Formulation of Complex Action Theory

TL;DR

This paper develops a complex action theory framework with non-hermitian operators and eigenstates, enabling the description of complex saddle points and non-hermitian Hamiltonians within a consistent formalism.

Contribution

It introduces a novel formalism for complex action theory using non-hermitian operators, eigenstates, and a phase space function-operator relation, extending quantum mechanics to complex variables.

Findings

Constructed non-hermitian operators and eigenstates with complex eigenvalues.

Demonstrated the formalism's ability to describe complex saddle points in path integrals.

Showed how to obtain a hermitian Hamiltonian from a non-hermitian one over time.

Abstract

We formulate a complex action theory which includes operators of coordinate and momentum $\hat{q}$ and $\hat{p}$ being replaced with non-hermitian operators $\hat{q}_{new}$ and $\hat{p}_{new}$, and their eigenstates ${}_m <_{new} q |$ and ${}_m <_{new} p |$ with complex eigenvalues $q$ and $p$. Introducing a philosophy of keeping the analyticity in path integration variables, we define a modified set of complex conjugate, real and imaginary parts, hermitian conjugates and bras, and explicitly construct $\hat{q}_{new}$, $\hat{p}_{new}$, ${}_m <_{new} q |$ and ${}_m <_{new} p |$ by formally squeezing coherent states. We also pose a theorem on the relation between functions on the phase space and the corresponding operators. Only in our formalism can we describe a complex action theory or a real action theory with complex saddle points in the tunneling effect etc. in terms of bras and kets in the functional integral. Furthermore, in a system with a non-hermitian diagonalizable bounded Hamiltonian, we show that the mechanism to obtain a hermitian Hamiltonian after a long time development proposed in our letter works also in the complex coordinate formalism. If the hermitian Hamiltonian is given in a local form, a conserved probability current density can be constructed with two kinds of wave functions.