Quasinormal Modes of Black Holes

and

Quantum Dissipative Dynamics

Thesis

Submitted in partial fulfillment of the requirements of

BITS C422T Thesis

By

Soham Pal

2008B5AA569H

Under the supervision of

Dr. S. Shankaranarayanan, Assistant Professor,IISER, Thiruvananthapuram

and

Dr. Sashideep Gutti, Assistant Professor,BITS, Pilani - Hyderabad Campus

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI HYDERABAD CAMPUS

April 30, 2013

CERTIFICATE

This is to certify that the Thesis entitled Quasinormal Modes of Black Holes and Quantum DissipativeDynamics and submitted by Soham Pal (ID: 2008B5AA569H ) in partial fulfillment of the requirementof BITS C422T Thesis embodies the work done by him under my supervision.

Date:

Dr. S. ShankaranarayananAssistant Professor

Dr. Sashideep GuttiAssistant Professor

ACKNOWLEDGMENT

I would like to extend my deepest gratitude to my supervisor, Dr. S. Shankaranarayan and my co-supervisor, Dr. Sashideep Gutti for all the motivation during the course of this work and for providingme the opportunity to work under their guidance. I would also like to thank Prof. A.P. Balachandranfor his valuable suggestions. I also thank BITS, Pilani - Hyderabad Campus for allowing me to dothis project and IISER, Thiruvananthapuram for providing me with all the facilities without which Icouldn’t have completed this project.

Abstract

Oscillating fields in a perturbed black hole background are characterized by quasinormal modes.These modes dominate a large portion of the time evolution of perturbed black holes. Study ofquasinormal modes can provide us with valuable data regarding the classical and quantum propertiesof black holes. Recent studies have shown that quasinormal modes are also present in damped systemsother than black holes. A quasinormal mode system is non-Hermitian in nature, thus making itdifficult to construct a quantum theory of quasinormal modes using the existing Hermitian quantummechanics.

In this thesis we have studied various methods of calculating quasinormal modes of black holesand have attempted to construct a pseudo-Hermitian quantum theory of quasinormal modes based ona biorthonormal basis expansion.

Contents

1 Introduction 2

2 What are Quasinormal Modes ? 42.1 Methods for calculating QNMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Quasinormal Modes of the BTZ Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Quantization via First Order Action 123.1 BCK Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 First order action quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Toy Models and Biorthonormal Basis Expansion 194.1 Toy Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Pseudo-Hermiticity and biorthonormal basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Quantum theory for QNMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3.1 Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3.2 Energy Expectation Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Conclusion 29

A Dirac Brackets: the Faddeev-Jackiw formalism 34

1

Chapter 1

Introduction

Quantum mechanics is the kernel around which much of modern physics has developed. It is, ar-guably, the most fundamental physical theory. Conventionally quantum systems have been treated asisolated systems. For example, we can consider the ubiquitous simple harmonic oscillator. It is un-doubtedly the quantum system with the most fan following and not unjustly so. The simple harmonicoscillator is the perfect tool to teach the newly-initiated, the intricacies of quantum mechanics. It isalso a very simple system, one that we understand completely, that can be used to model and studymore complex systems. However just like any classical system, a realistic quantum system is subjectedto a coupling to an environment, in a non-negligible fashion [1]. Perfect isolation of quantum systemsis impossible, and usually the degrees of freedom of the environment is not completely under ourcontrol. The theory behind realistic quantum systems would be more accurate if they are regarded asopen systems. The damped harmonic oscillator is therefore a better model for the real world quantumsystems, than the harmonic oscillator. However quantummechanics has been developed over the yearswith isolated systems in mind, and it cannot be applied just as it is to open systems, there are bothphysical and mathematical limitations to the application of conventional quantum mechanics to opensystems. Despite such limitations, many of the real-world systems, for example resonant systems likea ringing bell or a vibrating guitar string, can be studied using conventional quantum mechanics, upto a degree of practical accuracy, and it is usually more convenient to do so than develop a completelynew quantum theory for an open system. But dissipative systems like a black hole or a leaky opticalcavity, cannot, even in principle, be studied as isolated systems. For dissipative systems like these newapproach to quantum theory is required.

An isolated black hole in equilibrium is a very simple object, it can be described by just a fewparameters, like, mass , angular momentum and charge (“no-hair”conjecture)[2], but it is an idealobject and doesn’t exist in nature. Even if there were no matter in the vicinity of a black hole,it would interact with vacuum, producing particle-antiparticle pairs and would decay via Hawkingradiation[3]. Thus a black hole can never be described completely by it’s basic parameters and isalways in a perturbed state [3]. Therefore the perturbations of a black hole form an integral part ofany study involving the black hole.

The early 1970s saw a large number of physicists working on black hole perturbations. Includingthe ones mentioned in the previous paragraph, a black hole can be perturbed via various methods:scattering of waves by black hole, small bodies falling into black holes, non-spherical gravitational

2

Chapter 1: Introduction

collapse to form black holes, etc[4]. In 1970, C.V. Vishveshwara working on the scattering of gravita-tional waves by a Schwarzschild black hole, observed that the evolution of black hole perturbationsoccur in three stages: the first stage consists of a wave burst, that contains radiation emitted by thesource of the perturbations, the second stage is dominated by exponentially damped single frequencyoscillations (ringing), that are completely independent of the source of the perturbations, and thethird stage is just a power-law tail, that arises because of backscattering by long-range gravitationalfield[5]. The damped oscillations of the second stage depend only on the parameters that characterizethe black hole, it’s as if that the black hole has left it’s fingerprints on these oscillations. This is verysimilar to a bell sounding it’s last dying notes, hence the ‘ringing’. These characteristics oscillationshave been termed as ‘quasinormal modes’ (QNMs) and the corresponding’frequencies ‘quasinormalfrequencies’(QNFs).

QNMs have been named so because they are to a perturbed black hole what normal modes (NMs)are to isolated systems. Despite the similarity in names, there are some fundamental differences ofextreme importance, between the exponentially damped QNMs and the NMs. Unlike the NMs, theQNMs are not stationary. The non-stationary nature of the QNMs represent the fact that the blackhole is radiating energy away to spatial infinity in the form of gravitational waves. The other majordifference between NMs and QNMs involve completeness. NMs always form a complete set, QNMs,on the other hand, generally tend to shy away from forming a complete set, in the traditional sense[6],[7].

QNMs have been historically developed in the context of black holes. But there are other dissipativesystems, like the leaky optical cavity or a vibrating string attached to a medium. Works by Nollert[8],Leung et.al[9], Settimi et.al.[10] and others show that we won’t be wrong if we assume that QNMsalso exists for dissipative systems other than black holes.

The rest of the thesis deals with quasinormal modes of dissipative systems, like perturbing blackholes and optical cavities. the aim of this thesis is not to just calculate the quasinormal modes of a dis-sipative system but to develop quantum theories that can be used to describe such systems. In chapter2 we give a mathematical definition of QNMs and derive the QNMs of a BTZ (Banados-Teitelbom-Zanelli) black hole. In chapter 3 we present a theory, suggested by Baldiotti et.al[11], that can be usedto quantize dissipative systems. In chapter 4 we present some toy models to derive the QNMs of ablack hole and with the help of these toy models, develop a non-Hermitian quantum theory for theQNMs. We will use Planck’s units ( ħh = c =G = ke = kB = 1) in the rest of this report.

3

Chapter 2

What are Quasinormal Modes ?

As stated in the title of Nollert’s review article on QNMs [12], quasinormal modes are the char-acteristic sounds of black holes. They are single-frequency modes that dominate the perturbationsof damped systems. The damping might arise due to internal dissipation or radiating away energy.Because the system is damped, the frequencies of the modes are complex, ω = ωI + iωR, where theimaginary component is inversely proportional to the typical damping time [8]. Mathematically theQNMs of black holes are solutions to the partial differential equation of the form (Klein-Gordonequation)

∇ν∇ν +µ2

Φ= 0 (2.1)

subject to purely outgoing boundary conditions at spatial infinity

Φ∼ purely outgoing, r →+∞

and since nothing can come out of the black hole horizon, therefore also subjected to purely ingoingboundary conditions at the horizon

Φ∼ purely ingoing, r → rh

where r is the radial coordinate and rh represents the horizon. Over the years a lot of methods,both analytical and numerical, have been developed to find the QNMs of black holes. We will dis-cuss a few of these methods in the following sections and finally derive the QNMs of a BTZ black hole.

2.1 Methods for calculating QNMs

Mashhoonmethod This method was originally suggested by Blome andMashhoon [13] for a Schwarzschildblack hole, and was later expanded by Ferrari and Mashhoon [14] to include Kerr black holes. Herewe will describe the method for a spherical, symmetric black hole background.

We use the ansatz Φ= e iωt Y lm1...mD−1

(θ,ϕ1 . . .ϕD−1)ψ(x)

rin the Klein-Gordon equation

d 2

d x2+

ω2−U (x)

ψ= 0 (2.2)

4

Chapter 2: What are Quasinormal Modes ?

Here x is the tortoise coordinate ( r = r (x)) and U (x) is non-negative potential that falls off exponen-tially as x →−∞ and as 1/x2 as x → +∞. The boundary conditions for the radial wave function ψare

ψ∼ e∓iωx , x→±∞ (2.3)

We introduce a parameter p, such that the potential, the wave function, and the frequency are func-tions of p

U =U (x, p)

ψ=ψ(x, p)

ω =ω(p)

We consider the transformations x → −i x and p → p ′ = π(p) such that the potential remains un-changed, U (x, p) = U (−i x, p ′). The wave function and the frequency transform as ψ(−i x, p) =φ(x, p) and ω(p ′) = Ω(p). In terms of these new wave function and frequency, we rewrite equation(2.2) as

d 2

d x2+

−Ω2+U

φ= 0 (2.4)

and the corresponding boundary conditions are

φ∼ e∓Ωx , x→±∞ (2.5)

These boundary conditions correspond to a vanishing wave at the boundaries. The QNM problem isnow thus converted to a bounded states problem for the inverted potential U → −U . The invertedpotential is smooth and approaches some constant values at infinite boundaries. We can approximatethis potential gap, using a potential for which we already know the analytic solution to the waveequation. We will use the Pöschl-Teller potential since it’s bound states are well known [15].

UPT =U0

cosh2 p(x − x0)(2.6)

U0 is the height of the Pöschl-Teller potential and −2U0 p2 is the curvature of the potential at itsmaximum. The bound state frequencies of the potential are

Ω= p ′

−

n+1

2

+

1

4+

U0

(p ′)2

12

(2.7)

We can obtain the QNFs from these bound state frequencies by the transformation p ′ = i p.

ω =±

È

U0−1

4p2− i p

n+1

2

(2.8)

To find the QNFs of any arbitrary black hole, using this scheme, we will have to fit the black holepotential to the inverted Pöschl-Teller potential.

5

Chapter 2: What are Quasinormal Modes ?

Riccati equation and the “shooting”algorithm Chandrasekhar and Detweiler had originally pro-posed the shooting method in 1975[16] for a Schwarzschild black hole. We transform the wave equa-tion (2.2) to the Riccati equation

idφ

d x+ω2−φ2−U (x) = 0 (2.9)

using the logarithmic derivative of the radial wave function

φ=1

ψ

dψ

d x

The corresponding boundary conditions transform as

φ∼∓ω, x→±∞ (2.10)

Integrating equation (2.9) over the entire range of x and using the boundary conditions (2.10) we getthe identity

− 2iω+∫ +∞

−∞

ω2−φ2

d x =∫ +∞

−∞U (x)d x (2.11)

The boundary conditions (2.10) and the nature of the potential U (x) suggest that both the integralsin equation (2.11) are finite.Now we use the shooting procedure to calculate the QNFs from equation (2.9). We define similarfunctions ψ±(ω, x) and their logarithmic derivatives φ±(ω, x) at some general frequency ω as solu-tions to equations (2.2) and (2.9), respectively, but with each function satisfying only one boundarycondition, i.e. the ‘+’functions satisfy the corresponding boundary conditions at x → +∞ and the‘−’functions satisfy the corresponding boundary conditions at x →−∞. We now integrate equation(2.9) with φ+ from ∞ to some intermediate x0 and integrate equation (2.9) with φ− from −∞ tox0. We calculate the Wronskian of the two solutions at x0. The frequencies for which the Wronskianvanishes are the QNFs.

Integrating the wave equation We use the ansatz Φ = φ(t , x)Y lm1...mD−1

(θ,ϕ1 . . .ϕD−1), where x isagain the tortoise coordinate, in the Klein-Gordon equation for a spherical, symmetric black holebackground

∂ 2

∂ t 2−∂ 2

∂ x2+U (x)

φ= 0 (2.12)

Following the suggestion of Gundlach et.al[17],[3] we define the light-cone coordinates

d u = d t − d x and d v = d t + d x (2.13)

In the light-cone coordinates equation (2.12) becomes

4

∂ 2

∂ u∂ v+U (u, v)

φ= 0 (2.14)

6

Chapter 2: What are Quasinormal Modes ?

We will integrate equation (2.14) on a uv null grid. The time evolution operator in these coordinatesis

exp

h∂

∂ t

=exp

h∂

∂ u+ h

∂

∂ v

+h2

2

exp

h∂

∂ u

+ exp

h∂

∂ v

∂ 2

∂ u∂ v+O (h4) (2.15)

where h is a overall grid factor. Applying the time evolution operator to the wave function

φ(u + h, v + h) =φ(u + h, v)+φ(u, v + h)−φ(u, v)

−h2

8U (u, v) [φ(u + h, v)+φ(u, v + h)]+O (h4) (2.16)

Using initial data and equation (2.16) we can get the value of φ inside the rhombus bordered byu = u0 and v = v0. Using this values we can construct a time-domain profile of the wave function.Now to figure out which quasinormal frequency the time-profile data corresponds to we use the Pronymethod of fitting the profile data by superposition of damped exponential functions [3],[18]

φ(t ) =p∑

j=1

ci e−iω j t (2.17)

We assume that the quasinormal ringing period lasts from t = t0 = 0 to t =N h, where N is an integergreater than or equal to 2 p − 1. Then the Prony equation is valid for each value from the profile data

yn ≡φ(n, h) =p∑

j=1

c j e−iω j nh =

p∑

j=1

c j znj (2.18)

where n = p, . . . ,N . We define a polynomial

A(z) =p∏

j=1

z − z j

=p∑

m=0

αm z p−m , α0 = 1 (2.19)

Therefore the sum

p∑

m=0

αm yn−m =p∑

m=0

αm

p∑

j=1

c j zn−mj

p∑

j=1

c j zn−pj

p∑

m=0

αm z p−mj

p∑

j=1

c j zn−pj A(z j ) = 0 (2.20)

Since α0 = 1 thereforep∑

m=1

αm yn−m =−yn (2.21)

7

Chapter 2: What are Quasinormal Modes ?

which when rewritten in matrix form becomes

yp−1 yp−2 . . . y0

yp yp−1 . . . y1...

.... . .

...yN−1 yN−2 . . . yN−p

α1

α2...αp

=−

yp

yp+1...

yN

(2.22)

orYα=−y (2.23)

The matrix equation can be solved in the least-square sense

α=−

Y†Y−1Y†y

(2.24)

Now that we have the coefficients α j of the polynomial A(z), we can numerically calculate the rootsof the polynomial, z j . The QNFs are then calculated via

ω j =1

hln(z j ) (2.25)

QNMs via Laplace transform Nollert and Schmidt used the Laplace transform to give an alternatedefinition of QNMs and also suggest a method to calculate the QNFs[19]. We consider the equation(2.12) along with the initial conditions

φ(o, x) = f (x) and φ(0, x) = g (x)

Laplace transform of equation (2.12) gives

−∂ 2x + s2+U (x)

φ(s , x) = s f (x)+ g (x) = j (s , x) (2.26)

whereφ(s , x) =L [φ(t , x)] (s) =

∫ ∞

0e−s tφ(t , x)d t (2.27)

We solve equation (2.26) using Green’s function

−∂ 2x + s2+U (x)

G(s , x, x ′) = δ(x − x ′) (2.28)

The Laplace transformed wave function is derived from the Green’s function

φ(s , x) =∫ +∞

−∞G(s , x, x ′) j (s , x ′)d x ′ (2.29)

To construct the proper Green’s function we take the aid of the auxiliary functions, g±(s , x), thatsatisfy the homogeneous differential equation

−∂ 2x + s2+U (x)

g±(s , x) = 0 (2.30)

8

Chapter 2: What are Quasinormal Modes ?

and g+(s , x) satisfies the boundary condition at x→+∞ and g−(s , x) satisfies the boundary conditionat x→−∞, i.e.

∫ ±∞

a|g±(s , x)|2d x <∞, ∀a ∈R

Since the potential vanishes at infinity therefore

g±(s , x)∼ e∓s x , x→±∞ (2.31)

The Green’s function is constructed as

G(s , x, x ′) =−g−(s , x<)g+(s , x>)

W

g−, g+

(s)(2.32)

where x< =min(x, x ′) and x> =max(x, x ′) and W

g−, g+

(s) is the Wronskian of g− and g+. When-ever the Wronskian vanishes for some value of s it indicates that there exists a solution of the Laplacetransformed equation that satisfies both boundary conditions simultaneously. Therefore the zeros ofthe Wronskian (or the poles of the Green’s function) are the QNMs.

2.2 Quasinormal Modes of the BTZ Black Hole

The preceding section shows that the exact calculation of QNMs of black holes is extremely diffi-cult. The calculations are either based on some simpler approximations or use some numerical steps.However there’s an exception. Exact analytical solutions exist for the QNMs of the BTZ black hole.The BTZ black hole was introduced by Banados et.al. as a solution to Einstein’s equations in a 2+1asymptotically-AdS spacetime[20]. Cardoso and Lemos first presented the analytical solutions of theQNMs of a static BTZ black hole[21]. It was later generalized to rotating BTZ black holes by Birm-ingham et.al.[22].

The BTZ black hole metric

d s2 =− f (r )d t 2+ f (r )−1d r 2+ r 2(dφ+N (r )d t )2 (2.33)

where f (r ) =−M +r 2

l 2+

J 2

4r 2and N (r ) =−

J

2r. The zeros of f (r ) give

r = l

M

2

1±

s

1− J

M l

2

12

where the positive solution (r+) is the radius of the outer horizon of the black hole and the negativesolution (r−) is the radius of the inner horizon of the black hole.We consider a massive scalar field inthe background geometry of the BTZ black hole. Using the ansatz

Ψ= R(r )e i mφ−iωt (2.34)

9

Chapter 2: What are Quasinormal Modes ?

in equation (2.1)

−M +3r 2

l 2+

J 2

4r 2

d 2r R+

−M +3r

l 2−

J 2

4r 3

dr R

−

4mω

J+

ω2

M −r 2

l 2+

m2

r 2

R= 0 (2.35)

Introducing the new variable

z =r 2− r 2

+

r 2− r 2−

we rewrite the radial equation as

z(1− z)d 2z R+(1− z)dz R+

Az+B +

C1− z

R= 0 (2.36)

with

A =l 4

4(r 2+− r 2

−)2

ωr+−m

lr−2

B =−l 4

4(r 2+− r 2

−)2

ωr−−m

lr+2

C =−µ

4

We define the radial function in terms of a new functionF (z) as

R(z) = zα(1− z)βF (z)

thus transforming the radial equation into a standard hypergeometric equation

z(1− z)d 2zF +[c − (1+ a+ b )z]dzF − abF = 0 (2.37)

where

c = 2α+ 1

a+ b = 2(α+β) (2.38)

ab = (α+β)2−B

and

α2 = −A

(2.39)

β =1

2

1±p

1+µ

10

Chapter 2: What are Quasinormal Modes ?

We choose the following solutions of equation (2.39)

α=−ipA , β=

1

2

1−p

1+µ

The two linearly independent solutions of equation (2.36) are given by the F (a, b , c , z) and z1−c F (a−c + 1, b − c + 1,2− c , z). Therefore the radial function is

R1(z) = zα(1− z)βF (a, b , c , z) (2.40a)

R2(z) = zα(1− z)βz1−c F (a− c + 1, b − c + 1,2− c , z) (2.40b)

Since F (e , f , g , 0) = 1, therefore in the neighborhood of the outer black hole horizon, i.e. z = 0, boththe hypergeometric functions would be equal to unity. Since the quasinormal modes are solutionsthat are purely ingoing at the black hole horizon, i.e. Ψ∼ e−iωr∗ , where r∗ is the tortoise coordinate,therefore we choose equation (2.40a) as the solution of the radial wave equation. Therefore

R(z) = zα(1− z)βF (a, b , c , z)

⇒ R(z) = zα(1− z)β(1− z)c−a−b Γ(c)Γ(a+ b − c)

Γ(a)Γ(b )

× F (c − a, c − b , c − a− b + 1,1− z)+ zα(1− z)β

×Γ(c)Γ(c − a− b )

Γ(c − a)Γ(c − b )F (a, b ,a+ b − c + 1,1− z) (2.41)

Quasinormal modes also satisfy the vanishing boundary condition at spatial infinity. We implementthe vanishing boundary condition on equation (2.41) at z = 1. The first term in equation (2.41)vanishes. For the second term to vanish we need either Γ(c − a) =∞ or Γ(c − b ) =∞. Therefore

c − a =−n o r c − b =−n (2.42)

where n = 0,1,2, . . .. Thus the quasinormal mode frequencies from equation (2.38) and equation(2.42) are

ωmn =±m

l− i r+− r−

l 2

2n+ 1+p

1+µ

(2.43)

11

Chapter 3

Quantization via First Order Action

The general procedure of quantizing a classical system involves replacing the functions with the cor-responding operators and replacing the Poisson brackets with commutators. However this approachisn’t so effective when the system is damped or dissipative. Here we present a quantization procedure,proposed by Gitman and Kupriyanov[23] and Baldiotti et.al[11], based on a first-order action as a bet-ter alternative to the standard procedure. This procedure makes use of the Dirac brackets, in placeof the Poisson brackets. We will first give the general idea and then use a damped simple harmonicoscillator (DHSO) to elucidate. The DHSO serves dual purpose. A recent observation by Kim[24]

suggests that there is a connection between the QNMs of black holes and the Feshbach-Tikochinsky(FT) oscillator, the QNMs modes can be modeled using the FT oscillator. The FT oscillator can beconverted to a DSHO via a simple canonical transformation. This leads us to speculate that this quan-tization scheme can possibly be used to quantize QNMs .

Before presenting the first order action quantization we will study a established model of theDSHO, the Bateman-Calcidora-Kanai (BCK) model, to highlight the need for new approach to thequantization of dissipative systems.

3.1 BCK Model

We consider a DSHOwith unit mass, damping coefficient α and frequencyω. The equation of motionof this oscillator is

q + 2αq +ω2q = 0 (3.1)

This equation fails to satisfy the Helmholtz conditions. Therefore we cannot derive this as an Euler-Lagrangian equation from a Lagrangian. But we can consider an equivalent equation

(q + 2αq +ω2q)e2αt = 0 (3.2)

If f (x, t ) is a solution of equation (3.2) then it is also a solution of equation (3.1) because e2αt 6= 0The transformation q = e−αt y, changes equation (3.1) to the equation of motion of a simple harmonicoscillator with unit mass for which the Lagrangian is

LSH O =1

2

y2− γ 2y2

12

Chapter 3: Quantization via First Order Action

Using the inverse of the transformation we get

LBC K =e2αt

2

q2−ω2q2

(3.3)

Therefore, the Hamiltonian for this system, HBC K is

HBC K (q , p) =1

2

e−2αt p2− e2αtω2q2

(3.4)

To quantize the systemwe will replace q’s and p’s in the Hamiltonian by their corresponding operatorsq→ q , p→−i∂ /∂ q . The quantum Hamiltonian satisfies Schroedinger’s equation,

i∂

∂ tΨn =HBC KΨn (3.5)

To make the Hamiltonian in equation (3.4) time independent, we make a canonical transformationQ = qeαt , P = pe−αt ,T = t . This transformation does not change the phase space volume as it iscanonical. Therefore under this transformation equation (3.5) becomes

i

αQ∂

∂ Q+

∂

∂ T

Ψn =1

2

P 2+ω2Q2

Ψn (3.6)

⇒i∂

∂ tΨn =

P 2

2+ω2Q2

2+αQP

Ψn (3.7)

where P =−i∂

∂ Q=−i e−αt

∂

∂ q= e−αt p. To remove the QP term we again do a canonical transfor-

mation Q =Q, K = P +αQ and T = T .Therefore

i∂

∂ tΨn =

1

2

K2+ ω2Q2

Ψn (3.8)

where ω2 = (ω2−α2). Using the Baker- Hausdorff formula we have

K = P +αQ =

exp

−iαQ2

2

P exp

iαQ2

2

=−i∂

∂ Q+αQ =−i

∂

∂ Q+ iαQ

(3.9a)

and

K2 =

∂

∂ Q+ iαQ

2

= exp

−iαQ2

2

∂ 2

∂ Q2exp

iαQ2

2

(3.9b)

Using equations (3.9a) and (3.9b) we can write equation (3.8) as

i∂ Ψn

∂ t=

1

2

−i

∂

∂ Q+ iαQ

2

Ψn +ω2

2Q2Ψn

=−1

2

∂

∂ Q+ iαQ

2

Ψn +ω2

2Q2Ψ (3.10)

⇒ i∂ Ψn

∂ t=−

∂ 2Ψn

2∂ Q2− iαQ

∂ Ψn

∂ Q+ω2Q2Ψn

2(3.11)

13

Chapter 3: Quantization via First Order Action

Solving equation (3.11) we get

ΨBC Kn (q , t ) = (2n n!)−1/2

ω

π

1/4

exp

−i En t +αt

2− (ω+ iα)

q2

2e2αt

Hn(pωqeαt ) (3.12)

This is the wave equation of a quantum DSHO with unit mass, damping factor α and ω =p

ω2−α2.These are pseudo stationary states which are also the eigenstates of Hamiltonian HBC K +

α

2(pq+ q p)

with eigenvalues En .Therefore

⟨ΨBC Kn |HBC K |Ψ

BC Kn ⟩= ⟨Ψ

BC Kn |En −

α

2(pq + q p)|ΨBC K

n ⟩. (3.13)

Observe that (pq + q p) =−i

2q∂

∂ q+ 1

where 1 is the identity operator. Therefore

⟨ΨBC Kn |(pq + q p)|ΨBC K

n ⟩= ⟨ΨBC Kn | − i

2q∂

∂ q+ 1

|ΨBC Kn ⟩

=−i

(2n+ 1)(−(ω+ iα))

ω+ 2n

− i (3.14a)

which implies

⟨ΨBC Kn |

α

2( p q + q p)|ΨBC K

n ⟩=−α2

ω

n+1

2

(3.14b)

Also⟨ΨBC K

n |En |ΨBC Kn ⟩= ω

n+1

2

(3.15)

From equations (3.14b) and (3.15) we have

⟨ΨBC Kn |HBC K |Ψ

BC Kn ⟩= ⟨Ψ

BC Kn |En −

α

2(pq + q p)|ΨBC K

n ⟩

=ω2

ω

n+1

2

(3.16)

which shows that the expectation value of the Hamiltonian is constant though the system is damped.This does not reflect the fact that the system is losing energy to the surroundings. In the followingsection we will present the first order action quantization which gives a more accurate picture ofdissipative systems.

3.2 First order action quantization

For clarity, we will use ˆ to differentiate between quantum operators and their classical counterparts.We consider a system whose equations of motion are

x =A(t )x + j (t ) (3.17)

14

Chapter 3: Quantization via First Order Action

Given some initial conditions xα(0), the solution to the Cauchy problem for equation (3.17) is

x(t ) = Γ(t )x0+ γ (t ) (3.18)

where the matrix Γ(t ) is the fundamental solution of equation (3.18), that is

Γ =AΓ, Γ(0) = 1 (3.19)

and γ (t ) is a partial solution of equation (3.18). The fundamental solution can be easily computed bymaking use of matrix exponentials. We now construct a matrix Ω of the form,

Ω=ΛTΩ(0)Λ, Λ= Γ−1 (3.20)

where

Ω=

0 I−I 0

!

In this theory the action functional is given by

S[x] =1

2

∫

d t (xΩx − xB x − 2C x) (3.21)

where B =1

2

ΩA−ATΩ

and C = Ω j . Let xα = vα, J = xΩ and H =1

2xB x −C x. We therefore

construct the following action

Sv[x,π, v] =∫

[Jα(t , x)vα−H (t , x)+πα(xα− vα)]d t (3.22)

The action equation (3.22) is essentially just a modification of equation (3.21), but it gives the primaryconstraints of the system

δSv

δvα=Φα(t , x,π) =πα− Jα(t , x) = 0 (3.23)

This action therefore becomes the Hamiltonian action of a theory with primary constraints

SH =∫

d t (πα xα−H (1)), H (1) =H (t , x)+λαΦα(t , x,π) (3.24)

and the equations of motion areη=

¦

η, H (1)©

, Φ= 0 (3.25)

where η = (x,π). The Lagrange multipliers, λ-s, are determined from the consistency conditions ofthe primary constraints

λβ =ωβα ∂t Jα+ ∂αH

(3.26)

where ∂t = ∂ /∂ t , ∂α = ∂ /∂ xα and ωβα = Ω−1βα

Using the Lagrange multipliers we can write theequations of motion

η= η, HD(Φ)+ η,Φαωαβ∂t Jβ, Φ= 0 (3.27)

15

Chapter 3: Quantization via First Order Action

where · · · , · · · D(Φ) are the Dirac brackets with respect to the constraints Φ. For the canonical vari-ables, xα and πβ, the Dirac brackets are

xα, xβD(Φ) =ωαβ

πα,πβD(Φ) = ∂αJρωργ∂βJβ

xα,πβD(Φ) = δαβ+ωαγ∂βJγ (3.28)

If we now introduce a momentum ε conjugate to time t, and define the Poisson brackets, and thereforethe Dirac brackets, in an extended phase space,(η, t ,ε), then we can write the equations of motion as

η= η, H + εD(Φ), Φ= 0 (3.29)

To quantize this system, we assign time dependent operators to the phase space variables, which aresubject to equal time commutation relations and the constraints equations

[xα(t ), xβ(t )] = ixα, xβD(Φ)|x=x = iωαβ(t ) (3.30)

We get the time evolution of the operators as

d

d tx(t ) = x, H + εD(Φ)|x=x =−

1

2ωαβ(t )Ωβγ (t )x

γ (3.31)

and the operators x(0)= x0 obey

h

xα0 , xβ0i

= i

Ω(0)αβ

−1(3.32)

The quantum Hamiltonian is constructed as H =H (η, t ).

Damped simple harmonic oscillator The DSHO is described by the second order equation

q + 2αq +ω2q = 0

We convert it to a first order system of the form define by equation (3.17)

x = y, y =−ω2x − 2αy (3.33)

Following the procedure outlined above we construct a first order action for the DSHO

S =1

2

∫

d t

y x − xy −

y2+ 2αxy +ω2x2

e2αt (3.34)

The Euler-Lagrangian equations of motion are

(y + 2αy +ω2x)e2αt = 0, (x − y)e2αt = 0. (3.35)

To quantize the system we extend the initial phase space from (x, y, px , py ) to (x, y, px , py , t ,ε), whichwe refer to as η, to include time t and its associated momentum ε. The Poisson brackets can then be

16

Chapter 3: Quantization via First Order Action

defined as

F ,G=

∂ F

∂ x

∂ G

∂ px+∂ F

∂ y

∂ G

∂ py+∂ F

∂ t

∂ G

∂ ε

!

−

∂ G

∂ x

∂ F

∂ px+∂ G

∂ y

∂ F

∂ py+∂ G

∂ t

∂ F

∂ ε

!

(3.36)

Using Dirac brackets we can write equation (3.33) as

η= η, H (x, y, t )+ εD(φ), φx =φy = 0, (3.37)

whereH (x, y, t ) =

1

2

y2+ 2αxy +ω2x2

e2αt ,

andφx = px −

1

2ye2αt , φy = py +

1

2xe2αt .

Using these constraints we get

Ω= e2αt

0 −11 0

!

. (3.38)

Thereforex, yD(φ) = e−2αt . (3.39)

We introduce time-dependent operators η(t ), such that they satisfy the differential equations

d η

d t= iη, H (x, y, t )+ εD(φ)|n=η,

with initial conditions[n(0), η(0)] = iη,ηD(φ)|n=η(0),

and constraintspx (0)−

1

2y(0) = py (0)+

1

2x(0) = 0

As x, yD(φ) = e2αt , we can write [x, y] = i e2αt . As mentioned in the general procedure

d xα

d t=−

1

2ωαβ(t )Ωβγ (t )x

γ , (3.40)

where ωαβ(t ) = Ω−1αβ

.Therefore˙x =−αx, ˙y =−αy (3.41)

Let x(0) = q and y(0) = p. Therefore [q , p] = i . Solving the differential equations equation (3.41)

x = e−αt q , y = e−αt p.

The quantum states of the system are described by wave functions ψ following the Schrödinger equa-tion,

i∂ ψ

∂ t= Hψ(t )

17

Chapter 3: Quantization via First Order Action

TheHamiltonian is transformed via an unitary transformation , Hω = S−1H S, where S = exp

−iαq2

2

.

Using the Baker-Hausdorff expansion,

S−1H S = exp

iαq2

2

H exp

−iαq2

2

⇒ S−1H S = H +

iα

2q2, H

+ · · ·

∴ Hω =1

2

p2+ ω2 q2

(3.42)

Hence, the eigenfunctions of the modified Hamiltonian (which is essentially the Hamiltonian of asimple harmonic oscillator) are

ψωn (t , q) =

ω

π

1/4 1p

2n n!Hn

pωq

exp

−ωq2

2

exp (−i t En) (3.43)

Therefore eigenfunctions for the original Hamiltonian are

ψn(t , q) = Sψωn (t , q)

=

ω

π

1/4 1p

2n n!Hn

pωq

exp

−ω+ iα

2q2

exp

−i t En

(3.44)

The energy is

EM =1

2

y2+ω2x2

The corresponding Schrödinger operator is given by

EM =1

2

p2e−2αt +ω2e−2αt q2

= e−2αt

H −α

2(q p + p q)

(3.45)

The expectation value of EM in the energy eigenstates is

⟨ψn |EM |ψn⟩= e−2αt ω2

ω2

n+1

2

(3.46)

Unlike in the case of the BCK model, the expectation value of the energy as given by equation (3.46)has explicit time dependency thus indicating that energy is being exchanged between the system andthe surroundings.

18

Chapter 4

Toy Models and Biorthonormal BasisExpansion

As we have seen in Chapter 2, calculating the QNMs of a black hole is difficult. In this chapterwe will use some one dimensional toy models,that mimic the QNMs of black holes, to develop apseudo-Hermitian quantum theory[25],[26]. In the following sections we will first describe the toymodels[27], follow it with the basics of pseudo-Hermitian quantum mechanics and then develop thequantum theory.

4.1 Toy Models

Particle on an infinite string We consider an infinitely stretched string with constant linear massdensity ρ(x) = ρ0. We attach a point mass at x = 0. Then the linear density of the string becomes

ρ(x) = ρ0+mδ(x) (4.1)

The wave equation for the string displacements

ρ(x)∂ 2t − ∂

2x

φ(x, t ) = 0 (4.2)

We assume harmonic time dependency, φ(x, t ) = f (x)e−iωt . Therefore the wave equation for thespatial wave function

d 2x +ρ(x)ω

2

f (x) = 0 (4.3)

The boundary conditions for free oscillations are no incoming waves from both x → +∞ and x →−∞

∂xφ(x, t ) =−∂tφ(x, t ), x→+∞ (4.4a)

∂xφ(x, t ) = +∂tφ(x, t ), x→−∞ (4.4b)

Therefore f = fR(x) = ae iωx for x > 0 and f = fL(x) = b e−iωx for x < 0, satisfy equation (4.3).φR = fRe−iωt satisfies equation (4.4a) and φL = fLe−iωt satisfies equation (4.4a). Integrating (4.3)

19

Chapter 4: Toy Models and Biorthonormal Basis Expansion

over the entire real line∫ ε

−ε

d 2x f +ρ(x)ω2 f

= 0 (4.5)

⇒

d f

d x

0+

0−+mω2 f (0) = 0 (4.6)

The continuity of the string gives the relation between fL(x) and fR(x) at x = 0

fL(−0) = fR(+0) (4.7)

which leads tob = a (4.8)

From equations (4.6) and (4.8) we have

ω2m+ 2iω = 0 (4.9)

which gives the quasinormal frequencies ω = 0, −2i/m.

Particle on a semi-infinite string In principle this is very similar to the previous model, exceptthe fact that we will be working with the non-negative real line [0,∞) instead of the entire real line.We again consider the wave equation (4.2), now described on the half line with vanishing boundaryconditions at x = 0

φ(x = 0, t ) = 0 (4.10)

The mass is placed at x = c . The linear density of the string is

ρ(x) = ρ0+mδ(x − c) (4.11)

We impose the outgoing boundary condition

∂xφ(x, t ) =−∂tφ(x, t ), x > c (4.12)

We could have also considered this system as a cavity [0, c] coupled to a bath (c ,∞) such that energyexchange between the bath and cavity takes place at x = c . Again assuming harmonic time dependencyφ= f (x)e−iωt we get equation (4.3) for the spatial wave from (4.2). Therefore fL(x) = a sin(ρ1/2

0 ωx)

satisfies equation (4.3) and the vanishing boundary condition and fR(x) = b e iρ1/20 ωx satisfies equation

(4.3) and the outgoing boundary condition. From the continuity at x = c we have

b = a sin(ρ1/20 ωc)e−iρ1/2

0 ωc (4.13)

Integrating equation (4.3) over the half-line with the linear density defined by equation (4.11) we have

mω = ρ1/20 cot(ρ1/2

0 ωc)− iρ1/20 (4.14)

20

Chapter 4: Toy Models and Biorthonormal Basis Expansion

which gives us the quasinormal frequencies.

4.2 Pseudo-Hermiticity and biorthonormal basis

Pseudo-Hermitian quantum mechanics is a generalization of standard Hermitian quantum mechanics.Before developing a pseudo-Hermitian quantum theory for QNMs in the next section, we will presentsome mathematical concepts in this section, that are essential to the development of the pseudo-Hermitian quantum theory. These concepts have been discussed in details by Mostafazadeh in hisreview on pseudo-Hermitian quantum mechanics[28].

Hilbert space We consider a complex vector space, V and a function (· , · ) : V ×V → C that assignto any pair , ψ,φ ∈ V a complex number (ψ,φ). We demand that this complex number satisfy thefollowing properties

• It is positive definite, i.e. (ψ,ψ)≥ 0 and vanishes if and only if ψ= 0

• It is Hermitian, i.e. (ψ,φ)∗ = (φ,ψ)

• For any a, b ∈C,(ψ,aφ+ bξ ) = a(ψ,φ)+ b (ψ,ξ )

The complex number (· , · ) is called the inner product and (V , (· , · )) the inner product space. A quantitycalled the norm ||ψ|| is defined on the inner product space, ||ψ|| =

p

(ψ,ψ). The Hilbert space,H isan inner product space that satisfies that for any infinite sequence ψk of elements ψk ∈H , the con-dition that lim

j ,k→∞||ψ j−ψk ||= 0 implies that ψk converges to an element ψ ofH ; lim

k→∞||ψ−ψk ||= 0.

The Hilbert space can be of finite dimensions or of infinite dimensions. We will represent the dimen-sionality of the Hilbert space by N , N =∞ will mean that it is infinite-dimensional. A Hilbert spaceis said to be separable if and only if it has a countable orthonormal basis.

Biorthonormal basis Let ψn be a basis of an N -dimensional separable Hilbert space, N ≤∞. Weobtain the orthonormal basis ξn from ψn via Gram-Schmidt orthonormalization. Since ψn is abasis therefore for unique Bmn ∈C∀m ∈ 1,2, . . . ,N we have

ξm =N∑

n=1

Bnmψn (4.15)

Similarly

ψn =N∑

k=1

(ξk ,ψk )ξk =N∑

k=1

ckξk , ∀k ∈ 1,2, . . . ,N (4.16)

Substituting equation (4.16) in equation (4.15) and using the orthonormality of ξk we have

N∑

n=1

Bnm(xi j ,ψn) = δm j , ∀m ∈ 1,2, . . . ,N (4.17)

21

Chapter 4: Toy Models and Biorthonormal Basis Expansion

The N ×N matrix B = (Bnm) is invertible with the elements of B−1 being given by B−1mn = (ξm ,ψn).

Therefore

ψn =N∑

k=1

B−1knξk (4.18)

We define the vectors

φm =N∑

j=1

B∗m jξ j , ∀m ∈ 1,2, . . . ,N (4.19)

Therefore the vectors φm satisfies

(φm ,ψn) = δmn , ∀m, n ∈ 1,2, . . . ,N (4.20)

A sequence of ordered pairs (ψn ,φm) satisfying equation (4.20) is said to form a biorthonormal basisand (ψn ,φm) is said to be the biorthonormal expansion of ψn. Using Dirac’s bra-ket we write thecompleteness relation in the biorthonormal basis as

N∑

n=1

|ψn⟩⟨φn |=N∑

n=1

|φn⟩⟨ψn |= 1 (4.21)

Pseudo-Hermiticity A linear operator A is pseudo-Hermitian (or η-pseudo-Hermitian) if and onlyif there exists an everywhere-defined invertible Hermitian linear operator η :H →H such that

A† = ηAη−1 (4.22)

η is called a pseudo-metric operator. We generalize the notion of the inner product by replacing thefirst property of the inner product with

• (· , · ) is non-degenerate, i.e. given ψ ∈H , the condition (φ,ψ) = 0, ∀φ ∈H implies ψ= 0.

≺· , · :H ×H → C that satisfies this new property and the original second and third properties ofthe inner product is termed the pseudo-inner product. Therefore

≺· , ·= (· ,η· ) = (·η, · ) = (· , · )η (4.23)

Having presented all these concepts, we now consider a subspaceW of the total Hilbert spaceH ,on which we have define a non-Hermitian operator P and a conjugate linear duality transformationoperator D , D(α|α⟩+β|β⟩) = α∗|α⟩+β∗|β⟩, such that DP = P †D , i.e P is a D -pseudo-Hermitianoperator. We further define a biorthonormal basis (|U j ⟩, |V j ⟩) for P

P |U j ⟩= E j |U j ⟩ (4.24)

P †|V j ⟩=C j |V j ⟩ (4.25)

22

Chapter 4: Toy Models and Biorthonormal Basis Expansion

where the eigenvalues are in general considered to be complex. Due to the pseudo-Hermiticity wehave C j = E∗j . If D had been the identity operator then P would have been equal to P † and |U j ⟩would have coincided with |V j ⟩.

4.3 Quantum theory for QNMs

We take up themass on a semi-infinite string toy model to develop the quantum theory. We expand thewave function in terms of the QNMs

φ(x, t ) =∞∑

n=1

an fn(x)e−iωn t (4.26)

Substituting equation (4.26) in equation (4.2) we get equation (4.3) with f (x) = fn(x) and the linearmass density defined by equation (4.11). We define the momentum conjugate to the wave function

φ(x, t ) = ρ(x)∂tφ(x, t ) = ρ(x)∞∑

n=1

an(−iωn) fn e−iωn t (4.27)

Since φ(x) = φ(x, t = 0) =∑∞

n=1 an fn and φ(x) = ∂tφ(x, t )|t=0 = ρ(x)∑∞

n=1 an(−iωn) fn can beindependently specified, therefore using φ and φ we define a new two component wave function

|φ⟩=

φ

φ

!

We expand |φ⟩ in terms of the quasinormal modes and their associated ‘momenta’

|φ⟩=∞∑

n=1

an |Fn⟩ (4.28)

where

|Fn⟩=

1−iωnρ(x)

!

fn(x)

Using this two component wave function we recast the original wave equation as a Schrödingerequation[25]

i∂t |φ⟩=H |φ⟩ (4.29)

where the Hamiltonian is the non-Hermitian operator

H = i

0 ρ−1(x)∂ 2

x 0

!

(4.30)

The outgoing boundary condition (4.12) is represented in this two component formulation by theoperator equation

Q|φ⟩= 0 (4.31)

where

Q =

∂x 1−ρ(x)∂t 1

!

23

Chapter 4: Toy Models and Biorthonormal Basis Expansion

If Q|φ⟩ = 0 then QH |φ⟩ = 0. We now focus on the null space of Q, a subspace R : ∀|φ⟩ ∈H |Q|φ⟩ = 0 of the total Hilbert space H . Since the wave functions in the subspace R satisfy theoutgoing boundary condition therefore they can be expanded in terms of the quasinormal modes.Thussubstituting equation (4.28) in equation (4.29) we get the time-independent Schrödinger equation

H |Fn⟩=ωn |Fn⟩ (4.32)

Therefore the QNMs are the eigenfunctions of the Hamiltonian H and the eigenvalues are the cor-responding QNFs. Following the procedure outlined in the last section we proceed to define abiorthonormal expansion of |Fn⟩ using the D operator defined in the last section as the pseudo-metric operator

H †|Gn⟩=ω∗n |Gn⟩, |Gn⟩=D |Fn⟩ (4.33)

and we also define a pseudo-inner product

(Ξ,Ψ)D = ⟨DΞ|Ψ⟩= i∫ ∞

0

ξ ψ+ ξ ψ

(4.34)

where |Ξ⟩ = (ξ ξ )T and |Ψ⟩ = (ψ ψ)T are any two arbitrary vectors belonging to R . The pseudo-inner product so defined involves the wave functions from (c ,∞) (the bath). However we can definethe pseudo-inner product such that it depends only on the variables from [0, c] (the cavity)[29]

(Ξ,Ψ)D = i

(

∫ c+

0

ξ ψ+ ξ ψ

d x + ξ (c+)ψ(c+)

)

(4.35)

The completeness relations are

∞∑

n=1

|Fn⟩⟨Gn |=∞∑

n=1

|Gn⟩⟨Fn |= I (4.36)

We can write the completeness relation in the matrix notation

∞∑

n=1

−iρ(y)ωn 1−ω2

nρ(x)ρ(y) −iρ(x)ωm

!

fn(x) fn(y) = δ(x − y)

1 00 1

!

(4.37)

which gives us the sum rules

∞∑

n=1

fn(x) fn(y) = 0,∞∑

n=1

fn(x) fn(y)ω2n = 0 (4.38)

and∞∑

n=1

fn(x) fn(y)ρ(x)ωn = iδ(x − y) (4.39)

Following the standard procedure we can describe the time evolution of the state ket |φ⟩ by latchingexponential factors to each term in the QNM expansion

|φ(x, t )⟩=∞∑

n=1

an e−iωn t |Fn⟩ (4.40)

24

Chapter 4: Toy Models and Biorthonormal Basis Expansion

where

an = ⟨Gn |φ(x, 0)⟩ (4.41)

=∫ ∞

0

fn(x)φ(x)+ fn(x)φ(x)

d x

=∫ c+

0

fn(x)φ(x)+ fn(x)φ(x)

d x + fn(c+)φ(c+) (4.42)

We can similarly define the coefficients while expanding in the |Gn⟩ basis by replacing ⟨Gn | with ⟨Fn |in the above equations. We have therefore developed a quantum theory using the QNMs as the basis.In the following subsection we present the Green’s function approach to quantize QNM system as anapplication of the theory that we developed.

4.3.1 Green’s Function

A single-time propagator determines the wave function ψ(x, t ), at any instant t , from the initial wavefunction, ψ(x, t = 0), i.e.

ψ(x, t ) = ⟨x|ψ(t )⟩=∫

d yG(x, y; t )ψ(y, t = 0)

where G(x, y; t ) is the propagator. We define the propagator as the matrix element of the evolutionoperator in the coordinate representation

G(x, y; t ) = ⟨x|e i H t |y⟩

Using the biorthonormal completeness relations we write the quasinormal mode representation of theGreen’s function

G(x, y; t ) =∑

m,n⟨x|Fm⟩⟨Gm |e

−i H t |Fn⟩⟨Gn |y⟩ (4.43)

⇒ G(x, y; t ) =∑

m⟨x|Fm⟩⟨Gm |y⟩ (4.44)

=∑

m

−iρ(y)ωm 1−ω2

mρ(x)ρ(y) −iρ(x)ωm

!

× fm(x) fm(y)e−iωm t (4.45)

We define a component of the G matrix as the generic Green’s function for the wave equation (4.2)

G(x, y, t ) =∑

mfm(x) fm(y)e

−iωm t (4.46)

Therefore following the sum rule (4.39)

ρ(x)∂ 2t − ∂

2x

G(x, y, t ) = δ(x − y)δ(t ) (4.47)

25

Chapter 4: Toy Models and Biorthonormal Basis Expansion

In terms of this Green’s function we can write the propagator as

G(x, y; t ) =

ρ(y)∂t G(x, y, t ) G(x, y, t )ρ(x)ρ(y)∂ 2

t G(x, y, t ) ρ(x)∂t G(x, y, t )

!

(4.48)

and therefore the wave function at any time t in terms of the initial wave function at time t = 0 is

φ(x, t ) =∫ ∞

0d yh

G(x, y, t )φ(y)+ ∂t G(x, y, t )ρ(y)φ(y)i

(4.49)

or equivalently

|φ(x, t )⟩=∫ ∞

0d yG(x, y; t )|φ(y, t = 0)⟩ (4.50)

To determine the Green’s function for the wave equation we take the Laplace transform of equation(4.47)

ρ(x)ω2+ ∂ 2x

Gω(x, y,ω) =−δ(x − y) (4.51)

whereGω(x, y,ω) =

∫ ∞

0G(x, y, t )e−iωt dω

To solve for Gω(x, y,ω), we consider two auxiliary solutions (a±(ω, x)) of the following homogeneousdifferential equation

ρ(x)ω2+ ∂ 2x

k±(ω, x) = 0 (4.52)

such that they satisfy the boundary conditions

limx→∞

k+(ω, x)e−iωx

= 1 (4.53a)

k−(ω, x = 0) = 0 , k ,−(ω, x = 0) = 1 (4.53b)

i.e. k−(x,ω) satisfies the left boundary condition and k+(x,ω) satisfies the right boundary condition(outgoing wave condition). In terms of these auxiliary solutions the Laplace transformed Green’sfunction is

Gω(x, y,ω) =k−(x,ω)k+(y,ω)

W (ω)(4.54)

where W (ω) is the Wronskian of k−(x,ω) and k+(x,ω). The Laplace transformed Green’s func-tion is therefore independent of the normalization of the auxiliary solutions. For the toy model inconsideration

k−(x,ω) =A(ω) sin(ρ1/20 ωx) (4.55a)

k+(x,ω) = B(ω)e iρ1/20 ωx (4.55b)

where A and B are normalization constant. At the zeros of the Wronskian, W (ω) the auxiliarysolutions are no longer independent ,i.e. k+(x,ωn) = b (ωn)k−(x,ωn). Therefore k±(x,ωn) satisfyboth the left and right boundary conditions. Thus k±(x,ωn) determined at the zeros of theWronskianare the eigenfunctions of the operator, ρ(x)ω2 + ∂ 2

x . At these frequencies , the auxiliary function,k±(x,ω) satisfies the exactly same conditions as the quasinormal modes, fn(x). We can therefore saythat the zeros of the Wronskian are the quasinormal modes frequencies and the functions k(x,ωn) arethe corresponding quasinormal modes (The Laplace transformation method presented in chapter 2).

26

Chapter 4: Toy Models and Biorthonormal Basis Expansion

Since the discontinuity in our model is at x = c (energy interaction between system and medium takesplace at x = c ), we consider the zeros of the Wronskian at x = c .

k−(x = c ,ωn) = k+(x = c ,ωn) (4.56)

⇒B =Asin(ρ1/20 )e

−iρ1/20 ωn c (4.57)

Because of the delta function at x = c

∂x k+|c+ε− ∂x k−|c−ε =−mω2n k±|x=c (4.58)

⇒ mωn = ρ1/20 cot(ρ1/2

0 ωn c)− iρ1/20 (4.59)

This matches with the QNFs given by equation (4.14).

4.3.2 Energy Expectation Value

The expectation value of an operator with respect to the state |φ⟩ is

⟨A(t )⟩= ⟨φ(t )|A|φ(t )⟩ (4.60)

Applying the completeness relations of the biorthonormal basis to the above equation we get,

⟨A⟩=∑

m

∑

n⟨φ|Fm⟩⟨Gm |A|Fn⟩⟨Gn |φ⟩ (4.61)

Expanding ⟨φ| and |φ⟩ in terms of the biorthonormal basis we have

⟨φ|Fm⟩=∑

j

b ∗j ⟨G j |Fm⟩eiω∗j t = b ∗m e iω∗m t

and⟨Gn |φ⟩=

∑

k

ak⟨Gn |Fk⟩e−iωk t = an e−iωn t

where an and b ∗m are defined by equation (4.41). Therefore the expression for the expectation value ofthe operator is further simplified to

⟨A⟩=∑

m

∑

ne i(ω∗m−ωn )t an b ∗m⟨Gm |A|Fn⟩ (4.62)

If the operator is the Hamiltonian, H , the expectation value becomes

⟨H ⟩=∑

m

∑

ne−i(ωn−ω∗m )t an b ∗m⟨Gm |H |Fn⟩ (4.63)

=∑

m

∑

nωnan b ∗m e−i(ωn−ω∗m )t ⟨Gm |Fn⟩ (4.64)

=∑

nωnan b ∗m e−2ℑ(ωn )t (4.65)

where ℑ(ωn) refers to the imaginary part of ωn . The time dependency in the expectation value of theHamiltonian indicates that the system is dissipative.

27

Chapter 4: Toy Models and Biorthonormal Basis Expansion

Though we have presented this theory using the particle on a semi-infinite string model, this can beeasily extended to the entire real line by replacing the vanishing boundary conditions at x = 0 withthe ingoing boundary condition at x→∞ and defining a suitable operator to represent this boundarycondition in the two-component formalism. We are currently working on generalizing this formalismto spherically symmetric black hole backgrounds by making some minor changes in the Hamiltonian:ρ(x)→ 1 and ∂ 2

x → ∂2

x −U (x).

28

Chapter 5

Conclusion

Black hole space-times are probably the most fascinating objects whose existence is predicted byEinstein’s general theory of relativity. Numerous studies conducted over the decades show that quasi-normal modes carry a treasure trove of information concerning the classical and quantum propertiesof black holes. The possibility of detecting gravitational quasinormal mode radiation in the near fu-ture by the LIGO and VIRGO makes the study of quasinormal modes more interesting. If detected itwill probably enable us to observe interesting spacetime structures around black holes. The detectionof quasinormal modes will be direct proof of the existence of black holes. Even astrophysical objectslike neutron stars and other damped systems can be studied using quasinormal modes. However ourunderstanding of the physics behind quasinormal modes is far from complete. Quasinormal modesviolate many of the fundamental properties that define stationary modes, thus making it difficult toconstruct a quantum theory about quasinormal mode systems. In this thesis we have attempted to fillthis gap and tried to provide an understanding of the basic principle underlying quasinormal modes.

One thing that we have not discussed much about is the completeness issue. Quasinormal modesare not complete in general. However Price and Hussain have shown that quasinormal modes doform a complete set for some particular models of relativistic stellar oscillations[30]. Ching et.al laterderived the conditions that the scattering potential need to fulfill so that the quasinormal modes forma complete system. They also showed that even for systems that do not exhibit a complete set ofquasinormal modes, a minor modification to the scattering potential can make the set complete with-out altering the physics much. The pseudo-Hermitian quantum theory that we have presented here itbased on the fact that the quasinormal modes form a complete set in a sense defined by Ching et.al[7].The primary reason for the use of a biorthonormal basis is to satisfy this completeness. Researchcarried on a similar vein by Leung et.al showed that quasinormal modes a leaky optical cavity forms acomplete set [31].

Another thing to note is the fact that in general the eigenvalues of pseudo-Hermitian operatorsare either real or occur in complex conjugate pairs. Here we have assumed that the eigenvalues arecomplex. The assumption is based on Mostafazadeh’s work which gives the condition that determinesthe reality of the eigenvalues[32]. However we believe that further work is required to clarify this issue.

Regarding future plans concerning this topic we would like investigate the connection betweenquasinormal modes and the Feshbach-Tikochinsky oscillator and see if the quantization scheme pre-

29

Chapter 5: Conclusion

sented in chapter 2 can be used to develop a quantum theory for quasinormal mode systems and tryto relate that to the pseudo-Hermitian quantum theory that we have developed. A recent work byGalley[33] has provided us with a more promising scheme to quantize the Feshbach-Tikochinsky os-cillator. Our plans also include trying out Galley’s quantization scheme to construct the quantumtheory.

30

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33

Appendix A

Dirac Brackets: the Faddeev-Jackiwformalism

The Dirac brackets introduced in Chapter 3 are a generalization of the Poisson brackets. How-ever the method proposed by Dirac to calculate the Dirac bracket relations is cumbersome. Here wepresent a more elegant method proposed by Faddeev and Jackiw [34].

The Faddeev-Jackiw formalism starts with writing the Lagrangian of the system in the form of

L= ai (ξ )ξi −V (ξ ) (A.1)

where ξ is a 2N-component phase-space coordinate

ξ i =

q i , i = 1, . . . ,N

p i , i =N + 1, . . . , 2N(A.2)

The Euler-Lagrange equations for the phase-space variables are

fi j ξj = ∂ξ i V (ξ ) (A.3a)

fi j = ∂ξ i a j (ξ )− ∂ξ j ai (ξ ) (A.3b)

If the symplectic matrix f is nonsingular, then the Dirac brackets for the phase-space variables aregiven by

ξ i ,ξ j D = f −1i j (A.4)

However if the matrix is singular, then f −1i j does not exist. We then define the zero modes of the

matrix f asfi j v

jm = 0, m = 1,2, . . . , M (A.5)

The transposition of the zero modes (vTm) acting on equation (A.3a) from the left gives rise to the

constraintsφm(ξ ) = vT

m∂ξ i V (ξ ) (A.6)

34

Appendix A: Dirac Brackets: the Faddeev-Jackiw formalism

With λm being the Lagrange multipliers, the Lagrangian becomes

L= ai (ξ )ξi −V (ξ )+ λmφm(ξ ) (A.7)

The phase-space is therefore enlarged to include λ = (λ1, . . . ,λM ) as variables. Therefore the velocitycoefficients are

aλi (ξ ,λ) = ai (ξ )−λm∂ξ iφm(ξ ) , aλm = 0 (A.8)

Therefore the components of the matrix are

f λi j = ∂ξ i a j (ξ )− ∂ξ j ai (ξ ) , f λi ,2N+n = ∂ξ iφn(ξ )

(A.9)

f λ2N+m, j =−∂ξ jφm(ξ ) , f λ2N+m,2N+n = 0

Since the phase-space is enlarged therefore the new matrix may be nonsingular. If the matrix is stillsingular then we can repeat the above steps to enlarge the phase-space further, until the matrix becomesnonsingular. Once the matrix becomes nonsingular we can get the Dirac bracket relations from

ξ β,ξ αD = f λ−1

αβ(A.10)

Once we get the Dirac bracket relations, we can get the canonical commutation relations from

[ξ β,ξ α] = iξ α,ξ βD (A.11)

In this formalism the equations of motion are given by

ξ α =−∂ξ βV (ξ )ξ β,ξ αF J (A.12)

where the Faddeev-Jackiw bracket is defined by

ξ β,ξ αF J =−( f−1)αβ (A.13)

35