Unitarity of Symplectic Fermion in $\alpha$-vacua with Negative Central Charge

TL;DR

This paper investigates the symplectic fermion model with negative central charge, proposing a new inner product to resolve negative norm issues, and explores the properties of $ ext{CFT}_2$ with non-Hermitian Hamiltonians and real spectra.

Contribution

It introduces a new inner product for symplectic fermions with negative norm states and analyzes $ ext{CFT}_2$ with negative central charge and $ ext{alpha}$-vacua.

Findings

Negative norm states can be cured by a new inner product.

The model with $c=-2$ can have non-negative norm.

The $ ext{alpha}$-vacua have real energy spectra despite non-Hermiticity.

Abstract

We study the two-dimensional free symplectic fermion with anti-periodic boundary condition. This model has negative norm states with naive inner product. This negative norm problem can be cured by introducing a new inner product. We demonstrate that this new inner product follows from the connection between the path integral formalism and the operator formalism. This model has negative central charge, $c=-2$, and we clarify how CFT$_2$ with negative central charge can have the non-negative norm. We introduce $\alpha$-vacua in which the Hamiltonian is seemingly non-Hermitian. In spite of non-Hermiticity we find that the energy spectrum is real. We also compare a correlation function with respect to the $\alpha$-vacua with that of the de Sitter space.