TY - JOUR

T1 - Momentum and hamiltonian in complex action theory

AU - Nagao, Keiichi

AU - Nielsen, Holger Frits Bech

PY - 2012

Y1 - 2012

N2 - 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 $q$.

AB - 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 $q$.

KW - Faculty of Science

KW - Backward causation, quantum physics, nonhermitean Hamiltonian

U2 - 10.1142/S0217751X12500765

DO - 10.1142/S0217751X12500765

M3 - Journal article

VL - 27

SP - 1250076

JO - International Journal of Modern Physics A

JF - International Journal of Modern Physics A

SN - 0217-751X

IS - 14

ER -