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

2. 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.

3. 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.

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

5. 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.

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

7. 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)

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

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

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

11. 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)

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

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

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

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

16. (2)