Royal Society Publishing

Pseudo-Hermitian quantum mechanics with unbounded metric operators

Ali Mostafazadeh

Abstract

I extend the formulation of pseudo-Hermitian quantum mechanics to η+-pseudo-Hermitian Hamiltonian operators H with an unbounded metric operator η+. In particular, I give the details of the construction of the physical Hilbert space, observables and equivalent Hermitian Hamiltonian for the case that H has a real and discrete spectrum and its eigenvectors belong to the domain of η+ and consequently Embedded Image.

Pseudo-Hermitian quantum mechanics is a representation of conventional quantum mechanics that allows for describing unitary quantum systems using non-Hermitian Hamiltonian operators H whose Hermiticity can be restored by an appropriate change of the inner product [1].1 This theory has emerged [39] as a natural framework for examining the prospects of employing non-Hermitian Embedded Image-symmetric Hamiltonians [10], such as H=p2+ix3, in quantum mechanics. Since the publication of Mostafazadeh [3], many authors have studied particular examples and various aspects of pseudo-Hermitian operators, and a series of annual international conferences entitled ‘Pseudo-Hermitian Hamiltonians in Quantum Physics’ have been held (http://gemma.ujf.cas.cz/~znojil/conf/). These have led to rapid progress towards solving the basic problems of the subject, such as developing methods of constructing inner products, determining the observables of the theory, understanding the role and importance of the antilinear symmetries such as Embedded Image-symmetry, and the exploration of the classical limit of pseudo-Hermitian quantum systems. An important technical problem that has resisted a satisfactory resolution is the difficulty associated with the emergence of unbounded metric operators that define the inner product of the physical Hilbert space and the observables of the theory. Indeed, for the majority of the toy models studied in the context of pseudo-Hermitian quantum mechanics, the metric operator turns out to be unbounded, while the structure of the theory is developed and fully understood for systems involving bounded metric operators (with a bounded inverse) [1].

The importance of the complications caused by unbounded metric operators was initially noted in [11] where the authors considered a quasi-Hermitian Hamiltonian operator [12] with a discrete spectrum and proposed to construct the physical Hilbert space of the system by identifying the inner product with the one making the eigenvectors ψn of H orthonormal. This defines an inner-product space on the linear span2 of ψn which can then be (Cauchy-) completed into a Hilbert space [13], p. 7. The main difficulty with this procedure is that it does not give an explicit formula for the inner product of the physical Hilbert space in terms of the information provided by the Hamiltonian operator. Recall that bounded metric operators η+ admit a spectral representation involving a set of eigenvectors of H that form a Riesz basis [1], and the inner product is defined directly in terms of η+. The proposal of Kretschmer & Szymanowski [11] lacks a similar prescription for computing the inner product.

An alternative proposal for dealing with unbounded metric operators is the one given in [9,1]. It is based on the view that in pseudo-Hermitian quantum mechanics the (reference) Hilbert space Embedded Image, which is used to define the Hamiltonian and the metric operator, does not have a physical significance. This follows from the fact that all the states that can be prepared belong to the domain of the observables one can measure. It is well known that observables are generally represented by densely defined linear operators. This in turn means that the physical states are represented by vectors belonging to the intersection Embedded Image of a bunch of dense subsets of Embedded Image. In particular, the physical aspects of a given system are fully determined by a proper dense subset Embedded Image of Embedded Image and an inner product that defines the expectation value of the observables. The main message of pseudo-Hermitian quantum mechanics is that this inner product does not need to be the one that Embedded Image inherits from Embedded Image. The choice of a different inner product on Embedded Image turns it into an inner-product space that can be completed to a Hilbert space Embedded Image. In general, Embedded Image and Embedded Image are different as sets and in particular as topological vector spaces, but both of them include Embedded Image as a dense subset. A simple consequence of this fact is that a metric operator that is defined on Embedded Image may correspond to an unbounded operator with respect to the inner product of Embedded Image while it is a bounded operator with respect to the inner product of Embedded Image. The resolution of the issue of unbounded metric operators that is suggested in [9,1] rests on the idea of using Embedded Image in place of Embedded Image as the reference Hilbert space. In other words, it asserts that because the reference Hilbert space is an auxiliary mathematical construct, one can choose it in such a way that Embedded Image is a dense subset of the reference Hilbert space and the metric operator of interest acts as a bounded operator in it. This proposal is also difficult to implement in practice, because it does not provide the means for an explicit construction of a reference Hilbert space with these properties.

The purpose of the present paper is to outline a resolution of the problem of unbounded metric operators that gives an explicit construction for the physical Hilbert space and the observables of the system in terms of an unbounded metric operator.

I first list the basic assumptions upon which our proposal rests:

  • — Fix an infinite-dimensional reference Hilbert space Embedded Image in which the Hamiltonian and other linear operators of interest act as densely defined closed linear operators. Use the symbol 〈⋅|⋅〉 to denote the inner product of Embedded Image.

  • — Consider Hamiltonian operators Embedded Image with a real and discrete spectrum, so that the linear span of its eigenvectors ψn, which is denoted by Embedded Image, is an infinite-dimensional vector subspace of Embedded Image.

  • — Assume the existence of an unbounded positive-definite operator3 Embedded Image such that H is η+-pseudo-Hermitian [3], i.e. H and its adjoint4 H fulfil the condition Embedded Image 1 In particular, Hη+ and η+H have the same domain.5 An operator η+ with the above properties is called an unbounded metric operator. In view of the positive-definiteness of η+, equation (0.1) implies that H is a quasi-Hermitian operator [14].6

  • — Because η+ is a positive-definite operator, it has a unique positive square root Embedded Image that is also a positive-definite operator with a positive-definite inverse ρ−1 [15], p. 281. Require that the eigenvectors ψn of H belong to the domain of η+ and consequently ρ. This implies that Embedded Image is an infinite-dimensional vector subspace of Embedded Image.7

Similar to the proposal of Kretschmer & Szymanowski [11], I promote Embedded Image to an appropriate inner-product space in which the restriction of H to Embedded Image acts as a Hermitian operator. I do this by endowing Embedded Image with the inner product 〈〈⋅,⋅〉〉 that is defined by the metric operator η+ according to Embedded Image 2 Here, ϕ and ψ are arbitrary elements of Embedded Image, i.e. they are finite linear combinations of ψn, and I have used the fact that η+=ρ2 and ρ is a Hermitian operator. The right-hand side of (0.2) is finite, because according to the last assumption above, ψ and ϕ belong to the domain of η+ and consequently ρ. The inner-product space Embedded Image obtained in this way can be completed to a Hilbert space that I denote by Embedded Image. This is the physical Hilbert space of the pseudo-Hermitian quantum system that we wish to formulate.

We can consider the restriction of H onto Embedded Image and view it as an operator acting in Embedded Image. Then the domain and range of H coincide with Embedded Image, and in light of (0.1), for every pair of elements of Embedded Image, say ϕ and ψ, we have Embedded Image 3 This shows that Embedded Image is a densely defined symmetric operator [13], p. 255. It has also the appealing property of possessing a complete set of eigenvectors. To see this, recall that, because Embedded Image is a dense subset of Embedded Image, there is a complete set of eigenvectors of H in Embedded Image. Moreover, because Embedded Image is symmetric, the eigenvectors belonging to different eigenspaces are orthogonal. We can perform the Gram–Schmidt process on the eigenvectors belonging to eigenspaces of H to construct orthonormal bases for each eigenspace. The union of these is a complete orthonormal set of eigenvectors of H. This is an orthonormal basis of Embedded Image. If we label the eigenvectors belonging to such a basis by ψn with Embedded Image, and the corresponding eigenvalues by En, we have Embedded Image 4 where Embedded Image are arbitrary, |ψi〉〉〈〈ϕi| stands for the projection operator Embedded Image defined by Embedded Image and I denotes the identity operator acting on Embedded Image. Furthermore, because H is a symmetric operator, En are necessarily real.

In general, in order to define a quantum system, one needs a Hilbert space that determines the kinematic aspects of the system and a Hamiltonian operator that specifies its dynamics. The latter is required to be a self-adjoint (Hermitian) operator acting in the Hilbert space so that its expectation value in every state is a real number. In the above construction, I showed that H acts as a symmetric operator in the Hilbert space Embedded Image. But not every symmetric operator is self-adjoint. Indeed, the operator Embedded Image turns out not to be self-adjoint, as its adjoint has a larger domain than that of H [16], pp. 94–95. Therefore, we cannot identify it with the Hamiltonian operator for a unitary quantum system. What we can do is to use an appropriate self-adjoint extension of H for this purpose. This is actually very easy and natural to construct.

Recall that, because Embedded Image is an orthonormal basis of Embedded Image, every element of Embedded Image has the form Embedded Image, where {an} is a square-summable sequence of complex numbers, i.e. Embedded Image. Now, let Embedded Image be the subset of Embedded Image consisting of the elements Embedded Image that satisfy the condition Embedded Image, i.e. Embedded Image Clearly, Embedded Image. Therefore, Embedded Image is a dense subset of Embedded Image. Now, I define Embedded Image as the operator that has Embedded Image as its domain and satisfies Embedded Image 5 It is clear that H is the restriction of Embedded Image to Embedded Image. Furthermore, it is not difficult to show that Embedded Image is a self-adjoint operator [16], p. 94. Therefore, Embedded Image is a self-adjoint extension of H, and the pair Embedded Image defines a unitary quantum system. Again, the physical condition that the expectation values of observables must be real numbers demands that we identify the observables of this system with the self-adjoint operators acting in Embedded Image [1].

The self-adjoint operator Embedded Image is actually the closure of H. Therefore, H is essentially self-adjoint, and Embedded Image is its unique self-adjoint extension [16], p. 96. This shows that the unitary quantum system that I have constructed above is uniquely determined by the quasi-Hermitian operator H and the metric operator η+.

Next, consider the restriction of the operator ρ onto Embedded Image. This gives a one-to-one linear operator that maps Embedded Image into Embedded Image. Because Embedded Image is dense in Embedded Image, we can view this operator as a densely defined operator Embedded Image having Embedded Image as its domain. In view of equation (0.2), this is a bounded operator that can be extended to Embedded Image by continuity.8 According to (0.2), this bounded extension of Embedded Image that I denote by Embedded Image is an isometry [15], p. 257.

Let us use Embedded Image to label the range of Embedded Image, which is a vector subspace of Embedded Image. It is easy to show that Embedded Image is actually a closed subspace of Embedded Image. To see this, take a sequence {ξn} in Embedded Image that converges to some Embedded Image. Clearly, this is a Cauchy sequence, i.e. Embedded Image. Now, let Embedded Image. Then because Embedded Image is an isometry, we have Embedded Image Therefore, {ζn} is a Cauchy sequence in Embedded Image. Because Embedded Image is a Hilbert space, this sequence must converge to some Embedded Image. Now, recall that Embedded Image is a bounded (continuous) linear operator. This implies that the sequence Embedded Image must converge to Embedded Image, and as a result Embedded Image. This completes the proof that Embedded Image is a closed subspace of Embedded Image. The following are important consequences of this fact:

  • Embedded Image is a separable Hilbert space that I label by Embedded Image. In general, this is a Hilbert subspace of Embedded Image. It coincides with Embedded Image provided that the linear span of ρψn is dense in Embedded Image.

  • — The operator Embedded Image defined by Embedded Image is a unitary operator.

  • Embedded Image, which is the same as Embedded Image, is an orthonormal basis of Embedded Image.

We can use Embedded Image and Embedded Image to define a self-adjoint operator acting in Embedded Image that is related to Embedded Image via a similarity transformation. This is the operator Embedded Image. By construction the pairs Embedded Image and Embedded Image are unitary-equivalent, and therefore they represent the same quantum system. Following [8,9], I therefore refer to h as the equivalent Hermitian Hamiltonian to the quasi-Hermitian operator H, and call Embedded Image and Embedded Image the pseudo-Hermitian and Hermitian representations of the quantum system in question, respectively.9

Next, examine the application of our constructions for a very simple and well-known toy model with Embedded Image.

Let Embedded Image, V be a real and even confining potential, p:=−i(d/dx), and Embedded Image 6 This is one of the oldest examples of non-Hermitian Embedded Image-symmetric Hamiltonians that have a real spectrum. It was initially introduced for modelling certain localization effects in condensed matter physics [17], and is one of the earliest examples considered in the framework of pseudo-Hermitian quantum mechanics [18]. For definiteness, I will confine attention to the exactly solvable case where V (x):=ω2x2/2 and Embedded Image. Then H is η+-pseudo-Hermitian for η+:=e2αx. Both η+ and its positive square root, ρ:=eαx, are clearly unbounded positive-definite operators. It is easy to show that the following are eigenvectors of H: Embedded Image 7 where Embedded Image, Nn are normalization constants, and Hn are Hermite polynomials. Also note that Embedded Image 8 and Embedded Image 9 Because η+ψn are square-integrable functions, the ψn belong to the domain of η+, and our constructions apply.

For this model, the Hilbert space Embedded Image is defined by Cauchy-completing the inner-product space obtained by endowing the linear span of ψn with the inner product: Embedded Image 10 According to (0.9), Embedded Image is an orthonormal basis of Embedded Image. This implies that the Hilbert space Embedded Image coincides with Embedded Image, and Embedded Image is a unitary operator. We can also easily show that in this case Embedded Image. Therefore, Embedded Image is a pseudo-Hermitian representation of the simple harmonic oscillator that we usually represent by Embedded Image.

In conclusion, I have offered a mathematically rigorous construction of the physical Hilbert space, the observables and the equivalent Hermitian Hamiltonian for a pseudo-Hermitian quantum system defined by an unbounded metric operator. This construction that applies for quasi-Hermitian Hamiltonian operators H with a discrete spectrum relies on the natural assumption that the eigenvectors of H should belong to the domain of the metric operator. It generalizes the well-known constructions given originally in [9] for bounded metric operators and differs from the latter in the sense that, whenever the metric operator is unbounded, the physical Hilbert space is generally different from the reference Hilbert space not only as inner-product spaces but also as vector spaces and sets. This however does not cause any difficulty. On the contrary, as the above simple example shows, the results reported in this paper show that most of the unjustified and careless treatments of unbounded metric operators that are carried out in the literature on this subject can be put on solid grounds. One may try to extend the constructions given in this paper to indefinite metric operators. This would lead to indefinite-metric quantum theories with an unbounded metric operator whose study requires a separate investigation of its own.

Remark

After the submission of this paper, I was informed of Bagarello & Znojil [19], where the authors also consider unbounded metric operators. They postulate the existence of an equivalent Hermitian operator that in my notation corresponds to ρHρ−1, where Embedded Image acts in the original reference Hilbert space Embedded Image. They further demand that ρHρ−1 has a real discrete spectrum and a set of eigenvectors that form an orthonormal basis of Embedded Image. They call an operator H with the above property ‘well-behaved’ with respect to η+. In general, for a given densely defined closed linear operator H that is η+-pseudo-Hermitian and has a real and discrete spectrum, ρHρ−1 does not satisfy all of these properties. In fact, it may have a domain that is not even dense in Embedded Image. This raises the problem of characterizing these so-called well-behaved operators, an important problem that is not addressed in [19]. By restricting to these well-behaved operators, Bagarello & Znojil [19] essentially circumvent the real mathematical problems that one must face while dealing with unbounded metric operators. In particular, they overlook the need for considering Hilbert spaces that are different from Embedded Image even as sets.

Acknowledgements

I wish to thank Carl Bender for reminding me of the need for addressing the unbounded metric operators, particularly during my talk in the PT-Symmetric Quantum Mechanics Symposium held in Heidelberg, 25–28 September 2011. I would like to express my gratitude to the organizers of this symposium, particularly Maarten DeKieviet, for their hospitality. I am grateful to Ali Ülger and Gusein Guseinov for their most illuminating comments and suggestions. This work has been supported by the Turkish Academy of Sciences (TÜBA).

Footnotes

  • One contribution of 17 to a Theme Issue ‘𝒫𝒯 quantum mechanics’.

  • 1 Throughout this paper, I follow von Neumann’s terminology of using the term ‘Hermitian operator’ to mean ‘self-adjoint operator’ [2], p. 96.

  • 2 The linear span of ψn is the set of all finite linear combinations of ψn.

  • 3 A positive-definite operator, Embedded Image, is a self-adjoint operator such that for every non-zero element ξ of its domain the real number 〈ξ|πξ〉 is strictly positive.

  • 4 I do not identify linear operators with their matrix representations in some basis, and H does not mean the complex conjugate of the transpose of a matrix. I use the standard mathematical definition of the adjoint of a linear operator [13], p. 252. Namely, let Embedded Image denote the domain of H (which is supposed to be a dense subset of Embedded Image) and Embedded Image. Then Embedded Image is the linear operator with domain Embedded Image that satisfies the condition: Embedded Image and Embedded Image, 〈ϕ|〉=〈Hϕ|ψ〉. We say that H is Hermitian or self-adjoint if Embedded Image and for all Embedded Image, 〈ϕ|〉=〈|ψ〉. We say that H is a symmetric operator if the latter condition holds but Embedded Image. In general, Embedded Image may not coincide with Embedded Image. Therefore, not every symmetric operator is Hermitian.

  • 5 This means that for all Embedded Image the statement: ‘Embedded Image and Dom(η+)’ is equivalent to ‘ψDom(η+) and Embedded Image’.

  • 6 In [14] and other rigorous studies of the subject, quasi-Hermiticity is introduced using equation (0.1) except for the fact that η+ is taken to be a bounded (continuous) linear operator. I use the term quasi-Hermitian in the more general sense where η+ is allowed to be unbounded.

  • 7 One can slightly relax the conditions on η+ by requiring that it is the square of a given invertible (one-to-one) symmetric operator ρ with Embedded Image contained in the domain of η+, and (0.1) holds on Embedded Image.

  • 8 This is done by defining ρ(ξ) for each Embedded Image by taking a sequence {ξk} in Embedded Image that converges to ξ and identifying ρ(ξ) with the limit of the sequence {ρ(ξn)}.

  • 9 The elements of Embedded Image do not enter the formulation of the quantum system defined by the pair Embedded Image. They may be viewed as representing ‘unobservable states’, because they do not belong to the domain of the observables.

References

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