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Discrete Spacetime Symmetries and Particle Mixing in Non-Hermitian Scalar Quantum Field Theories

Discrete Spacetime Symmetries and Particle Mixing in Non-Hermitian Scalar Quantum Field Theories

PHYSICAL REVIEW D 102, 125030 (2020)

Discrete spacetime and particle mixing in non-Hermitian quantum theories

† ‡ Jean Alexandre ,1,* John Ellis ,1,2,3, and Peter Millington 4, 1Department of Physics, King’s College London, London WC2R 2LS, United Kingdom 2National Institute of Chemical Physics and Biophysics, Rävala 10, 10143 Tallinn, Estonia 3Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland 4School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom

(Received 19 June 2020; accepted 20 November 2020; published 28 December 2020)

We discuss quantization, discrete transformations, and inner products in free non-Hermitian scalar quantum field theories with PT symmetry, focusing on a prototype model of two complex scalar fields with anti-Hermitian mixing. Whereas the definition of the inner product is unique for theories described by Hermitian Hamiltonians, its formulation is not unique for non-Hermitian Hamiltonians. eigenstates are not orthogonal with respect to the conventional Dirac inner product, so we must consider additional discrete transformations to define a positive-definite norm. We clarify the relationship between canonical-conjugate operators and introduce the additional discrete symmetry C0, previously introduced for quantum-mechanical systems, and show that the C0PT inner product does yield a positive-definite norm, and hence is appropriate for defining the Fock in non- Hermitian models with PT symmetry in terms of energy eigenstates. We also discuss transformations between PT -symmetric non-Hermitian scalar quantum field theories and Hermitian theories, showing that they would require modification 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.

DOI: 10.1103/PhysRevD.102.125030

I. INTRODUCTION there are strong arguments for the consistency of PT - symmetric quantum field theory, a number of theoretical Recent have witnessed growing interest in non- issues merit further attention. These include the analysis of Hermitian quantum theories [1], particularly those with PT discrete symmetries, which requires in turn a careful symmetry, where P and T denote parity and reversal, analysis of the Fock of non-Hermitian quantum respectively [2]. It is known that a quantum system field theories with PT symmetry and their inner products.1 described by a non-Hermitian Hamiltonian has real ener- In this paper, we study and clarify these issues in the gies and leads to a unitary time evolution if this context of a minimal non-Hermitian bosonic field theory Hamiltonian and its eigenstates are under PT with PT symmetry at the classical and second-quantized symmetry [3]. This increasing interest has been driven in levels. We construct explicitly in the quantum version the part by theoretical analyses supporting the consistency of operators generating discrete symmetries and discuss the such theories in the context of both quantum mechanics and properties of candidate inner products in Fock space. We quantum field theory, and in part by the realization that also construct a similarity transformation between the free- such theories have applications in many physical contexts, field PT -symmetric non-Hermitian model and the corre- e.g., photonics [4,5] and phase transitions [6,7]. Although sponding Hermitian counterpart, showing explicitly that the correspondence would not hold without modification in the presence of interactions. *[email protected] † [email protected] As an application of this formalism, we discuss the ‡ [email protected] simplest nontrivial prototype quantum particle system, namely, mixing in models of noninteracting bosons— Published by the American Physical Society under the terms of building upon the study [9] that described how to interpret the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, 1A detailed description of the PT inner product in quantum and DOI. Funded by SCOAP3. mechanics can be found in Ref. [8].

2470-0010=2020=102(12)=125030(20) 125030-1 Published by the American Physical Society ALEXANDRE, ELLIS, and MILLINGTON PHYS. REV. D 102, 125030 (2020) the corresponding PT -symmetric Lagrangian.2 These sys- TABLE I. Summary of notational conventions used in this paper. tems appear in various physical situations of phenomeno- logical interest, such as coupled pairs of neutral mesons, and Complex conjugation T Operator/matrix transposition also appear in the PT -symmetric extension of supersym- C (Cˆ) Charge conjugation (operator) metry [14]. Issues arising in the formulation of such theories 0 C0 Cˆ C transformation (operator) include the roles of discrete symmetries, the relationship ( ) P Pˆ Parity transformation (operator) between the descriptions of mixing in the PT -symmetric ( ) T Tˆ Time-reversal transformation (operator) non-Hermitian case and the standard Hermitian case,3 and ( ) † ≡ ∘ T the status of unitarity, which has been questioned in non- Hermitian conjugation ‡ ≡ PT ∘ T PT conjugation Hermitian theories [19,20]. As an example, we exhibit a § ≡ C0PT ∘ T C0PT conjugation mechanism allowing oscillations between scalar and pseu- doscalar bosons, which is possible with a mass-mixing matrix that is anti-Hermitian, but with real eigenvalues, and II. PROTOTYPE MODEL we compare with results in the previous literature. For definiteness, we frame the discussions that follow in The layout of our paper is as follows. In Sec. II,we the context of a prototype non-Hermitian but PT -sym- introduce the minimal two-flavor non-Hermitian bosonic metric noninteracting bosonic field theory, comprising two field theory that we study, discussing in Sec. II A its flavors of complex spin-zero fields ϕ (i ¼ 1, 2 are flavor discrete symmetries P, T , and C0 [21] at the classical level i indices) with non-Hermitian mass mixing. The two com- as well as the similarity transformation relating it to a plex fields have 4 , the minimal number Hermitian theory, and mentioning a formal analogy with needed to realize a non-Hermitian, PT -symmetric field (1 þ 1)-dimensional in Sec. II B.We theory with real Lagrangian parameters. This should be discuss in Sec. III the second quantization of the theory contrasted with other non-Hermitian quantum field theories in both the flavor and mass bases. Then, in Sec. IV,we that have been discussed in the literature, which instead discuss the quantum versions of the discrete symmetries have fewer degrees of freedom but complex Lagrangian and various definitions of the inner product in Fock space. parameters [22–28]. It is understood that we are working in In particular, we discuss in Secs. IVA and IV B the parity 3 þ 1-dimensional Minkowski spacetime throughout. and C0 transformations, and we discuss the similarity The Lagrangian of the model is [9] transformation in Sec. IV C, emphasizing that the equiv- alence between the noninteracting non-Hermitian model ν 2 2 L ¼ ∂νϕ ∂ ϕ − m ϕ ϕ − μ ðϕ ϕ2 − ϕ ϕ1Þ; ð1Þ and a Hermitian theory does not in general carry over to an i i i i i 1 2 interacting theory, in the absence of modifications. 2 0 μ2 (Appendix A compares the similarity transformation dis- where mi > and are real squared-mass parameters. cussed in this paper with a previous proposal [15] in the The squared mass matrix literature.) In Sec. IV D, we distinguish the PT and C0PT m2 μ2 inner products from the conventional Dirac inner product, 2 ≡ 1 ð Þ 0 m 2 2 2 showing that only the C PT inner product is orthogonal −μ m2 and consistent with a positive-definite norm.4 Section IV E revisits the parity transformation, and, in Sec. IV F,we is skew symmetric. The squared eigenmasses are discuss time reversal in the light of our approach. As an qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi illustration, we discuss in Sec. V scalar-pseudoscalar m2 þ m2 1 m2 ¼ 1 2 ðm2 − m2Þ2 − 4μ4; ð3Þ mixing and oscillations in the non-Hermitian model, which 2 2 1 2 reflect the fact that the parity operator does not commute with the Hamiltonian. We compare with oscillations in a which are real so long as Hermitian model and emphasize that unitarity is respected. 2μ2 Our conclusions are summarized in Sec. VI. η≡ ≤ 1 ð Þ 2 2 ; 4 A summary of notation is provided in Table I, and some m1 − m2 useful expressions are gathered in Appendix B. which defines the PT -symmetric regime we consider here. η 1 PT 2Self-interactions of these scalar fields were considered in For > , symmetry is broken by the complex Ref. [10], their coupling to an Abelian gauge field in Ref. [11] eigenstates of the mass matrix; the eigenmasses are not and to non-Abelian gauge fields in Ref. [12]. See Ref. [13] for a real and time evolution is not unitary. At η ¼ 1, the study of ’t Hooft-Polyakov monopoles in a non-Hermitian model. eigenvalues merge and the mass matrix becomes defective; 3 – See Refs. [15 18] for an alternative description of these at this exceptional , the squared-mass matrix only has models in terms of similarity transformations that map to a Hermitian model. a single eigenvector (see, e.g., Ref. [12]). Hereafter, we take 4 2 2 For an alternative approach, see Ref. [8]. m1 >m2, without loss of generality, so that we can omit the

125030-2 DISCRETE AND PARTICLE MIXING … PHYS. REV. D 102, 125030 (2020) absolute value on the definition of the non-Hermitian and their “tilde” conjugates parameter η in Eq. (4). By virtue of the non-Hermiticity of the Lagrangian, L˜ ¼ ∂ ϕ∂νϕ˜ − 2ϕϕ˜ − μ2ðϕϕ˜ − ϕϕ˜ Þ ð Þ ν i i mi i i 1 2 2 1 ; 8a namely, that L ≠ L, the equations of obtained † by varying the corresponding with respect to ϕ ≡ L˜ ¼ ∂ ϕ˜ ∂νϕ − 2ϕ˜ ϕ þ μ2ðϕ˜ ϕ − ϕ˜ ϕ Þ ð Þ T 2 2 ν i i mi i i 1 2 2 1 ; 8b ðϕ1; ϕ2Þ and ϕ ≡ ðϕ1; ϕ2Þ differ by μ → −μ and there- fore differ except for trivial solutions. However, we are free ˜ differing by μ2 → −μ2, i.e., L˜ ðμ2Þ¼L˜ ð−μ2Þ. The fields to choose either of these to define the indicated by a tilde are defined by the action of parity, of the theory, since physical observables con- namely, sistent with the PT symmetry of the model depend only on 4 μ [9]. As we show in this paper, the choice of the equations 0 ˜ P∶ ϕ1ðt; xÞ → ϕ ðt; −xÞ¼þϕ1ðt; xÞ; of motion coincides with the choice of whether to take the 1 ˆ 2 ˆ 2 ˆ † 2 ˆ 2 0 ˜ Hamiltonian operator Hðμ Þ or Hð−μ Þ¼H ðμ Þ ≠ Hðμ Þ ϕ2ðt; xÞ → ϕ2ðt; −xÞ¼−ϕ2ðt; xÞ: ð9Þ (its Hermitian conjugate) to generate the time evolution. For definiteness, and throughout this work, the classical For these Lagrangians, the Euler-Lagrange equations are dynamics of this theory will be defined by varying with self-consistent, and Eq. (7a) yields respect to ϕ†, leading to the following equations of motion: □ϕ˜ ð Þþ 2 ϕ˜ ð Þ¼0 ð Þ i x mji j x ; 10a □ϕ ð Þþ 2 ϕ ð Þ¼0 ð Þ i x mij j x ; 5a □ϕ˜ ð Þþ 2 ϕ˜ ð Þ¼0 ð Þ i x mji j x : 10b □ϕð Þþ 2 ϕð Þ¼0 ð Þ i x mij j x : 5b Making use of Eq. (9) and the time-reversal transformations We reiterate that this choice amounts to no more than fixing in Eq. (6), we see that the Lagrangians in Eqs. (7) and (8) the irrelevant overall sign of the mass-mixing remain PT symmetric. in Eq. (1). In order to illustrate the flavor structure of this model, it is convenient to consider a matrix model with non- A. Discrete symmetries Hermitian squared Hamiltonian given by 2 2 At the classical level with c-number Klein-Gordon m1 μ PT H2 ¼ ; ð11Þ fields, the Lagrangian in Eq. (1) is symmetric under −μ2 2 the naive transformations m2

0 reflecting the squared-mass matrix of the model in Eq. (1). P∶ ϕ ðt; xÞ → ϕ ðt; −xÞ¼þϕ ðt; xÞ; 1 1 1 The Hamiltonian is (up to an overall sign) 0 ϕ2ðt; xÞ → ϕ2ðt; −xÞ¼−ϕ2ðt; xÞ; ð6aÞ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 H ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ∶ ϕ1ðt; xÞ → ϕ1ð−t; xÞ¼þϕ1ðt; xÞ; 2 2 2 2 4 m1 þ m2 2 m1m2 þ μ ϕ ð xÞ → ϕ0 ð− xÞ¼þϕð xÞ ð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2 t; 2 t; 2 t; ; 6b 2 2 2 4 2 m1 m1m2 þ μ μ × pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 2 2 4 if one of the fields transforms as a scalar and the other as a −μ m2 m1m2 þ μ pseudoscalar. As we show in this work, the Lagrangian of ð Þ this model is also PT symmetric at the quantum oper- 12 ator level. However, it is important to realize that the Lagrangian in with eigenvectors [9] Eq. (1), and the resulting equations of motion, is not pffiffiffiffiffiffiffiffiffiffiffiffi η 2 invariant under parity. In fact, the action of parity inter- −1 þ 1 − η eþ ¼ N pffiffiffiffiffiffiffiffiffiffiffiffi ; e− ¼ N ; changes the two possible choices of equation of motion −1 þ 1 − η2 η obtainable from Eq. (1). Taking this into account, there are ð13Þ a further two classical Lagrangians that are physically equivalent to Eq. (1) and for which the parity trans- where N is a normalization factor. We remark that it is formation can be consistently defined, necessary to take the positive root in Eq. (12) in L˜ ¼ ∂ ϕ˜ ∂νϕ − 2ϕ˜ ϕ − μ2ðϕ˜ ϕ − ϕ˜ ϕ Þ ð Þ order for the Hamiltonian to be well defined at the excep- ν i i mi i i 1 2 2 1 ; 7a tional points. Under a parity transformation, the squared Hamiltonian L˜ ¼ ∂ ϕ∂νϕ˜ − 2ϕϕ˜ þ μ2ðϕϕ˜ − ϕϕ˜ Þ ð Þ ν i i mi i i 1 2 2 1 ; 7b transforms as

125030-3 ALEXANDRE, ELLIS, and MILLINGTON PHYS. REV. D 102, 125030 (2020) m2 −μ2 gives the matrix similarity transformation that diagonalizes P∶ 2 ¼ 1 ¼ 2;T ð Þ PH P 2 2 H ; 14 the Hamiltonian, i.e., μ m2 2 2 m2 0 where the matrix P is a × matrix that reflects the 2 ¼ 2 −1 ¼ þ ð Þ h RH R 2 : 21 intrinsic parities of the scalar and pseudoscalar fields in 0 m− Eq. (1), We note that this similarity transformation leads to a 10 P ¼ : ð15Þ Hermitian Hamiltonian. Indeed, it is well established that 0 −1 for noninteracting non-Hermitian PT -symmetric theories the C0 transformation is directly related to the similarity An important difference from the Hermitian case is that transformation that maps the theory to a Hermitian one. the eigenvectors (13) are not orthogonal with respect to the Specifically, the matrix C0 can be written in the form7 Hermitian inner product, e− · eþ ≠ 0. Instead, they are PT orthogonal with respect to the inner product, C0 ¼ e−QP; ð22Þ

‡ PT ‡ PT e eþ ¼ e eþ ¼ −e−e− ¼ −e− e− ¼ 1; ð Þ þ þ · · 16a where the matrix Q has the property that ‡ PT ‡ PT e e− ¼ e · e− ¼ e−eþ ¼ e− · eþ ¼ 0; ð16bÞ 2 − 2 2 2 þ þ h ¼ e Q= H eQ= ; ð23Þ where ‡ ≡ PT ∘ T, with T indicating matrix transposition,5 and leading to the same Hermitian Hamiltonian. Using the identity PT e ¼ Pe; ð17Þ R ¼ PR−1P; ð24Þ and we choose the normalization constant [9] qffiffiffiffiffiffiffiffiffiffiffiffi we can confirm that Eq. (23) is consistent with Eq. (21), −1=2 2 2 i.e., N ¼ 2η − 2 þ 2 1 − η : ð18Þ 1 1 η e e−Q ¼ C0P ¼ RPR−1P ¼ R2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ; ð25Þ Notice, however, that one of the eigenvectors, viz. −, has 1 − η2 η 1 negative PT norm, as is expected for a non-Hermitian PT -symmetric theory. Note that the Hamiltonian is PT symmetric in the sense that ½H; ‡¼0. and it follows that As was first shown in Ref. [21], the PT symmetry of −2 ¯ the Hamiltonian allows the construction of an additional Q ¼ ln R ¼ −arctanhðηÞQ; ð26Þ symmetry transformation, which we denote by C0 and which can be used to construct a positive-definite norm: where C0PT 6 the norm. The C0 matrix for the squared Hamiltonian in Eq. (11) is 01 Q¯ ≡ : ð27Þ given by [10] 10 1 1 −η 0 C0 ¼ RPR−1 ≡ pffiffiffiffiffiffiffiffiffiffiffiffi ; ð19Þ The C PT conjugates of the eigenvectors are 1 − η2 η −1 1 η C0PT 0 1 where eþ ¼ C Peþ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi eþ 1 − η2 η 1 pffiffiffiffiffiffiffiffiffiffiffiffi ! ! η 1 − 1 − η2 η R ≡ N pffiffiffiffiffiffiffiffiffiffiffiffi ð20Þ ¼ N pffiffiffiffiffiffiffiffiffiffiffiffi ; ð28aÞ 1 − 1 − η2 η 1 − 1 − η2

5The ‡ notation was introduced in Ref. [29] and extended in Ref. [9]. 6As we discuss in Sec. IV B, the C0 transformation in a PT - 7There is a relative sign in the definition of the matrix Q symmetric quantum field theory cannot be identified with charge compared with Refs. [23,24], due to differing conventions for the conjugation. definition of the C0PT inner product.

125030-4 DISCRETE SPACETIME SYMMETRIES AND PARTICLE MIXING … PHYS. REV. D 102, 125030 (2020) 1 η C0PT 0 1 where e− ¼ C Pe− ¼ pffiffiffiffiffiffiffiffiffiffiffiffi e− 1 − η2 η 1  qffiffiffiffiffiffiffiffiffiffiffiffi ! 1 1 pffiffiffiffiffiffiffiffiffiffiffiffi v ≡ 1 − 1 − η2 and γ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi : ð35Þ 1 − 1 − η2 η 1 − 2 ¼ N ; ð28bÞ v η The PT -symmetric phase, characterized by 0 ≤ η ≤ 1, 0 corresponds to the “subluminal regime” 0 ≤ v ≤ 1, and it is easy to check that their C PT norms are positive whereas the PT symmetry-breaking phase corresponds definite, to the “superluminal regime” v>1. 0 § C PT As is known from special relativity, the Pauli matrix σ1 ee ¼ e · e ¼ 1; ð29Þ generates 1 þ 1-dimensional Lorentz boosts, and one can where § ≡ C0PT ∘ T, and that they are orthogonal, also write

C0PT e · e∓ ¼ 0: ð30Þ R ¼ expðασ1Þ with α ≡ arctanh v; ð36Þ 0 η → 0 ¯ We note that C reduces to P in the Hermitian limit , which is consistent with Eqs. (25)–(27), since Q ¼ σ1 and so that the C0PT inner product reduces to the Hermitian inner product. 1 ¼ η ð Þ It will prove helpful to note that we can also write the arctanh v 2 arctanh : 37 mass eigenstates and their C0PT conjugates in the follow- ing ways: The quadratic field invariants under a change of basis are † ϕ Pijϕj and ϕiPijϕj, as well as their complex conjugates. e ¼ −1e ¼ −1e ð Þ i þ R 1 R1j j; 31a e ¼ −1e ¼ −1e ð Þ III. QUANTIZATION − R 2 R2j j; 31b Having understood the flavor structure of this non- 0 C Peþ ¼ Re1 ¼ R1jej; ð31cÞ Hermitian model, we now turn our attention to its second quantization. 0 C Pe− ¼ Re2 ¼ R2jej; ð31dÞ A. Flavor basis where For the two-flavor model, the mass matrix is not 1 0 in the flavor basis, and the same is true of the energy, whose e1 ¼ and e2 ¼ ð32Þ 0 1 square is given by 2 ðpÞ¼p2δ þ 2 ð Þ are the flavor eigenstates. In addition, we can show that Eij ij mij: 38

T 0 T −1 0 −1 2 eþC Peþ ¼ e1 R C PR e1 ¼ e1 · e1; ð33aÞ Since the squared-mass matrix m is non-Hermitian, so too is the energy, i.e., E† ≠ E. T 0 T −1 0 −1 e− C Pe− ¼ e2 R C PR e2 ¼ e2 · e2; ð33bÞ As described earlier, and due to the non-Hermiticity of the action, we obtain distinct but physically equivalent † i.e., the Hermitian inner product of the flavor eigenstates, equations of motion by varying with respect to ϕˆ or ϕˆ 0 i i which is not problematic, is related to the C PT inner (see, e.g., Ref. [9]). Starting from the Lagrangian product of the mass eigenstates. Lˆ ¼ ∂ ϕˆ†∂νϕˆ − 2ϕˆ†ϕˆ − μ2ðϕˆ†ϕˆ − ϕˆ†ϕˆ Þ ð Þ ν i i mi i i 1 2 2 1 ; 39 B. Analogy with 1 + 1-dimensional special relativity The similarity transformation (23) between the flavor and choosing the equations of motion by varying with ϕˆ† and mass eigenbases is not a , since the original respect to i , we have mass-mixing matrix is not Hermitian. Interestingly, how- 1 þ 1 □ϕˆ þ 2 ϕˆ ¼ 0 ð Þ ever, it is analogous to a Lorentz boost in the - i mij j ; 40a dimensional field space ðϕ1; ϕ2Þ with metric P. Indeed, one can easily check that R can be written in the form □ϕˆ† þ 2 ϕˆ† ¼ 0 ð Þ i mij j : 40b 1 v † ¼ R ¼ γ ; ð34Þ Since Eij Eji, it follows that the - decom- v 1 positions of the operators are

125030-5 ALEXANDRE, ELLIS, and MILLINGTON PHYS. REV. D 102, 125030 (2020) Z ϕˆ ð Þ¼ ½2 ðpÞ−1=2½ð −ip·xÞ ˆ ð0Þþð ip·xÞ ˆ† ð0Þ factors must also be rank-two in flavor space, with i x E ij e jkak;p e jkck;p ; p the matrix-valued exponentials being understood in terms of their series expansions.8 ð41aÞ Z We have normalized the particle and antiparticle creation operators aˆ† and cˆ†, and the annihilation operators aˆ and cˆ, ϕˆ†ð Þ¼ ½2 ðpÞ−1=2½ð −ip·xÞ ˆ ð0Þþð ip·xÞ ˆ† ð0Þ i x E ij e jkck;p e jkak;p ; −3 2 p such that they have mass = . As a result, their canonical commutation relations (with respect to Hermitian ð Þ 41b conjugation) are isotropic both in the flavor and mass eigenbases at the initial time surface for the quantization, where we have used the shorthand notation ¼ 0 Z Z viz. t . Specifically, we have 3p ≡ d ð Þ ½ ˆ ð0Þ ˆ† ð0Þ ¼ ½ ˆ ð0Þ ˆ† ð0Þ ¼ ð2πÞ3δ δ3ðp − p0Þ 3 42 ai;p ; aj;p0 ci;p ; cj;p0 ij : p ð2πÞ ð43Þ for the three- . Since the energy is a rank-two in flavor space, it follows that the energy However, the nonorthogonality of the Hermitian inner factor in the phase-space measure and the plane-wave product becomes manifest at different ,

† † iET t ½ ˆ ð Þ ˆ ð Þ ¼ ½ ˆ ð Þ ˆ ð Þ ¼ ð2πÞ3ð −iEptÞ ð p0 Þ δ3ðp − p0Þ ai;p t ;aj;p0 t ci;p t ;cj;p0 t e ik e kj 8  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  > 4μ4 ðm2−m2Þ2−4μ4t >1 þ 1 − 1pffiffi 2 ¼ > ð 2− 2Þ2−4μ4 cos ¯ ;ij > m1 m2 2Ep <   3 3 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ð2πÞ δ ðp − p Þ 2μ2ð 2− 2Þ ð 2− 2Þ2−4μ4 4 ð 2− 2Þ2−4μ4 ; >− m1 m2 1 − m1pmffiffi 2 t þð−Þ 1 − 4μ m1pmffiffi 2 t > ðm2−m2Þ2−4μ4 cos 2 ¯ i ðm2−m2Þ2 sin 2 ¯ ; > 1 2 Ep 1 2 Ep :> i ¼ 1ð2Þ;j¼ð2Þ1 ð44Þ where ϕˆ ð Þ ϕˆ†ð Þ and are related to i x and i x by parity,  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2 1=2 ˇ ˆ ˆ ˆ −1 ¯ ¼ p2 þ m1 m2 þ ðp2 þ 2Þðp2 þ 2Þþμ4 P ϕ ðPxÞ¼Pϕ ðxÞP ; ð47aÞ Ep 2 m1 m2 ; ij j i ϕˇ†ðP Þ¼Pˆϕˆ†ð ÞPˆ −1 ð Þ ð45Þ Pij j x i x ; 47b and it is clear that the canonical-conjugate variables cannot cf. Eq. (9). Their plane-wave decompositions are be related by Hermitian conjugation. Z As identified earlier, the non-Hermitian terms of the ϕˇ ð Þ¼ ½2 TðpÞ−1=2½ð −ipT·xÞ ˇ ð0Þ i x E ij e jkak;p Lagrangian in Eq. (1) violate parity. In fact, parity acts to p transform the Lagrangian in Eq. (1) and the corresponding þð ipT·xÞ ˇ† ð0Þ ð Þ e jkck;p ; 48a Hamiltonian into their Hermitian conjugates. As a result, Z the field operators and their parity conjugates evolve with T ϕˇ†ð Þ¼ ½2 TðpÞ−1=2½ð −ip ·xÞ ˇ ð0Þ ˆ ˆ † i x E ij e jkck;p respect to H and H , respectively. To account for this, it is p convenient to introduce a second pair of field operators, T þð ip ·xÞ ˇ† ð0Þ ð Þ denoted by a check ( ˇ), which satisfy the alternative choice e jkak;p ; 48b of equations of motion, ½ T ¼ T 0 − δ p x where p · x ij Eij · x ij · , differing from □ϕˇ ð Þþð 2ÞT ϕˇ ð Þ¼0 ð Þ → T ϕˆ ϕˇ† i x m ij j x ; 46a Eq. (41) by E E . We emphasize that i and i evolve ˆ ˆ† ˇ ˆ † † † with H, whereas ϕ and ϕ evolve with H . The relations □ϕˇ ðxÞþðm2ÞT ϕˇ ðxÞ¼0; ð Þ i i i ij j 46b between the creation and annihilation operators are analo- gous to Eq. (47), 8 For a comprehensive discussion of flavor covariance, see ˇ ð Þ¼Pˆ ˆ ð ÞPˆ −1 ð Þ Ref. [30]. For notational simplicity, we do not distinguish in the Pijaj;−p t ai;p t ; 49a work between covariant and contravariant indices in ˇ† ð Þ¼Pˆ ˆ† ð ÞPˆ −1 ð Þ flavor space. Pijaj;−p t ai;p t ; 49b

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ˆ ˆ† may readily be confirmed that Eqs. (43), (51), and (52) lead and likewise for ci and ci . We emphasize, however, that the distinction between checked and hatted operators is neces- to canonical equal-time commutation relations sary only away from the initial time surface of the ½ϕˆ ð xÞ ϕˆ†ð yÞ ¼ 0 ð Þ quantization; namely, we have i t; ; j t; ; 53a ð†Þ ð†Þ ½ϕˆ ð xÞ πˆ ð yÞ ¼ δ δ3ðx − yÞ ð Þ ˇ ð0Þ¼ ˆ ð0ÞðÞ i t; ; j t; i ij ; 53b ai;p ai;p 50 ½ϕˆ†ð xÞ πˆ†ð yÞ ¼ δ δ3ðx − yÞ ð Þ and likewise for the antiparticle operators. Making use of i t; ; j t; i ij : 53c this identification, it is more illustrative to write the various field operators in the following forms: In addition, we have that Z ½ϕˆ ð xÞ ϕˇ†ð yÞ ¼ 0 ð Þ ϕˆ ð Þ¼ ½2 ðpÞ−1=2½ð −ip·xÞ ˆ ð0Þþð ip·xÞ ˇ† ð0Þ i t; ; j t; ; 54a i x E ij e jkak;p e jkck;p ; p ˇ ˇ† ½ϕiðt; xÞ; ϕ ðt; yÞ ¼ 0; ð54bÞ ð51aÞ j Z ˇ 3 ½ϕiðt; xÞ; πˇjðt; yÞ ¼ iδijδ ðx − yÞ; ð54cÞ ϕˆ†ð Þ¼ ½2 ðpÞ−1=2½ð −ip·xÞ ˇ ð0Þþð ip·xÞ ˆ† ð0Þ i x E ij e jkck;p e jkak;p ; p ½ϕˇ†ð xÞ πˇ†ð yÞ ¼ δ δ3ðx − yÞ ð Þ i t; ; j t; i ij ; 54d ð51bÞ Z where ϕˇ ð Þ¼ ½2 TðpÞ−1=2 i x E ij Z p ˆ† 1=2 −ip·x T T πˇ ðxÞ¼∂ ϕ ðxÞ¼−i ½2EðpÞ ½ðe Þ cˇ pð0Þ −ip ·x ip ·x † i t i ij jk k; × ½ðe Þ aˇ pð0Þþðe Þ cˆ ð0Þ; ð51cÞ p Z jk k; jk k;p − ð ip·xÞ ˆ† ð0Þ ð Þ ϕˇ†ð Þ¼ ½2 TðpÞ−1=2 e jkak;p ; 55a i x E ij p Z † 1 2 −ipT·x ipT·x † πˇ ð Þ¼∂ ϕˆ ð Þ¼− ½2 ðpÞ = ½ð −ip·xÞ ˆ ð0Þ × ½ðe Þ cˆ pð0Þþðe Þ aˇ pð0Þ; ð51dÞ i x t i x i E ij e jkak;p jk k; jk k; p − ð ip·xÞ ˇ† ð0Þ ð Þ where we draw attention to the fact that particle and e jkck;p : 55b antiparticle operators appear with opposing hats and checks. This convention makes manifest the necessity for both the We can now write down the Hamiltonian (density) particle annihilation operator aˆ and the antiparticle creation operator that generates the time evolution consistent with operator cˇ†, which appear in the field operator ϕˆ, to evolve the equations of motion in Eqs. (40) and (46), with the Hamiltonian Hˆ , and not with Hˆ and Hˆ †, respec- Hˆ ¼ πˇ†ð Þπˆ ð Þþ∇ϕˇ†ð Þ ∇ϕˆ ð Þþϕˇ†ð Þ 2 ϕˆ ð Þ tively, as one might have expected naively. i x i x i x · i x i x mij j x : ˆ A canonical-conjugate pair of variables, e.g., ϕi and πˆi, ð56Þ must evolve subject to the same Hamiltonian, i.e., they The corresponding Lagrangian density is must both evolve according to Hˆ or both according to Hˆ †. The conjugate momentum operators are therefore Lˆ ¼ ∂ ϕˇ†ð Þ∂νϕˆ ð Þ − ϕˇ†ð Þ 2 ϕˆ ð Þ ð Þ Z ν i x i x i x mij j x : 57 πˆ ð Þ¼∂ ϕˇ†ð Þ¼− ½2 TðpÞ1=2½ð −ipT·xÞ ˆ ð0Þ i x t i x i E ij e jkck;p Had we made the alternative choice for the equations of p motion, i.e., varying the Lagrangian in Eq. (39) with respect − ð ipT·xÞ ˇ† ð0Þ ð Þ ϕˆ e jkak;p ; 52a to i, the time evolution would instead be generated by Z ˆ † † ˆ† ˇ ˆ† 2 ˇ 1 2 T H ¼ πˆ ðxÞπˇ ðxÞþ∇ϕ ðxÞ ∇ϕ ðxÞþϕ ðxÞm ϕ ðxÞ; πˆ†ð Þ¼∂ ϕˇ ð Þ¼− ½2 TðpÞ = ½ð −ip ·xÞ ˇ ð0Þ i i i · i i ji j i x t i x i E ij e jkak;p p ð58Þ − ð ipT·xÞ ˆ† ð0Þ ð Þ e jkck;p : 52b but the physical results would be identical.

Were we instead to insist on the usual relationship between B. Mass basis the conjugate momentum operator and the time derivative † The transformation to the mass eigenbasis is effected by of the field operator, i.e., πˆ ¼ ∂ ϕˆ , we would ϕˆ and i t i i the similarity transformation πˆ to evolve with respect to Hˆ and Hˆ †, respectively, and i ˆ ˆ they would therefore not be canonical-conjugate variables. ξiðxÞ¼RijϕjðxÞ; ð59aÞ We recover the usual relationship between the field and ξˇ§ð Þ¼ϕˇ†ð Þ −1 ð Þ conjugate momentum only in the Hermitian limit μ → 0.It i x j x Rji : 59b

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By virtue of Eq. (31), or making use of the transformations A c-number complex scalar field transforms under parity as defined in the next section, we can readily convince ξˆ ξˇ§ C0PT ourselves that the variables i and i are the - 0 0 0 P∶ ϕðxÞ → ϕ ðx Þ¼ϕ ðt; −xÞ¼ηPϕðt; xÞ; ð61Þ conjugate variables of the mass eigenbasis. We infer from Eq. (59) that particle annihilation and 2 antiparticle creation operators have to transform in the same where ηP satisfies jηPj ¼ 1.Ifϕ ¼ ϕ is real, then ηP is way, under both the similarity transformation to the mass equal to þ1 if ϕ transforms as a scalar and equal to −1 if ϕ 0 eigenbasis and C (see Sec. IIA). transforms as a pseudoscalar.9 Requiring that the matrix elements of the quantum field IV. DISCRETE TRANSFORMATIONS ˆ operator ϕi transform as in Eq. (9) [see also Eq. (61)], we IN FOCK SPACE obtain We now turn our attention in this section to the definition of the discrete symmetry transformations of these non- −1 Pˆϕˆ ðxÞPˆ ¼ P ϕˇ ðPxÞ; ð62aÞ Hermitian quantum field theories in Fock space. In par- i ij j 0 ticular, we define the Cˆ operator, and show that the parity and time-reversal operators are uniquely defined, irrespec- Pˆϕˆ†ð ÞPˆ −1 ¼ ϕˇ†ðP Þ ð Þ tive of the choice of inner product. i x Pij j x ; 62b

A. Parity which are consistent with Eq. (47). As we show below, the We begin with the parity transformation, under which the definition of Pˆ and its action on the field operators do not x x → x0 ¼ −x spatial coordinates change sign, i.e., ,but depend on the choice of inner product that defines the not the time coordinate t, so that matrix elements. In terms of these creation and annihilation operators, the parity operator has the following explicit μ ≡ ð xÞ → P μ ¼ 0μ ¼ð0 x0Þ¼ð −xÞ ð Þ x t; x x t ; t; : 60 form [31]:

 Z  π Pˆ ¼ i ½ ˆ† ð0Þ ˆ ð0Þþ ˆ† ð0Þ ˆ ð0Þ − ˆ† ð0Þ ˆ ð0Þ − ˆ† ð0Þ ˆ ð0Þ ð Þ exp ai;p ai;−p ci;p ci;−p ai;p Pijaj;p ci;p Pijci;p : 63 2 p

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 ¼ 0.

B. C0 transformation 0 Using the Q matrix of the simplified model in Sec. II, it is straightforward to construct the Cˆ operator for the model, which is given by  Z  Cˆ0 ¼ η 𠈆 ð0Þ ¯ ˆ ð0Þ − ˆ† ð0Þ ¯ ˆ ð0ÞÞ Pˆ Pˆ ð Þ exp arctanh ai;p Qijaj;p ci;p Qijcj;p þ ; 64 p where the matrix Q¯ is given in the flavor basis in Eq. (27). The relative sign between the bracketed particle and antiparticle operator terms in the exponent of Eq. (64) ensures that the field operators transform appropriately and reflects the fact that 0 particle and antiparticle states must transform in the opposite sense (see below). We point out that the Cˆ operator is ill- defined at the exceptional point η ¼ 1, as is expected for this operator. Comparing with Eq. (22), we note the necessity of including an additional operator  Z  π Pˆ ¼ i ½ ˆ† ð0Þ ˆ ð0Þþ ˆ† ð0Þ ˆ ð0Þ − ˆ† ð0Þ ˆ ð0Þ − ˆ† ð0Þ ˆ ð0Þ ð Þ þ exp ai;p ai;−p ci;p ci;−p ai;p ai;p ci;p ci;p ; 65 2 p which implements the correct change of sign of the momentum in the C0PT inner product. For transformations in Fock 0 space, the Cˆ operator can be written in the form

9It is always possible to rephase the parity operator such that spin-0 fields transform up to a real-valued phase of 1,aswe assume here.

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ˆ0 −Qˆ ˆ ˆ C ¼ e PþP; ð66Þ where the operator Qˆ is discussed below. 0 In terms of the canonically conjugate field variables, the Cˆ operator can be written in the form  Z  Cˆ0 ¼ − η ðπˆ ð xÞ ¯ ϕˆ ð xÞ − πˇ†ð xÞ ¯ ϕˇ†ð xÞÞ Pˆ Pˆ ð Þ exp iarctanh i t; Qij j t; i t; Qij j t; þ : 67 x

We draw attention to the appearance of both hatted and checked operators, cf. Sec. III A, and the canonical algebra in Eqs. (53) and (54). 0 We emphasize that the Cˆ operator does not coincide with the usual charge-conjugation operator, which is [31]

 Z  π Cˆ ¼ i ½ ˆ† ð0Þ ˆ ð0Þþ ˆ† ð0Þ ˆ ð0Þ − 𠈆 ð0Þ ˆ ð0Þþ ˆ† ð0Þ ˆ ð0ÞÞ ð Þ exp ci;p ai;−p ai;p ci;−p ai;p Cijaj;p ci;p Cijcj;p : 68 2 p

The charge matrix Cij must be chosen such that Cij ¼ That aˆ and cˆ transform differently follows directly from C ˆ0 ˆ Pij in order for the Lagrangian to be symmetric, as a the fact that C and the usual charge conjugation operator C ˆ ˆ0 02 result of which C and C do not commute. We note do not commute. It is easy to confirm that Cˆ ¼ I and that 0 that the Cˆ operator depends on the non-Hermitian C0- and PT -conjugation commute.10 To see this, consider parameter η, whereas the usual charge-conjugation oper- the superposition of single-particle momentum states ator Cˆ does not. Z Cˆ0 The action of is as follows: jΨi¼ Ap;ijpii; ð72Þ p Cˆ0 ˆ† ð0ÞCˆ0;−1 ¼ 0 ˆ† ð0Þ ð Þ ai;q Cijaj;q ; 69a where the Ap;i are complex c-number coefficients. Acting ‡ Cˆ0 Cˆ0 ˆ ð0ÞCˆ0;−1 ¼ 0T ˆ ð0Þ ð Þ first with and then with , and making use of Eq. (69),we ai;q Cij aj;q ; 69b have Z Z Cˆ0 ˆ† ð0ÞCˆ0;−1 ¼ 0T ˆ† ð0Þ ð Þ ci;q Cij cj;q ; 69c jΨi‡Cˆ0 ¼ hp j Cˆ0 ¼ hp j 0 ð Þ j PjiAp;i k CkjPjiAp;i: 73 p p Cˆ0 ˆ ð0ÞCˆ0;−1 ¼ 0 ˆ ð0Þ ð Þ ci;q Cijcj;q ; 69d Conversely, we have with the fields transforming as Z Z ðCˆ0jΨiÞ‡ ¼ ð 0 jp iÞ‡ ¼ hp j 0T ð Þ Ap;iCij j k PkjCji Ap;i: 74 Cˆ0ϕˆ ð ÞCˆ0;−1 ¼ 0Tϕˆ ð Þ ð Þ p p i x Cij j x ; 70a Using the fact that Cˆ0ϕˇ†ð ÞCˆ0;−1 ¼ 0 ϕˇ†ð Þ ð Þ i x Cij j x ; 70b P:C0:P ¼ C0T; ð75Þ such that we see that C0- and PT -conjugation commute, as required. 1 ˆ ˆ ˆ Moreover, the Hamiltonian given by Eq. (56) (and the ϕ1ðxÞ → pffiffiffiffiffiffiffiffiffiffiffiffi ðϕ1ðxÞþηϕ2ðxÞÞ; 1 − η2 corresponding Lagrangian) is C0 symmetric, such that ½Cˆ0 ˆ ¼0 C0 ˆ 1 ˆ ˆ ; H . Since the transformation mixes the scalar ϕ2ðxÞ → − pffiffiffiffiffiffiffiffiffiffiffiffi ðϕ2ðxÞþηϕ1ðxÞÞ; ð71aÞ ˆ0 1 − η2 and pseudoscalar operators, we find that C does not commute with Pˆ. This is, in fact, a necessary consequence ˆ ˆ ˆ ˆ † 1 † † of the relation P Q P ¼ −Q, as we discuss in the next ϕˇ ðxÞ → pffiffiffiffiffiffiffiffiffiffiffiffi ðϕˇ ðxÞ − ηϕˇ ðxÞÞ; 1 1 − η2 1 2 section.

† 1 † † ϕˇ ð Þ → − pffiffiffiffiffiffiffiffiffiffiffiffi ðϕˇ ð Þ − ηϕˇ ð ÞÞ ð Þ 10 PT 2 x 2 2 x 1 x : 71b We reiterate that conjugation, denoted here by 1 − η ‡ ¼ PT ∘ T, includes operator/matrix transposition.

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ˆ 0 ˆ† ν ˆ ˆ† ν ˆ 2 ˆ† ˆ C. The similarity transformation L ¼ ∂νξ1ðxÞ∂ ξ1ðxÞþ∂νξ2ðxÞ∂ ξ2ðxÞ − mþξ1ðxÞξ1ðxÞ Qˆ 2 ˆ† ˆ The operator in Eq. (66) is given by − m−ξ2ðxÞξ2ðxÞ: ð80Þ Z Qˆ ¼ − η 𠈆 ð0Þ ˆ ð0Þþ ˆ† ð0Þ ˆ ð0Þ Note that the similarity-transformed Lagrangian is isospec- arctanh a1;p a2;p a2;p a1;p p tral to the original Lagrangian. Hence, the noninteracting † † non-Hermitian bosonic model is equivalent to a Hermitian − cˆ1 pð0Þcˆ2;pð0Þ − cˆ2 pð0Þcˆ1;pð0ÞÞ: ð76Þ ; ; theory. Oˆ → −Qˆ =2Oˆ Qˆ =2 We draw attention to the fact that we have used the form The similarity transformation e e has the Qˆ following effects on the particle and antiparticle annihila- of the operator extracted from Eq. (64) in terms of the tion and creation operators: creation and annihilation operators evaluated at the initial time surface and not from Eq. (67) in terms of the field ˆ arctanh η arctanh η operators at the finite time t. While both forms of the Q aˆ qð0Þ → aˆ qð0Þ cosh − aˆ ð0Þ sinh ; 0 i; i; 2 i;= q 2 operator give valid Cˆ transformations, only the former ð77aÞ choice, in terms of the creation and annihilation operators, gives a similarity transformation that both maps the

† † arctanh η † arctanh η Lagrangian to the Hermitian one and transforms the field aˆ ð0Þ → aˆ ð0Þ cosh þ aˆ ð0Þ sinh ; i;q i;q 2 i;= q 2 operators to those of the mass eigenbasis. Were we to take a Qˆ operator based on Eq. (67), it would map the Lagrangian ð77bÞ to the Hermitian one, but leave the field operators them- arctanh η arctanh η selves unchanged. This would therefore not represent a cˆ qð0Þ → cˆ qð0Þ þ cˆ ð0Þ ; consistent similarity transformation to the Hermitian i; i; cosh 2 i;= q sinh 2 “frame.” The reason for this discrepancy is the fact that, ð Þ 77c by virtue of the non-Hermitian nature of the evolution, the 0 Cˆ operators in Eqs. (64) and (67) are actually distinct. † † arctanh η † arctanh η cˆ ð0Þ → cˆ ð0Þ cosh − cˆ ð0Þ sinh ; We have constructed a similarity transformation that i;q i;q i;= q 2 2 maps the non-Hermitian free theory to a Hermitian one. If ð77dÞ we include interactions, however, it is not in general the case that this similarity transformation will map the full so that the fields transform as interacting Hamiltonian to a Hermitian one, even if those interactions respect the PT symmetry. For example, if one arctanh η arctanh η † 2 ϕˆ ð Þ → ξˆ ð Þ − ξˆ ð Þ ð Þ adds a Hermitian term λðϕ1ϕ1Þ to the i x i x cosh i= x sinh ; 78a 2 2 non-Hermitian bosonic model, as discussed in the context – η η of spontaneous symmetry breaking in Refs. [10 12], the ϕˇ†ð Þ → ξˆ†ð Þ arctanh þ ξˆ†ð Þ arctanh i x i x cosh x sinh ; similarity transformation converts it into a non-Hermitian 2 i= 2 † † combination of ξ1, ξ2, ξ1, and ξ2, ð78bÞ   λ λ 2 − 2 −2 ˇ† ˆ 2 m1 m2 2 2 4 ξˆ ðϕ ϕ1Þ → ðm − m Þ − μ where i are the field operators in the mass eigenbasis. 4 1 16 2 þ 2 Herein, i= ¼ 2 for i ¼ 1, and i= ¼ 1 for i ¼ 2. Using ½ð 2 − 2Þξˆ† þ μ2ξˆ†2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi × mþ m2 1 2 arctanh η 1 1 ½ð 2 − 2Þξˆ − μ2ξˆ 2 ð Þ cosh ¼ pffiffiffi 1 þ pffiffiffiffiffiffiffiffiffiffiffiffi; ð79aÞ × mþ m2 1 2 : 81 2 2 1 − η2 Hence, this interacting non-Hermitian bosonic model is not arctanh η 1 η 1 equivalent to a Hermitian theory according to the above sinh ¼ pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð79bÞ 2 2 1 − η2 1 þ pffiffiffiffiffiffiffi1 similarity transformation. Instead, it exhibits soft breaking 1−η2 of Hermiticity. On the other hand, were we to build interaction terms out of the quadratic field invariants one can show with some algebra that this indeed gives the discussed in Sec. II B, or powers of the mass term, the correct transformation to the Hermitian theory whose above similarity transformation would map these to 11 Lagrangian is Hermitian interactions. This observation does not necessarily indicate that the 11Note that both the kinetic terms have positive signs, unlike in eigenvalues of the interacting Hamiltonian are complex Ref. [15] (see also the Appendices). or preclude the possibility that there exists a different

125030-10 DISCRETE SPACETIME SYMMETRIES AND PARTICLE MIXING … PHYS. REV. D 102, 125030 (2020) similarity transformation that maps the full interacting ðjqiÞ†jq0i¼hqjq0i¼ð2πÞ3δ3ðq − q0Þ: ð84Þ Hamiltonian to a Hermitian one. It does, however, present a challenge for the perturbative treatment of non-Hermitian PT inner product: This indefinite inner product is defined theories, since the interaction pictures of the non-Hermitian via PT conjugation, which we denote by ‡ ≡ PT ∘ T, and and corresponding Hermitian theories would necessarily is written as have to be related by a similarity transformation that would involve a resummation of a series in the coupling constant ðjαiÞ‡jβi¼hαPT jβi¼hTˆ TPˆ Tαjβi¼hαjPˆ Tˆ jβi that may be nontrivial. We leave further study of this ¼hαjPˆ Tˆ βi ð Þ interesting point to work. : 85 Finally, we comment on the connection of this similarity PT transformation to the V norm considered in Ref. [8] at the For a scalar field, the inner product of single-particle level of the free theory. The V norm is constructed from the momentum eigenstates is Vˆ ¼ −Qˆ operator e , which maps the Hamiltonian to its ‡ 0 ˆ T ˆ T 0 3 3 0 † −1 ðjqiÞ jq i¼hT P qjq i¼ηPhqjqi¼ηPð2πÞ δ ðq − q Þ; Hermitian conjugate, i.e., Hˆ ¼ Vˆ Hˆ Vˆ . Since we also ˆ † ˆ ˆ ˆ −1 ˆ ˆ ˆ −1 ˆ ð Þ have that H ¼ P H P and PþHPþ ¼ H, it follows that 86 ˆ ˆ ˆ ˆ ˆ0 ˆ ˆ ˆ ½VPþP; H¼0, and we can identify C ¼ VPþP, so long ðVˆPˆ PˆÞ2 ¼ I Pˆ Qˆ Pˆ ¼ which is negative definite in the case of a pseudoscalar as þ . This is indeed the case, since þ þ η ¼ −1 ˆ ˆ ˆ ˆ ˆ ( P ), cf. the approach of Ref. [8]. Q and P Q P ¼ −Q. The latter identity follows immedi- C0PT ˆ inner product: This positive-definite inner product ately from Eq. (76), upon realizing that Q is bilinear in the is defined via C0PT conjugation, which we denote by ˆ ˆ 0 scalar and pseudoscalar operators. (We recall that P and Pþ § ≡ C PT ∘ T, and is written as are involutary). Since the Vˆ norm of Ref. [8] is positive ˆ0 § C0PT ˆ T ˆ T ˆ0T ˆ0 ˆ ˆ definite by construction, the same follows for the C Pˆ Tˆ ðjαiÞ jβi¼hα jβi¼hT P C αjβi¼hαjC P T jβi 0 norm constructed in this work, since they coincide, as we ¼hαjCˆ Pˆ Tˆ βi: ð87Þ will show below. An explicit comparison of the C0PT and V norms in the case of interacting theories warrants further With respect to this inner product, the norm of the single- investigation beyond the scope of this paper. particle momentum state is positive definite for both the scalar and pseudoscalar, D. Inner products § ˆ T ˆ T ˆ0T ˆ T ˆ T Before we can consider the definition of the time- ðjqiÞ jqi¼hT P C qjqi¼ηPhT P qjqi reversal operator in Fock space, we must first describe 2 ¼ ηPhqjqi¼1: ð88Þ the various inner products with respect to which it can be defined. For this purpose, it is convenient to define a Here, we have simply taken η → 0 in Eqs. (64) and (69) in variation of Dirac’s bra-ket notation in which the bra and † order to decouple the flavors. In this case, ϕˆ§ðxÞ¼ϕˆ ðxÞ, ket states are related by transposition rather than Hermitian trivially, i.e., in the Hermitian limit η → 0, C0PT con- conjugation. Specifically, we define jugation of the field operator coincides with Hermitian T conjugation. hαj ≡ ðjαiÞ ; ð82Þ Returning to the two-flavor case, we can take the single- particle mass eigenstate where T denotes transposition. Hermitian conjugation is indicated in the usual way by a superscript † denoting   qffiffiffiffiffiffiffiffiffiffiffiffi  2 the combination † ≡ ∘ T, where indicates complex jpþi¼N ηjp1iþ 1 − 1 − η jp2i ð89Þ conjugation. We can now distinguish the following inner products in Fock space: as an example. By our notation in Eq. (82),wehave Dirac inner product: In this notation, the usual Dirac   qffiffiffiffiffiffiffiffiffiffiffiffi  2 inner product, which is defined via Hermitian conjugation, hpþj¼N ηhp1jþ 1 − 1 − η hp2j : ð90Þ is written as

T ˆ0 ˆ ˆ ðjαiÞ†jβi¼hαjβi¼hKˆ αjβi¼hαjKˆ jβi¼hαjKˆ βi; ð83Þ In addition, it follows from the action of the C and P T operators that where the antilinear operator Kˆ is ∝ Tˆ and effects complex   qffiffiffiffiffiffiffiffiffiffiffiffi  ˆ0 ˆ ˆ 2 conjugation. For a spin-zero field, single-particle states of C P T jpþi¼N ηjp1i − 1 − 1 − η jp2i : ð91Þ momentum q and q0 have the usual Dirac normalization

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It is then easily verified that jPˆαi¼Pˆjαi ⇔ ðjPˆαiÞ§ ¼ðPˆjαiÞ§ ¼hαC0PT jPˆ § C0PT ˆ0 ˆ ˆ0 ˆ0 ˆ ˆ ¼hα jC PC ð98aÞ hpþjC P T jpþi¼1 > 0; ð92Þ ⇔ ðjαiÞ§Pˆ T ¼ðPˆjαC0PT iÞT ¼hαC0PT jPˆ −1 ð Þ as required. Moreover, we have that : 98b     qffiffiffiffiffiffiffiffiffiffiffiffi It is the matrix element involving the latter that leads to a ˆ ˆ 2 PþT jpþi¼N ηjp1iþ 1 − 1 − η jp2i ¼jpþi: definition of the parity operator consistent with Eq. (63), and we then have ð93Þ ˆ T C0PT ˇ ˆ C0PT ˆ −1 ˇ ˆ hP ðα ÞjϕiðPxÞjPβi¼hα jP ϕiðPxÞPjβi We can then confirm, as highlighted earlier, that the C0PT ¼! hαC0PT jϕˆ ð Þjβi ð Þ and V norms coincide for the free theory, Pij j x ; 99

ˆ0 ˆ ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ˆ giving the same transformation rules (62). hpþjC P T jpþi¼hpþjVPþP T jpþi¼hpþjVPþT jpþi ¼hp jVˆjp i ð Þ þ þ : 94 F. Time reversal Under a time-reversal transformation, the time coordi- nate t → t0 ¼ −t, and E. Parity revisited Having defined the various inner products, we can now xμ ≡ ðt; xÞ → T xμ ¼ x0μ ¼ðt0; x0Þ¼ð−t; xÞ: ð100Þ return to the parity operator and show explicitly that its definition does not depend on which inner product we use In this case, a c-number complex Klein-Gordon field to construct the matrix elements of the theory. transforms as Dirac inner product: In this case, the transformation 0 0 0 rules for the ket and bra states are T ∶ ϕðxÞ → ϕ ðx Þ¼ϕ ð−t; xÞ¼ηT ϕ ðt; xÞ; ð101Þ

ˆ ˆ ˆ † ˆ † ˆ † ˆ −1 2 jPαi¼Pjαi ⇔ ðjPαiÞ ¼ðPjαiÞ ¼hα jP ¼hα jP : where jηT j ¼ 1. When translating this transformation to the corresponding q-number field operator, we need to take ð Þ 95 into account the fact that time reversal interchanges the initial and final states. It is for this reason that the action of We note that parity and Hermitian conjugation commute, the time-reversal operator on field operators depends on the so that inner product used to determine the matrix elements. However, as we see below, the time-reversal operator hðPˆ TαÞjϕˇ ðP ÞjPˆβi¼hαjPˆ −1ϕˇ ðP ÞPˆjβi i x i x remains uniquely defined. ! Dirac inner product: In the case of the Dirac inner ¼ P hα jϕˆ ðxÞjβi; ð96Þ ij j product, the transformation rules for the ket and bra states are and we recover the results in Eq. (62). PT inner product: The situation is similar in this ˆ ˆ ˆ † ˆ † ˆ † ˆ −1 ˆ ˆ jT αi¼T jαi ⇔ ðjT αiÞ ¼ðT jαiÞ ¼hα jT ¼hα jT : case, because P and T commute (so long as ηP ∈ R). Specifically, the transformation rules for the ket and bra ð102Þ states are We note that time-reversal and Hermitian conjugation ˆ ˆ ˆ ‡ ˆ ‡ PT ˆ ‡ PT ˆ −1 ∈ R jPαi¼Pjαi ⇔ ðjPαiÞ ¼ðPjαiÞ ¼hα jP ¼hα jP ; commute (for Tij ), so that

ð97Þ T ˆ ˆ ˆ −1 ˆ ˆ hðT αÞ jϕiðT xÞjT βi¼hα jT ϕiðT xÞT jβi Pˆ ‡ ¼ðPˆ Tˆ ÞPˆ TðTˆ −1Pˆ −1Þ ¼! hβjϕˆ†ð Þjαi ð Þ where . We therefore recover the Tij j x : 103 same transformation rules (62) for the field operators as in the Hermitian case. This is perhaps not surprising, since Making use of the following identity that holds for an Hermitian conjugation is substituted by PT conjugation in antilinear operator: non-Hermitian theories. C0PT ˆ −1 ˆ ˆ ˆ −1 ˆ ˆ † inner product: This case is rather different, since hα jT ϕiðT xÞT jβi¼hβ jðT ϕiðT xÞT Þ jαi; ð104Þ the C0 and P transformations do not commute. The trans- formation rules for the ket and bra states are therefore we arrive at the familiar transformations

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ˆ ˆ ˆ −1 ˆ 13 T ϕiðxÞT ¼ TijϕjðT xÞ; ð105aÞ Taking matrix elements involving the latter, we require

T 0 0 −1 † −1 † hTˆ ðαC PT Þjϕˆ ðT ÞjTˆ βi¼hαC PT jTˆ ϕˆ ðT ÞTˆ jβi Tˆ ϕˆ ð ÞTˆ ¼ ϕˆ ðT Þ ð Þ i x i x i x Tij j x : 105b ! 0 ¼ T hβC PT jϕˆ§ðxÞjαi: ð111Þ Choosing both the scalar and pseudoscalar of our ij j þ1 prototype model to transform with a phase of under Making use of the identity time reversal, the explicit form of the time-reversal operator is (see, e.g., Ref. [32]) C0PT ˆ −1 ˆ ˆ C0PT ˆ −1 ˆ ˆ § hα jT ϕiðT xÞT jβi¼hβ jðT ϕiðT xÞT Þ jαi; ˆ ˆ ˆ ð Þ T ¼ KPþ; ð106Þ 112

Kˆ and we again recover the transformations in Eq. (105).We where is the operator that effects complex conju- ˆ0 ˆ ˆ see that C and T commute such that Eqs. (110a) and (110b) gation on c-numbers and Pþ is the operator defined are identical statements. in Eq. (65). PT inner product: For the PT -conjugate states, the G. PT conjugation transformation rules for the ket and bra states are Given the definitions of the parity and time-reversal jTˆ αi¼Tˆ jαi ⇔ ðjTˆ αiÞ‡ ¼ðTˆ jαiÞ‡ operators, we have PT ˆ ‡ PT ˆ −1 ˆ ˆ ˆ ˆ −1 ˆ −1 ˇ ¼hα jT ¼hα jT ; ð107Þ P T ϕiðxÞT P ¼ TijPjkϕkðPT xÞ; ð113aÞ

† −1 † ˆ ˆ T ˆ −1 ˆ † Pˆ Tˆ ϕˆ ð ÞTˆ Pˆ −1 ¼ ϕˇ ðPT Þ ð Þ where we have used T T T ¼ T . In this case, we i x TijPjk k x ; 113b have12 and, taking Tij ¼ δij, it follows that T −1 hðTˆ αÞPT jϕˆ ðT ÞjTˆ βi¼hαPT jTˆ ϕˆ ðT ÞTˆ jβi ‡ † i x i x ϕˆ ð Þ¼ ϕˇ ð Þ ð Þ i x Pij j x ; 114 ¼! hβPT jϕˆ‡ð Þjαi ð Þ Tij j x : 108 since ϕˇTðPT xÞ¼ϕˇ†ðxÞ. The PT symmetry of the Hamiltonians in Eqs. (56) and (58) is now readily con- Making use of the identity firmed. Note that, in Fock space, the requirement of PT symmetry is that ½H;ˆ ‡¼0, superseding the constraint of hαPT jTˆ −1ϕˆ ðT ÞTˆ jβi¼hβPT jðTˆ −1ϕˆ ðT ÞTˆ Þ‡jαi i x i x ; Hermiticity, i.e., ½H;ˆ †¼0. This should be compared with ð109Þ the classical, and quantum-mechanical requirement, that 0 ½H;ˆ Pˆ Tˆ ¼0. We can also easily check that ½Cˆ ; ‡¼0,as we quickly recover the transformations in Eq. (105). required. C0PT inner product: Without making any assumption as to whether the C0 and T transformations commute, the V. SCALAR-PSEUDOSCALAR MIXING transformation rules for the ket and bra states for the C0PT AND OSCILLATIONS inner product are We now illustrate the discussion in the previous sections by studying mixing and oscillations in the model with two ˆ ˆ ˆ ˆ C0PT ˆ § jT αi¼T jαi ⇔ ðjT αiÞ§ ¼ðT jαiÞ§ ¼hα jT spin-zero fields. As mentioned earlier, the Lagrangian (1) C0PT 0 0 and the corresponding Hamiltonian do not conserve parity. ¼hα jCˆ Tˆ Cˆ ; ð110aÞ We therefore anticipate the possibility of scalar-pseudo-

0 0 scalar mixing and oscillations, but issues of interpretation ⇔ ðjαiÞ§Tˆ T ¼ðTˆ jαC PT iÞT ¼hαC PT jTˆ −1 ð Þ : 110b arise (see Ref. [19,20]), as we now discuss in detail.

12 13 Taking Tij ¼ δij for simplicity, the action of an antilinear Taking Tij ¼ δij for simplicity, the action of an antilinear operator on the PT inner product is operator on the C0PT inner product is

hαPT jTˆ −1ϕˆ ðT ÞTˆ jβi¼hβjϕˆ†ð ÞjαPT i hαC0PT jTˆ −1ϕˆ ðT ÞTˆ jβi¼hβjϕˆ†ð ÞjαC0PT i i x i x i x i x ¼hβPT jKˆ Pˆ Tˆ ϕˆ†ð ÞKˆ Pˆ Tˆ jαi ¼hβC0PT jKCˆ 0Pˆ Tˆ ϕˆ†ð ÞKCˆ 0Pˆ Tˆ jαi i x i x ¼hβPT jϕˆ‡ð Þjαi ¼hβC0PT jϕˆ§ð Þjαi i x : i x :

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A. Issues in flavor oscillations in the the total probability is not conserved—instead it involves PT -symmetric model the amplitude and its C0PT conjugate. A straightforward In the mass eigenbasis (see Sec. II A), the classical calculation then leads to equations of motion take the form 2 η 2 1 Π ðtÞ¼− sin ðEþðpÞ − E−ðpÞÞt : ð122Þ 2 i→i= 1 − η2 2 □ξ þ mξ ¼ 0; ð115Þ which have the plane-wave solutions Alarmingly, this probability is negative, and the corre- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sponding survival probability is given by i½E pt−p·x 2 2 Z ξ ¼ Ae ; with E p ¼ p þ m; ð116Þ 1 ; ˇ ˆ0 ˇ0 ˆ Πi→iðtÞ¼ hp;i;tjp ;i;0ihp ;i;0jp;i;ti V p0 where A are constants. η2 1 The single-particle flavor eigenstates can be written in ¼ 1 þ 2 ð ðpÞ − ðpÞÞ ð Þ 2 sin Eþ E− t ; 123 terms of the mass eigenstates as follows: 1 − η 2  which can be larger than unity. Notice, however, that ˇ † jp; 1ð2Þ;ti¼aˇ1ð2Þ pðtÞj0i¼N ηjp; þð−Þ;ti Π þ Π ¼ 1 ; i→i i→i= , such that the total probability is    qffiffiffiffiffiffiffiffiffiffiffiffi conserved. − 1 − 1 − η2 jp; −ðþÞ;ti : ð117Þ It is interesting to note that the oscillation period obtained from the probability (122) diverges at the excep- η2 → 1 As per the discussions of Secs. II A and IV D, the mass tional points , where C0PT eigenstates are orthonormal with respect to the inner 2π 2π product. The conjugate flavor state is ¼ ≃ pffiffiffiffiffiffiffiffiffiffiffiffi T 2 with EþðpÞ − E−ðpÞ ðpÞ 1 − η  E0 2 2 hpˆ 1ð2Þ j ≡ h0j ˆ ð Þ¼ ηðjp þð−Þ iÞ§ m1 − m2 ; ;t a1ð2Þ;p t N ; ;t E0ðpÞ ≡ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð124Þ 2 p2 þð 2 þ 2Þ 2  qffiffiffiffiffiffiffiffiffiffiffiffi  m1 m2 = þ 1 − 1 − η2 ðjp; −ðþÞ;tiÞ§ ; ð Þ 118 since the eigenmasses become degenerate. Another way to understand this limit is to consider the similarity trans- which has been expressed in terms of the C0PT -conjugate formation (21) when η → ϵ ¼1, mass eigenstates by appealing to Eqs. (31) and (59). The   flavor and mass eigenstates obey the following orthonor- R ϵ 1 lim ¼ with N → ∞: ð125Þ mality relations: η→ϵ N 1 ϵ ˇ ˆ0 3 3 0 hp;i;tjp ;j;ti¼ð2πÞ δijδ ðp − p Þð119Þ We see that the eigenstates defined in Eq. (115) are in these limits. Therefore, in addition to having infinite and normalization, the similarity transformation is not invert- ible at the exceptional points, and one cannot define a map 0 3 3 0 ðjp; ;tiÞ§jp ; ;ti¼ð2πÞ δ ðp − p Þ; ð120aÞ back to the flavor states. It is illustrative to compare this oscillation probability for ðjp; ;tiÞ§jp0; ∓;ti¼0: ð120bÞ the non-Hermitian theory to the corresponding probability for the Hermitian theory with the Lagrangian Assuming for simplicity a localized initial state, the ¼ 0 Lˆ ¼ ∂ ϕˆ†∂νϕˆ − 2ϕˆ†ϕˆ − μ2ðϕˆ†ϕˆ þ ϕˆ†ϕˆ Þ probability for the scalar with flavor i at t to transition Herm ν i i mi i i 1 2 2 1 ; to the pseudoscalar with flavor j at t>0 is given naively by ð126Þ Z 1 ˇ ˆ0 ˇ0 ˆ 2 2 Πi→jðtÞ¼ hp;i;tjp ;j;0ihp ;j;0jp;i;ti; ð121Þ μ 0 where mi and are positive real-valued squared-mass V p 2 2 parameters, and we assume m1 >m2 as before. For this where V ¼ð2πÞ3δ3ð0Þ is a three-. We draw atten- theory, the oscillation probability is “ ” tion to the fact that this probability is not obtained from 1 the usual squared modulus with respect to Hermitian ΠHermð Þ¼ 2ð2αÞ 2 ð ðpÞ − ðpÞÞ ð Þ i→j t sin sin Eþ E− t ; 127 conjugation—were we to use this, we would find that 2

125030-14 DISCRETE SPACETIME SYMMETRIES AND PARTICLE MIXING … PHYS. REV. D 102, 125030 (2020)  qffiffiffiffiffiffiffiffiffiffiffiffi  α where is the mixing , which is given by ˆ ˆ 2 ˆ ϕ1 ¼ N ηξ1 − 1 − 1 − η ξ2 ; ð133aÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 1 1 4μ   sin ðαÞ¼ − 1 − : ð128Þ qffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 4 † † † 2 2 ðm1 − m2Þ þ 4μ ˇ ˆ 2 ˆ ϕ1 ¼ N ηξ1 þ 1 − 1 − η ξ2 ; ð133bÞ

We see that the probability (122) has the same form as in  qffiffiffiffiffiffiffiffiffiffiffiffi  the Hermitian case, provided one makes the identification ˆ ˆ 2 ˆ pffiffiffiffiffiffiffiffiffiffiffiffi ϕ2 ¼ N ηξ2 − 1 − 1 − η ξ1 ; ð133cÞ sinð2αÞ¼iη= 1 − η2. With this identification, we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffi  1 1 4μ4 ˇ† ˆ† 2 ˆ† 2ðαÞ¼ − 1 þ ð Þ ϕ2 ¼ N ηξ2 þ 1 − 1 − η ξ1 ; ð133dÞ sin 2 2 2 4; 129 2 2 ðm1 − m2Þ − 4μ

2 2 giving which is simply the analytic continuation μ → iμ of the Hermitian expression, and it is for this reason that the non-  qffiffiffiffiffiffiffiffiffiffiffiffi PT 2 2 Hermitian transition probability is negative in the - ΔF;21ðqÞ¼N η 1 − 1 − η symmetric regime. As we show in the next subsection, this   problem has arisen because of an attempt to treat the flavor i − i × 2 2 2 2 states as external states. q − Mþ þ iϵ q − M− þ iϵ μ2 ¼ i ð Þ B. Flavor mixing in scattering matrix elements 2 2 2 2 ; 134a ðq − Mþ þ iϵÞðq − M− þ iϵÞ The resolution of the above issues can be found by  qffiffiffiffiffiffiffiffiffiffiffiffi recalling that experimental observables are scattering Δ ð Þ¼ 2η 1 − 1 − η2 matrix elements, for which we now give a simplified D;12 q N   treatment. For this purpose, we introduce two complex i − i sources JA and JB, coupled to the fields ϕ1 and ϕ2 as × 2 2 2 2 q − Mþ − iϵ q − M− − iϵ L ¼ ϕˇ† þ † ϕˆ − ϕˇ† þ † ϕˆ ð Þ iμ2 int JA 1 JA 1 JB 2 JB 2: 130 ¼ ð Þ 2 2 2 2 : 134b ðq − Mþ − iϵÞðq − M− − iϵÞ L We draw attention to the important fact that int must be PT symmetric, giving rise to the relative minus sign Notice that, by virtue of the non-Hermiticity, we have between the last two terms [10,11]. The source terms are Δ ð Þ¼−Δ ð Þ ð Þ therefore not Hermitian. F;21 q D;12 q : 135 The matrix element for the process A → B is given by If the mass mixing were Hermitian, the Feynman and M ¼ð2πÞ4δ4ð − ÞΔ ð Þ ð Þ ð Þ i A→B pA pB F;21 q ; 131 Δ Herm ð Þ¼ Dyson would instead satisfy F;21 q ΔðHermÞð Þ with q ¼ pA ¼ pB, and the conjugate matrix element is D;12 q . The sign appearing in Eq. (135) is due to the skew symmetry of the squared-mass matrix. − MC0PT ¼ −ð2πÞ4δ4ð − ÞΔ ð Þ ð Þ i A→B pA pB D;12 q : 132 We therefore have  qffiffiffiffiffiffiffiffiffiffiffiffi We note the overall sign, which stems from the relative sign M ¼ð2πÞ4δ4ð − Þ 2η 1 − 1 − η2 Δ ð Þ i A→B pA pB N in Eq. (130). The Feynman and Dyson propagators F;ij q   Δ ðqÞ and D;ij are defined by i − i ð Þ × 2 2 ; 136a Z s − Mþ s − M− d4q Δ ð − Þ≡h ½ϕˇ †ð Þϕˆ ð Þi ¼ −iq:ðx−yÞΔ ð Þ F;ij x y T i x j y 4 e F;ij q ;  qffiffiffiffiffiffiffiffiffiffiffiffi ð2πÞ 0 −iMC PT ¼ −ð2πÞ4δ4ðp − p ÞN2η 1 − 1 − η2 Z 4 A→B A B † d q   Δ ðx−yÞ≡hT¯ ½ϕˇ ðxÞϕˆ ðyÞi ¼ e−iq:ðx−yÞΔ ðqÞ; D;ij i j ð2πÞ4 D;ij i − i ð Þ × 2 2 ; 136b s − Mþ s − M− where T and T¯ denote time and antitime ordering, respec- ¼ 2 ¼ 2 tively. They can be calculated directly by inverting the non- where s pA pB is the usual Mandelstam variable and Hermitian Klein-Gordon operator in momentum space, or by we have suppressed the pole prescription in the propaga- expressing the fields ϕ1 and ϕ2 in the mass eigenbasis, i.e., tors. The squared matrix element is then

125030-15 ALEXANDRE, ELLIS, and MILLINGTON PHYS. REV. D 102, 125030 (2020) qffiffiffiffiffiffiffiffiffiffiffiffi    2 1 1 2 MC0PT M ¼ ð2πÞ4δ4ð − Þ 4η2 1 − 1 − η2 − A→B A→B VT pA pB N 2 2 s − M s − M−  þ 1 η2 1 1 2 ¼ ð2πÞ4δ4ð − Þ − ð Þ VT pA pB 2 2 2 ; 137 4 1 − η s − Mþ s − M− where VT ≡ ð2πÞ4δ4ð0Þ is a four-volume factor. We observe that this result is positive for η2 < 1. However, it would seem naively that there is an issue with perturbative unitarity in the limit η2 → 1, due to the factor of 1=ð1 − η2Þ in (137), but this is not the case, since   1 1 2 ð 2 − 2 Þ2 ð 2 − 2Þ2 − ¼ Mþ M− ¼ð1 − η2Þ m1 m2 ð Þ 2 2 2 2 2 2 2 2 2 2 138 s − Mþ s − M− ðs − MþÞ ðs − M−Þ ðs − MþÞ ðs − M−Þ is proportional to 1 − η2. The final expression for the squared matrix element is

μ4 MC0PT M ¼ ð2πÞ4δ4ð − Þ ð Þ A→B A→B VT pA pB 2 2 2 2 ; 139 ðs − MþÞ ðs − M−Þ which is positive, vanishes in the limit μ → 0 (as it should), This offers a resolution of the problematic behavior in the and remains real and perturbatively valid all the way up to naive calculation of the oscillation probability presented in the exceptional point η2 ¼ 1. Sec. VA. Comforted by the example presented in this The matrix element for the corresponding flavor-con- subsection, we leave for future work the further detailed serving process A → A is study of this point.

M ¼ð2πÞδ4ð − 0 ÞΔ ð Þ ð Þ VI. CONCLUSIONS i A→A pA pA F;11 q ; 140 We have addressed in this paper some basic issues in the ¼ ¼ 0 with q pa pA, and the conjugate matrix element is formulation of non-Hermitian bosonic quantum field the- ories, discussing in particular the treatment of discrete − MC0PT ¼ð2πÞδ4ð − 0 ÞΔ ð Þ ð Þ i A→A pA pA D;11 q ; 141 symmetries and the definition of the inner product in Fock space. We have focused on PT -symmetric non-Hermitian where theories, commenting also on features at the exceptional points at the boundary between theories with PT sym- Δ ð Þ¼Δ ð Þ ð Þ metry and those in which it is broken. D;11 q F;11 q ; 142 As we have discussed, there is ambiguity in the formu- with lation of the inner product in a PT -symmetric theory. In this case, the conventional Dirac inner product ðjαiÞ†jβi¼  pffiffiffiffiffiffiffiffiffiffiffiffi  η2 ð1 − 1 − η2Þ2 hα jβi is not positive definite for the mass eigenstates, and Δ ð Þ¼ 2 i − i PT F;11 q N 2 2 2 2 is therefore deprecated, and the same is true of the inner q − M þ iϵ q − M− þ iϵ ‡ PT þ product ðjαiÞ jβi¼hα jβi, where ‡ ≡ PT ∘ T with T ð 2 − 2Þ ¼ i q m2 ð Þ denoting transposition. The appropriate positive-definite 2 2 2 2 : 143 C0PT ðq − M þ iϵÞðq − M− þ iϵÞ norm for the mass eigenstates is defined via con- þ 0 jugation: ðjαiÞ§jβi¼hαC PT jβi, where § ≡ C0PT ∘ T, where We therefore obtain the C0 operator was defined in Sec. IV B. As was explained there, the C0 transformation in a PT -symmetric quantum MC0PT M ¼ ð2πÞ4δ4ð − 0 Þ field theory cannot be identified with charge conjugation. A→A A→A VT pA pA We have formulated in Sec. IV C a suitable similarity ð − 2Þ2 s m2 ð Þ transformation between a PT -symmetric non-Hermitian × 2 2 2 2 ; 144 ðs − MþÞ ðs − M−Þ theory with two flavors of spin-zero fields and its Hermitian counterpart. The equivalence between the noninteracting which is again real, positive, and physically meaningful for PT -symmetric and Hermitian theories cannot, in general, all 0 ≤ η2 ≤ 1. be carried over to interacting theories with the same It is clear from these results that there is a subtlety arising similarity transformation. The Appendices contrast the from the factorization of the source-to-source probability similarity transformation we propose with the previous into production, oscillation, and detection probabilities. literature.

125030-16 DISCRETE SPACETIME SYMMETRIES AND PARTICLE MIXING … PHYS. REV. D 102, 125030 (2020)  Z  π As an illustration of this Fock-space discussion, we have ˆ ˆ ˆ† † S ¼ exp ðπˆ2ðt; xÞϕ2ðt; xÞþϕ2ðt; xÞπˆ2ðt; xÞÞ : considered mixing and oscillations in this specific model 2 x with two boson flavors, which is free apart from non- ðA2Þ Hermitian PT -symmetric mixing terms. The unmixed bosons are taken to be a scalar and a pseudoscalar, which Here, we have written the operator Sˆ in a manifestly mix via a non-Hermitian bilinear term. We have shown that Hermitian form. We note, however, that the similarity the resulting mass eigenvectors are not orthogonal with transformation is defined only up to a constant complex respect to the Dirac inner product, but are orthogonal with phase, such that one is free to reorder the operators in the 0 positive norm when the C PT inner product is used. We exponent by making use of the canonical equal-time have emphasized that the parity operator in this two-boson commutation relations. We note that, unlike the similarity model does not commute with the Hamiltonian, leading to transformation we propose in the main text, the trans- the appearance of scalar-pseudoscalar mixing and flavor formation (A2) does not depend on the non-Hermitian oscillations, which we have studied in Sec. V. These are of parameter η. similar form to the mixing between bosons in a Hermitian The similarity transformation (A2) has the following theory, respecting unitarity but presenting issues of inter- action on the field operators: pretation, which we show can be resolved by considering ˆ ˆ ˆ−1 ˆ physical scattering matrix elements, wherein flavor states Sϕ2ðt; xÞS ¼ −iϕ2ðt; xÞ; ðA3aÞ only appear internally. Sˆϕˆ†ð xÞSˆ−1 ¼ − ϕˆ†ð xÞ ð Þ The analysis in this paper has clarified the description of 2 t; i 2 t; ; A3b PT -symmetric non-Hermitian bosonic quantum field the- ories and provides a framework for formulating them off- and the transformed version of the Lagrangian (39) for the shell. Many of the features discussed here are expected to free scalar theory is therefore carry over to PT -symmetric non-Hermitian field theories ˆ ˆ† ν ˆ ˆ† ν ˆ 2 ˆ† ˆ 2 ˆ† ˆ LS ¼ ∂νϕ ∂ ϕ1 − ∂νϕ ∂ ϕ2 − m ϕ ϕ1 þ m ϕ ϕ2 of fermions [33], as we shall discuss in a following paper. 1 2 1 1 2 2 2 ˆ† ˆ ˆ† ˆ This programme constitutes an important step toward − iμ ðϕ1ϕ2 − ϕ2ϕ1Þ: ðA4Þ addressing deeper issues in field theory such as quantum loop corrections and , to which we also plan While this Lagrangian is Hermitian, we draw attention to to return in future work. the opposite relative signs of the kinetic and mass terms for ˆ ˆ the fields ϕ1;2, which imply that ϕ2 is a negative-norm ghost and is tachyonic. One should therefore suspect that ACKNOWLEDGMENTS the similarity transformation in Eq. (A2) is not directly 0 P. M. thanks Maxim Chernodub for helpful comments on related to the Cˆ operator needed to construct a positive the paper. The work of J. A. and J. E. was supported by the norm for these states. Moreover, one can readily confirm United Kingdom STFC Grant No. ST/P000258/1 and that that this similarity transformation, unlike the one defined of J. E. also by the Estonian Research Council via a in Sec. IVC, does not leave the Fock invariant. Mobilitas Pluss grant. The work of P. M. was supported The latter issue is most easily illustrated by decoupling by a Leverhulme Trust Research Leadership Award (Grant the two flavors, i.e., taking the Hermitian limit η → 0. The ˆ No. RL-2016-028) and a Nottingham Research Fellowship plane-wave decomposition of the field ϕ2 then takes a from the University of Nottingham. simple form, and we can immediately write  Z π † † 2 Sˆj ≡ Sˆ ¼ ð ˆ ð0Þ ˆ ð0Þ iE2;pt η→0 0 exp i a2;p c2;−p e APPENDIX A: AN ALTERNATIVE SIMILARITY 2 p TRANSFORMATION  −2iE2 pt − cˆ2 pð0Þaˆ2 −pð0Þe ; Þ : ðA5Þ 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 The creation and annihilation operators transform as comparison with the transformation detailed in Sec. IV C. follows: ˆ The Hamiltonian H of the two-flavor scalar theory can −1 2 † Sˆ ˆ ð0ÞSˆ ¼ − iE2;qt ˆ ð0Þ ð Þ ˆ 0a2;q 0 ie c2;−q ; A6a also be mapped to a Hermitian one hS (and similarly for the † −1 −2 Lagrangian) via the similarity transformation [15] Sˆ ˆ ð0ÞSˆ ¼ − iE2;qt ˆ ð0Þ ð Þ 0a2;q 0 ie c2;−q ; A6b −1 2 † ˆ ˆ ˆ ˆ−1 Sˆ ˆ ð0ÞSˆ ¼ − iE2;qt ˆ ð0Þ ð Þ hS ¼ S H S ; ðA1Þ 0c2;q 0 ie a2;−q ; A6c

† −1 Sˆ ˆ ð0ÞSˆ ¼ − −2iEqt ˆ ð0Þ ð Þ with 0c2;q 0 ie a2;−q ; A6d

125030-17 ALEXANDRE, ELLIS, and MILLINGTON PHYS. REV. D 102, 125030 (2020) which are consistent with the transformations of the fields in Eq. (A3). This transformation would lead to the following 0 candidate Cˆ operator:  Z  0 † † 2 −2 Cˆ ¼ π ð ˆ ð0Þ ˆ ð0Þ iEpt − ˆ ð0Þ ˆ ð0Þ iE2;ptÞ Pˆ ð Þ ? exp i a2;p c2;−p e c2;p a2;−p e : A7 p

However, we see immediately that this operator does not leave the Fock vacuum invariant. Instead, it is transformed to an infinite series of time-dependent multiparticle states,

2 Z 2 Z ˆ0 π ðiπÞ C?j0i¼ 1 þ þ j0iþiπ jp; 2;t; p¯ ; 2;tiþ jp; 2;t; p¯ ; 2;t; q; 2;t; q¯; 2;tiþ ; ðA8Þ 2! p 2! p;q

 qffiffiffiffiffiffiffiffiffiffiffiffi  wherein antiparticle states are indicated by a bar over the hpˆ 1 j¼ ηhp þ jþ 1 − 1 − η2 hp − j three-momentum with, e.g., p¯ ¼ −p. ; ;t N ; ;t ; ;t ; ðB3aÞ   APPENDIX B: SOME USEFUL EXPRESSIONS qffiffiffiffiffiffiffiffiffiffiffiffi hpˆ;2;tj¼N ηhp;−;tjþ 1 − 1 − η2 hp;þ;tj ; In this Appendix, we collect useful expressions for the various mass and flavor states. The ket states are as follows: ðB3bÞ  qffiffiffiffiffiffiffiffiffiffiffiffi   qffiffiffiffiffiffiffiffiffiffiffiffi  2 0 ˆ 2 jpˇ; 1;ti¼N ηjp; þ;ti − 1 − 1 − η jp; −;ti ; hp;þ;0jCˆ Pˆ T ¼ N ηhp;1;0j − 1 − 1 − η hp;2;0j ;

ðB1aÞ ðB3cÞ  qffiffiffiffiffiffiffiffiffiffiffiffi   qffiffiffiffiffiffiffiffiffiffiffiffi  0 2 hp;−;0jCˆ Pˆ Tˆ ¼ N ηhp;2;0j − 1 − 1 − η hp;1;0j ; jpˇ; 2;ti¼N ηjp; −;ti − 1 − 1 − η2 jp; þ;ti ; ðB3dÞ ðB1bÞ with  qffiffiffiffiffiffiffiffiffiffiffi  ˆ0 ˆ ˆ jp þ 0i¼ ηjp 1 0iþ 1− 1−η2 jp 2 0i ð Þ hp; þ;tjC P T ; ; N ; ; ; ; ; B1c  qffiffiffiffiffiffiffiffiffiffiffiffi  ¼ Ne−iEþt ηhp; 1; 0j − 1 − 1 − η2 hp; 2; 0j ;  qffiffiffiffiffiffiffiffiffiffiffi  jp;−;0i¼N ηjp;2;0iþ 1− 1−η2 jp;1;0i : ðB1dÞ ðB4aÞ hp; −;tjCˆ0Pˆ Tˆ   We also have that qffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ne−iE−t ηhp; 2; 0j − 1 − 1 − η2 hp; 1; 0j :  qffiffiffiffiffiffiffiffiffiffiffiffi  2 jp; þ;ti¼NeiEþt ηjp; 1; 0iþ 1 − 1 − η jp; 2; 0i ; ðB4bÞ

ðB2aÞ The relations for the mass eigenstates are therefore  qffiffiffiffiffiffiffiffiffiffiffiffi  ˆ0 ˆ ˆ 0 3 3 0 jp; −;ti¼NeiE−t ηjp; 2; 0iþ 1 − 1 − η2 jp; 1; 0i : hp; ;tjC P T jp ; ;ti¼ð2πÞ δ ðp − p Þ; ðB5aÞ

ˆ0 ˆ ˆ 0 ðB2bÞ hp; ;tjC P T jp ; ∓;ti¼0; ðB5bÞ

The distinction between checked and hatted operators is since not needed for the flavor eigenstates at the initial time or hp 0jp0 0i¼ð2πÞ3δ δ3ðp − p0ÞðÞ mass eigenstates for all times. The conjugate states are ;i; ;j; ij B6a

125030-18 DISCRETE SPACETIME SYMMETRIES AND PARTICLE MIXING … PHYS. REV. D 102, 125030 (2020) by virtue of the algebra in Sec. III. Moreover, we have that

jpˇ 1 i¼ð iETtÞ jpˇ 0i ; ;t e 1i ;i; 0  pffiffiffiffiffiffiffiffiffiffiffiffi 1 ηjp; þ; 0i − 1 − 1 − η2 jp; −; 0i T B C ¼ð iE tÞ   e 1iN@ pffiffiffiffiffiffiffiffiffiffiffiffi A ηjp; −; 0i − 1 − 1 − η2 jp; þ; 0i i T jp; þ; 0i iE t −1 ¼ðe Þ1 R i ij jp; −; 0i j jp; þ; 0i −1 iEdiagt −1 ¼ R1 ðe Þ R R l i ij jk k jp; −; 0i l eiEþtjp; þ; 0i ¼ −1 ð Þ R1i ; B7 iE−tjp − 0i e ; ; i which is consistent with Eq. (B3a).

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