1. Second Quantization and Quantum Field Theory
Kajal Krishna Dey
Assistant Professor in Physics
B. B. College, Asansol

2. Preliminary 1st quantization: 2nd quantization:
Systems with variable numbers of particles ~ Second quantization
High energy scattering and decay processes.
Relativistic systems.
Many body systems (not necessarily relativistic).
1st quantization:
Dynamical variables become operators;
E, L, … take on only discrete values.
2nd quantization:
Wave functions become field operators.
Properties described by counting numbers of 1-particle states being occupied.
Processes described in terms of exchange of real or virtual particles.

3. Canonical Quantization Procedure
FIRST QUANTIZATION
Particle Mechanics
q and p are coordinate & its conjugate momentum
Quantum Mechanics
q and p are operators in Hilbert space
Canonical Quantization Procedure
Particle aspect
of the object
Wave aspect
of the object
First Quantization

4. Quantum of the field (particle)
Why Quantum Field Theory?
Light is described in terms of
(1) electromagnetic field (classical description)
(2) photon (quantum mechanical description)
Quantum Field Theory establishes link between the two descriptions
Quantization of e m field photon (quantum of the field)
Structure to quantize the e m field is applicable to all elementary particles as quantum fields
Classical wave field
Quantum of the field (particle)
Second Quantization

5. What is Field? Road to Quantization of a Classical Field
Classical analogue: Weighted vibrating string with infinite number of beads along the string, we come to limit of a continuous string described by a displacement field which varies continuously with x and t.
Field associated with each kind of particle in nature satisfies an assumed wave equation and it has infinite number of degrees of freedom.
A wave field is specified by its displacement at all points of space and the dependence of these displacements on time.
Road to Quantization of a Classical Field
Start with the field equation
Seek a Lagrangian via Hamilton’s principle
Find the canonical momentum from the Lagrangian
Carry out quantization procedure

6. Klein-Gordon Equation
Klein-Gordon equation for free particle is given by
Energy eigen values are
(using natural units )
Problem: System will have no ground state and becomes unstable.
Hence wave function interpretation of K-G eqn is not possible.
Now we want to interpret as a field, so Klein-Gordon eqn for this field is
Consider an infinitesimal variation
Restriction: & system is localized in space

7. Lagrangian: Real Scalar Field
From Klein-Gordon eqn
Using the relation
Integrating by parts
This gives with or
Hamilton’s principle
In analogy with Hamilton’s principle
is called Lagrangian density

8. Euler-Lagrange Equation
The momentum conjugate to the real scalar field is given by
Hamilton’s principle with
This gives
using
Integrating by parts
This leads to Euler-Lagrange equation

9. Canonical Quantization
Particle Mechanics Quantum Mechanics
Canonical Quantization
Use the same prescription
So and are no more classical fields, they become operator
A real scalar field may be expanded as a Fourier integral
where and

10. Commutation Relations
The momentum conjugate to is
From
we have commutation relations
The Hamiltonian is
where
is called Hamiltonian density

11. Creation & Annihilation operators
This yields the Hamiltonian for the real scalar field
using &
It is easy to check &
For harmonic oscillator
Define
we have ,
, ,
= annihilation operator, = creation operator

12. Normal Ordering
Interpretation: as annihilation operator and as creation operator for a field quantum with momentum , the quantum is what we call a particle.
Ground state called the vacuum of the field is defined as
Hamiltonian
this gives = infinite!
Normal ordered Hamiltonian taking
Now the vacuum is the state of lowest energy, i.e., the ground state