YHy y [yiy i

\o put in normal order, all i/>( )'s to the right of any ?/>(+) must be moved to the left, and all V(+)'s to the left of any must be moved to the right. The terms not already in normal order are

Terms = ^ {x)xl)(+) {x)^ {y)ip{y) + ip(+) (x)ip{+} {x)i>(-) (y)ip(y)

These reduce to a sum of normal-ordered terms plus additional terms as follows: Terms

/ normal-ordered \ V terms J

ip{+){x)ip[+){y) + ^(xW^Hv) + il)(-\x)i)(+){y) - ^{y^+Hx)

Now, combining the tp terms with the A terms gives

H2 = {: A(x)A(y)-. +A(x)A(y)} |:${x)iP{x)ij>(y)il>{v):

Multiplying these out generates the eight terms given in Eq. (11.12) if we recall that [?//+)= j>{x)ip{y), etc.

Wick's theorem for normal-ordered products is a useful result, but what is really needed is an analogous theorem for the time-ordered products which occur in the perturbation expansion for Ui. We turn to this now.

Wick's Theorem for Time-ordered Products

First, consider the time-ordered product of a single pair of field operators. This is defined to be

[v<PHy)4>(x) x0<y0 , where 77 is -1 for the Fermi fields and +1 for the Bose fields. However, following our previous discussion this is just f : (f>{x)<f>Hy): + <p(x)^(y) x0 > y0

{ 77: </>t (y)(j)(x): (y)<f>(x) x0 < y0 .

However, the order of terms in a normal-ordered product can be changed (if we respect the anticommutation relations) and hence we can define a time-ordered contraction by

T (<j>(x)<f>Hy)) =:<l>(x)<l>Hy):+$&)l>Hv),

where the time-ordered contraction is distinguished from the normal-ordered contraction by placing the square brackets above the fields, and from Eqs. (11.16) and (11.17)

where 77 is ±1 depending on whether there are an even or odd number of interchanges of Dirac fields required to put the <f>'s in the desired order. Now apply the Wick normal order theorem to this product. The normal-ordered contractions will all be present with the correct time ordering. Next, using the fact that the order-ings of fields within a normal-ordered term can be interchanged (with the usual phase for Fermi interchanging), we can restore the normal-ordered terms to their standard order, fafa ■ • • 4>n- The resulting phase which will remain will be ±1, depending only on whether the normal-ordered contractions have an even or odd number of Fermi interchanges. Finally, for all terms with the normal contraction

corresponding to ¿1 > t2, there exists identical terms corresponding to tx < t2 which give

V^l X • These terms may be combined, giving

(<¿1^2 + rj <£2^1) X = &2X > which proves the theorem. I

We are now ready to apply these ideas to a systematic study of QED in second order.


We use QED to illustrate these ideas because it is the simplest, most successful quantum field theory for which perturbation theory works. (Recall a ~ 1/137). We will consider electron interactions only, so the Hamiltonian is a version of the one given in Eq.(10.8) and has the form

Note that there are no protons (there is only one fermion, the electron) and that we have restored the electron instantaneous self-energy term ignored in our previous discussions. Also, our treatment of Hinst will depart from the normal procedure.

Instead of normal ordering the entire term, we will only normal order each .7° in this term. That is, we use

J°e(r,t)J°e(r',t) = e2: ibVM)] : : [V>(r', i)/^', i)] : (11.25) instead of

0 0

Post a comment