Fundamentals of Quantum Electrodynamics MOHAMMAD IMRAN AZIZ Assistant Professor PHYSICS DEPARTMENT SHIBLI NATIONAL COLLEGE, AZAMGARH (India). aziz_muhd33@yahoo.co.in

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 . aziz_muhd33@yahoo.co.in

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. aziz_muhd33@yahoo.co.in

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 aziz_muhd33@yahoo.co.in

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: aziz_muhd33@yahoo.co.in

where the infinitesimal variation  of  is subject to the restrictions aziz_muhd33@yahoo.co.in

This is classical field equation derived from the Lagrangian density L(ψ, grad ψ, ,t). aziz_muhd33@yahoo.co.in

Functional Derivatives In order to pursue further the analogy with particle mechanics, it is desirable to rewrite Lagrangian equation in terms of L aziz_muhd33@yahoo.co.in

which closely resembles the Lagrangian equations . aziz_muhd33@yahoo.co.in

Hamiltonian Equations and The Hamiltonian aziz_muhd33@yahoo.co.in

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 aziz_muhd33@yahoo.co.in

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 aziz_muhd33@yahoo.co.in

Quantum equations for fields The quantum conditions for the canonical field variables then become The quantum conditions for the canonical field variables then become aziz_muhd33@yahoo.co.in

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. aziz_muhd33@yahoo.co.in

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Then the Hamiltonian density is Quantum Equations. As with the nonrelativistic Schrodinger equation, it is convenient to rewrite the Hamiltonian aziz_muhd33@yahoo.co.in

We therefore quantize the field by imposing anticommutation relations on the components of ψ aziz_muhd33@yahoo.co.in

Thus the four equations are equivalent to the Dirac equation and follows: aziz_muhd33@yahoo.co.in

QUANTIZATION OF ELECTROMAGNETIC FIELDS Lagrangian Equations: Maxwell's equations in empty space are obtained by setting  and  equal to zero aziz_muhd33@yahoo.co.in

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Hamiltonian Equations: The momentum canonically conjugate to A, is found The Hamiltonian equations of motion are aziz_muhd33@yahoo.co.in

The Hamiltonian is then aziz_muhd33@yahoo.co.in

Quantum equations: The classical electromagnetic field is converted into a quantum field in the following way aziz_muhd33@yahoo.co.in

Commutation Relations for E and H: The electric and magnetic fields are defined by the equations aziz_muhd33@yahoo.co.in

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Quantization of Schrodinger equations aziz_muhd33@yahoo.co.in

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Quantum equations: aziz_muhd33@yahoo.co.in

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