[Note the use of the screening factor e~£r, inserted to insure convergence of the integrals and removed after they have been done by letting e —» 0.] Now substitute (4.67) into (4.66), and assume Af2 is small enough so that all dependence of the integrand on the directions of kj can be ignored. This gives

(ZQ4TT)2

This is the Coulomb differential scattering cross section for a spin zero particle scattering from a fixed scattering center. Because there is no recoil, the behavior of the cross section is dominated by the familiar q~4 factor, where q2 — (kj — ki)2 = 2k2 (1 - cos9) = 4k2 sin2 - . The scattering is sharply peaked in the forward (9 = 0) direction.

4.8 NEGATIVE ENERGY STATES

The simple example we considered in Sec. 4.4 was sufficient to show that

• negative energy states cannot be ignored and

• they describe the production of particle-antiparticle pairs, which can occur virtually in higher order processes.

The (one-particle) KG equation can only do a limited job of describing pair production; a complete description of antiparticles must await the development of field theory (Chapter 7). In this section we lay the background for this study by developing the mathematical description of both positive and negative energy states, to the extent possible without the use of field theory.

To illustrate the techniques, we will calculate the matrix element for Coulomb scattering to second order in the electric charge e. From Eq. (3.24) for the time translation operator, the second order ¿»-matrix element is

The superscript (2) is to remind us that this is the contribution to the infinite sum (3.24) which is second order in the small electric charge e. While this formula was originally obtained for a field theory, it applies equally well, as noted in the previous section, to any quantum mechanical system described by an equation first order in time and which has been separated into an unperturbed Hamiltonian Ho and an interaction Hamiltonian Hi (written in the interaction picture). We may apply it to the two-component form of the KG theory, which casts the KG equation into a differential equation first order in the time.

The way to evaluate S(2) is to insert a complete set of states between ii/ (i2) and Hi{tx). Before we can do this, we must discuss the completeness relation for the KG states.

We are working in the interaction representation where the free states have been fixed in time (at t = 0 for convenience). The completeness relation for the KG states can be written

where 1 is a unit 2x2 matrix in the two-component space.

Proof: We can use the orthogonality relations to show that this has the correct properties. For any KG state </>(r')

where the minus sign in the second term compensates for the minus sign which comes from the norm of a negative energy state. However, it is instructive to prove (4.71) directly by construction. Substituting the solutions (4.47) directly

Next, introduce T = + <2) and t ~ {t2 - h) to get

JkfkJkik

The integral over T gives an energy conserving ¿-function, and the S-matrix reduces to the standard form with the reduced amplitude / given by

JkfkJkik

S4]tJ

Note that the negative energy states make a contribution to this sum, unless f'k~l or = 0, which is not generally the case.*

This confirms our conclusions from Sec. 4.4; the negative energy states cannot be ignored. Even if the initial and final states are restricted to positive energy, the full solution to any problem will usually include virtual contributions from negative energy intermediate states.

The next task is to give a physical interpretation to such contributions. At this point the single particle KG equation does not give a unique answer. First, observe that a-) a-)* ¿\ek+ei}t Jk,kJI

jj$J kfkJ fcjfc because the integral over t gives 6(Ek + Ei), which is always zero. Adding this term to Eq. (4.76a) and noting that 6(t) + 9(—t) = 1 give an alternative equation for the reduced amplitude

This equation gives the same mathematical result for the reduced amplitude / but suggests a very different physical interpretation. Later, we will see that field

*Note that these (—) matrix elements are zero if energy is conserved but are not zero in second order perturbation theory because the energy of the intermediate state is not the same as the energy of the initial (or final) state.

time time

Fig. 4.3 The left diagram illustrates the forward propagation of a positive energy intermediate state, while the right is the backward propagation of a negative energy intermediate state. The right-hand diagram is reinterpreted as the creation of a pair at time forward propagation of the antiparticle to time tj, followed by annihilation of the antiparticle at time ti.

theory naturally gives us the interpretation suggested by (4.76b), and this is the only picture which makes sense physically.

In the first of these two descriptions, Eq. (4.76a), both the negative energy and positive energy states propagate forward in time. In the second, Eq. (4.76b), the negative energy states propagate backward in time [because of the 9{-t) function which implies t2 < ¿i]. The meaning of this strange statement is illustrated in ' ig. 4.3. The second figure shows that the requirement t2 < h means that the line joining and t2 travels backward in time unless we turn the direction of motion around and think of a particle-antiparticle pair being created at time t2 and then annihilated at a later time ti. Thus the idea that negative energy states propagate backward in time, while at first very strange, actually enables us to reinterpret them as antiparticle states propagating forward in time. If the antiparticle states are the charge conjugates of the negative energy states, so that they carry opposite charge, opposite momentum, and have positive energy, then charge is conserved in both descriptions. Reinterpreting the virtual negative energy contributions as virtual antiparticle contributions shows how these contributions describe virtual pair production. This is consistent with the results we obtained in Sec. 4.4.

In order to reduce the amplitude further, we prove an important identity which will be used several times throughout this book:

Proof: Look at the complex ui plane. The integrand has only one pole at u> = E - it in the lower half plane. If t > 0, the contour must be closed in the lower half plane, while if t < 0, it must be closed in the upper half plane, in order that, in either case, the exponential has a negative real part and the contribution from the arc at oo converges (to zero). Therefore, the integral is e~lBt if t > 0 and 0 if t < 0. This agrees with the LHS of the identity. I

Using this identity (with n = 0) for the first term in (4.76b), and using it with t —> —t and u —► —w in the second term, gives the following reduction of (4.76b):

Discussion

The main results of this last section are:

• We will define the matrix elements so that positive energy states propagate forward in time, associated with 0(t2 - ti), and negative energy states propagate backward in time, associated with 6{t\ —¿2)- This is the Feynman prescription. There are two time-ordered diagrams, as shown in Fig. 4.3.

• By turning the negative energy line around and reinterpreting it as an antiparticle propagating forward in time, we see how pair production, a multi-particle process, is described by the one-particle KG equation.

For this interpretation to be consistent with the conventional rules of quantum mechanics, all incoming states with energy E must have the usual phase factor e~lEt and outgoing states the complex conjugate phase e+'Et. Using Ej = Eit it is easy to demonstrate that this is indeed true for both of the terms in Eq. (4.76b):

Furthermore, the energy denominators given in Eq. (4.78) are consistent with the rules of second order perturbation theory for positive energy intermediate states (with one intermediate particle for the first term and three for the second, as required by Fig. 4.3, and with a small negative imaginary part assigned to the ff — hf**"-™!:

1st term e_i(E|t_£;i)(t2_il) = e^Ej-Ek)t^e+i(Ek-Et)u 2nd term ^(E^E^ih-ti) _ e-i(E,+Ek)ti ei(Ef+Ek)t2

energy of the intermediate state in cases when the denominator might be zero, as discussed in Sec. 3.4):

1st term 2nd term

These same features will also arise in our study of the Dirac equation, which is the subject of the next chapter.

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