so that

which is the correct transformation law for a pseudoscalar if A e

(1) Normalization of Dirac wave functions. Note that the normalization integral, which involves ip^rp, can be expressed in terms of the following density:

"Iiis makes it clear that it is the fourth component of a four-current, which is conserved. We already wrote this conservation law in covariant form in Eq. (5.14); in terms of the Dirac adjoint it is a

The appearance of the factor y/E/m in the boost of a Dirac free particle state, Eq. (5.115), can now be understood. The free particle state ipp<3(x) has been normalized to unity using the normalization condition

Since this condition is the fourth component of a four-vector, the requirement that it be the same in all frames (i.e., behave like a scalar) is inconsistent with its Lorentz nature and must break covariance. We therefore expect a non-covariant factor in the transformation law which carries this state to the rest frame,

The normalization condition (5.135) requires that N = y/m/E, as already given in Eq. (5.115). To see this, observe that

J d3rxPPtS(x) ~i°iPp,s{x) = J d3ripPtS(x)S(L)S~1(L)'y0S(L)S~1(L)ipPtS(x) = N2J d3r {x') [7° coshi + 7 • £sinh tf „,«(x').

But, 4>o,s{x')y0ipo,s{x') = 1 /L3, and because has no lower components and 7 • £ is off-diagonal, t/>o,«(z')7 ' C^Po^ix) = 0. Hence

[ d3r \jjp 3(x)*y°tpp s(x) = N2 cosh£ = TV2- = 1 , J m which gives the desired result. Thus the extra non-covariantfactor l/N = E/m already incorporated in the definition of is just what is needed to insure the state is normalized to 1 in any frame. Because the normalization condition is non-covariant, the states ipPtS must include a non-covariant factor.

(2) Normalization and orthogonality relations for Dirac spinors. Because of the negative sign in 70, the covariant normalization and orthogonality relations satisfied by the u and v spinors are:

u(p, s)u(p, s') = 2m Sss> v(p,s)v(p,s') = —2m 6SS> u(p, s)v[p,s') - 0 .

Note that the negative energy v spinors now have negative norm. For convenience, the non-covariant versions of (5.136a) are u'(p, s)u(p,s') = 2E6SS vl(p,s)v(p,s') = 2E6ss< u^(p,s)v{-p,s') = 0 .

(3) Energy projection operators. It is useful to find projection operators which project out the positive and negative energy subspace. The matrices

are projection operators with the properties

If any state is expanded in terms of u and v spinors ip = ^2<isu(p,s)+ "^2bsv(p,s) , s s then the operators A± will project out the separate plus and minus parts

All of these results follow directly from the orthonormality relations (5.136a).

An alternative form for these projection operators is very useful and is conveniently expressed in terms of the Feynman notation for the scalar product of any four-vector p with the 7 matrices, i>= PnY

Then, if pß = (Ep,p), the equations satisfied by the u and v spinors, Eqs. (5.19) and (5.26), can be written in the following compact form:

Using these equations, it is easy to see that the projection operators can also be written__

These relations can also be obtained by direct construction from Eqs. (5.137). Using / p2 = m2, it is a simple matter to prove directly that A± = A± and

(4) Spin projection operators. The spin projection operator for a non-relativistic two-component spinor is i(l+<r.5)x(s> = x(s> ,

where s is the unit three-vector in the direction of the spin. For spins in the ¿-direction, for example, s = (0,0, ±1) for spin up (+) or spin down (-).

In this last section we discuss some of the special properties possessed by Dirac particles with zero mass. These particles are particularly fascinating and may very well exist in nature. If the masses of the neutrinos (see Appendix D) are not

5.2 An electron scatters from a repulsive spherical Coulomb potential of the form

(a) Calculate the unpolarized cross section in first Born approximation (lowest order in A°). Use the Dirac formalism.

(b) Compare your relativistic result [from (a) above] with the result you would obtain from the Schrodinger equation in first Born approximation.

5.3 Suppose the Coulomb potential transformed relativistically like a scalar field (rather than like the fourth component of a vector field) so that the interaction of the electron with the Coulomb potential would read eip(x)i/j(x)A0(x) (scalar case)

(vector case), where in both cases e A0(x) = -Ze2/A-n\r\ = -Za/r. Calculate the differential cross section in the Born approximation for the scalar case and show that, at high energies, both the angular and energy dependence are completely different from the vector case, even though the two differential cross sections are identical at nonrelativistic energies.

5.4 Prove that transforms like a pseudoscalar. If S(A)ip — ip', prove that

5.5 Consider the following Dirac matrix element:

(a) From the structure of M, guess how it transforms under LT's. Write down the transformation law explicitly, using the notation x' = Ax.

(b) Using the Lorentz transformation properties of the Dirac wave functions, Eq. (5.101), and the property Eq. (5.102), prove that your transformation law is correct or find the correct one.

5.6 [Taken from Bjorken and Drell (1964).] The Dirac equation describing the interaction of a proton or neutron with an applied electromagnetic field will have an additional magnetic moment interaction representing their observed anomalous magnetic moments:

where is the electromagnetic field tensor.

(a) For the proton, i = p, ep = |e|; for the neutron i = n, en = 0. Verify that the choice of kp = 1.79 and Kn = -1.91 corresponds to the observed magnetic moments and check that the additional interaction does not disturb the Lorentz covariance of the equation. Check also that the Dirac Hamiltonian is Hermitian and that probability is conserved in the presence of the additional interaction.

(b) Make a Foldy-Wouthuysen transformation for the neutron, keeping terms up to order (v/c)2. Give a physical interpretation of the individual terms.

(c) Suppose a negatively charged particle of mass m, charge -e, and anomalous moment k is captured by a nucleus of charge Ze. Suppose that m » me, so that screening by the other electrons can be ignored. Calculate the fine structure splitting of the energy levels, and comment on how the splitting depends on k.

5.7 New diagonal form for the Dirac equation. Paralleling the discussion following Eq. (5.55), we can introduce a FW transformation which will completely eliminate the lower components from the free positive energy solutions and the upper components from the free negative energy solutions. The advantage of such a representation is that it allows us to regard the mixing of upper and lower components as a dynamical consequence of the interaction; the free Dirac equation is fully diagonalized. A unitary transformation which accomplishes this is

(a) Show that U is unitary by direct computation. Show that

APPLICATION OF THE DIRAC EQUATION

Hence it is clear that both the upper and lower components can be expanded in terms of the generalized spherical harmonics yfm(r), which are constructed by vector addition* from the spatial spherical harmonics Yem(r) and the spin \ states

The ± superscript on the ^'s denotes the parity. With this notation, the states have the overall structure

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