Concepts in Theoretical Physics
Lecture 1: The Principle of Least Action
David Tong
Newtonian Mechanics
You've all done a course on Newtonian mechanics so you
know how to calculate the way things move.
You draw a pretty picture; you draw arrows representi

2. Classical Gases
Our goal in this section is to use the techniques of statistical mechanics to describe the
dynamics of the simplest system: a gas. This means a bunch of particles, ying around
in a box. Although much of the last section was formulated i

3. Quantum Gases
In this section we will discuss situations where quantum eects are important. Well still
restrict attention to gases meaning a bunch of particles moving around and barely
interacting but one of the rst things well see is how versatile the

5. Phase Transitions
A phase transition is an abrupt, discontinuous change in the properties of a system.
Weve already seen one example of a phase transition in our discussion of Bose-Einstein
condensation. In that case, we had to look fairly closely to s

Lent Term, 2011 and 2012
Preprint typeset in JHEP style - HYPER VERSION
Statistical Physics
University of Cambridge Part II Mathematical Tripos
Dr David Tong
Department of Applied Mathematics and Theoretical Physics,
Centre for Mathematical Sciences,
Wilb

4. The Hamiltonian Formalism
Well now move onto the next level in the formalism of classical mechanics, due initially
to Hamilton around 1830. While we wont use Hamiltons approach to solve any further
complicated problems, we will use it to reveal much mo

2. The Lagrangian Formalism
When I was in high school, my physics teacher called me down one day after
class and said, You look bored, I want to tell you something interesting.
Then he told me something I have always found fascinating. Every time
the subj

Michaelmas Term, 2004 and 2005
Preprint typeset in JHEP style - HYPER VERSION
Classical Dynamics
University of Cambridge Part II Mathematical Tripos
Dr David Tong
Department of Applied Mathematics and Theoretical Physics,
Centre for Mathematical Sciences,

Dynamics and Relativity: Example Sheet 2
Professor David Tong, February 2013
1. A particle moves in a xed plane and its position vector at time t is x. Let (r, )
be plane polar coordinates and let and be unit vectors in the direction of increasing
r
r and

Dynamics and Relativity: Example Sheet 1
Professor David Tong, January 2013
1. In one spatial dimension, two frames of reference S and S have coordinates (x, t)
and (x , t ) respectively. The coordinates are related by t = t and
x = f (x, t)
Viewed from f

5. Systems of Particles
So far, weve only considered the motion of a single particle. If our goal is to understand
everything in the Universe, this is a little limiting. In this section, we take a small step
forwards: we will describe the dynamics of N ,

3. Interlude: Dimensional Analysis
The essence of dimensional analysis is very simple: if you are asked how hot it is outside,
the answer is never 2 oclock. Youve got to make sure that the units agree. Quantities
which come with units are said to have dim

6. Non-Inertial Frames
We stated, long ago, that inertial frames provide the setting for Newtonian mechanics.
But what if you, one day, nd yourself in a frame that is not inertial? For example,
suppose that every 24 hours you happen to spin around an axis

4. Central Forces
In this section we will study the three-dimensional motion of a particle in a central
force potential. Such a system obeys the equation of motion
mx = V (r )
(4.1)
where the potential depends only on r = |x|. Since both gravitational and

7. Special Relativity
Although Newtonian mechanics gives an excellent description of Nature, it is not universally valid. When we reach extreme conditions the very small, the very heavy or
the very fast the Newtonian Universe that were used to needs repla

2. Forces
In this section, we describe a number of dierent forces that arise in Newtonian mechanics. Throughout, we will restrict attention to the motion of a single particle. (Well
look at what happens when we have more than one particle in Section 5). W

Dynamics and Relativity: Example Sheet 3
Professor David Tong, February 2014
1. In a system of particles, the ith particle has mass mi and position vector xi with
respect to a xed origin. The centre of mass of the system is at R. Show that L, the
total an

4. Classical Thermodynamics
Thermodynamics is a funny subject. The rst time you go through it, you
dont understand it at all. The second time you go through it, you think
you understand it, except for one or two small points. The third time you
go through

Statistical Physics: Example Sheet 3
David Tong, February 2012
1. A Wigner crystal is a triangular lattice of electrons in a two dimensional plane.
The longitudinal vibration modes of this crystal are bosons with dispersion relation
= k . Show that, at l

Concepts in Theoretical Physics
Lecture 8: Cosmology
David Tong
The Big Bang
This is not what the big bang looked like.
There is no bang in the big bang. There is no explosion.
Big bang theory has nothing to say about how the universe started.
We dont kno

Concepts in Theoretical Physics
Lecture 6: Particle Physics
David Tong
The
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Structure of
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Four fundamental particles
Repeated twice!
Four fundamental forces
All based on symmetry
which means group theory
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Quantum Field Theory: Example Sheet 2
Dr David Tong, October 2007
1. A string has classical Hamiltonian given by
12
p
2n
H=
22
+ 1 n qn
2
(1)
n=1
where n is the frequency of the nth mode. (Compare this Hamiltonian to the Lagrangian (3) in Example Sheet 1.

Quantum Field Theory: Example Sheet 3
Dr David Tong, October 2006
1. The Weyl representation of the Cliord algebra is given by,
0 =
01
10
i =
,
0 i
i 0
(1)
Show that these indeed satisfy cfw_ , = 2 , where 1 comes with an implicit 4 4
unit matrix. Find

5. Quantizing the Dirac Field
We would now like to quantize the Dirac Lagrangian,
/
L = (x) i m (x)
(5.1)
We will proceed naively and treat as we did the scalar eld. But well see that things
go wrong and we will have to reconsider how to quantize this the

6. Quantum Electrodynamics
In this section we nally get to quantum electrodynamics (QED), the theory of light
interacting with charged matter. Our path to quantization will be as before: we start
with the free theory of the electromagnetic eld and see how

4. The Dirac Equation
A great deal more was hidden in the Dirac equation than the author had
expected when he wrote it down in 1928. Dirac himself remarked in one of
his talks that his equation was more intelligent than its author. It should
be added, how

2. Free Fields
The career of a young theoretical physicist consists of treating the harmonic
oscillator in ever-increasing levels of abstraction.
Sidney Coleman
2.1 Canonical Quantization
In quantum mechanics, canonical quantization is a recipe that takes

3. Interacting Fields
The free eld theories that weve discussed so far are very special: we can determine
their spectrum, but nothing interesting then happens. They have particle excitations,
but these particles dont interact with each other.
Here well st

4. Linear Response
The goal of response theory is to gure out how a system reacts to outside inuences.
These outside inuences are things like applied electric and magnetic elds, or applied
pressure, or an applied driving force due to some guy sticking a s

3. Stochastic Processes
We learn in kindergarten about the phenomenon of Brownian motion, the random
jittery movement that a particle suers when it is placed in a liquid. Famously, it is
caused by the constant bombardment due to molecules in the surroundi