Formulation of Complex Action Theory

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.

29 pages, 2 figures, references added, presentation improved, typos corrected. (v5)The definition of ${\hat{q}}_{new}$ and ${\hat{p}}_{new}$ are corrected by replacing them with their hermitian conjugates. The errors and typos mentioned in the errata of PTP are corrected. arXiv admin note: substantial text overlap with arXiv:1009.0441

References in corpus (2)

• Complexified Path Integrals and the Phases of Quantum Field Theory
• Automatic Hermiticity

Cited by in corpus (8)

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• Theory Including Future Not Excluded -- Formulation of Complex Action Theory II --
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• Reality from maximizing overlap in the periodic complex action theory
• Reality from maximizing overlap in the future-included real action theory
• Automatic hermiticity for mixed states
• Momentum relation and classical limit in the future-not-included complex action theory