Branching Space-Time Author(s): Nuel Belnap Source: Synthese, Vol. 92, No. 3 (Sep., 1992), pp. 385-434 Published by: Springer Stable URL: http://www.jstor.org/stable/20117060 Accessed: 28/05/2009 14:37 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=springer. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact
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NUEL
BELNAP
BRANCHING
ABSTRACT. Postulates
'Branching and definitions
space-time'
SPACE-TIME
is a simple the
describe
rigorously
and indeterminism. blend of relativity 'causal order' relation between possible
events. is a version of 'everything has a causal origin'; The key postulate point key terms include but helpful defined Some elementary facts are 'history' and 'choice point'. to the status of causal contemporaries is made of indeterministic proved. Application events,
to how
'splitting'
Einstein-Podolsky-Rosen
of histories
happens, correlations.
distant
1.
to indeterminism
without
choice,
and
to
INTRODUCTION
can we combine in a rig relativity and indeterminism is difficult; The problem indeed some have presented a that it is in principle insoluble - Stein (1991) combines arguments refutation of those arguments with an account of their apparent force. I directly confront and offer a rigorously Here the problem framed Problem: How orous theory?1
to a solution. contribution The combinational both relativity and question evidently presupposes I hope that those who which is a lot to presuppose. indeterminism, find helpful the present reject one of these assumptions will nevertheless effort to devise a careful theory embodying them both. I further hope that those who reject one because of a belief that it is inconsistent with as flawed. the other will come to see the reason for their rejection there are many who take the following as given: the only way Lastly, to discuss relativity and determinism/indeterminism is by talking about or epistemology or the history of science, or about theories psychology or laws or models or other linguistic or quasi-linguistic I phenomena.2 hope that some of these people will find it helpful to have an additional approach worked out in some detail. The theory of this paper is simple in respect of vocabulary: although as revelatory, it involves several defined concepts intended its only are set events' the of and the causal primitives (i) (ii) 'possible point on relation them. The ideas will be ordering Disadvantage: developed remote
from
be fundamental,
?
'real' physics. Advantage: rigorous, and clear.
92: 385-434, 1992. Synthese 1992 Kluwer Academic Publishers.
Printed
Such
results
in the Netherlands.
as we
obtain
will
386
NUEL
BELNAP
idea is that a true description of our world requires underlying Einstein-Minkowski with Prior/Thomason space-time branching fusing seems as good time. For the resultant structure, branching space-time a name as any. Here is a historical reconstruction. at least inmy imagination, The 'old physicists', conceived of time as a The
linear ordering of spatially infinite but instantaneous Euclidean spaces. as I I all do this call their structure metrics, paper, Ignoring throughout linear time and I call its order linear temporal order. When I want a name for the individual instantaneous are Euclidean that spaces put in linear temporal order, I call them moments. In articulating the fundamental idea of Minkowski, special relativity, after Einstein, from the of was, view, to revise the very present point terms of the relation. Rather than spatially infinite moments, it is are events small that and what relates related, point infinitesimally them is termed a causal order. By 'causal order' I mean what could be or
order', with the addition of a sense or and theorists agree that causal 'direction'.3 (Old physicists relativity the 'influences' pass (only) along causal order, but they differ as to the nature of the terms of the relation.) The manifold of point events is as a set of point The called space-time. idea of viewing space-time events subject to a causal order seems to carry over from special to called
'time-like
light-like
it is often useful however, general relativity. For illustrative purposes, a particular metric is available. I to keep to special relativity, where to will use Minkowski denote this case, leaving plain space space-time4 time for general relativistic use. a history of physics might be able to obtain a As for indeterminism, and then a relativistic nonrelativistic version from quantum mechanics, from quantum field theory. I do not have the version of indeterminism for essaying such a history. Instead I draw on the work background some and McCall. To express logicians Prior and Thomason our as a features of world associated with indeterminism fundamental tense for modal foundation logic, Prior, and after him Thomason, started out as did the old physicists with moments. Then he generalized the linear temporal order to a branching temporal order. The manifold
of
the
of moments
ordered in this tree-like way the contrast. On the one hand, and quantum mechanics is of relativity wonder few persons claim to have much to in quantum field theory. But, on the other Observe
is called
time.5 branching the detailed physics of each No necessarily complicated. say about their 'combination' hand, neither the fundamen
BRANCHING
387
SPACE-TIME
nor the fundamental tal relativistic idea of Minkowski indeterministic is all that intricate. The hope arises that one idea of Prior/Thomason can say something indeterminism by simple but useful about relativistic result I these ideas. The is what call combining 'branching space-time'. has two parts: (1) the items related will The idea of the combination be point events, as required by Minkowski but not by Prior/Thomason; and (2) the ordering will be a branching (causal) order, as required by Prior/Thomason describes what
but not by Minkowski. is wanted.
The
: :branching
linear time/space-time
following
proportion
time/branching
thus
space
time.
Here
is a table for the above
'historical'
Structure linear time
Relata moments
space-time branching
time
branching
space-time
point events moments point events
jargon. Relation linear temporal causal order branching branching
order
temporal order causal order
The
the theory plan for the remainder of the paper is this. I develop in Sections 2-7 through a mixture of (i) rigorous postulates, definitions and facts (each of which is numbered), and (ii) informal motivation. Then I apply the theory in Sections 8-11 to four problem areas, the last being the Einstein-Podolsky-Rosen I in Sec summarize 'paradox'. tion 12. There is also an appendix suggesting a modest generalization.
2.
The the
OUR
WORLD
AND
ITS
CAUSAL
ORDER
1 below. Here with Postulate I begin theory commences a its name, gloss by introducing suggestive explaining with the clearest language I know. Let Our World be the set events that are 'in suitable external relations'6 to us. Accommo
rigorous informal
meaning o? point date indeterminism future possibilities future possibilities,
by including those point events that either are now or were future Of the ones that were possibilities.7 we might that 'could have been'. The say they
following preliminary words will serve, if you keep in mind that there are opposed possibilities ahead of us in a causal direction: include any are events that accessible from here-now by a possibly zigzagging point
388
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of causal combination over Our World. range
and reverse
BELNAP
causal
tracks. Let
e (often marked)
I have put in plain indexical English that I do not mean to be speaking or creatures of point events that are mere of belief or imagination I In what recombinational follows will try to otherworldly possibility. a I In not draw will avoid indexical distinction language. particular, indexical when not relational) between the actual and the (inevitably or in 'Possible point except motivating giving examples. possible events' are thus just 'point events'. These point events are to be taken concrete fil not as mere open for alternate spatiotemporal positions as concrete themselves but lings, particulars.8 Some possible point events are incompatible with others. Here is an is an ideally small event, em, at which a idealized illustration. There in a certain way. There are two possible certain electron is measured or outcomes: measured measured spin up spin down. Take a possible it is true to say, 'It has been measured point event, eu, at which spin at and which it is true to say 'It has been measured ed, another, up', spin down'. The point events eu and ed are incompatible, though each is compatible with em.9 Exactly how can two incompatible point events means both fit into Our World? Answer: of the causal order. By that e1 ^ e2 Let ^ be a relation on Our World having the significance a case with between and the former in there is causal order ex e2, just earlier than the latter (in the weak sense that allows identity). Given from the standpoint of e2 we should say that ex did occur, and ex^e2, from the standpoint of ex we should say that e2 might occur. Here are three paradigms. (If the indexical space-time annotation of the diagrams doesn't help, please pass on to the following text.) a given point Causal Causal order can hold between dispersion: two and future event, e3, space-like separated point events, ex and e2, in a single (e.g.) Minkowski just as you might expect: space-time, ^ ^ and Causal Causal order also can hold be ei e2. e3 e3 confluence: space-like separated point events, ex and e2, in a single a single future point event, e3, as you might and space-time, ^ e3 and e2 ^ e3. Causal branching: Causal order also equally expect: ex a given e3 and two possible can hold between future point events ex for occupying and e2 that might be said to be alternate possibilities the same 'spatiotemporal e3 ^ ex and e3 ^ e2.10 No backward position': is missing from Figure 1 correctly branching: That a fourth diagram tween
two given
Minkowski
suggests
that I am denying
that incompatible
point
events
can lie in the
BRANCHING
Outcome
Over
Left-Later
Outcome Over
389
SPACE-TIME
Outcome Here-Now
Right-Later
Event Here-Now
Causal dispersion Possible outcome #1,
Possible outcome #2,
There-Later
There-Later
?i
?3 ev Event Over
^2 Event Over
Left-Earlier
Right-Earlier
Causal confluence
?2
Event Here-Now
Causal branching Fig.
past, same
1. Causal
dispersion,
confluence,
and branching.
i.e., that some events could have incompatible sense that some have outcomes. incompatible
'incomes' in the No backward
is part of common sense, including that of scientists when branching some irrevers of measurements, probabilities, speaking experiments, scien ible phenomena, and the like. In many other contexts, however, a point of drawing no distinction backward and tists make between forward. Because this paper lacks space for discussion of this contro I hope the following is noncontentious: the assumption versial matter, to warrant of no backward branching is plausible clear making enough to the extent that one finds what it comes to. It will then be warranted helpful a theory of which it is a part. hold for the causal order? For Minkowski space postulates the 1914-36 results of Robb and gives time, Mundy (1986) describes additional results for the light-like order. That research, however, does as not immediately help here because aMinkowski I under space-time, What
shall need to it, never contains incompatible point events. We so more is natural and so vital that proceed slowly. The first postulate it I would not know what to say next. without stand
390
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1: Partial POSTULATE World: of Our ordering
Order.
BELNAP
The
relation
^
is a nontrivial
partial
is nonempty. Our World ^ e. e Reflexivity: if ex ^ e2 and e2 ^ e3, then ex ^ e3. Transitivity: if ex ^ e2 and e2 ^ ex, then ex - e2. Antisymmetry: Nontriviality:
an instructive argument for this I have not the slightest hope of making some For have questioned antisymmetry, asking postulate. example, us to consider I am 'causal chains' that double back upon themselves. am I to to do but the so, argue unwilling equally unwilling point. The without following discussion would surely be unintelligible antisymme in its favor.11 try which is perhaps after all not a bad argument are for convenience. The following simple definitions 2: I use < for the companion DEFINITION strict partial ordering: = e2. ex < e2 if ex ^ e2 but not ex I use 'causally earlier' and 'causally later,' etc., as English readings 'causal'. I mark of the weak relation, ^, but often drop the adjective the stronger relation with 'proper', as in 'ex is properly earlier than e2 . to use 'causal past' and 'causal On the other hand, it ismore convenient future' for the strong relation, again dropping 'causal' more often than not. Thus, if et
