U(1) symmetry breaking under canonical transformation in real scalar field theory

TL;DR

This paper demonstrates that U(1) symmetry is broken under Bogoliubov transformations in real scalar field theory, linking particle production phenomena to gravitational effects and showing the non-invariance of correlation functions and vacuum states.

Contribution

It reveals how Bogoliubov transformations break U(1) symmetry in scalar fields, connecting this to particle production in curved spacetime and gravitational effects.

Findings

Transformed Hamiltonian loses U(1) invariance after Bogoliubov transformation.

Correlation functions of field operators do not preserve U(1) symmetry post-transformation.

Vacuum state is not an eigenstate of the transformed Hamiltonian and changes under time evolution.

Abstract

In this article, I have considered a real scalar field theory and able to show that under Bogoliubov transformation in infinite volume limit or thermodynamic limit the transformed Hamiltonian no longer invariant under U(1) action defined appropriately as it was before doing transformation. We also have checked this fact by looking at the correlation functions under the action of U(1) group. We suitably defined field operators that are associated with particle production phenomena then we can also show that correlation functions of such field operators also don't follow U(1) invariance, shown in this article. This is a consequence of non-invariance of transformed Hamiltonian under U(1) action. Since, we know Bogoliubov transformation in curved spacetime is equivalent to doing a coordinate transformation, therefore this result directly shows the phenomena of particle production under the affect of gravity since changing coordinate is equivalent to turn on gravity according to Einstein's equivalence principle in GR. I also show that particle production does not take place out of vacuum state but it can happen out of other many-particle states and vacuum state is not an eigenvector of Hamltonian operator in transformed Fock space and vacuum state does not remain vacuum state under time evolution.