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a r X i v : q u a n t - p h / 0 7 0 3 0 6 0 v 1 7 M a r 2 0 0 7
A Topos Foundation for Theories of Physics:I. Formal Languages for Physics
A. D¨oring
1
andC.J. Isham
2
The Blackett LaboratoryImperial College of Science, Technology & MedicineSouth KensingtonLondon SW7 2BZ6 March 2007
Abstract
This paper is the ﬁrst in a series whose goal is to develop a fundamentallynew way of constructing theories of physics. The motivation comes from a desireto address certain deep issues that arise when contemplating quantum theoriesof space and time.Our basic contention is that constructing a theory of physics is equivalent toﬁnding a representation in a topos of a certain formal language that is attachedto the system. Classical physics arises when the topos is the category of sets.Other types of theory employ a diﬀerent topos.In this paper we discuss two diﬀerent types of language that can be attachedto a system,
S
. The ﬁrst is a propositional language,
PL
(
S
); the second is ahigher-order, typed language
L
(
S
).Both languages provide deductive systems with an intuitionistic logic. Thereason for introducing
PL
(
S
) is that, as shown in paper II of the series, it is theeasiest way of understanding, and expanding on, the earlier work on topos theoryand quantum physics. However, the main thrust of our programme utilises themore powerful language
L
(
S
) and its representation in an appropriate topos.
1
email: a.doering@imperial.ac.uk
2
email: c.isham@imperial.ac.uk
1 Introduction
This paper is the ﬁrst in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certaindeep issues that arise when contemplating quantum theories of space and time.A striking feature of the various current programmes for quantising gravity—including superstring theory and loop quantum gravity—is that, notwithstanding theirdisparate views on the nature of space and time, they almost all use more-or-lessstandard quantum theory. Although understandable from a pragmatic viewpoint(since all we have
is
more-or-less standard quantum theory) this situation is never-theless questionable when viewed from a wider perspective. Indeed, there has alwaysbeen a school of thought asserting that quantum theory itself needs to be radicallychanged/developed before it can be used in a fully coherent quantum theory of gravity.This iconoclastic stance has several roots, of which, for us, the most important isthe use in the standard quantum formalism of certain critical mathematical ingredientsthat are taken for granted and yet which, we claim, implicitly assume certain propertiesof space and time. Such an
a priori
imposition of spatio-temporal concepts would bea major error if they turn out to be fundamentally incompatible with what is neededfor a theory of quantum gravity.A prime example is the use of the
continuum
which, in this context, means the realand/or complex numbers. These are a central ingredient in all the various mathemat-ical frameworks in which quantum theory is commonly discussed. For example, this isclearly so with the use of (i) Hilbert spaces and operators; (ii) geometric quantisation;(iii) probability functions on a non-distributive quantum logic; (iv) deformation quan-tisation; and (v) formal (i.e., mathematically ill-deﬁned) path integrals and the like.The
a priori
imposition of such continuum concepts could be radically incompatiblewith a quantum gravity formalism in which, say, space-time is fundamentally discrete:as, for example, in the causal set programme.A secondary motivation for changing the quantum formalism is the peristalithicproblem of deciding how a ‘quantum theory of cosmology’ could be interpreted if onewas lucky enough to ﬁnd one. Most people who worry about foundational issues inquantum gravity would probably place the quantum cosmology/closed system problemat, or near, the top of their list of reasons for re-envisioning quantum theory. However,although we are certainly interested in such conceptual issues, the main motivation forour research programme is
not
to ﬁnd a new interpretation of quantum theory. Rather,the goal is to ﬁnd a novel structural framework within which new
types
of theory canbe constructed, and in which continuum quantities play no fundamental role.Having said that, it is certainly true that the lack of any external ‘observer’ of theuniverse ‘as a whole’ renders inappropriate the standard Copenhagen interpretationwith its instrumentalist use of counterfactual statements about what
would
happen
if
a certain measurement was performed. Indeed, the Copenhagen interpretation is1
inapplicable for
any
3
system that is truly ‘closed’ (or ‘self-contained’) and for which,therefore, there is no ‘external’ domain in which an observer can lurk. This problem hasmotivated much research over the years and continues to be of wide interest. Clearly,the problem is particularly severe in a quantum theory of cosmology.When dealing with a closed system, what is needed is a
realist
interpretation of the theory, not one that is instrumentalist. The exact meaning of ‘realist’ is inﬁnitelydebatable but, when used by physicists, it typically means the following:1. The idea of ‘a property of the system’ (i.e., ‘the value of a physical quantity’) ismeaningful, and representable in the theory.2. Propositions about the system are handled using Boolean logic. This requirementis compelling in so far as we humans think in a Boolean way.3. There is a space of ‘microstates’ such that specifying a microstate
4
leads to un-equivocal truth values for all propositions about the system. The existence of such a state space is a natural way of ensuring that the ﬁrst two requirementsare satisﬁed.The standard interpretation of classical physics satisﬁes these requirements, andprovides the paradigmatic example of a realist philosophy in science. On the otherhand, the existence of such an interpretation in quantum theory is foiled by the famousKochen-Specker theorem [4].What is needed is a formalism that is (i) free of
prima facie
prejudices about the na-ture of the values of physical quantities—in particular, there should be no fundamentaluse of the real or complex numbers; and (ii) ‘realist’, in at least the minimal sense thatpropositions are meaningful, and are assigned ‘truth values’, not just instrumentalistprobabilities.However, ﬁnding such a formalism is not easy: it is notoriously diﬃcult to modifythe mathematical framework of quantum theory without destroying the entire ediﬁce.In particular, the Hilbert space structure is very rigid and cannot easily be changed.And the formal path-integral techniques do not fare much better.Our approach includes ﬁnding a new way of formulating quantum theory which,unlike the existing approaches,
does
admit radical generalisations and changes. Arecent example of such an attempt is the work of Abramsky and Coecke who constructa categorical analogue of some of the critical parts of the Hilbert space formalism [5];see also the work by Vicary [6]. Here, we adopt a diﬀerent strategy based on theintrinsic logical structure that is associated with any topos.
5
3
Of course, the existence of the long-range, and all penetrating, gravitational force means that, ata fundamental level, there is really only
one
truly closed system, and that is the universe itself.
4
In simple non-relativistic systems, the state is speciﬁed at any given moment of time. Relativisticsystems (particularly quantum gravity!) require a more sophisticated understanding of ‘state’, butthe general idea is the same.
5
Topos theory is a sophisticated subject and, for theoretical physicists, not always that easy to
2
Our contention is that theories of a physical system should be formulated in a toposthat depends on both the theory-type and the system. More precisely, if a theory-type (such as classical physics, or quantum physics) is applicable to a certain class of systems, then, for each system in this class, there is a topos in which the theory is tobe formulated. For some theory-types the topos is system-independent: for example,conventional classical physics always uses the topos of sets. For other theory-types, thetopos varies from system to system: for example, this is the case in quantum theory.In regard to the three conditions listed above for a ‘realist’ interpretation, ourscheme has the following ingredients:1. The concept of the ‘value of a physical quantity’ is meaningful, although this‘value’ is associated with an object in the topos that may not be the real-numberobject. With that caveat, the concept of a ‘property of the system’ is also mean-ingful.2. Propositions about a system are representable by a Heyting algebra associatedwith the topos. A Heyting algebra is a distributive lattice that diﬀers from aBoolean algebra only in so far as the
law of excluded middle
need not hold,i.e.,
α
∨ ¬
α
1. A Boolean algebra is a Heyting algebra with strict equality:
α
∨¬
α
= 1.3. There is a ‘state object’ in the topos. However, generally speaking, there willnot be enough ‘microstates’ to determine this. Nevertheless, truth values can beassigned to propositions with the aid of a ‘truth object’. These truth values liein another Heyting algebra.This new approach aﬀords a way in which it becomes feasible to generalise quantumtheory without any fundamental reference to Hilbert spaces, path integrals, etc.; inparticular, there is no
prima facie
reason for introducing continuum quantities. As wehave emphasised, this is our main motivation for developing the topos approach. Weshall say more about this later.From a conceptual perspective, a central feature of our scheme is the ‘neo-realist
6
’structure reﬂected in the three statements above. This neo-realism is the conceptualfruit of the mathematical fact that a physical theory expressed in a topos ‘looks’ verymuch like
classical
physics.This fundamental feature stems from (and, indeed, is deﬁned by) the existence of two special objects in the topos: the ‘state object’
7
, Σ
φ
, mentioned above, and the‘quantity-value object’,
R
φ
. Then: (i) any physical quantity,
A
, is represented byan arrow
A
φ
: Σ
φ
→ R
φ
in the topos; and (ii) propositions about the system are
understand. The references that we have found most helpful in this series of papers are [7, 8, 10, 9,
11, 12]. Some of the basic ideas are described brieﬂy in the Appendix to this paper.
6
We coin the term ‘neo-realist’ to signify the conceptual structure implied by our topos formulationof theories of physics.
7
The meaning of the subscript ‘
φ
’ is explained in the main text. It refers to a particular topos-representation of a formal language attached to the system: see later.
3

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