1. Fundamentals of Quantum Electrodynamics
MOHAMMAD IMRAN AZIZ
Assistant Professor
PHYSICS DEPARTMENT
SHIBLI NATIONAL COLLEGE, AZAMGARH (India).

2. Coordinates of Fields
A wave field is specified by its amplitudes at all points of space and the dependence of these amplitudes on the time, in much the same way as a system of particles is specified by the positional coordinates q and their dependence on the time. The field evidently has an infinite number of degrees of freedom, and is analogous to a system that consists of an infinite number of particles. It is natural, then, to use the amplitudes ψ(r,t) at all points r as coordinates in analogy with the particle coordinates q .

3. PROPERTIES OF THE FIELDS:
1. In analogy with any physical system, the field is carrier of energy, momentum, angular momentum and other observable dynamical quantities.
2. It is to be ideal regarding the definition of the field that if the values of ψ are ordinary or complex numbers then it is called a classical field, and if the values of ψ are operators in some Hilbert space, then the fields in question is called a quantum field.
3. The degrees of freedom of the field is represented by amplitudes ψ(r,t) which are infinite in number corresponding to finite number of points in space.

4. Lagrangian Equations
The Lagrangian L is a function of the time and a functional of the possible paths q(t) of the system. The actual paths are derived from the variational principle

5. By analogy, we expect the field Lagrangian to be a functional of the field amplitude ψ(r,t). It can usually be expressed as the integral over all space of a Lagrangian density L:

6. where the infinitesimal variation  of  is subject to the restrictions

7. This is classical field equation derived from the Lagrangian
density L(ψ, grad ψ, ,t).

8. Functional Derivatives
In order to pursue further the analogy with particle mechanics, it is desirable to rewrite Lagrangian equation in terms of L

9. which closely resembles the Lagrangian equations .

10. Hamiltonian Equations
and
The Hamiltonian

11. It follows from the earlier discussion of functional derivatives that
We write H as the volume integral of a Hamiltonian density H, and assume that the cells are small enough so that the difference between a Volume integral and the corresponding cell summation can be ignored; we then have
It follows from the earlier discussion of functional derivatives that

12. The classical field equations in Hamiltonian form
This equation also serves to define the Poisson-bracket expression for two fuctionals of the field variables. In quantum mechanics Poission-bracket is

13. Quantum equations for fields
The
quantum conditions for the canonical field variables then become
The quantum conditions for the canonical field variables then become

14. QUANTIZATION OF THE DIRAC EQUATION
Our procedure consists in treating the one-particle equation as though it were a classical field equation. The resulting quantized field theory represents the motion of a number of non interacting free electrons.

16. Then the Hamiltonian density is
Quantum Equations. As with the nonrelativistic Schrodinger equation, it is convenient to rewrite the Hamiltonian

17. We therefore quantize the field by imposing anticommutation relations on the components of ψ

18. Thus the four equations are equivalent to the Dirac equation and follows:

19. QUANTIZATION OF ELECTROMAGNETIC FIELDS
Lagrangian Equations:
Maxwell's equations in empty space are
obtained by setting  and  equal to zero

21. Hamiltonian Equations:
The momentum canonically conjugate to
A, is found
The Hamiltonian equations of motion are

22. The Hamiltonian is then

23. Quantum equations:
The classical electromagnetic field is converted into a quantum field in the following way

24. Commutation Relations for E and H:
The electric and magnetic fields are defined by the equations

26. Quantization of Schrodinger equations

28. Quantum equations:

29. Thank You