TY - JOUR
T1 - Theory including future not excluded
T2 - Formulation of complex action theory II
AU - Nagao, K.
AU - Nielsen, H.B.
PY - 2013/2/4
Y1 - 2013/2/4
N2 - We study a complex action theory (CAT) whose path runs over not only past but also future. We show that, if we regard a matrix element defined in terms of the future state at time TB and the past state at time TA as an expectation value in the CAT, then we are allowed to have the Heisenberg equation, Ehrenfest's theorem, and the conserved probability current density. In addition,we showthat the expectation value at the present time t of a future-included theory for large TB - t and large t - TA corresponds to that of a future-not-included theory with a proper inner product for large t - TA. Hence, the CAT with future explicitly present in the formalism and influencing in principle the past is not excluded phenomenologically, because the effects are argued to be very small in the present era. Furthermore, we explicitly derive the Hamiltonian for the future state via a path integral, and confirm that it is given by the Hermitian conjugate of the Hamiltonian for the past state.
AB - We study a complex action theory (CAT) whose path runs over not only past but also future. We show that, if we regard a matrix element defined in terms of the future state at time TB and the past state at time TA as an expectation value in the CAT, then we are allowed to have the Heisenberg equation, Ehrenfest's theorem, and the conserved probability current density. In addition,we showthat the expectation value at the present time t of a future-included theory for large TB - t and large t - TA corresponds to that of a future-not-included theory with a proper inner product for large t - TA. Hence, the CAT with future explicitly present in the formalism and influencing in principle the past is not excluded phenomenologically, because the effects are argued to be very small in the present era. Furthermore, we explicitly derive the Hamiltonian for the future state via a path integral, and confirm that it is given by the Hermitian conjugate of the Hamiltonian for the past state.
UR - http://www.scopus.com/inward/record.url?scp=84876351632&partnerID=8YFLogxK
U2 - 10.1093/ptep/pts084
DO - 10.1093/ptep/pts084
M3 - Journal article
AN - SCOPUS:84876351632
SN - 2050-3911
VL - 2013
SP - 023B04
JO - Progress of Theoretical Physics Supplement
JF - Progress of Theoretical Physics Supplement
IS - 2
ER -