A Topos Foundation for Theories of Physics:
I.
Formal Languages for Physics
By A. Döring
and C.J. Isham
http://arxiv.org/abs/quant-ph/0703060v1
Edited by Andy Ross
Abstract. 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 certain deep issues that arise
when contemplating quantum theories of space and time. Our basic contention
is that constructing a theory of physics is equivalent to finding a
representation in a topos of a certain formal language that is attached to
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 attached to a system, S.
The first is a propositional language, PL(S); the second is a higher-order,
typed language L(S). Both languages provide deductive systems with an
intuitionistic logic. The reason for introducing PL(S) is that, as shown in
paper II of the series, it is the easiest way of understanding, and
expanding on, the earlier work on topos theory and quantum physics. However,
the main thrust of our programme utilises the more powerful language L(S)
and its representation in an appropriate topos.
***
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 certain deep 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 their disparate views on the nature of space and
time, they almost all use more-or-less standard quantum theory. Although
understandable from a pragmatic viewpoint (since all we have is more-or-less
standard quantum theory) this situation is nevertheless questionable when
viewed from a wider perspective. Indeed, there has always been a school of
thought asserting that quantum theory itself needs to be radically
changed/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 is the use in the standard quantum formalism of
certain critical mathematical ingredients that are taken for granted and yet
which, we claim, implicitly assume certain properties of space and time.
Such an a priori imposition of spatio-temporal concepts would be a major
error if they turn out to be fundamentally incompatible with what is needed
for a theory of quantum gravity.
A prime example is the use of the
continuum which, in this context, means the real and/or complex numbers.
These are a central ingredient in all the various mathematical frameworks in
which quantum theory is commonly discussed. For example, this is clearly so
with the use of (i) Hilbert spaces and operators; (ii) geometric
quantisation; (iii) probability functions on a non-distributive quantum
logic; (iv) deformation quantisation; and (v) formal (i.e., mathematically
ill-defined) path integrals and the like. The a priori imposition of such
continuum concepts could be radically incompatible with 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
peristalithic problem of deciding how a 'quantum theory of cosmology' could
be interpreted if one was lucky enough to find one. Most people who worry
about foundational issues in quantum gravity would probably place the
quantum cosmology/closed system problem at, 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 for our
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 can be constructed, and in which continuum quantities play
no fundamental role.
Our contention is that theories of a physical
system should be formulated in a topos that 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 to be
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, the topos varies from system to system: for example,
this is the case in quantum theory.
Topos theory is a remarkably rich branch of mathematics
which can be approached from a variety of different viewpoints. The basic
area of mathematics is category theory; where, we recall, a category
consists of a collection of objects and a collection of morphisms (or
arrows). In the special case of the category of sets, the objects are sets,
and a morphism is a function between a pair of sets.
From our perspective, the most relevant feature of a topos t is that it
is a category which behaves in many ways like the category of sets. An
important property for us is that, in any topos t, the collection of
subobjects Sub(A) of an object A forms a Heyting algebra.
***
Re: A
Topos Foundation for Theories of Physics
Peristalith is the
name of a stone circle, such as Stonehenge. Therefore, a 'peristalithic
debate' is one that is (i) very old; and (ii) goes round in circles :-) I
think that, as applied to the meaning of quantum theory, this is very
appropriate :-)
Best regards
Chris
Posted by: Chris Isham on March 4, 2007 4:17 AM
AR (2007) Wonderful! Physics without the continuum, generalized metatheory to cover classical and quantum physics, and set theory generalized in topos theory to allow nonclassical logic: these are the issues that exceeded my mental powers as a graduate student who would like to have solved just this set of problems. Must I now study these papers?
