

Biorthogonal Quantum Mechanics

[ Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, UK

The Hermiticity condition in quantum mechanics required for the characterisation of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called ’biorthogonal quantum mechanics’, is developed here in some detail in the case for which the Hilbert space dimensionality is finite. Specifically, characterisations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems.

: J. Phys. A: Math. Gen.

D. C. Brody]Dorje C. Brody

1 Introduction

In standard quantum mechanics observable quantities are characterised by Hermitian operators. The eigenvalues of a Hermitian operator represent possible outcomes of the measurement of an observable represented by that operator. Once the measurement of, say, the energy is performed and the outcome recorded, the system is in a state of definite energy, that is, there cannot be a transition into another state with a different energy. Hermitian operators conveniently encode this feature in the form of the orthogonality of their eigenstates.

The observed lack of transition into another state, however, can only be translated into the abstract ‘mathematical’ notion of the orthogonality of states in Hilbert space via the specification of the probability rules in quantum mechanics. When eigenstates of an observable are not orthogonal, however, there is an equally natural way of assigning probability rules so that the resulting quantum theory appears identical to the conventional theory. Evidently, in this case observables are not represented by conventional Hermitian operators, since otherwise the eigenstates are necessarily orthogonal. Nevertheless, if an operator has a complete set of eigenstates and real eigenvalues, then it becomes a viable candidate for representing a physical observable. The key mathematical ingredients required to represent physical observables are that the eigenvalues are real, and that eigenstates are complete; whereas the notion of orthogonality can be relaxed and substituted by a weaker requirement of biorthogonality. The resulting quantum theory will thus be called biorthogonal quantum mechanics.

There is a substantial literature on the idea of relaxing the Hermiticity requirement for observables in quantum mechanics. For example, Scholtz et al. [1, 2] proposes the introduction of a nontrivial metric operator in Hilbert space and defines physical observables as self-adjoint operators with respect to the choice of the metric. Viewed from the conventional ‘flat’ inner-product structure, therefore, observables are no longer Hermitian and their eigenstates are not orthogonal, but in the Hilbert space endowed with this nontrivial metric we recover the ‘standard’ quantum theory. Bender and others have developed PT-symmetric quantum theory where the Hermiticity condition is replaced by the invariance under simultaneous parity and time reversal operation. A PT-symmetric Hamiltonian is in general not Hermitian, but if the corresponding eigenstates are also PT symmetric, then the eigenvalues are real and eigenstates may be complete, and can be used to describe quantum systems [3, 4, 5]. Operators that are not Hermitian also play an important role in the physics of resonance, as discussed, for example, in [6, 7, 8]. The role of biorthogonal systems in PT-symmetric quantum theories is discussed in Curtright & Mezincescu [9].

The works mentioned here are detailed and substantial, and contain a large number of references. In spite of this, here we shall present ‘yet another account’ of the subject since a number of basic and foundational ideas of quantum mechanics, already required for the representation of quantum systems modelled on finite-dimensional Hilbert spaces, such as a detailed account of probabilistic interpretations, a characterisation of measurement processes, or a formulation of combined systems and the role of entanglements, have not been made completely transparent. It turns out that the approach based from the outset on the use of biorthogonal basis (as in [9]) allows us to develop these basic ideas in the most elementary manner. The purpose of the present paper therefore is to develop the formalism of biorthogonal quantum mechanics for systems modelled on finite-dimensional Hilbert spaces, and along the way clarify various issues in a transparent and accessible way.

The paper will be organised as follows. We begin in §2 with an overview of the biorthogonal system of basis in Hilbert space that arise from the eigenstates of a complex (i.e. not necessarily Hermitian) Hamiltonian and those of its Hermitian adjoint, for the benefit of readers less acquainted with the material. The effectiveness of the use of biorthogonal basis associated with operators that are not self adjoint has a long history and goes back to the work of Liouville [10], subsequently developed further by Birkhoff [11]. In the case of a real Hilbert space of square-integrable functions defined on a finite interval of the real line , properties of biorthogonal bases associated with operators that are not self adjoint have been worked out in detail by Pell [12, 13]. Many of the results, with suitable modifications, extend into the complex domain, as developed by Bari [14] (cf. [15]).

In §3 we establish the relation between the Hilbert space of states and its dual space , and this in turn leads to the identification of a consistent probability assignment for transitions between states. It will be shown that although eigenstates of a complex Hamiltonian are not orthogonal in , they nevertheless do correspond to maximally separated states in the ray-space, hence there cannot be transitions between these states. An analogous conclusion has been drawn previously (e.g., in [4]), but it will become evident that the biorthogonal method employed here leads to this result in the most elementary fashion, without referring to heavy-handed mathematical arguments. The construction of observables, their expectations, as well as the notion of general mixed states, are then developed in some detail in §4.

In §5 we discuss measurement-theoretic and further probabilistic aspects of complex Hamiltonians. It will be shown, in particular, that for unitary systems orthogonality of eigenstates in is not a condition that can be asserted from experiments, thus making any operator having a complete set of eigenstates and real eigenvalues a viable candidate for the representation of observable quantities. The construction of combined systems in biorthogonal quantum mechanics is then developed in §6, where we also define coherent states in this context. In §7 we describe how the Rayleigh-Schrödinger perturbation theory works in the case of complex Hamiltonians. Perturbation of complex Hamiltonians away from eigenstate degeneracies in fact has been known for some time [16, 17]. The purpose of this section is to give a brief review of the idea, partly for completeness and partly on account of the fact that the result provides an independent confirmation that the probability assignment rule of §3 is in some sense the ‘correct’ one. Properties of time evolution of quantum states generated by a complex Hamiltonian are described in §8, showing that reality and completeness lead to unitarity, without the orthogonality requirement. In §9 we turn to the discussion of PT-symmetric quantum mechanics, in particular how it ties in with the notion of biorthogonal quantum mechanics. We conclude in §10 with a brief discussion towards subtleties arising from the consideration of quantum systems described by infinite-dimensional Hilbert spaces.

2 Eigenstates of complex Hamiltonians and their adjoints

To begin the analysis of quantum mechanics using basis functions that are in general not orthogonal, we shall first review basic properties of eigenstates of generic complex Hamiltonians in finite dimensions. Let , with and , be a complex Hamiltonian with eigenstates and eigenvalues :


We shall assume for now that the eigenvalues are not degenerate. In addition to the eigenstates of , it will be convenient to introduce eigenstates of the Hermitian adjoint matrix :


Here and in what follows, a ‘Hermitian adjoint’ will be defined by the convention that denotes the complex-conjugate transpose of . The reason for introducing the additional states is because the eigenstates of are in general not orthogonal:


for , which follows from the facts that and that . An analogous result


holds for the eigenstates of . Of course, for a given some of its eigenstates can be orthogonal, but if is not Hermitian, then a typical situation that arises is where not all the eigenstates are orthogonal. Hence conventional projection techniques so commonly used in many calculations of quantum mechanics, for example, in measurement theory or perturbation analysis, are ineffective when dealing with the eigenstates of a complex Hamiltonian [16].

With the aid of the conjugate basis , let us first establish that the eigenstates of , although not orthogonal, are nevertheless linearly independent. To show this, suppose the converse that are linearly dependent. Then there exists a set of numbers such that , and that


Transvecting this relation with from the left, we find, for each , that , where we have made use of the facts that


and that . To see that (6) holds, we note that by definitions (1) and (2) we have


Hence if , and if . Since cannot hold for all , there has to be at least one such that . On the other hand, by assumption the eigenvalues are not degenerate, so there cannot be more than one for which . Without loss of generality we can label the states such that we have for all . It follows that if but , and this establishes (6). Now since when is nondegenerate, we must have for all , contradicting the hypothesis. It follows that the nondegenerate eigenstates of are linearly independent, and thus span the Hilbert space , since the number of linearly independent basis elements agrees with the Hilbert-space dimensionality. In other words, forms a complete set of basis for . Additionally, they are minimal in that exclusion of any one of the elements from the set spoils completeness. A set of basis elements that is both minimal and complete is called exact. In finite dimensions, the exactness of implies the exactness of , whereas in infinite dimensions this no longer is the case, as discussed below in §10.

Using the independence of the states we can establish the relation:


which hold in finite dimensions away from degeneracies. To show this, we remark that if has the property that holds true for an arbitrary vector , then it must be that . Writing for some we have


and this establishes the claim.

The operator defined by (cf. [18])


thus plays the role of a projection operator satisfying . Although is not Hermitian, its eigenvalues are all zero, except one which is unity, for which the eigenstate is . Writing for the eigenstate projector we have


It follows, in particular, that


While the complex Hamiltonian does not admit the representation , due to the fact that , it nevertheless can be expressed in the form (cf. [19]):


It follows, furthermore, that if we write, for an arbitrary state ,


then we have


A form of this result for real Hilbert-space vectors was obtained in [12].

3 Quantum probabilities

In the foregoing discussion we have not commented on the norm convention. In quantum theory, the norm of a state is closely related to probabilistic interpretations of measurement outcomes. Hence we wish to fix our norm convention so that it is consistent with probabilistic considerations of a quantum system when energy eigenstates are not orthogonal. Now in the literature on the use of biorthogonal basis for complex Hamiltonians, especially in quantum chemistry, the norm of the eigenvectors are often (but not always; cf. [20, 21] for a related discussion) assumed to take values larger than unity so as to ensure the following relation holds for all :


Under this convention, eigenvectors will no longer be normalised. In particular, if we assume that all eigenstates have the same Hermitian norm so that for all , then we have . This might at first seem a little odd from the viewpoint of traditional Hermitian quantum mechanics, however, for a range of analysis that follow, it turns out that the convention leads to considerable simplifications.

To begin, we recall that in standard quantum mechanics, the ‘transition probability’ between a pair of states and is given by the ratio of the form . Under the convention , however, we cannot maintain a consistent probabilistic interpretation from this definition. For instance, if the state of the system is in an eigenstate of a complex Hamiltonian , then on account of stationarity there cannot be a ‘transition’ into another state , , even though ; whereas according to the conventional definition the transition probability between these states is nonzero. To reconcile these apparent contradictions we need the introduction of the so-called associated state that defines duality relations between elements of the Hilbert space and its dual space .

For an arbitrary state , we define the associated state according to the following relations:


We shall let (17) determine the duality relation on the state space: . Putting the matter differently, the state dual to is given by of (17); the state associated to is then given by the Hermitian conjugate of . The quantum-mechanical inner product for a biorthogonal system is thus defined as follows: If and , then


Since we demand the convention that for all , we can assume that


It also follows that defines the transition probability between and :


provided that the Hilbert space pairing is defined by the convention (18). Here for definiteness we have expressed in a homogeneous form that is invariant under complex scale transformations of the states. The interpretation of the number is as follows: if a system is in a state characterised by the vector , and if a measurement is performed on the ‘complex observable’ , then the probability that the measurement outcome taking the value is given by .

More generally, the overlap distance between the two states and will be defined according to the prescription:


A short exercise making use of the Cauchy-Schwarz inequality shows that the right side of (21) is real, nonnegative, and lies between zero and one, thus qualifying the required probabilistic conditions. In particular, only if ; whereas only if where and .

In quantum mechanics the notion of probability is closely related to that of distance. To see this, suppose that is a neighbouring state to . Then expanding (21) and retaining terms of quadratic order, we obtain the following form of the line element, known as the Fubini-Study line element:


As an illustrative example, consider a two-dimensional Hilbert space spanned by a pair of states . Then an arbitrary normalised—in the sense of (19)—state can be expressed in the form


Evidently we have but , on account of (16). Taking the differential of and substituting the resulting expression in (22), making use of (17), we deduce that the line element is given by


It follows that the state space defined by the relation is a two-sphere of radius one half—the Bloch sphere of complex Hamiltonian systems. We shall have more to say about this.

4 Observables and states

We have shown in (13) that a complex Hamiltonian admits a spectral decomposition in terms of the complex projection operators . Evidently, for a fixed biorthogonal basis there are uncountably many such (commuting family of) operators for which eigenvalues are entirely real, even though they are not Hermitian in the sense that does not agree with . In fact, the class of such ‘real’ operators in this space is wider and contains those that do not commute with the Hamiltonian .

Given a fixed biorthogonal basis , a generic operator can be expressed in the form


Note that can likewise be expressed in terms of the nonorthogonal basis :


since the set is complete. However, in this case the array cannot be viewed as a matrix, whereas the array can, which shows the advantage of the use of biorthogonal basis. Thus, if is another operator with ‘matrix’ elements in the basis , then the matrix element of the product is just .

If and are nondegenerate Hermitian—in the usual sense—operators, the eigenstates of can always be transformed unitarily into those of . For complex operators, however, this is no longer the case. Nevertheless, two operators and will be said to belong to the same class of observables if there is a unitary transformation between the basis of and .

The expectation value of a generic observable in a pure state is defined by the expression


In particular, if the array in (25) is ‘biorthogonally Hermitian’ in the sense that , then defined by (27) is real for all states , even though is not real for most states. Thus, the notion of Hermiticity extends naturally to the biorthogonal setup, and we are able to speak about physical observables in the usual sense. This follows from the fact that although is not Hermitian in the sense that , its expectation value (27) in an arbitrary state is nevertheless real because the corresponding matrix in the biorthogonal basis is Hermitian. If we let and substitute this in (27), making use of (25), then we find


In particular, if are eigenstates of , then we can write , where are the eigenvalues of , hence


which is consistent with our probabilistic interpretation of the biorthogonal system.

The matrix interpretation here nevertheless requires further clarification. If a Hermitian ‘matrix’ is given without the information about the choice of basis, then there is no procedure to determine whether is Hermitian; whereas for orthogonal bases, the data is sufficient to determine whether is Hermitian, even though the choice of the orthogonal basis remains arbitrary. To make this transparent, suppose that is an orthonormal basis of such that


Then the matrix element of the observable in this orthonormal basis is given by


In this way we see more explicitly that while the reality of merely requires Hermiticity of , the Hermiticity of requires a more stringent condition that


In particular, if is Hermitian so that , then can be chosen to be so that and (32) reduces to the familiar condition ; if is symmetric, then the left side of (32) is invariant under the interchange of indices , and we have , i.e. components of are complex conjugates of the components of . The expansion coefficients are unique up to unitary transformations. The linear independence of implies that is invertible, and the orthonormality condition implies that the inverse of is given by . Phrased differently, if we write (30) in the form and , then we have ; if is real (biorthogonally Hermitian), then


where is an invertible positive Hermitian operator.

As an elementary illustrative example, consider the complex Hamiltonian with . A short calculation shows that the eigenstates of and , in the region for which the eigenvalues are real, are given by


where , and where we have written for , and so on. An arbitrary observable for which the expectation value defined by (27) is real can be expressed, up to trace, as a linear combination of the deformed Pauli matrices


These are obtained according to the prescriptions


It should be evident that the triplet fulfils the standard commutation relations, and that in the Hermitian limit we recover the standard Pauli matrices. The expectation values, in the sense of (27), of these Pauli matrices in a generic state (23) are thus given by


Note that the right-sides of these expectation values are independent of , on account of the -dependence of the eigenstates. Expectation values of Hermitian operators, such as the usual Pauli matrices, on the other hand, are in general not real since they do not represent physical observables in the biorthogonal system.

It should be evident, incidentally, that in the case of a two-level system, the choice of the biorthogonal system is uniquely determined by the overlap distance , up to unitarity. Physical observables constructed under the biorthogonal system therefore belong to the same class of observables as those constructed from another system , provided that .

We have spoken about pure states thus far, but the state of a physical system in quantum mechanics is, more generally, and perhaps more commonly, characterised by a mixed state density matrix:


A density matrix is thus not Hermitian in the usual sense so that , but it is ‘Hermitian’ with respect to the choice of biorthogonal basis so that . The eigenvalues of are nonnegative and add up to unity. The expectation value of a generic observable (25) in the state is thus defined by


It should be evident that a necessary and sufficient condition for the reality of , for an arbitrary , is that .

A simple example of a density matrix arises if a quantum system described by a complex Hamiltonian is immersed in a heat bath of inverse temperature . In particular, if the eigenvalues of are all real, then after a passage of time the system will reach an equilibrium state


if we assume the postulate that an equilibrium state should maximise the von Neumann entropy subject to the constraint that the system must possess a definite energy expectation . Here, denotes the partition function. The reality of all the eigenvalues of is crucial for the existence of a canonical distribution (40), owing to properties of the dynamics of the system, as described below in §8.

5 Measurement of spin- particle

We now wish to turn to the discussion about the Bloch sphere introduced in §3 above, in the context of a spin- particle system in quantum mechanics. To this end we recall first with the general discussion that in standard nonrelativistic quantum mechanics, the wave function of a particle splits into two components, one associated with its spacial symmetry and the other associated with its internal symmetry (such as spin, isospin, colour, flavour, etc.). Since in the nonrelativistic context these spacial and internal symmetries are independent, if one is interested only in the internal symmetry of a particle, then it is a common practice to ignore the spacial degrees of freedom of the wave function (belonging to an infinite-dimensional Hilbert space) and focus attention on the internal symmetries (belonging to a finite-dimensional Hilbert space). It follows, in particular, that internal symmetries of a particle, a priori, do not concern the spacial degrees of freedom.

In spite of the independence of these symmetries, one commonly speaks, for instance, about the spin of an electron in a certain spacial direction. The reason why this is permissible has its origin in the mathematical structure of the state space of a spin- particle system: The space of states for this system is a two-sphere—in the quantum context this is often referred to as the Bloch sphere—which can be embedded in a three-dimensional Euclidean space . The implication of this remarkable fact is that one may select an arbitrary point on the state space and declare this point to be, say, the ‘north pole’. In this manner, each spin degrees of freedom of a spin- particle is mapped, one-to-one, to a direction in three dimensions. This identification is sometimes referred to as the Pauli correspondence, and can be seen in different ways. For example, from (37) one sees that the expectation value of a spin operator (which is one-half of the Pauli matrices) takes a value on a sphere of radius one-half in (see [22, 23, 24] for further discussion on the relation between the spacial dimension of the space-time and the spin of quantum particles).

With this background of standard quantum mechanics in mind, let us now turn to a spin- particle characterised by a Hamiltonian whose eigenstates are not orthogonal. The relevant mathematical machineries have already been introduced above, but let us introduce them here in a slightly different order: Rather than starting from a Hamiltonian , let us start from the specification of the eigenstates. Specifically, suppose that a pair of distinct states is given in a two-dimensional Hilbert space such that . We then find the conjugate pair by solving the equations and , satisfying the norm convention ; solutions will be unique up to overall phases. We then identify the Hamiltonian according to


which, alternatively, can be expressed in the form for some choice of real vector , where is the Pauli-matrix vector obtained by use of the biorthogonal basis, in accordance with (36). This Hamiltonian, although not Hermitian, nevertheless has the interpretation of representing the energy of a spin- particle system immersed in an external magnetic field in .

This result follows from our probability assignment rule (21). To see this, we recall that a generic state of the particle can be expressed in the form (23). Now the spherical coordinates used in (23) show that the two eigenstates and are antipodal points on the Bloch sphere, even though they are not orthogonal in . We have explained that when an experimentalist performs a spin measurement, the direction of the measurement apparatus in is in one-to-one correspondence with the point on the Bloch sphere , not so much with the direction in Hilbert space as such, in the chain of abstraction . To put the matter differently, the data obtained from the Stern-Gerlach experiment (see [25] for a curious historical account of the experiment) does not provide information concerning whether the ‘spin-up’ state and ‘spin-down’ state correspond to orthogonal vectors in ; it merely tells us that they correspond to antipodal points on , whereas going from to requires further milages requiring more information than mere experimental data.

For sure the use of orthogonal bases—hence the use of Hermitian operators—simplifies the algebra, but apart from this ‘convenience’ argument, there is no need to require orthogonality in ; all that is needed is the completeness. We are therefore led to the following conclusion:

Proposition 1

In finite dimensions, the interrelation, i.e. the overlap distances, of the eigenstates of nondegenerate observables with real eigenvalues in Hilbert space cannot be determined from experimental data.

In other words, any operator possessing the relevant eigenvalue structure is a legitimate candidate for a physical observable. Hence Hermitian operators have no privileged status, apart from their ability in making calculations simpler. This conclusion, however, is not necessarily true in infinite dimensions; likewise in finite dimensions, one can identify differences between Hermitian and non-Hermitian observables if at least one of the eigenvalues is complex, or if there are degeneracies of eigenstates. We shall have more to say about these points.

6 Spin particles and combined systems

Particles with higher spin numbers can be formulated analogously. Of course, one might ask, even in the case of standard quantum mechanics with Hermitian observables, in which way spin measurements in can be related to points on the state space since the dimensionality of the state space for higher spin systems is larger than three and hence it cannot be embedded in . The way to realise the Pauli correspondence for higher spin systems is to note the fact that in the state space for each spin, there is a family of privileged quantum states, sometimes called the coherent states, that fully embody information concerning directional data in (see [26, 27] for a detailed discussion), and that the coherent state subspace is always a two sphere that can be embedded in . It is via this device that the idea of the Pauli correspondence for spin- particle can be extended to arbitrary spin particles. To put the matter differently, for higher spins there is a natural embedding of the directional data of in the state space of the system.

It should be evident from the discussion of the preceding section that a similar line of reasoning is applicable to biorthogonal quantum systems. As an example, consider a spin- state vector in , normalised as usual according to . We embed this state in by consideration of the product state:


This coherent state in is then identified as the spin- state in some direction of , which becomes more apparent if we choose the parameterisation and . Clearly is normalised in the sense of (19) since . If we call the positive -direction in , then the triplet of states

corresponds to the three spin- eigenstates of :

An arbitrary state of the spin- particle is therefore expressed as a liner combination of these basis states.

This line of construction extends to all higher spin particles. Thus, for example, for a spin- system we form the coherent state


in associated with , and identify the four states appearing here as the four eigenstates of the spin operator, and so on.

The formulation presented here is somewhat unduly rigid in that if we define a Hermitian matrix , then the Hermitian transition amplitudes—as opposed to the physical transition amplitudes specified by (21)—between the spin eigenstates for all higher spins are entirely specified by the matrix . In other words, the biorthogonal system for all higher spin systems are fixed once we fix that of the underlying spin- system. This rigidity, however, can in fact be relaxed, on account of Proposition 1, which shows that Hilbert space vectors play less prominent role than one might have thought. In particular, in biorthogonal quantum mechanics a coherent state can be constructed from incoherent Hilbert space vectors that are nevertheless projectively coherent. Thus, if is given as before and if we define , where so that and are inequivalent Hilbert space vectors, then we can still form an admissible coherent state according to . This follows on account of the fact that , , hence and are projectively equivalent under our scheme. In this way we see that the biorthogonal basis for each spin particle can be chosen arbitrarily, without constraints.

The observation made in the previous paragraph also shows that in biorthogonal quantum theory an arbitrary pair of systems can be combined without constraints. This, in turn, clarifies one of the outstanding issues of combined systems in PT-symmetric quantum mechanics, which we shall discuss later. For now it suffices to note that if one system represented by a Hilbert space and another system represented by a Hilbert space are combined, then the state vector of the combined system is an element of the tensor product space , just as in Hermitian quantum mechanics. Thus, for example, if is the state of one spin- particle, and is the state of another such particle, then a disentangled product state in takes the form


whereas a typical entangled state, such as the spin- singlet state, will be given by


This might appear paradoxical at first, since the singlet state has to be antisymmetric, which is not immediately apparent from the right side of (45). Indeed, and represent distinct states in , however, they are projectively equivalent, which in turn makes (45) antisymmetric in the projective Hilbert space.

For a combined system, the interaction Hamiltonian can also be represented in a manner analogous to that in standard quantum mechanics. Thus, in the case of a pair of biorthogonal systems represented by a pair of Hamiltonians and with , the quantum Ising spin-spin interaction Hamiltonian can be expressed in the form


whose eigenvalues are, of course, given by , independent of .

7 Perturbation analysis

We shall now turn to the perturbation analysis involving complex Hamiltonians, in the range where there are no degeneracies so that the Rayleigh-Schrödinger series is applicable. There is a substantial literature on perturbation theory involving complex Hamiltonians, even in the vicinities of degeneracies where not only eigenvalues but also eigenstates can be degenerate (see, for example, [16, 17, 28, 29, 30]). As such, we have little new to add in this section, except perhaps the discussion on the nature of the operator that generates the perturbation, which turns out not to be unitary.

Let be a complex Hamiltonian with distinct eigenvalues and biorthonormal eigenstates that are known. Suppose that we perturb the Hamiltonian slightly according to


where is the perturbation parameter, and represents perturbation energy, which may or may not be Hermitian. Under the assumption that there are no degeneracies, the eigenstates and the eigenvalues of the perturbed Hamiltonian can be expanded in a power series


As for the normalisation of the perturbed eigenstates, we shall assume that


Since , it follows that under this normalisation convention we require


It also means that , but the deviation from unity is negligible for .

If we substitute the series expansion (48) in the eigenvalue equation


and equate terms of different orders in , then we obtain


and so on. Transvecting from the left on the second equation of (52) we obtain


Thus, for we obtain the first-order perturbation correction to the eigenvalue:


On the other hand, for we obtain


and on account of the completeness condition we thus find