The action of the p-propagator U_R(0,t) is to advance the phase Γ, or a phase variable, forward in time from 0 to t. This must be equivalent to advancing time from 0 to s, then advancing time from s to t, whenever 0 < s < t. This implies that

(8.17)

The right hand side of (8.4.1) is a physical rather than mathematical statement. It is a statement of causality. If we wish to understand how we can generate B(t) from B(0) through an intermediate time s, we find that we will have to attack B first with the operator U_R(s,t) and then attack the resultant expression with U_R(0,s). The operator expression U_R(s,t)U_R(0,s)B cannot be equal to U_R(0,t), because its time arguments are not ordered from left to right. The correct operator equation is

(8.18)

To prove (8.18) we consider the product on the right-hand side and show that it is equal to U_R(0,t).

(8.19)

The first two terms are straightforward so we will consider in detail the three second order terms. In the second of these three terms the integration limits imply that

so that the time arguments of the operator product are correctly ordered, and we relabel them as follows:

The integration limits are independent, so we can interchange the order of integration, (8.8). After dropping the primes in the third term, all three terms have the same integrand so we need only consider the integration limits. The three second order terms are

In the second and third terms, the s₁ integrals are the same and the s₂ integrals add together to give

Now the s₂ integrals are identical and the s₁ integrals add together to give the required result

This is exactly the second order term in U_R(0,t). It may seem that we have laboured through the detail of the second order term, but it is now straightforward to apply the same steps to all the higher order terms and see that the result is true to all orders. Indeed it is a useful exercise for the reader to examine the third order term, as there are four integrals to consider, and after the same relabelling process is applied to the second and third terms, the four integrals obtained collapse from the right-hand side.

Combining equations (8.17) and (8.18) we see that the p-propagator U_R obeys an anti-causal associative law, (8.17). The fundamental reason for its anti-causal form is implicit in the form of the p-propagator itself, U_R. In applying the p-propagator to a phase variable it is, as we have seen, the latest times that attack the phase variable first.

Apart from the present discussion we will always write operators in a form which reflects the mathematical rather than the causal ordering. As we will see any confusion that arises from the anti-causal ordering of p-propagators can always be removed by considering the f-propagator form and then unrolling the operators in sequence to attack the phase variables. The f-propagators are causally ordered.