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C00002 00002	cause[w83,jmc]		Generalized causal systems
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cause[w83,jmc]		Generalized causal systems

	The theory of dynamical systems originated by George Birkhoff
provides a general setting in which to discuss causality.  However,
quantum mechanics raises such difficulties for causality, that it seems
worthwhile to cast our mathematical net wider and look for a mathematical
formalism more general than dynamical systems.  Here is the idea.

	1. A dynamical system involves a notion of time, usually continuous,
but discrete time is also treated.  There is a topological space of
states, and the state at a future time is determined by the state
at a given time.  The real line or the non-negative real line acts
on the space, so that we have a function  f: X x R → X   or
f: X x R+ → X  or  f: X x N+ → X, where this last can usually be
given by  f(x,n) = g↑n(x), where  g: X → X.

	2. A partial differential equation considers a function on
a space and connects the value of the function at a point with
its values in the immediate neighborhood of the point.  The main
kinds of partial differential equations are hyperbolic and elliptic.
Elliptic partial differential equations involve prescribing a function
on the boundary of a region and determining its value inside from
the equation.  That is, there are existence and uniqueness theorems
that determine a value from a prescription of the values on the
boundary.  (For present purposes we needn't distinguish further
the various kinds of elliptic boundary value problems).

	Hyperbolic equations involve the specification of the
function (and a normal derivative) on space-like
hypersurfaces and determine the values of the function on
"future" hypersurfaces.  Hyperbolic systems admit co-ordinate
systems in which one co-ordinate may be regarded as time and
in which the surfaces  t = constant  are all space-like and
hence appropriate for the prescription of initial values.
Therefore, a hyperbolic system can be considered to give rise to
a dynamic system.  Actually, we get a whole family of dynamical
systems, since the time co-ordinate as a function in the space
is not determined in a unique way.

	Causality is the notion of the present determining the
future.  Systems admitting causality seem essential if evolution
is to occur, if history is to occur and if purposeful action is
to occur.

	However, quantum mechanics offers great difficulties to
a causal interpretation, so we want to explore the possibility
of what might be called "weak causal systems" or "approximately
causal systems" or systems that map into approximately causal
systems.  This exploration is in a preliminary state, and I
don't now know that the proposed formalisms will correspond
to quantum mechanics in any way.  However, let's try.

	Suppose that the differential equations are not hyperbolic,
i.e. they do not admit space-like surfaces on which the values of
the functions can be prescribed.  What the appropriate boundary
conditions are I can't say.  In fact there needn't be appropriate
boundary conditions, i.e. there may be solutions of the equations,
but they may not be parametrizable by prescribing something on
the boundary of a region.

	However, suppose there is a time function on the space, and
suppose that there is a map from the space into 3-dimensional space,
although the space as a whole is more than 3-dimensional.  Suppose
further that the dependent variable may be approximately prescribed
on a surface t = constant.  ***** Stuck at this point.