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Introduction to Quantum Field Theory: Fermions and Bosons

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Mason Yu

on 3 April 2017

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Transcript of Introduction to Quantum Field Theory: Fermions and Bosons

Classically: we can always tell particles apart from one another
That is: nothing is truly indistinguishable,
in principle
We can for example stamp some form of identification on them, or track them continuously and follow their interactions
Here is an example of an interaction between 2 non-identical particles:
We now define an exchange operator
We can apply this to the probability and the wave function of the two particles:
Obviously, , so the eigenvalue of the exchange operator is +1 or -1
This is the
symmetrization requirement
:

In words:
identical particles have to occupy symmetric or antisymmetric states
Identical particles are required to satisfy this law, the ones that satisfy with the
plus sign
is called
bosons
, and the ones that satisfy with the
minus sign
is called
fermions
We now investigate the composite wave function of particles. That is:
For identical particles, we want the composite wave function to be
invariant under the swapping of particles
(up to a sign of +/-). Here is a way of constructing such a wave function:


Here the plus sign is for bosons (symmetric) and the minus sign is for fermions (antisymmetric)
For fermions, note that if we want to find 2 identical fermions at the same place:
So, this means the
probability amplitude of finding 2 identical fermions at the same place is 0
(there's no wave function at all), which is known as the
Pauli exclusion principle
:
Examples of bosons: photons (elementary) and mesons (composite)
They are force carriers
They obey Bose-Einstein statistics
There are currently 5 elementary bosons in the Standard Model: photon, gluon, Z boson, W boson (the gauge bosons) and the Higgs boson (the only scalar boson)
Introduction to QFT:
Fermions and Bosons

Two Particle Systems
Multiple Particles
But, in quantum mechanics, particles like electrons is fundamentally and utterly identical
Because we can't make any label on it, and any observation will unpredictably alter its state
Again, consider the previous situation, with identical particles like electrons
The probability of finding the particle at A and B must be identical, or else we would be able to distinguish between them.
The Symmetrization Requirement
The Symmetrization Requirement
Properties of Fermions and Bosons
What About Bosons then?
Some Examples
More Properties
There is a connection between spin and the statistics that we have discussed so far:
All particles with integer spin are bosons
All particles with half integer spin are fermions
It is taken as an axiom in non-relativistic quantum mechanics, but can be proven in relativistic quantum theories
Quantum Field Theory
Its Backgrounds
Schrodinger's equation was not completely "quantum" and it did not take relativity into account
It considers classical fields which are continuous instead of quantized
But, in the late 1920s, Paul Dirac was able to merge quantum mechanics and Einstein's theory of special relativity
He then initiated the second quantization revolution by finding the quantum formulation of the electric field surrounding the proton (canonical quantization)
Since then, the same ideas have been generalized to cover all atomic forces, with quantum electrodynamics and quantum chromodynamics
About Identical Particles
In a quantum field theory, a fundamental particle can be seen as a localized vibration of a corresponding field
Wave functions were thought of as fields too
These different fields were defined in the entire universe, and as particles are simply excitations in the same field, they must be all identical
These fields interact with each other by transferring energy between them, which we see as interactions of sub-atomic particles (an electron emitting a photon, for example)
The Pauli Exclusion Principle
A
B
A
B
We apply the same logic to bosons, and try to find the probability amplitude of finding 2 identical bosons in the same place.
For bosons:
So we have:
In fact, this is even larger than !
So bosons, in contrast to fermions, likes to be found together
Examples of fermions: quarks, leptons (elementary), protons, neutrons... (composite)
They are usually associated with matter
They obey Fermi-Dirac statistics
There are currently 24 different elementary bosons in the Standard Model: up, down, strange, charm, bottom and top quarks, electron, muon, tau particle and their respective neutrinos, along with the corresponding antiparticles
Some Examples
Quantum Field Theory: An Overview
Quantum Field Theory
Quantum Field Theory (QFT) in simple means, is the framework for construction of quantum mechanical models of subatomic particles in particle physics. Quantum Field theory treats particles in the physical field, so these are the field quanta.
Dynamics of QFT
Normal quantum mechanical systems have a fixed number of particles, with each particle having a finite number of amount of freedom. In QFT, the excited states of a QFT can represent any number of particles. This makes quantum field theories quite useful for describing systems where the particle number may change over time.
States of QFT
The interaction between particles in QFT are similar to the space between charges of electric and magnetic fields in Maxwell's theories. However, unlike the classical physics of Maxwell's theory, fields in QFT generally exist in superpositions of states.
Fields of QFT
The gravitational field and the electromagnetic field are the only fields in nature that have an infinite range and a corresponding classical low-energy limit, which greatly diminishes. Albert Einstein in 1905, attributed "particle-like" and discrete exchanges of momenta and energy, a characteristic of "field quanta", to the electromagnetic field.
Theories of QFT
There is currently no complete quantum theory for QFT, even though Edward Witten, a distinguished professor. describes QFT as "by far" the most difficult theory in modern physics, over QED and QCD. (Sort of contradicts, as really, is there s theory?)
History of QFT
1950s
Early 1970s
1975-
Early Foundations-1950s
The early development of the field involved Dirac, Fock, Pauli, Heisenberg and Bogolyubov. This phase of development started with the development of the theory of quantum electrodynamics in the 1950s.
Gauge Theory: A breakthrough-Early 1970s
Gauge theory was formulated and quantized, leading to the unification of forces embodied in the standard model of particle physics. This started in the 1950s with Yang and Mills, completed by the 1970s through the work of Gerard 't Hooft, Frank Wilczek, David Gross and David Politzer.
Grand Synthesis-1975-
This year the grand synthesis of theoretical physics was formed, which unified theories of particle and condensed matter physics through quantum field theory. This involved the work of Michael Fisher and Leo Kadanoff in the 1970s, which led to the seminal reformulation of quantum field theory by Kenneth G. Wilson in 1975.
By Ambrose Lee, Mason Yu, Alvin Tse
Harmonic Oscillators - Central Paradigm of Quantum Systems
A harmonic oscillator in classical mechanics can be thought of as just mass oscillating on a spring, with spring constant k
Displacement: x, momentum: , total energy is sum of kinetic energy and the potential energy
Now, quantum mechanically, we make the substitution to obtain the time independent Schrodinger equation:


Now, there are two different approaches to this problem, one is the power series method, which is a "brute force" method that is quite mathematically involved
And there is a neat algebraic technique with the use of ladder operators, which is what we are going to discuss
Harmonic Oscillators - Central Paradigm of Quantum Systems
We start by eliminating the spring constant in favor of the classical frequency , and rewriting the Schrodinger equation in the form:


We then try to factor the Hamiltonian using the identity:
Although we are working with operators which does
not commute
in general, we are still going to be interested in investigating the quantities:
Harmonic Oscillators - Central Paradigm of Quantum Systems
The product
But we all know the canonical commutation relation:
So we can make the substitution and obtain:


Using similar arguments, we also obtain: so we also have
We can now rewrite the time-independent Schrodinger equation like this:
Now, it can be proven that if satisfies the Schrodinger equation with energy E, then satisfies the Schrodinger equation with energy

Proof:
Harmonic Oscillators: Central Paradigm of Quantum Systems
We now have a wonderful machine which can
generate solutions
to the Schrodinger equation as long as we have one solution to get started
If we apply the lowering operator repeatedly, we will eventually reach a state which is not normalizable, which represent the "ground state"



We can then apply the raising operator repeatedly to obtain solutions for the excited states:


We can then do the normalization and obtain the normalization factor (which requires some complicated mathematics):
Harmonic Oscillators - Central Paradigm of Quantum Systems
So we have seen that the simple harmonic oscillator has some interesting property when we look at it quantum mechanically
It only has certain allowed energy levels, and in fact we find that the probability of finding the particle
outside of the classical amplitude
is non-zero, while for all
odd
states, the probability of finding the particle at the center is actually
zero
:
The Harmonic Oscillator
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