## Formulation of Complex Action Theory

Keiichi Nagao, Holger Bech Nielsen

We formulate the complex action theory from a fundamental level so that we can deal with a complex coordinate $$q$$ and a complex momentum $$p$$. We extend $$| q >$$ and $$| p>$$ to complex $$q$$ and $$p$$ by utilizing coherent states of harmonic oscillators. Introducing a philosophy to keep the analyticity in parameter variables of Feynman path integral, we define a modified set of complex conjugate, real and imaginary parts, hermitian conjugates and bras. They enable us to have both orthogonality and completeness relations for $$|q >$$ and $$|p >$$ with complex $$q$$ and $$p$$. We also pose a theorem on the relation between functions and operators to make it clear to some extent. Furthermore, extending our previous work \cite{Nagao:2010xu} to the complex coordinate case, we study a system defined by a diagonalizable non-hermitian bounded Hamiltonian, and show that a hermitian Hamiltonian is effectively obtained after a long time development by introducing a proper inner product. If the hermitian Hamiltonian is given in a local form, a conserved probability current density can be constructed with two kinds of wave functions.

http://arxiv.org/abs/1104.3381
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th)