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A Topos Model for Loop QuantumGravity
Tore Dahlen
Department of MathematicsThe Faculty of Mathematics and Natural SciencesUniversity of Oslo
Abstract
One of the main motivations behind so-called topos physics, asdeveloped by Chris Isham and Andreas Döring [4-7], is toprovide a framework for new theories of quantum gravity. In thisarticle we do not search for such theories, but ask instead how oneof the known candidates for a final theory, loop quantum gravity(LQG), fits into the topos-theoretical approach. In theconstruction to follow, we apply the ‘Bohrification’ methoddeveloped by Heunen, Landsman and Spitters [10, 11] to the C*-algebra version of LQG introduced by Christian Fleischhack [9].We then bring together LQG results and methods from toposphysics in a proof of the non-sobriety of the external state space
S
of the Bohrified LQG theory, and show that the constructionobeys the standard requirements of diffeomorphism and gaugeinvariance.
July 30th, 2011
1. The Topos-theoretical Approach to Quantum Physics
1.1. The Neo-realism of Döring and Isham
In a series of articles [4-7], Chris Isham and Andreas Döring have proposed a set of newmodels for quantum physics, dubbed as
neo-realism
. Neo-realism is conceived as analternative to the well-known
Copenhagen interpretation
, which introduces a separationof the measurement process for a physical magnitude into two components, a quantumsystem
S
and a classical observer
V.
The possible states of
S
are wave functions
Y
from aconfiguration space into the set of complex numbers, whereas the observer
V
alwaysregisters a real value as the outcome of his experiment. In Copenhagen terminology, thewave function collapses onto the registered value with probability
P
H
r
L
=
†X
r
Y
\§
2
.The physically meaningful (real) value
r
is not a value of the physical quantity before themeasurement is made. The interpretation breaks down for closed systems where no outside observer is to be found, such as quantum cosmology. In the topos schemesuggested by Isham and Döring, physical quantities
does
have a value independent of any
observer
V
. The scheme relies on non-standard representations of the states and quantityvalues of physics. It also turns out that a new, intuitionistic quantum logic supplants thefamiliar non-distributive logic of Birkhoff and van Neumann [2]. (It should be noted thatthe choice of the tag neo-realism would be protested by philosophers and logicians,such as Michael Dummett, who regard acceptance of the law of excluded middle as thehallmark of philosophical
realism
(e.g. [8], p. 130ff).)In this subsection, we only give a brief outline of the topos scheme introduced by Döringand Isham. In order to appreciate the scope of the models, it is necessary to read thesrcinal articles. Also, notions from topos theory are used without explanation or com-ment (see [14] for a proper introduction to this field.)Following earlier work by Isham and Butterfield, Döring and Isham [5] start theirapproach to quantum systems by assuming that the physical quantities
A
of a system
S
arerepresented by self-adjoint operators
Â
in the non-commutative von Neumann algebra,
(
), of all bounded operators on the separable Hilbert space
of the states of
S.
Theunital, commutative subalgebras
V
of
(
) are then considered as
classical contexts
or
perspectives
on the system
S
, and the
context category
(
) is defined with Ob(
(
))as the set of contexts
V
and Hom(
(
)) given by the inclusions
i
V
'
V
:
V'
Ø
V
.In general, a context
V
will exclude many operators. But, in a certain sense, excludedprojection operators
P
`
still have proxys in
V.
For note that
P
`
, even if not present in thecontext
V
, may be approximated by the set (where
(
V
) is the complete lattice of projec-tions in
V
, and the ordering
t
is defined as
Q
`
t
P
`
if and only if Im
P
`
Œ
Im
Q
`
, or, equiva-lently,
P
`
Q
`
=
P
`
)
(1)
d
H
P
`
L
V
:
=
Ì 8
Q
`œ
H
V
L
Q
`
t
P
<
`
.Truth values may now be assigned the projectors in each context
V
by using the Gelfandspectrum
S
V
. This is the set
(2)
S
V
:
=
8
l
:
V
Ø
l
isapositivemultiplicativelinearfunctionalofnorm1
<
.When
P
`
is a projection, the value
l
(
P
`
)
is either
0
(false)
or
1
(true)
, so
l
behaves like a“local” state for
V
: it answers “yes” or “no” to the “question”
P
`
. The construction of thestate object, the representation of the physical state space in the topos scheme, may nowbe undertaken. The
state object
(or
spectal presheaf
)
S
is the element in the class of objects of the topos of presheaves over the context category
(
),
t
:
=
Sets
H
L
op
,such that
S
V
:=
S
V
and, for morphisms
i
V
'
V
:
V’
Ø
V
in
(
)
,
S
(
i
V
'
V
) :
S
V
Ø
S
V
'
isdefined by
S
(
i
V
'
V
)(
l
) :=
l
V
'
(the restriction of
l
:
V
Ø
to
V’
Œ
V
).As a substitute for the notion of a state (that is, a global element of the state space, theexistence of which is excluded by the topos version of the Kochen-Specker theorem of quantum mechanics), Döring and Isham define a
clopen subobject S
of
S
as a subfunctorof
S
(in the standard sense) such that the set
S
V
is both open and closed as a subset of thecompact Hausdorff space
S
V
(with weak*-topology).There is now, from Gelfand spectral theory, a lattice isomorphism between the lattice
2
A Topos Model for Loop Quantum Gravity.nb
(
V
) of projections in
V
and the lattice of closed and open
subsets
, Sub
cl
H
S
V
L
, of
S
V
:
(3)
a
:
H
V
L
Ø
Sub
cl
H
S
V
L
where
a
H
P
`
L
:
=
9
l œ S
V
l
H
P
`
L
=
1
=
ª
S
P
`
.Both
H
V
L
andSub
cl
H
S
V
L
are Boolean algebras, so the law of excluded third holds ([14],p. 55). The commutative algebras
V
are classical contexts within the theory, so it is properthat these lattices are Boolean. Certainly, extraordinary logic is the last thing we wouldexpect to find when we are engaged in experimental physics. Extending this constructionto the total context category
(
), Döring and Isham ([4], th. 2.4) prove that, for eachprojection
P
`
œ
(
), there is a clopen sub-object
S
P
`
of the spectral presheaf
S
given by
(4)
S
P
`
:
=
:
S
d
H
P
`
L
V
Œ S
V
V
œ
Ob
H
H
LL>
.This leads to the main achievement of the Döring-Isham approach, the map which“throws” the observable into a world of classical perspectives:The
daseinisation
d
of projection operators
P
`
œ
(
) is the mapping
(5)
d
:
H
L
Ø
Sub
cl
H
S
L
P
`
#
S
P
`
.The importance of the definition of daseinisation rests on the mapping between a projec-tion, which in
quantum
physics is the representative of a proposition of the theory, and asub-object of the 'state object'
S
, the topos analogue of a subset of the state space, whichis the
classical
notion of a proposition in physics: projection
P
`
@
aquantummechanicalstatement
D
Ø
d
subobject
S
P
`
@
toposanalogueofasubspaceora classicalstatement
D
.The above constructions determine the logic appropriate for quantum physics in topoi, fornote that Sub
cl
H
S
L
(the clopen subobjects of
S
) is a
Heyting algebra
.
The distributive lawholds in all Heyting algebras:
(6)
x
Ï H
y
z
L
¨
H
x
Ï
y
L H
x
ﬂ
z
L
Hence, it is valid in propositional quantum logic. The well-known laws below, however,do not hold:
(7)
x
Ÿ
x
and
Ÿ Ÿ
x
Ø
x
.The quantum logic of topos physics is
intuitionistic
.
1.2. Bohrification
An alternative, mathematically sophisticated version of the topos-theoretical approach isfound in the work of Heunen, Landsman and Spitters [10, 11]. This alternative, known as'Bohrification', utilizes the topos-theoretical generalisation of the notion of space,
locales
(cf. sec. 4 below). The quantum logic is then read off from the Heyting algebra structureof the open subsets of a locale
L
(or, strictly, the frame
(
L
)), identified as the state spaceof the system. We review the main characteristics of Bohrification in this subsection.In the Döring-Isham approach, the context category was given by a family of commutativesubalgebras
V
of a non-commutative von Neumann algebra. The state object
S
was a
A Topos Model for Loop Quantum Gravity.nb
3
functor in the topos, with
S
V
the Gelfand spectrum in the context
V
. The constructionrelied on the rich supply of projections available in von Neumann algebras. In the Bohrifi-cation approach, a family of commutative subalgebras of a non-commutative C*-algebrais used instead of the von Neumann algebras. These algebras are generally poor inprojections, special cases (such as Rickart algebras or von Neumann algebras) excepted,so the former notion of a state is no longer useful. (The two approaches to topos physics,Döring-Isham and Bohrification, are compared in great detail in [17].)Bohrification starts from the topos
Sets
H
A
L
of
covariant
functors, where
(
A
) is the set of commutative C*-subalgebras of a C*-algebra
A
. The tautological functor
A
:
(
A
)
Ø
Sets
,which acts on objects as
A
(
C
) =
C
, and on morphisms
C
Œ
D
as the inclusion
A
(
C
)
Ø
A
(
D
), is called the
Bohrification
of
A
.Now consider the functor
:
CStar
Ø
Topos
, where
CStar
is the category of unital C*-algebras (with arrows defined as linear multiplicative functions which preserve theidentity and the *-operation), and
Topos
is the category of topoi (with geometric mor-phisms as arrows).
is defined by
(
A
) =
Sets
H
A
L
on objects and
(
f
)*(
T
)(
D
) =
T
(
f
(
D
))on morphisms
f
:
A
Ø
B
, with
T
œ
Sets
H
B
L
and
D
œ
(
A
). (
(
f
)* is the inverse image partof the geometric morphism
(
f
).) It can then be shown that
A
is a commutative C*-algebra in the topos
(
A
) =
Sets
H
A
L
. This crucial result rests upon a general fact fromtopos theory:
Fact.
If
Model
(
,
T
)
denotes the category of models of a geometric theory
in the topos
T
, there is an isomorphism of categories
(8)
Model
I
,
Sets
H
A
L
M
º
Model
I
,
Sets
L
H
A
L
.This is a special case of lemma 3.13 in [11]. (For a proof, see cor. D1.2.14 in [13]
.
)The proof of the commutativity of
A
appeals to Kripke-Joyal semantics for Kripke topoi[11]. It also makes use of the axiom of dependent choice (
DC
), which holds in
Sets
H
A
L
.Commutativity of
A
in
Sets
H
A
L
is proved by exploiting the proximity of the theory of C*-algebras to a
geometric
theory. In these theories, all statements have the form
(9)
H
x
”L@
y
H
x
”L
Ø f
H
x
”LD
.Here,
y
and
f
are positive formulae; i.e. formulae built by means of finite conjunctionsand existential quantifiers. Thus, geometric theories are formulae with finite verification (see [14], ch. X for more about this notion). If the theory of abelian C*-algebras (Banachalgebras with involution, and satisfying
∞
a*a
¥
=
∞
a
¥
2
) had been a geometric theory, wecould start from the following piece of information about
A
:
(10)
A
œ
Model
I
Thetheoryofabelian
C
*-
algebras,
Sets
L
H
A
L
.This is true by the definition of
A
as the tautological functor, and because
(
A
) containsonly commutative subalgebras. By the fact stated above, it would then seem follow that
4
A Topos Model for Loop Quantum Gravity.nb

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