Transcript Quantization of Dirac Field

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 equation of motion.
(1)
In deriving above, in first step on last slide we
used
Where,
(2)