J4iaaia 4a

Note that these relations are consistent with the commutation relations [Ai, Aj] 6ij, and if we make the identification

the operators Ql and Pi can be regarded as canonical coordinates, because they fully describe the degrees of freedom of the field and satisfy the required commutation relations [Qj, P3) = iStJ.

Because these coordinates have eigenfunctions (the coherent states) with eigenvalues a and a*, we will first express the path integral in terms of the c-numbers a and a* and then replace the integrations over a and a* with integrations over the c-number fields </>Q and na.

To prepare the way for the construction of the path integral, first observe that the matrix element of an operator built from normal-ordered products of the annihilation and creation operators is

where it is understood that a = {at} and a' = {a-}. Next, we show that the completeness relation for the coherent states takes the following form:

The presence of the exponential factor is related to the norm (14.63) of the states.

Proof: We will first show that (14.67) commutes with all creation and annihilation operators, which establishes that it must be a multiple of the identity. Then we will show that the constant of proportionality is unity. Using Eq. (14.64), the commutator is

where we integrated by parts in the last step, flipping the a" derivative over onto the exponential where it gives a factor which cancels a^ Taking the Hermitian conjugate shows that the right-hand side of (14.67) also commutes with the creation operators, and since the annihilation and creation operators are a complete set, (14.67) must be a multiple of the identity. To evaluate this multiple, compute its vacuum expectation value

If we introduce a^ = re'8, then dai da* = 2 irdrdO, and J 2m it J0 J0

and hence the quantity (14.69) is unity.

We are now ready to find the path integral for the "coherent state representation" of the time translation operator in field theory. We begin by introducing the field theory equivalent to the wave function (14.1). Since the annihilation operators depend on time, the coherent states will also depend on time, and by analogy with (14.1), the coherent state wave function for the state |s) is therefore ips(a*,t) = (a,t\s) .

Note that the wave function depends on a*, and not a. Using the completeness relation (14.67), the time evolution of the wave function for the state |s) can be written tps

X <a„,i„|ao,io)(ao,io|s) [daS][/«,in;a;,i0)^K,io) , (14.71) where ai]0 are the quantities a* at the time t0 and the time translation operator is U(a*n,tn-,a'0,tQ) = (an,tn\a0,t0)

= J{da0]e-a° a° {an,tn\a0)t0) , and the integration volumes are

Note the similarity between the definition of the time translation operator in field theory, Eq. (14.71), and the single particle time translation operator, which is the propagator given in Eq. (14.5). The principal difference is that now we have an infinite number of coordinates {a*} instead of a single coordinate q. There will be a close similarity between most of the following steps and the discussion in Sec. 14.1.

Next we divide the interval [in,io] into n intervals and use the completeness of the coherent states to obtain the following general formula for the time translation operator:

If the number of intervals n —> oc, so that £J+1 - tj = e —► 0, then we may estimate the overlap between the coherent states at neighboring times tj+\ and t.3 by expressing this in terms of coherent states in the Schrodinger representation (as we did in Sec. 14.1), and using Eq. (14.66)

= (aJ+i |exp{—zeH (Af, A) \ a3) - (oj+ila.,) - ie{a]+l\H (A], A) \a,)

[1 -ieH(a;+1,aj)] S e[a*+i aJ~ieH(a*+"a^)] . (14.75)

Inserting this into (14.74) gives the following expression for the time translation operator:

U(a'n,tn;a'0,t0)= f [] K] [da,] ^""K^)} .

Replacing a by ip and a* by q, as suggested by Eq. (14.65), gives

U(q, vhere now

The expression (14.77) is identical to its counterpart (14.17) if each q and p in (14.17) is replaced by the set q = {<?*} and p- {pi}.

To complete the derivation, we must replace the ql and pi in (14.77) by the field functions <pa and 7tq, the c-number equivalents of the operators $ and II. The connection between these averaged fields and the qt and Pi is

where the sum is over all energies i and fa,i is a plane wave averaged over the small volume centered at a.

The /'s satisfy the following completeness and orthogonality relations (see Prob. 14.1):

The relations (14.78) can be regarded as a canonical transformation of the coordinates {qi,Pi} {0q, 7rft}. The details are saved for Prob. 14.1. We obtain i/(0„, £n; 4>o, t0)

Note that, as in the one-particle case, all operators have been replaced by c-numbers.

For the 03 theory under discussion, the interaction terms do not involve any derivatives, and hence the rrQ may be integrated out, and the ira(d(pn/dt) -H(0Q,7rQ) may be replaced immediately by the Lagrangian. Adding a convergence factor and a source term, as we did before, gives the following generating function:

where the normalization constants (or factors) N which are encountered when ntegrating over the (j)a [recall Eq. (14.23)] are dropped because the overall normalization is not important in determining the vacuum expectation values (as we showed in the last section). It is somewhat more elegant (but perhaps less precise) to replace the sum over a by an integral over d3r, giving the following generating function, which is taken as our starting point:

where, for the symmetric theory introduced in Chapter 9,

We will now discuss the computation of propagators and scattering amplitudes from this generating function.

Generating Function for Free Fields

Begin by ignoring the interaction Lagrangian £int. Then the generating function is

Z0[J) = J Vl^j j d*x{CoW+Ui<t>2M + J(x)<l>(x)}

It is convenient to express the f d4x integral in terms of momentum space fields using

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