Momentum relation and classical limit in the future-not-included complex action theory

Abstract

Studying the time development of the expectation value in the future-not-included complex action theory, we point out that the momentum relation (the relation analogous to $p=\frac{\partial L}{\partial \dot{q}}$), which was derived via the Feynman path integral and was shown to be correct in the future-included theory in our previous papers, is not valid in the future-not-included theory. We provide the correct momentum relation in the future-not-included theory, and argue that the future-not-included classical theory is described by a certain real action. In addition, we provide another way to understand the time development of the future-not-included theory by utilizing the future-included theory. Furthermore, properly applying the method used in our previous paper to the future-not-included theory by introducing a formal Lagrangian, we derive the correct momentum relation in the future-not-included theory.

Latex 28 pages, 2 figures, typos corrected, the final version to appear in Prog.Theor.Exp.Phys. (v3) A missed $dt$-dependent normalization factor is properly considered in Section 3.1. The errors and typos mentioned in the erratum of PTEP are corrected. arXiv admin note: text overlap with arXiv:1205.3706

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