arXiv subjects
Publications and source records attributed to Keiichi Nagao.
We argue that we could make a scenario of deriving quantum mechanics, as a random dynamics project, in the sense of it being almost unavoidable. The basic idea is based on the weak value formulation.
We previously proposed a mechanism to effectively obtain, after a long time development, a Hamiltonian being Hermitian with regard to a modified inner product $I_Q$ that makes a given non-normal Hamiltonian normal by using an appropriately chosen Hermitian operator $Q$. We studied it for pure states. In this letter we show that a similar mechanism also works for mixed states by introducing density matrices to describe them and investigating their properties explicitly both in the future-not-included and future-included theories. In particular, in the latter, where not only a past state at the initial time $T_A$ but also a future state at the final time $T_B$ is given, we study a couple of candidates for it, and introduce a ``skew density matrix'' composed of both ensembles of the future and past states such that the trace of the product of it and an operator ${\cal O}$ matches a normalized matrix element of ${\cal O}$. We argue that the skew density matrix defined with $I_Q$ at the present time $t$ for large $T_B-t$ and large $t-T_A$ approximately corresponds to another density matrix composed of only an ensemble of past states and defined with another inner product $I_{Q_J}$ for large $t-T_A$.
The present work contains a review of some of the work we have done on complex action or non-Hermitian Hamiltonian theory, especially the result that the anti-Hermitian part of the Hamiltonian functions by determining the actual solution to the equations of motion, that should be realized; this means it predicts the initial conditions. It should be stressed that a major result of ours is that the effective equations of motion will in practice -- after long time -- be so accurately as if we had indeed a Hermitian Hamiltonian, and so there is at first nothing wrong in assuming a non-Hermitian one. In fact it would practically seem Hermitian anyway! A major new point is that we seek by a bit intuitively arguing to suggest some cosmologically predictions from the mentioned initial conditions predicted: We seek even by assumming essentially nothing but very general properties of the non-Hermitian Hamiltonian that we in practice should find a bottom in the (effective Hermitian) Hamiltonian and that the Universe at some moment should pass through a (multiple) saddle point very closely, so that the time spent at it would be very long.
We study the periodic complex action theory (CAT) by imposing a periodic condition in the future-included CAT where the time integration is performed from the past to the future, and extend a normalized matrix element of an operator $\hat{\mathcal O}$, which is called the weak value in the real action theory, to another expression $\langle \hat{\mathcal O} \rangle_{\mathrm{periodic}~\mathrm{time}}$. We present two theorems stating that $\langle \hat{\mathcal O} \rangle_{\mathrm{periodic}~\mathrm{time}}$ becomes real for $\hat{\mathcal O}$ being Hermitian with regard to a modified inner product that makes a given non-normal Hamiltonian $\hat{H}$ normal. The first theorem holds for a given period $t_p$ in a case where the number of eigenstates having the maximal imaginary part $B$ of the eigenvalues of $\hat{H}$ is just one, while the second one stands for $t_p$ selected such that the absolute value of the transition amplitude is maximized in a case where $B \leq 0$ and $|B|$ is much smaller than the distances between any two real parts of the eigenvalues of $\hat{H}$. The latter proven via a number-theoretical argument suggests that, if our universe is periodic, then even the period could be an adjustment parameter to be determined in the Feynman path integral. This is a variant type of the maximization principle that we previously proposed.
In a special representation of complex action theory that we call ``future-included'', we study a harmonic oscillator model defined with a non-normal Hamiltonian $\hat{H}$, in which a mass $m$ and an angular frequency $ω$ are taken to be complex numbers. In order for the model to be sensible some restrictions on $m$ and $ω$ are required. We draw a phase diagram in the plane of the arguments of $m$ and $ω$, according to which the model is classified into several types. In addition, we formulate two pairs of annihilation and creation operators, two series of eigenstates of the Hamiltonians $\hat{H}$ and $\hat{H}^†$, and coherent states. They are normalized in a modified inner product $I_Q$, with respect to which the Hamiltonian $\hat{H}$ becomes normal. Furthermore, applying to the model the maximization principle that we previously proposed, we obtain an effective theory described by a Hamiltonian that is $Q$-Hermitian, i.e. Hermitian with respect to the modified inner product $I_Q$. The generic solution to the model is found to be the ``ground'' state. Finally we discuss what the solution implies.
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 $T_B$ and the past state at time $T_A$ as an expectation value in the CAT, then we are allowed to have the Heisenberg equation, the Ehrenfest's theorem and the conserved probability current density. In addition we show that the expectation value at the present time $t$ of a future-included theory for large $T_B-t$ and large $t- T_A$ corresponds to that of a future-not-included theory with a proper inner product for large $t- T_A$. 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 Schrödinger equation and Hamiltonian for the future state via path integral, and confirm that the Hamiltonian is given by the Hermitian conjugate of the Hamiltonian for the past state.
Studying the time development of the expectation value in the future-not-included complex action theory, we point out that the momentum relation (the relation analogous to $p=\frac{\partial L}{\partial \dot{q}}$), which was derived via the Feynman path integral and was shown to be correct in the future-included theory in our previous papers, is not valid in the future-not-included theory. We provide the correct momentum relation in the future-not-included theory, and argue that the future-not-included classical theory is described by a certain real action. In addition, we provide another way to understand the time development of the future-not-included theory by utilizing the future-included theory. Furthermore, properly applying the method used in our previous paper to the future-not-included theory by introducing a formal Lagrangian, we derive the correct momentum relation in the future-not-included theory.
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 based on the complex coordinate formalism of our foregoing paper. After reviewing the formalism briefly, we describe in FPI with a Lagrangian the time development of a $ξ$-parametrized wave function, which is a solution to an eigenvalue problem of a momentum operator. Solving this eigenvalue problem, we derive the momentum, Hamiltonian, and Schrödinger equation. Oppositely, starting from the Hamiltonian we derive the Lagrangian in FPI, and we are led to the momentum relation 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 relation via the saddle point for $q$.
In quantum theory its action is usually taken to be real, but we can consider another theory whose action is complex. In addition, in the Feynman path integral, the time integration is usually performed over the period between the initial time $T_A$ and some specific time, say, the present time $t$. Besides such a future-not-included theory, we can consider the future-included theory, in which not only the past state $| A(T_A) \rangle$ at the initial time $T_A$ but also the future state $| B(T_B) \rangle$ at the final time $T_B$ is given at first, and the time integration is performed over the whole period from the past to the future. Thus quantum theory can be classified into four types, according to whether its action is real or not, and whether the future is included or not. We argue that, if a theory is described with a complex action, then such a theory is suggested to be the future-included theory, rather than the future-not-included theory. Otherwise persons living at different times would see different histories of the universe.
In the future-included complex and real action theories whose paths run over not only the past but also the future, we briefly review the theorem on the normalized matrix element of an operator $\hat{\cal O}$, which is defined in terms of the future and past states with a proper inner product $I_Q$ that makes a given Hamiltonian normal. The theorem states that, provided that the operator $\hat{\cal O}$ is $Q$-Hermitian, i.e. Hermitian with regard to the proper inner product $I_Q$, 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. Discussing what the theorem implicates, we speculate that the future-included complex action theory would be the most elegant quantum theory.
In the future-included real action theory whose path runs over not only past but also future, we demonstrate a theorem, which states that the normalized matrix element of a Hermitian operator $\hat{\cal O}$ defined in terms of the future state at the final time $T_B$ and the fixed past state at the initial time $T_A$ becomes real for the future state selected such that the absolute value of the transition amplitude from the past state to the future state is maximized. This is a special version of our previously proposed theorem for the future-included complex action theory. We find that though the maximization principle leads to the reality of the normalized matrix element in the future-included real action theory, it does not specify the future and past states so much as in the case of the future-included complex action theory. In addition, we argue that the normalized matrix element seems to be more natural than the usual expectation value. Thus we speculate that the functional integral formalism of quantum theory could be most elegant in the future-included complex action theory.
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.
We briefly review the correspondence principle proposed in our previous paper, which claims that if we regard a matrix element defined in terms of the future state at time $T_B$ and the past state at time $T_A$ as an expectation value in the complex action theory whose path runs over not only past but also future, the expectation value at the present time $t$ of a future-included theory for large $T_B-t$ and large $t-T_A$ corresponds to that of a future-not-included theory with a proper inner product for large $t-T_A$. This correspondence principle suggests that the future-included theory is not excluded phenomenologically.
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.
We study a diagonalizable Hamiltonian that is not at first hermitian. Requirement that a measurement shall not change one Hamiltonian eigenstate into another one with a different eigenvalue imposes that an inner product must be defined so as to make the Hamiltonian normal with regard to it. After a long time development with the non-hermitian Hamiltonian, only a subspace of possible states will effectively survive. On this subspace the effect of the anti-hermitian part of the Hamiltonian is suppressed, and the Hamiltonian becomes hermitian. Thus hermiticity emerges automatically, and we have no reason to maintain that at the fundamental level the Hamiltonian should be hermitian. If the Hamiltonian is given in a local form, a conserved probability current density can be constructed with two kinds of wave functions. We also point out a possible misestimation of a past state by extrapolating back in time with the hermitian Hamiltonian. It is a seeming past state, not a true one.
The domain wall fermion formalism in lattice gauge theory is much investigated recently. This is set up by reducing 4+1 dimensional theory to low energy effective 4 dimensional one. In order to look around other possibilities of realizing chiral fermion on the lattice, we construct vortex fermion by reducing 4+2 dimensional theory to low energy effective 4 dimensional one on the lattice. In extra 2 dimensions we propose a new lattice regularization which has a discrete rotational invariance but not a translational one. In order to eliminate doubling species in the naive construction we introduce the extended Wilson term which is appropriate to our model. We propose two models for convenience and show that a normalizable zero mode solution appears at the core of the vortex.
Recently in the authors' country Japan, the unpopularity of natural science among children has been a serious problem. Especially, physics is unpopular because physics requires mathematics. One of the reasons of this problem is that teachers themselves do not like physics. We focus our attention on the ``teachers in embryo'', namely the undergraduate students in a course for school teachers. We conducted a questionnaire and a quiz on the undergraduate students in the first grade of the Department of Science Education, Ibaraki University. We report the result of the questionnaire and the quiz, and also make suggestions to improve the present situation.
Ginsparg-Wilson relation and admissibility condition have the key role to construct lattice chiral gauge theories. They are also useful to define the chiral structure in finite noncommutative geometries or matrix models. We discuss their usefulness briefly.