Getting Over the Ideal Gas with Quantum Field Theory
1. RHODES UNIVERSITY
Department of Physics and Electronics
Honours Project
GETTING OVER THE IDEAL
Author:
Manqoba Hlatshwayo
Supervisor:
Dino Giovanonni
Abstract
In our physical world, it is rare to find systems with non-interacting
constituents yet when we do physics we often, to the first order approximation,
model systems assuming there are no interactions (ideal system). A good
theory must be able to provide a simple transition between the two systems.
In this project we will consider a quark-gluon plasma system, which we believe
describes the state of our early universe, and calculate the pressure of its gluon
gas using two theories; Statistical Mechanics and Thermal Field Theory. We
will find that both theories, using different mathematical tools, give the same
prediction for the ideal gas case but field theory is more favorable because it
provides an intuitively simple way to incorporate interactions.
October 27, 2015
3. CONTENTS 2
B Derivation of Feynman Path Integral 33
C Field Equations of Motion 37
D Field Wick Rotation 39
4. Chapter 1
Introduction
1.1 Overview
When ever one learns a superseding theory, one cannot resist the question that
do our teachers lie to us intentionally or do they not know better themselves?
Over 200 years physicists have been developing a theory of heat, perhaps to
have better cooked meals. We first had the theory of Thermodynamics which
explained heat from a macroscopic view guided by experiments. This brilliant
theory helps us to solve many thermal problems but it cannot account for the
origin of heat. This led us to the development of the theory of Statistical me-
chanics which explains the emergence of heat from a microscopic phenomena. It
considers the mechanical energy of each particle of the system and uses laws of
mechanics to describe the observed thermal properties. In most of the problems
we have encountered in undergraduate physics, we assume that the particles do
not interact with each other or their collisions are elastic such as an ideal gas.
If we waive this assumption, problems of interacting particles become difficult
to solve in the context of statistical mechanics. Motivated by this quest to find
a simpler method of dealing with interactions, in this project we will explore
tools from advanced superseding thermal theories and show how they can help
us “get over the ideal“.
In particle physics the hydrons (e.g protons and neutrons) are made of small
fundamental particles called quarks and they are held together by the strong
nuclear force which is mediated by 8 massless gauge bosons called gluons. In
analogy, gluons can be thought to be like the photon which mediates the elec-
tromagnetic force. Quarks carry colour which in analogy can be thought to be
the “charge” of the strong force. There are three colour charges for the strong
force; red, blue, and green. We know from experiments that quarks only exist
in colourless combinations of either quark and anti-quark pairs (q¯q - Mesons) or
three quarks (qqq - Baryons). This phenomena is called quark confinement and
so far we have no theoretical reason why this must always be true [2]. However,
if we consider the hydrons at extremely high densities or high temperatures,
the bond between the quarks is broken and asymptotically free quarks form
a quark-gluon plasma (QGP). This process is called quark deconfinement[2]
and we believe that this was the state of our early universe.
3
5. CHAPTER 1. INTRODUCTION 4
Figure 1.1: Formation of quark-gluon plasma [1]
In this project we are going to calculate the pressure of the gluon
plasma ( which is important in understanding the thermodynamics of the QGP)
using two methods. We will assume the gas is ideal, that is there are no inter-
actions between the particles, then use statistical mechanics and thermal field
theory to calculate the pressure. The goal is to demonstrate how the two the-
ories, using different set of mathematical tools, predict exactly the same value.
This is amazing! How does this happen? The connection is through a very im-
portant function in thermal physics, called the partition function, which encodes
all the thermal information we need [3].
Figure 1.2: Overview of the project
Since field theory is a conceptually technical subject, we choose to do the our
calculations using quantum mechanics path integral formalism and exploit the
6. CHAPTER 1. INTRODUCTION 5
fact that there are identical mathematical structures between the two theories.
1.2 Dirac Notation
The key insight of statistical mechanics is that if we consider the microscopic
view we can describe the origin of thermodynamics from normal laws of me-
chanics, we do not need to create new set of laws. Since statistical mechanics is
a quantum theory, we will briefly discuss the basics of Dirac’s notation used in
this report.
From quantum mechanics, we know that given any system, we can always impose
a wave function ψ that represents the probability amplitude of all the possible
states of the system. The evolution of the system is govern by Schrondiger’s
equation
− 2
2m
2
ψ + V ψ = i
∂ψ
∂t
(1.1)
which is simply an eigenvalue equation of the form
ˆHψ = Enψ (1.2)
where ˆH is the Hamiltonian and En is the quantized energy eigenvalues. In a
nutshell quantum mechanics is concerned with solving the eigenvalue problem
given a Hamiltonian. Dirac suggested a slightly different way of describing
this problem. Instead of only having a wave function that represents all the
possible states of the system he introduced state vectors to represent each state
of the system. These state vectors live in the Hilbert space, infinite dimensional
complex vector space. The wave function at a particular time now becomes an
inner product between two state vectors, interpreted as the projection of initial
state |αi to the final state αf |. The vector |αi , called the “ket”, live in the
vector space and vector αf |, called the “bra”, live in the dual vector space.
One can think of the bra as a row matrix and the ket as a column matrix such
that the inner product αf |αi gives a scalar number, say a ∈ C. It is important
to note that these are just representations, there is no real physical difference
between the state vector |β and β|. The role of operators are to transform
one state into another and base vectors form a representation of the system.
In particular, consider the time evolution of the state ket |α, t in the position
x-representation. The time dependent wave function is given by
ψ(x, t) = x|α, t (1.3)
where x| is a time-independent eigenbra with eigenvalue x and Schrondiger’s
equation is written as [5]
x| ˆH|α, t = i
∂
∂t
x|α, t . (1.4)
This is called Dirac’s “bra-ket” notation from the way the inner product is
written. This new representation has the advantage of being able use the tools
of linear algebra to solve the eigenvalue problem. Given any Hamiltonian the
problem is reduced to finding base kets that diagonalizes the matrix x| ˆH|α, t ,
therefore trivially give the eigenvalues. For most simple problems the position
7. CHAPTER 1. INTRODUCTION 6
|x or momentum |p state vectors are used as base kets depending on the
Hamiltonian.
Now consider a simple harmonic oscillator, with the Hamiltonian given by [5]
ˆH =
ˆp2
2m
+
1
2
mω2
ˆx2
. (1.5)
It turns out that the natural choice of position or momentum basis makes diag-
onalizing this Hamiltonian very difficult. To solve this problem we choose the
number state vectors |n as the basis, where n is the number of particles in the
system. The motivation1
for this choice is as follows [5]:
1. Define new non-Hermitian operators
• ˆa = mω
2 ˆx + iˆp
mω - annihilation operator
• ˆa†
= mω
2 ˆx − iˆp
mω - creation operator
• The annihilation operator destroys particles from the system
• The creation operator creates new particles into the system
2. The operators have the following properties
• [ˆa, ˆa†
] = 1 - the operators do not commute
• ˆa |n =
√
n |n − 1 - changes state of system to one less particle
• ˆa†
|n =
√
n + 1 |n + 1 - changes state of system to one more particle
• ˆa |0 = 0 - kills the vacuum state
• ˆa†
|0 = |1 - creates new particle from the vacuum
• |n = (ˆa†
)n
√
n!
|0
3. Define number operator ˆN = ˆa†
ˆa with the following properties
• ˆN, ˆa = −ˆa - the operators do not commute
• ˆN, ˆa†
= ˆa†
- the operators do not commute
•
n
n|n = 1 - completeness relation
• ˆN |n = n |n - eigenvalue equation, produces observable n
• ˆN = mω
2 ˆx2
+ ˆp2
m2ω2 + i
2 [ˆx, ˆp] =
ˆH
ω − 1
2
• ˆH = ω ˆN + 1
2 - ˆH and ˆN can simultaneously be diagonalized
We note that from the properties of ˆN it follows that
ˆH |n = ω n +
1
2
|n , (1.6)
therefore the matrix n| ˆH|n is diagonalized with eigenvalues En = ω n + 1
2 .
1This is actually a sketch of the motivation, full details are covered in Modern Quantum
Mechanics book by J.J Sakurai and Jim Napolitano
8. Chapter 2
Statistical Mechanics
Approach
2.1 Introduction
Think of a system of a container filled with a gas and you would like to study its
thermal properties. Let us say you want to know the pressure exerted by the gas
on the walls of the container, believe it or not this is a non-trivial problem. The
pressure of the system will depend on the properties of the gas, which intuitively
should make sense.
Figure 2.1: Isolated system with gas particles inside
We can make many copies of this system and measure the average pressure.
The formation of such many idealized copies that represent possible states of the
system is what J.W Gibbs defined as thermodynamic ensemble[3]. He identified
three principal ensembles:
1. Microcanonical ensemble: each copy of the ensemble is required to have the
same fixed total energy and particle number. This ensemble is useful for
describing totally isolated systems (unable to exchange energy or particles
with its environment) in order to stay in statistical equilibrium.
2. Canonical ensemble: for each copy of the ensemble the energy can vary
but the number of particles and temperature is fixed. This ensemble is
7
9. CHAPTER 2. STATISTICAL MECHANICS APPROACH 8
useful for describing a closed system (unable to exchange particles with
its environment) which is in, or has been in, weak thermal contact with a
heat bath in order to be in statistical equilibrium.
3. Grand canonical ensemble: for this ensemble neither the energy nor par-
ticle number are fixed. In their place, the temperature and chemical po-
tential are specified and kept fixed for each copy of the ensemble. This
ensemble is appropriate for describing an open system: one which is in,
or has been in, weak contact with a reservoir (thermal contact, chemical
contact, radiative contact, electrical contact, etc.). The ensemble remains
in statistical equilibrium if the system comes into weak contact with other
systems that are described by ensembles with the same temperature and
chemical potential.
The way to calculate the pressure of the system, will depend on the ensemble
and properties of the gas. We are interested in finding the pressure of the gluon
gas, thus we need choose an appropriate ensemble and identify key properties
of the gas. Since fluctuations of the vacuum energy can create and annihilate
particles, the gas is an open system described by the grand canonical ensemble
and the chemical potential can be ignore [4]. However, we will keep the chemical
potential only in this chapter just for calculation purposes but will then set it
to zero to get the desired result. A gluon is a boson, it has spin 1 therefore
follows the Bose-Einstein distribution, and it has zero mass [2]. Since the total
energy of the gas particles is due to its kinetic energy and a symmetry potential,
this system can be modeled by a simple harmonic oscillator. Our general goal
will be to determine the associated partition function of the system using the
different methods.
2.2 The Grand Potential
In this section, we will begin by deriving the partition function using statistical
mechanics and then determine the associated free energy, the grand potential.
We will later use the grand potential to extract useful thermal properties; pres-
sure, entropy density, number density and the equation of state. The general
partition function for the grand canonical ensemble is define to be [5]
Z = Tr e−β ˆH
(2.1)
where β = 1
kBT , where T is the temperature kB is Boltzmann constant. Using
the Hamiltonian for the simple harmonic oscillator, equation (1.6), and the
number basis into equation (2.1) we get
Z =
n
n| e− ωβ( 1
2 + ˆN)
|n
=
n
n| e− ωβ( 1
2 +n)
|n
=
n
e− ωβ( 1
2 +n)
n|n
= e− 1
2 ωβ
n
e− ωβn
.
(2.2)
10. CHAPTER 2. STATISTICAL MECHANICS APPROACH 9
We consider the asymptotic limit T → ∞ , which means β 1. Let x = e− ωβ
and since β is small we have x 1, therefore we can safely use the geometric
series formula for equation (2.2) such that the sum is given by
n
xn
=
1
1 − x
=
1
1 − e− ωβ
.
(2.3)
Substituting (2.3) into (2.2) we get the canonical partition function written as
Z =
e− 1
2 ωβ
1 − e− ωβ
. (2.4)
The partition function is related to the free energy Ω by [3]
Z = e−βΩ
, (2.5)
hence using equation (2.4) we can write the grand potential as
Ω = −
1
β
ln Z
= −T −
ω
2T
+ ln
1
1 − e− ωβ
=
ω
2
+ T ln(1 − e− ωβ
).
(2.6)
We drop the zero point energy term Ω0 = ω
2 , because it does not affect the
dynamics of the system. We take Ω0 as reference point which we a free to
change as the physics of the problem is encoded on the fluctuations around this
reference point. We can the write then grand potential as
Ω = T ln(1 − e− ωβ
). (2.7)
We note that in the exponent we have E = ω and the relativistic dispersion
relation is given by
E2
= k2
+ m2
(2.8)
where k is the momentum and m the mass in units where the speed of light
c = 1. Since a gluon is massless, we then have E = k = ω. Substituting this
result in equation (2.7), then summing over all possible momenta in a given
volume V , and taking account of the effect of the chemical potential µ, we
therefore have the grand potential to be given by
Ω = V T
∞
0
d3
k
(2π)3
ln 1 − e−β(k−µ)
(2.9)
We choose to drop the dash and denote this grand potential by Ω. Since the
grand potential is a function of T, V , and µ, that is Ω = Ω(T, V, µ), we have its
differential to be given by
dΩ =
∂Ω
∂T
dT +
∂Ω
∂V
dV +
∂Ω
∂µ
dµ. (2.10)
11. CHAPTER 2. STATISTICAL MECHANICS APPROACH 10
The grand potential is defined to be [3]
Ω = −ST − PV − Nµ (2.11)
therefore from this definition we can write its differential to be
dΩ = −SdT − PdV − Ndµ. (2.12)
Comparing equations (2.10) with (2.12) we can immediately deduce the follow-
ing relations
P = −
∂Ω
∂V
(2.13)
S = −
∂Ω
∂T
(2.14)
N = −
∂Ω
∂µ
(2.15)
for the pressure P, entropy S and particle number N.
2.3 Massless Bose Gas at µ = 0
In this section we will compute the thermal properties of the gas at non-zero
chemical potential.
2.3.1 Pressure
We are mainly interested in the pressure of the gas, therefore we begin by
evaluating it. Using equation (2.9) and (2.13) we have the pressure to be given
by
P = −T
∞
0
dk
2π2
k2
ln(1 − e−β(k−µ)
). (2.16)
We evaluate this integral using integration by parts by letting dv
dk = k2
, and
u = ln(1 − e−β(k−µ)
) then
v = dk
dv
dk
= dk k2
=
1
3
k3
,
and
du
dk
=
βe−β(k−µ)
1 − e−β(k−µ)
=
β
eβ(k−µ) − 1
,
(2.17)
such that
P = [uv]∞
0 −
∞
0
v
du
dk
dk
= −
T
6π2
ln(1 − e−β(k−µ)
)k3
∞
0
+
T
6π2β2
∞
0
dk
(βk)3
eβ(k−µ) − 1
.
(2.18)
12. CHAPTER 2. STATISTICAL MECHANICS APPROACH 11
If we assume that µ is finite then the first term becomes
lim
k→∞
ln(1 − e−β(k−µ)
)k3
− ln(1 − e−β(0−µ)
).0 = ln(1).k3
= 0.
(2.19)
To solve for the second term, we make the substitution x = βk then dx = βdk,
such that the pressure is given by
P =
T
6π2β3
∞
0
dx
x3
e−β(k−µ)ex − 1
. (2.20)
This integral is of the form (A.2)
Lis(z)Γ(s) =
∞
0
dx
xs−1
z−1ex − 1
. (2.21)
where Lis is a polylogarithm function1
and Γ(s) = (s − 1)! for s ∈ N is the
gamma function. Comparing equations (2.20) with (2.21) we see that z = eβµ
and s = 4 therefore the pressure can be written as
P =
1
π2
Li4(eβµ
)T4
. (2.22)
2.3.2 Entropy density
In this subsection we will calculate the entropy density of the gas, which will be
used to determine the equation of state. Using equations (2.14) and (2.9) the
entropy is given by
S = −V
∞
0
dk
2π2
k2
ln(1 − e−β(k−µ)
) − V T
∞
0
dk
2π2
k2 (µ − k)e−β(k−µ)
1 − e−β(k−µ)
. (2.23)
Let S1 and S2 represent the first and second term of equation (2.23) such that
the entropy is S = S1 +S2. The integral S1 is similar to the integral of equation
(2.16) of the pressure up to a factor of V
T hence S1 is given by
S1 =
V
π2
Li4(eβµ
)T3
(2.24)
To match polylogarithm functions for S2 we make the substitution x = βk such
that
S2 =
V
2π2T
∞
0
dk
k2
(k − µ)
eβ(k−µ) − 1
=
V
2π2Tβ3
∞
0
dk
(βk)3
− µβ(βk)2
e−βµeβk − 1
=
V
2π2Tβ4
∞
0
dx
x3
− µβx2
e−βµex − 1
=
V T3
2π2
Γ(4) Li4(eβµ
) −
V µT2
2π2
Γ(3) Li3(eβµ
)
=
3V
π2
Li4(eβµ
)T3
−
V µ
π2
Li3(eβµ
)T2
(2.25)
1Polylogarithm functions are standard functions in quantum statistics, see Appendix A to
read more about them.
13. CHAPTER 2. STATISTICAL MECHANICS APPROACH 12
Taking the entropy density s = S1+S2
V we get
s =
1
π2
4 Li4(eβµ
)T3
− µ Li3(eβµ
)T2
(2.26)
2.3.3 Particle density
We apply a similar procedure as in entropy density to calculate the particle
density. This quantity will be used to determine the equation of state of the
system. Using equations (2.15) and (2.9) we have the number density to be
given
n =
N
V
=
Tβ
2π2
∞
0
dk
k2
e−β(k−µ)
1 − e−β(k−µ)
=
T
2πβ
∞
0
dk
(βk)2
e−βµeβk − 1
=
T
2πβ2
∞
0
dx
x2
e−βµex − 1
=
1
2π2
Γ(3) Li3(eβµ
)T3
(2.27)
where Γ(3) = 2, hence
n =
1
π2
Li3(eβµ
)T3
. (2.28)
2.4 Massless Bose Gas at µ = 0
Since the chemical potential have negligable effects in the system, we set it to
zero which simplifies the polylogarithm functions in equations (2.22), (2.26),
and (2.28). We note that from equation (2.16) that Ω = −PV , hence the grand
potential is given by
Ω = −
V T
6π2β3
Γ(4)ζ(4)
= −
π2
90
V T4
(2.29)
where in the last line we have substituted2
ζ(4) = π4
90 and β = 1
T . Using
equations (2.13) and (2.29) we get the pressure to be
P =
π2
90
T4
(2.30)
which can also be found by substituting for µ = 0 into equation (2.22). Similarly
we can substitute for µ = 0 into equation (2.26) to get entropy density
s =
4π2
90
T3
. (2.31)
2ζ(s) is the Reimann Zeta function, see Appendix A on polylogarithm functions to read
more about it.
14. CHAPTER 2. STATISTICAL MECHANICS APPROACH 13
and into equation (2.28) to get the particle density
n =
1
π2
ζ(3)T3
. (2.32)
We have used statistical mechanics to get the thermal properties of the gluon
gas; pressure, entropy and particle density functions. In the next section we
will relate all the state variables of this problem into one equation called the
equation of state.
2.5 Equation of state
In thermodynamics the total energy of a system is split into mechanical work,
heat, and internal energy given by [3]
E = −PV + ST + µN. (2.33)
The total energy per unit volume of the system is given by
ε = −P + sT + µn (2.34)
and since this equation must hold for all cases, we substitute the values of P, s,
and n at µ = 0 given by equations (2.22), (2.26), and (2.28) respectively to get
ε =
3
π2
Li4(eβµ
)T4
. (2.35)
Comparing this energy density with the pressure given by equation (2.22), we
can write ε as
ε = 3P. (2.36)
This is an interesting result because it was not so obvious that the energy den-
sity of this system will only depends on the pressure. This result justifies why
we were more interested in calculating the pressure of the system than other
state variables. The physical significance of this equation of state is that it
shows us that the gluon gas behaves as ultra-relativistic particles and under lo-
cal thermodynamic equilibrium conditions it is equivalent to Plank’s blackbody
radiation [8] of photons.
15. CHAPTER 2. STATISTICAL MECHANICS APPROACH 14
2.6 Summary
In this chapter, we started with the grand canonical ensemble of a bose gas,
determined the associated partition function, and then extracted from it all
the thermal quantities we needed. This was to demonstrate that the partition
function is all that we need for any thermal problem. The following table is a
summary of all the key results from this chapter.
partition function Z = e− 1
2
ωβ
1−e− ωβ
grand potential Ω = V T
∞
0
d3
k
(2π)3 ln 1 − e−β(k−µ)
µ = 0 P = 1
π2 Li4(eβµ
)T4
s = 1
π2 4 Li4(eβµ
)T3
− µ Li3(eβµ
)T2
n = 1
π2 Li3(eβµ
)T3
µ = 0 P = π2
90 T4
s = 4π2
90 T3
n = 1
π2 ζ(3)T3
equation of state ε = 3P
In the following chapters our goal will be to use thermal field theory methods
to arrive at the same partition function or grand potential. We will see that it
is going to take a bit more work to get the partition function using the path
integral formalism than we have done in this chapter using statistical mechanics.
The advantage of path integral formalism is that we can systematically add an
interactions term and use perturbation theory to get higher order corrections to
the ideal case.
16. Chapter 3
Quantum Mechanics
Approach
3.1 Introduction
The great physicist Richard Feynman proposed a different way of looking at the
problems of quantum mechanics. Consider the following double slit experiment
using an electron.
Figure 3.1: Double slit experiment
We know that the electron forms an interference pattern on the screen and
to calculate the amplitude A at a certain point O in the screen, you would have
to (using superposition postulate of quantum mechanics) add the amplitude A1
for the electron to pass through the first slit then arrive at O and the amplitude
A2 for the electron to pass through second slit then arrive at O. There is an
apocryphal story, “The professor’s nightmare: wise guy in class” told by A.
Zee in his book Quantum Field Theory in a Nutshell which gives insights on
how Feynman extended this idea. In the story Feynman supposedly asked the
15
17. CHAPTER 3. QUANTUM MECHANICS APPROACH 16
following questions : what if you had
• three holes in the slit?
• infinitely many holes in the slit such that the slit wasn’t there?
• second screen behind the first screen which has two holes drilled on it?
• infinitely many screens of infinitesimal width and infinitely many holes,
stacked between the slit and final screen?
Clearly for all cases, to calculate the amplitude detected at point O, you would
sum all the amplitudes of all the possible paths the electron would take from
the gun to the point O, which is given by
A =
paths
Ap. (3.1)
Feynman proposed that even without the plate and screens, to find the proba-
bility for a quantum particle to move from point A to point B you needed to sum
over all the possible paths the electron can take to go from point A to B. That
is what is called Feynman path integral, the formal mathematical derivation of
this object is covered in Appendix B.
3.2 Partition Function
The partition function carries information about all the possible states of the
system, hence for finite temperatures it can be represented by a path integral
given by [11]
Z = dx x|e−β ˆH
|x . (3.2)
We can follow a similar procedure as in the Appendix B to derive path integral
partition function associated with the grand potential, and the result is given
by [12]
Z = C Dx[t] exp
i
dt
m
2
dx(t)
dt
2
− V (x(t)) (3.3)
where C is a constant, independent of the properties of the potential V (x(t)),
given by
C = exp
N
2
ln
mN
2π 2β
. (3.4)
Note that the exponent of equation (3.3) is complex whilst our usual partition
function from statistical mechanics is real. To solve this problem we make the
change of variables t → τ = it (Wick’s rotation) such that the partition function
becomes
Z = C Dx[τ] exp −
1
dτ
m
2
dx(τ)
dτ
2
+ V (x(τ)) (3.5)
where 0 < τ < β . This is the key insight that connects statistical mechanics
with the path integral formalism by making the identification
β = it. (3.6)
18. CHAPTER 3. QUANTUM MECHANICS APPROACH 17
Our goal is to evaluate equation (3.5) for the harmonic oscillator and show
that it is equal to the canonical partition function found in chapter 2, given by
equation (2.4).
3.3 Harmonic Oscillator
We will carry out this problem in the Fourier space with respect to the time
coordinate τ rather than in the configuration space and gather information
about the constant C without making use of its actual value in equation (3.5).
3.3.1 Fourier Representation
Let x(τ) be an arbitrary function, such that x(β ) = x(0) for 0 < τ < β , be
represented by the Fourier sum
x(τ) ≡ T
∞
n=−∞
xneiωnτ
(3.7)
where the factor T is a convention.
Periodic:
Using the periodicity of the Fourier function, x(β ) = x(0), we have the condi-
tion
eiβ ωn
= 1 (3.8)
which gives the values ωn = 2πTn/ for all n ∈ Z, and they are called the
Matsubara frequencies.
Real:
We also impose reality on x(τ) such that
x(τ) ∈ R ⇒ x∗
(τ) = x(τ) ⇒ x∗
n = x−n. (3.9)
Let xn = an + ibn then it follows from (3.9) that a−n = an and b−n = −bn. In
particular, b0 = 0 and x−nxn = a2
n + b2
n, thereby we have the representation
x(τ) = T a0 +
∞
n=1
(an + ibn)eiωnτ
+ (an − ibn)e−iωnτ
(3.10)
where a0 is called the amplitude of the Matsubara zero-mode.
3.3.2 Quadratic Forms
In this section we will evaluate the quadratic structures in configuration space
using the representation of equation (3.7). The general quadratic can be written
as
1 β
0
dτx(τ)y(τ) = T2
n,m
xnym
1 β
0
dτei(ωn+ωm)
. (3.11)
19. CHAPTER 3. QUANTUM MECHANICS APPROACH 18
Note that
1 β
0
dτ ei(ωn+ωm)
=
1
T δn,−m if n = −m
0 otherwise.
(3.12)
Therefore using this result (3.12), we can write equation (3.11) as
1 β
0
dτx(τ)y(τ) = T
n
xny−n. (3.13)
Let the argument of the exponential of equation (3.5) be given by
Q = −
1 β
0
dτ
m
2
dx(τ)
dτ
dx(τ)
dτ
+ ω2
x(τ)x(τ) (3.14)
where we have made the substitution V (x(τ)) = ω2
x(τ)x(τ) for a harmonic
oscillator. Using the result of (3.13) we have
Q = −
mT
2
∞
n=−∞
iωniω−n + ω2
xnx−n
= −
mT
2
∞
n=−∞
(ω2
n + ω2
)(a2
n + b2
n).
(3.15)
We simplify equation (3.15) by substituting for ω0 = 0 and b0 = 0 to get
Q = −
mT
2
ω2
a2
0 − mT
∞
n=1
(ω2
n + ω2
)(a2
n + b2
n). (3.16)
3.3.3 Integration Measure
In the previous section we have basically made a change of variables from x(τ)
for τ ∈ (0, β ) to the independent Fourier components {a0, an, bn} for n ≥ 1.
This change of variables introduces a Jacobian determinate in the integration
measure such that
Dx[τ] = det
δx(τ)
δxn
da0
n≥1
dandbn
. (3.17)
This change of bases is independent of the potential V (x(τ)) thus we can define
C ≡ C det
δx(τ)
δxn
(3.18)
as the unknown coefficient to be determined.
20. CHAPTER 3. QUANTUM MECHANICS APPROACH 19
3.3.4 Gaussian Integral
Using the results of the argument of exponential (3.16) and the integral measure
(3.17), we can write the partition function from equation (3.5) as
Z = C
∞
−∞
da0
∞
−∞
n≥1
dandbn
exp
−
mT
2
ω2
a2
0 − mT
n≥1
(ω2
n + ω2
)(a2
n + b2
n)
= C
∞
−∞
da0 e− mT
2 ω2
a2
0
∞
−∞
n≥1
dandbn
exp
−mT
n≥1
(ω2
n + ω2
)(a2
n + b2
n)
.
(3.19)
We make use of the Gaussian integral
∞
−∞
dx e−kx2
=
π
k
(3.20)
to evaluate the partition function of (3.19) to be given by
Z = C
2π
mTω2
∞
n=1
π
mT(ω2
n + ω2)
. (3.21)
Now all that remains to do is to determine the value of C and show that this
partition function is equal to the one found in chapter 2.
3.3.5 Regulator for C’
We will determine the value of C by investigating its properties. We note the
following:
1. Since C is independent of ω, therefore we can consider a particular value
of ω that simplifies the system. We determine C in the limit ω → 0.
2. However, in this limit the integral over the zero-mode a0 in equation (3.21)
is divergent. We call such a divergent an infrared divergence since the zero-
mode is the lowest energy mode.
3. We can get around this divergence problem and still be able to take ω → 0
if we momentarily regulate the integral over the zero-mode in some other
way.
4. We note that from the representation (3.10) that
1
β
β
0
dτx(τ) = Ta0, (3.22)
so that Ta0 represents the average value of x(τ).
5. We associate this average value with the “boundary conditions” of x over
which we integrate in equation (3.2). This allows us to regulate the system
by “putting it in a box”.
21. CHAPTER 3. QUANTUM MECHANICS APPROACH 20
6. This can be done by restricting the values of x to some (asymptotically
wide but finite) interval ∆x, and those of a0 to the interval ∆x/T.
Now with this setup we can proceed to match for C in the “effective theory
computation” with “full theory computation”.
Side A: “effective theory computation”
In the presence of the regulator equation (3.21) becomes
lim
ω→0
Zregulated = C
∆x/T
da0
∞
−∞
n≥1
dandbn
exp
−mT
n≥1
ω2
n(a2
n + b2
n)
.
= C
∆x
T
∞
n=1
π
mTω2
n
(3.23)
Side B: “full theory computation”
In the presence of a regulator and in the absence of a potential V (x), equation
(3.2) can be computed in the following way
lim
ω→0
Zregulated =
∆x
dx x|e− ˆp2
2mT |x
=
∆x
dx
∞
−∞
dp
2π
x|e− ˆp2
2mT |p p|x
=
∆x
dx
∞
−∞
dp
2π
e− p2
2mT x|p p|x
=
∆x
dx
∞
−∞
dp
2π
e− p2
2mT e
i
px
e− i
px
= ∆x
1
2π
√
2πmT.
(3.24)
Matching the two sides, equating (3.23) and (3.24), we get the coefficient to be
given by
C =
T
2π
√
2πmT
∞
n=1
mTω2
n
π
, (3.25)
where the regulator ∆x has dropped out.
22. CHAPTER 3. QUANTUM MECHANICS APPROACH 21
3.3.6 Final Result
We can now insert the value of C in equation (3.21) to get
Z =
T
2π
√
2πmT
∞
n=1
mTω2
n
π
2π
mTω2
∞
n=1
π
mT(ω2
n + ω2)
=
T
ω
∞
n=1
ω2
n
ω2
n + ω2
=
T
ω
∞
n=1
1
1 + ( ω
ωn
)2
=
T
ω
1
∞
n=1 1 + (ω /2πT )2
n2
(3.26)
where in the last line we have made the substitution for ωn = 2πnT/ . We
make use of
sinh(πx)
πx
=
∞
n=1
1 +
x2
n2
(3.27)
where x = ω /2πT in equation (3.26) to get the result
Z =
1
2 sinh ω
2T
. (3.28)
Using the hyperbolic function
sinh(x) =
ex
− e−x
2
(3.29)
we can simplify equation (3.28) to give
Z =
e− 1
2 ωβ
1 − e− ωβ
(3.30)
which is the same canonical partition function found in chapter 2. The following
chapter on thermal field theory will make use of the results found in this chapter
but will have a different interpretation.
23. CHAPTER 3. QUANTUM MECHANICS APPROACH 22
3.4 Summary
In this chapter we went through longer and a bit technical steps. We are going to
summarize the key results and concepts as they will have a similar mathematical
structure with the following field theory chapter.
Path integral A = paths Ap.
Path integral partition function Z = dx x|e−β ˆH
|x
Wick rotation t → τ = it exp −1 β
0
dτ m
2
dx(τ)
dτ
2
+ V (x(τ))
Fourier representation x(τ) = T a0 +
∞
n=1 (an + ibn)eiωnτ
+ (an − ibn)e−iωnτ
Matsubara frequencies ωn = 2πTn/
Quadratic forms 1 β
0
dτx(τ)y(τ) = T n xny−n
Integration measure Dx[τ] = C da0 n≥1 dandbn
Gaussian integration Z = C 2π
mT ω2
∞
n=1
π
mT (ω2
n+ω2)
Regulator for C’ C = T
2π
√
2πmT
∞
n=1
mT ω2
n
π
Final result Z = e− 1
2
ωβ
1−e− ωβ
24. Chapter 4
Field Theory Approach
4.1 Introduction
Quantum field theory (QFT) is a theory that combines quantum mechanics and
special relativity. In a broad sense it is concerned with small and fast (velocity
close to speed of light) moving particles. Before we talk about quantum field
theory, let us list key the concepts that we need from the theories to be merged:
Quantum mechanics
• Physical observables are mathematical operators in the theory.
• Heisenberg uncertainty principle, which forbids knowing two complimen-
tary variables of a system up to any precision.
• Commutation relation, in particular [ˆx, ˆp] = i
• Dynamics of a system is governed by Schrondiger’s wave equation
−
2
2m
∂2
ψ
∂x2
+ V ψ = i
∂ψ
∂t
(4.1)
Special relativity
• E = mc2
implies if we have enough energy that is proportional to the
particle’s mass then we can “create” a particle. We actually need twice
the particle’s mass, so to create a particle and its anti-particle because of
conservation laws.
• From the first point, this means the particle number is not fixed and the
type of particles present are not fixed.
The last fact of special relativity is in direct conflict with non-relativistic quan-
tum mechanics. There have been attempts to resolve this conflict in a relativistic
quantum mechanics by finding a corresponding wave equation to describe the
system. The first wave equation is the Klein-Gordon equation given by [10]
1
c2
∂2
φ
∂t2
−
∂2
φ
∂x2
=
m2
c2
2
φ (4.2)
23
25. CHAPTER 4. FIELD THEORY APPROACH 24
which was first written by Schrondiger but he discarded it because it allowed
negative energy states and appeared to give negative probabilities. The next
wave equation is the Dirac equation given by [10]
i
∂Ψ
∂t
= −i c−→α .
−→
Ψ + βmc2
Ψ (4.3)
where −→α and β are matrices. The problem with these relativistic wave equations
is in their interpretations. We transition to quantum field theory by discarding
the notion that φ and Ψ describe a single particle state and replacing with the
following ideas:
• The functions φ and Ψ are fields instead of wave functions.
• The fields are operators that can create new particles and destroy particles.
• In quantum mechanics time t is a parameter and position ˆx is an operator
but in special relativity time and space are in equal footing, space-time.
We therefore choose to demote position from operator to a parameter x.
• Momentum continues to be an operator as in quantum mechanics.
Generally in quantum field theory, we often use tools from classical mechanics to
deal with fields. Specifically, we use the Lagrangian because symmetries (such
as rotations) leave the form of the Lagrangian invariant. Thermal field theory
follows the same approach and shares the same tools as QFT to solve problems
of systems with finite temperatures.
In this chapter we will not do the full rigorous calculation of the pressure of
a gluon plasma using field theory, because the mathematical structures are
identical to path integral quantum mechanics discussed in chapter 3. We will
give the corresponding results and difference in interpretation. We start by
giving the corresponding commutation relation in field theory [10],
[ˆx(t), ˆp(t)] = i −→ [ ˆϕ(t, x), ˆπ(t, y)] = i δ(x − y) (4.4)
where ˆπ(t, y) is another field that plays the role of momentum. The state pro-
jection inner product is given by [11]
x|p =
1
√
2π
eip.x
−→ ϕ|π =
1
√
2π
ei d3
x ϕ(x)π(x)
(4.5)
where t = 0 and using the units where = 1. The completeness conditions are
given by
dx |x x| = 1 −→ dϕ |ϕ ϕ| = 1
dp
2π
|p p| = 1 −→
dπ(x)
2π
|π π| = 1.
(4.6)
Our goal in this chapter is to determine the associated field theory partition
function to the problem by matching mathematical structures with quantum
mechanics path integral formalism.
26. CHAPTER 4. FIELD THEORY APPROACH 25
4.2 Path Integral Partition
We start by the classical Lagrangian given by
L =
1
2
m ˙x2
− V (x) (4.7)
and re-interpret x as an “internal” degree of freedom φ situated at the origin 0
of a d-dimensional space such that
ˆφ(0) = ˆx. (4.8)
Using this interpretation of x, we can write equation (4.7) as
LQM
=
m
2
∂φ(t, 0)
∂t
− V (φ(t, 0)) (4.9)
where the superscript QM reminds us that this is the same Lagrangian used in
quantum mechanics.
The scalar field theory Lagrangian in d-dimensional space is given by [12]
LF T
M =
1
2
∂µ
φ∂µφ − V (φ) =
1
2
(∂tφ)
2
−
d
i=1
1
2
∂φ
∂xi
2
− V (φ). (4.10)
where in the subscript M reminds us that were are in Minkowski spacetime M4
with metric signature (+ - - -).
We consider the harmonic oscillator in quantum mechanics, then the scalar
field Lagrangian takes the same form as the quantum mechanics Lagrangian
but differs by the extra summation term. We interpret scalar field theory as a
collection of almost independent harmonic oscillators with unitary mass, m = 1,
at every point x. The derivative term (∂iφ)(∂i
φ) is the interaction term that
couples nearest neighboring oscillators:
∂iφ ≈
φ(t, x + ˆi) + φ(t, x)
(4.11)
where ˆi is a unit vector in the direction i. This means oscillators very far apart
in x can be considered independent of each other.
Note that the exponent of the path integral (3.3) only has a time derivative,
which means the extra coupling term will not change the derivation of the
path integral associated with the scalar field. This is because in the derivation
(covered in appendix B) it was only important that the Hamiltonian is quadratic
in the canonical momenta, p = m ˙x ↔ ∂tφ, therefore spatial derivatives play no
significant role. If we let
λ(φ) =
d
i=1
1
2
∂φ
∂xi
2
+ V (φ) (4.12)
then the scalar field Lagrangian can be written as
LF T
M =
1
2
(∂tφ)
2
− λ(φ) (4.13)
27. CHAPTER 4. FIELD THEORY APPROACH 26
which have an identical form as the harmonic oscillator in quantum mechanics.
To transition from the path integral to a scalar field partition function we make
the Wick rotation, t → τ = it 1
, and impose periodicity of the field, φ(0, x) =
φ(β , x) to compute the trace. We can then write the new Lagrangian as
LF T
E = −
1
2
(∂τ φ)
2
+ λ(φ) (4.14)
where the subscript E tells us that were are in the Euclidean space E4
. Compar-
ing with equation (3.5) we can directly write the scalar field partition function
as
ZF T
=
φ(0,x)=φ(β ,x) x
[C Dφ(τ, x)] exp −
1 β
0
dτ dd
x LF T
E (4.15)
Now that we have a partition function written in terms of a path integral, we
will follow the same strategy as in chapter 3 and solve for the integral in Fourier
space.
4.3 Fourier representation
In order to simply the notation, we will use units where = 1 and rewrite the
path integral in Fourier representation of φ given by
φ(τ, x) = g(τ)f(x) (4.16)
where the g(τ) and f(x) are functions containing information about the time
and spatial dependence respectively.
The τ dependence can be expressed as
g(τ) = T
∞
n=−∞
˜g(n)eiωnτ
(4.17)
where after imposing periodicity, 0 < τ < β, we get ωn = 2πTn for n ∈ Z.
For the spatial dependence we need to impose periodicity just like the time
coordinate but cannot do it directly since each spatial direction is allowed to be
infinite. We introduce a regulator by momentarily taking each direction to be
finite, 0 < xi < Li, then dependence on xi can be represented as
f(xi) =
1
Li
∞
ni=−∞
˜f(ni)eikixi
(4.18)
where ki = 2πni/Li and 1/Li plays the same role as T in the time direction. In
the infinite volume limit, Li → ∞, the sum becomes the usual Fourier integral,
1
Li ni
=
1
2π ni
∆ki −→
dki
2π
, (4.19)
1The full steps of the scalar field theory Wick rotation are covered in appendix D
28. CHAPTER 4. FIELD THEORY APPROACH 27
which shows that the finite volume was really an intermediate regulator.
The whole function becomes
φ(τ, x) =
T
V ωn k
˜φ(ωn, k)ei(ωnτ+k.x)
(4.20)
where V = L1L2L3. We also require φ(τ, x) to be real, therefore
˜φ(ωn, k)
∗
= ˜φ(−ωn, −k). (4.21)
Since half of the Fourier-modes are independent (knowing the positive gives the
negative mode), we can choose
˜φ(ωn, k), n ≥ 1; ˜φ(0, k), k1 > 0; ˜φ(0, 0, k2, ...), k2 > 0; ......; and ˜φ(0, 0) (4.22)
as the integration variables, where ˜φ(0, 0) is the zero-mode.
4.4 Quadratic Forms
Following the procedure in chapter 3, we note that from equation (3.13) that
we can write the scalar field quadratic structures as
β
0
dτ d3
xφ1(τ, x)φ2(τ, x) =
T
V ωn k
˜φ1(−ωn, −k)˜φ2(ωn, k). (4.23)
In particular, for a free scalar field the potential is given by V (φ) = 1
2 m2
φ2
[13],
then the exponent in equation (4.15) can be written as
exp(−SE) = exp −
β
0
dτ d3
x LE
= exp −
β
0
dτ d3
x
1
2
(∂µφ)
2
+
1
2
m2
φ2
.
(4.24)
Taking the derivatives with respect to τ and x will give ω2
n + k2
, and then
applying equation (4.23), we get the exponent to be
exp(−SE) = exp −
1
2
T
V ωn k
(ω2
n + k2
+ m2
) ˜φ2(ωn, k)
2
=
k
exp −
T
2V ωn
(ω2
n + k2
+ m2
) ˜φ2(ωn, k)
2
.
(4.25)
This exponent is exactly the same as the one given by equation (3.15) which
can be written as
exp(−SE) = exp −
mT
2
∞
n=−∞
(ω2
n + ω2
)|xn|2
(4.26)
29. CHAPTER 4. FIELD THEORY APPROACH 28
with the replacements
m →
1
V
, ω2
→ k2
+ m2
, |xn|2
= ˜φ2(ωn, k)
2
.
(4.27)
The scalar field partition function factorizes into a product of harmonic oscillator
partition functions, for which we know the answer already.
4.5 The Grand Potential
Using the result of the harmonic oscillator partition function from equation
(3.30), we can write the scalar field partition function as
Z =
k
e− 1
2 βEk
1 − e−βEk
(4.28)
where Ek = k2
+ m2. This partition function looks different but gives the
same physics and to see this, we compute the grand potential given by
Ω = −
1
β
ln
k
e− 1
2 βEk
1 − e−βEk
= −T
k
ln
e− 1
2 βEk
1 − e−βEk
= −T
k
−
Ek
2T
− ln 1 − e−βEk
=
k
Ek
2
+ T ln 1 − e−βEk
.
(4.29)
Note from (4.19) that the sum becomes an integral, therefore the grand potential
can be written as
Ω = V
d3
k
(2π)2
k
2
+ T ln 1 − e−βk
, (4.30)
where we have made the substitution Ek = k since a gluon is massless. This is
exactly the same result we found in chapter 2 using statistical mechanics.
30. CHAPTER 4. FIELD THEORY APPROACH 29
4.6 Summary
In this chapter we started with the relativistic wave equations and interpreted
φ and Ψ as fields instead of wave functions. Taking this field approach, we then
re-interpreted the position variable x in the classical Lagrangian as an “inter-
nal” degree of freedom φ situated at the origin of a d-dimensional space. This
allowed us to deduce the scalar field path integral partition function by match-
ing with the quantum path integral partition function found in chapter 3.
To solve this path integral, we used a Fourier representation for the field φ and
then matched mathematical structures to that of chapter 3 to get the solution.
Unlike chapter 3, we found that the partition function is a product harmonic
oscillator partition functions. We then use this partition function to calculate
the grand potential and found it to be the same as the one found in chapter 2
using statistical mechanics.
31. Chapter 5
Comments and Conclusion
The challenge for a theorist is to ensure that their theory provides a simple tran-
sition between a non-interacting (ideal) to interacting system. In this project we
have explored this theoretical challenge by calculating the pressure of an ideal
gluon gas using two theories; Statistical Mechanics and Thermal Field Theory.
For both theories the aim was to derive the associated partition function for
the system. The partition function of a system is defined to be the sum of all
possible states of the system given by Z = Tr e−β ˆH
. Since β is a parameter
independent of the mechanics of the system, the problem is computing Z for a
given Hamiltonian ˆH of the system. The ideal gluon gas system can be modeled
by the simple harmonic oscillator with the Hamiltonian ˆH = ˆp2
2m + 1
2 mω2
ˆx2
.
In the statistical mechanical approach, we use Dirac’s state vector interpretation
of quantum mechanics to find a basis that diagonalizes the Hamiltonian. This
basis is determined by inspection, we construct two non-Hermitian operators, ˆa†
and ˆa, and then use them to define the particle number basis |n where the the
Hamiltonian can be diagonalized. This makes calculating the trace trivial but
what about interactions? Can we add an interaction term in the Hamiltonian
and determine the associated basis that diagonalizes it? Maybe we can or maybe
we cannot, what matters is that we do not have a systematic way to determine
the basis that diagonalizes the Hamiltonian. When faced with interactions in
statistical mechanics we abandon this mechanism of finding a suitable basis and
turn to other methods like the cluster and virial expansions.
However, for the field theory approach, the theory allows us to add an interac-
tions term and follow a systematic procedure to deal with interactions. How is
this so? Note that the Hamiltonian is related to the Lagrangian by the Legendre
transforms H(p) = p ˙x − L( ˙x) [14], and field theory is formulated using the La-
grangian instead of the Hamiltonian. This make the problem of computing Z for
a given Hamiltonian ˆH be translated into computing Z for a given Lagrangian
L in field theory. We can add an interaction term λφ4
to the Lagrangian and
use perturbation theory to get the pressure of an interacting gluon gas, to the
30
32. CHAPTER 5. COMMENTS AND CONCLUSION 31
the second order approximation, to be given by [11]
P =
π2
90
T4
1 −
15λ
8π2
+ .... (5.1)
This is one of the key distinction between the two theories that makes thermal
field theory more favorable than statistical mechanics.
The thermodynamics of a quark-gluon plasma gives us key insights towards un-
derstanding the state of our early universe. Scientists at the Relativistic Heavy
Ion Collider at Brookhaven National Laboratory created QGP by smashing gold
atoms together, at nearly the speed of light, and at a temperature of about 4
trillion degrees [15]. The team at CERN used the Large Hydron Collider to
smash lead nuclei together and also discovered that early universe was not only
very hot and dense but behaved like a hot liquid [16].
Figure 5.1: A visualization of one of the first full-energy collisions between gold
ions at Brookhaven Lab’s Relativistic Heavy Ion Collider, as captured by the
Solenoidal Tracker At RHIC (STAR) detector .
There is no doubt that our technology is improving which enables us to probe
higher energies to test our theories. We live in the most exciting era of physics
where we have the capacity to answer deep questions about our universe.
33. Appendix A
Polylogarithm Functions
The polylogarithm function, Lis(z), is a special function of order s and argument
z formulated by the mathematician Alfred Jonquiere. In physics they it is mostly
used in quantum statistics and quantum electrodynamics. It is defined as the
infinite sum [9]
Lis(z) =
∞
n=1
zn
ns
. (A.1)
We can plot Lis(z) vs z for some values of s [6]
Figure A.1: plot of polylogarithm function for −3 ≤ s ≤ 3
There is also a integral representation of the functions and is given by [9]
Lis(z) =
1
Γ(s)
∞
0
dx
xs−1
z−1ex − 1
(A.2)
where Γ(s) = (n − 1)! is the gamma for n ∈ N. In the case where z = 1, the
polylogarithm function simplifies to
ζ(s) =
1
Γ(s)
∞
0
dx
xs−1
ex − 1
(A.3)
where for all s ∈ C, ζ(s) is the Reimann Zeta function defined by [9]
ζ(s) =
∞
n=1
1
ns
. (A.4)
32
34. Appendix B
Derivation of Feynman
Path Integral
Consider a system with one particle localized at an initial point at −→ri at t = 0
such that
|Ψ, 0 = |−→ri . (B.1)
The evolution of the system is governed by Schrondiger’s equation
i ∂t |Ψ =
ˆp2
2m
+ V (−→r ) |Ψ . (B.2)
The state of the system at time t is given by |Ψ, t and we are interested in
answering two question about this system:
1. The probability that the particle will propagate from initial position −→ri to
a final position −→rf ?
2. How will the particle get there?
The answers to these questions will lead us to the propagator and Feynman’s
path integral respectively.
The Propagator
The answer to the first question is given by the
Pi→f = | −→rf |Ψ, t |
2
(B.3)
therefore our task is to evaluate the matrix in this equation. Let ˆU(t) be a time
evolution operator such that
|Ψ, t = ˆU(t) |Ψ, 0 (B.4)
with the initial condition ˆU(0) = ˆ1, where ˆ1 stands for the identity operator.
Substituting equation (B.4) into equation (B.2) we get
i
∂
∂t
ˆU(t) = ˆH ˆU(t). (B.5)
33
35. APPENDIX B. DERIVATION OF FEYNMAN PATH INTEGRAL 34
Solving this first order separable differential equation with the given initial con-
dition gives
ˆU(t) = e− i ˆHt
. (B.6)
We define the propagator to be
−→rf |Ψ, t = −→rf | ˆU(t)|Ψ, 0 = −→rf |e− i ˆHt
|−→ri (B.7)
which gives the probability amplitude for the particle to move from initial po-
sition ri to final position rf in time t. We are now left with evaluating this
matrix, which will lead us to an answer of the second question, how will the
particle get there?.
Feynman Path Integral
We will consider a system where the particle being restricted in 1-D, say the
x-axis. Feynman postulated that the path taken by the particle from initial
position A = xi to B = xf is the weighted sum of all possible paths between the
two points. This is contrary to classical mechanics, which states that a particle
has one uniquely determined path between the two points obeying the principle
of least action. However, in the classical limit, → 0, we recover the principle
of least action as the path that minimizes the action has an associated high
measure in the path integral [5].
Figure B.1: Three possible paths from A to B
The time evolution operator has the composition property
ˆU(t) = ˆU(t − t1) ˆU(t1) (B.8)
for any t > t1. Assuming |Ψ is normalized, the identity operator is given by
ˆ1 = dx |x x| . (B.9)
If we let t1 = t/2 in equation (B.8) and insert the identity operator in between
the time evolution operators, the propagator is given by
xf | ˆU(t) |xi = dx xf | ˆU(t/2) |x x| ˆU(t/2) |xi . (B.10)
36. APPENDIX B. DERIVATION OF FEYNMAN PATH INTEGRAL 35
We can further divide the time t, N-times to give
ˆU(t) = ˆU(t/N) ˆU(t/N)........ ˆU(t/N) ˆU(t/N) (B.11)
and in the limit N → ∞ we define ∆t = t
N such that
e− i ˆH∆t ˆ1 −
i ˆH∆t. (B.12)
We Substitute expressions (B.11) and (B.12) into the propagator, and insert the
identity operator, dxi |xi xi|, between each product, to get
xf | ˆU(t) |xi = dx1 dx2..... dxN−1
N−1
n=1
xn+1| ˆ1 −
i ˆH∆t |xn . (B.13)
Given an Hamiltonian, ˆH = ˆp2
2m + ˆV (x), from equation (B.13) we need to
evaluate 3 inner products
xn+1|xn , xn+1|
ˆp2
2m
|xn , and xn+1| ˆV (x) |xn . (B.14)
We note that [5]
x|x = δ(x − x ) and δ(x) =
+∞
−∞
dk
2π
e−ikx
(B.15)
where k is the wave vector. We use De Broglie’s relation p = k, where p is the
momentum, to get the first inner product
xn+1|xn =
+∞
−∞
dpn
2π
e− i
pn(xn+1−xn)
. (B.16)
Similarly for the third inner product with the potential, it can be written as
xn+1| ˆV (x) |xn = V (xn) xn+1|xn = V (xn)
+∞
−∞
dpn
2π
e− i
pn(xn+1−xn)
.
(B.17)
To evaluate the remaining inner product we note that [5]
x|p =
1
√
2π
e
i
px
and dp |p p| = ˆ1, (B.18)
therefore it can be written as
xn+1|
ˆp2
2m
|xn = dpn xn+1|
ˆp2
2m
|pn pn|xn
=
dpn
2m
p2
n xn+1|pn pn|xn
=
dpn
2m
p2
n
1
√
2π
e
i
pnxn+1
1
√
2π
e− i
pnxn
=
dpn
4mπ
p2
n e
i
pn(xn+1−xn)
.
(B.19)
37. APPENDIX B. DERIVATION OF FEYNMAN PATH INTEGRAL 36
Using the three inner products, equation (B.13) becomes
xn+1| ˆ1 −
i ˆH∆t |xn =
1
2π
dpn 1 −
i p2
n
2m
+ V (xn) ∆t exp
i
pn∆x
(B.20)
where ∆x = xn+1 − xn. We make a similar approximation as (B.12) to change
the term in braces into an exponential, therefore
xn+1| ˆ1 −
i ˆH∆t |xn =
1
2π
dpn exp
i
−
p2
n
2m
∆t + pn∆x − V (xn)∆t .
(B.21)
This integral can be evaluated using the Gaussian integral given by [9]
+∞
−∞
exp(−ax2
+ bx + c)dx =
π
a
exp
b2
4a
+ c . (B.22)
We make the identification a = i∆t
2m , b = − i
∆x, and c = − i
V (xn)∆t, such
that the exponent of equation (B.21) becomes
exp
i (2m)∆x2
4∆t
− V (xn)∆t = exp
i 1
2
mv2
n − V (xn) ∆t . (B.23)
Note the classical Lagrangian term
Ln =
1
2
mv2
n − V (xn) (B.24)
appearing in the exponent. We can then write the nth
matrix element as
xn+1| ˆ1 −
i ˆH∆t |xn =
m
2πi ∆t
e
i
Ln∆t
. (B.25)
We define the infinite dimensional integral measure
Dx[t] := lim
N→∞
m
2πi ∆t
(N−1)/2 N−1
n=1
dxn, (B.26)
such that the path integral is given by
xf | e− i ˆHt
|xi =
xf
xi
Dx[t]
N−1
n=1
e
i
Ln∆t
=
xf
xi
Dx[t] exp
i
N−1
n=1
Ln∆t .
(B.27)
In the continuous limit N → ∞ the sum becomes an integral, therefore the final
path integral is given by
xf | e− i ˆHt
|xi =
xf
xi
Dx[t] exp
i t
0
L(x, ˙x, t)dt . (B.28)
38. Appendix C
Field Equations of Motion
General Field Equations
The field equation of motion follow and similar derivation as the classical Euler-
Lagrange equations of motion. The general field action is given by [10]
S = d4
x L(φ, ∂µφ), (C.1)
then applying the principle of least action, we get
δS = 0
= δ d4
x L(φ, ∂µφ)
= d4
x
∂L
∂φ
δφ +
∂L
∂[∂µφ]
δ(∂µφ)
= d4
x
∂L
∂φ
δφ +
∂L
∂[∂µφ]
∂µ(δφ) .
(C.2)
We will use integration by parts to evaluate the second term. Let U = ∂L
∂[∂µφ]
and V = ∂µ(δφ) then
U = ∂µ
∂L
∂[∂µφ]
and V = δφ, (C.3)
such that
d4
x
∂L
∂[∂µφ]
∂µ(δφ) = UV |b
a − d4
x V U
= − d4
x ∂µ
∂L
∂[∂µφ]
δφ
(C.4)
where UV |b
a = 0 since a and b are fixed ends. Putting all the terms together we
get
δS = d4
x
∂L
∂φ
− ∂µ
∂L
∂[∂µφ]
δφ. (C.5)
37
39. APPENDIX C. FIELD EQUATIONS OF MOTION 38
There are two cases that makes the integral zero: either the terms of the integral
take positive and negative terms that cancels out or the integral is zero over the
entire domain of integration. The later is preferred as the domain of integration
can vary which means the term inside the big square bracket must vanish. This
gives us the Euler-Lagrange equations for a field φ,
∂L
∂φ
− ∂µ
∂L
∂[∂µφ]
= 0. (C.6)
Free Particle
Now let us consider a free particle scalar field with the Lagrangian density given
by [13]
L =
1
2
(∂µφ)2
− m2
φ2
, (C.7)
written in natural units where c = = 1. Expanding the first term of this
equation we get
(∂µφ)2
= (∂µφ)(∂µ
φ) = (∂µφ)gµν
(∂νφ). (C.8)
where gµν
is the metric. To get the equations of motion from equation (C.6) we
take derivatives with respect to φ and ∂µφ treating ∂µφ as a parameter of the
Lagrangian density such that
∂L
∂φ
=
1
2
∂
∂φ
(∂µφ)2
− m2
φ2
= −m2
φ
(C.9)
and
∂L
∂[∂µφ]
=
1
2
∂
∂[∂µφ]
(∂µφ)2
− m2
φ2
=
1
2
∂
∂[∂µφ]
(∂µφ)gµν
(∂νφ)
=
1
2
[gµν
(∂νφ) + gνµ
(∂µφ)]
=
1
2
[2(∂µ
φ)]
= ∂µ
φ.
(C.10)
Putting everything together the Euler-Lagrange equations are given by
∂µ(∂µ
φ) + m2
φ = 0. (C.11)
Using ∂µ∂µ
= ∂2
∂t2 − 2
we can rewrite equation (C.11) as
∂2
φ
∂t2
− 2
φ + m2
φ = 0 (C.12)
which is known as the Klein-Gordon equation.
40. Appendix D
Field Wick Rotation
In this chapter we will connect a field theory object, a propagator, to a statistical
mechanics object, a partition function. This is done by imposing periodicity of
the field and making a Wick rotation. The partition function is define to be a
trace over all possible states,
Z = Tr e−β ˆH
(D.1)
which means φ must be periodic. The partition function is also real valued,
Z = dφ φ| e−β ˆH
|φ . (D.2)
hence the complex exponent of the field propagator,
φa| e− i ˆHt
|φa = D[φ] exp
i
dt d3
x L(φ, ∂µφ) (D.3)
must be transformed to a real exponent. This is done by making a change of
variable of the time coordinate t → τ = it (Wick rotation) such that the metric
transforms as gµν ∈ M4
−→ gab ∈ E4
.
We are going to show how the field action, given by
S[φ]M = dt d3
xL(φ, ∂µφ), (D.4)
transforms under this Wick rotation. Let us consider a free particle scalar field
with the associated Lagrangian given by [13]
L =
1
2
(∂µφ)2
− m2
φ2
. (D.5)
The derivative operators in M4
can be written as
∂µ =
∂
∂t
− and ∂µ
=
∂
∂t
+ . (D.6)
where µ = 0, 1, 2, 3. After a Wick’s rotation which only changes the time deriva-
tive, we get the derivative operators in E4
to be
∂a = −i
∂
∂τ
− and ∂a
= −i
∂
∂τ
+ . (D.7)
39
41. APPENDIX D. FIELD WICK ROTATION 40
where a = 1, 2, 3, 4. Using the Euclidean derivative operators, the Lagrangian
is given by
L =
1
2
(∂aφ∂a
φ) − m2
φ2
=
1
2
−
∂φ
∂τ
2
− ( φ)
2
− m2
φ2
.
(D.8)
Substituting this Lagrangian into equation (D.4) we get the associated action
to be given by
S[φ]A = −
1
2i
dτ d3
x
∂φ
∂τ
2
+ ( φ)
2
+ m2
φ2
. (D.9)
This functional is not real valued, but if we insert it into the equation (D.3) we
get
φa| e− i ˆHt
|φa = D[φ] exp −
1
dτ d3
xLE (D.10)
where
LE =
1
2
∂φ
∂τ
2
+ ( φ)
2
+ m2
φ2
. (D.11)
We define
SE[φ] dτ d3
xLE (D.12)
to be the Euclidean action therefore that the associated propagator is given by
φ| e− 1 ˆHτ
|φ = D[φ]e− 1
SE [φ]
(D.13)
where 0 < τ < β . The partition function can now be written as
Z = C
φ(0)=φ(β )
D[φ] exp −
1 β
0
dτ d3
x LE (D.14)
42. Bibliography
[1] http://hep.itp.tuwien.ac.at/ ipp/qgp.html
[2] David Griffiths, Introduction to Elementary Particles, Second Revised Edi-
tion 2008
[3] Henri J.F Jansen, Statistical Mechanics, Department of Physics, Oregon
State University, October 12, 2008
[4] Xiangdong Ji, A Modern Indroduction to Nuclear Physics
http://www.physics.umd.edu/courses/Phys741/xji/chapter2.pdf
[5] J.J Sakurai and Jim Napolitano, Modern Quantum Mechanics 2nd Edition,
2011
[6] http://www.mathworks.com/matlabcentral/fileexchange/23060-
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[7] W. Klein, Equilibrium Statistical Mechanics, 14 October 2007,
http://physics.bu.edu/ klein/chapter1.pdf
[8] Jill Knapp, AST 403 / PHY 402 Stars and Star Formation, Spring 2011
http://www.astro.princeton.edu/ gk/A403/state.pdf
[9] M. Abramowitz and I. A. Stegun, Handbook of Mathematical functions
[10] David McMahon, Quantum Field Theory Demystified, Scalar Fields
[11] Andreas Schmitt, Thermal Field Theory, WS 13/14
http://hep.itp.tuwien.ac.at/ aschmitt/thermal13.pdf
[12] M. Laine, Thermal Field Theory,
http://www.physik.uni-bielefeld.de/ laine/thermal/
[13] A. Zee, Quantum Field Theory in a Nutshell, Second Edition, 2009
[14] Mark Alford, Legendre transforms, Jan 2015
http://www.physics.wustl.edu/alford/physics/legendre.pdf
[15] http://news.mit.edu/2010/exp-quark-gluon-0609
[16] http://www.examiner.com/article/large-hadron-collider-shows-early-
universe-was-a-liquid
41
