Relativistic PT-symmetric fermionic theories in 1+1 and 3+1 dimensions

TL;DR

This paper explores PT-symmetric non-Hermitian fermionic theories in 1+1 and 3+1 dimensions, analyzing conditions for real spectra and the impact of PT symmetry and self-adjointness on the spectrum of such systems.

Contribution

It introduces new PT-symmetric non-Hermitian fermionic models in relativistic dimensions and examines their spectral properties, including the effects of additional constraints like self-adjointness.

Findings

Certain models exhibit real spectra in 1+1 dimensions under specific potentials.

In 3+1 dimensions, some PT-symmetric interactions lead to complex spectra as mass approaches zero.

The effect of PT inner product self-adjointness on the spectrum is analyzed.

Abstract

Relativistic PT-symmetric fermionic interacting systems are studied in 1+1 and 3+1 dimensions. The objective is to include non-Hermitian PT-symmetric interaction terms that give {\it real} spectra. Such interacting systems could describe new physics. The simplest non-Hermitian Lagrangian density is $L=L_0+L_{int}=\bar\psi(i\not\partial-m)\psi-g\bar\psi\gamma^5\psi$. The associated relativistic Dirac equation is PT invariant in 1+1 dimensions and the associated Hamiltonian commutes with PT. However, the dispersion relation $p^2=m^2-g^2$ shows that the PT symmetry is broken in the chiral limit $m\to0$. For interactions $L_{int}=-g(\bar\psi\gamma^5\psi)^N$ with N=2,3, if the associated Dirac equation is PT invariant, the dispersion relation gives complex energies as $m\to0$. Other models are studied in which x-dependent PT-symmetric potentials such as $ix^3$, $-x^4$, $i\kappa/x$, Hulth\'en, or periodic potentials are coupled to $\psi$ and the classical trajectories plane are examined. Some combinations of these potentials give a real spectrum. In 3+1 dimensions, the simplest system $L=L_0+L_{int}=\bar\psi(i\not\partial-m)\psi-g\bar\psi\gamma^5\psi$ resembles the 1+1-dimensional case but the Dirac equation is not PT invariant because $T^2=-1$. This explains the appearance of complex eigenvalues as $m\to0$. Other Lorentz-invariant 2-point and 4-point interactions give non-Hermitian PT-symmetric terms in the Dirac equation. Only the axial vector and tensor Lagrangian interactions $L_{int}=-i\bar\psi\tilde B_\mu\gamma^5\gamma^\mu\psi$ and $L_{int}=-i\bar\psi T_{\mu\nu}\sigma^{\mu\nu}\psi$ fulfil both requirements of PT invariance of the associated Dirac equation and non-Hermiticity. Both models give complex spectra as $m\to0$. The effect on the spectrum of the additional constraint of selfadjointness of the Hamiltonian with respect to the PT inner product is investigated.

Figures

Equations

$$\gamma^{0}=\left(\begin{array}[]{cc}0&1\\ 1&0\\ \end{array}\right),\qquad\gamma^{1}=\left(\begin{array}[]{cc}0&1\\ -1&0\\ \end{array}\right).$$

$$\big{[}i\gamma^{0}\partial_{0}+i\gamma^{1}\partial_{1}-m\big{]}\psi(t,x)=0.$$

$$\big{[}i\gamma^{0}\partial_{0}+i\gamma^{1}\partial_{1}-m\big{]}\gamma^{0}\psi(t,-x)=0.$$

$$\mathcal{P}:\psi (t,x)\to \mathcal{P}\psi (t,x){\mathcal{P}}^{-1}={\gamma }^0\psi (t,-x).$$

$$\big{[}i\gamma^{0}\partial_{0}+i\gamma^{1}\partial_{1}-m\big{]}\gamma^{0}\psi^{*}(-t,x)=0.$$

$$\mathcal{T}:\psi (t,x)\to \mathcal{T}\psi (t,x){\mathcal{T}}^{-1}={\gamma }^0{\psi }^{\ast }(-t,x).$$

$${\gamma }^0$$

$$\mathcal{P}:\psi (t,\mathbf{x})$$

$$\mathcal{T}:\psi (t,\mathbf{x})$$

$${\mathcal{L}}_0=\bar{\psi }(i/\partial -m)\psi ,$$

$$H(\mathcal{P}\mathcal{T}\psi )=H({\gamma }^0{\gamma }^0{\psi }^{\ast })=H{\psi }^{\ast },$$

$$\mathcal{PT}(H\psi)=\gamma^{0}\gamma^{0}\big{(}-i\gamma^{0}\gamma^{1}\partial_{1}+m\gamma^{0}\big{)}\psi^{*}=H\psi^{*}.$$

$$(i/\partial -m-g{\gamma }^5)\psi =0.$$

$$(i/\partial -m+g{\gamma }^5){\gamma }^0\psi (t,-x)=0,$$

$$(i/\partial -m+g{\gamma }^5){\gamma }^0{\psi }^{\ast }(-t,x)=0.$$

$$H=H_0+H_{\mathrm{int}}=-i{\gamma }^0{\gamma }^1{\partial }_1+m{\gamma }^0+g{\gamma }^0{\gamma }^5,$$

$$H^{\mathcal{P}\mathcal{T}}=\left(\begin{array}{cc}a& b\\ f& a\end{array}\right),$$

$$H_{\mathrm{int}}=g{\gamma }^0{\gamma }^5=g\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right).$$

$$H=m{\gamma }^0+g{\gamma }^0{\gamma }^5=\left(\begin{array}{cc}0& m+g\\ m-g& 0\end{array}\right).$$

$$(i/\partial -m-ig{\gamma }^5)\psi =0,$$

$$\mathcal{L}(N)={\mathcal{L}}_0+{\mathcal{L}}_{\mathrm{int}}=\bar{\psi }(i/\partial -m)\psi -g(\bar{\psi }{\gamma }^5\psi {)}^N.$$

$$\big{[}i\not{\partial}-m-Ng\gamma^{5}(\bar{\psi}\gamma^{5}\psi)^{N-1}\big{]}\psi=0,$$

$$\big{[}i\not{\partial}-m-2g\gamma^{5}(\bar{\psi}\gamma^{5}\psi)\big{]}\psi=0,$$

$$H=-i{\gamma }^0{\gamma }^1{\partial }_1+m{\gamma }^0+2g{\gamma }^0{\gamma }^5(\bar{\psi }{\gamma }^5\psi ).$$

$$p^2=-4g^2⟨\varphi {⟩}^2.$$

$$⟨\varphi ⟩=-iA,$$

$$\big{[}i\not{\partial}-m-3g\gamma^{5}(\bar{\psi}\gamma^{5}\psi)^{2}\big{]}\psi=0.$$

$$H_{\mathrm{int}}=3g{\gamma }^0{\gamma }^5(\bar{\psi }{\gamma }^5\psi {)}^2.$$

$$p^2=-9g^2⟨(\bar{\psi }{\gamma }^5\psi {)}^2{⟩}^2$$

$$\big{(}i\not{\partial}-ix^{3}\gamma^{0}\big{)}\psi(t,x)=0.$$

$$H=-i\alpha {\partial }_x+i{x}^3(\alpha \equiv {\gamma }^0{\gamma }^1),$$

Full text

$\mathcal{PT}$ symmetry, relativistic fermionic theories, Dirac equation

I Introduction

A non-Hermitian quantum-mechanical Hamiltonian $H$ that is invariant under combined parity (space reflection) $\mathcal{P}$ and time reversal $\mathcal{T}$ can have real eigenvalues r1 ; r1a . If the spectrum is entirely real, we say that $H$ has an unbroken $\mathcal{PT}$ symmetry. However, if $H$ has complex eigenvalues, we say that $H$ has a broken $\mathcal{PT}$ symmetry. Numerous theoretical studies of classical and quantum-mechanical $\mathcal{PT}$-symmetric systems have been done and many experiments on such systems have been performed. The remarkable features of $\mathcal{PT}$-symmetric include $\mathcal{PT}$ symmetry breaking in coupled wave guides, unidirectional invisibility, enhanced sensing at exceptional points, level bifurcation in superconducting wires, and robust wireless power transfer r2 ; r3 ; r4 ; r5 ; r6 ; r7 ; r8 ; r9 .

In quantum mechanics $x\to-x$ under parity $\mathcal{P}$ and $i\to-i$ under time reversal $\mathcal{T}$. Thus, the quantum-mechanical Hamiltonian $H=p^{2}+x^{2}(ix^{\varepsilon})$ ($\varepsilon$ real) is $\mathcal{PT}$ invariant; $H$ has a real positive discrete spectrum when $\varepsilon\geq 0$ r1 . This quantum theory generalizes to relativistic quantum field theory if the operator $x(t)$ is replaced by the pseudoscalar field $\phi(t,{\bf x})$ so that $\phi(t,{\bf x})\to-\phi(t,{-\bf x})$ under $\mathcal{P}$ and $\phi(t,{\bf x})\to\phi^{*}(-t,{\bf x})$ under $\mathcal{T}$. The analogous bosonic field-theoretic Hamiltonian density $(\partial\phi)^{2}+\phi^{2}(i\phi)^{\varepsilon}$ also appears to have a real spectrum; this was shown to first order in $\varepsilon$ for $0\leq D<2$ r10 .

While $\mathcal{PT}$-symmetric bosonic systems have been studied heavily (there are over 4,000 papers on such systems), only a few papers have been written on $\mathcal{PT}$-symmetric fermionic systems. Early work on matrix models of fermionic systems can be found in Refs. r11 ; r13 ; r12 ; r14 . The Lagrangian density for a free relativistic fermionic field with mass $m$ was extended by including a non-Hermitian axial mass term $\mathcal{L}_{\rm int}=-g\bar{\psi}\gamma^{5}\psi$, where $g$ is a real mass parameter r15 . Further developments were made in Ref. r16 in which quantum electrodynamics was extended to include such a term and the restoration of gauge symmetry was investigated. In Ref. R14 the relationship between conserved currents and invariances of the Lagrangian in the framework of non-Hermitian field theories was examined. An application of $\mathcal{PT}$-symmetric fermionic field theory to neutrino species oscillation was proposed in Ref. R15 in which an 8-dimensional Dirac equation was analyzed. Neutrino oscillations in the context of $\mathcal{PT}$ symmetry were studied further in Ref. R16 .

$\mathcal{PT}$-symmetric fermionic field theories in 1+1 dimensions share the property with quantum-mechanical and bosonic field theories that $\mathcal{T}^{2}=\mathds{1}$ r15 . However, in Ref. r12 it was noted that $\mathcal{PT}$-symmetric fermionic systems in 3+1 dimensions have the propery that $\mathcal{T}^{2}=-\mathds{1}$. To explain this we first examine what happens in 1+1 dimensions, where the gamma matrices are r19

[TABLE]

Note that \big{(}\gamma^{0}\big{)}^{2}=\mathds{1}, \big{(}\gamma^{1}\big{)}^{2}=-\mathds{1}, and $\gamma^{5}=\gamma^{0}\gamma^{1}=-\sigma_{3}$, where $\sigma_{3}$ is a Pauli matrix. Let us identify the discrete spatial symmetries of the free Dirac equation

[TABLE]

(Here $\partial_{0}=\partial_{t}$ and $\partial_{1}=\partial_{x}$.) To determine the effect of a space reflection we let $x\to-x$ and then multiply (2) on the left by $\gamma^{0}$ to get

[TABLE]

Because this equation has the same form as (2) we identify that the action of parity reflection $\mathcal{P}$ on the spinor $\psi(t,x)$ is given by

[TABLE]

Next, to determine the effect of time reversal $\mathcal{T}$ we let $t\to-t$ in (2), take the complex conjugate of the resulting equation, and again multiply on the left by $\gamma^{0}$. We get

[TABLE]

Again, from form invariance we conclude that time reversal for spinors in 1+1 dimensions is given by

[TABLE]

Since $\gamma^{0}$ is real we see that applying $\mathcal{P}$ or $\mathcal{T}$ twice leaves $\psi(t,x)$ invariant. Thus, $\mathcal{P}^{2}=\mathds{1}$ and $\mathcal{T}^{2}=\mathds{1}$. (Interestingly, this property of time reversal in 1+1 dimensions implies that the Dirac electron behaves like a boson thaller .)

In 3+1 dimensions the Dirac representation of the gamma matrices is R18

[TABLE]

where $\sigma^{i}$ are the Pauli matrices. The actions of parity and time reversal obtained similarly, are now R18

[TABLE]

If we apply $\mathcal{T}$ twice, we observe a change of sign: $\mathcal{P}^{2}=\mathds{1}$, but now $\mathcal{T}^{2}=-\mathds{1}$. This underscores the different nature of fermions in 3+1 dimensions.

The purpose of this paper is to investigate the behavior of 1+1- and 3+1-dimensional relativistic $\mathcal{PT}$-invariant fermionic theories. An exploratory study in Ref. R19 examined in part the properties of a $\mathcal{PT}$-symmetric fermionic Lee model. This paper begins by reexamining the results in r15 , where it was assumed that for real $g$ the Lagrangian $\mathcal{L}=\bar{\psi}(i\not{\partial}-m-g\gamma^{5})\psi$ is $\mathcal{PT}$ symmetric. We find that including the axial term gives a dispersion relation $p^{2}=m^{2}-g^{2}$ that yields a real value for the physical mass only when $m^{2}\geq g^{2}$. This implies that the spectrum is not real in the chiral limit $m\to 0$. This result holds for Lagrangians in both 1+1 and 3+1 dimensions. We ask, Why is this so and under what conditions is it not so? Obtaining spectral reality in the chiral limit is part of the motivation for this paper. One of our long-range goals to construct a $\mathcal{PT}$-symmetric version of the Thirring and Nambu-Jona-Lasinio models. The challenge is to identify additional non-Hermitian terms that are both $\mathcal{PT}$ symmetric and chiral and give rise to a real spectrum in the chiral limit R20 .

Two ingredients are required for a precise analysis of fermionic systems: (i) Care must be taken in analyzing time reversal, which is nontrivial for fermionic systems; (ii) care is needed in deciding on the form of $\mathcal{PT}$-adjoint operators. In this paper we focus first on time reversal and then address the constraint of selfadjointness with respect to the $\mathcal{PT}$ inner product for fermions.

For various interactions in 1+1 and 3+1 dimensions we use the Euler-Lagrange equations to construct the Dirac equation that results from a Lagrangian density and investigate whether this (quantum-mechanical) Dirac equation is form invariant under the actions of $\mathcal{P}$ and $\mathcal{T}$. This enables us to identify the transformation properties of the interaction term and also to calculate the dispersion relation associated with plane-wave solutions of the Dirac equation. In addition, by rewriting the Dirac equation in the form $i\partial_{t}\psi=H\psi$, we identify the effective Hamiltonian $H$ R18 associated with the interaction. We will see that the form invariance of the Dirac equation under $\mathcal{PT}$ is equivalent to the statement that $H$ commutes with $\mathcal{PT}$.

In analyzing the case of 1+1 dimensions, we find the surprising result that for complex fermionic fields, the bilinear interaction form $-g\bar{\psi}\gamma^{5}\psi$ gives a Dirac equation that is odd under time reversal and also odd under parity. Thus, the Dirac equation with the interaction term is form invariant under $\mathcal{PT}$. The $\mathcal{PT}$ symmetry can also be verified by determining the Hamiltonian $H$ associated with this Dirac equation $i\partial_{t}\psi=H\psi$. The $2\times 2$ matrix representation clarifies this result. Comparing with the general result for a $2\times 2$ $\mathcal{PT}$-symmetric fermionic Hamiltonian in 1+1 dimensions R19 , it becomes evident that in 1+1 dimensions the $\mathcal{PT}$ symmetry is broken when $m$ vanishes. In a second example, due to the similarities in the transformation properties of this interaction with those of $\phi(t,{\bf x})$, we surmise that higher integer powers of the interaction Lagrangian density $-g(\bar{\psi}\gamma^{5}\psi)^{N}$ might lead to a spectral relation that has real energies; we investigate this for $N=2$ and 3. We find that the $\mathcal{PT}$ symmetry is always broken if we assume that the expectation value $\langle\bar{\psi}\gamma^{5}\psi\rangle$ is negative imaginary. There are no other matrix potentials in 1+1 dimensions.

We then turn to further examples for which $x$-dependent $\mathcal{PT}$-symmetric potentials $ix^{3}$, $-x^{4}$, and $i\kappa/x$ introduced via vector or scalar coupling various combinations, as well as the complex $\mathcal{PT}$-symmetric lattice potentials $i\kappa\cot(x)+i\gamma^{0}\sin(x)$ and the Hulthén potential are included in the Dirac equation of motion. In order to gain some understanding of these systems, we construct the analogous classical systems for which a classical phase structure can be obtained.

The situation in 3+1 dimensions is different because $\mathcal{T}^{2}=-1$. Studying the algebra in 3+1 dimensions, we confirm that the interaction term $-g\gamma^{5}\psi$ in the equation of motion is even under time reversal. Since the parity transformation is still odd in 3+1 dimensions, we conclude that the interaction term in the Dirac equation is not invariant under $\mathcal{PT}$. While the dispersion relation is superficially the same as for the 1+1-dimensional case, which implies that there is no region in which the spectrum is real in the chiral limit, the associated interaction Hamiltonian is anti-$\mathcal{PT}$ symmetric, which is consistent with the complex nature of the spectrum.

In 3+1 dimensions, we search for other bilinear combinations of fermionic fields with the aim of determining all possible combinations that give a Dirac equation that is form invariant under $\mathcal{PT}$ and that are not Hermitian. We find two types of terms having either an axial vector or a tensor structure. The spectra of both of the non-Hermitian $\mathcal{PT}$-symmetric interactions are analyzed. Here too we find that the $\mathcal{PT}$ symmetry is always broken in the chiral limit. We also look at the consequences of imposing an additional condition that the Hamiltonian be selfadjoint under the $\mathcal{PT}$ inner product for fermions r13 ; r12 and investigate the restrictions that this implies. We demonstrate that the $\mathcal{PT}$ symmetry is always broken in the chiral limit, a feature that prevails in the analysis of the Dirac equation in the dimensions studied.

This paper is organized as follows: In Sec. II we investigate possible $\mathcal{PT}$-symmetric interactions in 1+1 dimensions. We analyze $\mathcal{L}_{\rm int}=-g\bar{\psi}\gamma^{5}\psi$ in Subsec. II.1 and extensions to this as $-g(\bar{\psi}\gamma^{5}\psi)^{N}$ in Subsec. II.2. We introduce the spatially dependent potentials $ix^{3}$, $-x^{4}$, and $i\kappa/x$, and the lattice and Hulthén potentials in Subsec. II.3. In Sec. III we analyze 3+1-dimensional interactions, starting with $\mathcal{L}_{\rm int}=-g\bar{\psi}\gamma^{5}\psi$ in Subsec. III.1 and other two-body (four-point) interactions in Subsec. III.2. Our conclusions and outlook are presented in Sec. IV.

II Non-Hermitian $\mathcal{PT}$-symmetric fermions in 1+1 dimensions

II.1 Axial bilinear fermionic interaction

We start with the Lagrangian density for a conventional Hermitian free fermionic field theory,

[TABLE]

where $\bar{\psi}=\psi^{\dagger}\gamma^{0}$ and $\psi^{\dagger}$ is the Hermitian conjugate of $\psi$. In 1+1 dimensions the gamma matrices are given in (1). We have shown that the free Dirac equation (2) associated with (14) is form invariant under the operation of $\mathcal{P}$ in (3) and of $\mathcal{T}$ in (4). Note that (2) is also form invariant under the combined operations of $\mathcal{P}$ and $\mathcal{T}$ because the functions $\psi(t,x)$ and $\mathcal{PT}\psi(t,x)=\gamma^{0}\gamma^{0}\psi^{*}(-t,-x)=\psi^{*}(-t,-x)$ both satisfy (2). A plane-wave solution to (2) gives the dispersion relation $E^{2}=p^{2}+m^{2}$. Finally, we read off the effective or quantum-mechanical Hamiltonian $H$ from the free Dirac equation $i\partial_{t}\psi=H\psi$ in (2): $H=-i\gamma^{0}\gamma^{1}\partial_{1}+m\gamma^{0}$. (This form is often written using the definitions $\alpha=\gamma^{0}\gamma^{1}$ and $\beta=\gamma^{0}$ R18 ).

We observe that the form invariance of the Dirac equation under $\mathcal{PT}$ is equivalent to the statement that $H$ commutes with $\mathcal{PT}$: $H(\mathcal{PT}\psi)=\mathcal{PT}(H\psi)$. This is so because the left hand side is

[TABLE]

and the right hand side is

[TABLE]

Next, we examine what happens if a pseudoscalar bilinear term in included in the Lagrangian density $\mathcal{L}=\mathcal{L}_{0}+\mathcal{L}_{\rm int}$, where $\mathcal{L}_{\rm int}=-g\bar{\psi}\gamma^{5}\psi$ and $g$ is a real parameter. Now the associated quantum-mechanical Dirac equation is altered to read

[TABLE]

Parity transforms this equation into

[TABLE]

and time reversal has the effect

[TABLE]

This Dirac equation is not invariant under $\mathcal{P}$ or $\mathcal{T}$ separately but it is invariant under $\mathcal{PT}$ because the axial interaction term changes sign twice; it is odd under both $\mathcal{P}$ and $\mathcal{T}$. So this axial non-Hermitian term is $\mathcal{PT}$ symmetric.

We can formulate this differently: We identify the effective quantum-mechanical Dirac Hamiltonian associated with the Dirac equation as

[TABLE]

where $H_{0}=-i\gamma^{0}\gamma^{1}\partial_{1}+m\gamma^{0}$ and $H_{\rm int}=g\gamma^{0}\gamma^{5}$. We have shown that $H_{0}$ commutes with $\mathcal{PT}$, and from the effect of $\mathcal{P}$ and $\mathcal{T}$ in 1+1 dimensions and the reality of $H_{\rm int}$, we see that $H_{\rm int}$ also commutes with $\mathcal{PT}$. Thus, the effective Hamiltonian $H$ reflects the symmetry of the Dirac equation.

For this case the dispersion relation is obtained from a plane-wave solution $\psi(t,x)$, and multiplying (15) by $(\not{p}+m+g\gamma^{5})$, where $\not{p}=\gamma^{0}p_{0}+\gamma^{1}p_{1}$, yields the result r15 $p^{2}=m^{2}-g^{2}$, which is positive only when $m^{2}\geq g^{2}$. Thus, in the chiral limit $m\to 0$ the spectrum is complex and the $\mathcal{PT}$ symmetry is broken in this limit.

The matrix representation makes this result clearer. Recall that a general two-dimensional $\mathcal{PT}$-symmetric fermionic Hamiltonian, which is selfadjoint with respect to the $\mathcal{PT}$ inner product for fermions and which commutes with $\mathcal{PT}$, can be written as R19

[TABLE]

where $a$, $b$, and $f$ are real numbers. The eigenvalues are $E_{\pm}=a\pm\sqrt{bf}$. Thus, if $b$ and $f$ have the same sign, the spectrum is real and the $\mathcal{PT}$ symmetry is unbroken.

Now, if the interaction Lagrangian density is $-g\bar{\psi}\gamma^{5}\psi$, the quantum-mechanical interaction Hamiltonian is

[TABLE]

Comparing this with (18), we confirm that $H_{\rm int}$ is $\mathcal{PT}$ symmetric and that this symmetry is always broken. Note that $H_{\rm int}$ is non-Hermitian.

If we add the conventional mass term to the interaction, the effective Hamiltonian in matrix form becomes

[TABLE]

We see immediately that it is $\mathcal{PT}$ symmetric and that the $\mathcal{PT}$ symmetry is unbroken if $g^{2}\leq m^{2}$.

Observe that the equation of motion resulting from the Dirac equation with an imaginary axial term,

[TABLE]

gives the dispersion relation $p^{2}=m^{2}+g^{2}$. So $m$ is real for all $g$, including the chiral limit $m\to 0$. However, this axial term is not $\mathcal{PT}$ symmetric. In fact, it is anti-$\mathcal{PT}$ symmetric, so that (19) is not form invariant under $\mathcal{PT}$.

II.2 Approximate solution for higher-power field-theoretic interactions

This section explores the effect that higher-power interaction terms have in $1+1$-dimensional systems. Our starting point is the general Lagrangian density

[TABLE]

The Euler-Lagrange equations give the corresponding Dirac or single-particle equation of motion as

[TABLE]

which is nonlinear if $N>1$. The case $N=1$ reduces to that examined in Sec. II.1. In the following we examine the cases $N=2$ and $N=3$.

II.2.1 $N=2$

When $N=2$, $\mathcal{L}(2)=\bar{\psi}(i\not{\partial}-m)\psi-g(\bar{\psi}\gamma^{5}\psi)^{2}$, and the interaction term is $\mathcal{PT}$ symmetric. The associated Euler-Lagrange equation is

[TABLE]

from which we deduce that the Hamiltonian $H$ satisfying $i\partial_{t}\psi=H\psi$ is

[TABLE]

To solve (20) approximately we replace $\bar{\psi}\gamma^{5}\psi$ by its average value $\langle\bar{\psi}\gamma^{5}\psi\rangle=\langle\phi\rangle$. Furthermore, by operating on the approximate version of (20) by $(i\not{\partial}-m-2g\gamma^{5}\langle\phi\rangle)$ we can solve for the spectrum. In the chiral limit $m\to 0$ this is

[TABLE]

Now, noting that the expectation value of a bosonic pseudoscalar field should be negative imaginary r10 ,

[TABLE]

where $A$ is a constant, it follows from (21) that $p^{2}=4g^{2}A^{2}$ is real. However, with this choice of $\langle\phi\rangle$, $H_{\rm int}=2g\gamma^{0}\gamma^{5}\langle\phi\rangle$ is anti $\mathcal{PT}$ symmetric, as is the interaction term in (20). Thus, the quantum-mechanical Dirac equation is no longer form invariant under $\mathcal{PT}$; also $\mathcal{PT}$ does not commute with $H$. Yet we obtain a real spectrum because now $H_{\rm int}$ is Hermitian. The opposite case, namely, when the Dirac equation is $\mathcal{PT}$ symmetric and $H$ commutes with $\mathcal{PT}$, can be simulated by letting $g\to ig$. Then $p^{2}<0$ so, as in Sec. IIA, $\mathcal{PT}$ symmetry is again realized in the broken phase.

II.2.2 $N=3$

When $N=3$, $\mathcal{L}(3)=\bar{\psi}(i\not{\partial}-m)\psi-g(\bar{\psi}\gamma^{5}\psi)^{3}$. This resembles the case for $N=1$. The Euler-Lagrange equation now reads

[TABLE]

It follows that the interaction part of the Hamiltonian is

[TABLE]

Again, to find an approximate solution we replace $(\bar{\psi}\gamma^{5}\psi)^{2}$ by its average value $\langle(\bar{\psi}\gamma^{5}\psi)^{2}\rangle$. Solving (22) we get

[TABLE]

in the chiral limit. We expect $\langle(\bar{\psi}\gamma^{5}\psi)^{2}\rangle$ to be real, so $p^{2}<0$ and the $\mathcal{PT}$ symmetry is always broken. We can confirm this explicitly by noting that (23) is simply proportional to $\gamma^{1}$ and thus only has off-diagonal values of opposite sign, see (1). Comparing this with (18), we note that (23) is manifestly $\mathcal{PT}$ symmetric.

We conclude that (i) If we construct a 1+1-dimensional Lagrangian density containing the axial $\mathcal{PT}$-symmetric interaction $(\bar{\psi}\gamma^{5}\psi)^{N}$ ($N$ odd), our approximation scheme shows that we obtain an equation of motion that is form invariant under $\mathcal{PT}$, and correspondingly a $\mathcal{PT}$-symmetric Hamiltonian. The $\mathcal{PT}$ symmetry is broken in the chiral limit. (ii) For even $N$ the equation of motion contains an anti-${\mathcal{PT}}$-symmetric term and the associated interaction Hamiltonian is also anti-$\mathcal{PT}$ symmetric but we obtain a dispersion relation that has real masses as a result of Hermiticity. If we modify the interaction by replacing $g\to ig$, we obtain a $\mathcal{PT}$-symmetric system but once again the $\mathcal{PT}$ symmetry is broken.

II.3 Dirac particle in $\mathcal{PT}$-symmetric potentials

In 1+1 dimensions there are no other $\gamma$-matrix-based interactions. However, in addition to these, we can include $\mathcal{PT}$-symmetric potentials having a spatial dependence such as $ix^{3}$, $-x^{4}$, $i\kappa/x$, or even periodic potentials into the relativistic Dirac equation and study the effects of these. Unlike nonrelativistic potentials, which are scalars and can only be included as such in the Schrödinger equation, in the Dirac equation, such potentials can be incorporated either as the nonvanishing scalar part of the 4-vector potential (which we refer to as vector coupling), or as pure scalar interactions, or as combinations thereof. We consider some examples below.

II.3.1 Vector coupling with $ix^{3}$

The 1+1-dimensional Dirac equation that includes the non-Hermitian $\mathcal{PT}$-symmetric vector-coupled potential $ix^{3}$ reads

[TABLE]

This is form invariant under $\mathcal{PT}$ and the associated relativistic Hamiltonian

[TABLE]

is also $\mathcal{PT}$ invariant. If we look for solutions of the form $\psi(t,x)=e^{-iEt}\psi(x)$, we arrive at the corresponding eigenvalue problem

[TABLE]

The eigenvectors $\psi_{1}(x)$ and $\psi_{2}(x)$ that solve this equation have the asymptotic behavior

[TABLE]

The convergence domain for $\psi_{1}(x)$ and $\psi_{2}(x)$ in the complex-$x$ plane are the $\mathcal{PT}$-symmetric Stokes sectors shown in Figs. 1 and 2 respectively. In these sectors $\psi_{1,2}(x)$ vanish exponentially as $|x|\to\infty$.

To obtain the selfenergy of the propagating particle we apply $i\not{\partial}$ to (24) and obtain the differential equation

[TABLE]

Since the matrix $\gamma^{1}\gamma^{0}={\rm diag}(1,-1)$ is diagonal, the two-component equations in (25) decouple. Although they are not Schrödinger-like, each is individually $\mathcal{PT}$ symmetric. We first examine the classical analog of these equations obtained by replacing $-i\partial_{x}$ by $p$,

[TABLE]

The classical Hamiltonian associated with $\psi_{1}$ is $H_{1}=\sqrt{p^{2}-2ix^{3}p-x^{6}-3x^{2}}$. The equation of motion of a classical particle described by $H_{1}$ is obtained by combining Hamilton’s equations $dx/dt=\partial H_{1}/\partial p$ and $dp/dt=-\partial H_{1}/\partial x$: $dx/dt=\pm\sqrt{1+3x^{2}/E^{2}}$. By rescaling both $x$ and $t$ this equation becomes

[TABLE]

We find that $x(t)$ forms open trajectories in the complex-$x$ plane, as shown in Fig. 3.

The open classical trajectories of the particle in the complex-$x$ plane reflects the behavior seen in the quantum case: By setting $p=0$, we observe that the selfenergy $\Sigma_{1}$ of the particle corresponding to $\psi_{1}$ is given by $\Sigma^{2}_{1}=-x^{6}-3x^{2}$, which implies that $\Sigma_{1}$ cannot be real BBH .

On the other hand, the trajectories of the classical particle in the complex-$x$ plane that are associated with the classical Hamiltonian $H_{2}=\sqrt{p^{2}+2ix^{3}p-x^{6}+3x^{2}}$, are closed, as can be seen in Fig. 4. In the quantum system, the selfenergy corresponding to $\psi_{2}$ is $\Sigma^{2}_{2}=-x^{6}+3x^{2}$. By parametrizing $x$ as $-i(\sqrt{1+ir}-1)$ where $r$ is real, $\psi_{2}$ vanishes exponentially as $r\to\pm\infty$. We note that the ends of this path lie in the left and right Stokes sectors of Fig. 2 as $|x|\to\infty$. When $-\sqrt[4]{3}<x<\sqrt[4]{3}$, $\Sigma^{2}_{2}$ is positive. Thus, the selfenergy associated with the particle is real.

II.3.2 Scalar coupling with $ix^{3}$ and vector coupling with $i\kappa/x$

In the previous subsection we treated the $\mathcal{PT}$-symmetric potential $ix^{3}$ in a vector-coupling scheme; now we consider it as a scalar potential, where, in addition, the Dirac particle is also under the influence of a complex $\mathcal{PT}$-symmetric Coulomb potential. The non-Hermitian $\mathcal{PT}$-symmetric Dirac equation now reads

[TABLE]

where $\kappa$ is a real parameter. The associated relativistic quantum-mechanical Hamiltonian is

[TABLE]

Again, looking for solutions of the form $\psi(t,x)=e^{-iEt}\psi(x)$ leads to an eigenvalue problem

[TABLE]

Writing the eigenfunction $\psi(x)$ in terms of its two spinor components, $\psi(x)=(\phi_{1}(x),\phi_{2}(x))$ 111We reserve the notation $\psi_{1},\psi_{2}$ for two different spinor solutions of the same equation. we find two coupled differential equations for the scalar functions $\phi_{1,2}(x)$,

[TABLE]

We can eliminate the second component $\phi_{2}$ from (33) by exploiting (34), and after rescaling $\phi_{1}$, and choosing $\kappa$ to be $-3/2$ for convenience, we obtain the simple form

[TABLE]

which is a Schrödinger-like equation with a $-x^{6}$ potential. On the real-$x$ axis this upside-down potential is unstable, but by imposing appropriate $\mathcal{PT}$-symmetric boundary conditions we can obtain a real spectrum. As in the previous subsection, we find that to have a convergent eigenfunction, we must treat the problem in the complex-$x$ plane.

The WKB approximation for the solutions of (35) to leading order is cmbbook

[TABLE]

where $Q(x)=E^{2}+x^{6}$. For large $|x|$ the exponential component of this asymptotic behavior is

[TABLE]

There are eight Stokes sectors in the complex-$x$ plane, each with an opening angle of $\pi/4$. To have a $\mathcal{PT}$-symmetric pair of Stokes sectors, we choose the minus sign in (37) for the right Stokes sector, which is located just below the positive-real-$x$ axis. For the left Stokes sector we choose the positive sign in (37), which determines a sector located just below the negative-real-$x$ axis. These two Stokes sectors are depicted in Fig. 5.

We can also approximate the eigenenergies of (35). To do so, we first find the two turning points which are determined by $E=-x^{6}$ and which lie in the Stokes sectors in Fig. 5. These two points are

[TABLE]

The WKB quantization condition is

[TABLE]

Thus,

[TABLE]

For $n=0$ or $1$, we obtain $E_{0}=\pm 1.0$ and $E_{1}=\pm 2.27$.

An exact calculation of the eigenvalues can be made on parametrizing $x$ as $-i(\sqrt{1+ir}-1)$, where $r$ is a real variable. As depicted in Fig. 6, the ends of this path lie inside the Stokes sectors as $|x|\to\infty$, so we pose the eigenvalue problem for the differential equation in (35) on this contour. We determine the ground-state and first-excited-state energies numerically as

[TABLE]

which illustrates the accuracy of the WKB approximation. Thus, the energy spectrum of the Dirac particle in the combined non-Hermitian $\mathcal{PT}$-symmetric potentials $ix^{3}$ and $i\kappa/x$ is real and discrete.

The trajectories of a classical particle in the complex-$x$ plane described by the classical Hamiltonian $H=\sqrt{p^{2}-x^{6}}$ obtained from (35) are shown in Fig. 7. These trajectories are closed, which reflects the reality and the discreteness of the spectrum at the quantum level.

II.3.3 Vector coupling with $-x^{4}$

Next, we consider a massless Dirac particle under the influence of the upside-down quartic potential $-x^{4}$. In the vector-coupling scheme, the relativistic Dirac equation is modified to read

[TABLE]

As in the previous examples, this equation is form invariant under $\mathcal{PT}$ and the associated Hamiltonian

[TABLE]

commutes with $\mathcal{PT}$. Looking for solutions of the form $\psi(t,x)=e^{-iEt}\psi(x)$ leads to an eigenvalue equation $H\psi=E\psi$, whose eigenvectors behave asymptotically as

[TABLE]

Note that $\psi_{1}$ vanishes exponentially in a Stokes sector with opening angle $\pi/5$. This sector contains the negative-imaginary-$x$ axis, so it vanishes exponentially as $x\to-i\infty$. The function $\psi_{2}$ also vanishes exponentially in the same Stokes sector, but one that has rotated upward; that is, $\psi_{2}\to 0$ as $x\to i\infty$.

Following the analysis given in Subsec. 1, we iteratively apply $i\not{\partial}$ to the corresponding Dirac equation and find the decoupled system of equations

[TABLE]

The selfenergies $\Sigma_{1}$ and $\Sigma_{2}$ of the particle corresponding to $\psi_{1}$ and $\psi_{2}$ are given by $\Sigma^{2}_{1}=x^{8}-4ix^{3}$ and $\Sigma^{2}_{2}=x^{8}+4ix^{3}$. As $\psi_{1}$ and $\psi_{2}$ converge on $x=-ir$ and $x=ir$, the selfenergies become real.

The trajectories of the classical particle described by both of the classical Hamiltonians obtained from (44) are closed in the complex-$x$ plane. In Fig. 8, this is shown for the classical Hamiltonian $H=\sqrt{p^{2}+2x^{4}p+x^{8}-4ix^{3}}$.

II.3.4 Scalar coupling with $-x^{4}$ and vector coupling with $i\kappa/x$

We now treat the upside-down potential $-x^{4}$ as a scalar potential and, in addition, we consider the effect of a complex $\mathcal{PT}$-symmetric Coulomb potential on the Dirac particle in a vector-coupling scheme, satisfying the modified Dirac equation

[TABLE]

where $\kappa$ is a real parameter. This equation is form invariant under $\mathcal{PT}$ and the associated Hamiltonian

[TABLE]

commutes with $\mathcal{PT}$. The search for solutions of the form $\psi(t,x)=e^{-iEt}\psi(x)$ requires solutions of the eigenvalue equation

[TABLE]

As in Subsec. 2, it is convenient to write $\psi(x)$ in terms of its (scalar) components, $\psi=(\phi_{1},\phi_{2})$ and derive the coupled equations that $\phi_{1}$ and $\phi_{2}$ satisfy. Following the procedure outlined in Subsec. 2, we eliminate $\phi_{2}$ and arrive at a Schrödinger-like equation for $\phi_{1}$,

[TABLE]

where, for convenience, we have set $\kappa=-2$. We have thus found an octic potential with positive sign. Hence, we pose the eigenvalue problem on the real-$x$ axis. As before we use the WKB approximation to obtain the eigenvalues for large $n$,

[TABLE]

From this equation we find that $E_{0}=\pm 0.87$ and $E_{1}=\pm 2.10$. A direct numerical calculation gives $E_{0}=\pm 1.11$ and $E_{1}=\pm 2.18$. Thus, once again, we find that the energy spectrum of a Dirac particle in the presence of combined non-Hermitian $\mathcal{PT}$-symmetric vector and scalar potentials $i\kappa/x$ and $-x^{4}$ is real and discrete.

Here again, we see that the reality and discreteness of the spectrum is evident at the classical level with closed trajectories in the complex-$x$ plane. We recognize the classical Hamiltonian of the system from (45) as being $H=\sqrt{p^{2}+x^{8}}$. Figure 9 shows that the classical trajectories described by this Hamiltonian $H$ are closed.

II.3.5 Complex $\mathcal{PT}$-symmetric lattice potentials

The methods in the previous subsections are general enough to be applied to a Dirac particle in complex $\mathcal{PT}$-symmetric lattices. The relativistic Dirac equation

[TABLE]

with $\kappa$ real, has non-Hermitian interaction terms, but is form invariant with respect to $\mathcal{PT}$. The associated Hamiltonian,

[TABLE]

commutes with $\mathcal{PT}$.

As before, we can search for time-independent solutions of (46). Writing $\psi(t,x)=e^{-iEt}\psi(x)$, we obtain coupled equations for the components of the spinor eigenfunction $\phi_{1}$ and $\phi_{2}$, where $\psi=(\phi_{1},\phi_{2})$. Eliminating $\phi_{2}$, we find a Schrödinger-like equation for the $\phi_{1}$, which after suitably rescaling, is

[TABLE]

where we have set $\kappa=-1/2$.

The spectrum of the operator $-d^{2}/dx^{2}-\sin^{2}(x)$ is real and consists of spectral bands separated by infinitely many spectral gaps r17 . The absence of discrete energies and the reality of the band-structure manifest itself via periodic, open trajectories of the classical particle described by $H=\sqrt{p^{2}-\sin^{2}(x)}$ in the complex-$x$ plane, as depicted in Fig. 10.

Before closing this subsection, we make a side remark: We note that the (quantum-mechanical, nonrelativistic) Hamiltonian $H=p^{2}+i\sin(x)$ describes a particle subject to the periodic potential $V(x)=i\sin(x)$ in a $\mathcal{PT}$-symmetric crystal. As was shown in Ref. r18 , by examining a discriminant, one can conclude that this Hamiltonian has real energy bands. However, to verify that the band structure is real, one can alternatively show that the eigenfunctions are $\mathcal{PT}$ symmetric; that is, that the $\mathcal{PT}$ symmetry of the Hamiltonian is unbroken. To this end we plot the absolute values of the eigenfunctions of the two states of $H=p^{2}+i\sin(x)$ in Fig. 11 and observe that both are in fact symmetric. The energy bands are real, and are shown in Fig. 12. We use this technique in the next subsection.

II.3.6 Scalar coupling with complex $\mathcal{PT}$-symmetric Hulthén

potential

The complex $\mathcal{PT}$-symmetric Hulthén potential is

[TABLE]

If we regard $V(x)$ as a potential in the nonrelativistic time-independent Schrödinger equation, $H\psi=E\psi$, with $H=p^{2}+V(x)$, we find that the band structure for the energies is entirely complex, and, as is the case with $\mathcal{PT}$-symmetric potentials in the broken-symmetry phase, the eigenvalues occur in complex-conjugate pairs. We illustrate this by plotting the absolute values of the eigenfunctions of the two states of the Hamiltonian that correspond to the complex-conjugate pairs of the band-edge energies $E=0.75\pm 0.59i$, see Fig. 13. Note that the eigenfunctions display no symmetry, which implies the complex nature of the band structure.

We now consider the relativistic Dirac equation that includes the $\mathcal{PT}$-symmetric Hulthén potential in a scalar-coupling scheme, together with an additional $\mathcal{PT}$-symmetric vector potential:

[TABLE]

with $\kappa$ being a real parameter. This equation has been constructed so as to be form invariant with respect to $\mathcal{PT}$ and the associated Hamiltonian

[TABLE]

once again commutes with $\mathcal{PT}$. Following the same procedure as in the last subsections, we search for time-independent solutions of the Dirac equation, and find the equations for the components of $\psi=(\phi_{1},\phi_{2})$. On eliminating $\phi_{2}$, we obtain a Schrödinger-like equation for the first component of the two-component spinor eigenfunction as

[TABLE]

where we have set $\kappa=-1/2$ for convenience.

By using spectral methods, we determine numerically that the band structure in (48) is entirely real; that is, the symmetry is unbroken. We have shown the absolute values of the first two eigenfunctions in Fig. 14, which are clearly symmetric as expected. The (real) energy bands corresponding to this potential are shown in Fig. 15.

The classical Hamiltonian associated with this system is $H=\sqrt{p^{2}+1/(1-e^{ix})^{2}}$. The trajectories of the classical particle, as shown in Fig. 16, are periodic and open. This appears to correspond to the fact that the quantum Hamiltonian has real energy bands but no discrete eigenvalues.

III Non-Hermitian $\mathcal{PT}$-symmetric fermions in 3+1 dimensions

III.1 Axial bilinear fermionic interaction

In 3+1 dimensions, we again start with the free fermionic Lagrangian $\mathcal{L}_{0}=\bar{\psi}(i\not{\partial}-m)\psi$ of (14) and the Dirac equation of motion,

[TABLE]

and recall that the actions of $\mathcal{P}$ and $\mathcal{T}$ are given in (13), where the gamma matrices are given in (12). Equation (49) is form invariant under the combined operations $\mathcal{P}$ and $\mathcal{T}$ because the functions $\psi(t,{\bf x})$ and $\mathcal{PT}\psi(t,{\bf x})=\gamma^{0}(i\gamma^{1}\gamma^{3})\psi^{*}(-t,-{\bf x})$ satisfy the same equation. For the free Dirac equation, this is true for $\mathcal{P}\psi=\gamma^{0}\psi(t,-{\bf x})$ and $\mathcal{T}\psi=i\gamma^{1}\gamma^{3}\psi^{*}(-t,{\bf x})$ individually. By setting ${\bf x}\to-{\bf x}$ in (49), it becomes

[TABLE]

where $i=1,2,3$ denote the spatial components. Multiplying this result from the left with $\gamma^{0}$ and using the anticommutation relations $\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}$, with $g^{\mu\nu}={\rm diag}(1,-1,-1,-1)$ results in

[TABLE]

On the other hand, taking the complex conjugate of (49) and replacing $t\to-t$ gives

[TABLE]

because \big{(}\gamma^{2}\big{)}^{*}=-\gamma^{2}. Multiplying this equation from the left by $i\gamma^{1}\gamma^{3}$ and using the anticommutation relations for the gamma matrices then gives

[TABLE]

The form invariance of the equation satisfied by $\mathcal{PT}\psi(t,{\bf x})$ then follows.

Next we include an axial non-Hermitian bilinear term into the Lagrangian density, ${\cal L}={\cal L}_{0}+{\cal L}_{\rm int}$, where ${\cal L}_{\rm int}=-g\bar{\psi}\gamma^{5}\psi$. The corresponding Euler-Lagrange equation

[TABLE]

superficially resembles the 1+1-dimensional case. However, here, while parity transforms this equation into

[TABLE]

time reversal transforms it into

[TABLE]

Note the minus sign before the last term in (52): While parity flips the sign of the axial term, time reversal in 3+1 dimensions does not. Parity is odd, but time reversal is even in 3+1 dimensions. So the combination of $\mathcal{PT}$ does not lead to a form-invariant Dirac equation. The axial term by itself is anti-$\mathcal{PT}$ symmetric. This differs from the 1+1-dimensional case [see (16) and (17)].

The dispersion relation that one obtains from (50) is formally the same as in the 1+1-dimensional case; assuming plane-wave solutions of the form $\psi=e^{-ip^{\mu}x_{\mu}}$ and multiplying (50) by $(\not{p}+m+g\gamma^{5})$, we arrive at the same spectral relation as in 1+1 dimensions,

[TABLE]

which is positive only when $m^{2}\geq g^{2}$ and is complex in the chiral limit $m\to 0$.

As before, the form invariance of the Dirac equation unter $\mathcal{PT}$ imples that $H(\mathcal{PT}\psi)=\mathcal{PT}(H\psi)$, where $H$ is the Dirac Hamiltonian identified through $i\partial_{t}\psi=H\psi$. Thus, we can ascertain the properties of various interaction terms by testing them with this commutation relation. For (50) the associated Hamiltonian is

[TABLE]

Let us check the symmetry of the axial interaction term $H_{\rm int}=g\gamma^{0}\gamma^{5}$ under $\mathcal{P}$ and $\mathcal{T}$. Using (13), we evaluate $\mathcal{PT}\psi(t,{\bf x})=\gamma^{0}i\gamma^{1}\gamma^{3}\psi^{*}(-t,-{\bf x})$ and apply $H_{\rm int}$:

[TABLE]

$H_{\rm int}$ anti-commutes with $\mathcal{PT}$, confirming that this term is not $\mathcal{PT}$ symmetric. It thus explains the complex nature of the dispersion relation in the chiral limit. By contrast, if $H_{\rm int}$ is imaginary, that is $H_{\rm int}=ig\gamma^{0}\gamma^{5}$, we have a $\mathcal{PT}$-symmetric Hamiltonian, which is also Hermitian, and does have a real spectrum for all $g$, $p^{2}=g^{2}$ in the chiral limit.

Once again, to clarify this point, we turn to an explicit matrix representation. Then $H_{\rm int}$ becomes

[TABLE]

which is not Hermitian. By comparison, a general four-dimensional $\mathcal{PT}$-symmetric fermionic Hamiltonian that is invariant under $\mathcal{PT}$ and also selfadjoint under the $\mathcal{PT}$ inner product has a matrix form r12 ; r13 ; abspk ,

[TABLE]

where $B_{\pm}=b_{1}\pm ib_{2}$ and $C_{\pm}=b_{3}\pm ib_{0}$. The parameters $a_{0}$, $b_{0}$, $b_{1}$, $b_{2}$, and $b_{3}$ are real. This matrix has twofold degenerate real eigenvalues

[TABLE]

for $a_{0}^{2}\geq\sum_{i=0}^{3}b_{i}^{2}$ footnote . Equation (54) is not a special case of (55), so it is does not represent a $\mathcal{PT}$-symmetric fermionic Hamiltonian.

Evidently, the symmetry properties of the axial term $-g\gamma^{5}\psi$ in the Dirac equation in 1+1 dimensions differ from those in 3+1 dimensions. The Dirac equation is form invariant in 1+1 dimensions under $\mathcal{PT}$, but not in 3+1 dimensions. This corresponds to a relativistic $\mathcal{PT}$-symmetric quantum-mechanical Hamiltonian in 1+1 dimensions, but not in in 3+1 dimensions. This difference is caused by the different effect of time reversal in 1+1 and 3+1 dimensions. The spectrum obtained in both cases is formally the same, so we conclude that the $\mathcal{PT}$ symmetry is always broken in 1+1 dimensions when $m\to 0$. However, in 3+1 dimensions the Hamiltonian is anti-$\mathcal{PT}$ symmetric in the chiral limit, which explains the complex nature of the spectrum when $m\to 0$.

Interestingly, if we include the conventional mass term $m\gamma^{0}$, (54) becomes

[TABLE]

which is neither Hermitian nor $\mathcal{PT}$-symmetric. However, $H_{\rm int}$ is pseudo-Hermitian in the sense of GWM because $H_{\rm int}^{\dagger}=\gamma^{0}H_{\rm int}(\gamma^{0})^{-1}$. Hence, this Hamiltonian can be used to describe pseudo-Hermitian fermions.

We can construct fermionic creation and annihilation operators which are quadratically nilpotent, and investigate their anticommutation relations. First, we note that the eigenvalues of (57) are

[TABLE]

with corresponding eigenvectors

[TABLE]

[TABLE]

The spectrum is twofold degenerate and is real if $g^{2}\leq m^{2}$. This degeneracy is the analog of the phenomenon of Kramer’s theorem in conventional Hermitian quantum mechanics, where the Hamiltonian is invariant under odd time reversal, as is the case with (57).

We introduce the annihilation operator for the Hamiltonian (57) as

[TABLE]

which is nilpotent ($\eta^{2}=0$) as required. We verify that

[TABLE]

[TABLE]

The creation operator reads

[TABLE]

One can now establish the anticommutation relations

[TABLE]

where $N$ is the number operator, $N=\eta^{\prime}\eta$, as well as the peculiar anticommutation relation $\eta\eta^{\prime}+\eta^{\prime}\eta=-\mathbbm{1}$. The minus sign indicates that the number operator gives the negative of the state occupation number. For further illustrations of this in the context of $\mathcal{PT}$ symmetry see Refs. r14 ; R19 .

Finally, we comment that in terms of the number operator $N$, we can write the four-dimensional pseudo-Hermitian fermionic Hamiltonian in (57) in the form of a free (bosonic) harmonic oscillator as

[TABLE]

where $\Delta\omega=\omega_{+}-\omega_{-}$ and $\mathbb{1}$ is the four-dimensional identity matrix.

III.2 Other matrix-type two-body (four-point) $\mathcal{PT}$- and

anti-$\mathcal{PT}$-symmetric interactions and the resulting $\mathcal{PT}$-symmetric Hamiltonians

Having determined that an axial non-Hermitian interaction Lagrangian density of the form $-g\bar{\psi}\gamma^{5}\psi$ in 3+1 dimensions does not give rise to a Dirac equation that is form invariant with respect to $\mathcal{PT}$, we seek other types of interactions that are $\mathcal{PT}$ symmetric but non-Hermitian. Usually, the standard method of analyzing two-body (four-point) interactions involves constructing the 16 independent bilinears from the 16 $4\times 4$ independent matrices and considering the Lagrangian density associated with each of these. The standard Hermitian combinations are (1) $\bar{\psi}\psi$, (2) $\bar{\psi}\gamma^{\mu}\psi$, (3) $\bar{\psi}\sigma^{\mu\nu}\psi$, (4) $\bar{\psi}\gamma^{5}\gamma^{\mu}\psi$, and (5) $i\bar{\psi}\gamma^{5}\psi$. This Lagrangian-density approach is suitable for a discussion of symmetries that lead to conserved currents through Noether’s theorem, but the analysis of $\mathcal{PT}$ symmetry is most simply done by examining the form-invariance of the appropriate Dirac-like equation that can be derived using the Euler-Lagrange equations. Since this in turn tranlates into a commutation relation of the Hamiltonian with $\mathcal{PT}$, in a form of reverse engineering, we only need to identify possible $\mathcal{PT}$-symmetric forms of the interaction Hamiltonians. Thus, we consider the five interaction Hamiltonians below and show that these combinations are all $\mathcal{PT}$ symmetric:

[TABLE]

where $g,\,B_{\mu},\,T_{\mu\nu},\,\tilde{B}_{\mu},$ and $g_{A}$ are taken to be real.

Using the procedure in (53) in which $H_{{\rm int},i}$ is applied to $\mathcal{PT}\psi$, we evaluate the commutator of $H_{{\rm int},i}$ and $\mathcal{PT}$ using (13), and where necessary, make use of the relation $\gamma^{\mu}i\gamma^{1}\gamma^{3}=i\gamma^{1}\gamma^{3}\gamma_{\mu}^{*}$. Then

[TABLE]

We conclude that

[TABLE]

Thus, the general form of a relativistic quantum-mechanical Dirac equation, which is form invariant under $\mathcal{PT}$ transformations, reads

[TABLE]

A brief analysis shows that $H_{{\rm int},3}$ and $H_{{\rm int},4}$ are anti-Hermitian, while $H_{{\rm int},1}$, $H_{{\rm int},2}$, and $H_{{\rm int},5}$ are Hermitian. So we have identified two types of terms that give rise to non-Hermitian but $\mathcal{PT}$-symmetric Hamiltonians. We consider each of these in turn.

III.2.1 $H_{{\rm int},3}=iT_{\mu\nu}\gamma^{0}\sigma^{\mu\nu}$

To understand the structure of $H_{{\rm int},3}$ we write it in matrix form:

[TABLE]

where the coefficients $q_{i}$, $i=1,\dots,6$, are abbreviations for combinations of the $T_{\mu\nu}$,

[TABLE]

The eigenvalues of (63) are

[TABLE]

where $Q^{2}=\sum_{i=1}^{6}q_{i}^{2}$. Thus, the eigenvalues are complex and the $\mathcal{PT}$ symmetry is broken. Including a finite mass term $m\gamma^{0}$ in general does not change this result. The eigenvalues of $H_{{\rm int},3}+m\gamma^{0}$ are modified to read

[TABLE]

As we have already argued, only if the spectrum is twofold degenerate, can the eigenvalues be real footnote .

If we compare (63) with (55), we see that both have a quaternionic structure. However, in addition to being $\mathcal{PT}$ symmetric, (55) fulfills the additional condition that this Hamiltonian is selfadjoint with regard to the $\mathcal{PT}$ inner product according to r12 . This means that, in addition, $H_{{\rm int},3}$ should fulfill the condition $H_{{\rm int},3}^{\mathcal{PT}}=PH_{{\rm int},3}^{\dagger}P=H_{{\rm int},3}$. If we construct $H_{{\rm int},3}^{\mathcal{PT}}$, we find that

[TABLE]

for this condition to hold. The eigenvalues are twofold degenerate and if a mass term is included, they are

[TABLE]

which is real provided that $m^{2}\geq q_{1}^{2}+q_{2}^{2}+q_{3}^{2}$. Thus, $\mathcal{PT}$ symmetry is broken in the chiral limit. The regions of unbroken $\mathcal{PT}$ symmetry for the Hamiltonian $H_{\rm int,3}+m\gamma^{0}$ for some specific parameters are shown in Fig. 17.

III.2.2 $H_{{\rm int},4}=i\tilde{B}_{\mu}\gamma^{0}\gamma^{5}\gamma^{\mu}$

We now consider the equation of motion resulting from the non-Hermitian $\mathcal{PT}$-symmetric Hamiltonian $H_{{\rm int},4}$ (as well as its corresponding Lagrangian $\mathcal{L}_{{\rm int},4}$),

[TABLE]

The spectrum associated with this equation can be obtained by calculating the poles of the associated Green function in momentum space, which satisfies

[TABLE]

Rationalizing this expression for $S(p)$, we identify the dispersion relation as

[TABLE]

This has no real solutions for all $p_{0}$. Thus, again we find that the $\mathcal{PT}$ symmetry of the Hamiltonian is broken. We also notice that an anti-$\mathcal{PT}$-symmetric but Hermitian Hamiltonian would give a real spectrum with dispersion relation $(p^{2}-\tilde{B}^{2})^{2}-4(p\cdot\tilde{B})^{2}=0$.

Note that the matrix form of the Hamiltonian $H_{{\rm int},4}$, with components $\tilde{B}_{\mu}=(\tilde{B}_{0},\tilde{B}_{1},\tilde{B}_{2},\tilde{B}_{3})$ is

[TABLE]

which has complex eigenvalues for all $\tilde{B}_{\mu}$ real,

[TABLE]

If, as in Subsec. IIIB1, we demand that the Hamiltonian $H_{{\rm int},4}$ satisfies the selfadjointness condition according to r12 ; r13 ; R19 , that is, $H_{{\rm int},4}^{\mathcal{PT}}=PH_{{\rm int},4}^{\dagger}P=H_{{\rm int},4}$, we calculate that $\tilde{B}_{0}\neq 0$ and $\tilde{B}_{1}=\tilde{B}_{2}=\tilde{B}_{3}=0$. The resulting twofold degenerate energies are

[TABLE]

where we have included a mass term. This implies a real spectrum for $m^{2}\geq\tilde{B}_{0}^{2}$. Once again, in the chiral limit the $\mathcal{PT}$ symmetry is broken.

IV Main conclusions and outlook

Our focus in this paper has been on investigating non-Hermitian $\mathcal{PT}$-symmetric extensions to fermionic systems in 1+1 and 3+1 dimensions. The main findings are the following:

a) Usually, we explore the symmetries of a field theory by examining the Lagrangian density. However, the properties associated with $\mathcal{PT}$ symmetry are more easily found by forming the Euler-Lagrange equations and demanding form-invariance of the relativistic equation of motion with respect to $\mathcal{PT}$. This is equivalent to constructing the quantum-mechanical relativistic Hamiltonian and investigating its commutation relation with $\mathcal{PT}$.

b) For a pure axial interaction the symmetry properties in 1+1 dimensions differ from those in 3+1 dimensions even though the formal structure of the energy relation is unchanged. This can be traced back to the different transformation properties of time reversal in 1+1 and 3+1 dimensions and is ultimately due to the fact that $\mathcal{T}^{2}=-\mathds{1}$ in 3+1 dimensions.

c) In 1+1 dimensions including a complex $\mathcal{PT}$-symmetric position-dependent potential in both scalar- and vector-coupling schemes and combinations thereof can result in real and discrete eigenvalues, when searching for plane wave solutions. For appropriately chosen combinations of scalar and vector couplings, a Schrödinger-like equation can be found and the spectrum can be determined numerically. The analogous classical systems give information about the nature of the spectrum. They display closed contours when the eigenvalues are real and discrete and they are periodic and open if there is a real band structure. If the eigenvalues are complex, the paths are open and nonperiodic.

d) In 3+1 dimensions only two possible Lorentz-invariant two-body combinations are $\mathcal{PT}$ symmetric and not Hermitian. These, however, give rise to a complex spectrum in the chiral limit. Including a mass term can result in a real spectrum. In addition, further constraints are placed on the parameters if the condition of selfadjointness with respect to the $\mathcal{PT}$ inner product is placed on the Hamiltonian. This does not change the conclusion.

It remains an open question as to whether including non-Hermitian $\mathcal{PT}$-symmetric terms can play a role in physical fermionic systems, for example, affecting chiral symmetry restoration within the Nambu-Jona-Lasinio or Thirring models, or in weak interactions.

Acknowledgements.

CMB thanks the Alexander von Humboldt Foundation for financial support and the Heidelberg Graduate School for Physics for its kind hospitality.