We now turn to the general question of how to calculate the higher order terms which arise in the perturbation expansion for the S-matrix. We have chosen to introduce this study by examining all the terms which arise in second order QED. The discussion therefore serves two purposes: the specific examples we study are of practical importance, and they also are rich enough to illustrate most of the issues which will arise in a general study.

A survey of all Feynman diagrams generated by second order processes shows that they are of two kinds. In some, the momenta of the internal (or virtual) particles is fixed by energy-momentum conservation. Such diagrams are referred to as "tree diagrams," and their "computation" involves no more that writing them down and evaluating all internal momenta using energy-momentum conservation. Examples of tree diagrams are scattering in the one-photon exchange approximator annihilation and pair production through a single intermediate photon, and Compton scattering, all of which were studied in the last chapter. The second kind of diagram has closed loops in which the momenta of all of the internal particles are not fixed by the four-momenta of the external particles. For each loop there is one four-momentum completely unspecified by energy-momentum conservation. Each different value of this internal four-momentum will give a different amplitude connecting the same initial and final states, and these amplitudes are therefore indistinguishable, and the rules of quantum mechanics tell us that these must be added together (by integrating over all possible values of the internal four-momentum) before we square the result to obtain predictions for physical observables. The evaluation of loop diagrams therefore requires that loop integrals be carried out, and this is much more difficult. Remarkably, there is a very powerful, standard method for evaluating loops. This method, referred to as dimensional regularizaron, will be introduced in Sec. 11.6. We will see that these integrations will sometimes produce infinities, which must be systematically removed if we are to obtain meaningful answers from the higher order terms. The infinities are first isolated through a process called regularization and then removed from the theory through a process referred to as renormalization. This involves systematically redefining the coupling constants and masses of the theory so that the infinities are systematically absorbed into these parameters, which are

then taken from experiment.

This chapter prepares the way for the more complete study of renormalization presented in Chapters 16 and 17. It also is a prerequisite for the study of bound states and unitarity presented in Chapter 12. However, a reading of this chapter in not necessary for the additional study of symmetries presented in Chapter 13, nor for a large part of Chapters 14 and 15, and the reader may prefer to turn to these topics first.

11.1 WICK'S THEOREM

We begin this chapter with a brief study of Wick's theorem, a standard tool for the study of higher order processes when quantum fields are treated as operators on a Fock space. Later, after we have introduced the path integral formalism in Chapter 14, we will be able to obtain the same results we obtain here using a completely different method.

The first problem we encounter in computation of higher order terms is the computation of the matrix elements of products of field operators, and Wick's theorem provides a systematic way to reduce these products. There are two theorems: one tells how to reduce a product of field operators to a sum of terms, each of which is a normal-ordered product, and the second tells how to reduce the product to a sum of terms, each of which is a time-ordered, product. In the following discussion, we will use the symbol <p to denote any quantum field: scalar, vector, or spinor.

Wick's Theorem for Normal-ordered Products

A contraction of two fields will be defined to be their vacuum expectation value and will be denoted by a square bracket connecting the two fields,

This contraction occurs naturally when normal ordering a product of two fields. Recall that fields can be broken into positive and negative frequency parts,

and since the (-) parts are always associated with creation operators,

11.1 WICK'S THEOREM

In order to treat commuting and anticommuting fields at the same time, we will use the notation

[a, b}± = ab ±ba . Then, it follows that, for both Bose and Fermi fields,

Also, since (¡>i+\x)\0) = 0 and <j>^(x)\0) = 0, it follows that and therefore, for both Bose and Fermi fields,

ct>(x)<pHy) =:<t>(x)<l>Hv)--+4{x)<l>i{y) .

This observation was previously encountered in Eq. (7.43), but now will be generalized to products of more than two fields.

When more than two fields are present, we will frequently encounter the product of a contraction (a c-number) multiplied by a normal-ordered product of fields (a q-number). For this purpose it is convenient to define a rearrangement of this product in the following way*:

(-1 Y<^jj>k : 4>\ ■ ■ ■ <t>j-\<t>j + \ ■ • • <Pk~l4>k+\ ■■■4>n-

= : 4> 102 ■ ■ • <pj ■ ■ ■ <pk ■ ■ ■ <t>n- , (11-8)

where = <fo(xj) and p is the number of interchanges of Fermi fields (even or odd) required to move 4'j and 4>k from their position on the RHS of the equation to their position on the LHS of the equation where they are in front of the product. This notation is convenient, but don't forget that no contractions are possible within a normal ordered product because the normal ordering insures that any vacuum expectation value of any two fields is zero. The product on the RHS of (11.8) should not be thought of as a contraction of fields within a normal-ordered product, but only as a shorthand for the LHS of the equation. Finally, note that the phase (-l)p is always +1 unless both <t>3 and <t>k are Fermi fields, and there are other Fermi fields "in the way" which must be passed in pulling <p3 and 4>k to the front.

With these definitions, the Wick theorem for normal-ordered products can be stated in a deceptively simple way:

Theorem: The ordinary product of field operators is equal to the sum of normal products with all possible contractions, including the normal product with no contractions.

"The discussion here follows Bogoliubov and Shirkov (1959).

Formally, we can write

<t>\ ' ' ' <Pn 4>1 ' ' ■ <f>n- += <£ljp2 ' ' ' 4>n- += <t>2<j>3 ' ' ' <t>n- + ' ' '

+ ; ^xj>2<^>i4>5 ■ ■ ■ <t>n-H----

+ : <£l</>2 <¿3^4 <$>S<j>6<t>7 •••</>« + •• • ■ (11-9)

Note that if all pairs of fields have non-zero contractions (which is not generally the case, of course), then for even n the above sum contains 1 uncontracted term n(n-l) 2

terms with one contraction terms with two contractions

2n^2"n/2)! fully contracted terms.

However, only fields with non-zero commutation or anticommutation relations can give a non-zero contraction, and thus in practice most of the fields will not contract. Hence, for example, ijnj)

^ ^ 0 ^ 0 if 0 is a charged field if <f> is self-conjugate .

^ ^ 0 ^ 0 if 0 is a charged field if <f> is self-conjugate .

The general proof of Wick's theorem (which is not difficult) can be found in many texts. Rather than present a general proof, we will work it out for second order QED. We limit ourselves to spinor QED, where the radiation part of the interaction Hamiltonian has the form H =:ip(x)ip(x)a(x):, where, in the interests of simplicity, the Dirac matrix and vector indices have been suppressed. The second order product of two radiation terms (time-ordered products will come later) which we will encounter is

Fields within a normal-ordered product must not be contracted, because they are already in normal order. Applied to the above product, Wick's theorem gives eight terms:

H2 =: ip{x)ip(x)A(x)ip(y)ip(y)A{y): +: rp(x)ip{x)A(x)ip{y)ip(y)A{y): +: ip{x)\l)(x)A{x)\j>{y)\{>{y)A{y): +: ■4>{x)ip{x)A{x)tp(y)-tp{y)A{y): +: ip(x)rp{x)^(x)4>{y)ip{y)^{y): +: 4>(x)ip(x)^(x)ip{y)ip(y)^(yy.

+: f(x)ip(x)A(x)tp{y)f{y)A{y): +: f{x)ip(x)A(x)xp{y)^(y)A{yy. .

11.1 WICK'S THEOREM Bringing all of the contracted fields to the front gives

H2 =:ip{x)ip(x)tp{y)ip{y)A{y): +A{x)A{y) :ip(x)ip{x)ip(y)'ip{y):

+ j>{x)ip(y) ■■ A{x)A(y)ip(x)ip(y): +ip(x)ip{y) : A{x)A(y)ip(x)ip(y):

+ Tp(x)j>{y) A(x)A(y) : ip(x)ip{y): A(x)A(y) : ip(x)ip{y) :

+ ip(x)i>(y) j>{x)j>{y) : A(x)A(y): +tp(x)ip(y) rp{x)y(y) A(x)A(y) .

In every case in this example the p of Eq. (11.8) is even.

We will now prove that Eq. (11.12) is the correct result by putting all of the fields into normal order, being careful to keep any commutators or anticommutators which may arise in the process. Since the ^4's commute with the ip's, we can place the X's in normal order independent of the ip's. The therefore give

Now, look at the xp terms. These are

= (^(~](x) Ui+){x) + V(_)(®)1 + ipi+){x)Tpi+)(x) - ^^(x^+^x)

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