Relativistic Quantum Mechanics Lecture 4 Books Recommended: Lectures on Quantum Field Theory by Ashok Das Advanced Quantum Mechanics by Schwabl Relativistic Quantum Mechanics by Greiner Quantum Field Theory by Mark Srednicki http://www.quantumfieldtheory.info/

Dirac Equation 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.

Dirac proposed equation of the form ----(1) Where, Hamiltonian = -----(2) Or Or (In natural units)

-----(3) (in natural units) Coefficients αi cannot be simply numbers because if is so then equation will not be invariant αi and β should be hermitian matrices so that H is hermitian say N by N matrix N component column vector

Following conditions will be imposed Klien Gordon Eq and hence Relativistic energy- moment relationship should be satisfied E2 = p2c2 + m2c4 Conserved four-current density with positive zero component probability density should exist Should be Lorentz invariant

Differentiating eq n (3) w.r.t. t ---------(4)

For eq (4) to satisfy KG eqn, we get -----(5) ------(6) --------(7) Thus, we have four anticommuting Hermitian Matrices.

Now we will show that above matrices should by 4 by 4 dimensions Matrices β and αi will be traceless From (6), we write Taking trace and using cyclic property, we have

As α and β are hermitian, they can be diagonalzed to get eigen values. As the square of these matrices is unity, eigen values should be ±1 As the trace of matrices is zero and hence matrices should be of even dimension Matrices cannot be of 2 by 2 dimensions, as there will not be 4 traceless ant commutating hermitian matrices. Only 3 e.g. Pauli matrices. Thus they will be of at least 4 by 4 dimension

Following representation is used for Dirac matrices -------(8) Where Pauli matrices are Exercise: Show that matrices in Eq (8) satisfy eq. (5) to (7)

Dirac Equation in the Covariant form Dirac Eq is ------(9) Multiplying by β/c, we get ----(10) Now we use representation ----------(11)

Properties of Gamma matrices: Proof: ------(12)

Also -----(13) Eq (12) and (13) gives ---------(14)

In terms of Gamma matrices, Dirac Eq (9) can be written as -----(15) Which is Covariant form of Dirac eq. Feynman slash notation -------(16) Where v is some arbitrary vector

In terms of Feynman slash notation, Dirac Eq will be ------(17) In terms of Dirac eq (15) can be written as (in natural units) -----(18) -----(19)

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,