Unitarity of Symplectic Fermions in & alpha; Vacua with Negative Central Charge

We study the two-dimensional free symplectic fermion theory with antiperiodic boundary condition. This model has negative norm states with a 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 a negative central charge, c = -2, and we clarify how two-dimensional conformal field theory with negative central charge can have a non-negative norm. Furthermore, 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.

Automated summary

We study the two-dimensional free symplectic fermion theory with antiperiodic boundary condition and address the issue of negative norm states. By introducing a new inner product, we demonstrate that this problem can be resolved. Moreover, we establish the connection between the path integral formalism and the operator formalism in deriving this new inner product. Additionally, we investigate α vacua in which the Hamiltonian appears non-Hermitian, yet the energy spectrum is found to be real. We also compare the correlation function between the α vacua and the de Sitter space.