In search of the true vacuum: natural ordering, $\gamma$ condensate and the last renormalization

TL;DR

This paper explores a novel vacuum state in $$ theory that incorporates a condensate with particle pairs, aiming to cancel divergences and investigate phase transitions, with implications for the true vacuum structure.

Contribution

It introduces a parameter interpolating between normal ordered and non-normal ordered Hamiltonians, leading to a new condensate ground state and modified propagator analysis.

Findings

Explicit expression for the condensate state derived

Conditions for canceling leading divergence analyzed

Potential phase transition identified at a critical coupling

Abstract

With the idea of canceling the leading divergence in vacuum energy of $\varphi^4$ field theory a parameter is introduced that interpolates between free Hamiltonian with or without normal ordering. This leads to a condensate ground state having an arbitrary number of particle-particle pairs. In addition to the usual states, the condensate supports the states of negative energy and negative norm. An explicit expression for the condensate state is derived and perturbation theory with this state investigated. The propagator is modified off the mass shell while unchanged on the mass shell. Lowest order correction to the vacuum energy is calculated and conditions for cancelation of the leading divergence investigated. One possible solution is that all radiative corrections in this formulation vanish. The other possible solution implies a phase transition above the coupling of $\frac{(2\pi)^2}{3}$ and the condensate non-analytical in the coupling constant. Possible implications are discussed.