Discrete spacetime symmetries and particle mixing in non-Hermitian scalar quantum field theories

We discuss second quantization, discrete symmetry transformations and inner products in non-Hermitian scalar quantum field theories with PT symmetry, focusing on a prototype model of two complex scalar fields with anti-Hermitian mass mixing. Whereas the definition of the inner product is unique for theories described by Hermitian Hamiltonians, it is not unique for theories with non-Hermitian Hamiltonians. Energy eigenstates are not orthogonal with respect to the conventional Dirac inner product, and the PT inner product does not correspond to a positive-definite norm. We clarify the relationship between canonical-conjugate operators and introduce a further discrete symmetry C', previously introduced for quantum-mechanical systems, and show that the C'PT inner product does yield a positive-definite norm, and hence is appropriate for defining the Fock space in non-Hermitian models with PT symmetry in terms of energy eigenstates. We also discuss similarity transformations between PT-symmetric non-Hermitian scalar quantum field theories and Hermitian theories, showing that they are not in general equivalent in the presence of interactions. As an illustration of our discussion, we compare particle mixing in a Hermitian theory and in the corresponding non-Hermitian model with PT symmetry, showing how the latter maintains unitarity and exhibits mixing between scalar and pseudoscalar bosons.


I. INTRODUCTION
Recent years have witnessed growing interest in non-Hermitian quantum theories [1], particularly those with PT symmetry, where P and T denote parity and time-reversal, respectively [2]. It is known that a quantum system described by a non-Hermitian Hamiltonian has real energies and leads to a unitary time evolution if this Hamiltonian and its eigenstates are invariant under PT symmetry [3]. This increasing interest has been driven in part by theoretical analyses supporting the consistency of such theories in the context of both quantum mechanics and quantum field theory, and in part by the realization that such theories have applications in many physical contexts, e.g., photonics [4,5] and phase transitions [6,7]. Although there are strong arguments for the consistency of PT -symmetric quantum field theory, a number of theoretical issues merit further attention. These include the analysis of discrete symmetries, which requires in turn a careful analysis of the Fock spaces of non-Hermitian quantum field theories with PT symmetry and their inner products. 1 In this paper, we study and clarify these issues in the context of a minimal non-Hermitian bosonic field theory with PT symmetry at the classical and quantum levels. We construct explicitly in the quantum version the operators generating discrete symmetries, and discuss the properties of candidate inner products in Fock space. We also construct a similarity transformation between the free-field PT -symmetric non-Hermitian model and the corresponding Hermitian counterpart, showing explicitly that the correspondence does not hold in general in the presence of interactions.
As an application of this formalism, we discuss the simplest non-trivial prototype quantum particle system, namely mixing in models of non-interacting bosons -building upon the study [9] that described how to interpret the corresponding PT -symmetric Lagrangian. 2 These systems appear in various physical situations of phenomenological interest, such as coupled pairs of neutral mesons, and also appear in the PT -symmetric extension of supersymmetry [14]. Issues arising in the formulation of such theories include the roles of discrete symmetries, the relationship between the descriptions of mixing in the PT -symmetric non-1 A detailed description of the PT inner product in quantum mechanics can be found in Ref. [8]. 2 Self-interactions of these scalar fields were considered in Ref. [10], their coupling to an Abelian gauge field in Ref. [11] and to non-Abelian gauge fields in Ref. [12]. See Ref. [13] for a study of 't Hooft-Polyakov monopoles in a non-Hermitian model.
Hermitian case and the standard Hermitian case, 3 and the status of unitarity, whose status in non-Hermitian theories has been questioned [19]. As an example of our approach, we exhibit a mechanism allowing oscillations between scalar and pseudoscalar bosons, which is possible with a mass-mixing matrix that is anti-Hermitian, but with real eigenvalues, and we compare with results in the previous literature.
The layout of our paper is as follows. In Sec. II, we introduce the minimal two-flavour non-Hermitian bosonic field theory we study, discussing in Sec. II A its discrete symmetries P, T and C ′ [20] at the classical level as well as the similarity transformation relating it to a Hermitian theory, and mentioning a formal analogy with (1+1)-dimensional Special Relativity in Sec. II B. We discuss in Sec. III the second quantization of the theory in both the flavour and mass bases. Then, in Sec. IV, we discuss the quantum versions of the discrete symmetries and various definitions of the inner product in Fock space. In particular, we discuss in Secs. IV A and IV B the parity and C ′ transformations, and we discuss the similarity transformation in Sec. IV C, emphasising that the equivalence between the non-interacting non-Hermitian model and a Hermitian theory does not carry over to an interacting theory, in general. (An Appendix compares the similarity transformation discussed in this paper with a previous proposal [15] in the literature.) In Sec. IV D, we distinguish the PT and C ′ PT inner products from the conventional Dirac inner product, showing that only the C ′ PT inner product is orthogonal and consistent with a positivedefinite norm. Section IV E revisits the parity transformation, and, in Sec. IV F, we discuss time reversal in the light of our approach. As an illustration of our approach, we discuss in Sec. V scalar-pseudoscalar mixing and oscillations in the non-Hermitian model, which reflect the fact that the parity operator does not commute with the Hamiltonian. We compare with oscillations in a Hermitian model and emphasize that unitarity is respected. Our conclusions are summarized in Sec. VI.

II. PROTOTYPE MODEL
For definiteness, we frame the discussions that follow in the context of a prototype non-Hermitian but PT -symmetric non-interacting bosonic field theory, comprising two flavours of complex spin-zero fields φ i (i = 1, 2 are flavour indices) with non-Hermitian mass mixing.
The two complex fields have four degrees of freedom, the minimal number needed to realize a non-Hermitian, PT -symmetric field theory with real Lagrangian parameters. This should be contrasted with other non-Hermitian quantum field theories that have been discussed in the literature, which instead have fewer degrees of freedom but complex Lagrangian parameters [21][22][23][24][25][26][27]. It is understood that we are working in 3 + 1-dimensional Minkowski spacetime throughout.
The Lagrangian of the model is [9] where m 2 i > 0 (i = 1, 2) and µ 2 are real squared-mass parameters. The squared eigenmasses are which are real so long as which defines the PT -symmetric regime we consider here. For η > 1, PT symmetry is broken by the complex eigenstates of the mass matrix; the eigenmasses are not real and time evolution is not unitary. At η = 1, the eigenvalues merge and the mass matrix becomes defective; at this exceptional point, the squared mass matrix only has a single eigenvector (see, e.g., Ref. [12]). Hereafter, we take m 2 1 > m 2 2 , without loss of generality, so that we can omit the absolute value on the definition of the non-Hermitian parameter η in Eq. (3).
By virtue of the non-Hermiticity of the Lagrangian, namely that L * = L, the equations of motion obtained by varying the corresponding action with respect to φ † ≡ (φ * 1 , φ * 2 ) and φ ≡ (φ 1 , φ 2 ) T differ by µ 2 → −µ 2 , and are therefore inconsistent except for trivial solutions.
However, we are free to choose either of these equations of motion to define the dynamics of the theory, since physical observables consistent with the PT symmetry of the model depend only on µ 4 [9]. As we show in this article, the choice of the equations of motion coincides with the choice of whether to take the Hamiltonian operatorĤ or its Hermitian conjugateĤ † =Ĥ to generate the time evolution. For definiteness, and throughout this work, the classical dynamics of this theory will be defined by varying with respect to φ † , leading to the equations of motion We reiterate that this choice amounts to no more than fixing the irrelevant overall sign of the mass mixing term in Eq. (1).

A. Discrete Symmetries
At the classical level with c-number Klein-Gordon fields, the Lagrangian in Eq. (1) is T : if one of the fields transforms as a scalar and the other as a pseudoscalar. As we show in this work, the Lagrangian of this model is also PT -symmetric at the quantum operator level.
However, it is important to realise that the Lagrangian in Eq. and for which the parity transformation can be consistently defined: The fields indicated by a tilde are defined by the action of parity, namely For these Lagrangians, the Euler-Lagrange equations are self-consistent, and Eq. (6a) yields Making use of Eq. (7) and the time-reversal transformations in Eq. (5), we see that the Lagrangians in Eq. (6) remain PT -symmetric.
In order to illustrate the flavour structure of this model, it is convenient to consider a matrix model with non-Hermitian squared Hamiltonian given by reflecting the squared mass matrix of the model in Eq. (1). The Hamiltonian is (up to an overall sign) with eigenvectors [9] e + = N where N is a normalization factor. We remark that it is necessary to take the positive square root in Eq. (10) in order for the Hamiltonian to be well defined at the exceptional points.
Under a parity transformation, the squared Hamiltonian transforms as where the matrix P is a 2 × 2 matrix that reflects the intrinsic parities of the scalar and pseudoscalar fields in Eq. (1): An important difference from the Hermitian case is that the eigenvectors (11) are not orthogonal with respect to the Hermitian inner product, e ⋆ − · e + = 0. Instead, they are orthonormal with respect to the PT inner product: where ‡ ≡ PT • T, with T indicating matrix transposition, 4 and and we choose the normalization constant [9] N = (2η 2 − 2 + 2 1 − η 2 ) −1/2 .
Notice, however, that one of the eigenvectors, viz. e − , has negative PT norm, as is expected for a non-Hermitian PT -symmetric theory. Note that the Hamiltonian is PT symmetric in the sense that [H, ‡] = 0.
As was first shown in Ref. [20], the PT symmetry of the Hamiltonian allows the construction of an additional symmetry transformation, which we denote by C ′ , which can be used to construct a positive-definite norm: the C ′ PT norm. 5 The C ′ matrix for the Hamiltonian in Eq. (9) is given by [10] where gives the matrix similarity transformation that diagonalizes the Hamiltonian: We note that this similarity transformation leads to a Hermitian Hamiltonian. Indeed, it is well-established that for non-interacting non-Hermitian PT -symmetric theories the C ′ transformation is directly related to the similarity transformation that maps the theory to a Hermitian one. Specifically, the matrix C ′ can be written in the form 6 where the matrix Q has the property that leading to the same Hermitian Hamiltonian. Using the identity we can confirm that Eq. (21) is consistent with Eq. (19), i.e., and it follows that whereQ The C ′ PT conjugates of the eigenvectors are and it is easy to check that their C ′ PT norms are positive definite: where § ≡ C ′ PT • T, and that they are orthogonal: We note that C ′ reduces to P in the Hermitian limit η → 0, so that the C ′ PT inner product reduces to the Hermitian inner product.
It will prove helpful to note that we can also write the mass eigenstates and their C ′ PT conjugates in the following way: where are the flavour eigenstates. In addition, we can show that i.e., the Hermitian inner product of the flavour eigenstates, which is not problematic, is related to the C ′ PT inner product of the mass eigenstates.

B. Analogy with 1+1-Dimensional Special Relativity
The similarity transformation (19) between the flavour and mass eigenbases is not a rotation, since the original mass mixing matrix is not Hermitian. Interestingly, however, it is analogous to a Lorentz boost in the 1+1-dimensional field space (φ 1 , φ 2 ) with metric P .
Indeed, one can easily check that R can be written in the form where The PT -symmetric phase, characterized by 0 ≤ η ≤ 1, corresponds to the "subluminal regime" 0 ≤ v ≤ 1, whereas the PT symmetry-breaking phase corresponds to the "superlu- As is known from Special Relativity, the Pauli matrix σ 1 generates 1+1-dimensional Lorentz boosts, and one can also write which is consistent with Eq. (23), sinceQ = σ 1 and The field invariants under a change of basis are then the quadratic terms φ † i P ij φ j and φ i P ij φ j , as well as their complex conjugates.

III. QUANTIZATION
Having understood the flavour structure of this non-Hermitian model, we now turn our attention to its second quantization.

A. Flavour Basis
For the two-flavour model, the mass matrix is not diagonal in the flavour basis, and the same is true of the energy, whose square is given by Since the squared mass matrix m 2 is non-Hermitian, so too is the energy, i.e., E † = E.
As described earlier, and due to the non-Hermiticity of the action, we obtain distinct but physically equivalent equations of motion by varying with respect toφ † i orφ i (see, e.g., Ref. [9]). Starting from the Lagrangian and choosing the equations of motion by varying with respect toφ † i , we have Since E † ij = E ji , it follows that the plane-wave decompositions of the scalar field operators where we have used the shorthand notation for the three-momentum integral. Since the energy is a rank-two tensor in flavour space, it follows that the energy factor in the phase-space measure and the plane-wave factors must also be rank-two tensors in flavour space, with the matrix-valued exponentials being understood in terms of their series expansions. 7 We have normalised the particle and antiparticle creation operatorsâ † andĉ † , and the annihilation operatorsâ andĉ, such that they have mass dimension −3/2. As a result, their canonical commutation relations (with respect to Hermitian conjugation) are isotropic both in the flavour and mass eigenbases at the initial time surface for the quantization, viz. t = 0.
Specifically, we have However, the non-orthogonality of the Hermitian inner product becomes manifest at different times: and it is clear that the canonical conjugate variables cannot be related by Hermitian conjugation.
As identified earlier, the non-Hermitian terms of the Lagrangian in Eq. (1) violate parity.
In fact, parity acts to transform the Lagrangian in Eq. (1) and the corresponding Hamiltonian into their Hermitian conjugates. As a result, the field operators and their parity conjugates evolve with respect toĤ andĤ † respectively. To account for this, it is convenient to introduce a second pair of field operators, denoted by a check (ˇ), which satisfy the alternative choice of equations of motion: and are related toφ i (x) andφ † i (x) by parity: cf. Eq. (7). Their plane-wave decompositions arě where The relations between the creation and annihilation operators are analogous to Eq. (45): and likewise forĉ i andĉ † i . We emphasise, however, that the distinction between checked and hatted operators is necessary only away from the initial time surface of the quantization; namely, we haveǎ and likewise for the antiparticle operators.
A canonical-conjugate pair of variables, e.g.,φ i andπ i , must evolve subject to the same Hamiltonian, i.e., they must both evolve according toĤ or both according toĤ † . The conjugate momentum operators are thereforê Were we instead to insist on the usual relationship between the conjugate momentum operator and the time derivative of the field operator, i.e.,π i = ∂ tφ † i , we would forceφ i and π i both to evolve with respect toĤ (orĤ † ), and they would not be canonical-conjugate variables. We recover the usual relationship between the field and conjugate momentum only in the Hermitian limit µ → 0. It may readily be confirmed that Eqs.
In addition, we have that We can now write down the Hamiltonian (density) operator that generates the time evolution consistent with the equations of motion in Eqs. (38) and (44): The corresponding Lagrangian density iŝ Had we made the alternative choice for the equations of motion, i.e., varying the Lagrangian in Eq. (37) with respect toφ i , the time evolution would instead be generated bŷ but the physical results would be identical.

B. Mass Basis
The transformation to the mass eigenbasis is effected by the similarity transformation By virtue of Eq. (29), or making use of the transformations defined in the next section, we can readily convince ourselves that these variables are the C ′ PT conjugate variables of the mass eigenbasis.
We infer from Eq. (56) that particle annihilation and anti-particle creation operators have to transform in the same way, under both the similarity transformation to the mass eigenbasis and C ′ .

IV. DISCRETE TRANSFORMATIONS IN FOCK SPACE
We now turn our attention in this section to the definition of the discrete symmetry transformations of these non-Hermitian quantum field theories in Fock space. In particular, we define theĈ ′ operator, and show that the parity and time-reversal operators are uniquely defined, irrespective of the choice of inner product.

A. Parity
We begin with the parity transformation, under which the spatial coordinates x change sign, i.e., x → x ′ = −x, but not the time coordinate t, so that A c-number complex scalar field transforms under parity as where η P satisfies |η P | 2 = 1. If φ = φ * is real then η P is equal to +1 if φ transforms as a scalar and equal to −1 if φ transforms as a pseudoscalar. 8 Requiring that the matrix elements of the quantum field operatorφ i transform as in Eq. (7) [see also Eq. (58)], we obtain which are consistent with Eq. (45). As we show below, the definition ofP and its action on the field operators do not depend on the choice of inner product that defines the matrix elements. In terms of these creation and annihilation operators, the parity operator has the 8 It is always possible to rephase the parity operator such that spin-0 fields transform up to a real-valued phase of ±1, as we assume here.
following explicit form [30]: We note that this operator is time-independent, and can therefore be written in terms of Hermitian conjugate creation and annihilation operators at the time t = 0.

B. C ′ Transformation
Using the Q matrix of the simplified model in Sec. II, it is straightforward to construct theĈ ′ operator for the model, which is given bŷ where the matrixQ is given in the flavour basis in Eq. (25). The relative sign between the bracketed particle and antiparticle operator terms in the exponent of Eq. (61) ensures that the field operators transform appropriately, and reflects the fact that particle and antiparticle states must transform in the opposite sense (see below). Comparing with Eq. (20), we note the necessity of including an additional operator which implements the correct change of sign of the momentum in the C ′ PT inner product.
For transformations in Fock space, theĈ ′ operator can be written in the form where the forms ofP + andQ are discussed below.
In terms of the canonically conjugate field variables, theĈ ′ operator can be written in the formĈ We draw attention to the appearance of both hatted and checked operators, cf. Sec. III A and the canonical algebra in Eqs. (50) and (51).
We emphasize that theĈ ′ operator does not coincide with the usual charge-conjugation operator, which is [30] The charge matrix C ij must be chosen such that C ij = P ij in order for the Lagrangian to be C-symmetric, as a result of whichĈ andĈ ′ do not commute. We note that theĈ ′ operator depends on the non-Hermitian parameter η, whereas the usual charge-conjugation operator C does not.
The action ofĈ ′ is as follows: with the fields transforming asĈ such that Thatâ andĉ transform differently follows directly from the fact thatĈ ′ and the usual charge conjugation operatorĈ do not commute. It is easy to confirm thatĈ ′2 = I, and that it commutes with ‡ and the Hamiltonian given by Eq. (53). Specifically, the Hamiltonian (and the Lagrangian) is C ′ symmetric. Since the C ′ transformation mixes the scalar and pseudoscalar operators, we see thatĈ ′ does not commute withP.

C. The Similarity Transformation
TheQ operator in Eq. (63) is given bŷ and the similarity transformationÔ → e −Q/2Ô eQ /2 has the following effects on the particle and antiparticle annihilation and creation operators: so that the fields transform aŝ whereξ i are the field operators in the mass eigenbasis. Using one can show with some algebra that this indeed gives the correct transformation to the Hermitian theory whose Lagrangian is 9 Note that the similarity-transformed Lagrangian is isospectral to the original Lagrangian.
Hence, the non-interacting non-Hermitian bosonic model is equivalent to a Hermitian theory.
However, this is not in general the case in the presence of interactions. For example, if one adds a Hermitian quartic interaction term λ φ † 1 φ 1 2 to the non-Hermitian bosonic model, as discussed in the context of spontaneous symmetry breaking in Refs. [10][11][12], the similarity transformation converts it into a non-Hermitian combination of ξ 1 , ξ 2 , ξ † 1 and ξ † 2 : Hence, the interacting non-Hermitian bosonic model is not equivalent to a Hermitian theory according to the above similarity transformation. Instead, it exhibits soft breaking of Hermiticity.

D. Inner products
Before we can consider the definition of the time-reversal operator in Fock space, we must first describe the various inner products with respect to which it can be defined. For this purpose, it is convenient to define a variation of Dirac's bra-ket notation in which the bra and ket states are related by transposition rather than Hermitian conjugation. Specifically, where T denotes transposition. Hermitian conjugation is indicated in the usual way by a superscript † denoting the combination † ≡ * • T, where * indicates complex conjugation.
We can now distinguish the following inner products in Fock space: 9 Note that both the kinetic terms have positive signs, unlike in Ref. [15] (see also the Appendix).
Dirac inner product: In this notation, the usual Dirac inner product, which is defined via Hermitian conjugation, is written as where the antilinear operatorK is ∝T and effects complex conjugation. For a spin-zero field, single-particle states of momentum q and q ′ have the usual Dirac normalization PT inner product: This indefinite inner product is defined via PT conjugation, which we denote by ‡ ≡ PT • T, and is written as For a scalar field, the PT inner product of single-particle momentum eigenstates is which is negative definite in the case of a pseudoscalar (η P = −1).
C ′ PT inner product: This positive-definite inner product is defined via C ′ PT conjugation, which we denote by § ≡ C ′ PT • T, and is written as With respect to this inner product, the norm of the single-particle momentum state is positive definite for both the scalar and pseudoscalar: Here, we have simply taken η → 0 in Eqs. (61) and (66) in order to decouple the flavours.

E. Parity Revisited
Having defined the various inner products, we can now return to the parity operator, and show explicitly that its definition does not depend on which inner product we use to construct the matrix elements of the theory.
Dirac inner product: In this case, the transformation rules for the ket and bra states are We note that parity and Hermitian conjugation commute, so that and we recover the results in Eq. (59).
PT inner product: The situation is similar in this case, becauseP andT commute (so long as η P ∈ R). Specifically, the transformation rules for the ket and bra states are whereP ‡ = (PT )P T (T −1P −1 ). We therefore recover the same transformation rules (59) for the field operators as in the Hermitian case. This is perhaps not surprising, since Hermitian conjugation is substituted by PT conjugation in non-Hermitian theories.
C ′ PT inner product: This case is rather different, since the C ′ and P transformations do not commute. The transformation rules for the ket and bra states are therefore It is the matrix element involving the latter that leads to a definition of the parity operator consistent with Eq. (60), and we then have giving the same transformation rules (59).

F. Time Reversal
Under a time-reversal transformation, the time coordinate t → t ′ = −t, and In this case a c-number complex Klein-Gordon field transforms as where |η T | 2 = 1. When translating this transformation to the corresponding q-number field operator, we need to take into account the fact that time reversal interchanges the initial and final states. It is for this reason that the action of the time-reversal operator on field operators depends on the inner product used to determine the matrix elements. However, as we see below, the time-reversal operator remains uniquely defined.
Dirac inner product: In the case of the Dirac inner product, the transformation rules for the ket and bra states are We note that time-reversal and Hermitian conjugation commute (for T ij ∈ R), so that Making use of the following identity that holds for an antilinear operator: we arrive at the familiar transformationŝ PT inner product: For the PT -conjugate states, the transformation rules for the ket and bra states are where we have usedTT TT −1 =T † . In this case, we have 10 Making use of the identity we quickly recover the transformations in Eq. (92).
C ′ PT inner product: Without making any assumption as to whether the C ′ and T transformations commute, the transformation rules for the ket and bra states for the C ′ PT inner product are (96b) 10 Taking T ij = δ ij for simplicity, the action of an antilinear operator on the PT inner product is Taking matrix elements involving the latter, we require 11 Making use of the identity and we again recover the transformations in Eq. (92). We see thatĈ ′ andT commute such that Eqs. (96a) and (96b) are identical statements.

G. PT conjugation
Given the definitions of the parity and time-reversal operators, we havê and, taking T ij = δ ij , it follows thatφ ‡ 11 Taking T ij = δ ij for simplicity, the action of an antilinear operator on the C ′ PT inner product is:

V. SCALAR-PSEUDOSCALAR MIXING AND OSCILLATIONS
We now illustrate the discussion in the previous sections by studying mixing and oscillations in the model with two spin-zero fields. 12 As mentioned earlier, the Lagrangian (1) and the corresponding Hamiltonian do not conserve parity. We therefore anticipate the possibility of scalar-pseudoscalar mixing and oscillations, as we now discuss in detail.

A. Mixing in the PT -Symmetric Model
In the mass eigenbasis (see Sec. II A), the classical equations of motion take the form which have the plane-wave solutions where A ± are constants.
The single-particle flavour eigenstates can be written in terms of the mass eigenstates as follows: As per the discussion of Sec. II A, the flavour states are orthonormal with respect to the C ′ PT inner product. However, some care has to be taken in determining the C ′ PT -conjugate states. This is most easily expressed by appealing to Eqs. (29) and (56), from which it follows that the relevant conjugate states are and (|p, Assuming for simplicity a localized initial state, the probability for the scalar with flavour i at t = 0 to transition to the pseudoscalar with flavour j at t > 0 is where V = (2π) 3 δ 3 (0) is a three-volume. We draw attention to the fact that the probability is not obtained from the usual squared modulus with respect to Hermitian conjugationwere we to use this, we would find a negative probability -instead it involves the C ′ PT norm of the mass eigenstates. A straightforward calculation then leads to It is interesting to note that the oscillation period obtained from the probability (108) diverges at the exceptional points η 2 → 1, where since the eigenmasses become degenerate in this limit. Another way to understand this limit is to consider the similarity transformation (19) when η → ǫ = ±1: We see that the eigenstates defined in Eq. (101) are parallel in these limits. Therefore, in addition to having infinite normalization, the similarity transformation is not invertible at the exceptional points, and one cannot define a map back to the flavour states.

B. Comparison with the Hermitian Case
It is illustrative to compare the oscillation probability for the non-Hermitian theory to the corresponding probability for the Hermitian theory with the Lagrangian where m 2 i and m 2 12 are positive real-valued squared mass parameters, and we assume m 2 1 > m 2 2 as before. For this theory, the oscillation probability is where α is the mixing angle, which is given by We see that the probability (108) has the same form as in the Hermitian case, provided one makes the identification sin(2α) = η/ 1 − η 2 . With this identification, we have and the maximum mixing angle π/4 is obtained for µ 2 → (m 2 1 − m 2 2 )/(2 √ 2), whereas it is obtained for 2m 2 12 ≫ m 2 1 − m 2 2 in the Hermitian case (113). As a corollary of the analogy between the Hermitian and non-Hermitian models, we note that unitarity is respected in our analysis of the latter case.

VI. CONCLUSIONS
We have addressed in this paper some basic issues in the formulation of non-Hermitian bosonic quantum field theories, discussing in particular the treatment of discrete symmetries and the definition of the inner product in Fock space. We have focused on PT -symmetric non-Hermitian theories, commenting also on features of theories at the exceptional point at the boundary between theories with PT symmetry and those in which it is broken.
As we have discussed, there is ambiguity in the choice of the inner product in a PTsymmetric theory. In this case, the conventional Dirac inner product (|α ) † |β = α * |β is not positive definite for the mass eigenstates, and is therefore deprecated, and the same is true of of the PT inner product (|α ) ‡ |β = α PT |β , where ‡ ≡ PT • T with T denoting transposition. The appropriate positive-definite norm for the mass eigenstates is defined via C ′ PT conjugation: (|α ) § |β = α C ′ PT |β , where § ≡ C ′ PT • T, where the C ′ operator was defined in Sec. IV B. As was explained there, the C ′ transformation in a PT -symmetric quantum field theory cannot be identified with charge conjugation.
We have formulated in Sec. IV C a suitable similarity transformation between a PTsymmetric non-Hermitian theory with two flavours of spin-zero fields and its Hermitian counterpart. The equivalence between the non-interacting PT -symmetric and Hermitian theories does not, in general, carry over to theories with quartic interactions. The Appendix contrasts the similarity transformation we propose with the previous literature.
As an illustration of this Fock space discussion, we have considered mixing and oscillations in this specific model with two boson flavours, which is free apart from non-Hermitian PTsymmetric mixing terms. The unmixed bosons are taken to be a scalar and a pseudoscalar, which mix via a non-Hermitian bilinear term. We have shown that the resulting mass eigenvectors are not orthogonal with respect to the Dirac inner product, but are orthogonal with positive norm when the C ′ PT inner product is used. We have emphasized that the parity operator in this two-boson model does not commute with the Hamiltonian, leading to the appearance of scalar-pseudoscalar mixing and flavour oscillations, which we have studied in Sec. V. These are of similar form to the mixing between bosons in a Hermitian theory, respecting unitarity but differ in their dependences on the squared mass parameters, and having the feature that the oscillation period diverges at an exceptional point.
The analysis in this paper has clarified the description of PT -symmetric non-Hermitian bosonic quantum field theories, and provides a framework for formulating them off-shell.
Many of the features discussed here are expected to carry over to PT -symmetric non-Hermitian field theories of fermions [32], as we shall discuss in a following paper. This programme constitutes an important step towards addressing deeper issues in field theory such as quantum loop corrections and renormalization, to which we also plan to return in future work.

APPENDIX
A different similarity transformation [15] has previously been applied to the boson model considered in this work. In this Appendix, we review it for completeness, and make a comparison with the transformation detailed in Sec. IV C.
The HamiltonianĤ of the two-flavour scalar theory can also be mapped to a Hermitian oneĥ S (and similarly for the Lagrangian) via the similarity transformation [15] h S =ŜĤŜ −1 , (A.1) withŜ = exp π 2 x π 2 (t, x)φ 2 (t, x) +φ † 2 (t, x)π † 2 (t, x) . (A.2) Here, we have written the operatorŜ in a manifestly Hermitian form. We note, however, that the similarity transformation is defined only up to a constant complex phase, such that one is free to reorder the operators in the exponent by making use of the canonical equal-time commutation relations. We note that, unlike the similarity transformation we propose in the main text, the transformation (A.2) does not depend on the non-Hermitian parameter η.
The similarity transformation (A.2) has the following action on the field operators: Sφ † 2 (t, x)Ŝ −1 = −iφ † 2 (t, x) , (A.3b) and the transformed version of the Lagrangian (37) for the free scalar theory is thereforê While this Lagrangian is Hermitian, we draw attention to the opposite relative signs of the kinetic and mass terms for the fieldsφ 1,2 , which imply thatφ 2 is a negative-norm ghost and is tachyonic. One should therefore suspect that the similarity transformation in Eq. (A.2) is not directly related to theĈ ′ operator needed to construct a positive norm for these states.
Moreover, one can readily confirm that this similarity transformation does not leave the Fock vacuum invariant.