Tense and Indeterminateness
Simon Saunders

1 Preamble

Is tense real and objective? Can the fact that something is past, say, be wholly
objective, consistent with modern physics? I believe that it can. But some hold
that for tense to be real, then a certain ontological doctrine must also hold.
There must be a fact of the matter as to what really, truly, exists at each time.
For example, it is held that only what is now really exists. This is the doctrine of
presentism. Alternatively, that only what is now or past really exists; following
Savitt, I shall call this possibilism. But on either view there is a problem with
special relativity.
There is another question that is closely related. Is the contrast between the

determinate and the indeterminate real and objective, consistent with physics?
I say that it is. But many will insist that it be ontologically grounded, taking
us back to presentism or possibilism. If so, and if it is true that the latter are
incompatible with relativity, then the contrast between the determinate and the
indeterminate is incompatible with it too. It would follow that quantum theory,
or at least relativistic quantum theory, cannot be interpreted in terms of this
distinction.
The two questions have been considered before (Maxwell 1985); but I shall

present them in a rather di¤erent light, and come to di¤erent conclusions.

2 The Problem With Presentism

I am construing presentism as an ontological doctrine; that all that is is what
is now. I take it that "what is now" is a 3-dimensional space, along with
sundry events; moments in the careers of objects. It is therefore a time-slice of
spacetime. If we are to take the position seriously, as a philosophical thesis, it is
a public space - for nothing else exists, this is the whole of reality; and I take it
that solipsism is not a serious position in philosophy. So we had better all agree
on how this public space is to be de�ned. The criterion will have to specify an
instant of time; over and above that, it had better not make any special reference
to a particular person, or a geographical location, or an everyday object. We
are after a view of cosmology, of all that exists; it had better not be a view
which is provincial.
From a 4-dimensional perspective, presentism is therefore saying that what

is real is a particular time-slice. But from the 4-dimensional perspective, this
is to single out a family of time-slices, a space-time foliation. It is a family and
not a time-slice because the presentist does not deny that there is change, that
one time is succeeded by another. In 4-dimensional terms, that can only mean
that the one time-slice is later than another; and so on with all the rest. No

1



wonder that the presentist position appears inconsequential; the clearest way of
understanding it is by casting it in 4- dimensional terms, but when that is done
it looks just like its rival.
Or so it does in the non-relativistic case. Given relativity theory, there is an

important di¤erence. For according to relativity, there are many inequivalent
way of slicing space-time into a family of time-slices, with no one of them privi-
leged. That is inconsistent with presentism. A single foliation is privileged, she
will have to insist, because at each time there is a fact of the matter as to all
that exists. In other words - again reverting to the 4-dimensional perspective
- there is a fact of the matter as to what "an instant of time" really is; and
likewise its successors. With that we have a preferred foliation.
How might this preference be made out? Not by appeal to a particular

worldline, surely. The world-line of what, or of whom? In fact this is our
normal procedure. We give special prominence to ourselves, speci�cally to the
Solar System, in the case of Ephemeris Time; and to Planet Earth, in the case of
the atomic clock standard (Temps Atomique International or TAI, the present
standard of SI units)1. Neither, in fact, gives us a foliation of space-time, but in
each case we do obtain a local system of time-slices. In the latter case, in e¤ect
we have dotted around a number of clocks, synchronized by the radar method,
on the surface of the Earth; which itself is functioning as an approximately rigid
body. Is it by reference to the Earth, or to the Solar System, that all that exists
at each time is decided? Surely not; surely, in adopting these time standards,
we do not suppose we are settling any grand metaphysical questions.
However far we might try to extend this local family of surfaces, in spacelike

directions, the presentist can hardly embrace it as the arbiter of all that there
is at each time. To do so would return us, not to Lorentz�s position, but to
Aristotle�s. The Earth is no longer at the centre of the universe; so how do we
de�ne a foliation without this sort of provincialism? Not, evidently, using the
radar method (Einstein synchrony). Einstein de�ned a natural relation between
points belonging to timelike worldlines. Denote such points pu for timelike
worldline u and let l(pu;qv) be the straight line connecting the point pu to the
point qv . Given lines u;v; write u ? vp i¤ u and vp intersect orthogonally at
the point p . Then the condition that qu is Einstein simultaneous with pv is:

Ein(qu;pv) i¤ l(qu;pv) ? uqu

Evidently Ein is not transitive. As pointed out by Putnam (1967), if we use this
to build up to a time-slice, either all events will satisfy Ein, or, in accordance
with Einstein�s original operational procedure, one particular �ducial world-line
u must be used throughout. The only remaining possibility is if all the world-
lines considered happen to be parallel (in which case Ein is transitive). This
is what happens for a system of inertial bodies which are all mutually at rest
(an inertial frame). With that, if we look for a realistic, approximately inertial
frame, we are back to TAI.
It is unfortunate that Putnam did not make it clear that not only transi-

tivity, but also symmetry, is required by the presentist. For Stein (1968) has

2



responded that there is a perfectly satisfactory, invariant, transitive relation,
namely past causal connectibility; and claimed, and has been widely thought to
have proved, that Putnam was mistaken. This relation, denote Con, holds for
events simpliciter, independent of world-lines:

Con(p;q) i¤ q lies in or on the past light-cone to p

If we build up a foliation in this way, it will be a nested sequence of past light-
cones. The tips of this sequence will lie on a time-like curve. Again we are back
to TAI; time as referred to a particular time-like curve.
But nothing less than an equivalence relation will do for the presentist.

An equivalence relation is distinguished from other kinds of relations precisely
because it partitions a set into subsets, without distinguishing any particular
element of each subset. But an equivalence relation is symmetric as well as
transitive (re�exivity is trivial); we have as yet no candidate in sight.
Maxwell did no better when he restated Putnam�s argument (Maxwell 1985);

provoking Stein to give a proof that Con is the only relation which is transitive,
de�ned in terms of the structure of Minkowski space (Stein 1991). Therefore
there is no equivalence relation, proving Putnam�s point. This is not to say
that none can be de�ned, period; only that none can be de�ned independent of
the matter distribution. Clifton and Hogarth (1995), thinking that they were
proving Stein right, went on to prove that no transitive worldline-dependent
relation could be de�ned, given an arbitrary collection of inertial lines. They
have strengthened Putnam�s point.
Of course, in a realistic model of a material system, there will be a lot

of structure to the system of worldlines, none of which will be inertial. But
the task of distinguishing a unique foliation, by any natural condition, seems
increasingly hopeless. That is the problem with presentism: it goes against the
grain of relativity theory. There was no such di¢ culty in classical mechanics2.

3 Relationism

I began with the claim that tense is perfectly real and objective. According
to presentism, that was supposed to require an ontological distinction between
past and future; but as we have seen, that cannot be drawn in simple and
intrinsic terms. Is there any alternative? Obviously possibilism fares no better;
if possibilism is true, there is a boundary between what is past and what is not.
Concerning this boundary, there must be intersubjective agreement; so we are
back to the same di¢ culty that we met before.
The alternative is to deny that the distinction between the past, present,

and future has to be drawn at all on the ontological level. Suppose that all
events exist; they may all have objective relations with one another, among
them spatio-temporal ones. On this view, Stein is exactly right to focus on
Con; there is indeed a natural and non-arbitrary de�nition of a time-order (in

3



McTaggart�s terms, of the B-series), so long as we allow it to be only a partial
ordering. It is perfectly objective and real for all that. Let me call this position
relationism3.
Of course, once one does adopt this position, there is no harm in choosing

a convenient foliation of space-time by reference to the matter distribution. By
all means let it be Ephemeris Time, or TAI. Nothing weighty hangs on it, not
in physics and not in metaphysics. The interesting metaphysical question has
already been settled; the physical ones have not been asked.

4 The Problem With Indeterminateness

Consider now the distinction between the determinate and the indeterminate.
I take it we have no very sharp pre-theoretical idea of what indeterminateness
means. It is a state of potentiality, of inde�niteness, of what is as yet unformed;
to be contrasted with what is actual, or what has become actual, or de�nite.
But vague as it is, I take it that this means something more than that the future
is in principle unpredictable; that is a necessary but not a su¢ cient condition
for indeterminateness.
Inquantummechanicswehaveamoreprecise ideaofwhat indeterminateness

might mean. It is represented by a superposition of states each corresponding to
a determinate outcome. Of course this is as yet only a mathematical expression
of the idea; it does not interpret indeterminateness, it is interpreted in terms of
it. But a precise language is worth something, and I will talk assume throughout
in terms of quantum mechanics; speci�cally, in terms of one or another variants
of quantum mechanics. I shall rule out deterministic hidden-variable theories
- speci�cally, the pilot-wave theory - but there remain the indeterministic in-
terpretations. These include stochastic state-reduction theories; this class of
theories accords well with the neo-classical, stochastic notion of indeterminism
(as in Brownian motion, for example). In other words, restricting ourselves to
quantum mechanics, we lose nothing in generality, and gain in clarity.
But we are concerned with the marriage of quantum mechanics to relativ-

ity - with relativistic quantum �eld theory (RQFT). The obvious di¢ culty, in
extending state-reduction theories to the relativistic domain, is that it appears
the process of reduction must pick out a unique frame of reference. This seems
to be true even in the case of "e¤ective" state-reduction, as it arises in the pilot-
wave theory (where "empty waves" are thrown away, when their contribution to
the quantum potential is negligible); in that theory too there is a problem with
covariance (Bohm and Hiley 1993, Ch.12). The problem arises when one tries
to de�ne a Lorentz-covariant stochastic dynamics. On making a Lorentz boost,
a transformation in velocity, one must transform data on one time-slice to data
on another. This involves the dynamics, because points on these two surfaces
are timelike separated; likewise the data. But given a stochastic dynamics, the
data at one time is not determined by the data at an earlier time; so we cannot
de�ne the boosts.
I know of two ways of avoiding this conclusion. The �rst is due to Philip

4



Pearle, who supposes that one can incorporate, into the generators of the boosts,
a white-noise function which e¤ectively codes the entire history of the universe
- including the future (Pearle 1990). In that case, insofar as in principle one
can de�ne a notion of Lorentz covariance - through possession of this white-
noise function - in principle the future can be speci�ed as well; so it is not in a
state of indeterminacy. The second arises in the consistent-histories approach to
quantum mechanics. Here one has a space of all possible histories, where each
history is considered as a 4-dimensional whole, without any preferred foliation.
A measure is de�ned on this space, and the histories are required to satisfy the
consistency condition with respect to it (essentially that distinct histories do not
interfere with one another). We suppose that one of these histories is ours, and
that it is "typical", as de�ned by this measure. Conditional probabilities for A,
conditional on B, can then be de�ned as the measure of all those histories which
contain B and A, divided by the measure of all those which contain B. This is
reminiscent of the situation in classical statistical mechanics; although we do not
have equations, for these histories taken separately, the future is already �xed.
As Grünbaum put it, responding to Reichenbach�s proposal that indeterminacy
can provide a criterion to distinguish the future from the past:

Nor does indeterminacy make for any di¤erence whatever at any
time in regard to the attribute-speci�city of the future events them-
selves. For in either kind of universe [deterministic or indeterminis-
tic], it is a fact of logic that what will be, will be! (Grünbaum 1973
p.324).

Indeterminacy - lack of predictability - is not to the point; indeterminateness
is. To say that the future is indeterminate is precisely to say that it is not
attribute-speci�c.
I come back to the more general argument. The contrast between the deter-

minate and the indeterminate, between the de�nite and the inde�nite, appears
to be an ontological one; a matter of what exists, in a universal and unquali�ed
sense. As such it is a matter of public record, the actually existent world which
encompasses us all. It follows that its boundary is a spacelike hypersurface; and
from this point on, we are familiar with all of the arguments from the case of
tense.
It seems that the idea that the future is indeterminate, and not just un-

predictable, is incompatible with relativity. Insofar as quantum mechanics has
provided a mathematical expression of it, in terms of the superposition of states,
it seems that this theory and relativity are pulling in diametrically opposite di-
rections. In fact, properly understood, they are pulling in tandem.

5 The Everett Interpretation

The Everett approach involves a kind of modal realism; in some sense all physical
possibilities are realized. The idea is fantastic, granted; but one should �rst see

5



what it means, and whether it is consistent with physical principles. Alternative
approaches to the problem of measurement call for their outright revision4.
Consider �rst non-relativistic quantum mechanics. Suppose, from the outset,

that a preferred basis is given once and for all (equivalently, that a preferred set
of projections is given once and for all). We have the following parallel: corre-
sponding to distinct times in non-relativistic classical theory, there are distinct
possibilities, or components of state, in NRQM (all at a �xed time). More-
over, corresponding to the binary relation of simultaneity between local events
a;b; ::: denote Sim, there is a binary relation between equal-time projectionsba;bb; ::: denote Def, depending on the universal state5, such that Def(ba;bb) if and
only if bb; is value-de�nite relative to ba. In the non-relativistic case Sim is an
equivalence relation; suppose that Def is too. One can then partition the set of
all events, respectively the set of all projections, into equivalence classes corre-
sponding to the di¤erent moments of time A;B;::: in spacetime, and to di¤erent
possible worlds bA; bB;::: in the universal state. Call the view that only one of
these possible worlds is real actualism. It is a natural extension of presentism6.
Clearly, on this view, the distinction between the determinate (the actual) and
the indeterminate (the non-actual) is drawn in terms of what exists.
It is not so clear that the alternative, that all possibly worlds (at a given time)

are equally real, has anything to do with relationism. The relationist accounted
for tense in terms of relations; nothing similar is so far on o¤er in the case of
indeterminateness. Is it possible to adopt relationism when it comes to times,
but actualism in the case of possibilities? Given determinism of course it is; that
is the picture we have had in mind all along in the classical case. But it cannot
be de�ned in terms of the relations available in quantum mechanics, assuming
the branching of the universal state with respect to the preferred basis. It is a
simple matter to extend Def to projections at di¤erent times. Doing this, one
�nds it is transitive7; but it cannot be symmetric in the presence of branching.
Given bifurcation, with no recombination of branches, then from any world bA a
unique path can be traced backwards in time to bB; so Def( bA; bB). If Def(bB; bA)
as well, then the transition amplitude for bA with respect to bB would be 1, and
there could be no branching from world bB, contrary to the hypothesis. So Def is
not an equivalence relation, and one cannot end up with the classical relational
picture of a 4-dimensional space-time without any distinguished present.
Neither, in that case, can one de�ne Lewis�s picture, where all possible 4-

dimensional worlds are supposed to be equally real. We should not want to.
According to Lewis, possible worlds bear no physical relations with one another.
In that case, since we are considering a physical theory, it is hard to see why a
single one of them, the actual world, is not enough. In any case Lewis has no
place for indeterminateness; his theory of probability is just like the one already
stated for the consistent histories approach (Lewis 1983).
But we can make sense of actualism combined with possibilism: a determi-

nate past, up to a certain moment of time, is all that exists; the future does
not exist, or is otherwise only potential. This is the view that will probably
have the broadest appeal. It has long been favored by Shimony, and appears

6



natural from the point of view of a state-reduction theory. Once again the dis-
tinction between the determinate and the indeterminate is drawn at the level of
existence.
There is an alternative. We can suppose that at this moment in time all such

possible worlds are equally real. It is still the case that in each the distinction
between the determinate and the indeterminate is drawn at the level of ontology,
along with the contrast between past and future; but the question of what the
determinate reality is, and the range of future possibles, becomes indexical; it
is the question of which world happens to be ours. This corresponds to the
many worlds view, understanding tense in accordance with possibilism. As
with Lewis�s metaphysics, the important question is whether, and how, these
worlds are related to each other; in the �rst instance, whether any parts of
them which are qualitatively identical are numerically identical. One of the
clearest exponents of this view is David Deutsch; according to him they are to
be distinguished (Deutsch 1985). In that case, again one wonders why we need
more than one of these worlds at each time.
More interesting is the case where one considers in addition all times. This

is the view championed by Storrs McCall (1994). McCall represents the inde-
terminate future as a set of branches, one for each potentiality, fanning out from
a trunk representing a unique and determinate past. The point of bifurcation
is the present. All possible tree-diagrams exist for every time; it only remains
to say which of them is ours. The distinction between past and future, and
between the determinate and indeterminate, is wholly indexical. Again, as with
the many-worlds view, there remains an ambiguity as to whether or not di¤erent
tree-diagrams are physically related.
There is a natural way to combine them all. Suppose that the local parts of

di¤erent diagrams are numerically identical if they are qualitatively identical.
The trunks of two diagrams, if they share an initial segment, are there fused into
one; yielding a new diagram, incorporating each of them as a branch, bifurcating
from their common part. Continuing in this way, one obtains a single dendritic
diagram, without any trunk8. Evidently the question of whether or not an event
is indeterminate or determinate has become relational, depending on its status
as future or past; as whether or not the question of whether an incompatible
event is indeterminate or determinate. In other words, which of two mutually
exclusive possibilities has happened, is likewise relational. This was Everett�s
original point of departure, his concept of the relative-state. This is the Everett
interpretation; it is the natural extension of relationism in the case of tense.
Just as one combines all moments of time into space-time - obtaining a

relational account of what is past and what is future - one combines all possible
tree-diagrams, obtaining a relational account of what is determinate and what
is indeterminate. The single diagram that results represents the unitary orbit
of the universal state, expressed as a system of correlations; a certain network
of relations. There are, of course, arbitrarily many other networks of relations;
just as, in classical spacetime, there are arbitrarily many distinct foliations. But
in both cases it is the local structure of the state, or of spacetime, that we are
interested in, that is better or worse exhibited by the choice of projections,

7



the dynamical variables; and better or worse by the choice of foliation, the
coordinates.
This point is worth emphasizing. If one chooses a di¤erent set of projec-

tions at each time, one obtains a di¤erent system of correlations; just as with
a di¤erent choice of coordinates on classical spacetime. But in both cases one
has exactly the same object: the universal state and spacetime, respectively.
Such di¤erent choices may be better or worse suited to describing the physics;
our understanding of it, using projections which are related in accordance with
quasiclassical equations of motion, is particularly suited to certain regions of the
universal state - those in which we are to be found. And they are particularly
suited to describing what we in fact observe - because we, our form of life, is built
up from just those kinds of correlations. Both the dynamical variables, in quan-
tum mechanics, and the coordinates, in relativity, used to describe these sorts
of patterns, are more or less approximately de�ned; for nothing fundamental,
at the ontological level, hangs on them.
But all of this is something of a fairy-tale, because, except for very special

choices of the universal state - or for special regions within it - Def will not be
symmetric at equal-times. It will be if the preferred basis at each time, the local
projections ba;bb; ::: are de�ned by the criterion of environmental decoherence9.
This does yield approximately classical histories, in favorable circumstances
(Diósi et al, 1994); but it does so only in an approximate sense, and only in
those favorable circumstances. In this sense the preferred basis may be de�ned
only locally. From it we cannot in general build up to global projections bA; bB
;::::without privileging a particular sub-system, or locality, in each. All the
strategies just discussed are still available, but the worlds all have a privileged
place, or a privileged sub-system. With that, actualism, as applied to presentism
and possibilism, is no longer credible.
There remain only the various forms of realism about possible worlds. When

we consider the relativistic case, we lose the symmetry of Sim as well, viewed
as an intrinsic geometrical relation. Deutsch�s approach is no longer available,
and nor is McCall�s, as he intended it. We cannot partition space-time into
times, on the basis of any intrinsic relation in the space- time metric; and we
cannot partition the universal state into worlds, using intrinsic relations in the
Hilbert-space norm. What one does have is a local version of McCall�s view:
many possible events, at di¤erent places and times, each with a locally-de�ned
determinate past and indeterminate future. But better is the relationist view,
where they are combined into a natural totality, and where our pre-theoretic
notions of tense and indeterminateness can be made out in terms of relations
within it.

6 Indeterminateness According to Everett

We learn better what form the Everett approach must take in RQFT. It is
an intrinsically local theory, supposing that the preferred basis is de�ned by
purely local criteria of decoherence; locally, in such decohering regions of the

8



universal state, Def will be approximately symmetric as well as transitive10.
But there will be no global extension of it. This point is important if we are
to understand how stochasticity can be reconciled with relativity; and, indeed,
the sense in which the Everett approach operates with Minkowski spacetime.
Callender (this volume) is right to call this in question; if spacetime points are
individuated by physical criteria, but referring to di¤erent branches, it follows
that spacetime cannot be Hausdorf11. It is of no use to point out that the
universal state is de�ned on Minkowski spacetime: whatever else relativity is,
it is a phenomenological theory; the universal state is not what is observed, but
only the relations which it codi�es. In what sense - at the observable level - is
our theory relativistic?
We should �rst verify that on the relational view Everett�s approach is in-

deterministic - at the same, phenomenological level. It surely is, because one
cannot solve the Cauchy problem, using data which concerns ordinary macro-
scopic events; not even if supplemented with data at the microscopic level, and
not even given the universal state. Knowing everything there is to know, one
cannot predict data of this sort for any future time-slice. One can predict all
possible data of this sort, with all the associated transition-amplitudes; but
nothing less.
For many that is not enough to ensure that the theory makes sense of prob-

ability; but I do not want to dispute that point here12. Let me grant that one
must be able to de�ne an e¤ective stochastic process, to govern this phenom-
enology (some will insist that is not enough either; but again I defer discussion
of that point). But we are agreed that the principles of relativity theory had
better apply to what is observed. So this e¤ective process had better be Lorentz
covariant; how are we to square this circle?
I come back to my opening remark: the approach is fundamentally a local

one. The process is to be de�ned at each space-time point, as an e¤ective state
reduction on a certain 3-dimensional surface in space-time. But this surface
is not a time-slice, a space-like hypersurface. It is the surface of the forward
light-cone of each point. As such, on making a Lorentz- transformation, this
surface, and the associated data, is invariant. We do not have to transform
between data at one space-time point, to data at another. Just as important,
these stochastic processes at these space-time points are all independent of each
other.
Of course, if we attempted to consider all of these operating in tandem, on a

space-like hypersurface, and thereby attempted to de�ne a stochastic process for
a global 3-dimensional world, we could never in this way obtain any EPR-type
correlations: the independence of the local processes implies factorizability, and
that in turn the Bell inequalities (which we know to be violated). But that is
only to say once again: the Everett relational approach is fundamentally local;
it is consistent with relativity, as goes the observed phenomenology, only in this
light. These correlations between measured outcomes are incorporated into local
records; and the amplitudes for such records, in which the correlations do not
obtain, are vanishingly small.
Is the Everett approach believable? But I know of no other that is. We

9



can do no better than seek a coherent and systematic interpretation of physical
theory. It should respect our pre-theoretical opinions that we are �rmly attached
to; and it should respect hard-won theoretical principles as well. Among our
pre-philosophical opinions I count the sense that the future is indeterminate, in
contrast to the present and past; and among theoretical principles I prize the
relativity principle, and the principles of quantum mechanics. Everett�s ideas
hold out a radical and austere way of combining them; that is reason to pursue
them. Revisionary approaches to the problem of measurement may yet yield
dividends in the laboratory; and they may yet theoretical dividends too, if they
further a theory of quantum gravity. But in the latter case, they will surely
have to be conservative with respect to particle physics - to RQFT. Therein is
the di¢ culty.

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Endnotes
1 The two techniques are in fact very di¤erent. For philosophical and historical
background, see my (1998a).
2 One can locally de�ne a foliation by reference to the cosmological microwave
background; but that there is no guarantee that it be extended globally, and
as COBE showed us, it is not perfectly isotropic. (See also my 1996a.) The
conclusion may be di¤erent on certain approaches to quantum gravity, however;
particularly the canonical approach. I have shall have nothing to say on that
here.
3 Since it is of the essence to this thesis that tense is perfectly objective and real,
albeit relational, the expressions �fatalism�, �eternalism�, and �the tenseless
view�are clearly inappropriate.
4 The point is widely acknowledged in the case of state-reduction theories. For
the pilot-wave theory, see my (1999); for the consistent histories approach, see
Dowker and Kent 1996).
5 If � is the universal state, Def(ba;bb) i¤ Tr(bbba�ba)=Tr(ba�) = 1; this is just
Everett�s relativization, written in terms of projections (see my 1995 for further
background).
6 Compare Hinchli¤ (this volume): �other times are like other possible worlds,
they are not real. As there is something special about what is actual - it is all
there is - so there is something special about the present - it is all there is.�
7 So long as the histories are consistent. See my (1996b).
8 I put to one side questions of the early history of the universe, and of an
absolute origin.
9 Partition space S at time t into volumes Sk, k 2 I, for a suitable index set I.
Let local projections bak;bbk ,.. be associated with Sk . Choose any j 2 I; take
the partial trace of the state over degrees of freedom external to Sj , and then
take the spectral decomposition of the reduced density matrix. If the basis that
results diagonalizes the projections baj;bbj , and this is so for every j 2 I, then
Def will be symmetric. (If I = f1;2g, this requirement is: local projections are
diagonal in the basis de�ned by the biorthogonal form.)
10 Stein has made a similar point in connection with Ein, in the classical case
(Stein 1991 p.161-2).
11 It is worth remarking that the Hausdorf property is a non-local topological
property. See Douglas (1995), who concludes it is suspect in consequence.

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12 See my (1998b), and references therein, for detailed discussion.

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