findings

Photon self-energy contribution violates Lorentz invariance.

No wave function renormalization for photon in exact equations.

Gauge condition for vacuum polarization tensor is based on a mathematical mistake.

Abstract

We investigate Tomonaga's conjecture that the self-energy of photon should vanish to zero. In fact, the contribution of photon's self-energy diagram violates the Lorentz invariance and therefore it is unphysical. In addition, there occurs no wave function renormalization of photon in the exact Lippmann-Schwinger equation for the vector potential and this confirms that the conjecture is correct. Further, it is shown that the gauge condition $ k_\mu \Pi^{\mu \nu}(k) =0$ of the vacuum polarization tensor does not hold and the relation is obtained simply due to a mathematical mistake in replacing the integration variables in the infinite integral.