SPACE-TIME-CHANCE
http://www.math.princeton.edu/~nelson/papers/cti.html
I would be surprised if anyone here pictured God as being above our northern
hemisphere sky, although such images used to be commonplace in religious
thought. We no longer attempt to locate God in space, but acknowledge God as
Sovereign of space, not subject to it. The scientific discoveries of the
seventeenth century helped to clarify our thinking about this.
But we all tend to locate God in time, imputing to the Sovereign of history
the now as we perceive it. The discovery of relativity should help to clarify
our thinking about this. In nonrelativistic Newtonian physics, the present
divided all events into past and future. But relativity has no invariant
concept of the present. According to relativity, a here-now divides events
into three types: the past cone, the future cone, and the rest -- for which
there is as yet no term in everyday language. (But when we begin seriously to
explore the solar system and beyond, we shall be forced to coin a term. We
shall be unable to converse with our explorers, and the concept of an
objective or common present will disappear.)
The past is unalterable; no one seriously doubts that. What we do can affects
the future; almost everyone believes this. But what about the rest -- the
vast region of space-time that according to relativity is neither past nor
future? Here physicists disagree. The consensus is that no signal can be sent
into the rest, and the reasons for this are overwhelming. Such a possibility
would lead directly to causality paradoxes implying the possibility of
altering the past: an Arcturan whose here-now is neither in my past nor my
future may have my here-now of last year in its rest of space-time, so if I
could send it a birthday greeting from my here-now to its, nothing would
preclude my receiving a thank you last year. But many physicists who struggle
to understand the world -- not content merely to predict and manipulate it --
are reluctantly led to believe that what we do must affect what happens in
the rest even if we can send no signals thereby. This is a fascinating domain
where relativity and quantum theory meet -- the world of Bell's theorem and
the Aspect experiments -- and I shall try to explain the problem by analogy
with a game.
Every day I lunch at McDonald's and get a game card with three slots A, B,
and C covered with a metallic film. I choose one slot and scrape off the
film; I find either WIN or LOSE. Every day my friend plays the same game in
another city. Over the years we have compiled statistics concerning one
question: when does it happen that just one of us wins? We have observed that
when we choose the same slot this always happens, and when we choose
different slots this happens one fourth of the time in the long run. The
question is: How does McDonald's prepare the game cards?
They have no way of knowing which slots we shall choose; we might choose the
same slot, but then it always happens that just one of us wins; consequently,
my card and my friend's card must always have opposite entries. Now there are
two possibilities. My card may have all of its entries identical; then, since
my friend's card is opposite, it must happen that just one of us wins. The
other possibility is that one of my entries is the opposite of the others.
There are six ways that I and my friend can choose different slots, and a
simple counting of the possibilities shows that two out of these six -- or
one third -- provide a win for just one of us. But we have observed that this
happens only one fourth of the time. The only conclusion is that the game
cards are not prepared in advance.
Precisely this situation occurs in the correlated polarization experiments of
Alain Aspect and collaborators, experiments suggested by the discoveries of
J. S. Bell. The game cards are photons, the slots are directions in which a
polarization measurement can be performed, the WIN and LOSE are opposite
polarizations. The conclusion both of quantum theory and of experiment is
that the photon cannot carry with it a game card telling it how to respond to
all of the polarization measurements that might be performed on it.
One delightful consequence is that one can say without exaggeration that
indeterminism is no longer merely a philosophical speculation; it is
established scientific fact. Einstein to the contrary, God plays dice with
the universe.
But another consequence is grimmer. The beauty of the Aspect experiments is
that the polarization measurements are performed so that no signal can be
transmitted from one to the other -- each is neither in the past nor the
future of the other, but in what I have called the rest. Most physicists are
led to the conclusion that the choice of which polarization measurement to
perform -- which slot on the game card to choose -- has a mysterious
influence on the state of affairs at the place and time of the other
measurement.
Let me dwell on this a bit longer, because it is deeply puzzling to everyone
who thinks about it and is leading some to very strange beliefs about the
nature of the world. No matter which slot I choose on the game card, my
friend's chances of winning are unaffected. No signal is sent, no information
is conveyed: my friend has no way of knowing which slot I choose until we
compare notes later (at a here-now that lies in both of our future cones).
But we have seen that the game cards are not prepared in advance. Are we not
led inescapably to the conclusion that my choice of slot A affects the
situation of my friend's card, forcing the corresponding entry to be
opposite? Perhaps not, but before discussing that let me explain why this
conclusion is disturbing. Since there is no invariant causal ordering of
events that are in each other's rest of space-time, rather than say that my
choice of a slot has caused a change in my friend's card, it is equally valid
to say that the change in my friend's card has caused my choice of a slot; in
fact, for some reference frames the change in the card occurs before my
choice of a slot. This is a grim conclusion.
I am one of a very small minority who cling to the hope that an objective
description of nature is possible in which the principle that only the future
can be affected still holds. I think that the resolution of the problem may
lie in thorough exploration of field-theoretic, as opposed to particle,
descriptions of nature, and that it bears a strong resemblance to the
snowflake problem. The snowflake problem is this: the inexhaustible variety
of snowflakes makes it evident that chance plays a major role in their
development, yet they always preserve hexagonal symmetry -- how does a
portion of the snowflake growing at random on one side know to grow in
precisely the same fashion as its partner all the way over on the other side?
This is mysterious but not beyond understanding, and the understanding has to
do with the study of the snowflake as a whole (a field-theoretic
description). It would be beyond comprehension were the two portions of the
snowflake disconnected (a particle description). But these are technical
issues, which I am studying with my colleague Eric Carlen, and I think it
best to put aside philosophical and religious beliefs in the day to day work
of doing science.
Quantum theory produced a crisis in human thought unlike any previous
scientific discovery, a crisis that, as John Bell has shown, is exacerbated
by the findings of relativity. There are problems in the interpretation of
quantum theory that simply refuse to go away. For a long time, it was thought
that the phenomena of organic chemistry were governed by principles
qualitatively different from those of inorganic chemistry. Vitalism, an early
form of holism, was introduced, but then with the synthesis of urea
understanding was achieved. It is my conviction that understanding in science
always comes from reduction. What is at stake in the investigation of these
questions is the very possibility of an objective description of nature.
Relativity and quantum theory taken together show that space, time, and
chance are inextricably linked together. Where am I? In Princeton, at the
Center of Theological Inquiry. What time is it? The morning of October 22,
1988, and time for me to stop talking. What will happen next? I don't know.
This is reality as I perceive it; this is my here-now-uncertainty. But I am a
creature, not the center of the universe and much less its Creator. Where is
God? We no longer ask that. What is God's present? If we take relativity
seriously, we must conclude that it is of a different order from my present,
for other creatures have a present that is incomparable with mine. Does God
know what will happen next? There are strong arguments from physics that any
postulation of a knowledge of all possible future outcomes is contradictory,
but I prefer a negative answer to this question on other grounds; I prefer to
think of God as a risk taker, one who when creating the world chooses to make
it alive. Preachers sometimes refer to chance as mere chance. But I believe
that chance is a mighty archangel, by which I mean that it is a deep part of
God's will, with immense power over our lives.
But I must confess to an inconsistency here. I began by pointing out that it
is primitive to picture God in space. Relativity tells us that space and time
are aspects of space-time, so it is primitive to picture God in time. And
quantum theory tells us that space, time, and chance are part of
space-time-chance, so it is primitive to picture God as subject to
uncertainty. But those of us who hold to faith in the Incarnation are not put
off by this; we know that even in the arena of space, time, and chance in
which we struggle, our Redeemer lives.
Bibliography
Alain Aspect, Jean Dalibard, and Gérard Roger, Experimental test of Bell's
inequalities using time-varying analyzers, Physical Review Letters 49, 1982,
1804--1807. -- Few experiments that confirmed what everyone already knew have
caused as much excitement as this.
J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge
University Press, Cambridge, 1987. -- See especially the funny and deep
article ``Bertlmann's socks and the nature of reality.''
N. D. Mermin, Bringing home the atomic world: quantum mysteries for anybody,
American Journal of Physics 49, 1981, 940--943. -- My formulation in terms of
game cards is based on this beautiful article.
Edward Nelson, Quantum Fluctuations, Princeton University Press, Princeton,
1985. -- A highly mathematical account of stochastic mechanics, an attempt at
a realistic particle picture of nature.
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Edward Nelson
Department of Mathematics
Princeton University
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