Beyond PT -symmetry: towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians

SILVA, L. F. A. Beyond PT -symmetry: towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians. 2022. 52p. Dissertation (Master in Science) Instituto de Física de São Carlos, Universidade de São Paulo, São Carlos, 2022. A new chapter in quantum mechanics has opened over the past 20 years with the fact that time-independent (TI) non-Hermitian Hamiltonians have a real spectrum and unitary time evolution when they exhibit an unbroken PT -symmetry and satisfy the pseudo-Hermiticity relation. In this Master’s thesis, we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for non-Hermitian Hamiltonian H(t) and metric operator ρ(t) in a TD scenario. Assuming H(t) to be simultaneously ρ(t)-pseudo-Hermitian and Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t) = Ξ−1(t)ρ(t), which retrieves an important result obtained by Mostafazadeh [J. Math, Phys. 43, 3944 (2002)] for the time-independent (TI) scenario. We apply our method for the derivation of the symmetry associated with TD non-Hermitian linear and quadratic Hamiltonians. In the TI scenario, we retrieve the well-known BenderBerry-Mandilara result for the symmetry operator: I2k = 1 with k odd [J. Phys. A 35, L467 (2002)]. The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians. From this relation, the TD metric is automatically derived from the TD symmetry of the problem. Our results reinforce the prospects of going beyond PT -symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising.


INTRODUCTION
In the last two decades since the seminal contributions of Bender and Boettcher (1) and Mostafazadeh (2)(3)(4), PT -symmetric Hamiltonians -invariant under parity (P) and time-reversal (T ) symmetry-have been investigated in practically all domains of physics, from low to high energies, revealing to be an increasingly autonomous and thoughtprovoking field. The PT -symmetry condition, weaker than Hermiticity, greatly expands the possibility of the Hamiltonian description of physical systems with real eigenvalues (1) and conservation of the norm (2), which is one of the strong calls for the field. And much has been done recently, such as the experimental realizations of Floquet PT -symmetric systems (5) and PT -symmetric flat bands (6), besides enhanced sensing based on PTsymmetric electronic circuits (7) and PT -symmetric topological edge-gain effect. (8) The linear response theory for a pseudo-Hermitian system-reservoir interaction was developed (9), as well as a protocol to approach non-Hermitian non-commutative quantum mechanics. (10) In this work, we propose a method to derive a general time-dependent (TD) continuous symmetry operator for a TD non-Hermitian Hamiltonian. This will be done in the broader scenario of non-autonomous Hamiltonians, and for this reason, we revisit the TD non-Hermitian Hamiltonians of a cavity field under linear (11) and parametric (12) amplifications. These Hamiltonians have been considered for approaching TD non-Hermitian Hamiltonians under TD Dyson maps, thus extending the method proposed by Mostafazadeh. (2) This extension was also undertaken in Refs. (13)(14) The consequences of allowing the Hilbert space to have a TD metric were posed in Ref. (15), while a treatment of a TD non-Hermitian Hamiltonian through a TI metric operator was done in Ref. (16) The investigation of TD metric operators for the treatment of TD non-Hermitian Hamiltonians was undertaken in Ref. (17) Many interesting contributions to the subject of TD non-Hermitian Hamiltonians have been presented. (18)(19)(20)(21)(22)(23) We mention, in particular, a method that adds to the achievements of Ref. (11), enabling the unitarity of the time-evolution and the observability of non-Hermitian Hamiltonians through particular TD Dyson maps that define time-independent (TI) metric operators (24). Moreover, on our understanding, we stress the introduction of the all-creation and all-annihilation TD pseudo-Hermitian bosonic Hamiltonians (25,26), able to generate an infinite squeezing degree at a finite time. A TD pseudo-Hermitian Hamiltonian for a cavity mode with complex frequency is also able to generate an infinite squeezing at a finite time. (27) Finally, we mention the enhancement of photon creation through the pseudo-Hermitian dynamical Casimir effect. (28) Our subject, pseudo-Hermiticity beyond PT -symmetry, is in fact at the foundations of pseudo-Hermitian quantum mechanics. A theorem by Mostafazadeh (4) -formulated for TI non-Hermitian Hamiltonians, symmetries and metric operators-asserts that a

diagonalizable (non-Hermitian) Hamiltonian is pseudo-Hermitian if and only if it has
an antilinear symmetry, i.e., a symmetry generated by an invertible antilinear operator. Moreover, Bender, Berry and Mandilara (29) have shown that a non-Hermitian Hamiltonian presents a real spectrum not only when invariant under PT -symmetry, but also under any antiunitary operator I satisfying I 2k = 1 with k odd. We also mention the demonstration that supersymmetry gives rise to non-PT -symmetric families of complex potentials with entirely real spectra (30), and also the proposition of chiral metamaterials with PT symmetry and beyond. (31) Despite the generality of the pseudo-Hermitian requirement, the particular case of PT -symmetric Hamiltonians gained prominence due to Bender and Boettcher's seminal work (1) and certainly due to the strong physical appeal of parity and time-reversal invariance.
Our objective is precisely to explore more general symmetries than PT by starting from the general scenario of TD non-Hermitian Hamiltonians. The method we propose for the derivation of TD symmetries for TD non-Hermitian Hamiltonians applies indistinctly to linear or antilinear, unitary or nonunitary symmetries. However, we assume the symmetry to be an antilinear operator aiming to retrieve the results by Mostafazadeh (4) and Bender-Berry-Mandilara (29) in the particular case of a TI scenario, i.e., TI non-Hermitian Hamiltonians, metrics and symmetries. The above-mentioned theorem by Mostafazadeh (4) is retrieved when considering anantilinear symmetry while the result by Bender-Berry-Mandilara is retrieved when considering a unitary antilinear or antiunitary symmetry.
After presenting our method to derive the symmetry operator, we then apply it for TD non-Hermitian linear and quadratic Hamiltonian, modeling a cavity field under linear and parametric amplifications. As expected, we have derived TD continuous antilinear symmetries far more complex than the spatial reflection and time reversal. These TD symmetries are then particularized to the equivalent TI non-Hermitian linear and quadratic Hamiltonians and their PT -symmetric restrictions. Then, in the TI scenario, the results in Refs. (2,3,29) are perfectly retrieved. The PT -symmetry is then a particular case of more general symmetries in which spatial reflection is generalized to continuous rotations followed by additional displacement and/or squeezing in phase space. Our results reinforce the prospects of going beyond PT -symmetric quantum mechanics making the field of pseudo-Hermitian quantum mechanics even more comprehensive and promising.
In addition, guided by the results in Refs. (2,3,29), here we explore the connection between antilinear symmetries and metrics. We derive a relation between the TD symmetry and a pair of TD metric operators, one linear and the other antilinear, which is analogous to Mostafazadeh's relation (4) for the TI scenario. This connection between symmetry and metric is explored a little further, leading us to propose a relation between symmetry (I) and metric (ρ) operators. Then, this symmetry-metric relation is put in perspective with the TD antilinear symmetry I(t) = Ξ −1 (t)ρ(t) we have derived, allowing us to finally compute the Ξ(t)-anti-pseudo-Hermitian operator.
We stress that from the 1990s and onwards, the field of radiation-matter interaction underwent extraordinary progress when experimentalists began to coherently control the process of the interaction of a single atom with a single photon of the radiation field. (32)(33) This control allowed probing fundamental aspects of quantum mechanics and implementing quantum logic operations, among other important achievements. We also remind the construction of the Bose-Einstein condensates, which allowed unprecedented control in the manipulation of many-body processes. (34)(35) For some time now, this control of the atom-field interaction has been sought towards time-dependent processes of radiation-matter interaction, as we know from the many advances made in the experimental verification of the dynamical Casimir effect. (36)(37)(38)(39)(40) We have reason to believe that not only the dynamical Casimir effect but other processes involving TD Hamiltonians -such as a TD Josephson-type coupling in two-mode Bose-Einstein condensates (41-42)-will soon be achieved. Therefore, the TD pseudo-Hermitian quantum mechanics, in its most general form, accounting for TD non-Hermitian Hamiltonians, symmetries and metric operators, must be studied to account for these TD processes.
We also note that our results shed light on the treatments already presented in the literature on TD non-Hermitian Hamiltonians. For example, our developments considerably broaden our understanding of those presented in Refs. (11)(12)(25)(26)(27)(28), where all the analysis on symmetry is reduced to the conditions for a TD Hamiltonian to be PT -symmetric. From our conclusions, we now know that there are close connections between the group algebra of the Hamiltonian, the symmetry and the metric operator. As concluded below, when considering a TD non-Hermitian and non-PT -symmetric Hamiltonian, we can now start by computing the symmetry operator I(t) of the system modeled by the Hamiltonian H(t), from which we automatically compute the metric operator, the ρ-pseudo-Hermitian and the Ξ-anti-pseudo-Hermitian operators.

Dissertation overview
The plan of the text is the following. In Chapter 2, we revisit the foundations of pseudo-Hermitian quantum mechanics for TI and TD Hamiltonians.
In Chapter 3, we present a method for the construction of a general TD symmetry operator for a TD non-Hermitian Hamiltonian. Besides that, in Section 3.3, we assume that the TD non-Hermitian Hamiltonian is simultaneously ρ(t)-pseudo-Hermitian and Ξ(t)-pseudo-Hermitian to derive the relation I(t) = Ξ −1 (t)ρ(t) for our TD antilinear symmetry operator. From this relation we retrieve Mostafazadeh's theorem for the TI scenario. (4) The TD non-Hermitian Hamiltonian describing a cavity field under linear amplification is introduced in Chapter 4. We then compute the TD symmetry operator for this non-Hermitian linear Hamiltonian using the method presented in Chapter 3. We demonstrate that this TD symmetry reduces to the PT operator when the non-Hermitian linear Hamiltonian is assumed to be PT -symmetric. An ansatz for the Dyson map is then proposed for the construction of the pseudo-Hermiticity relation. In Section 4.2 we address the TI equivalent of the TD non-Hermitian Hamiltonian introduced in Section 4.1. In this TI scenario we retrieve the Bender-Berry-Mandilara (29) result, and when considering a PT -symmetric TI non-Hermitian Hamiltonian, we verify that the TI symmetry again reduces to the PT operator.
In Chapter 5 we introduce the TD pseudo-Hermitian Hamiltonian of a cavity field under parametric amplification. The TD symmetry operator is then derived. Also in this chapter the TD Dyson map is presented and the Hermitian counterpart of the Hamiltonian is computed. In Section 5.2 we consider the TI equivalent of the Hamiltonian introduced in Section 5.1.
In Chapter 6 we present a symmetry-metric relation for TD non-Hermitian Hamiltonians by which the TD metric is automatically computed from the TD symmetry of the problem. By contrasting the symmetry-metric relation with the antilinear symmetry I(t) = Ξ −1 (t)ρ(t) derived in Section 3.3, we are finally able to compute the Ξ(t)-antipseudo-Hermitian operator. In Chapter 7 we present our conclusions.

PSEUDO-HERMITIAN QUANTUM MECHANICS
The state |Ψ(t) of a quantum system is described by a vector in Hilbert space whose dynamics is governed by Schrödinger equation ( = 1) where H(t) is the Hamiltonian modeling the system. The Hermiticity of Hamiltonians is one of the axioms of the quantum theory and guarantees two physical requirements: real energy spectrum and unitary time evolution (conservation of the norm). The interest in the non-Hermitian Hamiltonians has traditionally been in modeling dissipative processes (e.g. see (43-44)), but they are unacceptable (45, p. 105) and cannot provide a fundamental description of nature because they usually violate those physical requirements. Until the end of the 20th century, the non-Hermitian Hamiltonians were only phenomenological descriptions of physical processes.
The fact that non-Hermitian Hamiltonians can have a real spectrum when they exhibit an unbroken antilinear symmetry (1,29) opened, in the past twenty years, a new chapter in quantum mechanics. The most common of these antilinear symmetries is PT -symmetry which take many forms. For a Hamiltonian depending on momentum p and position x, the linear parity operator P performs spatial reflection (p → −p and x → −x), whereas the antilinear time-reversal operator T has the effect of the reversal of motion (p → −p, x → x, and i → −i). In this case, the PT operation takes the form If we adopt the standard product in Hilbert space ·|· , a non-Hermitian Hamiltonian, even though it has a real spectrum, does not describe a physical system as it conflicts with conventional quantum mechanics. We can calculate the expectation value of H and get an imaginary quantity, which disqualifies the Hamiltonian to represent an observable. Furthermore, according to the Schrödinger equation, since the Hamiltonian has non-real expectation values, it does not generate a unitary time evolution where "Im" denotes the imaginary part of their argument. However, it is possible to reformulate the standard definition of expectation values and matrix elements by introducing a metric ρ other than the identity operator 1, where ρ is a positive-definite Hermitian operator. In this way, the non-Hermitian Hamiltonian becomes the generator of a unitary quantum evolution.
The physical matrix elements of an operator O is defined as where, by convention, the metric operator must be placed on the left.

Pseudo-Hermiticity for TD observable non-Hermitian Hamiltonians
The requirement for a non-Hermitian diagonalizable Hamiltonian H(t) with inner product ·|· ρ to yield real eigenvalues and to generate a unitary time evolution is equivalent to requiring that it satisfies the ρ-pseudo-Hermiticity relation introduced by Mostafazadeh (in (2) for TI Hamiltonians and (17) for TD Hamiltonian) where the TI metric operator ρ is positive-definite and hence invertible. The dagger † stands for the adjoint of the corresponding operator. In the case of antilinear metric operators, let us denote by Ξ, the Eq. (2.6) is the Ξ-anti-pseudo-Hermiticity relation (4). The antilinear operators act according to the antilinearity property (50) The ρ-pseudo-Hermiticity of H(t) ensures that its expectation values are real and hence the conservation of norm since Eq. (2.3) holds for the new inner product with TI metric ρ: We note that if we choose the identity operator as the metric ρ = 1, the ρ-pseudo-

Hermiticity reduces to the Hermiticity condition H † (t) = H(t).
Because the non-Hermitian Hamiltonian H(t) is assumed to be diagonalizable, there is a complete biorthonormal set of eigenvectors {|Ψ n (t) , |Φ n (t) } obeying (17,43) where biorthonormality means Φ m (t)|Ψ n (t) = δ m,n with the completeness relation n |Ψ n (t) Φ n (t)| = 1. (2.11) Note that neither the eigenvectors of H(t) nor the eigenvectors of its adjoint H † (t) alone form an orthonormal basis when using the standard product, that is, the orthonormal relations Ψ m (t)|Ψ n (t) = δ m,n and Φ m (t)|Φ n (t) = δ m,n are not satisfied.
According to the ρ-pseudo-Hermiticity relation (2.6), the ket ρ −1 |Φ n (t) is an eigenvector of H(t) with eigenvalue E * n (t) Since ρ is positive-definite, we shall assume, without loss of generality, that Now, comparing equations (2.10) and (2.12), we can verify that the spectrum of the TD pseudo-Hermitian Hamiltonian H(t) is real and the biorthonormality reduces to Ψ m (t)|Ψ n (t) ρ = δ m,n . If ρ is no longer positive-definite, it may be invertible or not. In this case, the eigenvalues can come in complex conjugate pairs (E n , E * n ).

Dyson Representation for TD non-Hermitian Hamiltonians
In order to introduce an appropriate treatment of non-Hermitian Hamiltonians, we define the Dyson representation and what this term means. A first undergraduate course in quantum mechanics introduces the concept of the state of a quantum system |Ψ(t) as a solution to the Schrödinger equation already using the Schrödinger representation. The same system can be described by Heisenberg representation in which we perform a unitary transformation (46) upon the states and observables of the Schrödinger representation.
In contrast to others, the Dyson representation is the nonunitary transformation that maps the Hilbert space of the physical system with inner product ·|· ρ and described by the TD non-Hermitian Hamiltonian H(t), into the Hilbert space of the mathematical system with the standard inner product, described by the transformed Hamiltonian h(t). Under this mapping, performed by a nonunitary and possibly TD operator while the physical |Ψ(t) and mathematical |ψ(t) states are related as |ψ(t) = η(t) |Ψ(t) . The state vector |ψ(t) is the solution of transformed Schrödinger equation for h(t), The nonunitary operator η(t) is called Dyson map and we assume that it is positive-definite. We refer to Eq. (2.14) as the TD Dyson equation as it generalizes the similarity relation that first appears in a paper by Dyson (47) in the completely TI scenario.
For the states and operators in the Hilbert space of the non-Hermitian Hamiltonian we use capital letters (Ψ, H, and O), while for those in the Dyson representation we use lowercase (ψ, h, and o).
Observables in the physical system O(t) must be related to those in the Dyson representation o(t) by a similarity transformation (2.17) Since the metric operator is ρ(t) = η † (t)η(t), the matrix elements of O(t) are the same as in the Schrödinger representation

TD non-observable non-Hermitian Hamiltonians
We start this section following the Ref. (11), where a method is presented for approaching the quantum mechanics of TD non-Hermitian and non-observable Hamiltonians with TD metric operators. Alternative developments for the TD scenario are also given in Refs. (13)(14) From the particularization of these results for TI non-Hermitian Hamiltonians and metric operators, we automatically obtain the results presented by Mostafazadeh in Refs. (2) The Hermiticity requirement of h(t), as referred to in (11), yields the relation where ρ(t) = η † (t)η(t) can be interpreted as the TD metric operator and therefore Eq. (2.19) corresponds to TD ρ(t)-pseudo-Hermiticity relation, which in turn ensures the norm-conservation The observables O(t) satisfy a standard pseudo-Hermiticity relation Eq. (2.6) for TD metric operator, O † (t)ρ(t) = ρ(t)O(t). We notice that the TD Hamiltonian H(t) is non-observable due to the gauge-like term i (∂ t η(t)) η −1 (t) in Eq. (2.14).
The reason why a TD Dyson map is usually required for the construction of the Hermitian counterpart h(t) of an equally TD non-Hermitian H(t), is to avoid unwanted constraints between the parameters defining H(t). When considering a TI non-Hermitian H so that an equally TI Dyson map η can be considered, as in Ref. (2), the TD Dyson relation (2.14) simplifies to the similarity transformation Eq. (2.16), whereas the Eq. (2.19) simplifies to the well-known pseudo-Hermiticity relation Eq. (2.6).
We now analyze the consequences of the TD extension of the Mostafazadeh's method based on the two equations (2.6) and (2.16). First, within the TD extension and then the TD Dyson relation (2.14), we lose the similarity transformation (2.16) which ensures the observability of the Hamiltonian. However, the similarity transformation (2.17) remains valid for all operators O(t) other than the Hamiltonian, making the TD pseudo-Hermitian Hamiltonians as pertinent as their TI partners. It is worth noting that the observability of TD Hamiltonians is a sensitive point even in Hermitian quantum mechanics, where, as discussed in Ref. (24), the Hamiltonian acts essentially as the generator of the model's dynamics.
The Dyson map η(t) is not unique in general. For a particular TI Dyson map η and TD observable Hamiltonian H(t), the eigenvector of h(t) and H(t) can be qualitatively different while, according to the pseudo-Hermiticity relation, they share the same eigenvalues (isospectral partners) where |ψ n (t) = η |Ψ n (t) is a complete orthonormal basis, ψ m (t)|ψ n (t) = δ m,n , and the pair |Ψ n (t) = η −1 |ψ n (t) and |Φ n (t) = η † |ψ n (t) forms a complete biorthonormal basis.
Since h is Hermitian, then its eigenvalues as well as of those of the pseudo-Hermitian Hamiltonian H are real.
We end this review noting that in Ref. (24) a method is proposed for the derivation of particular TD Dyson maps which ensures the observability of TD pseudo-Hermitian Hamiltonians -as much as in Hermitian quantum mechanics-by restoring the similarity transformation between H(t) and h(t) from Eq. 2.14 where η(t) satisfy the Schrödinger-like equation The factor 2 arises from the substitution of the Schrödinger-like equation in Eq. 2.14 and is only a scale connecting the eigenvalus of h(t) and H(t).
In the treatment we have developed below, however, we are not considering the method in (24), and the Hamiltonians H(t) and h(t) must be transformed through the TD Dyson relation (2.14).

GENERAL SYMMETRY FOR TD HAMILTONIAN
In quantum mechanics, the concept of symmetry is related to a linear (or antilinear) transformation that acts on the states of the system and transforms them to new states, which are solutions to the Schrödinger equation (48)(49). In this chapter, we explore the generalizations of symmetries for quantum systems modeled by the TD Hermitian or non-Hermitian Hamiltonian.

General TD symmetry operator
In order to explore more general symmetries than PT for a TD non-Hermitian Hamiltonian H(t), we first propose a method to derive this symmetry I(t) which applies indistinctly to linear or antilinear, unitary or nonunitary symmetries. However, as anticipated in the Introduction, from now on we assume this symmetry to be antilinear so that we can retrieve the results in Refs. (4,29) for the particular case of TI Hamiltonians and symmetries. Moreover, as well as the Hamiltonian, we assume the symmetry to be a TD operator. Starting from the Schrödinger equation Eq. (2.1) for H(t), we apply the antilinear operator I(t) on both its left-hand sides and then replace t by −t, to obtain Therefore, for the transformation I(t) to be a symmetry of the system modeled by the Hamiltonian H(t), thus producing an independent solution I(−t) |Ψ(−t) of the Schrödinger equation from a given solution |Ψ(t) , we end up with the equation which defines an antilinear invariant for a Hermitian or non-Hermitian Hamiltonian. If we had considered a linear instead of antilinear transformation I(t), we would have obtained the equation and for the case where both the symmetry and the Hamiltonian are TI operators, the condition (3.4) is further simplified to the commutation Regarding the TI PT operation, the condition for a TD Hamiltonian to be PT -symmetric is given by We thus verify that the condition for the TD operator I(t) to be the symmetry associated with a TD Hamiltonian H(t), given by the differential equation (3.2), simplifies to the algebraic equation (3.4) for a TI symmetry operator. This represents a major reduction in the generality of the symmetry operator, which becomes even greater for a TI Hamiltonian.

Bender-Berry-Mandilara relation
In addition, Bender, Berry and Mandilara (29) have shown that a TI non-Hermitian Hamiltonian presents a real spectrum when invariant under any antiunitary symmetry I satisfying I 2k = 1 with k odd. The eigenvalues of the TI non-Hermitian Hamiltonians are real provided that the corresponding eigenvectors are also invariant under the action of antilinear symmetry I, that is (3) where |Ψ n is a eigenvector of the non-Hermitian Hamiltonian H |Ψ n = E n |Ψ n . This statement holds because I k |Ψ n is also an eigenvector of the non-Hermitian Hamiltonian with eigenvalue E * n for any k odd Thus, from equations (3.7) and (3.8) we obtain real eigenvalues E * n = E n . This is not true for k even, because I k with k even acts as a linear operator and hence the equation (3.8) is not hold.
Moreover, being I an antiunitary symmetry, its eigenvalues satisfy the following relation From this equation, applying I 2k with k odd to the eigenvector |Ψ n and using Eq. (3.7), we obtain the Bender-Berry-Mandilara relation (29) I 2k = 1 with k odd.

Method for the construction of a general symmetry operator
Following the reasonings in Ref. (52), where a method for the construction of nonlinear Lewis & Riesenfeld TD invariants is presented, we define the general symmetry operator as the product I(t) = Λ(t)U(t), with Λ(t) being either a unitary or nonunitary operator. Regarding U(t), from now on we assume it to be antilinear in accordance with the condition imposed in references (4,29), whose results we want to rescue in the scenario of TI Hamiltonian, symmetry and metric operators.

Considering the product I(t) = Λ(t)U(t), Eq. (3.2) can be rewritten in the form
By also rewriting the Hamiltonian as H(t) = H 0 (t)+V (t), with H 0 (t) being either a diagonal or nondiagonal operator with known eigenstates, we propose the ansatz U(t) = R(t)T , with T being the time-reversal operator and For H 0 (t) Hermitian, U(t) then becomes an antiunitary operator. We thus define the TD operator In summary, to obtain I(t) = Λ(t)U(t), we first compute the TD operator Θ(t) from Eq. (3.11), by taking the advantage of the known eigenstate basis of H 0 (t) which defines R(t). Next, starting from an ansatz for Λ(t), based on the symmetry group of V (t), we then compute this operator from Eq. (3.12), what finally gives us the symmetry I(t). It is evidently straightforward to derive the equivalent of Eq. (3.2) for a linear symmetry operator, with the same ansatz I(t) = Λ(t)U(t) applying for its solution.

Antilinear symmetry described by a couple of linear and antilinear metric operator
Let us consider a TD non-Hermitian Hamiltonian H(t) which obeys the TD pseudo-Hermiticity relation given by Eq. (2.19): , applying the linear metric operator ρ(t) on its l.h.s., and assuming the relation in Eq.
where we have defined |Φ(t) = ρ(t) |Ψ(t) because of Eq. (2.13). Next, we assume that H(t) also obeys a TD anti-pseudo-Hermiticity relation for the TD antilinear metric operator Ξ(t), the application of the operator Ξ(−t) on the l.h.s. of the Schrödinger equation for H(t), leads us again to the Eq. (3.13) once we define Considering both the TD pseudo-Hermiticity relations, in Eqs. (2.19) and (3.14), we derive H † (t) = ρ(t)H(t)ρ −1 (t) + iρ(t)ρ −1 (t) from the former and then substitute this adjoint Hamiltonian into the latter to obtain It is straightforward and remarkable to verify that the above expression recovers the Eq. (3.2) for the TD antilinear symmetry operator defined as We have thus verified that, for TD Hamiltonian, symmetry and metric operators, we have derived the TD counterpart of the important Mostafazadeh's relation for the symmetry operator, I = Ξ −1 ρ. We do not, of course, have a counterpart to the theorem proved by Mostafazadeh in the TI scenario, but verifying that the symmetry operator I(t) = Λ(t)U(t) we have derived through Eq. (3.2) can also be written in the form that generalizes Mostafazadeh's expression to the TD scenario, is significant and will be explored in later chapters.

The TD non-Hermitian Hamiltonian of a cavity field under linear amplification
The TD non-Hermitian Hamiltonian modeling a cavity field under linear amplification is given by with the TD parameters ω(t), α(t), and β(t) being complex functions. Here we just demand that H † (t) = H(t), such that ω * (t) = ω(t) and/or α * (t) = β(t). The usual requirement for the Hamiltonian (4.1) to be PT -symmetric, given by Eq.
where the condition of PT -symmetry with P as a reflection about whatever x 0 is (1) For the case of a TI Hamiltonian, x 0 = 0 implies a Hermitian Hamiltonian, whereas for a TD Hamiltonian, x 0 = 0 imposes constraints on the Hamiltonian parameters which do not occur for x = 0.

The TD antilinear symmetry operator
Considering the method proposed for deriving the symmetry operator, we rewrite the Hamiltonian (4.1) in the form Consequently, using Eq. (3.11) we obtain where the dot indicates a time derivative. Next, we consider, as an ansatz, the generalized displacement operator which becomes a unitary operator for λ(t) = −ν * (t) and µ(t) = −µ * (t), or a Hermitian operator for λ(t) = ν * (t) and µ(t) = µ * (t). We thus obtain from Eqs. (4.5) and (3.12): where From the r.h.s. of Eqs. (3.12) and (4.6), it follows that Note from Eq. (4.9d) that the parameter µ(t) is added to the generalized displacement operator to avoid undesirable constraints in the Hamiltonian's parameters. For the particular case of a unitary operator Λ(t), where λ(t) = −ν * (t), we use Eqs. (4.9b) and (4.9c) to obtain (4.10) Therefore, from Eqs. (4.9) we obtain the parameters defining the TD antilinear symmetry operator where we have replaced Λ for D, which, for a unitary Λ becomes the displacement operator. This symmetry operator describes the successive actions of a time-reversal operator T , a TD global rotation in phase space R(t) = e −iφ(t)a † a and, finally, let us say, a TD generalized displacement in phase space D(t) = e ν(t)a † +λ(t)a+µ(t) . For a unitary Λ, this TD symmetry

The Dyson map and pseudo-Hermiticity relation
For treating a TD non-Hermitian Hamiltonian we consider a TD Dyson map η(t) which results, in general (24), in a TD metric operator ρ(t) = η † (t)η(t). Otherwise, the TD pseudo-Hermiticity relation (2.19) imposes undesirable constraints on the TD parameters of the Hamiltonian. For the TD Dyson map, we consider the ansatz with (t) being a real function. To determine its time derivative we use the method of parameter differentiation (55), by which where Z = a † a + γa + γ * a † . We thus obtain the Hamiltonian where (4.15d) and the time dependence has been omitted for brevity.
To ensure the Hermiticity of h(t) we impose a complex TD frequency with ω R (t) being a real function, in addition to U = V * and F ∈ R, what demands thatγ Now, by substituting Eq. (4.16a) into Eq. (4.16b), and admitting momentarily the approximation 1, we obtain showing that the Hermiticity requirements in Eqs. (4.16) imposes no additional constraints on the Hamiltonian parameters, apart from the complex TD frequency ω(t) = ω R (t) − i˙ (t) coming from Eq. (4.15a). Otherwise, when we assume that ω(t) is real from Eq. (4.1), it follows that must be constant, Which leads to a new Hamiltonian h in Eq. (4.14), and consequently to a new system in Eq. (4.15) and a new hermitization condition in Eq. (4.16). When the simplified Dyson map η = e γa+γ * a † is considered, as in Ref. (11), with = 0, the pseudo-Hermiticity requirement of a complex frequency simplifies to that of a real one, ω(t) = ω R (t), still with no constraints on α(t) and β(t).
Therefore, when considering the symmetry operator for the TD pseudo-Hermitian Hamiltonian (4.14), with γ and following from Eqs. (4.17) and (4.19), we must necessarily assume the function ω(t) appearing in Eqs. (4.9) to be of the form ω(t) = ω R (t) − i˙ (t) (or ω = ω R for the particular Dyson map η = e γa+γ * a † ). Despite of the frequency constraint, the symmetry operator in Eq. (4.11), have no restrictions for the amplification parameters α(t) and β(t).

From I(t) in Eq. (4.11) to PT
The TI PT operator can be directly recovered from Eq. (4.11), starting with the constraints under which the Hamiltonian (4.1) is PT -symmetric: ω * (−t) = ω(t), α * (−t) = −α(t), and β * (−t) = −β(t). Assuming also a unitary Λ(t), we verify from Eqs. (4.9a) and (4.9b) that the rotation is reduced to a TI operator with φ(t) = φ 0 , while the parameter ν(t) of the displacement operator is simplified to For a TI symmetry operator, where I = DU must be a TI parameter, we must then assume, to avoid undesirable constraints on the Hamiltonian parameters, that φ 0 = (2n + 1) π, with n ∈ Z. From this assumption we obtain ν(t) = µ(t) = 0, and noting that the parity operator can be written in the form e −i(2n+1)πa † a , we finally recover the TI operator PT from Eq. (4.11), i.e., (4.21)

The TI antilinear symmetry operator
The condition for the TI Hamiltonian ωλ − α + α * e iφ = 0, (4.24b) For a unitary Λ (λ = −ν * ), and using the polar forms α = |α| e iϕα and β = |β| e iϕ β , it follows from Eqs. (4.24) that For |ν| = 0, we get the constraint which, considering ϕ α + ϕ β = ϕ, leads to the relation (4.30) in agreement with Eq. (4.25a). From the above results, we obtain from Eq. (4.25b) the expression together with |cos ϕ| > ω 2 − |α| 2 /2ω, and then p − < f < p + , with (4.33) The parameters in Eqs. (4.30) and (4.31), under the above constraints for ϕ and f, define the TI antiunitary symmetry operator Although the TI continuous symmetry operators in Eqs. (4.28) and (4.34) are particular cases of the TD symmetry operator in Eq. (4.11), they are generalizations of the discrete parity and time-reversal transformation. Differently from the operator I(t) in Eq. (4.11), whose TD parameters depend on the Hermiticity conditions only through the frequency requirement ω = ω R − i˙ (or ω = ω R for the particular Dyson map η = e γa+γ * a † ), the operators in Eqs. (4.28) and (4.34) take into account the constraints imposed on ω, α and β. The TI non-Hermitian Hamiltonians and consequently the associated symmetry operators are more vulnerable than their general TD equivalents to the constraints imposed by the pseudo-Hermiticity relation. This vulnerability to the constraints follows from the more stringent condition for the invariance of a TI Hamiltonian: [I, H] = 0.

Bender-Berry-Mandilara
Although it is straightforward to verify the validity of the relation I 2k = 1, with k odd (29), for the symmetry operator in Eq. (4.28), its validity for the operator in Eq. (4.34) demands a little algebra. In fact, for the operator in Eq. (4.34) we obtain since it follows from Eqs. (4.25c) and (4.31) that νe iφ = −ν * .
It is important to note that the TD symmetry operator I(t) does not obey the Bender-Berry-Mandilara relation. Although we still have the relation (4.36) the equality ν(t)e iφ(t) = −ν * (t) is no longer satisfied in the TD scenario.

The TD antilinear symmetry operator
Rewriting the Hamiltonian (5.1) in the form We now consider the ansatz Λ(t) = e ξ(t)a 2 +ζ(t)a †2 , where we have defined Next, we substitute Eq. (5.4) into Eq. (3.12) to obtain from which we compute the variables φ(t), ξ(t) = |ξ(t)| e iϕ ξ (t) and ζ(t) = |ζ(t)| e iϕ ζ (t) which define the operators U(t) and Λ(t). In order to find a possible solution for the system (5.7), we assume small amplification parameters, |ξ| 1 and |ζ| 1, in order to disregard their second order corrections, leaving us with the equationṡ For the particular case where Λ(t) is a unitary operator, i.e., ζ(t) = −ξ * (t), and Im ω(−t) = 0, the above system simplifies to the decoupled equationṡ For the case where Im ω(−t) = 0, the parameter ξ(t) follows from Therefore, from Eqs. (5.8) we obtain the parameters defining the TD antilinear symmetry operator which becomes an antiunitary operator, with Λ(t) being the squeezing operator S(t) = e ξ(t)a †2 −ξ * (t)a 2 when considering the parameters following from Eqs. (5.9). This symmetry operator describes the successive action of a time-reversal operator T , a TD global rotation in phase space R(t) = e −iφ(t)a † a and a TD operation performed by the generalized squeezing S(t) = e ξ(t)a †2 +ζ(t)a 2 . From Eq. (5.9b) we verify that the approximation |ξ| 1 corresponds to small amplification parameters |α| and |β| compared to |ω|.
Similarly to the symmetry operator derived above for the TD non-Hermitian linear Hamiltonian, for the case unitary Λ the operator in Eq. (4.28) resembles the evolution operator for the Hermitized counterpart of the TD Hamiltonian in Eq. (5.1) (12,(25)(26)(52)(53)(54), apart from the time-reversal operation. If applied to a given state of the Hermitized counterpart of our Hamiltonian, this symmetry operator moves the probability distribution across an upward spiral in phase space, continuously squeezing it.

Dyson map and pseudo-Hermiticity
For the TD quadratic Hamiltonian (5.1) we consider the TD quadratic Dyson map which is a positive Hermitian operator under the constraint θ 2 = 2 − 4 |γ| 2 > 0. This map provides us with the relations and consequently with the transformed Hamiltonian where, after defining the parameters For h in Eq. (5.14) to be Hermitian we require the constraints W = W * and V = U * , which results in the system where we have defined γ = |γ| e iϕ , Φ = λe iϕ , and with the initial values |z(0)| and ϕ(0) being two free parameters defining the Dyson Map. After deriving z, λ and ϕ from the system (5.17), we then compute from the relation = 1 As follows from the system (5.17), the requirement for the Hamiltonian h to be Hermitian does not imply any restrictions on the Hamiltonian parameters ω(t), α(t) and β(t), differently from what happens with the TD pseudo-Hermitian linear Hamiltonian, where a constraint is required on the frequency of the oscillator. This desired absence of constraints on the parameters defining the TD non-Hermitian quadratic Hamiltonian comes certainly from its SU(1,1) dynamical symmetry.

All-creation and all-annihilation Hamiltonians
We next consider, as defined in Ref. (24), the all-annihilation H aa (t) and all-creation H ac (t) non-Hermitian Hamiltonians, given by These peculiar Hamiltonians, which are extreme forms of the unbalanced Hamiltonian (5.1), follow when considering β(t) = 0 and α(t) = 0, respectively. To compute their symmetry operators we follow exactly the same procedure presented in subsection III.A. The same structure of the symmetry operator in Eq. (4.28) thus applies for these Hamiltonians.
Considering a unitary Λ(t), the Eqs. (5.9) simplifies, for the Hamiltonian H aa (t), tȯ whereas for H ac (t), we obtaiṅ The nature of the Hamiltonian -following from the group algebra describing the amplification process-is then what defines its symmetry operator. When considering the all-creation and the all-annihilation linear Hamiltonians, derived from the TD linear Hamiltonian introduced above, it can be directly verified that we also have the same structure I(t) = D(t)R(t)T for the symmetry operator.
Therefore, for the Heisenberg algebra of the linear amplification we have the symmetry operator I(t) = D(t)R(t)T , whereas for the SU(1,1) algebra of the parametric amplification, we have I(t) = S(t)R(t)T . Both symmetry operators share the time reversal and rotation operators. However, as it is well-known from the amplification processes of the radiation field (56)(57), while the Heisenberg algebra generates displacements in phase space, the SU(1,1) produces squeezing.

The TI antilinear symmetry operator for the TI Hamiltonian
After computing the TD symmetry in Eq. (4.28), we assume that its TI equivalent has the form I = SRT , with S being an antilinear operator. The condition for the Hamiltonian (5.25) to be invariant under this TI symmetry operator, given by [I, H] = 0, provides us with the system which, for an antiunitary S, i.e., ζ = −ξ * , simplifies to Considering a real frequency ω and, as in the TD case, a small squeezing parameter, |ξ| 1, to disregard its second-order corrections, we obtain from the above equations the squeezing parameter 28) and the rotation angle under the constraint |α| 2 + |β| 2 + 2αβ = 0. From the polar forms α = |α| e iϕα , β = |β| e iϕ β and ξ = |ξ| e iϕ ξ , it follows that For the TI non-Hermitian Hamiltonian (5.25) we then have the TI antiunitary symmetry operator defined by the parameters in Eqs. (5.28) and (5.29). For the case of a PT -symmetric Hamiltonian, where α and β are real parameters, the rotation angle following from Eq. (5.29) is given by φ = (2n + 1) π, which makes the squeezing degree null, |ξ| = 0, thus reducing the symmetry I = SRT to the PT operator.
Considering the pseudo-Hermiticity conditions in Eq. (5.37), we obtain from Eq. (5.30a) the squeezing degree showing that for φ = ϕ α , the symmetry operator reduces to the product I = RT , whereas for φ = ϕ α we still have I = SRT , with the squeezing operator being defined by the squeezing degree in Eq. (5.40) and direction

Symmetry breaking
Considering the bosonic operators b and b † defined by the transformations b = ua + va † and b † = v * a + u * a † , with |u| 2 − |v| 2 = 1 such that b, b † = 1, we can rewrite the Hamiltonian (5.25) in the form There are two different regimes of eigenvalues following from the parameters given by Eq. (5.37). One for an odd integer n, leading to a real frequency Ω, and the other for an even n, where Ω is real only under the additional constraint ω ≥ 2 |αβ|. For ω < 2 |αβ| the symmetry is broken; the frequency is no longer real and the eigenvalues become complex conjugate pairs.

CONNECTION BETWEEN THE SYMMETRY AND METRIC OPERATORS
If the symmetry operator stems from the group algebra of the Hamiltonian, the ansatz for the Dyson map must also be guided by this group algebra, as discussed in Ref. (12) where a strategy for constructing the Dyson map is presented. It is then reasonable to assume that there must be a close connection between both of these operators: the symmetry and the Dyson map (or equivalently the metric operator).
In chapter 4, we have derived the symmetry operator for the TD non-Hermitian linear Hamiltonian, which resulted in the form I(t) = D(t)R(t)T , with R(t) = e −iφ(t)a † a being a unitary operator and D(t) = e ν(t)a † +λ(t)a+µ(t) a unitary (λ = −ν * ) or a Hermitian nonunitary operator (λ = ν * ). The ansatz for the Dyson map we have used for the pseudo-Hermitization of this linear Hamiltonian, given by the Hermitian operator η(t) = e (t)a † a+γ(t)a+γ * (t)a † , bears resemblance to the generalized displacement D(t). In fact, we could have considered for the Dyson map the nonunitary operator D(t)R(t) instead of the η(t), possibly at the expense of constraints on the non-Hermitian Hamiltonian's parameters. We could also have associated the linear part of the symmetry directly to the metric operator, i.e., ρ(t) = D(t)R(t).
We observe however that, from Eqs.
From Eqs. (3.2) and (6.7), we obtain instead: Eqs. (111) or (112) can be solved by applying the same reasoning used in Section 3 to compute the symmetry operator.
We finally note that the formal simplicity of PT symmetry prevented us, so far, from exploring a relation between symmetry and metric as anticipated here. It is evident that our proposal for this relation was largely due to the complexity of the symmetry operators associated with the Hamiltonians discussed here. This fact should motivate us to persist in the investigation of symmetries that are more general than PT , through the consideration of other Hamiltonians with different symmetry groups. We believe that we can further deepen our understanding of pseudo-Hermitian systems and even the symmetry-metric relation through this investigation.

CONCLUSIONS
In this work, we have proposed a method for the derivation of general TD continuous symmetry operators for TD non-Hermitian Hamiltonians. Although our method applies indistinctly to linear or antilinear, unitary or nonunitary symmetries, we then assume an antilinear symmetry to retrieve the results by Mostafazadeh (4) and Bender-Berry-Mandilara (29) for the case of TI Hamiltonian and symmetry operators. In fact, assuming that the TD non-Hermitian Hamiltonian is simultaneously ρ-pseudo-Hermitian and Ξ-antipseudo-Hermitian, we derive the relation I(t) = Ξ −1 (t)ρ(t) for our TD antilinear symmetry operator. From this relation we recover the Mostafazadeh's theorem, for TI Hamiltonian and symmetry operators, asserting that the pseudo-Hermiticity of a Hamiltonian implies the existence of an antilinear symmetry of the form I = Ξ −1 ρ. We also retrieve the Bender-Berry-Mandilara result that a non-Hermitian Hamiltonian presents a real spectrum when invariant under any antiunitary operator I satisfying I 2k = 1 with k odd.
Our method is also based on a proposal in Ref. (52), for the construction of Lewis & Riesenfeld TD nonlinear invariants, and we have applied it for the case of TD non-Hermitian linear and quadratic Hamiltonians modeling a cavity field under linear and parametric amplifications. We have thus derived TD continuous symmetry operators, given in Eqs. (4.11) and (5.11), for the linear and quadratic cases respectively. These operators describe the successive actions of a time-reversal operator T , a TD rotation and a generalized displacement or squeezing in phase space, respectively. This TD continuous symmetry automatically reduces to the TI discrete PT operator when we restrict our TD Hamiltonian to be PT -symmetrical.
After computing the symmetry operators we then consider the pseudo-Hermitization of our TD linear and quadratic Hamiltonians. For the case of the TI equivalents of our non-Hermitian Hamiltonians, the stringent invariance requirement [I, H] = 0, imposes TI continuous symmetry operators which are very particular case of the TD symmetry operators in Eq. (4.11) and (5.11), even though they are generalizations of the discrete parity and time-reversal transformation. The TI non-Hermitian Hamiltonians and the associated symmetry operators are more vulnerable, as expected, than their general TD equivalents to the constraints imposed by the pseudo-Hermiticity relation.
The TD general symmetries in Eq. (4.11) strongly resemble the evolution operator for the Hermitized counterparts of our TD non-Hermitian linear and quadratic Hamiltonians, except for the time-reversal operation. (25)(26)(27)(53)(54) If applied to a given state of these Hermitized counterparts of our Hamiltonians, these peculiar symmetry operators cause the probability distribution to trace an upward spiral in phase space, with TD rotation and translation rates, for the linear case, or translation plus squeezing rates for the quadratic case.
We have also computed the TD symmetry operators for the all-creation and all-annihilation Hamiltonians, in Eqs. (5.20a) and (5.20b), which are extreme cases of the parametric amplification in Eq. (5.1), where only creation or annihilation quadratic operators take place in the interaction picture. We verify, as expected, that the TD symmetry operators for these peculiar Hamiltonians have the same form as the symmetry operator for our non-Hermitian parametric amplification. The group algebra describing the Hamiltonian is what defines its symmetry operator.
From the results we have derived for the linear and the quadratic non-Hermitian Hamiltonians, we have proposed a relation between the symmetry and the metric operators which enables us to automatically derive the metric operator ρ(t). In fact, from the symmetry of the problem, we are able to derive both the ρ-pseudo-Hermitian and the Ξ-anti-pseudo-Hermitian operators, as introduced by Mostafazadeh in Ref. (4). In order to consolidate the symmetry-metric relation which we have derived, our next step is to approach other Hamiltonians, associated with different groups of symmetries.
Therefore, in the present work, we have explored more general symmetries than parity and time-reversal for TD non-Hermitian Hamiltonians. The TD continuous symmetry operators we have derived follow evidently from the symmetry group of the Hamiltonians we have studied, and the symmetry-metric relation we have proposed will certainly act as a guide for the study of pseudo-Hermitian quantum mechanics for symmetry groups far broader than PT .