Chapter III Dirac Field Lecture 2 Books Recommended: Lectures on Quantum Field Theory by Ashok Das A First Book of QFT by A Lahiri and P B Pal

Solution for Dirac Equation Plane wave solution ------(1) Using this, Dirac Eq -----(2) where

We use following two component form We can write -----(3) We use following two component form for 4-component spinor (also known as bispinor) --------(4) For upper two components For lower two components

From Eq (22), we can write now ---------(5) Above eq lead to following coupled equations -----(6)

From 2nd relation in Eq (26), we have Using above Eq. in 1st relation of (26), we get Which is relativistic energy momentum relationship.

Note that -----(7) Now from Dirac Eqs., (28), we have --------(8)

We consider first solution --------(9) Using (28) and (29) in (24), we can write ------(10) With p = 0, above Eqns. reduces to free particle Solution with E> 0 .

Now we use ------(11) and this give ----(12) which is for E<0.

Thus, we write -----(13) Exercise: Discuss the non-relativistic limit of above Solutions.

Normalisation method Defining ----(1) We write solution as ----------(2) Where, α and β are normalization constants.

are normalized as ----(3) which is for same spin components. For different spin components it vanish.

We now calculate -----(4)

Negative energy solutions ---------(5) Also -----(6)

Wave function (adjoint spinor) ---(7) e.g. ----(8) ----(9)

Using (8) -----(10)

Similarly, using (9) -------(11) For relativistic normalization, we will not have normalization condition -----(12) Probability density transform like time component of a four vector

For relativistic covariant normalization, we need ---(13) In rest frame Independent free particle wave function With above normalization condition (eq 13), ----------(14)

Using (4), (5) and (13) --------(15) --------(16)

Normalized +Ve and –Ve energy solutions are ----(17) Also ---(18) Which is Lorentz scalar.

Positive and negative energy solutions are orthogonal = 0. ----------(19)

Note that --------(20) Normalization discussed above is for massive particle Only. Alternative, normalization condition which work Well for massive and mass-less particles is ------(21)

From this, we have -----(22) --------(23)

Also ---(24) Which is again scalar.

More on Solutions and orthogonality relations Positive energy sol of Dirac Eq satisfy ----(25) where ----(26) Negative energy sol satisfy ---(27)

We write positive and negative energy sol as ----(28) Using above from (25), ----(29)

And for Eq (27), we have ----(30) Which is for negative energy sol. Adjoint Eq corresponding to (29) (take hermitia -n conjugate and multiply by on right) : ------(31)

Adjoint Eq corresponding to (30) is written as ----(32) Two +Ve and two –Ve energy solutions can be Denoted a ----(33) r actually represent spin projection.

Each sol. is a component spinor. For spinor index we use α Each sol. is a component spinor. For spinor index we use α. Thus, α = 1, 2, 3, 4. We can write the Lorentz invariant conditions studied earlier in Eq. (18) using above notations as --(34)

Compare last Eq of (34) with Eq (20). Is there anything wrong? From (34) we can write ---(35) Also ---(36)

Projection operator and Completeness Conditions: We define the operators ----(37) ----(38)

Consider the operation of above operators on solutions ---(39) ---(40) ---(41) ---(42)

Note --(43) ---(44)

Also ----(45) ---(46)

We now consider the outer product of the solutions. Consider the elements of P matrix ----(47)

Acting matrix P on positive spinor give the --(48) ----(49)

Also -----(50) Thus, we can write ---(51)

For negative sol, we define outer product ---(52) Operating on spinors, we get ----(53) ----(54)

Also ---(55) Matrix Q project on to space of –Ve energy sol ---(56)

Completeness condition ---(57) Or in Matrix form ---(58)