Momentum and hamiltonian in complex action theory
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Momentum and hamiltonian in complex action theory. / Nagao, Keiichi; Nielsen, Holger Frits Bech.
In: International Journal of Modern Physics A, Vol. 27, No. 14, 2012, p. 1250076.Research output: Contribution to journal › Journal article › Research › peer-review
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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 -
ID: 33454918