Chapter III Dirac Field Lecture 1 Books Recommended: Lectures on Quantum Field Theory by Ashok Das A First Book of QFT by A Lahiri and P B Pal

Dirac Equation (For details on Dirac Eq and its Sol see lectures of QM-II, III Semester ) Klein Gordon lead to negative energy solutions. Probability density can be negative. This is because it has first derivative of wave function w.r.t. t. We need to have first derivative of ϕ w.r.t time in wave equation. But then, spatial derivative should. Also be of first order because of Lorentz invariance. We can do this if energy and momentum are linearly dependent.

In terms of Gamma matrices, Dirac Eq is given by -----(1) Which is Covariant form of Dirac eq. Feynman slash notation -------(2) Where v is some arbitrary vector

In terms of Feynman slash notation, Dirac Eq will be ------(3) In terms of Dirac eq (1) can be written as (in natural units) -----(4) -----(5)

Properties of Gamma matrices: Gamma matrices are even dimensional Matrices (Pauli-Dirac representation) ---(6) Where sigma Pauli matrices are ---(7)

Gamma matrices satisfy the Clifford algebra ----(8) From above we have -----(9)

-----(10) --(11)

---(12) ---(13) We define six antisymmetric matrices ----(14)

--------(15) ----(16)

Matrices represent spin operator for Dirac particle. We can identify using -----(17) as spin operator.

We constructed following 16 matrices ----(18)

Pauli’s fundamental theorem: If satisfy Clifford algebra, then they must be related through the similarity transformations. i.e. if -----(20) ------(21) Then there exist non-singular matrix S such that ------(22)

Starting from We have Where,

Other representations for Dirac matrices Majorana Representation: In this are Purely imaginary. We have for Useful for study of Majorana fermions which Are charge neutral particle.

Dirac matrices in Dirac Pauli representation and Majorana representation are related through the unitary transformation

Weyl Representation: Used for study of massless fermions where Dirac Pauli representation and Weyl representation are related through