From: THEORM::STAPP 25-AUG-1997 15:18:54.64
To: @KLEINDIS.DIS
CC: STAPP
Subj: Reply Re: Causality/Hume/Hilbert-Space
Pat, Gregg, and Stan sent comments on August 23.
Pat wrote:
P1. Special relativity might not be absolutely accurate.
[Gravity causes problems!]
P2. How did causality get in there?
[Hume shot it down!]
Reply to P1: I mentioned in an earlier posting that we were
talking about huge (25% of the outcomes opposite to expected)
effects that hold in principle independently of separation
distance. Gravity effects should be negligible on this this
scale.
Reply to P2: In the example Pat cited (the reaction of the
wall to my pressing my hand against it) the reaction does
come come after the cause, which is my pressing of my hand
against the wall.
In relativistic physics, both classical and quantum mechanical,
the idea of causality is introduced in the following way:
We have some (putative) law of evolution of the universe. It is
specified by picking a Lagragian.
The evolution is specified by the boundary conditions + these laws.
The boundary condition can initially be specified as the complete
description of "everything" before some "initial time" T_in.
The laws are then supposed to determine in principle "everything"
for all later times.
By "everything" one means, in classical mechanics, the values of all of the
physical variable that are supposed to describe the physical universe.
By "everything" one means, in quantum mechanics, all of the
"expectation values" of all the of conceivable possible physical
observables. ["Expectation value" means predicted average value over
an (in principle) infinite set of instances.]
To bring in the notion of causality one extends the idea of the boundary
conditions. It is possible in both classical and quantum theory to
imagine the changing incrementally the Lagrangian that specifies the laws:
The change might correspond to changing an external magnetic field in some
small spacetime region R that lies later than time T_in. This change might
be regarded as being introduced whimsically by some outside agent. But, in
any case, one can compare the values of "everything" at times later than
time T_in in the new modified world (i.e., controlled by the new modified
Lagrangian) to the values generated from the laws specified by the original
Lagrangian. If one is dealing with an idealized world without
gravity, or at least with no distortion of the `flat' Minkowsky spacetime,
then it is a mathematical property of relativistic field theories, both
classical and quantum mechanical, that
"nothing" will be changed outside the forward cone of the region R
in which the Lagrangian was changed!
In other word, "everything" will be exactly the same in the two cases
at all points that cannot be reached from the spacetime region R without
moving faster than the speed of light.
This property of relativistic field theories is called a causality
property. The intuition is that this change in the Lagrangian can be
regarded or identified as a "cause" because it can be imposed whimsically
from outside the physical system. The mathematical property just described
says that the effects of this "cause" are confined to its forward light
cone; i.e., to spacetime points that can be reached from the spacetime
region R of the cause without ever traveling at a speed greater than
light.
This relativistic causality property is a key feature of relativistic
field theories in flat Minkowsky spacetime: it is all the causality
that the orthodox pragmatic quantum philosophy calls for.
But notice that by "everything" one means in the quantum case,
merely the "expectation values", which are averages over (in
principle) an infinite ensembles of instances.
Now one might suspect that since this relativistic causality property
holds for these averages it ought to be *at least conceivably possible*
that it could hold also for the individual instances, particularly in
cases where the observed outcome is one or the other of two very
distinct and distinguishable possibilities.
But the amazing thing is that this is not true! It is not logically
possible to impose the no-faster-than-light condition on the individual
results, if one accepts as true the predictions of quantum
theory, which, as just emphasized, are themselves only about average
values.
Isn't this a fine kettle of fish?
I use this result in two ways:
1. To rigourize the contention of the founders that classical ideas
are definitely inadequate.
2. To justify the attempt to ontologicalize the orthodox interpretation.
It had been believed that this was a nonsensical thing to try, because
it would entail faster-than-light influences on the individual-instance
level, which is "unacceptable". But now we know that such influences are
entailed just by the validity of the (unchallenged) *predictions* of
the orthodox theory, without making ontological assumptions beyond the
rejection of the many-minds (Everett) approach.
Gregg said: Questions of temporal precedence or spatial proximity
are entirely secondary.
Well, it may be that they are entirely secondary. But the
causality issue I was talking about, which relativistic quantum
field theories maintain, do have to do precisely with the direction
of the spacetime separation between cause and effect.
Stan asks whether the familiar fact that the quantum mechanical
system is represented by a vector in Hilbert space continues to be
true in, for example, string theories.
String theories are relativistic quantum field theories on a Hilbert
space.
Maybe you are wondering about deep mathematical issues of convergence
etc. that might give trouble if one tries to be very rigorous.
I am not aware of anyone's being worried about such deep technical
questions at the moment: it would, I think, be premature to do so,
since details remain to be worked out. But the fact that string
theories are supposed to be finite "should" mean that no pathology
that kicks you out of Hilbert space will occur.
Henry P. Stapp