Ps

The derivation of these results for the Dirac theory is identical to that for the KG theory [review the arguments which led from Eq. (4.73) to Eq. (4.78)]. Each of these two expressions for differs from its KG counterpart [which is (4.76b) for the first and (4.78) for the second] only in the sign of the negative energy term. And here, as in the KG theory, the propagation of the negative energy states backward in time is interpreted as the propagation of the corresponding antiparticle states forward in time (recall Fig. 4.3).

The difference in sign of the negative energy contributions to the KG and Dirac expressions for /(2) will appear again in field theory. In that discussion the sign difference will come from the fact that Dirac particles satisfy Fermi-Dirac statistics (i.e., their field operators anticommute) and that when the time ordering of the interactions is changed, as it is for the negative energy states, there is an extra minus sign for fermions.

5.7 NONRELATIVISTIC LIMIT

We now investigate the non-relativistic limit of the Dirac equation. As we did for the Klein-Gordon equation, we will work out the expansion to order (v/c)2 ~ (p/m)2x leading terms. In making our estimates, we assume all potentials V° and V to be of the same order as the kinetic energy term (justified by the virial theorem). Since all of these leading terms are of order p2/m, we want all terms up to order p4/m3.

Assume a positive energy solution of the form where E = m + T. Then, using the Dirac equation, the coupled equations for \{r) and 77(7-) become

In the non-relativistic limit, T, \p\, and all components of |VM| = |e.AM| are assumed to be very much smaller than m. Hence, the second of the two equations (5.52) shows that the lower components of the Dirac spinor are very much smaller than the upper components, and therefore the equations are solved approximately by eliminating the lower components, as we did for the KG equation. However, if we proceed directly by solving the lower equation for 77 and substituting the solution into the equation for x< we obtain

Since T is of the same order as V°, which is ~ p2/m, it is necessary to expand the denominator of the second term if we want to collect all terms of order p4/m3.

Tx = V0x + <r-(p- V)v (2m + T)n = V°r] + (T ■ (p - V) x ■

This expansion gives

4 7TI2<T

Note the presence of the energy T in the last term on the right-hand side. This means that the effective Hamiltonian defined by Eq. (5.54) is dependent on the energy, and an energy-dependent Hamiltonian leads to many complications which should be avoided, if possible. The explicit dependence on the energy should be eliminated. Since the T dependence occurs only in the highest order term, it might seem that it could be removed by replacing it by an estimate obtained from the solution of the lower order equation, i.e.,

However, this method will not give a unique answer because T is a number and commutes with <r • (p — V), while V°, part of the above estimate for T, does not. It is better to attack the problem from a different direction.

A better method, known as the Foldy-Wouthuysen (FW) transformation [FW 50], is to transform the equations to a new form in which the off-diagonal elements of the Hamiltonian are so small that the leading order estimate of the lower components (which does not depend on the energy T) is sufficient to get the effective Hamiltonian to the desired order of accuracy. For example, in this problem where we want the Hamiltonian to order p4/m3, it would be sufficient to reduce the off-diagonal elements to order p2/m. If they were that small, the leading contribution from the lower components would be of order p2/m2, and their contribution to the equation for x would therefore be of order p4/rri3, sufficient for our purposes. In the KG case treated in the last chapter, the off-diagonal elements were initially that small, so we were able to get the desired result immediately. Here, the off-diagonal elements of the Dirac equation are of (larger) order p, so the simplest approach did not work.

To prepare for the application of the FW transformation, return to the matrix equations (5.52), and write them in terms of Dirac matrices

The off-diagonal terms are those involving the Dirac matrices a, and they are large (of order m°). We want to transform the equation so that they are of order m~l. Then, when the equation is solved, T will not enter into the m~3 term.

The equation will be transformed using a general unitary transformation constructed from the Dirac matrices. Since the large off-diagonal terms we wish to

Hence, choosing A = ~ gives

•^off-diag — a ■ V , which is O (m-1) by assumption.

With these approximations, the coupled equations (5.52) become

where only the largest (leading) terms have been retained in every element but H'n, which is yet to be reduced. We may now neglect Trf in the second equation, giving

Noting that the large

The remaining task is to reduce H'n using A = terms proportional to m occur in the combination (-1 + /3), which makes no contribution to the H'n matrix element, we have, to O (m~3),

where the first three terms on the RHS are the expansion of A V°A, the first two in the second line are the expansion of the contributions from Ua-(p — V)U'"1, and the last is the combined contribution from Um0U~l. To further reduce these terms we will use the identity

Using this identity gives

=■ 2p2 -p ■ V - V p - ier ■ (p x V) - ier ■ (V x p) = iP - V)2 +P2 ~ V2 ~ tr ■ [V x V] = (p-V)2+p2-V2-etr-B , (5.68)

where the use of square brackets will mean that p or V operates only within the brackets. Note the new term describing a magnetic moment interaction, which

*For an introductory discussion of these topics see, for example, Gottfried (1966) or Sakurai (1985).

Darwin term

8m2 v 8m2 8m2 w

Because of the 63(r), this term is non-zero for S-states only. Physically, it comes from quantum fluctuations in the position of the electron, referred to as Zitterbewegung (jittering motion), which make the electron sensitive to the average potential in the vicinity of its average position. The average of the potential is proportional to V20 ~ <53(r), and this accounts for the general structure of the Darwin term.

where S = <r/2 is the electron spin operator. This term is due to the interaction of the electron's magnetic moment with the magnetic field it sees due to its motion and automatically includes the Thomas precession, which reduces the result naively expected by a factor of 2. It is zero in S-states, because L = 0.

The Darwin term contributes only to L = 0 states, and the spin orbit term only to states where L ^ 0, but when both corrections are taken into account, the spin orbit splitting is given by a single formula which depends only on the principal quantum number and the total angular momentum j of the state,

2 n2

The first term is the familiar nonrelativistic result, and the second is the fine structure, which splits states with the same n but different j. In the next chapter we will show that the exact solutions of the Dirac equation also predict levels which depend on n and j only. This gives a good account of the main features of the hydrogen atom spectrum, but the additional ¿-dependent Lamb shift can only be explained by field theory, as we discussed in Chapter 3.

Zeeman Effect (Dirac)

The full Zeeman effect comes from two terms. The orbital part is the same as the result obtained from the KG equation and was calculated in Eq. (4.60). The result is

-¿(P.A+A.PH--ZTL . Combining this with the spin part, —eB ■ cr/2m, gives

Note the factor of 2 for the electron's intrinsic gyromagnetic ratio. This factor has no classical explanation but was discovered empirically before the Dirac equation was discovered. Its automatic appearance in the Dirac theory is one of its major successes and provides the only "explanation" for this effect that we have.

5.8 THE LORENTZ GROUP

The Dirac space is four-dimensional but is otherwise an abstract space unrelated to physical space-time. To discuss the Lorentz transformation (LT) of a Dirac wave function, the Dirac equation, or a Dirac matrix element requires that we first construct a representation of each Lorentz transformation on the Dirac space and then show that the wave functions and matrix elements transform in such a way that the Dirac equation is invariant in form and the matrix elements transform as scalars, four-vectors, or tensors, depending on their structure. In this section the properties of the Lorentz group will be reviewed, and in the next two sections the representation of the Lorentz group on the Dirac space will be worked out and the construction and transformation of Dirac matrix elements will be discussed.

In Sec. 2.1 we discussed how Lorentz transformations change the space-time coordinates. Any transformation which leaves the metric tensor invariant is, by definition, a LT. In the matrix notation, Eq. (2.8), this was written

The set of all transformations which satisfy this constraint form a group, which is called the homogeneous Lorentz group. The four group properties are easily demonstrated:

• If Ai and A2 are members of the group, then AXA2 is also, because

• The multiplication law (matrix multiplication in this case) is associative:

• There exists an identity A = 1 which is a Lorentz transformation.

• For each A, there exists an inverse A-1 because

and hence det A = ±1, and since it is not zero, A-1 exists. Multiplying Eq. (2.8) by (AT) 1 from the left and A-1 from the right gives

G = (A_1)t GA"1 showing that A-1 is a Lorentz transformation.

Fig. 5.3 Diagrammatic representation of the four classes of the homogeneous Lorentz group connected by the discrete transformations T, P, and TP.

Figure 5.3 illustrates this continuity by showing the four classes as disconnected regions, with a continuous distribution of transformations within each region (class). The figure and the above equations show that to study the homogeneous Lorentz group, it is sufficient to study the group of continuous transformations L\ and the two discrete transformations T and P.

The complex LT's must also have detA = ±1, but the restriction (5.78) on Aoo no longer holds [because (Aj0)2 need no longer be positive]. Therefore

Table 5.2 Properties of the four classes of homogeneous Lorentz transformations.

Label

Properties

Class

Continuity with

Aoo > 1 detA = +1

orthochronous, proper restricted group

1

Aoo < -1 detA = +1

non-orthochronous, proper

TP

Ll

Aoo > 1 detA = -1

orthochronous, improper

P

Li

Aoo < -1 detA = -1

non-orthochronous improper

T

5.8 THE LORENTZ GROUP

the complex LT's separate into only two classes, depending on the sign of the determinant, and the transformations in L\ and L\j_ can now be connected by a continuous path. As an example of such a "path," consider the transformations

cos 6

¿sin0

COS0

- sinö

sin 0

cos 8

i sin#

which depend on the continuous parameter 9. These transformations satisfy (2.8) for all values of 9 and hence map out a continuous path of transformations in the space of complex LT's. By varying 9 continuously from 0 to 7r, we are able to connect the transformations 1 and -1. This fact will be of crucial importance to our discussion of the PCT theorem in Sec. 8.7.*

Infinitesimal Transformations in l\

Consider the real LT's in the subgroup L\. Because they can be continuously connected to the identity, they can be written

where 9 is a number and A is said to be the generator of the transformation A. [This is assumed without proof. It is a general property of a continuous group.] The structure of the group can be inferred from the structure of the generators A.

To study this structure, it is sufficient to consider those transformations for v hich 9 = e is infinitesimally small. In this case, the transformations can be expanded and only the first order terms retained, so that

It is easy to determine the structure of A from this equation, which looks like A31 A41 \

/An

A21

A12

A22

A23

\ ...

-An

A12

A21

— A22

A31

"For more discussion of these issues see Sweater and Wightman (1964).

From this equation we draw the following conclusions:

• All diagonal elements of A are zero.

• There are three independent A's which are symmetric. These have space-time components and are the generators of boosts.

• There are three independent A's which are antisymmetric. These have spacespace components and are the generators of rotations.

The generators therefore span a six-dimensional vector space. The six independent generators which will be taken to be basis vectors for this space are denoted wM„, where /i ^ v and w^ = 1 in the /ith column and ¡4h row, is symmetric or antisymmetric depending on the indices, and has zeros for all other elements. Explicitly,

LJ12

0

1

0

0

0

0 0 1

0

0

0

1

0

-1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

-1

0

1

0

0

0

0

1

0

0

0

0

-1

0

These generators can be written in the following compact form:

{Unv)0 0 = —\tnv\o£Xaa0 , where eM„Q/3 is the four-dimensional antisymmetric symbol normalized to

The w's are the basis for a six-dimensional space of 4 x 4 traceless matrices, so that any generator is now described by six continuous parameters. The most general infinitesimal LT is then

The continuous parameters are & and 9X. By considering a succession of infinitesimal transformations, we can exponentiate this expression and write the finite transformations as in Eq. (5.83),

Equation (5.92) is the explicit characterization of the LT's in L\ which we have been seeking. To better understand this equation, we look at a few examples.

Then, using the familiar relations between the hyperbolic functions leads to the correspondence cosh£

and the familiar form for the active boost 1

The active boost in the i-direction propels a particle of mass m from rest into motion along the x-axis with momentum px.

5.9 COVARIANCE OF THE DIRAC EQUATION

Now we are ready to study the covariance of the Dirac theory. To establish covariance we must construct a representation of the Lorentz group on the four-dimensional Dirac space. In general, a representation of a group is a mapping of each element of the group A into a matrix 5(A) which preserves the group multiplication law. This means that if Ai A2 = A3, then 5(Ai)5(A2) = 5(A3). Since each group element has an inverse, the matrices which represent the group must also be non-singular, and the identity of the group is represented by the Hentity matrix.

The representation 5(A) we seek should operate on the four-dimensional Dirac space in such a way that the Dirac equation is invariant in form. For this purpose we use the covariant form, Eq. (5.12), with the matrices defined in Eq. (5.8),

Then, for any LT A which transforms the coordinates and four-vector potential from an unprimed frame to a primed frame,

we seek a representation, 5(A), which transforms the Dirac wave function from the unprimed to the primed coordinate system i{i'{x') = S(A)ip(x) .

Covariance is the requirement that this transformation leave the Dirac equation (5.12) invariant in form, so that in the primed frame,

5.9 COVARIANCE OF THE DIRAC EQUATION

This requirement determines S(A). To find the equation which defines S{A), substitute (5.101) into the above equation and multiply by 5_1(A). Recall that p'ii _ hn^p" implies that p' = (A-1)17 M and obtain

S-\k) (A-1)^ - eA„(x)) - m} 5(A)t/,(x) = 0 . This equation is invariant in form if which implies

This equation will tell us how to construct the 5(A).

Each A € L\ has the form given in Eq. (5.92) and is defined by six numbers The existence of a representation of the Lorentz group on the Dirac space implies that, for every choice of the six parameters, there exists a corresponding 5(A) of the form

with the same six parameters but with new generators which describe how the transformations act on the Dirac space. To find all of the representations, we need only construct the six generators.

To find the generators, it is sufficient to apply (5.102) to all infinitesimal transformations. If the parameters are infinitesimal, then

S(A) = l+^lBt + ^eieljkRjk A = 1 + \6l tijk^jk , and Eq. (5.102) becomes

5_1(A)7m5(A) = (1 - \eiiljkR]k) 7" (1 + freijkRjk)

= 7M-& [Bwy^-^k [Rjk, 7"] = [1 + + fatijkWjk]11 „ 7" • (5-104)

Since the parameters & and are independent, we must equate their coefficients, giving

-[Bi, 7"] = (Wio)" ,7" -[Rjk,Y) = (ujkyi,l'J ■

Substituting the specific forms for the w's given in Eq. (5.88) gives the following results for the boosts, Bi.

0 0

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