Physics – Quantum Physics
Scientific paper
2011-04-18
Prog. Theor. Phys. 126 (2011), 1021-1049
Physics
Quantum Physics
29 pages, 2 figures, references added, presentation improved, typos corrected. (v5)The definition of $\hat{q}_{new}$ and $\hat
Scientific paper
10.1143/PTP.126.1021
We formulate a complex action theory which includes operators of coordinate and momentum $\hat{q}$ and $\hat{p}$ being replaced with non-hermitian operators $\hat{q}_{new}$ and $\hat{p}_{new}$, and their eigenstates ${}_m <_{new} q |$ and ${}_m <_{new} p |$ with complex eigenvalues $q$ and $p$. Introducing a philosophy of keeping the analyticity in path integration variables, we define a modified set of complex conjugate, real and imaginary parts, hermitian conjugates and bras, and explicitly construct $\hat{q}_{new}$, $\hat{p}_{new}$, ${}_m <_{new} q |$ and ${}_m <_{new} p |$ by formally squeezing coherent states. We also pose a theorem on the relation between functions on the phase space and the corresponding operators. Only in our formalism can we describe a complex action theory or a real action theory with complex saddle points in the tunneling effect etc. in terms of bras and kets in the functional integral. Furthermore, in a system with a non-hermitian diagonalizable bounded Hamiltonian, we show that the mechanism to obtain a hermitian Hamiltonian after a long time development proposed in our letter works also in the complex coordinate formalism. If the hermitian Hamiltonian is given in a local form, a conserved probability current density can be constructed with two kinds of wave functions.
Nagao Keiichi
Nielsen Holger Bech
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