Anan

where { , } denotes the anticommutator.

We next discuss a remarkable fact: the Schrôdinger theory can be quantized equally well by imposing commutation or anticommutation relations on the operators a and a*. As a demonstration of how this works, we show that the Hamiltonian satisfies (7.15), a condition which must hold if the Hamiltonian is to be interpreted as the generator of time translations.

These matrices have the following properties:

as required by their definition.

Finally, we summarize the main result: The Schrôdinger theory is consistent with either commutation relations (Bose-Einstein statistics) or anticommutation relations (Fermi-Dirac statistics) and hence provides no connection between spin and statistics. The same statement does not hold for relativistic field theories. One of the major triumphs of relativistic quantum field theory is that it does provide such a connection. It can be shown that

Integer spin <=> Bose-Einstein statistics Half (odd) integer spin <=> Fermi-Dirac statistics

We will show this for spin 0 and spin \ systems now.

7.3 CHARGED KLEIN-GORDON THEORY

A charged KG field must be complex [otherwise the current defined in Eq. (4.11) would be zero]. The Lagrangian density for a classical complex Klein-Gordon field is

where (j> and <j>* are regarded as independent fields (corresponding to two independent real fields required to describe the two charge states of the field). The equation of motion which follows from this Lagrangian is the KG equation

The generalized momenta are dC

so that the Hamiltonian density becomes

7.3 CHARGED KLEIN-GORDON THEORY

Therefore the total Hamiltonian is

H = J d3r H{r, i) = J d3r [tt* tr + V^'V^ + m2<p*(t>] .

A convenient formula for the total energy can be obtained from this expression if we substitute for n, integrate by parts (dropping surface terms), and use the KG equation to simplify the final expression:

dr^dl\8t

where 3/dt = d/dt - d/dt is familiar from the KG norm, Sec. 4.2.

Now expand the KG field in terms of positive and negative energy solutions of the free KG equation. As before, the field is quantized by imposing commutation (or anticommutation) relations on the expansion coefficients, a step which turns them into particle creation and annihilation operators. As we saw in Chapter 4, the complete expansion of a relativistic field requires both the positive and negative energy solutions and therefore has the form

where (t>l^\x) are the normalized ± energy states, defined in Eq. (4.19), and we will show that the operators a and c have the following interpretation:

an destroys a particle with momentum pn and positive charge c„ destroys an antiparticle with momentum p„ and negative charge al creates a particle with momentum pn and positive charge cl creates an antiparticle with momentum pn and negative charge .

The interpretation assigned to the a„'s is a straightforward application of our previous study. That the cn's should destroy and create antiparticles (instead of negative energy particle states) is necessary if the field theory is to describe only positive energy states, which is clearly the goal. That the coefficient of cn should be fan(x) instead of <p(n'{x) is required because the charge conjugation transformation shows us that the antiparticle states of momentum pn are related to negative energy states with momentum ~pn. An additional, desirable feature of the expansion (7.38) is that fa^ (x) has the covariant form of the scalar product p-x in its exponent, instead of the clumsy form Ent+pn-r. The operator cj, (instead of cn) must accompany an for two reasons. First, both an and c)n lower the charge of a state by one unit; an does this by destroying a particle with -I- charge, while does this by creating an antiparticle with - charge. Thus the field operator r-t

"n

(j> always destroys one unit of charge, and by a similar argument, the operator $ creates one unit of charge. Hence the operator conserves charge. Had cn been chosen to accompany an, the operator would not conserve charge; it would include terms like aj,c„, which creates two units of charge, and cj,an, which destroys two units of charge. A second consequence of this assignment is that if the particles were neutral, then the particles and antiparticles might be identical (but not necessarily); if they were, then a = c 0 = tf. A charged field requires that a ^ c and (f> $.

Next we use the orthogonality relations satisfied by free KG wave functions to reduce the Hamiltonian (7.37). These relations, previously given in Sec. 4.3, are , , d3rfa)*{x)~fa]{x)^8n i J d3r fa i J d3r fa

Hence the Hamiltonian becomes

where we have been careful to preserve the order of the operators an, aj,, cn, 4. Note that the second term is +cncjl, with the + sign coming from the negative norm of the states. This term has the cc* in the wrong order to be a number operator. If the c's (and hence the a's) satisfy either commutation or anticommutation relations, we may write

We now discuss the second quantization of the Dirac theory.

We now turn this classical theory into a quantum field theory by turning the field ip into an operator. This is done by expanding ip in a complete set of positive and negative energy eigenfunctions, introducing annihilation operators 6„iS and dn,g as follows:

where the ip^ are the positive and negative energy wave functions defined in Sec. 5.3. The structure of (7.51) is similar to the one we introduced for the KG theory and is justified in precisely the same way. The operators are interpreted as follows:

bis annihilates a particle of spin projection s, momentum pn annihilates an antiparticle of spin projection s, momentum pn creates a particle of spin projection s, momentum pn creates an antiparticle of spin projection a, momentum p„ , and, as required by charge conjugation, V'-«,-« (instead of ifihj) must go with

The Hamiltonian can be re-expressed in terms of annihilation and creation operators following a now standard method. Using the fact that the 1/>(±) are eigenfunctions of the Dirac operator and orthonormal relations satisfied by the states ip^K the Hamiltonian reduces to

Note that if d and b were complex numbers instead of operators, the second term would be negative, so H could not be positive definite, and there is no classical Dirac theory. Ironically, the positive definite norm gives us trouble with negative energy states since there is no way to change the sign of the energy.

However, if the b's and d's satisfy anticommutation relations, H can be made positive definite. If we require

This completes our discussion of the construction of free field theories and the connection between spin and statistics. We are now ready to study how interactions are added to these field theories.

7.5 INTERACTIONS: AN INTRODUCTION

The first step in finding the Lagrangian which correctly describes a given physical system is to identify the fundamental degrees of freedom, or particles, which are needed to describe the system. For atomic systems these are electrons and nuclei. The nuclei are complex composites, but at the energy scales probed by atomic physics they may be treated as fundamental. For nuclear systems, the choice of the fundamental constituents is still very much at issue; at energies of a few MeV, most physicists would agree that the neutrons and protons can be chosen to be the constituents, but at higher energies, when the structure of the nucleon begins to become evident, the basic constituents are the quarks and gluons which are the ultimate building blocks from which hadronic matter is formed. Nuclear physicists are currently trying to understand precisely how to incorporate quarks and gluons into the description of nuclei and at what energy scales the structure of nuclei becomes sensitive to the presence of these fundamental constituents. Finally, particle physics now has the very successful Standard Model, which includes six flavors of quarks and six leptons organized into three generations as shown in Appendix D, five kinds of gauge bosons, and the Higgs. Particle physicists continue to look for the sixth flavor of quark and for the Higgs and to search for evidence for the existence of possible additional particles, which would signal the 'jreakdown of the Standard Model.

In general, each fundamental constituent is described by a separate quantum field, and the full Lagrangian density is a sum of the free particle Lagrangian densities for each of the constituents plus an interaction term:

The free Lagrangians can describe particles of any type. If the interaction term £i„t is zero, the general solutions to the problem are states which are direct products of the free particle states described by each of the free Lagrangians Ci, and the particles are all free particles which do not interact. The interaction term contains fields which enter more than one £, and hence couples the fields together and produces the interaction.

How are we to determine the structure of the interaction Lagrangian? Much of the rest of this book will discuss this question. There is no simple answer, and the art of finding the correct interaction is at the heart of modern research in particle physics. In the last 15-20 years, interactions have been constructed to obey certain symmetries, and this appears to be the "correct" way to find the interaction. Gauge invariance appears to be one of the key symmetries, and the

properties of gauge invariant theories will be the subject of much of the latter half of this book. For now we note that £int must satisfy the following constraints:

• it must be Lorentz invariant;

The last requirement means that all of the fields in £int are evaluated at the same point in space-time. Interactions for which this is not the case are said to be nonlocal, and if used, care must be taken to insure that the non-locality is constructed in such a way that Lorentz invariance is not violated. Non-local theories will not be discussed in this book.

In addition to the rules outlined above, the interaction Lagrangian should be simple with as few free parameters as possible. The fewer the number of parameters, the greater the predictive power of the resulting theory. Another criterion, motivated more by simplicity that by any compelling physical requirement, is that £;nt should, if possible, contain no time derivatives. If this is the case, the generalized momenta are not changed by the interaction, and the Hamiltonian can be calculated from

H = ^ Hi + 'Hint i i where Hi are the free Hamiltonian densities corresponding to £, and Hmt — —£int. This condition makes the theory simpler, but it is sometimes not possible (for example, the EM interactions of scalar fields and quantum chromodynamics (QCD) involve time derivatives). In practice, interaction Lagrangians are usually polynomials in the fields, with a single parameter which defines the strength and s referred to as a coupling constant.

As an example, consider a system with two fundamental constituents described by the Dirac field rp and a Hermitian (therefore charge zero) scalar field (p. A simple interaction between these fields which satisfies the above requirements is

£int - -Hint = -9 ■ j(x)il>{x)4>(x): (7.60)

where g is a real constant (in order that £ be Hermitian). We will refer to this as a theory with <ps structure because the interaction Lagrangian is a third order polynomial; the term tj>3 theory will be applied exclusively to theories with an interaction involving three scalar fields interacting at a point.

It is important to get a physical feeling for the meaning of an interaction like (7.60), and this is fortunately very easily done. First recall that each field operator must create or annihilate only one particle. Hence if n fields act at a point (a theory with 0" structure), the interaction always describes a situation in which n-l particles come into the point and I leave (for any 0 < I < n). For example, to first order in a perturbative treatment of the interaction, a theory with a structure, such as that described in Eq. (7.60), describes the eight elementary processes shown in Fig. 7.1. The reason for this is that \¡>%jj(j> contains precisely time

£ = 0; 3 particle annihilation i = 3; 3 particle production

£ = 0; 3 particle annihilation i = 3; 3 particle production

£ = 1 ; particle absorption and 2 body annhilation

£ = 1 ; particle absorption and 2 body annhilation

£=2\ particle emission and 2 body decay rig. 7.1 Diagrams showing the possible interactions which result from the single term given in Eq. (7.60). The antifermion lines have arrows pointing in a direction opposite to the flow of time (see the discussion in Sec. 10.3).

three annihilation or creation operators and therefore has non-zero matrix elements between the following states:

where p\ and p2 are momenta of fermions, pi and p2 are momenta of antifermions (all described by the Dirac field tp), and k is the momenta of the scalar particle, which is its own antiparticle. All other matrix elements of Hmt are zero. In higher orders, other processes are possible (as we shall soon see), but they must all be built up out of the eight elementary processes above. Six of these (corresponding to £ — 1 or 2) describe emission and absorption of one particle at a time.

£=2\ particle emission and 2 body decay

t = 1 (p2\nint\pi k) <*|Wi„tbiP2} (p2\nint\pï k)

£ = 2 (pip2\Hmt\k) (j>2k\nint\pi) (pafc|Wi»t|px>

Many other interactions can be constructed. Examples of other types are: structure:

<p2n structure:

derivative:

non-linear:

where in all of these cases <p is a Hermitian field and <f>i is a complex (non-Hermitian) field. The so-called "non-linear" interaction derives its name from the fact that it contains polynomial interactions of all orders.

The simplest and most commonly encountered interactions have a <p3 structure, and these will occupy our attention in Chapter 9. Later, in Chapter 13, we will see that interactions with a <f>A structure occur in QCD and the standard electroweak theory and that the nonlinear interaction arises in the nonlinear sigma model. But before we begin our study of the dynamics of interacting theories, we take a first look at consequences which can be derived solely from the presence of a symmetry of the theory.

PROBLEMS

7.1 Show that the charged KG field discussed in Sec. 7.3 satisfies the CCR

and that all other commutators are zero.

7.2 Neutral KG theory. Construct the theory for a neutral KG particle (i.e., where the field <j> is Hermitian) from the following arguments:

(a) If the charged field <j> = (</>i 4- ifo)/-^, where <f>i and 4>2 are commuting Hermitian fields, and if the charged field 4> satisfies the CCR's worked out in Prob. 1 above, show that

= iS3(r — r') , where is either 4>i or </>2-

(b) Show that the Lagrangian density for the charged field <f>.

[which is just the operator form of Eq. (7.32)], can be written as the sum of tv/o independent Lagrangian densities

£ = £i+£2 , where each density is multiplied by an overall factor of

compared with the Lagrangian density for its charged counterpart.

(c) Using the density £j, find the momentum 7Tj conjugate to fa, and find the Hamiltonian density Hi for a neutral theory. Express the Hamiltonian Hi in terms of the annihilation operators ai„ = (an + Cn)/\/2 and the corresponding creation operators a\n = (an + c„)t/v/^.

(d) Discuss the significance of your results. What is the Lagrangian density for a neutral scalar theory?

7.3 Using the ideas developed in Prob. 2 above, and working from the Lagrangian density for a neutral massive vector field [given in Eq. (2.39) in Sec. 2.5], find the Lagrangian density for a charged massive vector field.

7.4 Consider a <i>3 theory with a charged scalar field and a neutral scalar field <j> and an interaction Lagrangian density of the form

(a) Write out the full Lagrangian density for the theory.

(b) Evaluate the following matrix element:

0 0

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