Chapter III Dirac Field Lecture 4 Books Recommended: Lectures on Quantum Field Theory by Ashok Das Advanced Quantum Mechanics by Schwabl

Pauli Exclusion Principle Quantized Klein Gordon Field theory is Used for the Spin 0 boson particles. To construct the field theory for Fermions we need to incorporate the Pauli exclusion principle. As per Pauli principle: at the most be one fermions in a given state.

Consider an oscillator having annihilation And creation operator . Corresponding number operator -----(1) Above oscillator will obey Fermi Dirac Statistics if annihilation and creation operators obey anti-commutation relation.

We have anti-commutation relations -----(2) We write, ---(3)

From (3), we can write -----(4) Eigenvalues of number operator ----(5) which is Pauli exclusion principle. With anti-commutation relations, wave function will be antisymmetric and therefore, describe fermions.

Quantization of Dirac Field Dirac Eq ----(6) Adjoint Eq ---(7) Where adjoint spinor ---(8)

Dirac field operators belong to the Spin ½ representation of Lorentz group and hence, are fermions and should be described by Anti-commutation relations. The Lorentz invariant Lagragian density for Dirac field ------(9)

Using (9) and Euler Lagrange Eq., we can find Eqs (6) and (7) ----(10) ----(11)

Lagrangian given by (9) is not hermitian. First Term of (9) is not hermitian 2nd term of (9) is hermitian -------(12)

We can write a Hermitian and Lorentz invariant Lagrangian ---- (13) Lagrangian (9) and (13) are differ by total divergence only: -----(14)

Dynamical Eqns derived using (9) and (13) will be same and we will use Lagrangian given by (9). Momenta conjugate to and will be ----(15) --------(16)

Equal time anti-commutation relation will be ---(17) WE can also write using (15): ----(18) ----(19)

Hamiltonian density -----(20) Total Hamiltonian ---(21)

Using Heisenberg Eq, we can derive Dirac equat- ion of motion. (1)

In deriving above, in first step on last slide we used Where,

(2)