Poincar´e symmetries and representations in pseudo-Hermitian quantum ﬁeld theory

This paper explores quantum ﬁeld theories with pseudo-Hermitian Hamiltonians, where P T -symmetric Hamiltonians serve as a special case. In speciﬁc regimes, these pseudo-Hermitian Hamiltonians have real eigenspectra, orthogonal eigenstates, and unitary time evolution. So far, most pseudo-Hermitian quantum ﬁeld theories have been constructed using analytic continuation or by adding non-Hermitian terms to otherwise Hermitian Hamiltonians. However, in this paper, we take a diﬀerent approach. We construct pseudo-Hermitian scalar and fermionic quantum ﬁeld theories from ﬁrst principles by extending the Poincar´e algebra to include non-Hermitian generators. This allows us to develop consistent pseudo-Hermitian quantum ﬁeld theories, with Lagrangian densities that transform appropriately under the proper Poincar´e group. By doing so, we establish a more solid theoretical foundation for the emerging ﬁeld of non-Hermitian quantum ﬁeld theory. This is an author-prepared post-print of Phys. Rev. D 109 (2024) 065012, published by the American Physical Society under the terms of the CC


INTRODUCTION
In "standard" quantum mechanics, the description of a physical system relies on two crucial elements: a Hilbert space H of states and a Hamiltonian operator Ĥ : H → H that determines the time evolution.The Hamiltonian and real-valued physical observables correspond to Hermitian operators.This guarantees that their expectation values are real and that their eigenstates are orthogonal.Moreover, the time evolution operator generated by a Hermitian Hamiltonian is unitary, ensuring that probability is conserved.However, it is now well established that operators do not need to be Hermitian to produce real expectation values [1][2][3][4].Instead, an operator, say Â, must satisfy a condition known as pseudo-Hermiticity: Â † = η Âη −1 [5] (where † is the usual composition of complex conjugation and matrix transposition) for some Hermitian operator η † = η.We define a new inner product •|• η := •|η• , which yields real expectation values of Â.Unlike Hermitian operators, whose eigenvalues are always real, pseudo-Hermitian operators exhibit eigenvalues that are either real, come in complex-conjugate pairs, or so-called exceptional points, where eigenvalues merge and the operator becomes defective [6].When the eigenvalues of the Hamiltonian are real, there exists an additional discrete symmetry of the Hamiltonian that ensures unitary time evolution [7], and the pseudo-Hermitian theory can be made Hermitian via a similarity transformation [8].However, the similarity transformation becomes singular at the exceptional points, and there is no Hermitian counterpart when the eigenenergies are complex [9,10].
Many of these theories are constructed by appending non-Hermitian operators to an otherwise Hermitian Lagrangian, or by analytic continuation of Hermitian theories, for instance, by rotating coupling constants into the complex plane.Often times, these theories are analyzed in their P T -unbroken regimes (when the energy eigenvalues are real) by transforming to a Hermitian theory.However, if we instead consider the non-Hermitian theory directly, we run into issues with physical consistency, as detailed below.
In quantum field theory, the physical system is described by a Fock space F consisting of multiparticle states, rather than a single-particle Hilbert space.The system dynamics is governed by a Hamiltonian operator Ĥ : F → F acting on this Fock space.Similar to non-Hermitian quantum mechanics, if the Hamiltonian operator is pseudo-Hermitian Ĥ † = η Ĥ η−1 with respect to some Hermitian operator η : F → F , it will exhibit real eigenspectra, complex-conjugate pairs of eigenvalues, or exceptional points.As in pseudo- In "standard" quantum field theory, the Hamiltonian is a functional of field operators and their canonical momenta ( ψ, ψ † , π, π † ), instead of position and momentum operators (x, p), as it is in quantum mechanics.The time evolution of the field operators is governed by Hamilton's equations: [ ψ( x, t), Ĥ] = i∂ 0 ψ( x, t) and [ ψ † ( x, t), Ĥ † ] = i∂ 0 ψ † ( x, t) . (1.1) Most notably, the field operator ψ and its Hermitian conjugate ψ † do not evolve with the same Hamiltonian, since it is non-Hermitian Ĥ † = Ĥ.This leads to mutual inconsistency in the Euler-Lagrange equations for ψ and ψ † , a common feature observed in various non-Hermitian quantum field theory models, first pointed out in Ref. [49].One option is to fix the dynamics with respect to only one of the Euler-Lagrange equations [49].For noninteracting theories, it can be argued that physical observables remain unchanged regardless of the chosen equation [49].However, in the context of interacting theories, this method leads to distinct physical results [37,39,40] compared with approaches based on transforming the non-Hermitian theory to the Hermitian, i.e., standard quantum field theory [38,[41][42][43].It is also an open question as to how to consistently introduce gauge symmetries in non-Hermitian quantum field theory [50].Moreover, it has been observed that the conserved currents in a non-Hermitian Yukawa theory are not invariant under proper Lorentz transformations [31].
In this paper, we show that these inconsistencies naturally arise and are to be expected due to the quantum fields ψ and ψ † not transforming in the same representation of spacetime symmetry transformations (the proper Poincaré group).
Attempts have been made [51,52] to resolve these issues in some P T -symmetric models by redefining the conjugate field in terms of the parity transformation.In these models, the field and its parity conjugate evolve with the same Hamiltonian.In this work, we confirm that the conjugate field must be determined with care in order to build self-consistent pseudo-Hermitian quantum field theories.As was noted in Ref. [53], this requires us to consider non-Hermitian generators of the proper Poincaré group, which is clearly the case for the generator of time translations: the Hamiltonian P 0 = Ĥ.This marks a significant difference between pseudo-Hermitian quantum mechanics and pseudo-Hermitian quantum field theory, where, for the latter, spacetime symmetries play a pivotal role.Moreover, we will see that the non-Hermiticity of the generator of time translations implies that other generators are, in general, non-Hermitian.Using group transformation properties, we construct conjugate field operators transforming consistently under the full proper Poincaré group.
The paper is organized as follows.In Sec. 2, we reexamine time evolution in the case of pseudo-Hermitian quantum mechanics in terms of representations of time translations.
In Sec. 3, we turn our attention to pseudo-Hermitian quantum field theory and, by demanding Poincaré invariance, we show that pseudo-Hermiticity of the Hamiltonian implies pseudo-Hermiticity of the remaining group generators.In Subsec.3.1, we identify the relevant representations that act on the quantum field operators by considering their matrix elements with respect to the inner product •|η• .In Sec. 4, we examine how quantum fields behave under spacetime translations and proper Lorentz transformations when Fock-space representations are non-Hermitian.
In Subsec.4.2, we present the key result of this paper.Therein, we define the "dual" quantum field operator ψ † , which transforms in the dual representation of the proper Poincaré group of the field operator ψ.Hence, the Lagrangian composed of the field operator ψ and its "dual" ψ † transforms as one object in a single representation of the proper Poincaré group.This also leaves any bilinear combinations ψ † ψ Poincaré invariant.Additionally, as the "dual" field operator evolves with the Hamiltonian Ĥ instead of Ĥ † , the Euler-Lagrange equations are automatically consistent, unlike those formulated in terms of the Hermitian conjugate field operator ψ † .
In Sec. 5, we show how to construct pseudo-Hermitian finite-dimensional representations of the proper Lorentz group, which are necessary to define a "dual" field for fields higher than spin-0.We consider the specific cases of: spin-0 scalars, for which the finite-dimensional representations are trivial; spin-half Weyl spinors, which are the smallest nontrivial representation of the Lorentz Lie algebra; and spin-half Dirac fermions.Finally, in Sec.6, we apply our discussion to a specific example of a P T -symmetric 2-component complex scalar field theory, correctly determining the dual field and identifying the relevant discrete symmetries.

MECHANICS
Before we turn to the case of non-Hermitian quantum field theory, it is helpful to first reexamine the concept of unitary time evolution in non-Hermitian quantum mechanics and its connection to representations of time translations.This will prove useful when considering representations in non-Hermitian quantum field theory.
As mentioned in the Introduction, the properties of real expectation values and unitary time evolution are not unique to Hermitian operators.In fact, a more general condition for an observable Â to yield real expectation values is for it to be pseudo-Hermitian [5]: An operator Â : H → H is η-pseudo-Hermitian if and only if Â † = η Âη −1 with respect to some operator η : H → H that is Hermitian η † = η.
In contrast to a Hermitian Hamiltonian, an η-pseudo-Hermitian Hamiltonian allows for both real and complex energy eigenvalues [3].In each case, the expectation value of an η-pseudo-Hermitian Hamiltonian is always real with respect to the inner product If the energy eigenvalues are real, then it is possible to find a similarity transformation that maps this non-Hermitian Hamiltonian to a Hermitian operator [8].However, the non-Hermitian Hamiltonian has no Hermitian counterpart at the exceptional points or when the eigenenergies are complex.Hence, a pseudo-Hermitian Hamiltonian may describe a unique physical system.
When we say that an η-pseudo-Hermitian Hamiltonian has unitary time evolution, it is not exactly the same as the unitary time evolution in "standard" quantum mechanics.
Instead, the time evolution is "pseudounitary", which means it is unitary with respect to the operator η: An operator Û : Indeed, in pseudo-Hermitian quantum mechanics, we can check that, given an η-pseudo-Hermitian Hamiltonian Ĥ, the time-evolution operator Û (t) = e −i Ĥt is η-pseudo-unitary.
As a result, the time evolution of the "ket" states |ψ ∈ H in our Hilbert space is governed by the η-pseudo-unitary operator Û (t): In the context of representations, the operator Û(t) = e −i Ĥt is the representation of time translations on the Hilbert space H, and the Hamiltonian Ĥ is the generator of this representation.
The "bra" states ψ| ∈ H * , however, are governed by Û (t) † : Hence, the dynamics of "bra" and "ket" states is not governed by the same Hamiltonian as it is non-Hermitian.They evolve under different representations of time translations.
Similarly, the wave function ψ( x, t) := x|ψ(t) ∈ F(H) evolves with the Hamiltonian function H : F(H) → F(H) acting on the vector space of wave functions F(H) of the Hilbert space H: However, the complex-conjugate wave function ψ * ( x, t) ∈ F(H * ) evolves with the Hermitian conjugate H † : F (H * ) → F (H * ), where F(H * ) is the vector space of wave functions of the dual Hilbert space H * : As the Hamiltonian H † = H is non-Hermitian, the complex-conjugate wave function transforms in a different representation of time translations, generated by H † instead of H.
This has significant implications for the probability density, which is composed of two components: viz. the wave function ψ, evolving with the Hamiltonian H, and its complex conjugate ψ * , evolving with H † .Consequently, the overall object, i.e., the probability density, undergoes transformations in two distinct representations of time translations and is not conserved.
In pseudo-Hermitian quantum mechanics, this is fixed by defining the probability density with respect to the new inner product •|η• : This inner product remains invariant under time translations generated by a η-pseudo-Hermitian Hamiltonian.Thus, the probability density is conserved.
However, another interpretation of this result, which will be relevant in the next section, is to introduce a new object called the "dual" wave function ψ * , which evolves with the same Hamiltonian as ψ.This enables us to redefine the probability density in terms of the wavefunction ψ and its "dual" ψ * , so that the probability density transforms in a single representation of time translations.
To find the dual wave function, we consider (U, H) to be a representation of time translations on the Hilbert space H.We define a "dual" representation (U (2.9) Hence, we see that its time evolution is governed by H and not H † .
For the special case where η is itself a coordinate transformation (e.g., parity), its action on the position eigenstates can be described as follows: η| x = η| x η , where η ∈ C is the phase. (2.10) The dual wave function simplifies to ψ * ( x, t) = ψ * ( x, 0)e iHt where ψ * ( x, 0) = ψ * ( x η , 0)η , Finally, we define the probability density in terms of the wave function ψ and its dual ψ * : which transforms in a single representation generated by H and is now conserved.
The above procedure is equivalent to defining a new inner product •|η• on the Hilbert space H: which is invariant under time translations.
The above considerations are nonrelativistic.In the relativistic case of quantum field theory, we are not only concerned with time translation invariance, but also with Lorentz invariance.Specifically, the symmetry group of Minkowskian quantum field theory is the . It is composed of the proper Lorentz transformations SO(1, 3) ↑ and spacetime translations R 1,3 .Instead of position and momentum operators, we have quantum field and canonical momentum field operators, and, as we will show in the next section, we face similar issues when trying to find conjugate field operators that transform correctly in the dual representation of proper Poincaré transformations.

POINCAR É INVARIANCE IN NON-HERMITIAN QUANTUM FIELD THE-ORY
We now turn our attention to quantum field theory.In the canonical operator formulation, the Hamiltonian operator Ĥ : F → F acts on the Fock space F .It is a function of field operators and their canonical momenta, i.e., Ĥ = Ĥ ψ, ψ † , π, π † .Through the rest of the paper, we assume that the Hamiltonian Ĥ † = η Ĥ η−1 is η-pseudo-Hermitian with respect to some Hermitian operator η : F → F .
The "ket" states in Fock space |α(t) ∈ F evolve subject to the Schrödinger equation Just as in pseudo-Hermitian quantum mechanics, their time evolution is η-pseudo-unitary.
We define our Fock space F = ( ⊗ H, •|• η), with respect to the inner product which is invariant under time translations and yields real energy expectation values.
The time evolution of the field operators ψ and ψ † is governed by Hamilton's equations of motion.However, by considering the Hermitian conjugate of the Hamilton's equation for ψ, namely, we observe that while ψ evolves with the Hamiltonian Ĥ, its Hermitian conjugate ψ † evolves with Ĥ † .As the Hamiltonian is non-Hermitian, this implies that the two fields are subject to different Hamiltonians.We also see this in the Heisenberg picture: ψ( x, t) = e i Ĥt ψ( x)e −i Ĥt and ψ † ( x, t) = e i Ĥ † t ψ † ( x)e −i Ĥ † t . (3.3) In the language of representations, the conjugate field ψ † does not transform in the dual representation of ψ.This implies that the kinetic and mass terms are not invariant under time translations, namely, ψ † ( x, t) ψ( x, t) = e i Ĥ † t ψ † ( x)e −i Ĥ † t e i Ĥt ψ( x)e −i Ĥt = e i Ĥt ψ † ( x) ψ( x)e −i Ĥt . (3.4) Our aim is to find the "dual" field operator that transforms in the dual representation.
However, quantum field theory in Minkowski spacetime has to be invariant under proper Poincaré transformations ISO(1, 3) ↑ , where time and space transformations are intermixed.
The Hamiltonian Ĥ is the generator of time translations, but the non-Hermiticity of Ĥ will turn out to translate into the non-Hermiticity of other generators of the proper Poincaré group.This observation is the primary focus of this section.
Let Û , F be a η-pseudo-unitary representation of ISO(1, 3) ↑ on the Fock space F .Any element of the proper Poincaré group can be expanded in terms of generators of the Poincaré Lie algebra iso(1, 3), i.e., Herein, Ĵ0i are the generators of boosts, Ĵij are the generators of rotations, the Hamiltonian P 0 = Ĥ is the generator of time translations, and the 3-momentum operator P i is the generator of space translations.
This can be seen directly by considering the Poincaré Lie algebra Ĵµν , Ĵρσ = i(g µσ Ĵνρ + g νρ Ĵµσ − g µρ Ĵνσ − g νσ Ĵµρ ) where g µν = diag 1, −1, −1, −1 is the Minkowski metric.Taking µ = 0 in the second bracket, we have which also implies that the generators cannot be Hermitian, unless they commute with η.

Connection to classical fields
It is important to consider the connection between quantum and classical fields, at the very least to identify the relevant representations for our subsequent discussions.
Let us consider an n-component quantum field ψa (x), a ∈ {1, • • • , n}.In Hermitian quantum field theory, the matrix elements of this field would be defined as an n-component In this way, we define the expectation value of the quantum field operator as Note that the operator η and the field operator ψa (x) do not, in general, commute.
By postulating that the expectation values of quantum fields possess the same transformation properties as classical field functions under the Poincaré group, we establish a connection between the transformation properties of quantum fields and their classical counterparts.In the quantum field theory literature, this is known as the "correspondence principle" [54], [55].
Under these transformations, the expectation value of the quantum field operator undergoes changes in three distinct representations of the proper Poincaré group: 1.The infinite-dimensional "coordinate representation" acting on the spacetime coordinates: (3.12) 2. The infinite-dimensional "Fock-space representation" Û , F , acting on states in Fock space: mixing components of an n-component field: where a, b ∈ 1, • • • , n.

FOCK-SPACE REPRESENTATIONS OF THE PROPER POINCAR É GROUP
In the previous section, we showed that the conjugate field ψ † evolves with a different Hamiltonian to ψ.As a result, the bilinear combination ψ † ψ in the Lagrangian is not invariant under time translations (3.4).Moreover, if we look at the full proper Poincaré group ISO(1, 3) ↑ , we see that the non-Hermiticity of the Hamiltonian implies non-Hermiticity of other generators (3.6), unless they commute with the Hermitian operator η.Thus, if we wish to construct a quantum field theory that is invariant under ISO(1, 3) ↑ , we cannot construct it from ψ and ψ † .Instead, we need to find a new quantum field operator ψ † that transforms in the "dual" representation of ψ, just as we did with the "dual" wave function in Sec. 2.
To find the "dual" field operator, we consider ( Û , F ) to be a representation of the proper Poincaré group ISO(1, 3) ↑ on the Fock space F .We define a "dual" representation ( Û * , F * ) on the "dual" Fock space F * as Definition 4.1 (Dual Fock-Space Representation).
However, as noted in the previous section, if the quantum field ψa is multicomponent, its components will mix under finite-dimensional representations (D, C n ) of the proper Lorentz group SO(1, 3) ↑ .We define a "dual" representation (D * , C n * ) on the "dual" vector space Given that the quantum field operator ψ transforms in a Fock-space representation ( Û, F ) and a finite-dimensional representation (D, C n ), we define the "dual" field operator ψ † as the field operator that transforms in a "dual" Fock-space representation ( Û * , F * ) and a "dual" finite-dimensional representation (D * , C n * ).We then say that the quantum field ψ † transforms in the "dual" representation of ψ.
We postulate that, under the proper Poincaré group ISO(1, 3) ↑ , the expectation value of an n-component quantum field (3.11) transforms as an n-component classical function: Here, Ψ a (x) := α| ψa (x)|α is the expectation value of the quantum field ψ transforming in the finite-dimensional representation (D, C n ), while Ψ †a (x) := α| ψ †a (x)|α is the expectation value of the "dual" quantum field ψ † transforming in the "dual" finite-dimensional Considering that the states |α ∈ F transform under the Fock-space representation ( Û, F ) and their "dual" states α| ∈ F * under the "dual" Fock-space representation ( Û * , F * ): we derive the transformation laws for the quantum field ψ and its "dual" ψ † : Taking the Hermitian conjugate of Eq. (4.5), we obtain the transformation law for the conjugate field ψ † : which clearly shows that the Hermitian-conjugate field ψ † does not transform in the "dual" representation of ψ, unless Û and D are both unitary representations, i.e., the Hamiltonian Ĥ is Hermitian.In fact, in Subsec.4.2, we demonstrate that nonunitary Fock-space representations ( Û, F ) of the proper Lorentz group SO(1, 3) ↑ imply nonunitarity of the finite-dimensional representations (D, C n ) and vice versa.For now, let us assume that the Fock-space representation is η-pseudo-unitary Û † η Û = η, which is the case for an η-pseudo-Hermitian Hamiltonian Ĥ.We also assume that the finite-dimensional representation is π-pseudo-unitary D † πD = π with respect to some n × n Hermitian matrix π : Using this, we rearrange Eq. (4.7): This is exactly the transformation law of the "dual" field (4.6).Hence, in general, the "dual" field operator will be of the form Since η : F → F can in general be a coordinate transformation (e.g., parity), we need to include its action on the coordinates x η (e.g., x P ) in the definition of the "dual" field.The field (4.9) transforms in the "dual" representation of the proper Poincaré group.As we did not assume anything about the spin of the quantum field ψ, the definition in (4.9) holds for fields of any spin.Thus, we will use it to define the "dual" fields for non-Hermitian scalar and fermionic quantum field theories in Secs. 5 and 6.
In the following subsections, we examine how quantum fields behave under the generators of spacetime translations and proper Lorentz transformations.In particular, in Subsec.4.1, we derive Hamilton's equations, confirming that they are inconsistent for the quantum field ψ and its Hermitian conjugate ψ † , but are in agreement with the "dual" field ψ † defined above.In Subsec.4.2, we show that non-Hermiticity of the Fock-space generators, originating from the non-Hermitian Hamiltonian Ĥ, directly leads to non-Hermitian generators of finitedimensional representations of the proper Lorentz group SO(1, 3) ↑ .

Spacetime translations
Consider a translation by a constant 4-vector x → x ′ = x + ε.The corresponding transformation of a quantum field ψa , according to Eq. (4.5), is Since the components of a multicomponent field do not mix under spacetime translations, the only representations acting on the quantum field are the Fock-space and coordinate representations.Both of these can be expanded in terms of their generators: where P µ are the four generators of spacetime translations in the Fock-space representation and ∂ µ are the generators of spacetime translations in the coordinate representation.
Expanding each side in Eq. (4.10) gives us the relationship between the generators of spacetime translations for Fock-space and coordinate representations: Notably, for µ = 0, we recover Hamilton's equation of motion: Taking a Hermitian conjugate of the above, we find that the conjugate field ψ † evolves with Hermitian conjugates of these generators: In particular, for µ = 0, the conjugate field ψ † evolves with Ĥ † instead of Ĥ: In Sec. 3, we showed that if the Hamiltonian is η-pseudo-Hermitian, then the 3-momentum operator P i is also η-pseudo-Hermitian, i.e., P i † = η P i η−1 for i = 1, 2, 3. Hence, it is Hermitian if and only if it commutes with η.Using this, we rearrange the commutator: We see that the 'dual' field defined by Eq. (4.9) will evolve with the same set of generators of spacetime translations as the quantum field ψ: In particular, the 'dual' field ψ † evolves with the same Hamiltonian Ĥ: Thus, a Lagrangian composed of the "dual" field ψ † and the quantum field ψ will yield consistent equations of motion, and the bilinear terms ψ † ψ will be invariant under spacetime translations.

Proper Lorentz transformations
Consider a proper Lorentz transformation x → x ′ = Λ • x.Unlike spacetime translations, proper Lorentz transformations mix the components of multicomponent fields (both classical and quantum).Hence, the corresponding transformation of an n-component quantum field ψa according to Eq. (4.5) is Here, Û (Λ) = Û (0, Λ), for some proper Lorentz transformation Λ ∈ SO(1, 3) ↑ .The mixing of field components is described by an n × n matrix D(Λ), given by an n-dimensional matrix representation of the proper Lorentz group SO(1, 3) ↑ .
The representations acting on a quantum field are the Fock-space, finite-dimensional and coordinate representations.All of these can be expanded in terms of their generators: where Ĵµν are the six generators of rotations and boosts in the Fock-space representation, Expanding each side in Eq (4.20) gives us the relationship between the generators of proper Lorentz transformations SO(1, 3) ↑ for the Fock-space, finite-dimensional and coordinate representations: Taking a Hermitian conjugate of the above equation, we find that the conjugate field ψ † evolves with Hermitian conjugates of these generators: For a unitary finite-dimensional representation (D, C n ), the generators M µν † = M µν are Hermitian.This would imply that the Fock-space generators Ĵµν † = Ĵµν are also Hermitian, meaning the Fock-space representation ( Û, F ) is unitary.However, according to Sec. 3, if the Hamiltonian operator Ĥ is η-pseudo-Hermitian, i.e., Ĥ † = η Ĥ η−1 , then other generators of the proper Lorentz transformations are also η-pseudo-Hermitian: Ĵµν † = η Ĵµν η−1 .Hence, the Fock-space generators Ĵµν are Hermitian if and only if they commute with η.Thus, if the Fock-space generators are not Hermitian, neither are the generators of finite-dimensional representations and vice versa.
Using the pseudo-Hermiticity of the Fock-space generators, we can rewrite Eq. (4.22) as We can use a biorthonormal basis [3,5] to construct a Hermitian n × n matrix π : C n → C n such that the generators M µν † = πM µν π −1 are π-pseudo-Hermitian.This means that the finite-dimensional representation D † πD = π is π-pseudo-unitary.In the next section, we will show how to construct pseudo-Hermitian finite-dimensional representations from the representation theory of SO(1, 3) ↑ .
Further, applying the pseudo-Hermiticity of matrix generators, we find the field which evolves with the same set of generators as the field operator ψ: This is exactly the "dual" field defined in Eq. (4.9).Thus, the "dual" field ψ † evolves with the same generators of proper Lorentz transformations as ψ in Eq. (4.21): Hence, bilinear operators of the form ψ † ψ will be invariant under the proper Lorentz transformations.

PSEUDO-HERMITIAN FINITE DIMENSIONAL REPRESENTATIONS
If we consider a multicomponent quantum field ψa (e.g., a fermion field), its components We use these pseudo-Hermitian representations to construct the dual quantum field operator for the simplest trivial representation (0, 0) of the spin-0 scalar field and a more complicated case of the smallest nontrivial representations ( 1 2 , 0) and (0, 1 2 ), which represent spin- 1  2 left-and right-handed spinors, respectively.Finally, we show how to construct pseudo-Hermitian representations and the dual quantum field for Dirac fermions, which are the direct sum (0, 1  2 ) ⊕ ( 1 2 , 0) of right-and left-handed spinors.The complex special linear group SL(2, C) is the group of complex 2 × 2 matrices with a unit determinant: It has the Lie algebra sl(2, C), which is a complex vector space of traceless complex 2 × 2 matrices: The Lie algebra sl(2, C) ∼ = su(2) C is isomorphic to the complexified Lie algebra su(2) C of the special unitary group SU(2).In simpler terms, sl(2, C) is the vector space su(2) spanned over complex numbers instead of real numbers.Thus, in the literature, it is common to write the Lorentz Lie algebra as a direct sum su(2) ⊕ su(2).
The generators of sl(2, C) are J a = σa 2 , where σ a are the Pauli matrices.The Lie brackets for this basis are (5.4) The Lie algebra of sl(2, C) possesses the structure of "ladder operators", which we obtain by defining a new basis: (5.5) Here, J ± are the raising and lowering operators with the Lie brackets: Now, let (ρ, V) be a representation of sl(2, C) on some finite-dimensional vector space V: (5.7) The dimension of a representation is defined as the dimension of the vector space dim(ρ) = dim(V).It is also common to refer to the vector space V as the representation space of a representation ρ.In the context of "standard" quantum field theory, we assume representations of sl(2, C) to be Hermitian: Note that the representation ρ(X) † = ρ(X) on a general element X ∈ sl(2, C) is not, in general, Hermitian, as sl(2, C) is a complex vector space.Hence, the definition of a Hermitian representation is basis dependent.
However, in general, the representations of sl(2, C) need not be Hermitian.Given that ρ(J 3 ) is finite dimensional and diagonalizable, we use the ladder operators (5.5) to obtain all irreducible representations of sl(2, C) [57].An irreducible representation is a representation that cannot be broken down into smaller subset representations while preserving properties of the Lie algebra.
All of the irreducible representations (ρ j , V j ) of sl(2, C) can be classified by an integer or half-integer number j ∈ {0, 1 2 , 1, ...} = N/2.The dimension of each representation is dim(ρ j ) = dim(V j ) = 2j + 1 and the representation space Let us consider a representation (ρ j , V j ) and let ρ j (J 3 ) be non-Hermitian.As we assumed ρ(J 3 ) to be diagonalizable, we can use its eigenvectors to construct a biorthonormal basis [3,5], from which we obtain a Hermitian matrix π : V j → V j such that ρ j (J 3 ) † = πρ j (J 3 )π −1 is π-pseudo-Hermitian.As representations preserve the Lie bracket: we have that the other generators are also π-pseudo-Hermitian: (5.10) We call this representation a π-pseudo-Hermitian representation of sl(2, C).Note that, for a general element X ∈ sl(2, C) the representation ρ j (X) † = πρ j (X)π −1 is not π-pseudo-Hermitian.Thus, the definition of a pseudo-Hermitian representation is basis dependent.

Irreducible representations of the proper Lorentz group
Having classified all irreducible representations of sl(2, C), we can obtain all irreducible representations of the proper Lorentz group SO(1, 3) ↑ .This follows from the fact that the complexified Lorentz Lie algebra so(1, The Lorentz Lie algebra so(1, 3) is a real vector space of traceless 4 × 4 matrices: It has six generators: three rotation generators R a and three boost generators B a , which form a basis of so(1, 3).The complexified Lorentz algebra is just the complex vector space: so(1, 3) C = span C {R a , B a , a = 1, 2, 3} . (5.15) Most literature on the representation theory does not distinguish between the Lie algebra and its complexification.This is because the majority of the results from the complexified Lie algebra can be directly applied to the original Lie algebra by taking the vector space to span over the real numbers instead of the complex numbers.
Let us define a new basis for the Lorentz Lie algebra so(1, 3) C : (5.16) The Lie bracket for this basis is exactly that of sl(2, C) ⊕ sl(2, C): Hence, the Lorentz Lie algebra so(1, 3) C is isomorphic to the direct sum sl(2, C) ⊕ sl(2, C).Now consider two irreducible representations (ρ j , V j ) and (ρ k , V k ) of sl(2, C).We can construct a corresponding irreducible representation of sl(2, C) ⊕ sl(2, C): All irreducible representations of sl(2, C) ⊕ sl(2, C) are obtained from (κ jk , V jk ), where the representation space of κ jk is a tensor product ) are classified by two integer or half-integer numbers (j, k), where The relationship between the elements of the complexified Lorentz Lie algebra so(1, 3) C and sl(2, C) ⊕ sl(2, C) is given explicitly through an isomorphism: κ jk (T a ) ∼ = κ jk (J a , 0) = ρ j (J a ) ⊗ ½ 2k+1 and κ jk (K a ) ∼ = κ jk (0, J a ) = ½ 2j+1 ⊗ ρ k (J a ) .(5.19)This allows us to obtain all irreducible representations (κ jk , V jk ) of the Lorentz Lie algebra so(1, 3) by going back to the basis of rotation and boost generators: (5.20) Thus, the (j, k) representation for any element M = α a R a + β a B a ∈ so(1, 3) with α a , β a ∈ R of the Lorentz Lie algebra so(1, 3) is given by where we defined the coefficient ) is an element of the Lie algebra sl(2, C).Hence, the Eq.(5.21) explicitly maps an element of sl(2, C) to an element of the Lorentz Lie algebra so(1, 3).
We use Eq.(4.9) to define the dual scalar field operator: For a single-component scalar field, π ∈ R can be any real number.However, this is not true if we consider multicomponent scalar/pseudoscalar fields.In the case of an n-component Returning to the construction of pseudo-Hermitian quantum field theories, we require the Lagrangian operator L (x) to be 1.Invariant under proper Poincaré transformations: In the case of noninteracting theories, the kinetic and diagonal mass terms are of the form φ † (x) φ(x), which are both Poincaré invariant and η-pseudo-Hermitian.More generally, however, we can include a mass mixing.In this case, there will be a nondiagonal mass term of the from φ † (x)B φ(x), where B is some n × n nondiagonal matrix.Under the action of the proper Poincaré group, we have (5.34) For n-component scalars/pseudoscalars, the n-dimensional proper Lorentz transformation is just the n × n identity matrix, i.e., (5.35) Hence, the mass mixing terms are Poincaré invariant.However, the mixing term is η-pseudo-Hermitian: if and only if B † = ΠBΠ −1 is Π-pseudo-Hermitian.This places restrictions on the matrices B and Π.
In Sec. 6, we study an example of this type of Lagrangian.We consider a 2-component scalar field with mass mixing term of the form L (x) = φ †1 (x) φ2 (x) − φ †2 (x) φ1 (x).In matrix form this term is (5.37) However, the matrix B is Π-pseudo-Hermitian if and only if π 2 = −π 1 , which restricts Π to the form: (5.38) In the example, this is a parity matrix that reflects the intrinsic parities of a scalar component

Two-component spinors
Unlike for the case of scalar fields, the representations ( 1 2 , 0) and (0, 1 2 ), which correspond to left-and right-handed spinors, are nontrivial.The dual left-and right-handed field operators are obtained via Eq.(4.9).However, introduction of the dual field alone is insufficient to formulate a Lorentz-invariant spinor Lagrangian.This is due to the dependence of the kinetic terms on the Pauli matrices, which do not transform in the same representation of SO(1, 3) ↑ as the quantum field operators.Our goal is to write down the spinor Lagrangian which transforms in the same representation of the proper Poincaré group ISO(1, 3) ↑ .Having the spinor Lagrangian in hand, we obtain a pseudo-Hermitian representation for Dirac fermions (0, 1  2 ) ⊕ ( 1 2 , 0).However, a comprehensive study of a specific example of a non-Hermitian fermionic quantum field theory is beyond the scope of this work.

If the representation ρ1
2 , C 2 ) is non-Hermitian, we have seen that we can use a biorthonor-mal basis [3,5] to construct a 2 × 2 Hermitian matrix π : C 2 → C 2 such that the representation is π-pseudo-Hermitian: This leads to Lorentz Lie algebra representations being anti-π-pseudo-Hermitian under an interchange of left-handed ( 1 2 , 0) and right-handed (0, 1 2 ) representations: Hence, the representations of the proper Lorentz group are π-pseudo-unitary under the interchange of left-handed and right-handed representations: (5.44) As shown in Sec. 4 we can write down the dual field operator, provided the finitedimensional representations are pseudounitary.Thus, we define the dual left-and right handed quantum field operators using Eq.(4.9): where η : x → x η is the coordinate transformation of x ∈ R 1,3 under η.
Indeed, we can check that, under the proper Lorentz group SO(1, 3) ↑ , the left-handed dual field ψL transforms under the dual representation of the right-handed spinor ψR and the right-handed dual field ψR transforms under the dual representation of the left-handed (5.46) Hence, the typical mass terms in the non-Hermitian spinor Lagrangian ψ † L (x) ψR (x) and ψ † R (x) ψL (x) (5.47) are both η-pseudo-Hermitian and invariant under proper Poincaré transformations.Moreover, in the Hermitian limit, where η = 1 and π = ½ 2 , we recover the mass terms of the Hermitian spinor Lagrangian: ψ † L (x) ψR (x) and ψ † R (x) ψL (x) . (5.48) However, even with the dual spinor fields, the kinetic terms will not be Lorentz invariant.This is because the kinetic terms involve Minkowski 4-vectors in the Hermitian representation of sl(2, C).Consider the kinetic terms in the Hermitian spinor Lagrangian [56] ψ † Here, [∂] is a map from Minkowski 4-vectors to 2 × 2 matrices defined as (for more details see Sec. 7 in Ref. [56]): (5.50) We have defined ∂ µ P = (∂ 0 , −∂ i ) to be the parity transformation of the 4-vector ∂ µ , and the four-Pauli matrices are σ µ = (½ 2 , σ i ) and σµ = (½ 2 , −σ i ).
The map [ ] gives an explicit relationship between the proper Lorentz group SO(1, 3) ↑ and the complex special linear group SL(2, C): (5.51) Here, A ∈ SL(2, C) is a 2×2 matrix in the complex special linear group, see Eq. (5.2).Indeed, we can write any element of SL(2, C) in terms of its Lie algebra generators X ∈ sl(2, C): where we have used Eq.(5.41), which relates the j = 1 2 representation of sl(2, C) to the (0, 1 2 ) representation of so (1,3).We also note that (ρ (J a ) = J a .Thus, we also denote (D H 0 1 2 , C n ) to be the unitary representation of SO(1, 3) ↑ with respect to the interchange of (0, 1  2 ) and ( 1 2 , 0), such that D H Hence, the derivative matrices [∂] and [∂] P transform in the unitary representation of the proper Lorentz group, while the spinor fields transform in the pseudounitary representation.Due to this discrepancy, the kinetic terms will not be Lorentz invariant: However, the π-pseudo-Hermitian representation (ρ1 , C 2 ) by a similarity transformation (5.11): Thus, the π-pseudo-unitary representation of the proper Lorentz group SO(1, 3) ↑ is related to the unitary one via a similarity transformation: Using this property, we define a new map from Minkowski 4-vectors to 2 × 2 matrices: We can check that this map indeed transforms in the π-pseudo-unitary representation of the proper Lorentz group: (5.60) Hence, the kinetic terms in a non-Hermitian spinor Lagrangian, which are both η-pseudo-Hermitian and Lorentz invariant, will be of the form ψ † L (x){∂} ψL (x) and ψ † R (x){∂} P ψR (x) .
(5.61) Thus, combining the mass terms in Eq. (5.47) with the kinetic terms in Eq. ( 5.61) we can write down the noninteracting part of the non-Hermitian spinor Lagrangian as This Lagrangian is both η-pseudo-Hermitian and invariant under proper Poincaré transformations ISO(1, 3) ↑ , as required.Moreover, in the Hermitian limit, it reduces to the free Hermitian spinor Lagrangian.

Four-component fermions
Having found the pseudo-Hermitian representations of right-and left-handed spinors, we are able to obtain the corresponding pseudo-Hermitian representation of Dirac fermions.The Dirac fermion is a 4-component field, which transforms under the direct sum (0, 1 2 ) ⊕ ( 1 2 , 0) of right-handed and left-handed spinor representations of the proper Lorentz group [56].
We begin by defining a new representation on the Lorentz Lie algebra so(1, 3) by taking a direct sum of the right-and the left-handed spinor representations s := κ 0 1 2 ⊕ κ1 2 0 over the direct sum of their representation spaces (5.63) As s(M) is a direct sum, it can be written as 4 × 4 block-diagonal matrix acting on a 4-component quantum field composed of 2-component spinors: Here, we used Eq.(5.41), which relates the j = 1 2 representation of sl(2, C) to the (0, 1 2 ) and ( 12 , 0) representations of the Lorentz algebra so(1, 3).A more familiar basis is defined by acting on the rotation and boost generators: It obeys the Lorentz Lie bracket in Eq. (3.7) and can be written as the commutator of the gamma matrices: However, here, the gamma matrices γa (a = 1, 2, 3) are not in the Hermitian representation of sl(2, C).Instead, they are in the (ρ1 This is due to the pseudo-Hermitian nature of the representation (s, C 4 ) defined in Eq. (5.64).
Hence, in the Weyl (chiral) basis, the gamma matrices are given by γµ = (5.68) The π-pseudo-Hermitian representation will be related to the Hermitian representation via a similarity transformation: (5.69) Thus, the gamma matrix γ0 is the same for Hermitian and π-pseudo-Hermitian representation.We can also check that the gamma matrices γµ obey the Clifford algebra where g µν = diag(1, −1, −1, −1) is the Minkowski metric.
The Poincaré invariant, pseudo-Hermitian spinor Lagrangian in Eq. ( 5.62) can be written in the 4-component Weyl (chiral) basis as we can write the Lagrangian in terms of gamma matrices: (5.73) As we see, while the mass term is of the usual form of the Dirac Lagrangian, the kinetic term has picked up a matrix V V † π.Here, π is the 2 ×2 matrix constructed using a biorthonormal basis for sl(2, C) and V is the matrix diagonalizing the representation of ρ1 (J 3 ), C] = 0, which we find from the biorthonormal basis for our system [3,5].
We call this matrix C due to convention across the literature of non-Hermitian quantum mechanics [7]; however, it has nothing to do with charge conjugation.

EXAMPLE: P T -SYMMETRIC SCALAR FIELD THEORY
Before concluding this work, we consider a concrete example of a P T -symmetric scalar field theory.The theory is composed of two complex scalar fields with a non-Hermitian mass mixing matrix.This archetypal model, introduced in Ref. [49], has been considered in a number of existing works (see, e.g., Refs.[37,51]).In this section, we show how it can be formulated consistently based on our preceding discussions.

Naive Lagrangian
Following the strategy of appending non-Hermitian terms to an otherwise Hermitian Lagrangian, it is tempting to write the Lagrangian density, as was done in Ref. [49]: Here, Φ =  is a 2-component complex scalar field composed of a scalar component Φ 1 and a pseudoscalar component Φ 2 .The Lagrangian's non-Hermiticity arises from the presence of a non-Hermitian mass matrix: Naively, we might assume the standard parity P and time-reversal T transformation properties of scalar and pseudoscalar fields as follows: Here, P is the "parity" matrix: reflecting the intrinsic parity +1 of the scalar and −1 of the pseudoscalar field.Furthermore, we observe that the mass matrix is P -pseudo-Hermitian, i.e., M 2 † = P M 2 P −1 , with respect to the parity matrix P .Indeed, the classical Lagrangian is P T symmetric under the standard parity and time-reversal transformations in Eq. ( 6.3): Let us now consider the corresponding quantum Lagrangian operator L : F → F , acting on the Fock space F : where P : F → F is the parity operator and T : F → F is the time-reversal operator, both acting on Fock space F .Indeed, the quantum Lagrangian is P-pseudo-Hermitian under the standard parity transformation in Eq. (6.8): L (x P ) † = P L (x) P−1 .(6.10) We can check that the corresponding Hamiltonian operator: is also P-pseudo-Hermitian, i.e., Ĥ † = P Ĥ P−1 .
However, for a non-Hermitian Hamiltonian, the standard scalar/pseudoscalar parity transformation is incorrect.This can be seen implicitly by parity-transforming Hamilton's equation: P : [ φ(x), Ĥ] = i∂ 0 φ(x) → [ P φ(x) P−1 , P Ĥ P−1 ] = i P∂ 0 φ(x) P−1 =⇒ [P φ(x P ), Ĥ † ] = iP ∂ 0 φ(x P ) . (6.12) Multiplying the left-hand side by P −1 , and since the Hamiltonian Ĥ is independent of x, we can relabel x P to x, giving By virtue of the non-Hermiticity of the Hamiltonian, Ĥ † = Ĥ, we see that the field operator cannot transform under parity in the usual way.
We can also see this explicitly by acting with parity on the momentum decomposition of the field φ.Varying the Lagrangian with respect to φ † , gives the Euler-Lagrange equation for φ: The general solution to this equation can be written in momentum space as where E p = p 2 ½ 2 + M 2 is the 2 × 2 energy-momentum matrix, which is nondiagonal and non-Hermitian due to the non-Hermiticity of M 2 .Here, â and ĉ † are the 2-component annihilation and creation operators.
Hence, if we act with parity on the momentum decomposition of the field φ in Eq. (6.15), we find that its parity transformation is indeed nontrivial:

.19)
Now consider the Euler-Lagrange equations of motion.If we vary the Lagrangian with φ, then we get the Euler-Lagrange equation for φ † : However, it is not the Hermitian conjugate of the Euler-Lagrange equation for φ given in Eq. (6.14) by virtue of the non-Hermiticity of the mass mixing matrix, These issues are a direct consequence of the fields φ and φ † being governed by two different Hamiltonians, Ĥ and Ĥ † , respectively.As described in the preceding section, we fix this by determining the dual field φ † and using it to construct a consistent Lagrangian operator.

Poincaré-invariant Lagrangian
Our prescription for a self-consistent non-Hermitian quantum field theory suggests that we should begin with a Lagrangian built from the field operator φ and its dual φ † : Here, we have the same non-Hermitian mass mixing matrix as in Eq. (6.2).The dual field for an n-component scalar field is given by Eq. (5.33).In this example, n = 2, and the dual At this point, the operator η and the matrix Π are yet to be determined.
By construction, this Lagrangian is η-pseudo-Hermitian and the mass matrix is Π-pseudo-Hermitian: In the previous subsection, we found that the "naive" Lagrangian and the Hamiltonian operator are P-pseudo-Hermitian with respect to the parity operator P and the mass matrix is P -pseudo-Hermitian with respect to the parity matrix P given in Eq. (6.5).Hence, this suggests natural choices for the operator η to be the parity operator P and Π to be the parity matrix P .
Indeed, if we vary the Lagrangian in Eq. (6.22) with φ, we get the Euler-Lagrange equation for the dual field φ † : In momentum space, it has a solution φ † This is exactly the parity transformation of φ found in Eq. (6.19), which can be rewritten as φ † (x) = P−1 φ † (x P ) P P .(6.27) We note that this matches the construction proposed in Ref. [51].Given the dual field above, the Lagrangian in Eq. (6.22) operator is indeed P-pseudo-Hermitian: L † (x P ) = P L (x) P−1 .(6.28) Having chosen η to be the parity operator P, a natural choice for the inner product yielding real eigenspectra would be However, if we wish to keep the 3-momentum operator Hermitian, it will not commute with the parity operator, and the theory with η = P will then not be Poincaré invariant.It turns out that such a symmetry exists in any discrete pseudo-Hermitian system, provided it is diagonalizable.In our example, this system is the P -pseudo-Hermitian mass mixing matrix.It is diagonalizable with an eigenspectrum M 2 |ψ 1,2 = m2 1,2 |ψ 1,2 , where m1 is the "−" and m2 is the "+" root, respectively [49]: We have chosen m 2 1 > m 2 2 so that ν > 0. The mass eigenvalues m1,2 ∈ R are real for ν ≤ 1 and the mass eigenstates will be "pseudo" orthonormal with respect to the indefinite inner product Here, the normalized mass eigenstates are [49] |ψ ) As mentioned, this system will have an additional discrete symmetry [M 2 , C] = 0 with C 2 = ½ 2 .While this symmetry exists for both real and and complex mass eigenvalues, in the case of real eigenspectrum, it will be related to the parity matrix via [51] As the pseudo-Hermiticity of our Lagrangian arises entirely from the P -pseudo-Hermitian mass mixing matrix, the P-pseudo-Hermitian Hamiltonian operator Ĥ is also diagonalizable.
Thus, it also contains a discrete symmetry [ Ĥ, Ĉ] = 0 with Ĉ2 = 1.A difficult, but possible, way to find this symmetry is to take an infinite sum of multiparticle eigenstates of Ĥ similar to Eq. (6.35).However, a more straightforward approach would be to notice that the C matrix is just the parity matrix P in the basis where M 2 is nondiagonal [37]: In the eigenbasis of M 2 , C is just the parity matrix P [37]: For a quantum field theory to be Poincaré invariant, the Lagrangian should transform as a single object under the proper Poincaré group.However, we observe that the field operator ψ and its Hermitian conjugate ψ † transform in two different representations, leading to the lack of Poincaré invariance in both the Lagrangian and the theory as a whole.This prompts us to search for a new conjugate field, which we refer to as the "dual" field.This dual field operator, denoted by ψ † , transforms in the dual representation of ψ.In Sec. 4, we found it to have a general form, which holds for fields of any spin j: Herein, the operator η is such that the Hamiltonian is η-pseudo-Hermitian, x η is the coordinate transformation with respect to the operator η (e.g., parity x P ), and the matrix π is such that the finite-dimensional Lorentz representations are π-pseudo-Hermitian.Equation (7.1) is the central result of this work, and we have demonstrated its importance using a simple model of two complex scalar fields with non-Hermitian mass mixing, where the Hamiltonian is pseudo-Hermitian with respect to the parity operator P.
With this result, we have established a fundamental framework for developing selfconsistent non-Hermitian quantum field theories.By laying down these foundational prin-ciples, we pave the way for future applications, in particular in the challenging context of interacting non-Hermitian quantum field theories.
Note added.After the preprint of this work was posted, Ref. [58] appeared, which brought the earlier work [59] to our attention.The construction of the dual field for the second-order fermionic theory described in these works, while not motivated by Poincaré invariance, bears some similarities with the construction described here, and we leave a detailed application of the present approach to this theory for future work.
Hermitian quantum mechanics, we choose a new inner product •|• η := •|η• , instead of the Dirac inner product •|• , as the former yields real expectation values of the Hamiltonian.
using the Dirac inner product •|• : M a αβ (x) = α| ψa (x)|β (3.9) for given Fock-space states |α , |β ∈ F .However, in the case of a non-Hermitian Hamiltonian, the energy eigenstates are not orthogonal with respect to the Dirac inner product •|• .Instead, if the Hamiltonian is ηpseudo-Hermitian, the eigenstates with real eigenvalues become orthogonal with respect to the inner product •|η• .As a result, the matrix elements are defined in terms of this inner product as follows: Ma αβ (x) := α| ψa (x)β η = α|η ψa (x)|β .(3.10) µν are the n × n matrix generators of rotations and boosts in the n-dimensional matrix representation, and m µν = x µ ∂ ν − x ν ∂ µ are the generators of rotations and boosts in the coordinate representation.
will mix under proper Lorentz transformations SO(1, 3) ↑ .The action of the proper Lorentz group on an n-component field is given by an n-dimensional matrix representation (D, C n ):D(Λ) : C n −→ C n ψa −→ D a b (Λ) ψb .(5.1) Here, D(Λ) is an n×n matrix determined by a proper Lorentz transformation Λ ∈ SO(1, 3) ↑ .In Eq. (4.22), we noted that if the generators of the Fock-space representation are non-Hermitian Ĵ †µν = Ĵµν , then the generators of finite-dimensional representations are not, in general, Hermitian either M µν † = M µν .Nonetheless, if we can find a Hermitian n×n matrix π : C n → C n , such that these generators are π-pseudo-Hermitian M µν † = πM µν π −1 , then we can derive the dual quantum field ψ † , which transforms in the dual representation of the quantum field ψ.In this section, we explore how the pseudo-Hermitian finite-dimensional representations naturally emerge in the representation theory of the proper Lorentz group SO(1, 3) ↑ .The (complexified) Lorentz Lie algebra so(1,3) C ∼ = sl(2, C) ⊕ sl(2, C) is a direct sum of two Lie algebras of the complex special linear group SL(2, C)[56].This allows us to obtain all finitedimensional representations, both Hermitian and non-Hermitian, of the Lorentz Lie algebra so(1, 3) from the finite-dimensional representations of sl(2, C)[57].We then exponentiate these to obtain all finite-dimensional representations of the proper Lorentz group SO(1, 3) ↑ .

φ1 and a pseudoscalar component φ2 in the 2 -
component scalar field φ =

2 (J 3 ). Hence, if the eigenstates of ρ1 2 (J 3 ) 2 (
have a positive norm with respect to the inner product •|π• , thenV V † π = ½ 2 isjust the identity.On the other hand, if the eigenstates have indefinite norm with respect to •|π• , then V V † π = C is the C matrix, i.e., the discrete symmetry of ρ1

6. 3
. C operator In P T -symmetric quantum mechanics, where the Hamiltonian is non-Hermitian, but P Tsymmetric, i.e., [H, P T ] = 0, we discover an additional discrete symmetry, denoted as C such that [H, C] = 0 and C 2 = ½ [7].Despite its name, this symmetry is unrelated to charge conjugation.In the P T -unbroken phase, where the Hamiltonian has real eigenvalues, we use this symmetry to construct a positive-definite metric P C.This metric ensures that the eigenstates of the Hamiltonian are orthogonal with respect to the positive-definite inner product •|P C• .

. 38 )
Hence, the Ĉ operator is just the parity operator P in the eigenbasis of the Hamiltonian operator Ĥ:Ĥ = R D R−1 −→ Ĉ = R P R−1 ,(6.39)whereD : F → F is the diagonalized Hamiltonian operator.Since, the diagonalized mass mixing matrix commutes with parity, i.e., [D, P ] = 0, so does the diagonalized Hamiltonian operator [ D, P] = 0.It follows that Ĉ acts on the annihilation/ creation operators just as the parity operator P, but with C instead of P as given in Eq. (6.16):Ĉâ(0, p) Ĉ−1 = Câ(0, − p) , Ĉĉ † (0, p) Ĉ−1 = Cĉ † (0, − p) , Ĉâ † (0, p) Ĉ−1 = â † (0, − p)C , Ĉĉ(0, p) Ĉ−1 = ĉ(0, − p)C .(6.40)Indeed, if we expand the Hamiltonian in terms of creation and annihilation operators, then we find that it commutes with Ĉ:p) + ĉ †i (0, p)E ij p ĉj (0, p) =⇒ Ĉ Ĥ Ĉ−1 = Ĥ .(6.41)This is due to the matrix C commuting with E p = p 2 ½ 2 + M 2 .Also, the operator Ĉ will square to unity:Ĉ2 â(0, p) Ĉ−2 = â(0, p) =⇒ Ĉ2 = 1 .(6.42)The easiest way to find the operators P and Ĉ is to use the Baker-â p := â(0, p) and ĉ p := ĉ(0, p) are creation/annihilation operators at t = 0.The important property of the Ĉ operator is that it flips the sign of the 3-momentum operator:Ĉ ˆ P Ĉ| p, k = Ĉ ˆ P C jk | − p, j = (− p)C jk Ĉ| − p, j = (− p)C jk C ij | p, i = − p| p,k , the previous subsection, the parity operator also flips the sign of the 3-momentum operator.Thus, if we wish to keep the 3-momentum operator Hermitian, the theory with η = P will not be Poincaré invariant.However, the existence of the discrete symmetry Ĉ, allows us to construct a new positive-definite metric operator P Ĉ, such that the eigenstates of the Hamiltonian operator are orthogonal with respect to the positive-definite inner product •|• P Ĉ = •| P Ĉ• and have positive norms.But more importantly, if we choose η = P Ĉ it commutes with the 3-momentum operator: ( P Ĉ) P i ( P Ĉ) −1 = P i .(6.47) Hence, the 3-momentum can simultaneously be Hermitian and P Ĉ-pseudo-Hermitian, leaving the theory Poincaré invariant.Therefore, in Eq. (5.33), we take η = P Ĉ, Π = P C and the coordinates remain unchanged under this transformation, i.e., x P C = x, so that the dual field is φ † (x) = ( P Ĉ) −1 φ † (x)( P Ĉ)P C .(6.48)Both the Lagrangian and the Hamiltonian are P Ĉ-pseudo-Hermitian: L † (x) = ( P Ĉ) L (x)( P Ĉ) −1 and Ĥ † = ( P Ĉ) Ĥ( P Ĉ) −1 .(6.49)The inner product •| P Ĉ• yields a theory that is invariant under proper Poincaré transfor-mations.7.CONCLUSIONIn this work, we have considered a generalization of the Poincaré group when the generator of time translations, i.e., the Hamiltonian operator, is non-Hermitian.The time evolution of the quantum field operator ψ and its Hermitian conjugate ψ † are then governed by distinctHamiltonians, Ĥ and Ĥ † , respectively.As a consequence, a theory built from ψ and ψ † exhibits inconsistent equations of motion and lacks Poincaré invariance.We have also shown that when the Hamiltonian is non-Hermitian, its non-Hermiticity extends to the other group generators, such as space translations, rotations, and boosts.Specifically, if the Hamiltonian is diagonalizable, we can always find an operator η such that Ĥ † = η Ĥ η−1 , which implies that the Hamiltonian is η-pseudo-Hermitian.The pseudo-Hermiticity of the Hamiltonian leads to pseudo-Hermiticity of the other generators in the Poincaré algebra.These generators become Hermitian if and only if they commute with η.
As we established in Eq.(3.6), to preserve Poincaré invariance, all of the generators must be η-pseudo-Hermitian.Hence, they are Hermitian if and only if they commute with η.