Field-Dependent Particle Excitations in Null-Shifted Rindler Wedges

Abstract

The Unruh effect establishes that the Minkowski vacuum, when restricted to a Rindler wedge and described in terms of boost eigenmodes, satisfies the Kubo-Martin-Schwinger (KMS) condition and is therefore perceived as a thermal state by uniformly accelerated observers. This result is robust and remains valid for both massless and massive fields. In this work, we investigate a distinct geometric setup involving two null-shifted, nested Rindler wedges $R_2\subset R_1$, where the standard assumptions underlying the Unruh effect are not directly applicable. We analyse how the Rindler vacuum defined with respect to $R_1$ is represented in terms of mode functions adapted to $R_2$ by computing overlap coefficients defined through projection of field modes along null directions. Owing to the absence of a common global Cauchy surface for the nested wedge pair, these coefficients are defined in a formal sense and do not correspond to a unitary Bogoliubov transformation between complete Fock spaces. Our results show that the projected mode content exhibits nontrivial mixing, but this does not constitute a violation of the KMS condition nor a modification of the standard Unruh effect. Instead, it reflects the limitations of defining particle content through partial, null-based projections in a restricted geometric setting. The analysis is carried out for both scalar and Dirac fields, with particular attention to the role of mode structure, inner products, and consistency conditions.