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arXiv:1902.01424 · Progress of Theoretical and Experimental Physics

Formalism of a harmonic oscillator in the future-included complex action theory

Keiichi Nagao, HOLGER BECH NIELSEN

Published Apr 26, 2019• 2 citations
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Abstract

Abstract 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 $\omega$ are taken to be complex numbers. In order for the model to be sensible some restrictions on $m$ and $\omega$ are required. We draw a phase diagram in the plane of the arguments of $m$ and $\omega$, 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 $\hat{H}^\dagger$, 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.

DOI
10.1093/ptep/ptz047

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Europhysics Letters (EPL)•May 1, 2012•14 citations
Abstract 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 H^\hat{H}H^, in which a mass mmm and an angular frequency ω\omegaω are taken to be complex numbers. In order for the model to be sensible some restrictions on mmm and ω\omegaω are required. We draw a phase diagram in the plane of the arguments of mmm and ω\omegaω, 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 H^\hat{H}H^ and H^†\hat{H}^\daggerH^†, and coherent states. They are normalized in a modified inner product IQI_QIQ​, with respect to which the Hamiltonian H^\hat{H}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 QQQ-Hermitian, i.e. Hermitian with respect to the modified inner product IQI_QIQ​. The generic solution to the model is found to be the “ground” state. Finally we discuss what the solution implies.
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 O^\hat{\cal O}O^, which is defined in terms of the future and past states with a proper inner product IQI_QIQ​ that makes a given Hamiltonian normal. The theorem states that, provided that the operator O^\hat{\cal O}O^ is QQQ-Hermitian, i.e. Hermitian with regard to the proper inner product IQI_QIQ​, the normalized matrix element becomes real and time-develops under a QQQ-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 complex action theory whose path runs over not only past but also future, we study a normalized matrix element of an operator |O^\hat {\mathcal O}O^| defined in terms of the future state at the latest time |TBT_BTB​| and the past state at the earliest time |TAT_ATA​| 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 |O^\hat {\mathcal O}O^| is |QQQ|-Hermitian, i.e., Hermitian with regard to the proper inner product, the normalized matrix element becomes real and time-develops under a |QQQ|-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 |QQQ|-Hermitian Hamiltonian in terms of |QQQ|-Hermitian coordinate and momentum operators, and construct a conserved probability current density.
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 |O^\hat{\mathcal O}O^| defined in terms of the future state at the final time |TBT_BTB​| and the fixed past state at the initial time |TAT_ATA​| 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.
We formulate a complex action theory which includes operators of coordinate and momentum q and p being replaced with non-hermitian operators qnew and pnew, 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 qnew, pnew, 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 11) 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 discuss the quantum theory of an harmonic oscillator with time-dependent mass and frequency submitted to action of a complex time-dependent linear potential with [Formula: see text] symmetry. Combining the Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having [Formula: see text] symmetry and linear invariants, we solve the time-dependent Schrödinger equation for this problem and use the corresponding quantum states to construct a Gaussian wave packet solution. We show that the shape of this wave packet does not depend on the driving force. Afterwards, using this wave packet state, we calculate the expectation values of the position and momentum, their fluctuations and the associated uncertainty product. We find that these expectation values are complex numbers and as a consequence the position and momentum operators are not physical observables and the uncertainty product is physically unacceptable.
\n We have formulated a model of a complex (two-dimensional) quantum harmonic oscillator. All dynamical physical variables are expressed in terms of the creation and annihilation operators, viz., $a_{z}, \\bar{a}_{z}\\ {\\rm and} \\ a_{\\bar{z}}, \\bar{a}_{\\bar{z}} $. The Hamiltonian of the system is $H_{z\\bar{z}} = (\\bar{a}_{z}a_{z} + 1) + \\omega L_{z} $, where ω is the oscillator frequency and $L_{z} = \\frac{\\hbar }{2}\\,(\\bar{a}_{\\bar{z}} a_{\\bar{z}}-\\bar{a}_{z}a_{z}) $ is the orbital angular momentum. The oscillator is found to be described by a conserved orbital angular momentum (Lz) besides energy. While the ground-state wave function is real, all excited states are complex and degenerate. The oscillator in these states carry a quantum of charge of $e^{{\\ast}} = \\frac{n}{2n\\pm1} e $. These degenerate wave functions are eigenstates of the orbital angular momentum with eigenvalues nℏ and −nℏ, where h=2πℏ is the Planck's constant and n=1, 2, … . The two wave functions are degenerate with energy En=(n+1)ℏω. The comparison with Landau level reveals that in the presence of the magnetic field, B, where ω is equal to the cyclotron frequency, the current moment is quantized and is proportional to the square root of the magnetic field, i.e., ${\\cal I}_{n}\\propto n e \\sqrt{B} $.\n