Abstract
We formulate a complex action theory which includes operators of coordinate and momentum q̂ and p̂ being replaced with non-hermitian operators q̂new and p̂new, and their eigenstates |q)new and |p)new 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 q̂new, p̂new, |q)new and |p)new 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 paper [Prog. Theor. Phys. 125 (2011), 633] 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.
| Original language | English |
|---|---|
| Journal | Progress of Theoretical Physics |
| Volume | 126 |
| Issue number | 6 |
| Pages (from-to) | 1021-1049 |
| ISSN | 0033-068X |
| Publication status | Published - 1 Dec 2011 |