and dividing by the various factors, we see that the ground state expectation value is proportional to {q', oo|g, -00),

Ti—*i oo Tt —» — too where Mz(q', q) is a function of proportionality which insures that the product on the right-hand side is independent of q and q'. The ground state expectation value is therefore proportional to the original functional z[J], provided we let T/ and —T} —► 00 along a line which passes from the second to the fourth quadrant, as shown in Fig. 14.4.

As discussed above, an alternative way to insure convergence is to give the energy a small negative imaginary part, E —+ 2£(1 — ie). With applications to field theory in mind, where the non-interacting Hamiltonian has the form of a simple harmonic oscillator, it is sufficient to approximate the energy by |q2, where q2 is the square of the coordinate. This is a simple positive definite quantity which is a minimum for the ground state. This is the procedure which we will follow, and hence the ground state expectation values will all be obtained from

where Z differs from 2 only in the limits of the time integration and in the small negative real part (proportional to q2 to insure convergence in q as well as t and coming from the negative imaginary part given to the energy).

The ground state expectation values are proportional to derivatives of the generating functional Z[J}. How are we to determine the unknown constant A/z? From Eq. (14.57) this appears to be a very difficult task, and indeed it would be if it was necessary. However, all the physics can be extracted from Z[J) without knowing the proportionality constant. This is because the vacuum state must be normalized to unity, and if J - 0, Eq. (14.57) tells us that

Hence the vacuum expectation values are obtained from the normalized generating function Z,

This gives Z[0] = 1, as required. From now on we will freely ignore any multiplicative constants which may emerge in the computation of Z[J], anticipating the fact that the physics comes finally from Z[J], where all constants cancel.

We now apply the previous ideas to field theory. For simplicity we first treat the symmetric <f>3 theory introduced in Sec. 9.1. The results will be extended to spinor fields in Sec. 14.6 and to gauge theories in the next chapter.

The central idea in extending the path integral formalism to field theory is to replace the generalized coordinates Q of the previous discussion by the fields 4>, which will be the new generalized coordinates. However, two problems prevent us from carrying this over directly. The first problem is that the <p's themselves have an uncountable number of degrees of freedom (the values of <j) at each space-time

*I thank Michael Frank for helpful conversations on the definition of path integrals in field theory. See [Fr 91], point), but this is easily handled by dividing up space into N3 non-overlapping cells, centered at the points a = (xa,ya,za). We then average the fields over each small cell (with volume Va) centered at a:

V Ct Jva

The averaged quantities 4>a(t) are now a countable number of independent coordinates which are also better behaved than the original fields (recall Prob. 1.5).

The second problem is more serious and requires some discussion. The key to the development of the path integral is the introduction of eigenstates of the operators which correspond to the generalized coordinates, but the quantum field operators have no eigenstates. However, the coherent states, which we introduced briefly in Sec. 1.7, are eigenstates of the annihilation operators Ai (in this chapter, these operators will be denoted by capital letters in order to distinguish them from their eigenvalues, ai). These states will be denoted |a), where ia>=nz^ Mr io>,

where the product is over all frequencies i, m is the number of quanta with frequency i, and the state is described by the numbers a — {at}. These states are not normalized; their norm is

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