---
abstract: 'The Hilbert space in [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric quantum mechanics is formulated as a linear vector space with a dynamic inner product. The most general [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric matrix Hamiltonians are constructed for $2\times 2$ and $3\times 3$ cases. In the former case, the [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian represents the most general matrix Hamiltonian with a real spectrum. In both cases, Hermitian matrices are shown to be special cases of [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric matrices. This finding confirms and strengthens the early belief that the [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric quantum mechanics is a generalization of the conventional Hermitian quantum mechanics.'
author:
- 'Qing-hai Wang'
- 'Song-zhi Chia'
- 'Jie-hong Zhang'
date: 'April 16, 2010'
title: '**[$\mathcal{P}$]{}[$\mathcal{T}$]{} Symmetry as a Generalization of Hermiticity**'
---

Introduction
============

The seminal paper by Bender and Boettcher in 1998 has lead to an alternative formulation of quantum mechanics (QM): non-Hermitian [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric QM [@BB]. Although the Hamiltonians involved appear to be not Hermitian ($H\neq H^\dag$), they yield only real spectra and the time evolution is unitary when the [$\mathcal{P}$]{}[$\mathcal{T}$]{} symmetry is not broken [@BB; @BBJ; @Review; @AliReview].

Since the introduction of [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric QM, there has been a debate about whether [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric QM is more general than the conventional QM. In the first paper about [$\mathcal{P}$]{}[$\mathcal{T}$]{} symmetry, Bender and Boettcher stressed that [$\mathcal{P}$]{}[$\mathcal{T}$]{} symmetry is a weaker condition than Hermiticity and a [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric theory can be considered as a complex extension of a conventional Hermitian theory [@BB]. In 2002, Bender [*et al.*]{} constructed a positive-definite norm of [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric QM using the newly discovered [$\mathcal{C}$]{} operator [@BBJ]. The claim that [$\mathcal{P}$]{}[$\mathcal{T}$]{} symmetry is more general than Hermiticity was restated in the paper’s title: [*“Complex Extension of Quantum Mechanics.”*]{} Interestingly, in the same year, Mostafazadeh pointed out that all pseudo-Hermitian Hamiltonians are Hermitian (self-adjoint) with respect to a positive-semidefinite inner product [@AliJMP2002]. As such a [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian can be considered as a special case of a pseudo-Hermitian Hamiltonian with a real spectrum. In 2003, Mostafazadeh showed that diagonalizable pseudo-Hermitian Hamiltonians are extensions of [$\mathcal{P}$]{}[$\mathcal{T}$]{}- or [$\mathcal{C}$]{}[$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonians [@AliJMP2003]. A less general result was obtained independently by Bender [*et al.*]{} in Ref. [@BMWparity]. But they took a different point of view which can be seen from the title of the paper: [*“All Hermitian Hamiltonians Have Parity.”*]{}

The study of finite-dimensional matrix Hamiltonians may shed some light on the debate. The original $2\times 2$ [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric matrix Hamiltonian introduced in Ref. [@BBJ] has three real parameters. In 2003, Bender [*et al.*]{} extended it to a four-parameter class and to higher dimensions [@BMW]. From their results, they identified [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonians and Hermitian Hamiltonians as two distinct extensions of real-symmetric Hamiltonians. For matrix dimension higher than two, [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonians have less real parameters than Hermitian Hamiltonians in the same dimension. This view was challenged immediately by Mostafazadeh in a paper titled [*“Exact [$\mathcal{P}$]{}[$\mathcal{T}$]{}-Symmetry Is Equivalent to Hermiticity”*]{} [@AliJPA2003]. In this paper, the author constructed a $2\times 2$ [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian with five real parameters. In 2007, Mostafazadeh and Özçelik explicitly constructed the most general $2\times 2$ quasi-Hermitian Hamiltonian with six real parameters [@AliTurk2007].

In this paper, we are trying to settle the debate by constructing the most general [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric matrix Hamiltonians. Part of the reason for the debate is due to the different usage of the terminology. To avoid further confusion, we restrict the term “Hermitian conjugate” or “$\dag$” in the Dirac sense: complex conjugate and transpose. In the present paper, we formulate [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric QM slightly different from the literature in two aspects. First, we define the time reversal operator as the Dirac conjugation rather than just the complex conjugation used in Ref. [@BB; @BBJ; @BMW]. Second, we do not introduce any biorthonormal basis as in Ref. [@AliJMP2002; @AliJPA2003]. We link the [$\mathcal{C}$]{}[$\mathcal{P}$]{}[$\mathcal{T}$]{}-inner product to the general inner product in a linear vector space with a weight function. The latter is the standard notation which can be found in modern quantum mechanics textbooks such as Ref. [@BalBook].

With this formulation, we solve for the general [$\mathcal{P}$]{} operator and present the solutions explicitly in the case of $2\times 2$ and $3\times 3$. Using the general [$\mathcal{P}$]{} operators, we construct the general [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric matrix Hamiltonians. We confirm that the most general $2\times 2$ [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian has six real parameters as shown in Ref. [@AliTurk2007]. We find that the general $3\times 3$ [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian has thirteen real parameters. In the case of $2\times 2$, the general [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian represents the most general matrix Hamiltonian with a real spectrum. Interestingly, this is not true in the case of $3\times 3$. In both cases, we show clearly that all Hermitian Hamiltonians are just special cases of [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonians. From these finite dimensional results, we conjecture that [$\mathcal{P}$]{}[$\mathcal{T}$]{} symmetry is a generalization of Hermiticity in general.

The paper is organized as follows. In Sec. \[sec:PTQM\], we give our formulation of [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric QM. In Sec. \[sec:2x2\], we illustrate our ideas in the case of $2\times 2$. All relevant operators or matrices are calculated explicitly. Special cases discussed in the literature are analyzed. In Sec. \[sec:3x3\], we construct the [$\mathcal{P}$]{} operator and the [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian in the case of $3\times 3$. Finally, in Sec. \[sec:conclusion\], we give some concluding remarks.

[$\mathcal{P}$]{}[$\mathcal{T}$]{}-Symmetric Quantum Mechanics {#sec:PTQM}
==============================================================

In this section, we formulate [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric QM. We start with a brief summary on the inner product in QM. We adopt the notation used in Ref. [@BalBook].

Inner Product {#sec:inner}
-------------

In quantum mechanics, the Hilbert space can be considered as a linear vector space associated with an inner product. The inner product between two quantum states, denoted $(\cdot,\cdot)$, must satisfy the following conditions [@BalBook]:

1.  $(\psi,\phi)$ is a complex number,

2.  $(\psi,\phi)=(\phi,\psi)^*$, where $*$ denotes complex conjugate,

3.  $(\psi,c_1\phi_1+c_2\phi_2)=c_1(\psi,\phi_1)+c_2(\psi,\phi_2)$, where $c_1$ and $c_2$ are complex numbers,

4.  $(\phi,\phi)\geq 0$, with equality holding if and only if $\phi=0$.

In general, we may define the inner product as $$(\psi,\phi)\equiv {\langle}\psi|W|\phi{\rangle},
\label{eqn:inner}$$ where $W$ is the weight function and the bra state is defined as the Hermitian conjugate of the ket state, ${\langle}\cdot|\equiv |\cdot{\rangle}^\dag$ [@BalBook].

From [Eq. (\[eqn:inner\])]{} and the first three properties of the inner product, it can be easily shown that $W$ must be a Hermitian operator: $W=W^\dag$. From the fourth property of the inner product, $W$ has to be positive definite. That is, all the eigenvalues of $W$ are positive.

In the Hilbert space defined above, a self-adjoint operator, such as the Hamiltonian $H$, satisfies $$(\psi,H\phi)=(H\psi,\phi)
\label{eqn:self-a}$$ for arbitrary states $\phi$ and $\psi$. All eigenvalues of a self-adjoint operator are real. And the eigenstates corresponding to different eigenvalues are orthogonal [@BalBook].

Plugging the self-adjoint condition [(\[eqn:self-a\])]{} into the definition of the inner product in [(\[eqn:inner\])]{}, we obtain $$WH=H^\dag W.
\label{eqn:weight}$$ We may consider this equation as the definition of the weight function $W$ for a given Hamiltonian $H$. Thus, the inner product is dynamic (Hamiltonian dependent) in general.

In conventional QM, the weight function can be chosen as the identity operator. In this case, the self-adjoint condition in [(\[eqn:weight\])]{} reduces to the Hermiticity condition: $H=H^\dag$. Since the identity operator is independent on Hamiltonians, the inner product is no longer dynamic.

[$\mathcal{P}$]{}[$\mathcal{T}$]{} Symmetry {#sec:PT}
-------------------------------------------

We define the time reversal operator, [$\mathcal{T}$]{} as Dirac conjugate. That is, for an operator $A$, $${\ensuremath{\mathcal{T}}}A {\ensuremath{\mathcal{T}}}= A^\dag.$$ It follows that ${\ensuremath{\mathcal{T}}}^2=\openone$, here $\openone$ is the identity matrix. Note that our definition of the [$\mathcal{T}$]{} operator differs from Refs.[@BB; @BBJ; @BMW]. This definition allows us to have a more general parity operator.

For the parity operator [$\mathcal{P}$]{}, we demand it to commute with the time reversal operator and to be an involution. That is,

1.  $[{\ensuremath{\mathcal{P}}},{\ensuremath{\mathcal{T}}}]=0$, or equivalently, ${\ensuremath{\mathcal{P}}}={\ensuremath{\mathcal{P}}}^\dag$,

2.  ${\ensuremath{\mathcal{P}}}^2=\openone$.

There are obviously two trivial solutions to these constraints, $${\ensuremath{\mathcal{P}}}_0=\pm\openone.
\label{eqn:P0}$$ We will discuss the non-trivial $2\times 2$ solutions of [$\mathcal{P}$]{} in Sec. \[sec:2x2\] and $3\times 3$ in Sec. \[sec:3x3\].

A [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian commutes with the combination operator [$\mathcal{P}$]{}[$\mathcal{T}$]{}: $$[H,{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{T}}}]=0 \qquad \Leftrightarrow \qquad {\ensuremath{\mathcal{P}}}H^\dag {\ensuremath{\mathcal{P}}}= H.
\label{eqn:PT-H}$$ Using the trivial solutions of ${\ensuremath{\mathcal{P}}}_0$ in [(\[eqn:P0\])]{}, we get that a ${\cal P}_0{\cal T}$-symmetric Hamiltonian is Hermitian $$[H,{\cal P}_0{\cal T}]=0 \qquad \Leftrightarrow \qquad H=H^\dag.$$ In this sense, a Hermitian Hamiltonian is ${\ensuremath{\mathcal{P}}}_0{\ensuremath{\mathcal{T}}}$-symmetric with ${\ensuremath{\mathcal{P}}}_0$ to be the plus or minus identity matrix.

In general, the eigenstates of a [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian with different eigenvalues are [*not*]{} orthogonal with respect to the Dirac inner product. To solve this problem, one may define a [$\mathcal{P}$]{}[$\mathcal{T}$]{} inner product as $$(\psi,\phi)_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{T}}}} \equiv {\langle}\psi|{\ensuremath{\mathcal{P}}}|\phi{\rangle}.$$ From the commutation relation between $H$ and [$\mathcal{P}$]{}[$\mathcal{T}$]{}, it can be shown that the eigenstates of $H$ with different eigenvalues are orthogonal with respect to the [$\mathcal{P}$]{}[$\mathcal{T}$]{} inner product [@BB; @Review].

However, the norm with respect to the [$\mathcal{P}$]{}[$\mathcal{T}$]{} inner product is not positive definite. This is simply because [$\mathcal{P}$]{} has negative eigenvalues. One has to normalize the eigenstates to $\pm 1$. To overcome this difficulty, one needs to find a positive-definite norm by introducing the [$\mathcal{C}$]{} operator [@BBJ]. Here, we define the [$\mathcal{C}$]{} operator as $${\ensuremath{\mathcal{C}}}\equiv \sum_i |E_i{\rangle}{\langle}E_i|{\ensuremath{\mathcal{P}}},
\label{eqn:C}$$ where $|E_i{\rangle}$ are the eigenstates of $H$ with the [$\mathcal{P}$]{}[$\mathcal{T}$]{}-norm $+1$ or $-1$.

Our definition of the [$\mathcal{C}$]{} operator has the same properties as the one constructed in Ref. [@BBJ]. From the orthogonality and the (non-positive-definite) normalization of the [$\mathcal{P}$]{}[$\mathcal{T}$]{} inner product, it can be shown that $|E_i{\rangle}$ are eigenstates of [$\mathcal{C}$]{} with eigenvalue equal to the [$\mathcal{P}$]{}[$\mathcal{T}$]{}-norm: $${\ensuremath{\mathcal{C}}}|E_i{\rangle}={\langle}E_i|{\ensuremath{\mathcal{P}}}|E_i{\rangle}|E_i{\rangle}.
\label{eqn:CE}$$ Thus, [$\mathcal{C}$]{} commutes with $H$. The [$\mathcal{C}$]{} operator also commutes with [$\mathcal{P}$]{}[$\mathcal{T}$]{}: $[{\ensuremath{\mathcal{C}}},{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{T}}}]=0$. This fact can be verified by using ${\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{C}}}^\dag{\ensuremath{\mathcal{P}}}={\ensuremath{\mathcal{C}}}$. Because the eigenvalues of ${\ensuremath{\mathcal{C}}}^2$ are all unity, the [$\mathcal{C}$]{} operator is an involution just like the [$\mathcal{P}$]{} operator.

Note that we do not require $|E_i{\rangle}$ to be simultaneous eigenstates of [$\mathcal{P}$]{}[$\mathcal{T}$]{} and of $H$. In fact, this can only be achieved in the special cases of symmetric matrices, such as Hamiltonians constructed in Refs. [@BBJ; @BMW]. We will discuss more details about these two examples in Sec.\[sec:2x2\].

Equipped with the [$\mathcal{C}$]{} operator, we are ready to construct an inner product with the positive-definite norm. We define the [$\mathcal{C}$]{}[$\mathcal{P}$]{}[$\mathcal{T}$]{} inner product as $$(\psi,\phi)_{{\ensuremath{\mathcal{C}}}{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{T}}}} \equiv {\langle}\psi|{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{C}}}|\phi{\rangle}.$$ Comparing to the general inner product in [(\[eqn:inner\])]{}, we recognize the weight function for the [$\mathcal{C}$]{}[$\mathcal{P}$]{}[$\mathcal{T}$]{}-inner product is $$W={\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{C}}}.$$

Since the weight function is positive-definite, we may find its square root $W=\eta^2$, where $\eta$ is Hermitian, $\eta^\dag=\eta$. Using the operator $\eta$, we may define a Hermitian Hamiltonian $h$, which has same spectrum as a [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian $H$ [@AliJPA2003], $$h \equiv \eta H\eta^{-1}, \quad {\rm where}\quad h=h^\dag.
\label{eqn:h}$$ We would like to emphasize that the transformation $\eta$ is Hermitian rather than unitary, hence the above relation is not unitary equivalence in the usual sense.

$2\times 2$ Case {#sec:2x2}
================

In this section, we illustrate the ideas in the previous section by using $2\times 2$ matrices. We find that any $2\times 2$ Hermitian matrix is a special case of the general [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric matrices. Furthermore, it is found that the most general matrix with a real spectrum must coincide with the general [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric matrix we constructed.

Since the [$\mathcal{P}$]{} operator is an involution, it is a square root of the identity matrix. In $2\times 2$ matrices, other than the trivial roots in [(\[eqn:P0\])]{}, there is a non-trivial root with the form $${\ensuremath{\mathcal{P}}}=\left(
\begin{array}{cc}
\cos\theta & \sin\theta~ {{\rm e}}^{-{{\rm i}}\varphi}\\
\sin\theta~ {{\rm e}}^{{{\rm i}}\varphi} & -\cos\theta
\end{array}
\right),
\label{eqn:P_2x2}$$ where $\theta$ and $\varphi$ are two real parameters. In terms of the Pauli matrices, the [$\mathcal{P}$]{} operator can be written as ${\ensuremath{\mathcal{P}}}={\bf n}^r\cdot\bm{\sigma}$, where ${\bf n}^r\equiv (\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$ is a unit vector.

To find the general [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian, we use the following [*ansatz*]{}: $$H=\varepsilon\openone + \bm{\alpha}\cdot\bm{\sigma},
\label{eqn:H_ansatz}$$ where $\varepsilon$ and $\bm{\alpha}\equiv(\alpha_x,\alpha_y,\alpha_z)$ are complex numbers. Plugging the above [*ansatz*]{} into [(\[eqn:PT-H\])]{}, we get the equations satisfied by $\varepsilon$ and $\bm{\alpha}$: $$\varepsilon = \varepsilon^*, \qquad
\bm{\alpha}+\bm{\alpha}^* = 2 \left(\bm{\alpha}^*\cdot{\bf n}^r\right) {\bf n}^r.
\label{eqn:alpha}$$ The first equation simply says that $\varepsilon$ is real. The second equation can be written as $$\sum_i M_{ki}\alpha_i = \alpha_k^*,\quad {\rm with}\quad M_{ki} \equiv -\delta_{ki} +2 n^r_k n^r_i.$$ In this form, the searching for a [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian becomes an eigenvalue problem. If we separate the real part and the imaginary part as $\bm{\alpha}={\bf A}+{{\rm i}}\,{\bf B}$, then ${\bf A}$ and ${\bf B}$ can be considered as the eigenvectors of the matrix $M$ with eigenvalues $+1$ and $-1$, respectively. The matrix $M$ always has one eigenvector with eigenvalue $+1$, and two eigenvectors with eigenvalue $-1$. It is easy to show that the eigenvector with positive eigenvalue is parallel to ${\bf n}^r$ and the eigenvectors with negative eigenvalue are perpendicular to ${\bf n}^r$. Thus, the solutions to the eigenvalue problem are $${\bf A} = \gamma {\bf n}^r,\qquad
{\bf B} = \mu {\bf n}^\theta +\nu {\bf n}^\varphi,
\label{eqn:AB}$$ where $\gamma$, $\mu$, and $\nu$ are real parameters and ${\bf n}^\theta \equiv (\cos\theta\cos\varphi,\cos\theta\sin\varphi,-\sin\theta)$ and ${\bf n}^\varphi \equiv (-\sin\varphi,\cos\varphi,0)$ are two unit vectors.

Thereafter, plugging $\bm{\alpha} = \gamma {\bf n}^r + {{\rm i}}\mu {\bf n}^\theta + {{\rm i}}\nu {\bf n}^\varphi$ into the [*ansatz*]{} in [(\[eqn:H\_ansatz\])]{}, we get the general [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric $2\times 2$ Hamiltonian matrix, $$\begin{aligned}
H&=&\varepsilon\openone + \left(\gamma {\bf n}^r + {{\rm i}}\mu {\bf n}^\theta + {{\rm i}}\nu {\bf n}^\varphi\right)\cdot\bm{\sigma}{\nonumber}\\
&=&\left(
\begin{array}{cc}
\varepsilon + \gamma\cos\theta - {{\rm i}}\mu\sin\theta &
(\gamma\sin\theta + {{\rm i}}\mu\cos\theta+\nu) {{\rm e}}^{-{{\rm i}}\varphi}\\
(\gamma\sin\theta + {{\rm i}}\mu\cos\theta-\nu) {{\rm e}}^{{{\rm i}}\varphi} &
\varepsilon - \gamma\cos\theta + {{\rm i}}\mu\sin\theta
\end{array}
\right).
\label{eqn:H}\end{aligned}$$ This Hamiltonian has six real parameters: $\varepsilon$, $\gamma$, $\mu$, $\nu$, $\theta$, and $\varphi$. All $2\times 2$ Hermitian matrices can be recovered as special cases with $\mu=\nu=0$. In other words, the [$\mathcal{P}$]{}[$\mathcal{T}$]{} symmetry is a generalization of Hermiticity.

The eigenvalues of the Hamiltonian $H$ are $$E_\pm=\varepsilon \pm \sqrt{\gamma^2-\mu^2-\nu^2}.
\label{eqn:Epm}$$ For eigenvalues to be real, it requires $\gamma^2 \geq \mu^2 + \nu^2$. For simplicity, we only consider the non-degenerated case with $\gamma^2 > \mu^2 + \nu^2$ in this paper. The corresponding eigenstates are $$|E_\pm{\rangle}= \frac{u}{\sqrt{2}} \left(
\begin{array}{c}
{{\rm e}}^{{{\rm i}}\kappa_0-{{\rm i}}\varphi} \sqrt{1 + \frac{\nu}{\gamma}\sin\theta \pm \frac{1}{\gamma}\sqrt{\gamma^2-\mu^2-\nu^2} \cos\theta}\\
{{\rm e}}^{{{\rm i}}\kappa_\pm} \sqrt{1 - \frac{\nu}{\gamma}\sin\theta \mp \frac{1}{\gamma}\sqrt{\gamma^2-\mu^2-\nu^2} \cos\theta}
\end{array}\right),
\label{eqn:eigenstates}$$ where we have defined three angles and a normalization constant as $$\begin{aligned}
\kappa_0 &\equiv& \arg \left(\gamma\sin\theta  + \nu + {{\rm i}}\mu \cos\theta \right),{\nonumber}\\
\kappa_\pm &\equiv& \arg \left(-\gamma\cos\theta \pm \sqrt{\gamma^2-\mu^2-\nu^2}  + {{\rm i}}\mu \sin\theta \right),{\nonumber}\\
u &\equiv& \sqrt{\frac{\gamma^2}{\gamma^2-\mu^2-\nu^2}}.\end{aligned}$$ The normalization is chosen such that $${\langle}E_\pm|{\ensuremath{\mathcal{P}}}|E_\pm{\rangle}=\pm {\rm sign}(\gamma).$$

The six-parameter-class of matrix in [(\[eqn:H\])]{} with the condition $\gamma^2 \geq \mu^2 + \nu^2$ coincides with the most general $2\times 2$ matrix with only real eigenvalues. Qualitatively, this can be seen from parameter counting. A general complex $2\times 2$ matrix has eight real parameters. The reallity of all eigenvalues puts two constraints on the matrix. Therefore, the most general $2\times 2$ matrix with only real eigenvalues should consist of six real parameters.

This coincidence can also be proved rigorously. By direct computation, the eigenvalues of an arbitrary $2\times 2$ matrix with the form of the [*ansatz*]{} in [(\[eqn:H\_ansatz\])]{} are $E_\pm=\varepsilon \pm \sqrt{\bm{\alpha}\cdot\bm{\alpha}}$. Imposing reality condition on the eigenvalues leads to $$\varepsilon = \varepsilon^*, \qquad \bm{\alpha}\cdot\bm{\alpha} \geq 0.
\label{eqn:MostGeneral}$$ The first condition is the same constraint on the parameter $\varepsilon$ as in the [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian in [(\[eqn:H\])]{}. The second condition in [(\[eqn:MostGeneral\])]{} implies that the real part and the imaginary part of $\bm{\alpha}$ are perpendicular to each other, ${\bf A}\cdot{\bf B}=0$, and that the real part vector is not shorter than the imaginary part vector, ${\bf A}\cdot{\bf A}\geq {\bf B}\cdot{\bf B}$. Without loss of generality, we may parametrize the real part vector as ${\bf A} = \gamma {\bf n}^r$. Then the above conditions lead to a unique solution for the imaginary part vector which can be parametrized as ${\bf B} = \mu {\bf n}^\theta +\nu {\bf n}^\varphi$. Since ${\bf A}$ is not shorter than ${\bf B}$, we have $\gamma^2 \geq \mu^2 + \nu^2$. Clearly then, the [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian in [(\[eqn:H\])]{} represents the most general $2\times 2$ matrices with only real eigenvalues.

For the [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian $H$ in [(\[eqn:H\])]{}, the [$\mathcal{C}$]{} operator can be calculated directly from its definition in [(\[eqn:C\])]{}. It is also straightforward to construct it from the relation in [(\[eqn:CE\])]{}. Either way, the [$\mathcal{C}$]{} operator is found to be $${\ensuremath{\mathcal{C}}}= \frac{u}{\gamma}\bm{\alpha}\cdot \bm{\sigma}.$$ The form of the [$\mathcal{C}$]{} operator is not a surprise because it is defined as an involution. Therefore, it must be a square root of the identity matrix just like the [$\mathcal{P}$]{} operator. This fact can be easily verified by observing that $(u/\gamma)\bm{\alpha}$ is a unit vector.

Because the [$\mathcal{C}$]{} operator has eigenvalues ${\ensuremath{\mathcal{C}}}|E_\pm{\rangle}= \pm {\rm sign} (\gamma)|E_\pm{\rangle}$, the eigenstates in [(\[eqn:eigenstates\])]{} are normalized to unity with respect to the [$\mathcal{C}$]{}[$\mathcal{P}$]{}[$\mathcal{T}$]{}-inner product. Thus, we have a set of orthonormal eigenstates, $${\langle}E_i|{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{C}}}|E_j{\rangle}=\delta_{ij}, \qquad i,j=\pm.$$

The weight function has the form $$W = {\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{C}}}= u\left(\openone + \bm{\beta}\cdot\bm{\sigma} \right), \qquad {\rm where}\quad \bm{\beta} \equiv \frac{\nu}{\gamma} {\bf n}^\theta - \frac{\mu}{\gamma} {\bf n}^\varphi.$$ Note that $\bm{\beta}$ is a unit vector and it is perpendicular to $\bm{\alpha}$. Interestingly, the square root of $W$ has the form $$\eta_\pm=\frac{1}{\sqrt{2(u\pm 1)}}(W\pm\openone).
\label{eqn:eta}$$ There are two solutions for $\eta$, which is [*not*]{} because of the arbitrary overall sign in the square root. Rather, it is corresponding to two choices of mapping the eigenstates during the similarity transformation in [(\[eqn:h\])]{}. In the Hermitian limit, $W\to\openone$, $\eta$ is once again a square root of the identity matrix. The “$+$” sign in [(\[eqn:eta\])]{} has the limit $\eta_+\to\openone$ and the “$-$” sign has the limit $\eta_-\to \bm{\beta}\cdot\bm{\sigma}.$

Using the operator $\eta$, we find the Hermitian equivalence of [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian $H$ in [(\[eqn:H\])]{}, $$h = \varepsilon \openone \pm \frac{\gamma}{u} {\bf n}^r\cdot \bm{\sigma}.$$

Now let us consider some special cases. We show that both Hermitian Hamiltonians and several [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric $2\times 2$ Hamiltonians studied in the literature can be reduced from our general [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian in [(\[eqn:H\])]{}.

Special Case 1: Hermiticity {#sec:Hermitian}
---------------------------

If we set $\mu=\nu=0$, $H$ becomes Hermitian, $H=H^\dag$, $$H_{\rm Hermitian}=\varepsilon\openone + \gamma {\bf n}^r\cdot\bm{\sigma}=\left(
\begin{array}{cc}
\varepsilon + \gamma\cos\theta & \gamma\sin\theta~ {{\rm e}}^{-{{\rm i}}\varphi}\\
\gamma\sin\theta~ {{\rm e}}^{{{\rm i}}\varphi} & \varepsilon - \gamma\cos\theta
\end{array}
\right).$$ This matrix Hamiltonian has four real parameters, and it includes all $2\times 2$ Hermitian matrices.

In this case, the weight function reduces to the identity matrix and the [$\mathcal{C}$]{} operator coincides with the parity operator: $$W_{\rm Hermitian}=\openone, \qquad {\ensuremath{\mathcal{C}}}_{\rm Hermitian}={\ensuremath{\mathcal{P}}}.$$

All these observations are consistent with the conventional QM. We may say that the Hermitian Hamiltonian is a special case of the [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian with ${\ensuremath{\mathcal{C}}}={\ensuremath{\mathcal{P}}}$. Or, equivalently, [$\mathcal{P}$]{}[$\mathcal{T}$]{} symmetry is a generalization of Hermiticity.

Special Case 2: Bender-Brody-Jones Hamiltonian {#sec:BBJ}
----------------------------------------------

In Ref. [@BBJ], Bender [*et al.*]{} studied a [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian with real and symmetric off-diagonal matrix elements. Their choice of the parity operator is ${\ensuremath{\mathcal{P}}}_{\rm BBJ}=\sigma_x$. This case can be reduced from our general case by setting $\nu=\varphi=0$ and $\theta=\pi/2$. In particular, we have $$H_{\rm BBJ}=\left(
\begin{array}{cc}
\varepsilon - {{\rm i}}\mu & \gamma\\
\gamma & \varepsilon + {{\rm i}}\mu
\end{array}
\right), \qquad
{\ensuremath{\mathcal{C}}}_{\rm BBJ}=\sqrt{\frac{\gamma^2}{\gamma^2-\mu^2}}\left(
\begin{array}{cc}
 -{{\rm i}}\frac{\mu}{\gamma} & 1\\
1 & {{\rm i}}\frac{\mu}{\gamma}
\end{array}
\right).$$ These expressions are equivalent to those in Ref. [@BBJ] by mapping our parameters $\varepsilon$, $\mu$, and $\gamma$ to $r\cos\theta$, $-r\sin\theta$, and $s$ therein.

Special case 3: Bender-Meisinger-Wang Hamiltonian {#sec:BMW}
-------------------------------------------------

In Ref. [@BMW], Bender [*et al.*]{} generalized the Hamiltonian matrix in Ref. [@BBJ] by choosing a one-parameter class of parity operator, $${\ensuremath{\mathcal{P}}}_{\rm BMW}=\left(
\begin{array}{cc}
\cos\theta & \sin\theta\\
\sin\theta & -\cos\theta
\end{array}
\right).$$ This can be recovered by setting $\varphi=0$ in [(\[eqn:P\_2x2\])]{}. If we consider only symmetric Hamiltonian as in Ref. [@BMW], we can further set $\nu=0$ in [(\[eqn:H\])]{}. With this choice of parameters, we have $$\begin{aligned}
H_{\rm BMW}&=&\left(
\begin{array}{cc}
\varepsilon + \gamma\cos\theta - {{\rm i}}\mu\sin\theta& \gamma\sin\theta + {{\rm i}}\mu\cos\theta\\
\gamma\sin\theta+ {{\rm i}}\mu\cos\theta & \varepsilon - \gamma\cos\theta + {{\rm i}}\mu\sin\theta
\end{array}
\right),{\nonumber}\\
{\ensuremath{\mathcal{C}}}_{\rm BMW}&=&\sqrt{\frac{\gamma^2}{\gamma^2-\mu^2}}\left(
\begin{array}{cc}
\cos\theta -{{\rm i}}\frac{\mu}{\gamma}\sin\theta& \sin\theta + {{\rm i}}\frac{\mu}{\gamma}\cos\theta\\
\sin\theta+ {{\rm i}}\frac{\mu}{\gamma}\cos\theta & - \cos\theta + {{\rm i}}\frac{\mu}{\gamma}\sin\theta
\end{array}
\right).\end{aligned}$$ Once again, these are the same formulas as in Ref. [@BMW] by properly mapping the parameters.

The symmetric Hamiltonian, $H$ in [(\[eqn:H\])]{} with $\nu=\varphi=0$, has additional properties. In this case, the eigenstates of $H$ are also the eigenstates of [$\mathcal{P}$]{}[$\mathcal{T}$]{}. By choosing a proper phase, the eigenvalue of [$\mathcal{P}$]{}[$\mathcal{T}$]{} can be set to unity: $${\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{T}}}|E_\pm{\rangle}\equiv {\ensuremath{\mathcal{P}}}|E_\pm{\rangle}^* = |E_\pm{\rangle}.$$

One may think that our definition of [$\mathcal{C}$]{}[$\mathcal{P}$]{}[$\mathcal{T}$]{}-inner product is slightly different from the one in the literature. In Refs. [@BBJ; @BMW] the [$\mathcal{C}$]{}[$\mathcal{P}$]{}[$\mathcal{T}$]{}-inner product was defined as $$(\psi,\phi)_{\rm BBJ-BMW} = ({\ensuremath{\mathcal{C}}}{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{T}}}|\psi{\rangle})^T|\phi{\rangle}= ({\ensuremath{\mathcal{C}}}{\ensuremath{\mathcal{P}}}|\psi{\rangle}^*)^T|\phi{\rangle}={\langle}\psi|{\ensuremath{\mathcal{P}}}^T{\ensuremath{\mathcal{C}}}^T|\phi{\rangle},$$ where $T$ denotes matrix transpose. This definition leads to a weight function $W_{\rm BBJ-BMW}={\ensuremath{\mathcal{P}}}^T{\ensuremath{\mathcal{C}}}^T$. Since both [$\mathcal{P}$]{} and [$\mathcal{C}$]{} are symmetric in this case, the two definitions of the inner product are actually the same.

Special case 4: Mostafazadeh Hamiltonian {#sec:Ali2x2}
----------------------------------------

In Ref. [@AliJPA2003], Mostafazadeh introduced a five-parameter-class of Hamiltonians with the form $$H_{\rm Mostafazadeh}=\left(
\begin{array}{cc}
r + t\cos\phi - {{\rm i}}s\sin\phi& t\sin\phi + {{\rm i}}(s\cos\phi-u)\\
t\sin\phi+ {{\rm i}}(s\cos\phi+u) & r - t\cos\phi + {{\rm i}}s\sin\phi
\end{array}
\right).$$ This is a special case of our general Hamiltonian in [(\[eqn:H\])]{} by the following parameter mapping: $$\begin{aligned}
\varepsilon \to r, &\qquad& \gamma \to \sqrt{t^2+u^2},{\nonumber}\\
\mu \to \frac{s\sqrt{t^2+u^2}\sin\phi}{\sqrt{t^2\sin^2\phi+u^2}},
&\qquad& \nu \to -\frac{su\cos\phi}{\sqrt{t^2\sin^2\phi+u^2}},{\nonumber}\\
\cos\theta \to \frac{t\cos\phi}{\sqrt{t^2+u^2}}, &\qquad&
\tan\varphi \to \frac{u}{t\sin\phi}.\end{aligned}$$

Special case 5: Mostafazadeh-Özçelik Hamiltonian
------------------------------------------------

In Ref. [@AliTurk2007], Mostafazadeh and Özçelik constructed a six-parameter-class of Hamiltonians with a very elegant form $$H_{\rm MO}=q\openone_{2\times 2} + E \left(
\begin{array}{cc}
\cos\Theta & {{\rm e}}^{-{{\rm i}}\Phi} \sin\Theta\\
{{\rm e}}^{{{\rm i}}\Phi} \sin\Theta & \cos\Theta
\end{array}
\right),
\label{eqn:MOH}$$ where $q$ and $E$ are real and $\Theta$ and $\Phi$ are complex. In principle, this Hamiltonian is equivalent to our Hamiltonian in [(\[eqn:H\])]{} by the following parameter mapping, $$\begin{aligned}
\varepsilon \to q, &\qquad& \pm\sqrt{\gamma^2-\mu^2-\nu^2} \to E,{\nonumber}\\
\gamma\cos\theta-{{\rm i}}\mu\sin\theta \to E\cos\Theta, &\qquad&
\frac{\gamma\sin\theta + {{\rm i}}\mu\cos\theta - \nu}{\gamma\sin\theta + {{\rm i}}\mu\cos\theta + \nu}{{\rm e}}^{2{{\rm i}}\varphi} \to {{\rm e}}^{2{{\rm i}}\Phi}.\end{aligned}$$ However, at the degenerate point, $E=0$ in [(\[eqn:MOH\])]{} or $\gamma^2=\mu^2+\nu^2$ in [(\[eqn:H\])]{}, these two parametrizations are no longer equivalent. In this special point, $H_{\rm MO}$ is proportional to the identity matrix but $H$ in [(\[eqn:H\])]{} is not.

$3\times 3$ Case {#sec:3x3}
================

In this section, we reveal the general form of [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric $3\times 3$ matrix Hamiltonians. In this case, we use the Gell-Mann matrices, which are the generalization of the Pauli matrices in $3\times 3$. Any $3\times 3$ matrix can be written as a linear combination of the identity matrix and the eight Gell-Mann matrices. If a matrix is Hermitian, the expansion coefficients are all real.

There are two types of solutions for the [$\mathcal{P}$]{} operator as well. The first type is the trivial solutions ${\ensuremath{\mathcal{P}}}_0=\pm\openone$. The second type is the non-trivial solutions. If we expand the non-trivial solutions of [$\mathcal{P}$]{} as $${\ensuremath{\mathcal{P}}}_{3\times 3}=\pm\left(P_0 \openone + \sum_{i=1}^8 P_i \lambda_i\right),
\label{eqn:Pexp}$$ where $\lambda_i$ are Gell-Mann matrices for $i=1,\cdots,8$. We find that $$P_0=\frac{1}{3}$$ and that coefficients $P_i$ depend on four independent parameters. We thus choose the parametrization as $$\begin{aligned}
P_4&=&\sin 2\chi\sin\theta\cos\varphi,{\nonumber}\\
P_5&=&\sin 2\chi\sin\theta\sin\varphi,{\nonumber}\\
P_6&=&\sin 2\chi\cos\theta\cos\rho,{\nonumber}\\
P_7&=&\sin 2\chi\cos\theta\sin\rho.
\label{eqn:P4567}\end{aligned}$$ The other four components, $P_1$, $P_2$, $P_3$, and $P_8$ depend on the sign of $\cos 2\chi$. For the case of $\cos 2\chi \geq 0$, we have $$\begin{aligned}
P_1 &=& -\sin^2\chi\sin 2\theta \cos(\rho-\varphi),{\nonumber}\\
P_2 &=& \sin^2\chi\sin 2\theta \sin(\rho-\varphi),{\nonumber}\\
P_3 &=& \sin^2\chi\cos 2\theta,{\nonumber}\\
P_8 &=& \frac{1}{2\sqrt{3}}(1+3\cos 2\chi).
\label{eqn:P1238}\end{aligned}$$ Plugging $P_0$ and $P_i$ into [(\[eqn:Pexp\])]{}, we get the non-trivial four-parameter solutions of the [$\mathcal{P}$]{} operator with the form $${\ensuremath{\mathcal{P}}}_{3\times 3}=\pm\left(
\begin{array}{ccc}
\cos 2\chi \sin^2\theta + \cos^2\theta&
-\sin^2\chi \sin 2\theta~{{\rm e}}^{{{\rm i}}(\rho-\varphi)}&
\sin 2\chi \sin\theta~{{\rm e}}^{-{{\rm i}}\varphi}\\
-\sin^2\chi\sin 2\theta~{{\rm e}}^{-{{\rm i}}(\rho-\varphi)}&
\cos 2\chi \cos^2\theta + \sin^2\theta&
\sin 2\chi \cos\theta~{{\rm e}}^{-{{\rm i}}\rho}\\
\sin 2\chi\sin\theta~{{\rm e}}^{{{\rm i}}\varphi} &
\sin 2\chi \cos\theta~{{\rm e}}^{{{\rm i}}\rho}&
-\cos 2\chi
\end{array}
\right).
\label{eqn:P_3x3}$$ For the case of $\cos 2\chi<0$, the correct results are obtained by replacing $\cos 2\chi$ by $-\cos 2\chi$ in [(\[eqn:P1238\])]{} and [(\[eqn:P\_3x3\])]{}.

Just like in the $2\times 2$ case, [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonians can be found by solving an eigenvalue problem. If we use the [*ansatz*]{} of the Hamiltonian as $$H_{3\times 3}=\varepsilon\openone + \sum_{i=1}^8 \alpha_i \lambda_i,
\label{eqn:H3x3}$$ the [$\mathcal{P}$]{}[$\mathcal{T}$]{} symmetry leads to the conditions of $\varepsilon=\varepsilon^*$ and that $\alpha_i$ satisfy the eigenvalue equation $$\sum_{i=1}^8 M_{ki}\alpha_i=\alpha_k^*,$$ where $$M_{ki} = P_0^2\delta_{ki} + 2P_0 \sum_j P_j d^{ijk}+\frac{2}{3} P_k P_i + \sum_{jml}P_j P_m\left( d^{ijl}d^{lmk} + f^{ijl}f^{lmk}\right)
\label{eqn:Mki3x3}$$ with $d^{ijk}$ and $f^{ijk}$ being symmetric and antisymmetric structure constants of the SU(3) group.

Once again, the real part of $\alpha_i$ forms eigenvectors of $M$ with eigenvalue $+1$, and the imaginary part of $\alpha_i$ forms eigenvectors with eigenvalue $-1$. There are always four eigenvectors with eigenvalue $+1$ and four eigenvectors with eigenvalue $-1$.

It is straightforward to show that the vector $P_i$ with components defined in [(\[eqn:P4567\])]{} and [(\[eqn:P1238\])]{} is an eigenvector of $M$ with eigenvalue $+1$. All other eigenvectors can be constructed by the derivatives of $P_i$. The set of all four first order derivatives, $\{\partial_\chi P_i, \partial_\theta P_i, \partial_\rho P_i, \partial_\varphi P_i\}$, forms a subspace with eigenvalue $-1$. The remaining three eigenvectors with eigenvalue $+1$ can be constructed from the second derivatives. Below is a choice of orthonormal set of eigenvectors with eigenvalue $+1$, $$\begin{aligned}
A_i^{(1)} &=& \frac{\sqrt{3}}{2}P_i {\nonumber}\\
          &=& \frac{\sqrt{3}}{2}\Biggl(
    -\sin^2\chi\sin 2\theta\cos(\rho-\varphi),
    \sin^2\chi\sin 2\theta\sin(\rho-\varphi),
    \sin^2\chi\cos 2\theta,{\nonumber}\\
    &&\quad \sin 2\chi\sin\theta\cos\varphi,
    \sin 2\chi\sin\theta\sin\varphi, \sin 2\chi\sin\theta\cos\rho,
    \sin 2\chi\sin\theta\sin\rho, \frac{1+3\cos 2\chi}{2\sqrt{3}}\Biggr), {\nonumber}\\
A_i^{(2)} &=& \frac{1}{2} \partial_\chi^2 P_i + \frac{3}{2} P_i {\nonumber}\\
    &=&\frac{1}{2} \left(-\frac{3+\cos 2\chi}{2}\sin 2\theta\cos(\rho-\varphi),
    \frac{3+\cos 2\chi}{2}\sin 2\theta\sin(\rho-\varphi),
    \frac{3+\cos 2\chi}{2}\cos 2\theta, \right.{\nonumber}\\
    && \quad
    -\sin 2\chi\sin\theta\cos\varphi,
    -\sin 2\chi\sin\theta\sin\varphi,
    -\sin 2\chi\cos\theta\cos\rho,
    -\sin 2\chi\cos\theta\sin\rho,{\nonumber}\\
    &&\quad \left.
    \frac{\sqrt{3}}{2}\sin^2\chi\right), {\nonumber}\\
A_i^{(3)} &=& \Bigl(-\cos\chi\cos 2\theta\cos(\rho-\varphi),
    \cos\chi\cos 2\theta\sin(\rho-\varphi),
    -\cos\chi\sin 2\theta,{\nonumber}\\
    &&\quad 
    -\sin\chi\cos\theta\cos\varphi,     
    -\sin\chi\cos\theta\sin\varphi,
    \sin\chi\sin\theta\cos\rho,\sin\chi\sin\theta\sin\rho,0
    \Bigr),{\nonumber}\\
A_i^{(4)} &=& \Bigl(\cos\chi\sin(\rho-\varphi), \cos\chi\cos(\rho-\varphi),
    0, -\sin\chi\cos\theta\sin\varphi,
    \sin\chi\cos\theta\cos\varphi, {\nonumber}\\
    &&\quad 
    \sin\chi\sin\theta\sin\rho,-\sin\chi\sin\theta\cos\rho,0\Bigr).\end{aligned}$$ Likewise, a set of eigenvectors with eigenvalue $-1$ can be chosen as $$\begin{aligned}
B_i^{(1)} &=& \frac{1}{2} \partial_\chi P_i {\nonumber}\\
    &=& \frac{1}{2} \Bigl(-\sin 2\chi\sin 2\theta\cos(\rho-\varphi),
    \sin 2\chi\sin 2\theta\sin(\rho-\varphi),
    \sin 2\chi\cos  2\theta,2\cos 2\chi\sin\theta\cos\varphi,{\nonumber}\\
    &&\quad 2\cos 2\chi\sin\theta\sin\varphi, 2\cos 2\chi\cos\theta\cos\rho,
    2\cos 2\chi\cos\theta\sin\rho,
    -\sqrt{3}\sin 2\chi\Bigr),{\nonumber}\\
B_i^{(2)} &=& \frac{1}{2\sin\chi} \partial_\theta P_i {\nonumber}\\
    &=& \Bigl(-\sin\chi\cos 2\theta\cos(\rho-\varphi),
    \sin\chi\cos 2\theta\sin(\rho-\varphi),
    -\sin\chi\sin 2\theta,{\nonumber}\\
    &&\quad \cos\chi\cos\theta\cos\varphi,
    \cos\chi\cos\theta\sin\varphi, -\cos\chi\sin\theta\cos\rho,
    -\cos\chi\sin\theta\sin\rho,0\Bigr),{\nonumber}\\
B_i^{(3)} &=& \frac{\partial_\phi P_i+\partial_\varphi P_i}{\sin 2\chi} {\nonumber}\\
    &=& \Bigl(0,0,0,-\sin\theta\sin\varphi, \sin\theta\cos\varphi,
    -\cos\theta\sin\rho,\cos\theta\cos\rho,0\Bigr), {\nonumber}\\
B_i^{(4)} &=& \Bigl(\sin\chi\sin(\rho-\varphi),
    \sin\chi\cos(\rho-\varphi),0,
    \cos\chi\cos\theta\sin\varphi,{\nonumber}\\
    &&\quad -\cos\chi\cos\theta\cos\varphi,
    -\cos\chi\sin\theta\sin\rho, \cos\chi\sin\theta\cos\rho,0\Bigr).\end{aligned}$$

With the solution of $\alpha_i=A_i+{{\rm i}}\,B_i$, we can construct the general $3\times 3$ [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian by plugging the above results into the [*ansatz*]{} in [(\[eqn:H3x3\])]{}: $$H_{3\times 3}=\varepsilon \openone + \sum_{i=1}^8 \left[\gamma_1 A_i^{(1)} +
\gamma_2 A_i^{(2)} + \gamma_3 A_i^{(3)} + \gamma_4 A_i^{(4)} + {{\rm i}}\left(\mu_1 B_i^{(1)} + \mu_2 B_i^{(2)} + \mu_3 B_i^{(3)} + \mu_4 B_i^{(4)}\right)\right]\lambda_i.
\label{eqn:HH3x3}$$ This construction has thirteen real parameters, four in [$\mathcal{P}$]{}, four $\gamma$’s, four $\mu$’s, and one $\varepsilon$. Any $3\times 3$ Hermitian Hamiltonian can be considered as a special case with all $\mu$’s vanishing, which also has the correct number of parameters: nine. Unlike the $2\times 2$ case, the general [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric matrix Hamiltonian in [(\[eqn:HH3x3\])]{} does not present all $3\times 3$ matrices with all real eigenvalues. This can be seen by a simple parameter counting. A $3\times 3$ matrix with all real eigenvalues should have fifteen real parameters but $H_{3\times 3}$ in [(\[eqn:HH3x3\])]{} only has thirteen parameters.

Conclusion {#sec:conclusion}
==========

In this paper, we find the general [$\mathcal{P}$]{} operator and construct the general [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric matrix Hamiltonians in $2\times 2$ and $3\times 3$. In both cases, [$\mathcal{P}$]{}[$\mathcal{T}$]{} symmetry can be considered as a generalization of Hermiticity. We conjecture that this statement is true in general.

We convert the searching for a general [$\mathcal{P}$]{}[$\mathcal{T}$]{}-symmetric Hamiltonian problem to an eigenvalue problem. This method also applies to higher dimensions. For example, the definition for the matrix $M_{ki}$ in [(\[eqn:Mki3x3\])]{} can be easily generalized to $N$ dimensions by replacing $3$ by $N$.

QW is very grateful to Professor Carl M. Bender for helpful comments and correspondence. QW would also like to thank Dr. Jiangbin Gong for many useful discussions and some help on writing.

[99]{}

C.M. Bender and S. Boettcher, “[*Real Spectra in Non-Hermitian Hamiltonians Having [$\mathcal{P}$]{}[$\mathcal{T}$]{} Symmetry*]{},” Phys. Rev. Lett. [**80**]{}, 5243-5246 (1998).

C.M. Bender, D.C. Brody, and H.F. Jones, “[*Complex Extension of Quantum Mechanics*]{},” Phys. Rev. Lett. [**89**]{}, 270401 (2002) \[Erratum: [*ibid.*]{} [**92**]{} 119902 (2004)\].

C.M. Bender, “[*Making Sense of Non-Hermitian Hamiltonians*]{},” Rep. Prog. Phys. [**70**]{}, 947-1018 (2007) \[arXiv:hep-th/0703096\].

A. Mostafazadeh, “[*Pseudo-Hermitian Quantum Mechanics*]{},” arXiv:0810.5643.

A. Mostafazadeh, “[*Pseudo-Hermiticity for a Class of Nondiagonalizable Hamiltonians,*]{}” J. Math. Phys. [**43**]{}, 6343-6352 (2002) \[Erratum: [*ibid.*]{} [**44**]{} 943 (2003)\].

A. Mostafazadeh, “[*Pseudo-Hermiticity and Generalized [$\mathcal{P}$]{}[$\mathcal{T}$]{}- and [$\mathcal{C}$]{}[$\mathcal{P}$]{}[$\mathcal{T}$]{}-Symmetries,*]{}” J. Math. Phys. [**44**]{}, 974-989 (2003).

C.M. Bender, P.N. Meisinger, and Q. Wang, “[*All Hermitian Hamiltonians Have Parity*]{},” J. Phys. A: Math. Gen. [**36**]{}, 1029-1031 (2003) \[arXiv:quant-ph/0211123\].

C.M. Bender, P.N. Meisinger, and Q. Wang, “[*Finite-Dimensional [$\mathcal{P}$]{}[$\mathcal{T}$]{}-Symmetric Hamiltonians*]{},” J. Phys. A: Math. Gen. [**36**]{}, 6791-6797 (2003) \[arXiv:quant-ph/0303174\].

A. Mostafazadeh, “[*Exact [$\mathcal{P}$]{}[$\mathcal{T}$]{}-Symmetry Is Equivalent to Hermiticity,*]{}” J. Phys. A: Math. Gen. [**36**]{}, 7081-7092 (2003) \[arXiv:quant-ph/0304080\].

A. Mostafazadeh and S. Özçelik “[*Explicit Realization of Pseudo-Hermitian and Quasi-Hermitian Quantum Mechanics for Two-Level Systems*]{}” Turk. J. Phys. [**30**]{}, 437-443 (2006) \[arXiv:quant-ph/0607120\].

L.E. Ballentine, [*Quantum Mechanics: A Modern Development*]{} (World Scientific, Singapore, 1998).
