Abstract
In the complex action theory whose path runs over not only past but also future we study a normalized matrix element of an operator $\hat{\cal O}$ defined in terms of the future state at the latest time $T_B$ and the past state at the earliest time $T_A$ with a proper inner product that makes normal a given Hamiltonian that is non-normal at first. We present a theorem that states that, provided that the operator $\hat{\cal O}$ is $Q$-Hermitian, i.e., Hermitian with regard to the proper inner product, the normalized matrix element becomes real and time-develops under a $Q$-Hermitian Hamiltonian for the past and future states selected such that the absolute value of the transition amplitude from the past state to the future state is maximized. Furthermore, we give a possible procedure to formulate the $Q$-Hermitian Hamiltonian in terms of $Q$-Hermitian coordinate and momentum operators, and construct a conserved probability current density.
Highlights
The complex action theory (CAT) is a trial to extend quantum theories so that the action is complex at a fundamental level, but effectively looks real
In Ref. [18], we proposed a complex coordinate and momentum formalism by explicitly constructing non-Hermitian operators of complex coordinate q and momentum p and their eigenstates, so that we can deal with complex q and p properly
Using the complex coordinate and momentum formalism [18] in the Feynman path integral, we found that the momentum relation is given by the usual expression p = mq, where m is a complex
Summary
The complex action theory (CAT) is a trial to extend quantum theories so that the action is complex at a fundamental level, but effectively looks real. That is to say, according to the choice of the inner product used in the normalization of the initial and final states, two slightly different versions could be defined. In the large T ≡ TB − TA case, only terms associated with the largest imaginary parts of the eigenvalues of the Hamiltonian would dominate, and even with random initial and final states the dominant term would give the biggest value. We call this thinking the maximization principle. It is very important to obtain a real expectation value and a Hermitian Hamiltonian in the CAT so that it can survive as
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