# Classical and quantum analysis of 3D electromagnetic pp-wave spacetime

## TL;DR

This paper derives classical solutions for 3D electromagnetic pp-wave spacetimes, classifies them, and explores their quantum counterparts, revealing differences between classical and quantum behaviors through a Bohm-like analysis.

## Contribution

It provides the first known classical solutions for one of the classes of 3D electromagnetic pp-wave spacetimes and investigates their quantum and semi-classical properties.

## Key findings

- Classical solutions depend on an arbitrary function, leading to infinite geometries.
- Quantum analysis shows the classical-quantum equivalence breaks at the quantum level.
- Semi-classical trajectories approach classical ones as the wavepacket spreads widely.

## Abstract

The general classical solution of the 3D electromagnetic pp-wave spacetime has been obtained. The relevant line element contains an arbitrary essential function providing an infinite number of in-equivalent geometries as solutions. A classification is presented based on the symmetry group. To the best of our knowledge, the solution corresponding to only one of the Classes is known. The dynamics of some of the Classes was also derived from a minisuperspace Lagrangian which has been constructed. This Lagrangian contains a degree of freedom (the lapse) which can be considered either as dynamical or non-dynamical (indicating a singular or a regular Lagrangian correspondingly). Surprisingly enough, on the space of classical solutions, an equivalence of these two points of view can be established. The canonical quantization is then used in order to quantize the system for both the singular and regular Hamiltonian. A subsequent interpretation of quantum states is based on a Bohm-like analysis. The semi-classical trajectories deviate from the classical only for the regular Hamiltonian and in particular for a superposition of eigenstates (a Gaussian initial state has been used). Thus, the above mentioned equivalence is broken at the quantum level. It is noteworthy that the semi-classical trajectories tend to the classical ones in the limit where the initial wavepacket is widely spread. Hence, even with this simple superposition state, the classical solutions are acquired as a limit of the semi-classical.