Bicoherent-state path integral quantization of a non-hermitian hamiltonian

Highlights

• Path integral quantization of non-hermitian systems.

• Novel bicoherent state path integration.

• Feynman path integrals for non-hermitian systems.

• Generalization of conventional coherent state path integration.

• Pseudo-boson and bicoherent state quantization of Swanson’s model.

Abstract

We introduce, for the first time, bicoherent-state path integration as a method for quantizing non-hermitian systems. Bicoherent-state path integrals arise as a natural generalization of ordinary coherent-state path integrals, familiar from hermitian quantum physics. We do all this by working out a concrete example, namely, computation of the propagator of a certain quasi-hermitian variant of Swanson’s model, which is not invariant under conventional $PT$-transformation. The resulting propagator coincides with that of the propagator of the standard harmonic oscillator, which is isospectral with the model under consideration by virtue of a similarity transformation relating the corresponding hamiltonians. We also compute the propagator of this model in position space by means of Feynman path integration and verify the consistency of the two results.

Introduction

In this paper we focus on the quantum-mechanical oscillator described by the non-self-adjoint hamiltonian

$$H_{\theta }=\frac{1}{2}\left(p^2+x^2\right)-\frac{i}{2}(tan2\theta )\left(p^2-x^2\right).$$

acting on wave-functions in the standard Hilbert space $\mathcal{H}={\mathcal{L}}^2\left(\mathbb{R}\right)$. Here $\theta$ is a real parameter restricted to the range

$$-\frac{\pi }{4}<\theta <\frac{\pi }{4}$$

and $x$ and $p$ are the usual (hermitian) canonical position and momentum operators satisfying^{3 4} $[x,p]=i\mathbb{1}$.

For our paper to be self-contained, and for the sake of pedagogical clarity, in this section we shall review in some detail all the relevant facts about (1.1) and the dynamics associated with it, which are necessary for path-integral quantization of this system.

The hamiltonian (1.1) is a variant [1] of Swanson’s model [2], which unlike the latter, is not invariant under conventional $PT$-symmetry [3]. In fact, under $PT$-transformation, $H_{\theta }\to H_{-\theta }=H_{\theta }^{†}$. Note, however, that the canonical transformation $x\to p,p\to -x$, transforms $H_{-\theta }$ to $H_{\theta }$. Thus, $H_{\theta }$ and $H_{-\theta }$ are isospectral.

We can rewrite $H_{\theta }$ in yet a different form as

$$H_{\theta }=\frac{1}{2cos2\theta }\left[{\left(e^{-i\theta }p\right)}^2+{\left(e^{i\theta }x\right)}^2\right].$$

Thus, $H_{\theta }$ is obtained from the standard harmonic oscillator $H_{\theta =0}$ (with spectrum $E_n=n+\frac{1}{2}$) by a complex canonical transformation

$$x\to X_{\theta }=e^{i\theta }x,p\to P_{\theta }=e^{-i\theta }p,$$

such that

$$[X_{\theta },P_{\theta }]=i\mathbb{1},$$

followed by multiplication by an overall scale factor

$${\omega }_{\theta }=\frac{1}{cos2\theta }.$$

(Note that ${\omega }_{\theta }$ is well defined for all $\theta$ in the range (1.2).) We thus conclude that the eigenvalues of $H_{\theta }$ are

$$E_n={\omega }_{\theta }\left(n+\frac{1}{2}\right)n=0,1,2,\dots .$$

This spectrum is an even function of $\theta$. Consequently, once again, we see that the spectrum of $H_{\theta }$ is invariant under $PT$-transformation. It obviously coincides with the spectrum of the conventional hermitian harmonic oscillator

$$h_{\theta }={\omega }_{\theta }\left(a^{†}a+\frac{1}{2}\mathbb{1}\right),$$

where $a=\frac{1}{\sqrt{2}}(x+ip)$ and $a^{†}=\frac{1}{\sqrt{2}}(x-ip)$ are the standard annihilation and creation operators. In fact, $H_{\theta }$ and $h_{\theta }$ are isospectral because they are related by the similarity transformation [4]

$$H_{\theta }=T_{\theta }{h}_{\theta }{T}_{\theta }^{-1},$$

where

$$T_{\theta }=e^{\frac{i\theta }{2}(a^2-a^{†2})}$$

is an unbounded, self-adjoint positive-definite operator, which we readily recognize as the squeezing operator [5] with imaginary squeezing parameter $i\theta$.Consistency of (1.3), (1.9) then requires that

$$X_{\theta }=T_{\theta }x{T}_{\theta }^{-1}\mathrm{and}{P}_{\theta }=T_{\theta }p{T}_{\theta }^{-1},$$

as well. This is indeed the case, because (1.4) is nothing but the result of the non-unitary squeezing transformation with $T_{\theta }$.

For the benefit of readers not familiar with squeezed coherent states, we note that in the position representation

$$T_{\theta }=e^{-\frac{\theta }{2}(xp+px)}=e^{i\frac{\theta }{2}}{e}^{i\theta \frac{d}{dlogx}}.$$

Thus, $T_{\theta }$ acting on any sensible function $\psi \left(x\right)$ shifts $logx$ by $i\theta$, followed by a multiplication by an overall phase: $T_{\theta }\psi \left(x\right)=e^{i\frac{\theta }{2}}\psi \left(e^{i\theta }x\right)$. Using this representation of $T_{\theta }$ and the fact that $T_{\theta }^{-1}=T_{-\theta }$, we can prove that $T_{\theta }\left(x\left(T_{\theta }^{-1}\psi \left(x\right)\right)\right)=e^{i\theta }x\psi \left(x\right)$ in a straightforward manner. In a similar way, we can prove that $P_{\theta }\psi \left(x\right)=e^{-i\theta }p\psi \left(x\right)=-i{e}^{-i\theta }\frac{d\psi \left(x\right)}{dx}$.

The similarity transformation (1.11) implies that $X_{\theta }$ and $P_{\theta }$ are isospectral to $x$ and $p$. That is, $X_{\theta }$ and $P_{\theta }$ are diagonalizable with purely real spectra. At the same time, (1.4) tells us that they are also proportional to $x$ and $p$, up to complex phases. However, this does not lead to a contradiction, because the similarity transformation (1.11) maps $\mathcal{H}$ onto the Hilbert space ${\mathcal{H}}_{\theta }$ defined following Eq. (1.14) below. The relations (1.4) are only valid in ${\mathcal{H}}_{\theta }$, where $x$ and $p$ are not hermitian. On the other hand, the operators $x$ and $p$ which appear on the right-hand sides of the two equations in (1.11), do operate on $\mathcal{H}$, where they are hermitian.

It follows from (1.9) that $H_{\theta }^{†}=T_{\theta }^{-1}{h}_{\theta }{T}_{\theta }=T_{\theta }^{-2}{H}_{\theta }{T}_{\theta }^2$. That is, $H_{\theta }^{†}$ and $H_{\theta }$ are also related by a similarity transformation, or equivalently, satisfy the intertwining relation⁵

$$H_{\theta }^{†}{g}_{\theta }=g_{\theta }{H}_{\theta },$$

with

$$g_{\theta }=T_{\theta }^{-2}=e^{i\theta (a^{†2}-a^2)}=e^{\theta (xp+px)}.$$

The intertwining relation (1.13) means that $H_{\theta }$ is actually hermitian with respect to the metric $g_{\theta }$, namely, in the Hilbert space ${\mathcal{H}}_{\theta }$ equipped with inner product

$${〈{\psi }_1|{\psi }_2〉}_{\theta }=〈{\psi }_1\left|g_{\theta }\right|{\psi }_2〉.$$

That is,

$${〈H_{\theta }{\psi }_1|{\psi }_2〉}_{\theta }={〈{\psi }_1|H_{\theta }{\psi }_2〉}_{\theta },$$

by virtue of (1.13). As $\theta \to 0$, this Hilbert space turns, of course, into the standard Hilbert space ${\mathcal{H}}_{\theta =0}\equiv \mathcal{H}(\equiv {\mathcal{L}}^2\left(\mathbb{R}\right))$ with metric $g_0=\mathbb{1}$.

Clearly, $H_{\theta }$ is an observable in our theory, as are $X_{\theta }$ and $P_{\theta }$. More generally, any operator $A$ in this theory, which is hermitian with respect to $g_{\theta }$, that is, satisfies the intertwining relation

$$A^{†}{g}_{\theta }=g_{\theta }A,$$

is an observable. Such observables are sometimes referred to as a quasi-hermitian operators [6], because $A$ is related to a hermitian operator by a similarity transformation, in a manner analogous to (1.9), (1.11).

The Swanson model [2] mentioned above, namely, the $\mathcal{PT}-$symmetric relative of (1.1), offers yet another example of a quasi-hermitian hamiltonian. This well-studied model also demonstrates the fact that given the hamiltonian $H$, the metric $g$ is not unique [2], [6], [7], [8]. Various choices of $g$ lead to different quantizations of the system, with different irreducible sets of observables, in different Hilbert spaces.

More generally, the hamiltonian of any $\mathcal{PT}-$symmetric quantum mechanical system, with unbroken $\mathcal{PT}$ symmetry, is hermitian with respect to the $\mathcal{CPT}$-inner product of that system [3], and therefore possesses real spectrum.

Path integration is a method for computing matrix elements of the time evolution operator (or propagator) $e^{-i{H}_{\theta }t}$, which governs dynamics of the system. Thus, a few words are in order concerning dynamics of observables in our model.

Time evolution in the Hilbert space ${\mathcal{H}}_{\theta }$ is unitary, namely,

$$e^{i{H}_{\theta }^{†}t}{g}_{\theta }{e}^{-i{H}_{\theta }t}=g_{\theta },$$

which is a trivial consequence of (1.13).

Consequently, Quantum Mechanics formulated in ${\mathcal{H}}_{\theta }$ is essentially not different from conventional hermitian Quantum Mechanics. To start with, the inner product of any two states $|{\psi }_{1,2}\left(t\right)〉=e^{-i{H}_{\theta }t}|{\psi }_{1,2}\left(0\right)〉$, which evolve in time according to the Schrödinger equation, is time-independent: ${〈{\psi }_1\left(t\right)|{\psi }_2\left(t\right)〉}_{\theta }={〈{\psi }_1\left(0\right)|{\psi }_2\left(0\right)〉}_{\theta }$. This implies, of course, probability conservation in the case of identical states.

By forming matrix elements of an observable $A=A\left(0\right)$ in the Schrödinger picture, we immediately deduce from (1.17) that its Heisenberg picture counterpart $A\left(t\right)$ evolves according to

$$A\left(t\right)=e^{i{H}_{\theta }t}A\left(0\right)e^{-i{H}_{\theta }t}.$$

Equivalently, the Heisenberg equation of motion resulting from (1.19) is

$$i\dot{A}\left(t\right)=[A\left(t\right),H_{\theta }].$$

Thus, observables which commute with $H_{\theta }$ are conserved in time.

It is trivial to verify that $A\left(t\right)$ fulfills the intertwining relation (1.17) at all times, and is therefore an observable. This is consistent with the fact that the spectrum of $A\left(t\right)$ is conserved in time, because formally, (1.19) is a similarity transformation.⁶

The Heisenberg equations of motion (1.20) for $X_{\theta }$ and $P_{\theta }$,

$${\dot{X}}_{\theta }={\omega }_{\theta }{P}_{\theta }\mathrm{and}{\dot{P}}_{\theta }=-{\omega }_{\theta }{X}_{\theta }$$

are similar in form to the corresponding equations of motion in the hermitian problem. We can readily solve them and find

$$X_{\theta }\left(t\right)=e^{i{H}_{\theta }t}{X}_{\theta }\left(0\right)e^{-i{H}_{\theta }t}=X_{\theta }\left(0\right)cos{\omega }_{\theta }t+P_{\theta }\left(0\right)sin{\omega }_{\theta }t$$$$P_{\theta }\left(t\right)=e^{i{H}_{\theta }t}{P}_{\theta }\left(0\right)e^{-i{H}_{\theta }t}=P_{\theta }\left(0\right)cos{\omega }_{\theta }t-X_{\theta }\left(0\right)sin{\omega }_{\theta }t.$$

These solutions preserve the equal-time canonical commutation relation

$$[X_{\theta }\left(t\right),P_{\theta }\left(t\right)]=i\mathbb{1}$$

at all times.

Formally, by invoking the similarity transformations (1.9), (1.11) to the middle term in each of the equations in (1.22), we can easily show that

$$X_{\theta }\left(t\right)=T_{\theta }\left(e^{i{h}_{\theta }t}x\left(0\right)e^{-i{h}_{\theta }t}\right)T_{\theta }^{-1}=T_{\theta }x\left(t\right)T_{\theta }^{-1}$$$$P_{\theta }\left(t\right)=T_{\theta }\left(e^{i{h}_{\theta }t}p\left(0\right)e^{-i{h}_{\theta }t}\right)T_{\theta }^{-1}=T_{\theta }p\left(t\right)T_{\theta }^{-1}.$$

Thus, (1.11) holds for operators in the Heisenberg representation as well. $X_{\theta }\left(t\right)$ and $P_{\theta }\left(t\right)$ are (formally) similar to their hermitian counterparts $x\left(t\right)$ and $p\left(t\right)$. They are therefore diagonalizable, and their spectra are real as well.

We shall now derive the spectral decompositions of ${\hat{X}}_{\theta }\left(t\right)$ and ${\hat{P}}_{\theta }\left(t\right)$ in terms of complete biorthogonal bases of right- and left-eigenvectors.⁷ To this end, we start by substituting the standard spectral decompositions

$$\hat{x}\left(0\right)={\int }_{-\infty }^{\infty }dxx|x〉〈x|,{\int }_{-\infty }^{\infty }dx|x〉〈x|=\mathbb{1},〈x|x^{\prime }〉=\delta (x-x^{\prime })$$$$\hat{p}\left(0\right)={\int }_{-\infty }^{\infty }dpp|p〉〈p|,{\int }_{-\infty }^{\infty }dp|p〉〈p|=\mathbb{1},〈p|p^{\prime }〉=\delta (p-p^{\prime })$$

of $\hat{x}\left(0\right)$ and $\hat{p}\left(0\right)$, valid in the Hilbert space $\mathcal{H}$, in (1.24). Therefore,

$${\hat{X}}_{\theta }\left(t\right)={\int }_{-\infty }^{\infty }dxx\left(T_{\theta }{e}^{i{h}_{\theta }t}|x〉\right)\left(〈x|e^{-i{h}_{\theta }t}{T}_{\theta }^{-1}\right)$$$${\hat{P}}_{\theta }\left(t\right)={\int }_{-\infty }^{\infty }dpp\left(T_{\theta }{e}^{i{h}_{\theta }t}|p〉\right)\left(〈p|e^{-i{h}_{\theta }t}{T}_{\theta }^{-1}\right).$$

From this equation we can read-off the desired spectral decompositions in terms of complete biorthogonal bases of right- and left-eigenvectors for the time-dependent operators as follows. For ${\hat{X}}_{\theta }\left(t\right)$ we obtain

$${\hat{X}}_{\theta }\left(t\right)={\int }_{-\infty }^{\infty }dxx{|x;t〉}_R{}_L〈x;t|$$$${|x;t〉}_R=T_{\theta }|x;t〉=T_{\theta }{e}^{i{h}_{\theta }t}|x〉=e^{i{H}_{\theta }t}{T}_{\theta }|x〉$$$${|x;t〉}_L=T_{\theta }^{-1}|x;t〉=T_{\theta }^{-1}{e}^{i{h}_{\theta }t}|x〉=g_{\theta }{e}^{i{H}_{\theta }t}{T}_{\theta }|x〉=g_{\theta }{|x;t〉}_R,$$

where we used (1.9), (1.14), and invoked hermiticity of (1.10). Completeness and biorthogonality follow immediately:

$${\int }_{-\infty }^{\infty }dx{|x;t〉}_R{}_L〈x;t|=\mathbb{1},{}_L{〈x;t|x^{\prime };t〉}_R=\delta (x-x^{\prime }).$$

Similarly, for ${\hat{P}}_{\theta }\left(t\right)$ we obtain

$${\hat{P}}_{\theta }\left(t\right)={\int }_{-\infty }^{\infty }dpp{|p;t〉}_R{}_L〈p;t|$$$${|p;t〉}_R=T_{\theta }|p;t〉=T_{\theta }{e}^{i{h}_{\theta }t}|p〉=e^{i{H}_{\theta }t}{T}_{\theta }|p〉$$$${|p;t〉}_L=T_{\theta }^{-1}|p;t〉=T_{\theta }^{-1}{e}^{i{h}_{\theta }t}|p〉=g_{\theta }{e}^{i{H}_{\theta }t}{T}_{\theta }|p〉=g_{\theta }{|p;t〉}_R,$$

and

$${\int }_{-\infty }^{\infty }dp{|p;t〉}_R{}_L〈p;t|=\mathbb{1},{}_L{〈p;t|p^{\prime };t〉}_R=\delta (p-p^{\prime }).$$

These spectral decompositions, into eigenstates with purely real eigenvalues $x$ and $p$, will be crucial for us in constructing the Feynman path integral for this system in Section 5.

The classical counterparts of the operators $X_{\theta }\left(t\right)$ and $P_{\theta }\left(t\right)$ in (1.22) are real variables, corresponding to their parent quantum observables with their real spectra. This, together with the classical limit of (1.4), means that $e^{i\theta }x$ and $e^{-i\theta }p$ should be taken as real variables as well. That is, in the classical limit of the system originally defined in ${\mathcal{H}}_{\theta }$, the particle actually moves along the line in the complex $x$-plane making an angle $-\theta$ with the real axis. (This is precisely the anti-Stokes line mentioned in Section 2.1.1 below.)

The classical variables $X_{\theta }\left(t\right)$ and $P_{\theta }\left(t\right)$ obviously have canonical Poisson brackets, and any function $A(X_{\theta }\left(t\right),P_{\theta }\left(t\right))$ of these phase-space variables satisfies the canonical Hamiltonian equation of motion

$$\dot{A}=\{A,H_{\theta }\}=\frac{\partial A}{\partial X_{\theta }}\frac{\partial H_{\theta }}{\partial P_{\theta }}-\frac{\partial A}{\partial P_{\theta }}\frac{\partial H_{\theta }}{\partial X_{\theta }},$$

obtained from (1.20) in the classical limit.

The main objective of this paper is to compute the propagator associated with (1.1) by means of path integration based on bicoherent states [9]. Bicoherent states are a powerful tool in studying non-hermitian systems such as (1.1), and the present paper is the first application of bicoherent states to path integration. We shall also compute the transition amplitude of (1.1) in position space, by means of Feynman path integration, and verify the consistency of the two results. As a byproduct of this computation, we shall gain insight into what Feynman path integrals of non-hermitian quantum systems really mean.

The object of main interest in this paper is the probability amplitude

$$A_{fi}\left(t\right)={〈{\psi }_f\left|e^{-i{H}_{\theta }t}\right|{\psi }_i〉}_{\theta }=〈{\psi }_f\left|g_{\theta }{e}^{-i{H}_{\theta }t}\right|{\psi }_i〉$$

for unitary time evolution of an initial state $|{\psi }_i〉$ into a final state $|{\psi }_f〉$. Time evolution is unitary, because (1.32) is defined in the Hilbert space ${\mathcal{H}}_{\theta }$. In this evolution, the metric $g_{\theta }$ acts as a boundary term, at the end of the process. Its sole function is to ensure unitarity of the process, so that ${\left|A_{fi}\left(t\right)\right|}^2$ is the probability to start from $|{\psi }_i〉$ and end up at $|{\psi }_f〉$ after time $t$. Now, imagine sandwiching the propagator $e^{-i{H}_{\theta }t}$ between two resolutions of unity associated with the position operator, as in the first equation in (1.25):

$$A_{fi}\left(t\right)=〈{\psi }_f|g_{\theta }{\int }_{-\infty }^{\infty }d{x}_1|x_1〉〈x_1|e^{-i{H}_{\theta }t}{\int }_{-\infty }^{\infty }d{x}_2|x_2〉〈x_2|{\psi }_i〉={\int }_{-\infty }^{\infty }d{x}_{1}d{x}_2〈{\psi }_f\left|g_{\theta }\right|x_1〉〈x_1\left|e^{-i{H}_{\theta }t}\right|x_2〉〈x_2|{\psi }_i〉$$

The factors $〈{\psi }_f\left|g_{\theta }\right|x_1〉$ and $〈x_2|{\psi }_i〉$ are fixed position-dependent wave-functions, determined by the initial and final states. The interesting term, encoding the dynamics of our system, is the matrix element

$$G(x_1,x_2;t)=〈x_1\left|e^{-i{H}_{\theta }t}\right|x_2〉$$

of the propagator. It is this object (or its analog, with $|x_{1,2}〉$ replaced by a pair of bicoherent states to be defined below) which we shall derive path-integral representations for. As it stands, we can think of it simply as a matrix element of a non-hermitian operator acting on the standard Hilbert space $\mathcal{H}$. This is the approach we shall adopt henceforth throughout the rest of this paper.

In contrast to the hermitian case, (1.34) by itself is of course not a probability amplitude, but this should not prevent us from representing it as a path integral. After computing (1.34) by what-ever method we choose, we can plug it back into (1.33) and compute the relevant probability amplitude.

The rest of this paper is organized as follows: Section 2 is devoted to a pedagogical review of the application of pseudo-bosons and the bicoherent states associated with them to studying the non-hermitian oscillator (1.1). In Section 3 we compute the matrix elements of the propagator $e^{-i{H}_{\theta }t}$ both in the basis of bicoherent states and between position eigenstates, and verify the consistency of these matrix elements. Sections 4 The bicoherent-state path integral, 5 The Feynman path integral lie at the heart of this paper and contain our main results: In Section 4, based on the detailed expositions of all the preceding sections, we derive, for the first time ever, the bicoherent-state path integral for the bicoherent propagation amplitude derived in Section 3. Similarly, Section 5 is devoted to deriving the corresponding Feynman path integral. We draw our conclusions and give brief outlook in Section 6. Some technical details are relegated to the Appendix.