Download A summary on Solitons in Quantum field theory

TVE 16 030 maj
Examensarbete 15 hp
Juni 2016
A summary on Solitons in Quantum
field theory
Hugo Laurell
Abstract
A summary on Solitons in Quantum field theory
Hugo Laurell
Teknisk- naturvetenskaplig fakultet
UTH-enheten
Besöksadress:
Ångströmlaboratoriet
Lägerhyddsvägen 1
Hus 4, Plan 0
Postadress:
Box 536
751 21 Uppsala
Telefon:
018 – 471 30 03
Telefax:
018 – 471 30 00
Hemsida:
http://www.teknat.uu.se/student
This text is a brief summary on solitons in
quantum field theory. An introduction to quantum
field theory is made through which certain
notions in the theory of solitons in quantum field
theory is possible to understand. The explicit
time-independent kink solutions are computed
for the phi^4- and the sine-Gordon theory. The
time-dependent kink solutions were then found
by Lorentz transformation of the timeindependent solutions. Interesting quantities
such as the kink mass, the charge and the
energy density is computed in both phi^4- and
the sine-Gordon theory. An investigation of
geometrical methods for finding the integrals of
motions for the sine-Gordon field is made.
Additionally the equivalence between the sineGordon model and the massive Thirring model is
presented.
Handledare: Luigi Tizzano
Ämnesgranskare: Gopi Tummala
Examinator: Martin Sjödin
ISSN: 1401-5757, TVE 16 030 maj
Contents
1 Introduction
2
2 Introduction to Quantum field theory
2.1 The transition from a discrete system to a continuous
2.2 Generalization for higher dimensions
2.3 The Euler-Lagrange equations of motion
2.4 the Noether theorem: Conservations laws in field theories
2.5 Scalar Field Theories and the Klein-Gordon field
2.6 Second Quantization
2.7 Lorentz invariant normalization
3
3
4
5
6
7
8
10
3 Soliton theory in scalar field theories
3.1 Non-trivial time-independent finite energy solutions for single scalar fields
3.1.1 The explicit time independent sine-Gordon solution
3.1.2 The explicit time independent 4 -field solution
3.2 Stability of the kink
3.3 Taylor expansion of the sine-Gordon potential around a minimum
3.4 Energy density calculation for the sine-Gordon and the 4 -model
3.5 Non-trivial time-dependent finite energy solutions for single scalar fields
3.6 The Topological charge
3.7 Derrick’s theorem
3.8 Soliton Quantization
3.8.1 The weak coupling criterion
3.8.2 Perturbation around a minimum of the potential
3.8.3 Recapitulation of the diatomic molecule
3.8.4 Transition to the Infinitely polyatomic molecule
3.8.5 Quantized kink energy levels
3.9 Soliton interactions
3.9.1 Soliton-soliton interaction
3.9.2 Soliton-anstisoliton interaction
3.9.3 Breather interaction
10
11
12
13
14
16
16
17
18
19
20
20
22
23
25
25
27
28
29
30
4 Geometrical methods in the theory of solitons
4.1 Parallel transport and Gauge transformations
4.2 An interesting duality for the sine-Gordon model
31
32
35
5 Summary
37
–1–
Populärvetenskaplig sammanfattning
Den här texten sammanfattar och förenklar litteratur kring solitoner i kvantfältteori. Kvantfältteori är den delen av fysiken där man både behandlar mikroskopiska och makroskopiska
fenomen samtidigt. Dessutom betraktar man systemet som ett fält istället för punktvisa
generaliserade koordinater. Fördelen med att komprimera oändligt många koordinater i en
fält konfiguration är att det blir mycket lättare att räkna med system av högre dimension
och komplexitet. En soliton är en ensam vågpuls som inte minskar i styrka med tiden. Om
två solitoner kolliderar överlever båda solitonerna kollisionen och fortsätter rörelsen med en
fasvridning. Solitonen är inte ett oberoende objekt utan är starkt sammanbunden med sin
motpart anti-solitonen. Mellan solitonen och anti-solitonen verkar attraktiv växelverkan
samt mellan en soliton och soliton verkan repulsiv växelverkan. Den här texten behandlar
främst solitoner i skalära fältteorier, fält där partiklarna som existerar är bosoner. Ett
exempel på en boson är fotonen. En speciell egenskap hos bosoner är att man kan placera
oändligt många bosoner på varandra utan att de repellerar vandra. Bosoner verkar med
svag växelverkan i relation till till exempel fermioner. Ett exempel på en fermion är elektronen. Man kan inte placera flera elektroner på varandra. Om man försöker bildas en mycket
stor kraft som gör att elektronerna repelleras från varandra. Kraften genereras som en konsekvens av Paulis uteslutningsprincip. Det visar sig att sine-Gordon modellen som beskriver
ett skalärt fält och massiva Thirring modellen som beskriver ett Dirac fält med fermioner
som elementarpartiklar är ekvivalenta under en indentifikation. Solitoner i kvantfältteori
modellerar oupptäckta partiklar så som kosmiska strängar. Kosmiska strängar är endimensionella topologiska defekter bildade tidigt i universums historia. Om kosmiska strängar
skulle detekteras skulle detta vara ett hinder för att förena generell relativitetsteori med
kvantfysiken en förening som är en av de största utmaningarna i modern teoretisk fysik.
1
Introduction
Finite energy solutions are crucial to understand the interplay between the topology of
space-time and physical phenomena. It is very important to deepen our understanding
of these kind of solutions because they might be useful in the discovery of new physical
phenomena. The study of one (1+1)-dimensional solitons can be thought of as a laboration
where a much easier analysis of the properties and consequences of the solitons can be made.
For these experiments to say something about the real world the dimension of the solitons
can then be inflated. The real-time solitons in quantum field theory model undiscovered
particles such as the magnetic monopole and can be extended to model cosmic strings.
The hypothetical cosmic string is a one-dimensional topologic defect formed in the early
universe. If such a hypothetical particle would be measured it could constrain the unifying
of the two major disciplines in modern physics the quantum physics and general relativity.
Soliton theory is extremely important in applications such as non-linear optics where selfenforcing solitary waves can be used to increase performance of information transmission
in optical fibres. With a continuously increasing demand on the data rate in the optical
fiber network the optical pulses are made shorter and shorter. However, with shorter pulses
–2–
the spectrum broadens, as understood from the uncertainty principle. The redshifted light
of the pulse travels faster than the blue shifted, due to dispersion, which make the pulse
stretch out in time. After some propagation distance the pulses in the fiber will become
superimposed on each other making the information inaccessible, which ultimately limits
the length over which the fiber can be used. However, by increasing the amplitude of the
pulse a self-stabilizing pulse can be formed, an optical soliton, where the refractive index
of the fiber changes due to non-linear effects such that the speed of the redshifted part of
the pulse becomes equal to the blue shifted part. This means that the pulse becomes self
enforcing making the information accessible for high frequencies [1]. This is just but one
of the applications of solitons in modern science and there is a lot more to unveil in the
subject.
2
2.1
Introduction to Quantum field theory
The transition from a discrete system to a continuous
In this section I will briefly introduce the basics of classical field theory and continue with
quantum field theory. I will follow closely the method of [2], [3]. To develop a classical field
theory first we must look at the transition from a set of discrete generalized coordinates qn ,
into a continuous field coordinate (x). An example is coupled oscillations in a infinitely
long and thin elastic rod. As Goldstein describes the generalized coordinates, qn , are placed
such that qn measures the amplitude at x and qn+1 measures the amplitude at x + a and
so forth. The total Lagrangian for this system looks as follows.
L=
n
X
Li =
i=1
n
X
Ti
Ui =
i=1
n
X
1
i=1
2
1
k(qi
2
mq˙i 2
qi
1)
2
Where k is the spring constant and m is the mass of the rod. This expression can also be
written as
"
n
1X
m 2
L=
a
q˙i
2
a
1
ka
2
i=1
✓
qi
qi
a
1
◆2 #
The the following five limits describes how the relevant quantities change as the separation between the equally spaced coordinates go to zero and the discrete system becomes
continuous.
lim
a!0
n
X
i=1
a
!
=
Z
dx,
lim qn = (x),
a!0
lim
a!0
✓
qi
qi
a
1
◆
=
@
,
@x
m
= ⇢,
a!0 a
lim
lim ka = Y
a!0
Here Y is the Young’s modulus for the material and ⇢ is the density. Using the limits above
it is possible to transform the Lagrangian quantity to a Lagrangian density. A continuous
–3–
quantity described by the limit below.
lim L = lim
a!0
a!0
1
2
Z
"
n
1X
m 2
a
q˙i
2
a
1
ka
2
i=1
✓
qi
@ (x) 2
Y
dx =
@x
⇢ ˙ (x)2
Z
qi
a
1
◆2 #!
=
Ldx
where L is the Lagrangian density. Now the formalism have transformed from n discrete
coordinates to one continuous field coordinate which have compactified the problem significantly. Equivalently the Hamiltonian can be described as the space integral over the
Hamiltonian density. If we consider the example of the continuous rod we get the following
expression describing the Hamiltonian.
pi =
H=
n
X
pi q˙i
i=1
@L
@Li
=a
@ q̇
@ q˙i
n
X
@Li
L=
a
q˙i
@ q˙i
aLi
i=1
Under the limit a ! 0 the Hamiltonian transforms to
✓
◆
Z
@L ˙
H = dx
L
@˙
Another interesting quantity is the canonical momentum. It is the momenta of the field
in an infinitesimal section of the field. Therefore the canonical momenta is defined as the
following limit.
lim
a!0
pi
@L
⌘⇡=
a
@˙
(2.1)
Which gives the following final expression for the continuous Hamiltonian in the example
of coupled oscillations in an infinitely long and thin rod.
Z
⇣
⌘ Z
˙
H = dx ⇡
L = dx H
2.2
Generalization for higher dimensions
To generalize the classical field theory for higher dimensions we simply transform the scalar
quantity x to a vector. More specifically a four-vector,
x 7! xµ
where xµ is the four-vector and spans µ = {0, 1, 2, 3}. The differentiation operator then
transforms to
@
@
7! @ µ ⌘
@x
@xµ
–4–
Including Lorentz invariance in our description we can define the metric in Minkowski space
with the following convention ( , +, +, +) and defining the one-form.
0
1
1000
B 0 1 0 0C
B
C
gµ⌫ = B
C
@ 0 0 1 0A
0 001
@µ = gµ⌫ @ µ
This means that we consider time and space as equals with the exception of a minus-sign.
Time and space are just extensions of an four-dimensional euclidian geometry. For example
if one walks 1 positive unit distance in space the separation is equivalent to standing still and
waiting one positive unit time. This is of course only true when all fundamental constants
of nature are set to 1. The Lagrangian density is now a function of the four-gradient of
the field and the field configuration, L(@µ , ), so the action over the Lagrangian density
becomes
Z
Z
S = L dt = L(@µ , ) d4 x
This action is perhaps the most important quantity in field theories. It has the unit of
[Energy⇥time] and as we will se later on using the Hamilton’s principle the variation of the
action can give us the integrals of motion.
2.3
The Euler-Lagrange equations of motion
Hamilton’s principle is a central part of all physics. It states that the variation of the action
must be zero. Marpertuis who is credited for the formulation of the principle of least action,
the forerunner of the Hamilton’s principle, said the following regarding the principle of least
action [4].
“The laws of movement and of rest deduced from this principle being precisely
the same as those observed in nature, we can admire the application of it to all
phenomena. The movement of animals, the vegetative growth of plants ... are
only its consequences; and the spectacle of the universe becomes so much the
grander, so much more beautiful, the worthier of its Author, when one knows
that a small number of laws, most wisely established, suffice for all movements”.
-Pierre Louis Maupertuis
Applying Hamilton’s principle on the action S will generate the Euler-Lagrange equations
of motion for the field. Hence if the variation of the action is zero the Lagrangian will be
an extremum, most commonly a minimum.
Z
S=
d4 x L(@µ
✓
Z
@L
4
= d x
@
S=0
✓
◆
Z
@L
@L
4
, )= d x
(@µ ) +
@(@µ )
@
✓
◆
✓
◆◆
@L
@L
@µ
+ @µ
@(@µ )
@(@µ )
–5–
The third term vanishes since according to the divergence theorem.
✓
◆ I
Z
@L
@L
4
d x @µ
= d3 x
=0
@(@µ )
@(@µ )
Since the 3-integral is on the boundary of the 4-volume the variation of the field configuration, , is zero by definition on the entire 3-surface. Hence the 3-integral is equal to zero.
This gives the following variation of the action.
✓
◆
Z
@L
@L
4
S= d x
@µ
=0
@
@(@µ )
Since
is arbitrary and non-zero inside of the 3-surface the expression in the brackets
must be zero.
✓
◆
@L
@L
@µ
=0
@
@(@µ )
Giving the Euler-Lagrange equations for the 4-dimensional Lorentz invariant field.
2.4
the Noether theorem: Conservations laws in field theories
Emmy Noether’s theorem generalizes conservation law’s for conformal field theories under
special transformations called symmetries. A symmetry is a transform that leaves the
equations of motion unchanged, where the equations of motion are given by the EulerLagrange equations. Consider the following transformations.
7!
0
=
+ ↵@µ
0
L 7! L = L + ↵@µ J µ
As we saw earlier a variation of a four-divergence will not change the equations of motions
since the four-divergence term will vanish when we consider the variation of the action. The
transformations above are therefore symmetries. Consider the variation of the transformed
Lagrangian density.
✓
◆
✓
◆
@L
@L
@L
@L
@L
L(@µ , ) =
+
(@µ ) =
@µ
+ @µ
@
@(@µ )
@
@(@µ )
@(@µ )
✓
✓
◆◆
✓
◆
@L
@L
@L
=
@µ
+ @µ
@
@(@µ )
@(@µ )
The first term is the Euler-Lagrange equation term and is zero by the formulation. As the
transformation is a symmetry up to a four-divergence the variation of the Lagrangian is
equal to ↵@µ J µ , which gives.
✓
◆
@L
L(@µ , ) = @µ
= ↵@µ J µ
@(@µ )
Inserting the variation of the field parameter the expression can be written as.
✓
◆
@L
µ
@µ
@µ
J
=0
@(@µ )
@L
jµ =
@µ
Jµ
@(@µ )
–6–
These two equations are the Emmy Noether conservation law’s. If the transformation is a
translation the resulting conserved current will be the Stress-Energy tensor. Consider the
following 4-space translation.
xµ 7! xµ0 = xµ
7!
0
=
aµ
+ aµ @ µ
L 7! L0 = L + aµ @µ L = L + a⌫ @µ ( ⌫µ L)
These transformations are clearly symmetries which means we can consider the variation
of the translated Lagrangian density
✓
◆
✓
◆
@L
@L
@L
@L
@L
L(@µ , ) =
+
(@µ ) =
@µ
+ @µ
@
@(@µ )
@
@(@µ )
@(@µ )
✓
◆
@L
= @µ
= a⌫ @µ ( ⌫µ L)
@(@µ )
✓
◆
@L
µ
@µ
@⌫
⌫L = 0
@(@µ )
@L
µ
T⌫µ =
@⌫
⌫L
@(@µ )
Where T⌫µ is the Stress-Energy tensor. The conserved charges and the total momentum are
given by
Z
Z
00 3
H = T d x = Hd3 x,
(2.2)
Z
Z
P i = T 0i d3 x =
⇡@i d3 x
(2.3)
2.5
Scalar Field Theories and the Klein-Gordon field
The Klein-Gordon field is a scalar field described by the Klein-Gordon equation
(⇤ + m2 ) (x̄, t) = 0
The ⇤ operator is the D’Alembertian which can also be written as @µ @ µ . The m in the
equation can be interpreted as the mass of the particle. For simplicity fundamental constants of physics are often chosen to 1 in QFT, further on the values of the propagation
speed c and the Planck constant ~ will be set equal to 1. Writing the wave function (x̄, t)
in momentum space, Fourier-expanded with momentum as the Fourier-variable, the KleinGordon equation will transform to a much simpler equation where the solution is a simple
harmonic oscillator.
Z
d3 p ip̄·x̄
(x̄, t) =
e
(p̄, t)
(2⇡)3
✓ 2
◆
@
2
2
+ p̄ + m
(p̄, t) = 0
@t2
Since the solution to the Klein-Gordon equation in momentum space is a harmonic
poscillator
we can interpret the result as quantum harmonic oscillator with frequency ! = p̄2 + m2 .
–7–
2.6
Second Quantization
For the quantum harmonic oscillator the following commutator relations holds.
[qi , pj ] = i
ij
[qi , qj ] = [pi , pj ] = 0
The Hamiltonian of the harmonic oscillator can be written as
1
1
H = p2 + ! 2 q 2
2
2
(2.4)
By inferring creation and annihilation operators the generalized coordinate and the momentum operator can be expressed as follows.
r
r
!
i
!
i
†
a=
q + p p,
a =
q p p
2
2
2!
2!
r
1
!
q = p (a + a† ),
p= i
(a a† )
2
2!
Inserting into the commutator relation between the momentum operator and the generalized
coordinate the momentum operator expressed as creation and annihilation operator gives
the following commutator relation between the creation and annihilation operator.
[a, a† ] = 1
Inserting the representation as creation and annihilation operators into the Hamiltonian
gives
✓
◆
1
†
H =! a a+
2
Now we want to look at a generalization of the commutator relations for the continuous field
theory as the generalized coordinate transforms to the wave function and the momentum
operator to the canonical momentum operator. It is reasonable that the Kroenecker delta
function transforms into the Dirac delta distribution.
[ (x̄), ⇡(ȳ)] = i
(3)
(x̄
ȳ)
[ (x̄), (ȳ)] = [⇡(x̄), ⇡(ȳ)] = 0
Writing the wave functional in momentum space as a linear combination of the creation and
annihilation operator gives the form of the wave functional and the canonical momentum
in scalar field theories.
Z
d3 p
1
p
(x̄) =
(ap eip̄·x̄ + a†p e ip̄·x̄ )
3
(2⇡)
2!p
r
Z
!p
d3 p
⇡(x̄) =
( i)
(ap eip̄·x̄ a†p e ip̄·x̄ )
3
(2⇡)
2
–8–
The commutation relations for the creation and annihilation operators become
[ap , a†q ] = (2⇡)3
[ap , aq ] =
(3)
(p̄
[a†p , a†q ]
q̄)
=0
Inserting the quantized wave functional and canonical momentum in the expression of the
Hamiltonian 2.4 for the harmonic oscillator gives.
Z
Z 3 3 3
⇣
⌘⇣
⌘
d pd qd x
1
3
2
ipx
†
ipx
iqx
†
iqx
p
d x =
a
e
+
a
e
a
e
+
a
e
p
q
p
p
(2⇡)6
4!p !q
Z 3 3 3
⇣
⌘
d pd qd x
1
† ix(p q)
†
ix(p q)
p
=
a
a
e
+
a
a
e
p
q
q
p
(2⇡)6
4!p !q
Z 3 3
⇣
⌘
1
d pd q
3 (3)
†
†
p
a
=
(2⇡)
(p
q)
a
a
+
a
p q
p q
(2⇡)6
4!p !q
Z
⌘
d3 p 1 ⇣ †
†
=
a
a
+
a
a
p p
p p
(2⇡)3 2!p
Z
r
⌘⇣
!p !q ⇣ ipx
d3 pd3 qd3 x
2
†
ipx
d x⇡ =
(
i)
a
e
a
e
aq eiqx
p
p
(2⇡)6
4
Z 3 3 3 r
⌘
d pd qd x !p !q ⇣ † ix(p q)
†
ix(p q)
=
a
a
e
+
a
a
e
p
q
q
p
(2⇡)6
4
Z 3 3 r
⇣
⌘
d pd q !p !q
3 (3)
†
†
=
(2⇡)
(p
q)
a
a
+
a
a
p
q
q
p
(2⇡)6
4
Z
⌘
d3 p !p ⇣ †
†
=
a
a
+
a
a
p p
p p
(2⇡)3 2
3
Z
2
1
H=
2
Z
h
i
d3 p
†
†
!
a
a
+
a
a
=
p
p
p
p
p
(2⇡)3

d3 p
1
!p a†p ap + (2⇡)3
(2⇡)3
2
Z
a†q e
(3)
iqx
⌘
(0)
The last term in the Hamiltonian corresponds to an infinite energy. This infinity will not
be a problem since we are interested in energy differences. It can easily be verified that
[H, a†p ] = !p a†p
[H, ap ] =
!p ap
This means that we can construct a particle with momentum p with the following operation
|p̄i = a†p |0i
Where |0i is thep
empty state. The eigenvalue of the Hamiltonian acting on a state p is the
frequency !p = p̄2 + m2 .
H |p̄i = !p |p̄i
The total momentum operator can be defined from the expression from the Noether theorem
Z
Z
d3 p
P =
⇡@i d3 x =
p̄a† ap
(2⇡)3 p
–9–
2.7
Lorentz invariant normalization
The vacuum is normalized as.
h0| 0i = 1
The normalization between two particle states are.
hp̄| q̄i = (2⇡)3
(3)
(p̄
q̄)
To construct a Lorentz invariant normalisation we must find the Lorentz invariant measure.
The Lorentz invariant measure is given by.
Z 3
d p
2Ep
Where Ep is given by the relativistic dispersion relation.
p
Ep = p̄2 + m2
This can be shown rather easily. The following measure is obviously Lorentz invariant.
Z
d4 p
Using the relativistic dispersion relation.
pµ pµ = m2 ! p20 = Ep2 = p̄2 + m2
Combining these two quantities we can construct a third Lorentz invariant quantity.
Z
Z 3
Z 3
d p
d p
4
2
2
2
d p (p0 p̄
m )
=
=
2p0
2Ep
p0 >0
This gives us the Lorentz invariant delta function.
p
2Ep
Since
Z
3
d3 p
2Ep
2Ep
(3)
(p̄
(3)
(p̄
q̄)
q̄) = 1
Soliton theory in scalar field theories
A soliton solution to a field equation is a non-dissipative non-trivial finite energy solution.
They are a subset of kinks which are also non dissipative non-trivial finite energy solutions
what distinguishes the kinks from the solitons is that solitons remain unperturbed in collisions with other solitons, while this is not the case for kinks. A necessary condition for the
existence of soliton and kink solutions to a field equation is that the potential energy has
– 10 –
at least two degenerate minimas. This is a consequence of the bounded energy of the kink.
The standard Lagrangian for scalar field theories is as follows
1
L = (@µ )(@ µ )
2
(3.1)
U( )
Where U ( ) is any potential. In this text two different scalar theories will be examined:
the 4 -theory and the sine-Gordon theory. The finite energy solutions are called Kinks
by Weinberg [5] and lumps by Coleman [6]. Coleman tells us that he will not call them
solitons since a soliton is a precisely defined mathematical object which some kinks/lumps
does not fulfil. The difference between solitons and kinks as stated above is that solitons
are unperturbed by collisions with other solitons while kinks are not [7].
3.1
Non-trivial time-independent finite energy solutions for single scalar fields
The kinetic energy is the term in the scalar field Lagrangian density (3.1) associated with the
square of the time derivative of the field. The potential energy is defined as the remaining
terms, the space derivatives of the field and the field potential. For there to exist non-trivial
finite energy solutions the potential U (x) must have at least two degenerate minimas. The
time-independent hamiltonian density is given by.
1
H = (@x )2 + U ( )
2
For the energy to remain finite the potential U ( ) must go to a minimum as x ! ±1. So if
the potential has only one zero this means that the solution (x) must be at the minimum
for x = ±1 if the energy is to be finite. This is of course a trivial finite energy solution.
However if U ( ) has at least two zeros it is possible for the solution to be at one minima of
the potential at x = 1 and another minima of the potential at x = 1. This means that
all non-trivial finite energy solutions travel from one minima of the potential at x = 1
to another minima at x = 1. If the Noether theorem is applied on the field the equation
of motion is found.
@L
@L
= @µ ,
=
@(@µ )
@
@U
,
@
) @⌫ @ µ +
@U
=0
@
Considering the (1 + 1)-dimensional field where µ = {0, 1}. For the (1 + 1)-dimensional
field the equations of motion becomes.
@t2
@x2 +
@U
=0
@
First the static solutions will be studied, the solutions where @t = 0. After the analysis of the time independent solutions through Lorentz transformation the time-dependent
solutions will be found. For the time independent scalar field the action becomes.
– 11 –
S=
Z
dx
✓
1
(@x )2 + U ( )
2
◆
Under the change of variables 7! x, x 7! t, it is possible to recognise the time-independent
problem as the problem in classical mechanics of a particle with unit mass moving in the
potential U (x).
S=
Z
dt
✓
1
(@t x)2 + U (x)
2
◆
=0
From classical mechanics we know that the particle’s motion has zero total energy which
means that.
1
(@t x)2
2
U (x) = 0
Making the inverse change of variables, x 7! , t 7! x, gives the very important relation
in soliton theory, the relation from which the explicit time independent solution can be
computed.
1
(@x )2 = U ( )
2
p
@
= ± 2U ( )
@x
(3.2)
(3.3)
By integration equation 3.3 gives the explicit formula for finding the time independent kink
solutions. The plus-minus sign on the right hand side has some interesting consequences. If
one follows the positive branch when computing the explicit time independent solution the
kink solution is found. If the negative branch is followed the anti-kink solution is found.
Z
3.1.1
2
1
p
d
2U ( )
=±
Z
x2
(3.4)
dx = ± x
x1
The explicit time independent sine-Gordon solution
The sine-Gordon scalar field is interesting in many aspects. It is periodic and has infinite
degenerate minimas. The potential of the sine-Gordon field is given by.
U( ) =
m2
2
(1
cos
)
This means that the Lagrangian for the sine-Gordon system is defined as follows.
1
L = ⇡2
2
1
(@x )2
2
– 12 –
m2
2
(1
cos
)
(3.5)
Remember that the Hamiltonian is a function of the Lagrangian density
H=
Z
Z
dxT00 =
dx(@0 @ 0
0
0 L)
The only aspect of this Hamiltonian that differs from another scalar field hamiltonian is
the last term of the integral, the field potential term. The Hamiltonian for the sine-Gordon
field is defined as follows.
H=
Z ✓
R
1 2 1
m2
⇡ + (@x )2 + 2 (1
2
2
cos
)
◆
The sine-Gordon potential is periodic and has degenerate minimas at
2n⇡
=
, n2Z
Inserting the sine-Gordon potential into the explicit integral formula 3.4 gives a way to
compute the time independent kink’s for the sine-Gordon field.
Z
d
2
1
r ⇣
2
2 m2 (1
cos
)
⌘ = 2m
Z
d
2
sin
1
=
2
⇢
=
2
arcsin t,
d =
2
p
1
1
t2
dt
(
)
p
Z
Z
p
1 t2 1
1
1 t2
1 s2 1
2
p
=
dt = s = 1 t , dt =
ds =
ds
m t 1 t 1 t2
t
m s1 s 2 1
◆
Z
Z s2 ✓
1 s2
1
1
1
1
1
1 s s2
=
ds =
+
ds =
ln
=± x
m s1 (s 1)(s + 1)
2m s1
s 1 s+1
2m
1 + s s1
)
1
cos
1 + cos
2
2
=
sin2
cos2
4
= e±2m
x
4
) tan
4
= e±m
x
) (x) =
4
arctan e±m
x
Hence the solution to the time-independent sine-Gordon equation is
(x) =
4
arctan e±m
x
(3.6)
Remember that the plus-minus determines if the solution is a kink or a anti-kink. A plus
sign means the solution is a kink and a minus sign an anti-kink.
3.1.2
For the
The explicit time independent
4-
4 -field
solution
field the potential is given by.
U( ) =
4
(
2
2 2
v ) , v=
– 13 –
r
m2
Where the relationship between the weak-coupling ,
2
v
2
=
=
and v is as follows.
(3.7)
m2
The integral equation becomes.
,
r Z
2
p
+v
= e⌥m 2
v
r Z
2
2
1
v2
1
x
2
d
2
,
d
2
=v
=
r
e
m
±p
e
m
±p
2
v2
=± x
2 1
ln
2v
2
x
x
This means that the time independent for the
v
+v
e
m
⌥p
x
+e
m
⌥p
2
x
2
4 -field
0
=± x
m
= v tanh ± p x
2
is.
±m
(x) = v tanh p
x
2
The kink is the solution obtained by following the positive branch of the ±-sign.
m
(x) = v tanh p x
2
The anti-kink is found by following the negative branch.
m
(x) = v tanh p
x
2
3.2
Stability of the kink
To analyse the stability of the kink we study the first order perturbation around a minima
of the potential under the assumption that there are at least two degenerate minimas of the
potential enabling non dissipative finite energy solutions. The expansion of the potential
around the minima min look’s as follows where (x, t) is an arbitrary function of small
oscillations.
U( ) =
U(
min )(
min )
0
0!
+
U 0(
min )(
min )
1
+
1!
U 00 (
min )(
min )
2
2!
+ O(
3
)
O(
3
)
The definition of a minima of U ( ) is defined as follows.
U 0(
min )
=0
The Lagrangian for the scalar field becomes.
1
L = (@µ )(@ µ )
2
U(
min )
U 0(
min )(
1!
– 14 –
min )
1
U 00 (
min )(
2!
min )
2
Through the Noether theorem the equation’s of motion for the Lagrangian is found.
@v @ µ + U 00 ( ) + O(
2
)=0
In first order perturbation theory this becomes.
@v @ µ (x, t) + U 00 (
min )
(3.8)
(x, t) = 0
This equation is invariant under time translation since it is a equation generated from the
Noether theorem which means that the solutions to equation (3.8) can be written as.
(x, t) = Re
X
an ei!n t
(3.9)
n (x)
n
and !n and
n
is determined by.
@x2
+ U 00 (
n
min ) n
= !n2
(3.10)
n
The solutions (3.9) are stable if and only if the eigenfrequencies are real. This can be
showed by the following proof. Consider the real eigenfrequencies
!n 2 R
For the solution to be stable means that it does not diverge as t ! ±1. This can be
written as
lim | (x, t)| < ⌦
t!±1
Where ⌦ is a finite constant. For the real eigenfrequencies this becomes.
lim
t!±1
Re
X
an ei!n t
n (x)
n
Using the Triangular inequality this is less or equal to
lim
t!±1
an
n (x)
X
an ei!n t
n
n (x)
 lim
t!±1
X
n
|an
n (x)|
ei!n t = lim
t!±1
is bounded by definition. This means that.
X
lim
|an n (x)| < ⌦
t!±1
X
n
|an
n (x)|
n
This shows that if the eigenfrequencies are real the solitons are stable.
lim | (x, t)| < ⌦
t!±1
Now consider the arbitrary complex eigenfrequencies
!n = an + ibn , (an , bn ) 2 R2
Inserting this expression in the stability condition gives
lim | (x, t)| = lim
t!±1
t!±1
Re
X
an e(ian
bn )t
n (x)
n
The limit above is either zero or infinite, hence the eigenfrequencies must be real for stability.
– 15 –
3.3
Taylor expansion of the sine-Gordon potential around a minimum
One minimum of the periodic sine-Gordon potential is = 0. This is a good choice to
expand around since the Taylor expansion becomes the simpler MacLaurin expansion.
U( ) =
m2
2
(1
cos
@U
m2
=
sin
@
@3U
=
m2 sin
@ 3
@5U
= 3 m2 sin
@ 5
m2
U( ) =
2!
2
m2
)
@2U
= m2 cos
@ 2
@4U
=
@ 4
@6U
=
@ 6
2 4
4!
+
m2
4 6
+ O(
6!
8
)=
2
m2 cos
4
m2 cos
1
m2 X (
2
n=1
)2n
(2n)!
The first term in the expansion is the classical harmonic potential.
3.4
Energy density calculation for the sine-Gordon and the
4 -model
The total energy and the hamiltonian are the same quantities. The energy integral is given
by the hamiltonian from the Noether theorem 2.2.

Z
Z
1
1
E = dx E = dx (@0 )2 + (@1 )2 + U ( )
2
2
Where E is the energy density. Using the identity in 3.2 gives the total energy as.
Z
Z
2
E = dx(@1 ) = dx 2U ( )
For the sine-Gordon model the time-independent energy density becomes
E = 2U ( ) =
2m2
2
(1
cos
)=
2m2
2
sin2
2
The simplest way to compute the total energy of the sine-Gordon field is through the
integral of the space derivative of the analytic solution.

Z
Z
1
1 16 4m2 e±m x
16m
1
16m
2
E = dx(@1 ) =
dx
=
±
= 2
2
±2m
x
2
±2m
x
2 (1 + e
)
(1 + e
) 1
So the total energy for a single soliton in the sine-Gordon model is 16m
2 . This quantity is
4
sometimes referred to as the soliton mass. For the -model the energy density is defined
as.
E = 2U ( ) =
2
(
– 16 –
2
v 2 )2
The total energy or the soliton mass for a single soliton becomes.
E=
3.5
Z
Z
✓
◆
2 v
dx2U ( ) = dx
(
v ) = ⌥p
2
2m
p 3
✓
◆
3
2 v
2v
2 2m
= ⌥p
⌥
=
3
3
2m
2
2 2
Z
±v
d
2
v2
0
Non-trivial time-dependent finite energy solutions for single scalar fields
To find the time-dependance of the soliton solutions we simply Lorentz transform the timeindependent solutions.
x ut
x 7! p
1 u2
The solution for the sine-Gordon field transforms to
(x, t) =
The solution for the
4 -field
4
arctan e
✓
±m px
ut
1 u2
◆
transforms to
"
±m( x ut)
(x, t) = v tanh p
2(1 u2 )
#
The soliton masses transforms under the Lorentz transformation as follows.
E 7! p
E
1
u2
So when the soliton velocity approaches one the mass diverges, as would be expected from
special relativity. The solitons are similar to particles. Since they are Lorentz invariant the
obey the standard relativistic energy relation.
E=
p
p2 + m2
(3.11)
But there is some differences from particles. If we have one particle solution it is possible
to extend it and get a asymptotic many particle solution i.e if all particles are placed at
the edge of infinity their interaction goes to zero and superposition is trivial. However the
same does not hold for solitons. Consider n equally space points a1 < ... < an with the
corresponding solutions fi . If the spacing goes to infinity does there exist a solution where
⇡ fi (x ai )? The answer is yes if and only if.
fi (1) = fi+1 ( 1)
– 17 –
This has interesting consequences to the symmetry of the fields. Since the sine-Gordon
potential is 2⇡ -symmetric the only observable functions in the sine-Gordon model are the
functions unchanged by the following transformation, i.e are a symmetry with respect to.
7!
For the
4 -field
+
2⇡
the only observables are the functions symmetric with respect to
7!
Figure 1 shows the graph of the sine-Gordon and the
4 -potential.
The symmetry of the
4-
Sine-Gordon potential
2
U(φ)
m 2 /β 2
1.5
1
0.5
0
-6
-4
-2
0
2
4
6
βφ
φ 4 potential
2
U(φ), a = 1
λ/2
1.5
1
0.5
0
-1.5
-1
-0.5
0
0.5
1
1.5
φ
Figure 1. Graph over the sine-Gordon and the
4
-potential
field has some extraordinary consequences. Solitons and anti-solitons are not independent
objects. A soliton must always be followed by an anti-soliton and so forth.
3.6
The Topological charge
The topological charge is an integral of motion and is defined as [5].
µ
Jtop
= ✏µ⌫ @⌫
– 18 –
Where ✏µ⌫ is anti-symmetric. As we saw in the Noether theorem, if we have a conserved
current the conserved charge is the space integral of the conserved current.
✏µ⌫ =
✏⌫µ , ✏01 = 1
For the sine-Gordon model the topological charge becomes
Z
Z
4⇣ ⇡
01
Qtop =
dxJ =
dx(@1 ) = [ (1)
( 1)] =
±
2
R
R
For the
4 -model
Qtop
the topological charge becomes.
Z
Z
⇣ ⇡
=
dxJ 01 =
dx(@1 ) = [ (1)
( 1)] = v ±
2
R
R
⇣ ⇡ ⌘⌘
4⇡
⌥
=±
2
⇣ ⇡ ⌘⌘
⌥
= ±v⇡
2
The soliton has positive charge and the anti-soliton has negative charge.
3.7
Derrick’s theorem
An important question to ask is whether it is possible to find time-independent soliton
solutions in a scalar field and if so how does the solutions change as the spatial dimension
increases. As we have seen before it is possible to find time-independent soliton solutions in
scalar field theories with one spatial dimension if the potential has at minimum two degenerate minimas. However we have not investigated what happens if the spatial dimension is
grater than one. Derrick’s Theorem says something very important regarding the existence
of non-trivial time-independent finite energy soliton solutions in scalar field theories. If the
spatial dimension is greater than 1 they cannot exist. I should however emphasise that this
hold only for time-independent solutions.
Derricks’s theorem. Let be a set of scalar fields with one time dimension and D spatial
dimensions that follows the dynamics determined by the Lagrangian density.
1
L = @µ @ µ
2
U( )
The potential U is non-negative and zero for the ground states. Then for D
non-trivial finite energy solutions are the ground states [6].
2 the only
Proof. Define
1
V1 =
2
V2 =
Z
Z
dD x(r )2
dD xU ( )
Where (x) is a time-independent solution. Consider the one-parameter family of field
configurations given by
(x, ) ⌘ ( x),
– 19 –
2 R+
Since the solution is time-independent the energy for this family becomes
H( ) =
(2 D)
D
V1 +
V2
According to Hamilton’s principle this must be stationary at
@H
@
= (2
D)V1
DV2 =
((D
= 1. This is equivalent to
2)V1 + DV2 ) = 0
=1
We see that V1 and V2 must vanish for D > 2. For D = 2, V2 must vanish. The Hamiltonian
becomes.
H2 = V 1
Applying Hamilton’s principle on the Hamiltonian for D = 2 implies that V1 is zero.
@H
@
= V1 = 0
=1
This is an remarkable result and quite discouraging [6]. It means that we will never find
time-independent two or higher dimensional non trivial finite-energy solutions.
3.8
Soliton Quantization
In this section we will quantize the soliton of the scalar field theory. We will follow the
methods of [5]. Two things are especially important to think about when quantizing the
soliton. Firstly, how does the explicit form of the classical field configuration affect the
quantum field configuration and secondly how is the soliton reflected in the states of a
quantum theory. When considering a quantum wave-function the uncertainty of the wavefunction increases as the scale diminishes according to the uncertainty principle. In fact
the fluctuations diverges as the scale goes to zero.
~
2
~
lim
=1
x !0 2 x
x p
lim
x !0
p
We want to choose a distance scale L such that this limit does not diverge at the same time
as the distance scale is small with respect to the classical limit. This is ensured by the weak
coupling criterion.
3.8.1
The weak coupling criterion
The variation of the classical field is proportional to.
(
L )classical
– 20 –
⇠L
@
@x
We know the soliton solution so we can differentiate it for both the 4 -model and the
sine-Gordon model and find that the derivative of the wave-function is proportional to the
mass and the weak coupling.
@
vm
= p (1
@x
2
(
L )classical
m x
tanh p ) ⇠ vm
2
@
Lm2
⇠L
⇠ Lmv ⇠ p
@x
If we define a smeared out quantum field
L (x̄).
1
n
L (x̄) =
(2⇡L2 ) 2
Z
dn ye
(ȳ x̄)2
2L2
(x̄)
Then we are able to estimate the variation of the quantum fluctuations of the free field
around a vacuum with the definition.
Z
ȳ 2 z̄ 2
1
2
2
n n
2L2
( L )quantum ⌘ h0| L (0) |0i =
d
yd
z
e
h0| (ȳ) (z̄) |0i
(2⇡L2 )n
It is possible to compute h0| (ȳ) (z̄) |0i remembering that h0| ap a†q |0i = (2⇡)n (p̄ q̄).
Z n n
⇣
⌘⇣
⌘
d pd q
1
ipy
†
ipy
iqz
†
iqz
p
h0| (ȳ) (z̄) |0i = h0|
a
e
+
a
e
a
e
+
a
e
|0i
p
q
p
q
(2⇡)2n 4!p !q
Z n n
Z n n
⇣
⌘
d pd q
1
d pd q
1
† i(py qz)
p
p
= h0|
ap aq e
|0i =
hp| qiei(py qz)
(2⇡)2n 4!p !q
(2⇡)2n 4!p !q
Z n n
Z
d pd q
1
dn p
1
(n)
i(py qz)
p
p
=
(p
q)e
=
eip(y z)
n
n
(2⇡)
(2⇡) 2 p2 + m2
4!p !q
This gives.
(
2
L )quantum
1
⌘
(2⇡L)2n
Which is equivalent to.
(
Z
1
dn ydn zdn p p
e
2
2 p + m2
2
L )quantum
⌘
Z
2
dn p e p L
(2⇡)n 2!p
ȳ 2 z̄ 2
2L2
eip(y
z)
2
(3.12)
The variation of the field is primarily in the region of m 1 surrounding the kink for both
the sine-Gordon model and the 4 -model [5]. Therefore the length scale should be very
small in relation to the inverse mass.
L⌧m
1
, mL ⌧ 1
The behaviour of the integral 3.12 in this interval tells us
8q
< ln 1
if n = 1
mL
( L )quantum ⇠
(n 1)
:L 2
if n > 1
– 21 –
In the case of one dimensional solitons we can now set up the weak coupling criterion. We
want the fluctuation of the classical model to be a lot bigger than the fluctuations of the
quantum model.
r
1
Lm2
ln
⌧ p
mL
, 1 ⌧ mLe
The weak coupling criterion forces
3.8.2
L2 m 4
to be very small if the criterion is to hold for L ⌧ m
1.
Perturbation around a minimum of the potential
Perturbation around a minimum of the potential is a classic approach in physics. In this
section we will see that small perturbations around the ground state of the kink will in fact
describe interaction with the kink. In section 3.1 we encountered the classical sine-Gordon
Lagrangian density (3.5). If the following substitution is made 0 = , @µ 0 = @µ the
Lagrangian density transforms to
✓
◆
1 1
0
µ 0
2
0
L= 2
(@µ )(@ ) m (1 cos )
2
is the weak coupling and is an arbitrary constant.
is therefore not important when
considering solutions of the sine-Gordon equation since if we can find solutions without
considering it is possible to find solutions when considering by scaling the solution.
However the situation is different when considering the quantum version of the theory. As
explained by [6]; in quantum physics the relevant quantity is not the Lagrangian density,
L, but the Lagrangian density divided by the reduced Planck’s constant L~ . This gives the
following form of the Lagrangian density for the sine-Gordon model.
✓
◆
L
1
1
0
µ 0
2
0
=
(@
)(@
)
m
(1
cos
)
µ
~
~ 2 2
We can now ask ourselves what happens to other interesting quantities such as the canonical
momentum. The canonical momentum is defined as in (2.1).
⇡0 =
@L
@0 0
=
2
@(@0 0 )
To generalize the quantum Lagrangian density for all scalar field’s the sine-Gordon potential
is simply interchanged with an abstract potential U ( 0 ). At the same time as ~ ! 1 and
standard quantum field theory units is adopted.
✓
◆
1 1
0
µ 0
0
L= 2
(@µ )(@ ) U ( )
2
Let’s consider the quantum Hamiltonian for such a system.
✓
◆ Z
✓
◆
Z
@L
1 4 02 1
2
0
0 2
0
H = 2 dx
@
L
=
dx
⇡
+
(@
)
+
U
(
)
x
@(@00 ) 0
2
2
✓
◆
Z
1 4 02
= dx
⇡
+ V ( 0)
2
– 22 –
(3.13)
(3.14)
This Hamiltonian is rather curious. The infinitesimal coupling constant is multiplied with
the kinetic energy of the system. A rather common approach in physics is now to see if
we can recognise such a system. In the section 2.5 regarding second quantization we did
a similar approach when we realised the Klein-Gordon equation in momentum space was
in fact an equation with standard harmonic oscillators as solutions. Then we choose to
interpret the wave functional as quantum harmonic oscillators which turned out to be the
correct interpretation. This time we realise that a similar model is the Hamiltonian for the
diatomic molecule.
Hdi =
p̂2
+ U (r)
2µ
Where µ is the reduced mass of the molecule
m1 m2
µ=
m1 + m2
The exact solution is however a polyatomic with infinitely many atoms.
identified with the individual mass of the atoms.
Hpoly =
4
= mi
1
is
1
X
p̂2i
+ U (ri )
2mi
i=1
3.8.3
Recapitulation of the diatomic molecule
One approach to find the energy perturbations of the scalar field theories is to use the
resemblance between the diatomic molecule and soliton solutions single scalar fields. The
energy levels of the diatomic model were found in the early twentieth century and have been
studied rigorously ever since. After finding the energy levels for the diatomic molecule it
is possible to proceed by generalizing the diatomic energy levels to the polyatomic system
and finally apply the field language to the description of our system. Beginning with the
diatomic molecule the potential as a function of distance between the atoms looks as follows.
The asymptotic behaviour of the Morse potential towards zero as r ! 1 is a consequence
of that the Coulomb potential dominates at large length scales. See figure 2. The Pauli
exclusion principle is the reason for the sharp increase of the potential as the radius goes
to zero. Between these two asymptotes there are a minima, this is the ground state energy.
The zeroth order of the perturbation of the energy is exactly this ground state energy. If we
consider harmonic oscillations around this minima we get the first level approximation. The
energy eigenvalue for the first order perturbation is the quantum harmonic oscillator eigenvalue. The second order approximation is including the rotational energy levels of the diatomic molecule. The energy degeneracy is lifted yet one level with the corresponding eigenvalue of the angular momentum operator energy eigenvalue, L̂2 |n, l, mi = l(l + 1) |n, l, mi.
In the first order perturbation we can consider the solutions to follow
(x, t) =
Considering the example of the
4-
vac
+ ⌘(x, t)
model the vacuum,
vac
⌘ ±v
– 23 –
vac ,
would be defined as.
U (r)
U (r) = De (1
e
(r r0 )2
1)
r
r0
E0
Figure 2. Graph over the potential for the diatomic molecule i.e. the Morse potential. The dashed
line is the harmonic first order perturbation of the potential. The first order approximation is
accurate if the domain is close to the domain of the minima of the Morse potential.
Order of approximation
0
1
Energy eigenstate
|r0 , ✓, 'i
|n, ✓, 'i
2
|n, l, mi
Energy eigenvalue
E0
q
00
U (r0 )
+ n + 21
m
+ l(l+1)
+ ...
2mr2
0
Table 1. Energy approximations up to second order for the diatomic molecule.
Looking at the partitions of the total Lagrangian of the system gives
L = LCoulomb + Lvibrational + Linteractions + ...
The small oscillations around the ground state of U ( ) would have to follow the equations
of motion determined by the following harmonic Lagrangian found by Taylor expanding the
potential U (r) to second order approximation around the minima r0 .

Z
1
1 00
Lvib. = dx (@µ ⌘)(@ µ ⌘)
U (r0 )⌘ 2 ,
(3.15)
2
2!
Finding the Euler-Lagrange equation’s for this Lagrangian gives.
@⌫ @ µ ⌘ + U 00 (r0 )⌘ = 0
– 24 –
We want to diagonalize the Lagrangian. This is done by considering the time independent
solutions, assume periodic solution’s in time and doing the change of variables that follows
the time independent Schrödinger equation.
@x2 + U 00 (r0 )
⌘i (x, t) =
i (x)
= !i2
i (x)e
i!i t
i (x)
Generalizing this further gives
⌘(x, t) =
1 ⇣
X
ci (t) i (x)e
i!i t
+ ci (t) i (x)† ei!i t
i=1
⌘
Inserting the expanded solution into equation 3.15 gives the energy eigenvalues of the quantum harmonic oscillator.
✓
◆
n
X
1
En =
! i ni +
2
i=1
Now to the rotational energy levels. The diatomic molecule can be viewed as a rigid rotator
with an angular momentum. The rotational energy levels are given by l(l+1)
, where the
2mr02
numerator is the moment of inertia for the diatomic molecule and l are the angular quantum
numbers.
3.8.4
Transition to the Infinitely polyatomic molecule
To make the transition to the polyatomic molecule is simple. We have to remember that for
each energy level there are an infinite number of atom interactions to include. This means
that the integers in the diatomic model becomes infinite string’s of integers in the infinitely
diatomic case. Consider the first perturbation energy for the ground state. The following
holds.
✓
◆ r 00
◆r
1 ✓
X
1
U (r0 )
1
ki
n+
!
ni +
(3.16)
2
m
2
m
i=1
Where ki is the individual spring constants between the atoms.
3.8.5
Quantized kink energy levels
The transition to the infinitely polyatomic molecule at the same time as incorporation our
kink description produces the following table. Defining the finite energy solutions as
0
(x) = f (x
b)
Where b is the centre of mass of the soliton. The ground state energies for the hamiltonian
3.13 is defined as.
E0 ⌘ V (f )
– 25 –
Order of approximation
0
Energy eigenstate
|bi
1
Energy eigenvalue
E0
|n1 , n2 , ...; bi
2
+
|n1 , n2 , ...; bi
P
2
ni +
2 p2
1
2
!i
+ 2E0 + ...
Table 2. Quantized kink energy levels.
Considering the ground state the zeroth order approximation for the kink gives the first line
in the table. The first order approximation is achieved by letting the integers n transform to
the infinite string of integers {ni }1
i=1 and taking the sum over the entire string as described
by 3.16. The !i is the same as from the stability analysis of the kink 3.10. The occupation
numbers is interpreted as interaction with the kink. The state where all the n’s vanish is
an alone kink, the state where n = 1 is an kink interacting with one meson and so forth.
The ground state with the alone kink can be written as.
|0, bi
The standard normalization of two such ground state is.
h0; b0 | 0; bi = (b0
b)
Which implies that.
h0; b0 |
0
(x) |0; bi = (b0
b)f (x
b)
As in second quantization 2.6 the solution can be written in momentum space as.
Z
db
p eipb |n1 , ...; bi
|n1 , ...; pi =
2⇡
As was shown in section 3.5 the solitons obey the standard relativist energy relation (3.11).
The relativistic energy has a minimum at p = 0. Maclaurin expansion around that minimum
gives the following expression.
@2E
m2
=
3
@p2
(m2 + p2 ) 2
1
E 00 (0) =
m
@E
p
=p
2
@p
p + m2
E 0 (0) = 0
E(p) = m +
p2
+ O(m
2m
In the field formalism this becomes by replacing m with
E0
2
+
2 p2
2E0
+ O(
– 26 –
6
)
3
)
E0
2
So the degeneracy has been lifted yet another level. The frequencies in table 1 are discrete
but they form a continuous set which means that the sum is in fact an integral. Up to this
point we have studied the system in a box a now we shall let the box go to infinity. From
the stability analysis we remember that equation (3.10) holds for the kink. This means that
the squares of the eigenfrequencies are the eigenvalues to the K-operator.
K=
@x2 + U 00 (f )
For the ground state the K-operator is defined as
K0 =
@x2 + U 00 (
0)
This means that we can write the energy difference between the ground state and the kink
as
X
i
(!i |kink
p
!i |vacuum ) = tr( K
p
K0 )
(3.17)
Since the trace operator is the sum of all the diagonal elements. Letting the box go the
infinity the right hand side of equation (3.17) becomes the energy value in table 2. To reach
a model of the quantized soliton we started off from the model of the diatomic molecule to
the infinitely polyatomic molecule incorporating the field formalism and finally letting the
box the system was placed in go to infinity. The leading term in the expansions of the kink
is the classical term. Higher order terms correspond to quantum corrections such as the
quantum harmonic oscillator.
3.9
Soliton interactions
The main property separating a soliton from a kink is that a soliton will be unperturbed in
a collision with another soliton [7]. A kink on the other hand will not conserve it’s shape in
collisions. In figure 3 and 4 showing soliton interaction it is clearly visible that the soliton
survives the collision and maintains it’s shape.
– 27 –
3.9.1
Soliton-soliton interaction
1.5
t = -3
t = -2
t = -1
t=0
t=1
t=2
t=3
1
φβ/4
0.5
0
-0.5
-1
-1.5
-6
-4
-2
0
2
4
6
X-axis
Figure 3. Soliton-soliton collision for u = 0.1
ss (x, t)
=
4
tan
1

u sinh mx
cosh mut
The soliton-soliton interaction seen in figure 3 begins at t = 1. The soliton with initial
position x = 1 has a positive velocity and the soliton with initial position x = 1 has
a equal but negative velocity. The two solitons moves towards each other and collide at
t = 0, the force interacting between the solitons is repulsive and conservative meaning that
it will be a totally elastic collision and the kinks will return the way the came and at t = 1
find themselves at their initial position.
– 28 –
3.9.2
Soliton-anstisoliton interaction
1.5
t = -3
t = -2
t = -1
t=0
t=1
t=2
t=3
1
φβ/4
0.5
0
-0.5
-1
-1.5
-6
-4
-2
0
2
4
6
X-axis
Figure 4. Soliton/Anti-soliton collision for u = 0.1
sa (x, t)
=
4
tan
1

sinh mut
u cosh mx
The soliton/anti-soliton interaction visible in figure 4 also begins at t = 1. The soliton’s
initial position is at x = 1 with positive initial velocity and the anti-soliton begins at
x = 1 with negative velocity. The interaction between the soliton and anti-soliton is
attractive meaning that they will speed up as they get close to one another. They collide
at t = 0 and pass through each other unperturbed with the exception of a phase shift. At
t = 1 the soliton is placed at the anti-soliton’s initial position and vice versa.
– 29 –
3.9.3
Breather interaction
1
t = -4
t = -3
t = -2
t = -1
t=0
t=1
t=2
t=3
t = -4
0.8
0.6
0.4
φβ/4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-6
-4
-2
0
2
4
6
X-axis
Figure 5. Breather soliton: A soliton and a anti-soliton oscillation around a common center of
mass.
v (x, t) =
4
0

sin mvt 0
tan
v cosh mx
p
= 1 + v2
1
0
The analytic solution of the breather soliton can be found by letting the real velocity u
become a imaginary velocity in sa (x, t).
u = iv, v 2 R
The period of the breather soliton is given by
⌧=
2⇡
mv
0
The breather soliton in figure 5 is composed of a soliton and a anti-soliton oscillating around
a common centre of mass. The oscillation is stable. There also exist moving breather solitons
where the common centre of mass has a velocity.
– 30 –
4
Geometrical methods in the theory of solitons
Here we will introduce the notion of the no curvature condition representation as a method
of determining the equations of motion for a model. As in many cases there exist interesting
ways to think about certain constants of motions. For the sine-Gordon model it is possible
to use the no curvature condition construction. We will follow [8]. Consider the auxiliary
linear problem.
@
=U
@x
@
=V
@t
Where U and V are two 2 ⇥ 2 differential matrix operators and functions of space, time and
the spectral parameter. For the sine-Gordon model U and V is defined as follows
U (x, t, ) =
i @t
4
3
ik0 ( ) sin
V (x, t, ) =
i @x
4
3
ik1 ( ) sin
2
2
m
1
k0 ( ) = ( + )
4
m
1
k1 ( ) = (
)
4
1
ik1 ( ) cos
1
ik0 ( ) cos
2
2
2
2
Where a is the Pauli spin matrices. The compatibility condition for the auxiliary linear
problem xt = tx is equivalent to the no curvature condition.
Ut
(4.1)
Vx + [U, V ] = 0
The first diagonal element of the no curvature condition gives the sine-Gordon equation of
motion
2
Ut,1,1
Vx.1,1 + [U, V ]1,1 = Ut
+ik12 cos
2
sin
ik02 cos
=
k0 k1 cos2
2
2
sin
2
Vx +
2
2
16
+ k0 k1 cos
i @t2 + i @x2 + i sin
4
4
, @t2
16
(k12
2
2
k0 k1 sin2
@t @x
@x @t + k0 k1 sin2
= Ut
k02 ) =
@x2 +
m2
Vx
2
2
ik02 sin
i @t2 + i @x2
4
4
sin
iko2 sin
+ ik12 sin
+
2
2
cos
cos
2
2
ik12 sin
im2
sin
4
=0
=0
So a question that can be asked is how does the other elements of the no curvature condition
look. As will be shown the only elements that survive are those in the main diagonal and
they are generators of the equations of motion. In the following computations the canonical
– 31 –
momenta and the derivative of the soliton with respect to x is identified below for simplicity.
⇡ = @t
⇧=
@x
Let’s begin by consider the off diagonal elements.
Element {1, 2}
Ut,1,2
✓
◆
⇡
2
✓
◆
⇧
2
✓
◆
⇡
2
✓
◆
⇧
2
Vx,1,2 + [U, V ]1,2 =
ik0 cos
+ k1 sin
ik1 cos
+ k0 sin
2
2
2
2
✓
◆
✓
◆
⇡
⇧
+
k1 sin
+ ik0 cos
+
k0 sin
+ ik1 cos
=0
2
2
2
2
2
2
Element {2, 1}
Ut,2,1
Vx,2,1 + [U, V ]2,1 =
ik0 cos
k1 sin
ik1 cos
k0 sin
2
2
2
2
✓
◆
✓
◆
⇡
⇧
+
k1 sin
+ ik0 cos
+
k0 sin
+ ik1 cos
=0
2
2
2
2
2
2
So the off diagonal elements are identically equal to zero.
Element {2, 2}
Ut,2,2
Vx.2,2 + [U, V ]2,2 = i @t2
4
= i @t2
4
i @x2
4
i @x2 + i(k02 sin
4
im2
+
sin
4
k12 sin
)
This shows that the only elements that survive are the diagonal elements and the diagonal
elements generate the sine-Gordon equation’s of motion.
4.1
Parallel transport and Gauge transformations
If we transform the wave function as follows the corresponding change happens to U and
V.
(x, t, ) 7! G(x, t, ) (x, t, )
@G 1
U 7!
G + GU G 1
@x
@G 1
V 7!
G + GV G 1
@t
These transformations acting on U and V can be though of as Gauge transformations.
In vector calculus the only determining property of a vector is it’s end point, so if we
would transport an arbitrary vector from an initial position to a final position we would
only need to transport the end point. However in the notion of parallel transport it is
– 32 –
different. A parallel transport can be visualised as a transformation of a vector tangent to
a sphere preserving the vectors property as a tangent. We want to translate the tangent
on the surface of the sphere while still letting it be a tangent to the sphere. This means
that we need to do a translation and a rotation of the tangent. If this were in ordinary
vector calculus we would only need a translation since we would have no condition that the
vector should be tangent to the sphere. The translation is done by the Monodromy matrix
TL ( , t0 ) and the rotation via the rotational matrix Q(✓).
Paralell transport induced by the (U,V)-connection. Let be a curve in R2 with
initial point (x0 , t0 ) and final point (x, t). The parallel transport from the initial point to
the final point is defined as
Z
⌦ = exp (U dx + V dt)
If the parallel transport is acted on by a gauge transformation it transforms as follows
⌦ 7! G(x, t)⌦ G
1
(x0 , t0 )
The superposition of two curves follow the classical exponential formula
⌦
1+ 2
= ⌦ 1⌦
2
(4.2)
If the curvature is zero it means that ⌦ only depend on the end point of the transformation.
If additionally is a closed curve the parallel transport is trivial.
(4.3)
⌦ =I
Faddeev and Takhtajan shown in [9] that that models that accept a no curvature representation have infinitely many integrals of motion. I will go through the argument briefly.
Consider a parallel transport along the space-axis symmetric with respect to the time axis
in 2-dimensional space time.
is covariantly constant if
@
= U (x, t0 , )
@x
Using the analogy of the tangent of the sphere we know that we need to rotate the the
tangent under a translation. This is represented in matrix form as
U (x + 2L, t, ) = Q
1
(✓)U (x, t, )Q(✓)
(4.4)
V (x + 2L, t, ) = Q
1
(✓)V (x, t, )Q(✓)
(4.5)
These expressions are called the quasi periodicity conditions. Where 2L is the entire distance of the symmetric translation. Q(✓) is defined as.
i✓
Q(✓) =
e2 0
i✓
0 e 2
– 33 –
!
The monodromy matrix is the matrix of parallel transport symmetric with respect to the
time axis at constant time and is defined as
TL ( , t0 ) = exp
Z
L
U (x, t0 , )dx
L
Consider a closed rectangular curve in 2 dimensional space-time symmetric with respect
to the time axis as in figure 6. From the superposition property (4.2) combined with the
consequence of no curvature (4.3) gives.
x
(t1 , x2 )
(t2 , x2 )
t
(t1 , x1 )
(t2 , x1 )
Figure 6. Closed rectangular curve in (1 + 1)-dimensional spacetime symmetric with respect to
the time axis.
S
1
TL 1 (t2 )S+ TL (t1 ) = I
(4.6)
Z
(4.7)
Where
S± ( , t1 , t2 ) = exp
t2
V (±L, t, )dt
t1
Using the quasi-periodic condition (4.4) on S± shows that S+ is conjugate to S .
S+ = Q
1
(✓)S Q(✓)
We can therefore rewrite equation (4.6) as.
S
1
TL 1 (t2 )S+ TL (t1 ) = I , TL 1 (t2 )S+ TL (t1 ) = S
, S+ TL (t1 ) = TL (t2 )S = TL (t2 )Q(✓)S+ Q
1
(✓)
1
, TL ( , t2 )Q(✓) = S+ (t1 , t2 )TL ( , t1 )Q(✓)S+ (t1 , t2 )
Which in turn says that TL ( , t2 )Q(✓) is conjugate to TL ( , t1 )Q(✓). This implies that the
traces of the transports are equal.
tr TL ( , t2 )Q(✓) = tr TL ( , t1 )Q(✓)
– 34 –
This means that the traces are time independent and we have found our integrals of motion.
The functional FL is hence a generation function for infinitely many integrals of motion.
FL ( ) = tr TL ( )Q(✓)
4.2
An interesting duality for the sine-Gordon model
A duality in physics is when a physical phenomena can be described by two different models without loss of information when switching between the two models. In a paper by
Coleman [10] a duality between the massive Thirring model and the sine-Gordon model
was discovered. I will begin by briefly go through the basics of the massive Thirring model
and then continue with describing the duality with the sine-Gordon model. The massless Thirring field is a one dimensional Dirac field defining the massless Thirring model
Lagrangian density as.
L = ¯i
µ@
1
gjµ j µ
2
µ
With the current defined as.
jµ = ¯
where g is a coupling constant and
are defined as follows.
0
1
0
10 0 0
0
B0 1 0 0 C
B0
B
C
B
0
=B
C, 1 = B
@0 0 1 0 A
@0
00 0
1
1
µ
0
0
1
0
µ
are the Dirac matrices. The Dirac gamma matrices
1
1
0C
C
C,
0A
0
0
1
0
0
2
0
B
B
=B
@
0
0
0
i
The Dirac matrices follows the anti-commutator relation
{
µ
,
⌫
}=
µ ⌫
+
⌫ µ
0
0
i
0
1
0 i
i 0C
C
C,
0 0A
0 0
3
0
B
B
=B
@
0
0
1
0
0
0
0
1
1
1 0
0 1C
C
C
0 0A
0 0
= 2g µ⌫
Where g µ⌫ is the metric. The fifth Dirac matrix is defined as
5
=i
0 1 2 3
Both the fifth and the zeroth Dirac matrix are hermitian matrices. The hamiltonian density
for the massless Thirring model becomes. From the Noether theorem the Hamiltonian
density can be found.
⇡=
H=
@L
= ¯i 0
@(@0 )
¯i i @ i + 1 gjµ j µ
2
Remember that the latin indices span {1, n} where n counts the total spacetime dimension.
The massive Thirring model is constructed by adding a mass term to the hamiltonian
density for the massless Thirring model.
– 35 –
H!H+M
Where the mass term M is given by.
1
M = m0 (
2
And the eigendensities
±
+
+
)
is given by.
±
= Z ¯(1 ±
5)
Where Z is a cutoff dependent constant. A sine-Gordon breather has the following mass
[5].
"
#
n⇡ 8⇡m2
Mn = 2Mkink sin
2 1 8⇡m2
This expression tells us that.
n<
1
8⇡m2
This means that no breather states can exist if.
4⇡m2
(4.8)
Since it would force n to be zero yielding a zero mass for the breather. If we assume that
the breather is in fact in the weak coupling regime, m2 ⌧ 1 , the ground state mass of the
breather can be expanded as.
"
M1 = m 1
1
6
✓
16m2
◆2
+O
✓
3
m6
◆#
Rewriting equation (4.8) on the following form makes it possible to analyse the coupling
for values slightly below the boundary.
1
4⇡m2
4⇡m2
=
1
1+
⇡
Then the ground state breather mass transforms to.

M1 = Mkink 2
2
+
– 36 –
4 3
+ O( 4 )
⇡
(4.9)
Now consider the mass of a fermion-antifermion bound state in the massive Thirring model.
It has the following mass [5].

g2 +
Mbound = M 2
4g 3
+ O(g 4 )
⇡
(4.10)
The mass of the ground state breather (4.9) has exactly the same mass as a fermionantifermion bound state (4.10) if is equal to g. Suppose that is identified with g. Then
it is possible to express the breather coupling as a function of the fermion-antifermion
coupling and vice versa.
g
✓
m2
◆
4⇡m2
=
=
4⇡ 2 m2
1
1+
⇡,
g
⇡
m2
(g) =
4⇡
1 + ⇡g
For the weak coupling limit this becomes
lim g
m2
!0
lim
g!0
✓
m2
m2
◆
=1
(g) = 4⇡
This suggest the following correspondence between the massive Thirring model and the
sine-Gordon model [5].
1. kink () , fermion
2. anti-kink () ¯, anti-fermion
3. boson
()
¯, fermion-antifermion pair
4. topological charge () fermion number
5
Summary
Solitons are interesting objects with several applications in modern science. Solitons in
quantum field theory can act as a method for the discovery of new physical phenomena. This
text begins with a brief introduction to quantum field theory to give the tools needed for
understanding solitons in quantum field theory and continues with a deeper analysis of finite
energy solution to field equations. Explicit computation of the time-independent non-trivial
finite energy solution for the 4 - and the sine-Gordon field is made using the relation between
the field potential and the position. After finding the time-independent solution the timedependent solution is found by Lorentz transformation of the time-independent. A brief
walkthrough of the interesting quantities of the soliton is made such as the energy density
and the soliton mass. Solitons in a low dimensional quantum field theory is an interesting
– 37 –
tool for theorists for finding new physical phenomena. The soliton in low dimension can
be extended to higher dimensions however the is a limitation for this extension. Derrick’s
theorem states that the spatial dimension can not transgress two if non-trivial finite energy
solutions are to exist. The stability analysis of the kink shows that the kink is stable for
all real eigenfrequencies. The same eigenfrequencies used in the energy eigenvalue for first
order perturbation of the quantized kink. In the examination of soliton interaction it was
visualised the repulsive force in soliton-soliton interaction and the attractive force in the
interaction between a soliton and a anti-soliton and that the solitons are unperturbed in
collisions with other solitons. It was shown breather soliton, a soliton and a anti-soliton
oscillating around a common centre of mass, was identical to the soliton for the soliton antisoliton collision except with an imaginary eigenfrequency. The period of the breather soliton
was also found. The fact that the sine-Gordon model is a no curvature representation was
shown. With a theorem regarding parallel transport it was shown that models that accept
a no curvature representation have infinite integrals of motion. Finally the equivalence
between the sine-Gordon model and the massive Thirring model was presented.
Acknowledgments
Thanks to my project mentor Luigi Tizzano for his patience, his large time investment in
this project, since the autumn of 2015, and his ability to make me work harder. And to
my father, Fredrik Laurell, who helped me understand spatial solitons in non-linear optics.
Additionally thanks to Erik Orvehed Hiltunen for helping me understand math I had not
yet mastered. And of course thanks to my girlfriend Gjertrud Louise Langaas.
References
[1] B. Kath, B. Kath, “Making Waves: Solitons and Their Optical Applications”. SIAM News,
Volume 31, Number 2.
[2] M. E. Peskin and D. V. Schroeder, “An Introduction to quantum field theory,” Reading,
USA: Addison-Wesley (1995) 842 p
[3] H. Goldstein, “Classical Mechanics, third edition”, Columbia, South Carolina, July (2000)
[4] O. Temkin, M.D., “Forerunners of Darwin, 1745-1859”, Johns Hopkins University Press,
(1968)
[5] E. J. Weinberg, “Classical Solutions in Quantum Field Theory", Cambridge University
Press, (2012)
[6] S. Coleman, “Aspects of Symmetry," Cambridge University Press, (1985)
[7] R. Rajaraman, “Solitons and Instantons," Amsterdam, North-Holland, (1987)
[8] V. Caudrelier, “Multisymplectic approach to integrable defects in the sine-Gordon model”,
arXiv:1411.5171, J. Phys. A 48 (2015) 195203
[9] L. D. Faddeev and L. Takhtajan, “Hamiltonian Methods in the Theory of Solitons”,
doi:10.1007/978-3-540-69969-9, (1987)
– 38 –
[10] S. Coleman, “The quantum sine-Gordon equation as the massive Thirring model”, Phys. Rev.
D 11, 2088 (1975).
– 39 –