an introduction and brief topical review of non-hermitian quantum mechanics with real spectra

THE UNIVERSITY OF HULL

AN INTRODUCTION AND BRIEF TOPICAL REVIEWOF NON-HERMITIAN QUANTUM MECHANICS

WITH REAL SPECTRA

being a Thesis submitted for the Degree ofMaster of Science (by research)

in the University of Hull

by

David Richard Gilson, MPhys

March 2008

AN INTRODUCTION AND BRIEF TOPICAL REVIEW OF NON-HERMITIAN QUANTUM MECHANICS WITH REAL SPECTRA

Acknowledgements

This work originally began as a PhD project in 2001 with the Department of Mathematics at the

University of Hull, under the supervision of Professor Paul Busch and Dr. Stefan Weigert. I extend my

gratitude and respect to both of them for their guidance at that time.

After my first year with the department, I developed Chronic Fatigue Syndrome (CFS), which led to

an indefinite intercalation from my PhD. Later, the Department of Mathematics closed and reopened

as the “Centre for Mathematics” (CfM). During this transition phase, I decided that I would not let

CFS stop me achieving my ambitions. I contacted the university to enquire if I could resume my

research, by reducing my PhD to a MSc degree. I extend my gratitude to both Dr. Nigel Shaw (Hull

University Graduate School) and Dr. Tim Scott (head of the CfM) for having agreed to this. Hence,

from September 2005 I was able to embark upon rehabilitative study.

Since I resumed my research, I have faced many difficulties. Not least, that of learning to live with

CFS. Despite all of this, I was determined to see this project through. However, I must give the

following thanks to those people who have supported and guided me in this endeavour. Dr. Stefan

Weigert for continued academic and intellectual support; and proof reading my first draft; all despite

having moved on from Hull University. Dr. Tim Scott for administrative support and proof reading.

Avril Johnson for learning support. Finally, I would also like to extend thanks and admiration to my

mother, Carol Gilson, who has given unwavering moral support and many hours proof reading

throughout this project, despite her own ill health.

I also wish to cite the computer software used in the production of this thesis. All of the software listed

here is free and open source, released under either the GNU Public Licence (GPL) or the Limited GPL

(LGPL). All diagrams were produced with “The GIMP” (GPL, http://www.gimp.org/) and “Inkscape”

(GPL, http://www.inkscape.org/). The bibliographic database was created in “OpenOffice.org Base”,

all mathematical objects were written in “OpenOffice.org Math” and the thesis was completely written

and managed in “OpenOffice.org Writer” (LGPL, http://www.openoffice.org/).

David R Gilson 2 of 77

http://www.gimp.org/

http://www.openoffice.org/

http://www.inkscape.org/


Abstract

This thesis begins by introducing the postulates of standard quantum theory. It then recounts

historical attempts to break away from the limitation of Hermiticity, while preserving the

requirements of a real spectrum and unitary time evolution. It goes on to report how non-

Hermitian theories have been put to use in recent years, as academic interest has grown

markedly. The thesis proceeds to examine in detail the main non-Hermitian frameworks (PT

symmetry and pseudo-Hermiticity), their similarities, and their differences. It goes on to

investigate the limitations of such theories, by examining how the reality of their

eigenspectrum can breakdown. Finally, an original contribution is given, demonstrating a

method to calculate “exceptional points” of a PT-symmetric version of the Lipkin model.



Table of contentsChapter 1. Introduction............................................................................. 5

1.1. Organisation of thesis......................................................................................................51.2. Mathematical language and notation...............................................................................61.3. Physical motivation for quantum mechanics...................................................................7

Chapter 2. Postulates of quantum mechanics........................................ 112.1. Quantum states.............................................................................................................. 112.2. Time evolution.............................................................................................................. 122.3. Quantum measurements................................................................................................ 152.4. Composite systems........................................................................................................ 18

Chapter 3. Breaking away from Hermiticity.............................................193.1. Historical background of non-Hermitian quantum mechanics......................................193.2. Applications of non-Hermitian theories........................................................................ 223.3. Requirements for a non-Hermitian Hamiltonian...........................................................25

Chapter 4. Non-Hermitian frameworks of quantum mechanics.............. 264.1. PT and CPT symmetry.................................................................................................. 264.2. Pseudo-Hermiticity........................................................................................................344.3. Comparisons of CPT symmetry and pseudo-Hermiticity............................................. 404.4. Comparisons of PT symmetry of Hermiticity............................................................... 43

Chapter 5. Studies on spontaneous breakdown of PT symmetry...........575.1. PT symmetry breakdown...............................................................................................575.2. Example: calculating exceptional points in the Lipkin model...................................... 59

Chapter 6. Conclusion............................................................................ 706.1. Summary....................................................................................................................... 706.2. Comments and outlook..................................................................................................72

Appendix A. Computer code................................................................... 74References............................................................................................. 75



Chapter 1. Introduction

1.1. Organisation of thesis

In a study of the literature on non-Hermitian frameworks of quantum theory, a large number

of papers could have been included in a review. Therefore, I set the following criteria:

1. Papers published before 1998 were included if they were useful in setting the

historical context of the field of study.

2. Papers published since 1998 were included if they:

(a) Introduced original ideas to the field.

(b) Provided a basis on which original ideas were later brought to the field.

(c) Provided retrospective proof of earlier original ideas.

The organisation of this thesis is as follows:

Chapter two gives a detailed mathematical statement of the postulates of quantum mechanics.

The postulates are written with the presumption that the reader has a solid grasp of linear

algebra but is not familiar with the physical concepts of quantum mechanics.

Chapter three gives a historical account of non-Hermitian quantum mechanics research since

1959, going as far as research in 2006. The historical picture shows how non-Hermitian

theories were used before the current frameworks were developed and how this was mostly

done on phenomenological grounds. The chapter goes on to state the requirements that a non-

Hermitian quantum mechanics theory must meet, with reference to the postulates given in

chapter two. Finally, chapter three reports on how non-Hermitian theories have been put into

practical application.

David R Gilson Chapter 1. Introduction 5 of 77


Chapter four gives a detailed examination of the non-Hermitian frameworks developed since

1998. In particular, these frameworks involve “PT/CPT symmetry” (Bender et al, 1998) [1],

and “pseudo-Hermiticity” (Mostafazadeh, 2002) [2]. To ensure that a firm understanding of

these frameworks is conveyed, the chapter reviews papers in which they were introduced. The

chapter continues by detailing two crucial criticisms of PT-symmetric quantum mechanics

(PTSQM), the first being that pseudo-Hermiticity is an encompassing set of theories to

PTSQM, and the second being that PTSQM is not an extension to the condition of

Hermiticity.

Chapter five looks more closely at PTSQM and discusses methods for identifying valid

PTSQM theories. It includes reviews of suggested ways to check for broken or unbroken PT

symmetry and conservation of probability. The chapter continues by discussing a method of

detecting symmetry breaking by locating “exceptional points”. I conclude the chapter by

giving a report of a method which I created for analytically calculating exceptional points of a

PT-symmetric version of the Lipkin model.

Finally, chapter six gives a summary of the thesis.

1.2. Mathematical language and notation

The notation used for various mathematical objects will now be defined. This thesis will make

exhaustive use of the Dirac bra-ket notation, since this is a useful way of writing inner

products, although it does differ in principle from the traditional mathematical way of writing

vectors and inner products.

A n-dimensional vector in a Hilbert space H is defined as the “ket” vector, where 1,, n∈ℂ :



∣ ⟩=1

⋮n (1)

The “bra” vector is the complex-conjugate and matrix transpose of the ket vector (the

superscript * denotes complex conjugation):

⟨∣=1* , , n

* (2)

Locking the bra Eq.(2) and the ket Eq.(1) together forms a bra-ket, an inner product. As an

example, here is the inner product of with itself:

⟨∣ ⟩ =1

* n* 1

⋮n = ∑

i=1

n

∣i∣2 = ∥∥2 (3)

Having introduced this form of notation, one can assume all the normal axioms of linear

algebra to be valid.

Symbols for linear operators are distinguished from vector space symbols by a hat, such as H

. Operators have an adjoint H † , often referred to as their “Hermitian conjugate”, produced by

complex conjugation and transposition of their matrix representation.

1.3. Physical motivation for quantum mechanics

At the turn of the 20th century most fundamental problems in physics were thought to have

been addressed. Around that time, physicists were considering the theoretical problem of

“black bodies”. A black body is defined as an object that absorbs all incident electromagnetic

radiation, reflecting none, and is capable of emitting any wavelength, with the energy of any

wavelength being a function of temperature. It was this study that would sow the seeds of

quantum theory. Physicists then broadly identified themselves as either “energeticists” or

“atomists”, a distinction that would persist into the era of quantum theory.

The first attempt at calculating the energy output of a black body was by Wien in 1896 [3].



His result was widely accepted because it agreed with contemporary experimental

measurements, and was named “Wien's Law”. However, it did depend on two empirical

parameters and hence it was a purely phenomenological formula. Owing to the lack of a

theoretical basis, the formula was soon modified by Max Planck, and became known as the

“Wien-Planck Law”. This work was ultimately in vain, since experiments showed that the

Wien-Planck Law did not agree with experimental measurements at low frequencies.

Max Planck set about formulating a new law to account for the low frequency observations.

In 1901, he published a paper [4] showing a function that became known as “Planck's Law”,

quoted here for interest [3].

I ,T =2h3

c21

eh/k T−1 (4)

The key concept behind this function is the quantization of energy. This was the first time that

energy had been treated as discrete and “quantized”; this was something inexplicable in

classical physics. Planck regarded it as merely a mathematical device to fit experimental

evidence. He was quite vague as to the nature of how or why energy was quantized, despite

many references to “Hertzian ocillators” [4]. Planck's law enjoyed great success with

experimental predictions, unlike the Wien-Planck Law.

It was not until 1905 that anyone suggested that nature itself might be quantized. Albert

Einstein took Planck's idea of quantized energy, and suggested that this would apply to light.

He proposed that packets of energy would come in discrete quanta of light (named

“photons”). This was the basis of the explanation of the “Photoelectric Effect” [5], for which

Einstein later received a Nobel Prize in 1921.

Until Einstein, the quantization of energy and “energy jumps” had only been considered as a

mathematical construct, not as something to do with the actual physical interaction between



matter and radiation [3]. The deep philosophical issue raised by Einstein's paper, which

caused it to be rejected by some, was the contradiction of the traditional picture of light as a

continuous wave. Suddenly light had begun to seem more like a particle; it was no longer a

non-localised wave of energy.

After 1905 it was only a matter of time before more physicists would embrace the idea of

quantization. In 1913, Neils Bohr revolutionised atomic physics by introducing a quantized

model of the hydrogen atom. In this model, the orbiting electron could only have discrete

“orbitals” around the proton [6]. It would require a specific quantum of light to kick the

electron from one orbital to a higher one, in a process of absorption.

In 1924, Louis DeBroglie wrote a PhD thesis [7] which further blurred the boundary between

waves and particles. He proposed that, since photons had both momentum and wavelength,

then one could assume that particles would also have some form of wavelength, a “matter

wave”. This became known as “wave particle duality”. The mathematical expression of this is

quoted below:

=hp (5)

In 1925, two complete forms of quantum theory were published, each one having been

developed independently of the other. Erwin Schrödinger published his “wave mechanics”

[8], which can be thought of as naturally leading on from the DeBroglie matter waves. The

“Schrödinger Equation” is a second order differential equation. From this, quantum states are

related to “wave functions” which obey this equation. A much more abstract theory called

“matrix mechanics” was published by Werner Heisenberg [9]. Matrix mechanics works by

describing quantum systems in terms of matrices. It is the linear algebra of matrices and

vectors that give rise to the characteristic “quantum jumps” between states which are

described using “eigenvalues” and “eigenvectors”. The study of matrix mechanics led to the



“Copenhagen Interpretation”, which says that one should interpret quantum theory as a

probabilistic theory. In wave mechanics the square of the amplitude of the “wave function”

should be interpreted as a probability density. After closer analysis it can be shown that these

two different early forms of quantum theory are actually equivalent. This thesis will be

focused on the matrix mechanics approach.



Chapter 2. Postulates of quantum mechanics

2.1. Quantum states

Quantum states are associated with state vectors, which exist within a state space. The length

of a state vector is always equal to one; it is the orientation (“phase”) of a vector that defines

the state. State spaces in quantum mechanics are defined over complex Hilbert spaces, which

often have infinite dimensions. The use of dimensions should not be confused with spatial

dimensions. In a state space each dimension represents a distinct state of the system and an

infinite dimensional space simply represents a system with an infinite number of possible

states.

The basis over the state space is always orthonormal, this allows states to be clearly defined.

It is always possible to form an orthonormal basis in any Hilbert space by applying the Gram-

Schmidt process. If V is some Hilbert space which is spanned by the set of vectors

{∣v1⟩ , ,∣vn⟩} , then we can construct another set of spanning vectors {∣u1⟩ , ,∣u n⟩} in which all

vectors are mutually orthogonal i.e., ⟨ui∣u j ⟩=i , j . (The Kronecker Delta function, i , j is equal

to one for i= j and equal to zero for i≠ j .) Then, each member of the set {∣u1⟩ , ,∣u n⟩} can

be normalised, thus creating an orthonormal basis set {∣e1⟩ , ,∣en⟩} . In general, we can

express each member of the normalised set as:

∣e i⟩=∣ui ⟩

⟨ui∣u j⟩ (6)

Each member of the orthogonal set ∣ui ⟩ is expressed in terms of the ∣v i ⟩ via the Gram-

Schmidt equation:

∣ui ⟩=∣vi ⟩−∑j=1

i−1 ⟨vi∣e j⟩⟨e j∣e j ⟩

∣v i ⟩ (7)

David R Gilson Chapter 2. Postulates of quantum mechanics 11 of 77


There are two types of states, pure states and mixed states. The distinction between these is a

matter of statistics. Both types may be assigned a “density operator”, which is denoted by .

A pure state is a clearly defined state ∣⟩ , whose density operator is a simple projection

operator, and as such is an idempotent transformation. Therefore, for the state ∣⟩ , let the

corresponding density operator be:

=∣ ⟩ ⟨∣ (8)

This has the trivial effect of projecting the state ∣⟩ back onto itself:

∣⟩ = ∣⟩ ⟨∣ ⟩= ∣⟩ (9)

Mixed states represent systems whose state is not precisely known. The state is then expressed

as a linear combination of pure states ∣ ⟩mixed=∑ pi∣i ⟩ , where the coefficients pi are the

corresponding probability for each state. Similarly, the mixed state density operator is a linear

combination of all the possible pure state density operators:

mixed=∑ pi∣i ⟩ ⟨i∣ (10)

Since the pi are probabilities, we may define them as members of the set:

{pi∈ℝ+∣∑ pi=1} (11)

Given the outer product structure of mixed in Eq.(10), we can infer that it has a diagonal

matrix, where the set {pi } are its eigenvalues. Given Eq.(11), we can see that mixed has a

normalised trace. Because mixed would be unaffected by complex conjugation and matrix-

transposition it is a self-adjoint operator.

2.2. Time evolution

The time dependent Schrödinger equation is the most general way of describing how a state

changes with time, where ℏ is Planck's constant h divided by 2 :



i ℏ ddt ∣t ⟩= H∣t ⟩ (12)

This is the key equation in the Schrödinger picture of quantum mechanics in which the time

dependence of a dynamic system is determined by the system's state vector. This is in contrast

to the Heisenberg picture, where the time dependence is determined by that of the linear

operators which act as the physical observables of the system (observables are explained in

the third postulate).

In Eq.(12), H is the Hamiltonian of the system. This is the operator that represents the total

energy of a quantum system. As with all other observable quantities in quantum mechanics, it

has a spectral decomposition (see section 2.3, the third postulate, below):

H=∑ En∣En⟩ ⟨En∣ , {En∈ℝ∣En≥0} (13)

The Schrödinger equation, Eq.(12), is a first order differential equation with respect to time.

Its solution can be used to show how the state of the system changes between the times t1 and

t2 for a time independent H :

∣t 2 ⟩=e−

iℏ

H t 2− t 1

∣t 1⟩ (14)

This special case can be described as a discrete evolution, which is the change of state

between two distinct points in time. The exponential factor acts as a unitary operator. Unitary

operators are an important class of operators which preserve the norm of vectors and can be

pictured as acting to rotate a vector about its point of origin. They have the following

property:

U U †= U † U=I (15)

where U † is the Hermitian conjugate (complex-conjugate and matrix transpose) of U . Given

Eq.(15), we can see that unitary operators are invertible and that the Hermitian conjugate of a

unitary operator is equivalent to its inverse.



We therefore define:

U :=e−

iℏ

H t 2−t 1 (16)

and can rewrite Eq.(14) as:

∣t 2 ⟩= U∣ t 1⟩ (17)

So far, the explanation of this postulate has been implicitly restricted to pure states. However,

it is possible to expand the time dependent Schrödinger equation to account for mixed states.

Imagine that we have a mixed state which is varying with time. Recalling Eq.(10), we can

write the mixed state at two points in time:

1=∑ pi∣t 1 ,i ⟩ ⟨t 1, i∣ (18)

2=∑ pi∣t 2 , i⟩ ⟨t 2 , i∣ (19)

Substituting Eq.(17) into Eq.(19) gives:

2=∑ piU ∣t 1 ,i ⟩ ⟨t 1, i∣ U † (20)

If we now substitute Eq.(18) into Eq.(20) we obtain the relation:

2= U 1U † (21)

If we let t1 be zero, and t2 be a general time t, then we can make the definition:

t := U U † (22)

Let us now differentiate Eq.(22) with respect to time and substitute in Eq.(16):

ddt

= iℏ

e− i H t

ℏ H ei H tℏ − i

ℏH e

− i H tℏ e

i H tℏ (23)

If we then collect terms and re-substitute Eq.(16) we obtain:

ddt

t = iℏ U H U †− H U U † (24)

Eq.(24) can be rewritten with the commutator of H and ; this is possible because U and

H commute. The total energy of a closed system has to be a conserved quantity, and



therefore invariant with any transformation over time:

ddt

t = iℏU [ H , ] U † (25)

We can now make use of Eq.(22):

ddt

t = iℏ[ H , t ] (26)

This is the extension of the Schrödinger equation for mixed states. However, another

interesting observation can be drawn from this. Considering that the density operator of a

mixed state is an operator composed of states (as outer products), we have constructed an

operator that has time dependence, even though we have been looking at the Schrödinger

picture of quantum mechanics where time dependence is meant to be within state vectors!

Considering Eq.(26), we can see that t is a Hermitian operator, and could be replaced by

any other Hermitian operator. This almost reconstructs the Heisenberg picture of quantum

mechanics. The only term missing is the classical time derivative, because a quantum state

has no classical analogue:

ddt

At = 1i ℏ

[ H , At ]∂ A∂ t classical

(27)

2.3. Quantum measurements

The third postulate explains how quantum states are affected by measurement. The second

postulate shows how states change with time in a closed system. However, the act of

measurement introduces an external system, and any measuring apparatus is simply another

physical system. Therefore, the act of measurement involves an interaction between the

apparatus and the system of interest. This is a concept that does not exist in classical physics

and is not observable for macroscopic objects.

In the formalism of linear algebra the act of measurement is expressed by using linear



operators, which are called observables. As would be expected of such an operator class, they

have eigenvectors (also called eigenstates) and eigenvalues. Observables are required to

correspond to physically measurable quantities, which means that their eigenvalues must be

entirely real. Therefore, text book quantum mechanics requires that observables are self-

adjoint (Hermitian) operators, so that their eigenspectrum is guaranteed to be real.

Given that observables are Hermitian, they can be expressed in terms of their spectral

decomposition:

M =∑mi∣mi ⟩ ⟨mi∣ (28)

From Eq.(28), the set {∣m1, , n⟩∈H ∣⟨mi∣m j⟩=ij} is the set of eigenstates of M , which could also

be called an eigenbasis of the state space. Each possible state of the system can be thought of

as a one-dimensional subspace of the state space. Therefore, ∣mi ⟩ ⟨mi∣ is a projector onto the ith

eigenstate. Hence, for a state ∣⟩ , the probability p mi of measuring the ith eigenvalue mi is

defined as:

pmi:=⟨∣mi⟩ ⟨mi∣ ⟩ (29)

If we were to rewrite a state in terms of the eigenbasis of a measurement operator (an

observable), then we would find the state to be in a superposition (linear combination) of the

operator's eigenstates. Measurement of the state by this operator would then project the

superposition into a single eigenstate; this is how a state is changed by the act of

measurement.

After the measurement the state becomes:

∣new⟩=∣mi⟩ ⟨mi∣ ⟩

⟨∣mi⟩ ⟨mi∣ ⟩ (30)

As we can see, the state becomes an eigenstate of the observable, to within a factor of minus

one.



For measurement operators, we define a quantity called the expectation value, which can be

thought of as a weighted mean over a series of repeated measurement results, it is denoted by

⟨ M ⟩ :

⟨ M ⟩ := ∑mi p mi= ∑mi ⟨∣mi⟩ ⟨mi∣ ⟩= ⟨∣∑mi∣mi ⟩ ⟨mi∣∣ ⟩

⟨ M ⟩ := ⟨∣ M ∣ ⟩

(31)

Similarly, we follow the same process for mixed states but treat it as a linear combination of

expectation values by recalling Eq.(10):

tr M = ∑ pi ⟨i∣ M ∣i⟩= ∑ pi ⟨ M ⟩i

(32)

Given the mean measurement, we can calculate the standard deviation (or error) for

measurements:

M =⟨ M 2⟩−⟨ M ⟩2 (33)

Furthermore, the Heisenberg uncertainty principle shows how the uncertainty (i.e., error) of

two non-commuting observables are related. The result can be derived by use of the Cauchy-

Schwarz inequality, as shown in many text books; the result is quoted here:

A B≥ 12∣⟨∣[ A , B ]∣⟩∣ (34)

Eq.(34) is applied to quantum theory by substituting into the equations the various canonical

commutation relations for quantum observables. For example, the commutator of the position

and momentum operators is:

[ x , p ]=i ℏ (35)

By substituting Eq.(35) into Eq.(34), we can calculate the product of the standard deviations

of repeated position and momentum measurements:



x p≥ℏ2 (36)

2.4. Composite systems

When we wish to join two or more systems together, we define the new state space as the

tensor product of all of the individual state spaces. State vectors in the composite space are

made up of tensor products of state vectors from the component state spaces. This way of

expressing joint systems leads to the definition of entangled states.



Chapter 3. Breaking away from Hermiticity

3.1. Historical background of non-Hermitian quantum mechanics

In 1959, Wu published a paper calculating the ground state energy of “Bose spheres” [10].

According to the paper, a common problem with that type of calculation was that the ground

state energy was “divergent”. Wu found that using a non-diagonalisable non-Hermitian

Hamiltonian avoided this problem. Remarkably though, this Hamiltonian possessed real

eigenvalues. However, the paper offered no justification for introducing such a Hamiltonian,

other than that it gave the required solution. In particular, it gave real numbers representing

low-lying energy levels of a Bose system. This paper was the earliest use of a non-Hermitian

Hamiltonian found in the literature survey for this thesis.

In 1967, Wong published a paper about “Physically reasonable non-Hermitian Hamiltonians”

[11]. He made the point that Hamiltonians of closed systems are Hermitian, but that when an

external interaction is considered the Hamiltonian loses its Hermiticity. Therefore, his class of

physically reasonable Hamiltonians would be of a perturbed type:

H= H 0g H (37)

Here, H 0 is a Hermitian Hamiltonian, H is a non-Hermitian Hamiltonian, and g is simply a

parameter to vary the influence of H . H has the restriction that it may only have a discrete

spectrum unless part of its spectrum coincides with that of H 0 . He calls this class

“dissipative” but does not fully define the term. The paper derives several propositions

relating to Hamiltonians in this class. However, at no point in the paper is the reality of the

spectrum mentioned. Moreover, complex eigenvalues seem to be admitted, yet there is no

explanation of how they could possibly be physically reasonable.

A paper in 1969 by Wu and Bender [12] discusses an example of the anharmonic oscillator.

David R Gilson Chapter 3. Breaking away from Hermiticity 19 of 77


The paper does not explicitly discuss Hermitian or non-Hermitian Hamiltonians. The

Hamiltonian for this model is (see appendix A of [12]):

H=12̇21

2m224 (38)

In [12], the parameter is extended into the complex plane, so the non-Hermiticity of their

Hamiltonian is implicit. Coincidentally, this approach was taken because perturbation

methods of calculating the ground state of this system were also divergent. (Later

renormalised perturbation theory cured this problem.)

In 1975, Haydock and Kelly [13] published a letter reporting that they had used the “recursion

method” with non-Hermitian Hamiltonians to calculate the electronic structure of crystalline

Arsenic. According to this letter, “chemical pseudo-potential theory” often gave rise to a non-

Hermitian representation of interactions between localised electron orbits. (See [13] for

further references.) This was one of the earliest papers to say explicitly that the condition of

Hermiticity was sufficient but not necessary to ensure a real eigenspectrum.

In 1980, Stedman and Butler [14] published a paper reviewing material in the field of time

reversal and point group theory. In particular, focussing on the effect and selection rules for

time reversal in the rotation group SO3. This was the earliest reference found to the term “time

reversal operator” in relation to complex conjugation, as it is used in PT/CPT symmetry.

In 1981, Faisal and Moloney [15] studied a quantum decay process with a non-Hermitian

Hamiltonian and Schrödinger equation. They claimed that, as a consequence of the

uncertainty principle, a decaying state could not have a sharp energy, and that the width of

such an energy level could be represented by an imaginary energy component. Furthermore,

such complex energies can be shown to be eigenvalues of a non-Hermitian Hamiltonian

associated with the decay process. The paper also includes a non-unitary yet self-consistent



(probability conserving) algorithm for time evolution.

These are just some examples of work from 1959 to 1998, which use non-Hermitian theories.

The common factor in all of these papers is that such theories were introduced on

phenomenological or heuristic grounds. In other words, such theories were chosen because

they fitted with experimental observation. From a pragmatic point of view, merely fitting

theories to observations could be considered reasonable. After all, nature is always correct!

However, papers which allowed the use of a complex spectrum (e.g., energy eigenvalues)

never appear to justify or elucidate the validity of such spectra. Therefore, since the prediction

of eigenvalues is the crucial link between theory and experimental observation, it would seem

highly questionable to allow eigenvalues which could not be physically measured.

A 1997 paper by Hatano and Nelson [16], did however, justify the use of a complex spectrum.

The relevant passage is quoted here (context: applying depinned field lines of a non-

Hermitian external magnetic field to a type II semiconductor):

“We can interpret the appearance of the imaginary part of the energy in the

following way. A depinned flux line in a periodic system has a spiral

trajectory and hence periodicity in the imaginary time direction. (See Fig.

6.) Because the single-line partition function at ‘‘time’’ t may be written

Z =∑n cnnRx e-n /ℏ where the constants cn are coefficients depending

on ‘‘initial conditions’’ at, say, the bottom surface of the sample, the period

of motion in the imaginary time direction associated with the nth eigenstate

is given by ℏ/ Im n , where n denotes the wave function describing the

depinned flux line. Thus a complex energy, as well as an imaginary part of

the current (2.19), is an indicator of the depinning transition in periodic

systems.”



Throughout the papers mentioned here, there was no fundamental or analytical basis on which

non-Hermitian theories were chosen. Non-Hermitian theories with real spectra which did have

any sort of analytical basis did not appear in the literature until 1998.

3.2. Applications of non-Hermitian theories.

The following are examples of ways in which non-Hermitian theories have been used.

As mentioned above, Hatano and Nelson [16] used a non-Hermitian magnetic field

Hamiltonian for type II semiconductors. In Bender's first publication on PTSQM [1] he

mentions that non-Hermitian theories had even been used for a theoretical model of

population biology.

In 1998, Cannata et al [17] published a paper about Schrödinger operators with complex

potentials having real spectra. The introduction of the paper makes many references to noted

uses of non-Hermitian theories; it is quoted here for reference (please refer to the source for

their references):

“quantum systems characterized by non-Hermitian Hamiltonians are of

interest in several areas of theoretical physics. For example, in nuclear

physics one studies standard Schrödinger Hamiltonians with complex-

valued potentials, which in this connection are called optical or average

nuclear potentials. Non-Hermitian interactions are also discussed

in field theories, for example, when studying Lee-Yang zeros. Even in recent

studies on localization-delocalization transitions in superconductors and in

the theoretical description of defraction of atoms by standing light waves

non-Hermitian Hamiltonians are of interest.”



In 2003, Killingbeck and Jolicard wrote an extensive two part topical review of Bloch wave

operators. Part one [18] references the use of perturbation series for a non-Hermitian

“effective” Hamiltonian in nuclear theory. It also references non-Hermitian Hamiltonians

being used in iterative methods to calculate a wave operator with second order convergence.

Part two [19] mentions an optical potential which produces a non-Hermitian Hamiltonian with

an associated bi-orthogonal basis set. It also references an iterative integration method which

can use non-Hermitian Hamiltonians. Although the Bloch effective Hamiltonian is non-

symmetric it can give real eigenvalues.

In 2005, Znojil [20] wrote extensively about experiments in PTSQM. He cited several

phenomenological uses of PTSQM: Field theory, Nuclear physics, Many body theory,

Condensed matter physics, Cosmology and Magneto-Hydrodynamics. He noted the reluctance

of experimentalists to weaken their reliance on Hermitian phenomenological models. He

suggested that “we” should look for new observables and that PTSQM is enriching the

“laboratory of solvable models”. He summarised by saying that new horizons are opening up,

both for relativistic quantum mechanics and for interpretations of the Einstein-Podolsky-

Rosen paradox.

In a 2004 paper, Ben-Aryeh [21] discusses the use of non-Hermitian operators to treat

scattering phenomena in atomic and molecular physics. The paper actually applies both

PTSQM (Bender et al) and pseudo-Hermitian Hamiltonians (Mostafazadeh) to Rabi-

oscillations of two-level atomic systems.

Bender et al published a paper in 2004 [22], in which they stated that it was unknown whether

any non-Hermitian Hamiltonians can be used to describe any experimentally observable

phenomena, although they had already been used to describe interacting systems [10]. In this

paper they suggested that “an experimental signal of a complex Hamiltonian might be found



in the context of condensed matter physics”. Elaborating on this, they suggested the example

of a complex crystal lattice for which the potential would be expressed as:

V x=i sin x (39)

The Hamiltonian would then be:

H= p 2 V x (40)

It would have a real energy spectrum but would be PT-symmetric, not Hermitian. In

Hermitian lattices, the wave function for a particle at the edge of the lattice would be 4

periodic (Fermionic). For the PT-symmetric case, such a wave function would be 2

periodic (Bosonic).

In 2004 Weigert [23] noted some applications of non-Hermitian Hamiltonians to the

description of absorptive optical media, inelastic scattering from nuclei, and other loss

mechanisms on the atomic or molecular level. Weigert also noted that non-Hermitian theories

had recently been “rediscovered” within particle physics (refer to the paper for these

references).



3.3. Requirements for a non-Hermitian Hamiltonian

The following are the requirements for a physically acceptable non-Hermitian theory:

1. The eigenspectrum must be entirely real.

2. Time evolution should be unitary, to prevent “Probability leakage” (Bender et al [24],

[22]).

• Weigert [23] suggested an exception in which imaginary potentials can allow for

consistent, but locally non-unitary, time evolution.

3. The set of eigenstates must be complete.

• This would be determined by the diagonalisability of the Hamiltonian (see

Weigert: 2005 [25] and 2006 [26]).



Chapter 4. Non-Hermitian frameworks of quantum mechanics

4.1. PT and CPT symmetry

PT and CPT-symmetric quantum mechanics is an alternative formulation of quantum

mechanics which has been principally developed by Bender et al. It has so far been the most

popular non-Hermitian form of quantum mechanics, as indicated by the number of

publications on the subject.

PTSQM was first introduced in a 1998 paper by Bender and Boettcher [1]. Their paper was

based on a conjecture, made in a private communication from Bessis, that the following

Hamiltonian, while not being Hermitian, would have a real and positive spectrum:

H= p 2x2−i x3 (41)

where p is the momentum operator and x is the position coordinate.

The paper went on to make the claim that switching to PT symmetry promised “new infinite

classes of complex Hamiltonians whose spectra are real and positive”.

The verification of the claims made about the spectrum of Eq.(41) relied on numerical

methods, since there was no theoretical proof. Analytic methods were only used briefly to

show how the eigenvalues could change from real to complex values.

The meaning of “PT” symmetry comes from the invariance of PT-symmetric Hamiltonians

under both parity and time reversal transformation. The parity operator, P effects spatial

reflections [1]:

P : x −xp − p (42)

The time-reversal operator T has the following effects [1]:

David R Gilson Chapter 4. Non-Hermitian frameworks of quantum mechanics 26 of 77


T : x xp − pi −i

(43)

Clearly from these definitions we can obtain the following identities:

P 2=1 (44)

T 2=1 (45)

In PTSQM, it is not necessary for a Hamiltonian to be invariant under either P or T

individually. As can be seen, Eq.(41) is invariant under neither of these, although it is

invariant under both operations combined. According to [1], Eq.(42) and Eq.(43) show that

the canonical commutation relation, [ x , p ]=i I still holds. From [27], this must also be true if

we were to have a complex position and momentum, in which case, Eq.(42) and Eq.(43)

would respectively become:

P : Re x −Re x Im x −Im xRe p −Re p Im p −Im p (46)

T : Re x Re x Im x −Im xRe p −Re p Im p Im p

(47)

Moving on from Eq.(41), one can define a whole class of Hamiltonians of the form [1]:

H= p 2m2 x2−i x N (48)

where m is mass and N∈ℝ .

For this set of theories it was found that there are critical values of the parameter N at which

PT symmetry would spontaneously break. At such points, the spectra would change from

having all real eigenvalues to having a decreasing number of real eigenvalues and an

increasing number of complex eigenvalues. The complex eigenvalues appear as complex

conjugate pairs. In the case of [1], this behaviour was determined by numerical methods; there

was no theoretical way to predict this behaviour.



The way in which a PT-symmetric Hamiltonian can have a real spectrum was further

elucidated by Bender in [28]. He pointed out that if the P T operator commutes with the

Hamiltonian H and if we assume that the two operators can be simultaneously diagonalised,

then we can prove that the eigenvalues of the Hamiltonian are real. For elucidation, we will

expand on this:

If P T and H commute, then we can write

P T H= H P T (49)

Post-multiplying both sides by T P demonstrates how the Hamiltonian would be PT

invariant.

P T H T P= H (50)

We now write the time-independent Schrödinger equation:

H∣ ⟩n=En∣⟩n (51)

Now applying the P T transformation to Eq.(51) gives:

P T H T P ∣⟩n= P T EnT P ∣⟩n (52)

Given that T EnT =En

* and that P 2=1 we can then write:

P T H T P∣⟩ n=En* ∣⟩n (53)

Eq.(50) and Eq.(51) imply the result:

En∣⟩ n=En* ∣⟩n (54)

which therefore means that:

En=En* (55)

Hence the eigenspectrum is real, En∈ℝ .

In [27] Bender continued to focus on the phase transitions in PT-symmetric Hamiltonians.

This paper introduced several new classes of PT-symmetric theories, all of which had a real



parameter , which allowed exploration of the phase transition properties. The first

Hamiltonian was:

H= p 2x2ix (56)

For ≥0 the spectrum was determined to be real and positive. For 0 the spectrum had an

increasing number of complex values; PT symmetry had been spontaneously broken. Again,

there was no proof for this, just observation based on numerical tests. Further observation

showed that as decreased the eigenvalues paired off into complex conjugate pairs. A

similar observation was made for Eq.(41), see [1]. As the value of continually decreased,

there were eventually no real eigenvalues left.

The paper [27] introduced a more general Hamiltonian which was shown to have the same

characteristics as Eq.(56):

H= p 2x2K ix (57)

In 2001 Dorey et al [29] (Appendix B) wrote a paper which included the first elementary

attempt at a fundamental proof of the Bessis conjecture for the class of Hamiltonians given in

Eq.(57). Their proof made use of Bethe Ansatz equations, which are beyond the scope of this

study.

Also in 2001, Handy [30] developed a theoretical method for reproducing the results of

Bender et al [1] for the V x=−ix3 potential, applying a method known as the “eigenvalue

moment method”. His results were reported to be in agreement with Bender's [1]. This was

another good step towards providing a fundamental theoretical grounding for a PTSQM

theory, although it was limited in that it only applied to the pure imaginary cubic potential,

while there were many other possible PT-symmetric theories.

A further theoretical study of the eigenspectrum of PT-symmetric non-Hermitian

Hamiltonians was published by Weigert in 2003, [31]. In this paper he offered an alternative



view on the “spontaneous breakdown of PT symmetry”. He points out that previous works

gave no mechanism to explain PT symmetry breaking. He argued that the change in

eigenspectrum can be explained by taking into account that the P T operator is not unitary,

but is an anti-unitary symmetry of a non-Hermitian operator. From this understanding he

claims, “The properties of PT-symmetric systems are explained in a natural way” [32].

In 2002 Bender et al [33] extended the idea of PT symmetry. Previous attempts at formulating

a PT-symmetric quantum mechanical theory lacked a fundamental proof and had relied

heavily on numerical methods to check that the spectrum was positive-definite. The

development shown in [33] was the discovery of an underlying symmetry in PT-symmetric

systems which would allow a dynamically determined inner product to be defined. The

underlying symmetry involved a new symmetry operator, denoted by C .

The following is a summary of how Bender et al determined C .

In [33], for the Hamiltonian H= p 2x 2 i x ≥0 , which is PT-symmetric and has an

associated Sturm-Liouville differential equation eigenvalue problem,

−n' ' xx2 i x n x=Enn x , Bender et al established the following facts. Eigenfunctions

nx of the Hamiltonian are simultaneously eigenstates of the P T operator, for which the

eigenvalue may be “absorbed into n x by multiplicative rescaling” (see [33]) so that it is

unity:

P T n x=n*−x=n x (58)

From this they then gave the completeness relation associated with Eq.(56) as:

∑n−1nnxn y∣{ x, y∈ℝ}= x− y (59)

This relation was noted as “a non-trivial result that has been verified numerically to

extremely high accuracy” [33]. However, there was no explanation of how it was reached and



no references were given to its derivation, which is regrettable. Also, the precise nature of x

and y was not clarified; they seem to form some orthonormal position basis, such that

⟨ x∣y ⟩= x−y .

They went on to show an inner product for eigenfunctions which applied only for the case of

≥0 in Eq.(56):

⟨m∣n ⟩=−1nmn (60)

This raises a crucial question as to whether a PT-symmetric quantum theory is physically

viable or not. A theory that has an inner product such as that in Eq.(60) is describing a space

spanned by states, half of which having a norm of +1 and the other half having a norm of -1.

Because of the Copenhagen school of thought, we attach a probabilistic interpretation to these

norms; therefore Bender et al admitted that having a negative norm created a serious issue of

interpretation. Another example of the difficulty this would cause is that associated with

expectation values. Expectation values are expected to be real and bounded below. However,

in this case they would be unbound, again causing a serious difficulty for the interpretation of

the theory.

Bender's solution to this problem was in turn one of interpretation, capitalising on the fact that

there are an equal number of positive and negative norms spanning its Hilbert space. This in

itself is a kind of symmetry. Therefore the linear operator C was introduced to describe this

symmetry (it was so named because its properties were similar to that of the charge

conjugation operator in quantum field theory [33]). It was then shown that C commutes with

both the Hamiltonian and P T . Most importantly, C had the property that C 2=I , which meant

that its eigenvalues were ±1 , precisely what was needed to counter the problem of indefinite

norms.

They constructed C directly from the energy eigenstates/eigenfunctions of the Hamiltonian.



For example, in the position representation C is written as [33]:

C x , y=∑nn xn y (61)

Weigert (2003) [34] wrote a paper to elucidate the meaning of the C operator. Here it was

pointed out that we are dealing with non-Hermitian operators and as a result we must take into

account biorthonormal bases.

In general, for a non-Hermitian operator with a complete basis we may write the eigenvalue

problem:

H∣n⟩=En∣n⟩ (62)

The Hermitian conjugate of the operator then has a different eigenbasis whose eigenvalues are

the complex conjugate of the operators eigenvalues:

H †∣n⟩=En*∣n⟩ (63)

These then have the following completeness and orthonormality relations:

∑i=1

i=n

∣i⟩ ⟨ i∣=I (64)

⟨i∣ j ⟩=ij (65)

For such a system, nothing can be known of the inner product. However, in the case of

PTSQM and CPTSQM the inner product structure is of the form:

⟨∣⟩ := C P T ⋅ (66)

Weigert showed that the C operator could still be expressed as the sum of a series of outer

products of basis vectors from the biorthonormal basis of the Hamiltonian (refer to [34] for

proof):

C=∑j=1

j=n

s j∣ j ⟩ ⟨ j∣ (67)

with:



s j=−1 j (68)

Weigert went on to give a simple expression for the C operator, to stress its relationship to

the Hamiltonian of a system. First, take the biorthonormal diagonal representation of the

Hamiltonian [34]:

H=∑j=1

j=n

E j∣ j⟩ ⟨ j∣ (69)

Then define a function that takes the eigenvalues of the Hamiltonian into those of the C

operator:

f E j =s j (70)

Being able to relate Eq.(67) to Eq.(69) by means of Eq.(70), Weigert stated:

C= f H (71)

Bender's proof for C 2=I comes from the completeness relation Eq.(59) (as quoted from [33]):

∫dy C x , y C y , z= x−z (72)

For clarity, rewriting Eq.(72) in Dirac notation would give us:

⟨ x∣ C∣y ⟩ ⟨ y∣ C∣z ⟩= x−z (73)

To demonstrate the relationship between C and P T Bender et al went on to show that the

parity operator could be constructed from eigenstates, as with C (shown here in position

representation) [33]:

P x , y = xy =∑n−1 nn xn−y (74)

From this it can be shown in [33] that C and P do not commute:

C P = P C * (75)

It is clear from Eq.(75) that C will commute with P T , which means that the linear operator

C P T will solve the problem of indefinite norms, thus we seek Hamiltonians which are CPT-

symmetric.



Relating CPTSQM to Hermitian quantum mechanics, Bender [33] notes that as the parameter

in Eq.(57) tends to zero, C becomes identical to P . Since P 2=I , C P T will simply tend

to T . However, as we go to this limit, the H in Eq.(57) tends to Hermiticity, so that complex

conjugation has no effect. That is why Bender et al [33] claimed that CPTSQM is the natural

complex extension of Hermitian quantum mechanics. They further claimed that this would

also imply that unitary dynamics would also be preserved.

The paper [33] concluded by giving a worked example of the construction of the C operator

for a given Hamiltonian, showing how the inner product structure of CPTSQM should be

written, in general:

⟨∣ ⟩ C P T= C P T ⋅ (76)

Here ⟨⋅∣ denotes the CPT Conjugate of ∣⋅⟩ .

For examples of explicit calculations of PT-symmetric observables and physical quantities,

please refer to Mostafazadeh and Batal [35], Mostafazadeh [36] and Jones [37].

4.2. Pseudo-Hermiticity

In 2002, Mostafazadeh published a trio of papers [38], [39], [40]. In which he laid out the

framework for an alternative form of quantum mechanics, in which the Hamiltonians are not

self-adjoint but still have a real positive-definite and complete spectrum. As observed in the

paper, it was already known that there were plenty of PT-symmetric Hamiltonians which did

not have a real spectrum and that there certainly were Hamiltonians with real spectra which

were not PT-symmetric. Therefore, he reasoned that PT symmetry was “neither sufficient or

necessary” to guarantee a real spectrum. The aim of this new framework of “pseudo-

Hermiticity” was hence intended to supply an explanation of the underlying structure of

theories with complete, real positive-definite spectra.



In the first paper, [38], Mostafazadeh began by listing the properties of pseudo-Hermiticity.

The whole framework depends on an invertible operator , a “Hermitian Linear

Automorphism”. It is defined to obey:

⟨∣w ⟩=⟨ ∣w ⟩ (77)

for arbitrary ∣v ⟩ and ∣w ⟩ .

An operator H is said to have a -pseudo-Hermitian adjoint, denoted by H # , which is

defined as:

H #=-1 H † (78)

An operator is then said to be -pseudo-Hermitian if:

H= H # (79)

Next, Mostafazadeh [38] went on to give a series of propositions to demonstrate the properties

of this pseudo-Hermitian framework.

The first proposition was:

“The Hermitian indefinite inner product ⟨∣⟩ defined by , i.e.

⟨1∣2⟩ := ⟨1∣∣2⟩ ,∀∣1⟩ ,∣2 ⟩∈H , is invariant under the time-translation generated by the

Hamiltonian H if and only if H is -pseudo-Hermitian” [38]

This is proved by first noting that if the Hamiltonian is -pseudo-Hermitian then we may

write:

H †= H -1 (80)

Then, inserting the pseudo-Hermitian inner product into the Schrödinger equation and using

the product-rule one may write:

i ddt ⟨1t ∣2t ⟩ =⟨1t ∣ H− H † ∣2t ⟩ (81)

The inner product in Eq.(81) can only be time invariant if its rate of change with time is zero.

However, given the form of Eq.(81), it may only be zero if Eq.(80) is true. The generalisation



of ordinary quantum mechanics can also be seen here; if we choose =I , then we obtain the

ordinary definition of Hermiticity, H †= H .

In my view, the definition given by Eq.(80) has some remarkable consequences. Starting with

Eq.(80), right-multiply both sides by an eigenvector ∣E j⟩ , then left-multiply both sides by

another eigenvector ⟨E i∣ . Then substitute the Hamiltonian with the respective eigenvalues.

Finally, rewrite in terms of the -pseudo-Hermitian inner product:

H † = HH † ∣E j ⟩ = H∣E j ⟩

⟨E i∣ H † ∣E j ⟩ = ⟨E i∣ H∣E j⟩E i

* ⟨E i∣∣E j ⟩ = E j ⟨E i∣ ∣E j⟩E i

* ⟨E i∣E j⟩ = E j ⟨E i∣E j ⟩ E i

*−E j ⟨E i∣E j⟩ = 0

(82)

The result of Eq.(82) leads directly to Mostafazadeh's second proposition, quoted below [38]:

An -pseudo-Hermitian Hamiltonian has the following properties:

(a) The eigenvectors with a non-real eigenvalue have vanishing -semi-norm, i.e.;

E i∉ℝ implies ∥∣E i ⟩∥2 :=⟨E i∣E i⟩ =0 (83)

(b) Any two eigenvectors are -pseudo-orthogonal unless their eigenvalues are

complex conjugates,

E i≠E j* implies ⟨E i∣E j ⟩ =0 (84)

Following on from these results, Mostafazadeh explored more properties of pseudo-

Hermiticity, which are quoted here for reference (proofs for all of the following are given in

[38]):

1=1# (85)

O##= O (86)

z1O1z2

O 2#=z1

* O 1#z2

* O2#

For {z 1 , z2∈ℂ} , O :V V (87)

For inner product spaces V 1,2,3 having Hermitian linear automorphisms 1,2,3 and



linear maps O1:V 1V 2 , O2 :V 2V 3 , one can write:

O 2O1

#=1

-1 O 2O1

†3=1

-1 O 1†22

-1 O 2†3= O1

# O2# (88)

Given a unitary operator U :V V , a linear operator O :V V and a linear

automorphism , we can then define U := U † U and O U := U † O U . From these it

follows that pseudo-Hermiticity is unitary-invariant:

U-1 O U

† U= U † -1 U U † O† U U † U= U † -1 O† U (89)

Let an inner product space V have two Hermitian linear automorphisms 1 , 2 , and

have a linear operator O :V V . Then the 1 and 2 -pseudo-Hermitian adjoints are

only equal if 2-1 1 commutes with O .

1-1 O† 1 = 2

-1 O† 2

O† 1 = 1 2-1 O† 2

O† 1 2-1 = 1 2

-1 O†

2-1 1

O = O 2-1 1

[ O , 2-1 1 ] = 0

(90)

The paper [38] went on to show that a biorthonormal basis is the necessary condition for the

eigenspectrum of a pseudo-Hermitian Hamiltonian to change from being entirely real to being

a mixture of a decreasing amount of real eigenvalues and an increasing amount of complex

eigenvalues (in the form of conjugate pairs). This phase transition is an important observation

requiring explanation; it has been observed by Bender et al in numerical models [1] and

theoretically determined by Dorey et al [29].

The essence of the biorthonormality proposition in [38] (Prop. 7) is that the Hermitian linear

automorphism and its inverse can be expressed by using outer products of elements from

the two sets of eigenvectors which form the complete biorthonormal basis of the Hamiltonian.

Furthermore, and −1 are mappings between those two sets of eigenvectors. The full proof

of the following results are contained in [38], but are summarised here for interest and

reference.



According to equations (62), (63) and (80) we may write (where ∣n⟩ and ∣n⟩ are the

biorthonormal sets of eigenvectors):

H -1∣n⟩= -1 H †∣n⟩=En* -1∣n⟩ (91)

From this we can see that -1 maps between the ∣n⟩ eigensubspace and the ∣n⟩

eigensubspace.

We use subscript labels: “0” to denote eigenvectors associated with real eigenvalues and “±”

to denote eigenvectors associated with conjugate complex eigenvalues having ± imaginary

parts. We can express the above relationship by writing:

∣n0⟩=∣n 0⟩ , ∣n± ⟩= ∣n∓⟩ (92)

Combining Eq.(92) with Eq.(65) gives us the pseudo-Hermitian orthonormality relations:

⟨ ⟨n0∣m0⟩⟩ =n0 , m0, ⟨⟨n±∣m∓⟩⟩ =n± ,m∓

⟨ ⟨n0∣m0 ⟩⟩=n0 , m0, ⟨⟨n±

∣m∓⟩⟩=n± , m∓

(93)

Given equations (64) and (69) we may write the completeness identity and the Hamiltonian in

their spectral form:

I=∑n0

∣n0⟩ ⟨n0∣∑n+

∣n+ ⟩⟨n+∣∣n- ⟩ ⟨n-∣ (94)

H=∑n0

En0∣n 0⟩ ⟨n0∣∑n +

En+∣n+⟩ ⟨n +∣E n-

* ∣n- ⟩ ⟨n-∣ (95)

This leads to the calculation of and its inverse, in spectral form:

=∑n0

∣n0 ⟩⟨n0∣∑n+

∣n-⟩ ⟨n+∣∣n+⟩ ⟨n -∣ (96)

-1=∑n 0

∣n0 ⟩⟨n0∣∑n+

∣n-⟩ ⟨n+∣∣n+⟩ ⟨n-∣ (97)

These results were claimed [38] to give the necessary and sufficient condition for the presence

of pseudo-Hermiticity. By construction a pseudo-Hermitian theory may only have real or

complex-conjugate pairs for its spectrum, just as with PTSQM.

It should be noted that in the original paper [38] all of these results were even more general,

because they allowed for degeneracy. In fact, could not be invertible if this were not the



case. We have omitted this complication in our account of the theory.

As claimed in the paper [38], this was the only theory to explain the underlying structure of

the real and/or conjugate-pair spectra, which in my opinion would seem to suggest that

pseudo-Hermitian quantum mechanics is a more fundamental theory than PTSQM. At the

time there was no corresponding explanation in the PTSQM literature. However, there was a

later explanation for the spectrum of PT-symmetric systems published by Weigert in 2003,

[31].

Mostafazadeh gave two more corollaries of his results on the pseudo-Hermitian framework:

1. Every non-Hermitian Hamiltonian with a discrete real spectrum and a complete

biorthonormal system of eigenbasis vectors is pseudo-Hermitian.

2. Every PT-symmetric Hamiltonian with a discrete spectrum and a complete

biorthonormal system of eigenbasis vectors is pseudo-Hermitian.

The first paper of his three papers [38] showed that having eigenvalues that are either real or

complex-conjugate pairs is sufficient and necessary for pseudo-Hermiticity and vice versa. As

such, pseudo-Hermiticity is a necessary but not a sufficient condition for the existence of an

entirely real spectrum. The second paper of the three [39] thus gave a further characterization

of Hamiltonians which have a complete biorthonormal eigenbasis and a real spectrum.

Mostafazadeh [39] proves a short but elegant theorem which provides the added condition

required to ensure the reality of a Hamiltonian's spectrum: “For a Hamiltonian H acting in a

Hilbert space H having a discrete spectrum and complete set of biorthonormal eigenvectors

{∣n ⟩ ,∣n⟩} , the spectrum of H is real if and only if there is an invertible linear operator

O :H H such that H is O O † -pseudo-Hermitian.”.

Defining O as follows, where the ∣n ⟩ are basis vectors:

O :=∑n∣n⟩ ⟨n∣ (98)



H 0=∑n

En∣n ⟩ ⟨n∣ (99)

the equations (65) to (68) allow us to write:

O-1 :=∑n∣n⟩ ⟨n∣ (100)

We also have:

O-1 H O= H 0 (101)

If we assume by hypothesis that the spectrum of H is real and hence H 0 is then Hermitian,

we could write:

O-1 H O= O† H † O-1† (102)

Which rearranges to:

H= O O† H † O O† -1 (103)

H is then O O † -pseudo-Hermitian. As shown in the final of the three papers [40], = O O† . In

[35], Mostafazadeh and Batal demonstrated how to calculate pseudo-Hermitian observables

and a new pseudo-Hermitian quantization scheme.

4.3. Comparisons of CPT symmetry and pseudo-Hermiticity

As indicated in the previous section, Mostafazadeh made the claim that PT symmetry is

effectively a sub-class of pseudo-Hermiticity. This is an important point to consider when

weighing the relative merits of each framework and in identifying the most fundamental

theory. Accordingly, the summaries that follow in this section will outline the arguments

made to support Mostafazadeh's claim. No counter arguments are included in this section,

because none were found in the literature.

In [40] Mostafazadeh explored the relationship between pseudo-Hermiticity and anti-linear

symmetries. Let us recall that an operator is called anti-linear if it acts as follows, where a and

b are complex numbers:



a∣⟩b∣⟩ =a *∣⟩b*∣⟩ (104)

is an anti-linear operator if it obeys the relationship:

⟨∣ ∣⟩=⟨∣∣ ⟩ (105)

Then a linear operator H is said to be -anti-pseudo-Hermitian if it obeys the following

equation, where is described as an anti-linear anti-Hermitian automorphism:

H †= H -1 (106)

The central theorem proved in the paper [40] is that every diagonalisable linear operator with

a discrete spectrum possesses an anti-linear symmetry, as described in equations (104)-(106).

This then leads to the corollaries given in [40]:

1. “Every diagonalisable linear operator H :H H with a discrete spectrum is anti-

pseudo-Hermitian”.

2. “Every diagonalisable pseudo-Hermitian linear operator H :H H with a discrete

spectrum has an anti-linear symmetry.”.

The key observation made by this paper is that PT symmetry is an anti-linear symmetry.

Therefore, according to this line of reasoning, every PT-symmetric Hamiltonian implies the

presence of -pseudo-Hermiticity. This is encapsulated in the result:

= P T (107)

In [35], Mostafazadeh showed how to construct PT-symmetric physical quantities based on a

pseudo-Hermitian quantisation scheme. The same paper also made an interesting calculation

of the C operator in terms of the Hermitian linear automorphism + , where the plus-sign

relates to the set of operators which ensure a positive-definite pseudo-Hermitian inner

product. The result is quoted here:

C=+-1 P (108)



In my view, this is most interesting because it is also the C operator in CPTSQM that

determines the inner product of the physical Hilbert space, as do the Hermitian linear

automorphisms in pseudo-Hermitian quantum mechanics.

Mostafazadeh also claimed in [35] that performing calculations in the pseudo-Hermitian

framework in order to describe PT-symmetric systems has practical and conceptual

advantages. If correct, this would seem to further suggest the greater generality of the pseudo-

Hermitian framework.

In 2005, Mostafazadeh published a paper [41] dealing with a method of practical application

of the pseudo-Hermitian operators to PT-symmetric systems defined on a complex contour.

PT-symmetric systems defined on the real line have the familiar space H =L2 ℝ as their

Hilbert space. However, this is not so for systems defined on a complex contour. Instead, the

Hilbert space is obtained by Cauchy completing the span of the eigenfunctions of the

Hamiltonian with respect to an arbitrary positive-definite inner product [41]. Mostafazadeh

notes in [41] that there is a view by some (but does not cite a reference) that the possibility of

this construction rules against the application of the pseudo-Hermitian framework. He also

discusses PT-symmetric theories defined on complex contours for which there have been no

“natural or useful” choices for the reference Hilbert space. He goes on to show that one can

construct equivalent PT-symmetric Hamiltonians with the reference Hilbert space H =L2 ℝ .

He claims that giving such theories a “real description” facilitates a better understanding of

their physical content. As a particular example, he shows that the Bender/Bessis Hamiltonian,

Eq.(56), defined on a complex contour can be recast as a Hamiltonian in the H =L 2 ℝ space.

According to [41] doing this allows for simple application of the theory of pseudo-Hermitian

operators. It provides a description of the Hamiltonians which reveals the source of the

discreteness of their spectrum. It also has the practical advantage that approximation schemes



for solving differential equations on the real line can be used.

In my view, whether one would agree with these points or not, it is noteworthy that a PT-

symmetric theory was constructed in terms of the pseudo-Hermitian framework. It certainly

strengthens the argument that pseudo-Hermitian quantum mechanics is the more fundamental

theory. Even if that were not so, it must at least point to the idea of the two frameworks being

equivalent, in the sense that one may transform between the two.

4.4. Comparisons of PT symmetry of Hermiticity

In 2003 two papers ([42] and [43]) from Bender et al explored how PT symmetry relates to

Hermiticity, with an approach based on the following established properties:

i. P is linear and Hermitian.

ii. P & H commute: [ H , P ]=0 (If H is P invariant).

iii. P 2=1

iv. The nth eigenstate of H is also the nth eigenstate of P with eigenvalue, −1 n

The first paper [42] aimed to show that PT symmetry is a generalisation of Hermiticity by

showing that all Hermitian Hamiltonians have PT symmetry.

The paper works on the general class of Hamiltonians:

H= p 2V x (109)

For this H to be Hermitian V x needs to be real. Showing that Eq.(109) is T invariant is

trivial because it is entirely real, so that complex conjugation has no effect. Therefore, as the

title of the paper implied, the rest of the paper sought to prove that “All Hermitian

Hamiltonians have Parity”. This was done by exploiting the process developed in their

previous paper [33] for constructing operators. By hypothesis, if one makes the assumption



that the parity operator for a given system commutes with the system's Hamiltonian, then one

can construct the parity operator in terms of the simultaneous eigenfunctions. Since the set of

eigenfunctions for a given Hamiltonian are a complete set, then one may write the

Hamiltonian as a matrix represented in coordinate space:

H x , y=∑n=0

∞

Enn xn* y (110)

or using Dirac notation:

⟨ x∣ H∣y ⟩=∑n=0

∞

En ⟨ x∣n⟩ ⟨n∣y⟩ (111)

Then by analogy, one may construct the parity operator as shown in Eq.(74). Because the

eigenfunctions of any quantum mechanical system are complete and orthonormal, it can be

confirmed that the parity operator in Eq.(74) obeys the properties i-iv, listed above.

Based upon this result, Bender et al made the claim that this proved that all Hermitian

Hamiltonians are PT-symmetric. This is a proposition which is elegant in its simplicity, but

neither numerical or analytic verification was given.

The next paper by Bender et al in 2003 [43], would actually contradict the previous paper's

conclusion that all Hermitian Hamiltonians have parity. In this paper they argued that PTSQM

was not a generalisation of Hermitian quantum mechanics. Instead, one should start with a

real symmetric Hamiltonian, then extend its matrix elements into the complex domain in such

a way that the appropriate requirements to determine the spectra and the time evolution

(unitary quantum mechanics) are satisfied. The class of real symmetric Hamiltonians can be

thought of as the overlap of the class of Hermitian Hamiltonians and the class of PT-

symmetric Hamiltonians (see fig. 4.1). That being so, there are then two ways to generalise

the real symmetric Hamiltonian matrix elements, either by moving into the complex domain

so that the matrix is still Hermitian and thus has real spectra, or by similarly generalising for



the case of PT-symmetric Hamiltonians, on the condition that the PT operator commutes with

the Hamiltonian, thus ensuring real spectra, as has been shown before. Following either of

these recipes will construct self-consistent theories of quantum mechanics, if one accepts

PTSQM.

In the paper [43], Bender et al compared these cases on the basis of how many parameters are

required to describe each case for a finite D dimensional system, which led to a quantitative

comparison of the size of each class. This comparison was stated as follows: for large D, a

Hermitian matrix would have D2 parameters, a PT-symmetric matrix would have

asymptotically ¾D2 parameters and a real symmetric matrix would have asymptotically ½D2

parameters. See [43] for derivations of the latter two results. This comparison was illustrated

in a figure which is reproduced here [43]:

The paper [43] showed that the most general (i.e. parametric) Hamiltonian operator is:

H 0=A iBiBT C (112)

In general, when the Hamiltonian is D-dimensional, the parity operator has m+ positive


Figure 4.1: Venn Diagram illustrating the intersection

between the class of Hermitian and PT-symmetric

Hamiltonians


eigenvalues and m- negative eigenvalues such that D=m+m- . Therefore, in Eq.(112) A is a

real symmetric m+×m+ matrix, C is a real symmetric m-×m- matrix, and B is a real

symmetric m+×m- matrix [43].

Having constructed the Hamiltonian, parity and charge conjugation operators, Bender et al

then concluded by considering alternative ways of generalising the theories [43]. What if the

Hamiltonian was not symmetric? In this case, eigenstates would not be orthogonal. In order to

correct this Bender et al proposed that a weight matrix would be required to enforce

orthonormality. (Interestingly, Mostafazadeh [44] suggested the same thing, seemingly

independently.) Such a matrix would be performing the same role as the charge conjugation

operator. However, because of the Hamiltonian's lack of symmetry, this weight matrix would

not commute with the Hamiltonian, which is one of the key requirements for the charge

conjugation operator. Then, because the weight matrix would not commute with the

Hamiltonian, the inner products would become time dependent. Therefore, we must reject

such a theory because it is non-unitary. They then considered the consequences of the parity

operator being an asymmetric matrix. Since a PT-symmetric Hamiltonian must commute with

the P T operator, then the Hamiltonian would need to be asymmetric also, which brings us

back to the previous case. Finally, they considered the following questions: What if the time-

reversal operator is more than just complex conjugation? What if it was the combined

operation of some complex matrix B and complex conjugation? Because of the property

T 2=1 , we would need to have B B*=1 ; since we also must have [ P , T ]=0 then we would

have to have [ P , B ]=0 . Bender et al [43] stated that these conditions are so strong that no

other parameters would appear in a general PT-symmetric Hamiltonian.

All of this provided strong evidence that Hermitian and PT-symmetric Hamiltonians are two

distinct unitary classes and that neither is a generalisation of the other. Close examination of



the paper [43] shows a much stronger argument than Bender et al made for their opposite

claim in [42]. In my opinion, the proof is particularly convincing because the authors retain

generality by keeping their Hamiltonian and parity matrices parametrised. Even though the

proof is limited to finite dimensional cases the authors conjecture that their conclusion

extends to infinite dimensions.

In the latter part of 2003, Mostafazadeh [44] published a detailed critique of PTSQM, in

which he set out to show that PTSQM was neither an alternative to or an extension of

ordinary quantum mechanics. His initial argument was that the physical content of PTSQM is

obscured by ambiguities, the principle ones being the unclear nature of the eigenvalue x in

PTSQM and the use of the terms “self-adjoint” and “non-Hermitian”.

PT symmetry is usually thought of as a physical requirement, because it is meant to refer to a

symmetry under space-time reflections. However, this relies on the eigenvalue x to actually

represent the position of a particle. Mostafazadeh [44] claimed that, for an eigenvalue to have

meaning it has to be associated with a linear operator representing a physical observable.

Mostafazadeh argued that one could not a priori justify relating the eigenvalue x to physical

position. In this case, one would have to construct PTSQM as a physical theory without any

prior assumptions, which would require asking the following questions [44]:

1. What is the nature of the state vectors?2. What are observables?3. How are the observables measured?4. How does the theory relate to known theories?

In my opinion, it is not clear that the PTSQM literature covered in this review sufficiently

answers these questions.

The next ambiguity involves the use of the terms “Hermitian” and “self-adjoint”. In ordinary



quantum mechanics, these terms are almost universally interchangeable and denote the

following property:

⟨∣ H ⟩=⟨ H∣ ⟩ (113)

The vectors are elements of some Hilbert space and the angled brackets denote an inner

product on that space.

It is common to refer to Hermitian matrices as “self-adjoint” matrices represented in an

orthonormal basis. Then for any Hamiltonian satisfying Eq.(113), we can say of its matrix

elements:

H ij= H ji* (114)

This definition is commonly used because in ordinary quantum mechanics the inner product is

always the same, i.e. ⟨∣ ⟩=∑i=0i=n i

*i . Also, there is always an orthonormal basis of

commuting observables. However, Mostafazadeh makes the point that quite clearly CPTSQM

has no such fixed inner product or observables. As discovered by Mostafazadeh in 2002 [39],

it can be shown that PT-symmetric Hamiltonians obey the general Hermiticity property

Eq.(113) with respect to other inner product structures. Therefore, there is good reason not to

use the term “non-Hermitian” and to be more strict with terminology, so that these

ambiguities do not arise.

In [44], Mostafazadeh pointed at a “naive definition of a Hermitian operator” as the cause of

this seemingly poor terminology in PTSQM. This “naive definition” is simply an extension of

the condition in Eq.(113) to apply to an infinite matrix:

H x , y*= H y , x (115)

However, this is explicitly dependent on the position basis, which cannot be an orthonormal

basis for CPTSQM, because it is not orthonormal with respect to the C P T inner product. This

shows that the correspondence between the term “Hermitian” and “self-adjoint” does not hold



in all cases, as shown here for the case of the position-basis. Therefore, the terms cannot be

used interchangeably as they are in ordinary quantum mechanics. The same argument was

noted to apply to the term “symmetric operators”.

Mostafazadeh [44] went on to determine the physical observables for the CPT-symmetric

systems defined by the widely quoted Bessis/Bender Hamiltonian:

H= p 2x2 i x (116)

His aim was to show that the CPT-symmetric model Eq.(116) can be recreated as a self-

adjoint model via a “similarity transformation” and that in finite dimensions CPT-symmetric

observables are inconsistent within the Heisenberg picture.

To identify CPT-symmetric observables we must employ the C P T inner product [44]:

,+ :=∫ℂdx [ C P T x] x (117)

Immediately, the common position and momentum operators are ruled out as physical

observables because they are not Hermitian with respect to the inner product defined by

Eq.(117). This leaves one asking the question “what are the PT-symmetric position and

momentum operators?”.

To answer this question Mostafazadeh quotes a text book result that “up to isomorphisms

there is a unique infinite-dimensional separable Hilbert space”, see [44] for this reference.

This means that there is a linear transformation U which maps from a Hilbert space H to the

space L 2ℝ , which admits the standard Euclidean L 2ℝ inner product ⟨⋅∣⋅⟩ :

,+=⟨ U ∣ U ⟩{ ,∈H }

(118)

This link between two Hilbert spaces provides the answer to the question of CPTSQM

observables. For any observable in a CPTSQM system, we may write:



O C P T= U -1 o U (119)

where O C P T is a CPT-symmetric observable and o is a Hermitian (self-adjoint) observable

acting in L 2ℝ .

Mostafazadeh [44] noted that a limitation to this “similarity transformation” is that it is “non-

local” in that it maps differential operators to non-differential operators. Therefore, a self-

adjoint Hamiltonian obtained through similarity transformation of a PT-symmetric

Hamiltonian will lack the familiar form of kinetic+potential; observables would also be non-

local.

The consequence of this is that one may describe any CPT-symmetric Hamiltonian which has

a real eigenspectrum as a self-adjoint Hamiltonian in the Hilbert space L 2ℝ . Based upon

this, Mostafazadeh stated that the claim of CPTSQM to be a fundamental physical theory

which extends quantum mechanics is not valid [44].

Another limitation reported by Mostafazadeh [44] was that the construction of an explicit

form of the operator U was not possible because of the closed form of the C P T inner

product Eq.(117). However, Mostafazadeh pointed out that Bender et al [43] showed there are

finite dimensional analogues of the CPT-symmetric Hamiltonians of Eq.(116), for which the

C operator has a simple matrix form. For such systems, Mostafazadeh has published a way of

obtaining explicit forms of similarity transformed operators in the framework of pseudo-

Hermiticity [38].

In [44], Mostafazadeh went on to re-examine the two-dimensional model produced by Bender

et al [33]. The matrix form of the Hamiltonian is repeated here for reference:

H C P T :=r ei ss r e− i (120)

This model was associated with a PTSQM theory over the space ℂ2 (which we shall refer to



as H ) endowed with the C P T inner- product:

,+= C P T ⋅ (121)

where:

⋅ :=∑i=1

i=2

ii (122)

Here Mostafazadeh criticised the use of the standard basis {10 ,0

1 } for avoiding the problem of

basis dependence in the context of symmetric operators. However, in my opinion the strongest

argument made by Mostafazadeh was that the standard ℂ2 basis {10 ,0

1 } is not even an

orthonormal basis with respect to the inner product Eq.(121).

The definitions of the parity and charge conjugation operators, as stated in [44] are: P :=1

and C :=sec 1i tan 3 , where 13 are the Pauli matrices and ∈−/2 ,/2 .

Therefore, observables in this system are written as:

O C P T=a0bc sin 1 c−bsin 3 , {a ,b ,c∈ℝ } (123)

Mostafazadeh [44] then went on to show that a system equivalent to Bender's CPT-symmetric

model, Eq.(120), can be constructed from a self-adjoint Hamiltonian. Again, the space ℂ2

was used, but with the Euclidean inner product ⟨∣ ⟩ :=∑i=1i=2 i

*i . We will refer to this space

as . He then introduced the unitary similarity transformation, in other words, the map U

which maps from the space H to the space :

U := 12 cos e i /2 e−i /2

−i ei /2 i ei /2 (124)

For this U it can be shown that:

h := U H C P TU -1=+ 0

0 -=r cos 0s cos 3 (125)

This shows that the CPT-symmetric Hamiltonian Eq.(120) is changed into a self-adjoint

Hamiltonian by means of the unitary similarity transformation Eq.(124). The observables for



this Hamiltonian are linear combinations of the spin operators. Mostafazadeh defined the spin

operators as s= /2 and the observables are written as:

o=∑=0

=3

c s , {c∈ℝ } (126)

Given Eq.(119) and setting S := U−1 s U , one may write:

O C P T=∑=0

=3

cS (127)

Mostafazadeh then gave the following solutions for S in terms of the Pauli spin matrices

[44]:

S 0=12 0

S 1=− 12 2

S 2=12 tan 1−sec3 S 3=1

2 sec 1tan 3 (128)

This then shows that the O of Eq.(123) is a sub-set of the O of Eq.(127), suggesting that the

self-adjoint formulation is actually more general than the CPT-symmetric model from [33].

Given the requirement for observables to be C P T invariant, we find an interesting limitation

for the observables listed in Eq.(128); measurement in the S 1 direction is excluded! To prove

this we need to show that 2 is not C P T invariant, as I have presented below. Taking the

above definitions for parity and charge conjugation operators, setting A=sec and

B=tan for convenience, we can write:

C P T 2T P C = C P 2

* P C

= A iB−iB A 0 i

−i 0 A −iBiB A

= i 0 A2−B2B2−A2 0

= 0 i−i 0

= 2*

≠ 2

(129)

Having such a limitation has no physical justification, which is clearly a problem with the

formulation for the Hamiltonians of Eq.(116) in the CPTSQM framework.



Finally, Mostafazadeh [44] highlighted a problem for CPTSQM within the Heisenberg picture

of dynamics. For a general observable, we can relate its Heisenberg picture O H , to its

Schrödinger picture, OS , as follows:

O H t =eit H OS e−it H (130)

We now set OS= S 2 , which is both symmetric and CPT-symmetric according to Eq.(123) and

Eq.(128). By means of a similarity transformation and the properties of the Pauli matrices, the

Heisenberg picture of S 2 would be found to be [44]:

O H t =sin 2 s cos t S 1cos 2s cos t S 2 (131)

As discussed, S 1 is not CPT-symmetric and consequently O H t inherits this lack of CPT

invariance. Therefore, the CPT invariant Hamiltonian Eq.(116) would seem to lose its CPT

invariance as soon as one considers the dynamics. Specifically, the observable Eq.(131)

becomes unobservable through time, albeit with periodic exceptions!

In his critique of CPTSQM [44], Mostafazadeh covered details of terminology and basic

“book keeping” which, in my view, were overlooked or ill-considered elsewhere. From this,

several short-comings were uncovered which CPTSQM must overcome if it could ever be

adopted as a true replacement for ordinary quantum mechanics.

In many of their papers, Bender et al, criticised the condition of Hermiticity as having an

unclear physical meaning and promote CPTSQM as having a clearer physical meaning. In

turn, Mostafazadeh rejected the notion that the condition of CPT symmetry has a clear

physical meaning. In my view though, pseudo-Hermiticity also lacks any physical

interpretation, despite its mathematical merits. Bender has since stated that it is unknown

whether “non-Hermitian” PT-symmetric Hamiltonians could be used to describe

experimentally observable phenomena [22]. Although, this use of the term “non-Hermitian”

could be misleading.



Mostafazadeh made a clear argument that CPTSQM may be reduced to ordinary quantum

mechanics by means of similarity transformations. From this, he concludes that CPTSQM is

merely equivalent to ordinary quantum mechanics. However, it is my view that his argument

against CPTSQM would have been more complete if he had also demonstrated the reverse of

a similarity transformation. For example:

• Take a self-adjoint Hermitian Hamiltonian already used to describe physically

observable phenomena.

• Perform the inverse of a similarity transformation to create a CPT-symmetric model.

• Then test to see if it would produce the same results as it did prior to transformation.

This suggestion was addressed to some extent by the next paper of Mostafazadeh, discussed

below.

In 2004 Mostafazadeh [45] published further work on PTSQM to demonstrate the theory's

unitary equivalence to ordinary “Hermitian” quantum mechanics. The central theme of the

paper is to apply the rigid mathematical structure that ordinary quantum mechanics possesses

to a study of PTSQM. The rigorous mathematical work done in this paper provides a much

more concise recasting of his critique of PTSQM, [44].

The paper shows that the physical Hilbert space, the Hamiltonians and the observables of a

PTSQM theory are unitarily equivalent to the L 2ℝ space, Hamiltonians and observables of a

Hermitian theory, respectively. It should also be noted that this formulation is completely

independent of the framework of pseudo-Hermitian operators.

The work in this paper somewhat addresses my suggestions for further study given above. By

demonstrating unitary equivalence between PTSQM and ordinary quantum mechanics, the

idea of “proving the reverse” is satisfied. On a similar note (also in 2004) Mostafazadeh [35]



demonstrates that a Hermitian Hamiltonian can be recovered from every PT-symmetric

Hamiltonian.

In 2006 Bender et al published their own account of PTSQM versus Hermitian quantum

mechanics [46]. In this paper they acknowledge that PT-symmetric systems are equivalent to

Hermitian (self-adjoint) systems. Therefore, the aim of their paper was to determine which

framework was the most efficient one for calculations.

Initially, the paper recapitulates some established recipes, repeated here for reference. The

inner product structure depends on the C operator, which must satisfy the following

conditions:

C 2=1{ C , P T }=0C=e

Q P (132)

where Q is an antisymmetric Hermitian operator.

We also recall that for a Hermitian Hamiltonian h and a non-Hermitian Hamiltonian H we

can state the following relationship:

h=e−Q /2 H e

Q/ 2 (133)

As a brief exercise, they show the Hermiticity of h in the CPTSQM framework, as

demonstrated here, quoting [46]:

The adjoint of Eq.(133) can be written in terms of PT symmetry:

h† = eQ /2 H † e−

Q/2

= eQ /2 P H P e−

Q /2 (134)

Noting the last line of Eq.(132) we have:

e−Q /2 C=e

Q /2 P (135)

Substituting into Eq.(134):



h†=e−Q /2 C H C e

Q /2 (136)

Then, because the Hamiltonian commutes with C and C 2=1 , we can see that C H C= H C 2= H .

Therefore, Eq.(136) reduces to Eq.(133), completing the proof of Hermiticity for h .

Bender et al found that calculations, based on Feynman rules, of the ground state energy of a

quantum field theory Hamiltonian H= 12 p2 1

2 x2igx3 were simpler to perform in the PT-

symmetric framework rather than with a similarity transformed Hermitian Hamiltonian. In

relation to comments made by Mostafazadeh [44], they agreed that all PT-symmetric theories

can be similarity transformed to Hermitian theories. However, they disagreed with the

statement by Mostafazadeh, “A consistent probabilistic PT-symmetric quantum theory is

doomed to reduce to ordinary quantum mechanics” [44]. Their paper [46] states that a PT-

symmetric model is free from all the difficulties and complications associated with a

similarity transformed Hermitian theory. Therefore, they assert that a PT-symmetric theory

does have something to offer over a corresponding Hermitian theory.

Even if PT-symmetric theories are free of difficulties encountered with similarity transformed

Hermitian theories, it is my opinion that Bender's counter argument does seem to miss the

point of Mostafazadeh's criticism. Mostafazadeh's criticism refers to the status of PT

symmetry as a fundamental theory, not to which theory gives the simplest calculation. In the

work presented by Bender et al [46], CPT-symmetric quantum mechanics does indeed appear

to lose its status as a fundamental theory, but does offer itself as a more efficient calculation

scheme, at least in some cases.



Chapter 5. Studies on spontaneous breakdown of PT symmetry

5.1. PT symmetry breakdown

A quantum mechanical model must have certain universal properties in order for it to be

physically reasonable. We require that the eigenspectrum of a Hamiltonian is entirely real, so

that we can compare theoretical predictions with real world measurements in the laboratory.

The time evolution of the system must be unitary in order to avoid “probability leakage”. We

require that the set of eigenstates of the Hamiltonian should be mathematically complete.

Consequently, the Hamiltonian of the system is then diagonalisable in its eigenbasis. Self-

adjoint Hermitian operators possess all of the properties cited above, and therefore were the

most suitable choice of operator class. However, natural curiosity and a sufficient number of

mathematical oddities have led us to ask whether other types of operator have the same

characteristics.

One of the alternative formalisms has been PTSQM, although only a limited set of all PT-

symmetric Hamiltonians have the desired characteristics for a physically reasonable theory.

Therefore, if we wish to apply PTSQM, we need to undertake an assessment of each and

every model which we might use in order to check for the required characteristics.

There have been many publications dealing with the structure of PTSQM. However, little has

been published on the parametric limits within which PT-symmetric systems would remain

physically reasonable. Dorey related the spontaneous breaking of PT symmetry to “super

symmetry” [47]. Prior to 2005, the only other investigations of how PT-symmetric

Hamiltonians maintain a real spectrum were made by Bender et al, see section 4.1. These

investigations were limited to numerical observations only; at that time, there had been no

David R Gilson Chapter 5. Studies on spontaneous breakdown of PT symmetry 57 of 77


analytical analysis.

In 2005 and 2006 Weigert published papers ([25], [26] and [48]) which proposed algorithmic

tests to characterise the nature of PT-symmetric theories. Two of his papers ([25] and [48])

made extensive use of the minimal polynomial of a PT-symmetric matrix (the minimal

polynomial is the polynomial of least degree that annihilates the matrix). For a given matrix

M , we have a characteristic polynomial of degree P, p M =det I− M . We also have a

minimal polynomial m M M =0 of degree N≤ P for which no polynomial of degree less than

N can annihilate M . p M is related to m M by p M =d M m M . In general,

m M =∏=1=N −M where is the multiplicity of each root M . M is only

diagonalisable if it has distinct roots, i.e. {∀∣=1} .

In these papers, Weigert also discussed an algorithm for constructing the minimal polynomial

by using the Euclidean algorithm, rather than by needing to compute the characteristic

polynomial. This method is most useful because it is guaranteed to be completed in a finite

number of steps. However, it has yet to be proven that these methods can be applied to

systems with an infinite number of countable dimensions. These are the most practically

useful methods related to PTSQM which I have found in the literature.

The location of “exceptional points” (EPs) is an alternative way of assessing the physical

validity of a PT-symmetric theory. As discussed previously, the eigenspectrum of a PT-

symmetric Hamiltonian can undergo phase transitions at which pairs of real numbers coalesce

to give a single real number, then split to give complex conjugate pairs as the value of some

perturbation parameter varies. The point at which eigenvalues coalesce is what we call an

exceptional point (see Heiss 2004 [49] for an interesting discussion of EPs).



As demonstrated by Weigert in [25] and [48], the minimal polynomial of a Hamiltonian

which has such a perturbation parameter is a polynomial in two variables. That is to say, the

perturbation parameter becomes its second variable. The exceptional points of the

Hamiltonian are found at the roots of the minimal polynomial in the perturbation parameter.

5.2. Example: calculating exceptional points in the Lipkin model

Here I report on work which I undertook to create a PT-symmetric model, from which I could

analytically determine its EPs. This involved taking a modified form of the “Lipkin model”;

while not possessing Hermiticity or PT symmetry, this model does have analytically solvable

EPs. Therefore the aim of this exercise was to introduce PT symmetry into the model and then

re-solve for its new EPs.

Heiss et al (2005) published a paper [50] which used a modified version of the Lipkin model.


Figure 5.1: Transition of eigenvalues as perturbation parameter varies


The original Lipkin model seemed suitable for adaptation to a PT-symmetric system because

it has been used for demonstrations of quantum phase transitions and spontaneous symmetry

breaking. The original Lipkin model as quoted in [50] is:

H = S z

2 N S +2 S -

2 (137)

where: N∈ℕ , ∈ℝ , S ±= S x±i S y and S x , S y , S z are the familiar Pauli spin matrices. For the

H of Eq.(137) one finds that the eigenvalues “decouple” into two sets, corresponding to

even and odd values of N . Therefore, in order “to facilitate technicalities” Heiss et al

modified Eq.(137) to the following form (a direct quote of their Eq.(2)):

H = H 0 H 1H 0=k k , k '

H 1=N8 1−1− 2k

N 2 k , k '−1h.c. , k=1 , , N

(138)

Here, the Kronecker deltas are denoted with hats since they represent a matrix form, as

demonstrated below:

For k , k '∈{1,2,3}

k ,k '=1 0 00 1 00 0 1 k k ,k '=1 0 0

0 2 00 0 3

k , k '−1=0 1 00 0 10 0 0 k , k '1=0 0 0

1 0 00 1 0

Therefore k ,k '−1=k , k '1†

and k ,k '1=k−1, k '

(139)

In Eq.(138) “h.c.” means “Hermitian conjugate”; therefore we can rewrite H 1 as:

H 1=N8 1−1− 2k

N 2 k , k '−1k , k '1 (140)

Heiss et al, solve for the value at the phase transition, i.e. the exceptional point. Their

result was:

k=±i NN−2k−1

k=1, , N2 (141)



It was suggested to me by Weigert [51] that the Heiss model Eq.(138) could be modified, so

as to produce an analytically solvable PT-symmetric model. By following the same process as

Heiss [50], the aim would be to determine for which values of the PT symmetry of the

model breaks down, i.e. the exceptional point.

Therefore, the first step is to determine how to make Eq.(138) PT-symmetric. To start with,

let us remind ourselves of the PT transformations of the Pauli spin operators. Both S x and S y

are PT invariant, but P T S zT P=−S z . From this, one may make S z PT invariant by

multiplication with the imaginary unit, which would compensate for the negative term, giving

P T i S z T P=i S z . The S ± operators are not PT invariant, although their squares are, as

follows from the definition S ±= S x±i S y . These identities hold for cases beyond the familiar

2x2 cases. A 1978 paper by Jeffery [52] shows how one may calculate n-dimensional spin

matrices, e.g. the 3x3 spin matrices:

S x=0 12 0

12 0 1

2

0 12

0 S y=0 −i 1

2 0i 1

2 0 −i 12

0 i 12

0 S z=1 0 0

0 0 00 0 −1

(142)

One can see by inspection that the PT invariances for the 3x3 cases are the same as those for

the 2x2 cases.

From these observations, it follows that a trivial modification of the Lipkin model as given in

Eq.(137) will endow it with PT symmetry:

H =i S z

2 N S+2 S -

2 (143)

This is instructive in indicating how the Heiss/Lipkin Hamiltonian Eq.(138) can be made PT-



symmetric. Since the S z term is playing the role of H 0 , we can compare their respective

matrix structures.

First, consider the matrix representation of H 0=k k ,k ' from Eq.(138):

H 0=1 ⋱N (144)

Secondly, we wish to study the general form of S z , which is more complicated. Consider the

following NxN matrix representation:

S z=−n−n2

⋱n−2

n {For N even, n=N−1

For N odd, n=N12

(145)

Here, one must consider whether N is even or odd. When N is even we can simply shift the

diagonal terms of H 0 by subtracting a multiple of the identity matrix to obtain S z . Therefore,

we can write:

S z=2 H 0−1N even I (146)

For example, we would write the following matrix sum for the 6x6 case:

S z=212

34

56−71

11

11

1=−5

−3−1

13

5 (147)

However, if N is odd, such as for the 5x5 case, Eq.(146) does not work:

S z≠212

34

5−61

11

11=−4

−20

24 (148)

We see that Eq.(148) contradicts Eq.(145).



Further inspection reveals that when N is odd the relationship between S z and H 0 is:

S z= H 0−1N odd

2I (149)

Rewriting the 5x5 case according to Eq.(149) gives:

S z=1 23

45−31 1

11

1=−3

−10

13 (150)

This demonstrates seemingly different expressions for the cases when N is even or odd.

However, given that Eq.(146) & Eq.(149) only differ by a factor of 2, there seems no

justifiable reason to let this decoupled space persist. Therefore, my solution was to use the

greatest common divisor (GCD) of N and 2 (for which the Euclidean algorithm is the most

efficient method [53]). If N is even, the GCD of N and 2 is 2. If N is odd, the GCD of N and 2

is 1.

Hence, we can reunite the even and odd spaces by using the equation:

S z=gcd N ,2 H 0−N1

2I (151)

Rewriting Eq.(151) to substitute H 0 for k k , k ' , as stated in Eq.(138) we obtain:

S z=gcd N , 2 k−N12 k , k ' (152)

Now, let us recall H 1 from Eq.(140). We can see that the k , k '−1k , k '1 term possesses PT

symmetry, according to the definitions given in Eq.(139). The matrix structure is very similar

to that of S +2S -

2 . Here is a 6x6 example of H 1 to show how the off-diagonal elements

change, but maintain P symmetry:



H 1=0 5/12 0 0 0 05 /12 0 2 /3 0 0 00 2 /3 0 3 /4 0 00 0 3/4 0 2 /3 00 0 0 2 /3 0 5/120 0 0 0 5/12 0

(153)

For H 1 we can also make the simplification:

N8 1−1− 2k

N 2= k N−k

2 N (154)

Therefore, we can finally rewrite H 1 as:

H 1=k N−k

2 N k , k '−1k , k '1 (155)

If we now take S z from Eq.(152), multiplied by the imaginary unit and then added to the H 1

of Eq.(155) multiplied by , we finally obtain the PT-symmetric version of the Lipkin/Heiss

model:

H =i gcd N ,2 k− N12 k , k ' k N−k

2 N k , k '−1k , k '1 , For k , k '=1, , N (156)

The next step that Heiss et al took was to find the simultaneous roots belonging to both the

characteristic polynomial and to its derivative; the corresponding values of are exceptional

points. It is useful to think of the characteristic polynomial in the form:

P E=∏i=1

N

E−i (157)

Hence, one may inductively calculate the derivative of P E .

Take a general polynomial P x=∏i=1n P i x ; according to the product rule we have:

ddx

P x=dP1

dxP2 P3

dP2

dxP1 P3

dP3

dxP1 P2 (158)

Now factorise this equation with respect to ∏i=1n P i x :

ddx

P x=∏i=1

n

P i x 1P1

dP1

dx 1

P2

dP2

dx 1

P3

dP3

dx (159)



Rewriting Eq.(159) in a compact form gives:

ddx P x=P x∑

i=1

n 1Pi

dPi

dx (160)

Applying this result to the P E of Eq.(157) gives:

ddE

P E=P E ∑i=1

N 1E−i (161)

The logic behind finding the simultaneous roots of P E and P ' E is that as the

perturbation term approaches an exceptional point, pairs of real eigenvalues of the

Hamiltonian coalesce into single real values, as illustrated in figure 5.1. The roots of the

characteristic polynomial will do the same because they are the eigenvalues. At this stage one

may imagine the graph of a polynomial, plotted on the same axes as its derivative, moving

vertically so that its turning points and/or stationary points coincide with the roots of the

derivative graph. This is demonstrated in figure 5.2:



Figure 5.3 illustrates the behaviour of the eigenspectrum in the N=6 case of the H of

Eq.(156) as varies. We can see that as increases from negative values, pairs of

eigenvalues come together until they are no longer defined as real values. Eventually, all the

eigenvalues have become complex or imaginary. It is not until arrives at its respective

positive exceptional points that we regain single real eigenvalues, which then go on to split

into pairs of real eigenvalues as continues to increase. Eventually, we recover the full

eigenspectrum as real numbers.


Figure 5.2: A representation of the change in the characteristic polynomial

P(E) as pairs of real eigenvalues coalesce into single eigenvalues and coincide

with the roots of P'(E), prior to a spontaneous breaking of PT symmetry.


Following the procedure reported in [50], I used the computer program Mathematica to

calculate the resultant of the characteristic polynomial and its derivative. The resultant

eliminated the energy variable E, leaving only the roots in to be solved for. (See Appendix

A for the list of Mathematica commands required.) Because of a technical limitation I could

not input an expression for a general case (i.e., a matrix of indefinite size). As an alternative, I

took particular cases and sought an inductive solution.

In each case, the resultant polynomial R for the PT-symmetric Lipkin model was of order

N N−1 . This is the same as for the resultant calculated by Heiss et al [50], so we see that the

degree of R is unaffected by PT symmetry. The resultant for the PT-symmetric Lipkin

model factorised into the following form, just as it did in the calculations of Heiss et al:

R=S T 2 (162)

where S was of the form:


Figure 5.3: Real energy eigenvalues for the N=6 case of the PT-symmetric Lipkin Hamiltonian as a

function of .


S = ∏k=1

k=N /2

k (163)

This factorised form was desirable because inspection showed that the position of the

exceptional points were encoded as the roots of S . Therefore, T was of no interest

other than to make a further comparison of polynomial degrees. S was of order 2N, and

T was of order N−1 2−1 /2 . These are the same as the orders of the corresponding

polynomials given in [50]. Hence, we see that even by incorporating PT symmetry the

fundamental structure of the exceptional point problem is unaltered. However, there is one

important difference between the Heiss/Lipkin Hamiltonian and the PT-symmetric Lipkin

Hamiltonian. The EPs of the Heiss model are imaginary, as shown in Eq.(141), whereas the

EPs of the PT-symmetric Lipkin Hamiltonian are all real, as shown below.

Table 5.1 shows the roots of S for various values of N, as calculated by the computer

program Mathematica. I wrote a MatLab script to calculate the PT-symmetric Lipkin

matrices, from which the polynomials were computed in Mathematica. I eventually wrote a

Mathematica equivalent to the MatLab script (see Appendix A for all computer codes). Since

the case of odd values of N gave more complicated roots I limited my analysis to even values

of N.



N k

2 ±4

4 ±83

,±8

6 ±125

,±4 ,±12

8 ±167

,±165

,±163

,±16

10 ±209

,±207

,±4 ,±203

,±20

12 ±2411

,±83

,±247

,±245

,±8 ,±24

14 ±2813

,±2811

,±289

,±4 ,±285

,±283

,±28

16 ±3215

,±3213

,±3211

,±329

,±327

,±325

,±323

,±32

18 ±3617

,±125

,±3613

,±3611

,±4 ,±367

,±365

,±12 ,±36

20 ±4019

,±4017

,±83

,±4013

,±3611

,±409

,±367

,±8 ,±403

,±40

22 ±4421

,±4419

,± 4417

,±4415

,±4413

,±4 ,±449

,±447

,± 445

,±443

,±44

24 ±4823

,±167

,±4819

,±4817

,±165

,± 4813

,±4811

,±163

,±487

,±485

,±16 ,±48

26 ±5225

,±5223

,±5221

,±5219

,±5217

,±5215

,±4 ,±5211

,±529

,±527

,±525

,±523

,±52

28 ±5627

,±5625

,±5623

,± 87

,±5619

,±5617

,±5615

,±5613

,±5611

,±569

,±8 ,±565

,±563

,±56

30 ±6029

,±209

,±125

,±6023

,±207

,±6019

,±6017

,±4 ,±6013

,±6011

,±203

,±607

,±12 ,±20 ,±60

32 ±6431

,±6429

,±6427

,±6425

,±6423

,±6421

,±6419

,±6417

,±6415

,±6413

,±6411

,±649

,±647

,±645

,±643

,±64

Table 5.1: Exceptional points of the PT-symmetric Lipkin model, for various values of N.

From this, a general expression for the kth exceptional point of even dimensional cases can be

seen by inspection:

k=2 N

N−2k−1 , k=1 , , N (164)

While this is not a rigorous proof, it shows that the Lipkin model can be modified to produce

a PT-symmetric model, which in turn allows all of the exceptional points to be calculated

analytically. In my research, I have not found any other such PT-symmetric model.



Chapter 6. Conclusion

6.1. Summary

This thesis has endeavoured to give an overview of quantum mechanics: setting out the

original motivations, the established text book postulates and the latest attempts to generalise

the mathematical framework. The latter attempts effectively tried to modify the mathematical

constraints to be satisfied in order to model a physical system.

However a quantum theory is expressed there are certain requirements that must be fulfilled

in order for the theory to be physically reasonable (see section 5.1). In the past, these

requirements have been fulfilled by using the mathematics of Hermitian operators in a

complex Hilbert space, with the Euclidean inner product, ⟨∣ ⟩ :=∑i=1i=n i

*i . In this traditional

formalism, the terms “Hermitian” and “self-adjoint” are synonymous; if an operator is

“Hermitian” then it satisfies both ⟨ H∣ ⟩=⟨∣ H ⟩ for the Euclidean inner product, and

H= H † where † represents complex conjugation and matrix transposition. However, we have

since learnt that those terms are not synonymous when we venture into a formalism using

exotic inner products!

This thesis has discussed two alternatives to the Hermitian framework, namely that of CPT

symmetry introduced by Bender et al, and that of pseudo-Hermiticity introduced by

Mostafazadeh.

PTSQM replaces the condition of Hermiticity for an operator with the condition of PT

symmetry. An example of this is the Bessis conjecture [1], for which the PT invariant

Hamiltonian p2i x3 has a real and positive spectrum. However, one should always be aware

that not all PT invariant operators have a completely real spectrum. Later investigation of

various PT-symmetric systems showed that for any given Hamiltonian only half of its

David R Gilson Chapter 6. Conclusion 70 of 77


eigenvectors had positive norms, while the other half had negative norms. Norms are usually

required to be positive so as to permit an interpretation in terms of probabilities. However, the

fact that there was an equal share of positive and negative norms was in itself a kind of

symmetry. Therefore, a third operator called C was introduced in order to compensate for the

presence of negative norms.

The operators described above are defined as follows. T is the operation of complex

conjugation. The parity operator P , in its simplest form can be pictured as a square matrix

with ones along its minor diagonal and zeroes elsewhere. The C operator is more

complicated; it is composed by using the entire biorthonormal eigenbasis of the Hamiltonian,

C=∑ j=1j=n −1 j∣ j ⟩ ⟨ j∣ . Thus in defining transformations and inner products we use the

compound operator C P T .

The other alternative framework is that of pseudo-Hermiticity. In this framework,

Mostafazadeh replaces the traditional condition of Hermiticity with that of being “Hermitian”

with respect to a Hermitian linear automorphism, . In this case a Hamiltonian is called -

pseudo-Hermitian if it satisfies H=−1 H † . This approach can be seen to give a direct

generalisation of Hermitian quantum mechanics, to which it reduces when becomes the

identity operator, since the condition of -pseudo-Hermiticity then reduces to that of normal

Hermiticity.

To calculate , one needs to use the entire biorthonormal eigenbasis of the Hamiltonian (see

Eq.(96) and Eq.(97)). This property is similar to that of the C operator in CPTSQM, in that

one needs access to the entire eigenbasis before even attempting to test for CPT invariance. It

was later shown that pseudo-Hermitian quantum mechanics could be a parent class to

CPTSQM, given the relationship, = P C .



6.2. Comments and outlook

Prior to writing this final chapter, I thought that I should revisit the literature to look for any

new developments. I found that Mostafazadeh had published a time dependent model of the

imaginary cubic potential v x=i x3 , using a pseudo-Hermitian framework [54]. Until now, a

time dependent model had been missing from the pseudo-Hermitian literature. I also found

that Bender had written a review of non-Hermitian Quantum Mechanics [55].

The development of pseudo-Hermiticity has been encouraging, in that it has not given rise to

the same conceptual difficulties that PTSQM has produced. According to [35], applications of

the approach have been found in quantum cosmology, relativistic quantum mechanics,

statistical mechanics, and magnetohydrodynamics. Furthermore, the structure of the theory

seems to be intuitively correct, being based on a simple relaxing of the original condition of

Hermiticity. However, PT symmetry has the honour of being the first framework to open our

eyes to the possibility of having a non-Hermitian Hamiltonian with a real eigenspectrum.

In the future, I would hope that an experimental signature to characterize pseudo-Hermiticity

can be devised, or at least that the quantum cosmological applications of pseudo-Hermiticity

find some relationship to astronomical observations. Also, it would be of interest if anyone

could find an -pseudo-Hermitian Hamiltonian which has a complex spectrum, while still

satisfying the condition for a real spectrum (i.e. disprove the theorem given in equations (97)

to (102)). A significant achievement for PT symmetry would be the possibility of finding

signatures of its existence in experimental results [22]. Bender et al have already suggested

this for lattices with complex potentials. As of yet, it is still unknown whether PT symmetry

will relate to any experimental or practical test.

From my point of view, the major question for PT symmetry relates to its applicability. There



are good reasons to say that it is not a universally applicable theory, although there are cases

in which it has indeed been productive to apply it, (see [46]). Weigert has written algorithms

to check for the viability of potential PT-symmetric systems, on a one by one basis (see [25]

and [48]). However, what is not known and would be an exciting discovery, is to find some

criterion determining when PT symmetry is the appropriate tool to use. I believe that a rich

area to search for this answer is in particle physics; this idea is inspired by the complex lattice

potential suggestion of Bender, [22]. Also, Weigert has mentioned uses of non-Hermitian

Hamiltonians in particle physics [23].

It is also unknown whether PT symmetry could be applied to areas of physics other than

quantum theory. However, several uses of non-Hermitian theories have been noted by

Weigert [23], all of them providing possibilities for applications of PT symmetry.

Within the research community there has been a marked difference in the respective

acceptance of PTSQM and pseudo-Hermitian quantum mechanics. This has been all the more

surprising, given the many technical drawbacks of PTSQM, whereas pseudo-Hermitian

quantum mechanics has suffered no such problems.

Finally, quantum mechanics has (somewhat notoriously) given rise to a great deal of

philosophical debate about its interpretation, more so than any other physical theory. A rich

avenue of philosophical and metaphysical discussion could well be found in the implications

of expanding the condition of Hermiticity. Such a study would provide many linguistic and

logical problems. As the study for this thesis has certainly shown, when dealing with such

mathematical subtleties we need to be as strict with our language as we are with our

mathematics.



Appendix A. Computer codeThis is a copy of the Matlab script I wrote to generate the matrix for particular cases of the

PT-symmetric Lipkin model. The argument N is the dimension of the matrix:

Function M = heiss(N)k=1:N;h0=zeros(N); h1=zeros(N);m=zeros(N); M=zeros(N);A=sqrt(-1)*gcd(N,2); B=-((N+1)/2)*eye(N);syms L realfor i=1:N;h0(i,i)=i;

endfor i=1:N;for j=1:N;

m(i,j)=((N/8)*(1-(1-(2*k(i)/N))^2))*isequal(k(i),k(j)'-1);end;

end;h1=m+m';M=A*(h0+B)+(L*h1);sym(M);return;

This is an equivalent function written for Mathematica; the input argument n is the number of

dimensions:

Heiss[n_] := Table[I*(GCD[n, 2]*(i - ((n + 1)/2)))*KroneckerDelta[i, j], {i, n}, {j, n}] + Table[\[Lambda]*((i*(n - i))/(2*n))*KroneckerDelta[i, j - 1], {i, n}, {j, n}] + Table[\[Lambda]*((j*(n - j))/(2*n))*KroneckerDelta[i - 1, j], {i, n}, {j, n}] // MatrixForm

The chain of Mathematica commands to compute the exceptional points were as follows

(italics are to be read as instructions to the reader):

H:= (Matrix output from the Heiss function, see above)P = CharacteristicPolynomial[H, x];Q = D[P, x];R[\[Lambda]] = Resultant[P, Q, x];Factor[R[\[Lambda]]S[\[Lambda]]:= (Manually selected output from the previous command)Solve[S[\[Lambda]] == 0, \[Lambda]]

David R Gilson Appendix A: Computer code 74 of 77


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an introduction and brief topical review of non-hermitian quantum mechanics with real spectra

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