## Correspondence between future-included and future-not-included theories

Keiichi Nagao, Holger Bech Nielsen

We briefly review the correspondence principle proposed in our previous paper, which claims that if we regard a matrix element defined in terms of the future state at time $$T_B$$ and the past state at time $$T_A$$ as an expectation value in the complex action theory whose path runs over not only past but also future, the expectation value at the present time $$t$$ of a future-included theory for large $$T_B-t$$ and large $$t- T_A$$ corresponds to that of a future-not-included theory with a proper inner product for large $$t- T_A$$. This correspondence principle suggests that the future-included theory is not excluded phenomenologically.

http://arxiv.org/abs/1211.7269

Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th)

## Momentum and Hamiltonian in Complex Action Theory

Keiichi Nagao, Holger Bech Nielsen

In the complex action theory (CAT) we explicitly examine how the momentum and Hamiltonian are defined from the Feynman path integral (FPI) point of view. In arXiv:1104.3381[quant-ph], introducing a philosophy to keep the analyticity in parameter variables of FPI and defining a modified set of complex conjugate, hermitian conjugates and bras, we have extended $$| q >$$ and $$| p >$$ to complex $$q$$ and $$p$$ so that we can deal with a complex coordinate $$q$$ and a complex momentum $$p$$. After reviewing them briefly, we describe in terms of the newly introduced devices the time development of a $$\xi$$-parametrized wave function, which is a solution to an eigenvalue problem of a momentum operator $$\hat{p}$$, in FPI with a starting Lagrangian. Solving the eigenvalue problem, we derive the momentum and Hamiltonian. Oppositely, starting from the Hamiltonian we derive the Lagrangian in FPI, and we are led to the momentum again via the saddle point for $$p$$. This study confirms that the momentum and Hamiltonian in the CAT have the same forms as those in the real action theory. We also show the third derivation of the momentum via the saddle point for $$q$$.

http://arxiv.org/abs/1105.1294
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th)

## 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)

## Automatic Hermiticity

Keiichi Nagao and Holger Bech Nielsen

We study the Hamiltonian that is not at first hermitian. Requirement that a measurement shall not change one Hamiltonian eigenstate into another one with a different eigenvalue imposes that an inner product must be defined so as to make the Hamiltonian normal with regard to it. After a long time development with the non-hermitian Hamiltonian, only a subspace of possible states will effectively survive. On this subspace the effect of the anti-hermitian part of the Hamiltonian is suppressed, and the Hamiltonian becomes hermitian. Thus hermiticity emerges automatically, and we have no reason to maintain that at the fundamental level the Hamiltonian should be hermitian. We also point out a possible misestimation of a past state by extrapolating back in time with the hermitian Hamiltonian. It is a seeming past state, not a true one.

http://arxiv.org/abs/1009.0441
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); General Physics (physics.gen-ph)