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

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