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IV. CAUSALITY

 

The past influences the future but the future does not influence the past; that is the essence of causality.

 

A fixed spacetime geometry whose causal structure defines "future" and "past" is needed just to ask whether or not a theory is consistent with causality.

 

A fixed background spacetime has been assumed for the field theories that are the concern of this paper, but the future and past cannot be unambiguously distinguished for points inside nonchronal regions connected by closed timelike curves.

 

However, we can ask whether the probabilities of a set of alternatives defined entirely outside such regions are independent of the geometry of spacetime to their future.

 

It is straightforward to see that the generalized quantum mechanics of matter fields described in the previous section is not causal in this sense if the evolution through nonchronal regions is not unitary.

 

Suppose that spacetime contains a single nonchronal region that is to our future and we are concerned with

the probabilities of a chain of alternatives C all occurring before the nonchronal region. If these alternatives decohere, then their probabilities p(a) are given, according to (3.1) and (3.12), by

region and N = Tr(XpXt).

 

Were X unitary, … there is no dependence on the geometry of spacetime to the future of the surface … whether or not it contains nonchronal regions. In this sense, unitary evolution leads to causality.

 

If X is not unitary, then the probabilities defined by (4.1) depend on the future geometry of spacetime.

 

Experiments could, in principle, detect the existence of nonchronal regions in our future by testing whether

present data is better fit by (4.2) or (4.1) with the appropriate.

 

We shall return to some simple considerations of such experiments in Sec. V.

 

Another way of seeing that information about the future is required to calculate present probabilities is …  Equation (4.3) is the formula for the probabilities of a generalized quantum mechanics with both an initial condition … and a final condition …. both the future and the past is required to make predictions in the present. In the example under discussion, that information concerns the failure of unitarity in the future arising from nonchronal regions of spacetime.
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