arXiv:quant-ph/0612227v1 29 Dec 2006
A topological study of contextuality and modality
in quantum mechanics
Graciela Domenech
∗1,3
, Hector Freytes
2
and Christian de Ronde
3,4
March 26, 2008
1. Instituto de Astronom´ıa y F´ısica del Espacio (IAFE)
Casilla de Correo 67, Sucursal 28, 1428 Buenos Aires, Argentina
2. Dipartimento di Scienze e Pedagogiche e Filosofiche - Universita degli Studi di
Cagliari - Via Is Mirrionis 1, 09123, Cagliari - Italia
3. Center Leo Apostel (CLEA)
4. Foundations of the Exact Sciences (FUND)
Brussels Free University - Krijgskundestraat 33, 1160 Brussels - Belgium
Abstract
Kochen-Specker theorem rules out the non-contextual assignment
of values to physical magnitudes. Here we enrich the usual ortho-
modular structure of quantum mechanical pro positions with modal
operators. This enlargement allows to refer consistently to actual and
possible properties of the s ystem. By means of a topological argu-
ment, more precisely in terms of the existence of se c tio ns of sheaves,
we give an extended version of Kochen-Specker theorem over this new
structure. This allows us to prove that contextuality remains a central
feature even in the enriched propositional system.
Keywords: contextuality, sheaves, modal, quantum logic
1 Introduction
Modal interpretations of quantum mechanics (Dieks, 1988; Dieks, 1989;
Dieks, 2005; van Fraassen, 1991) face the problem of finding an objective
∗
Fellow of the Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET)
1
reading of the accepted mathematical formalism of the theory, a reading
“in terms of properties possessed by physical systems, independently of con-
sciousness and measurements (in the sense of human interventions)” (Dieks,
2005). These interpretations intend to consistently include the possible prop-
erties of the system in the discourse looking for a new link between the s tate
of the system and the probabilistic character of its p roperties, thus sus-
taining that the interpretation of the quantum state must contain a modal
aspect. The name modal interpretation was for the first time used by B.
van Fraassen (van Fraassen, 1981) following modal logic, precisely th e logic
that deals with possibility and necessity. Within this frame, a physical prop-
erty of a sys tem means “a definite value of a physical quantity belonging
to this system; i.e., a feature of physical reality” (Dieks, 2005). As usual,
definite values of physical magnitudes correspond to yes/no propositions
represented by orthogonal projection operators acting on vectors belonging
to the Hilbert space of the (pure) states of the system (Jauch, 1996).
Formal studies of modal interpr etations of quantum logic exist which are
similar to the modal interpretation of the intuitionistic logic (Dalla Chiara,
1981; Goldblatt, 1984). A description of that those kind of approaches m ay
be found in (Dalla Chiara et al., 2004). At first sight, it may be thought
that th e enrichment of the set of (actual) propositions with modal ones
could allow to circumvent the contextual character of quantum mechanics.
We h ave faced the study of this issue and given a Kochen-Specker type
theorem for the enriched lattice (Domenech et al., 2006). In this paper, we
give a topological version of that theorem, i.e. we study contextuality in
terms of sheaves.
2 Basic Notions
We recall from (Goldblatt, 1986; Mac Lane and Moerdijk, 1992) and (Maeda
and Maeda, 1970) some notions of sheaves and lattice theory that will play
an important role in what follows. First, let (A, ≤) be a poset and X ⊆ A.
X is a decreasing set iff for all x ∈ X, if a ≤ x then a ∈ X. For each a ∈ A we
define the principal decreasing set associated to a as (a] = {x ∈ A : x ≤ a}.
The set of all decreasing sets in A is denoted by A
+
, and it is well known
that (A
+
, ⊆) is a complete lattice, thus hA, A
+
i is a topological space. We
observe that if G ∈ A
+
and a ∈ G then (a] ⊆ G. Therefore B = {(a] : a ∈ A}
is a base of the topology A
+
which we will refer to as the canonical base.
Let I be a topological sp ace. A sheaf over I is a pair (A, p) where A is a
2
topological space and p : A → I is a local homeomorphism. This means that
each a ∈ A has an open set G
a
in A that is m apped homeomorph ically by p
onto p(G
a
) = {p(x) : x ∈ G
a
}, and the latter is open in I. It is clear that p
is a continuous and open map. Local sections of the sheaf p are continuous
maps ν : U → I defined over open proper subsets U of I such that pν = 1
U
.
In particular we use the term global section only when U = I.
In a Boolean algebra A, congruences are identifiable to certain subsets
called filters. F ⊆ A is a filter iff it satisfies: if a ∈ F and a ≤ x then
x ∈ F and if a, b ∈ F then a ∧ b ∈ F . F is a proper filter iff F 6= A or,
equivalently 0 6∈ F . If X ⊆ A, the filter F
X
generated by X is the minimum
filter containing X. A proper filter F is maximal iff the quotient algebra
A/F is isomorphic to 2, being 2 the two elements Boolean algebra. It is
well known that each proper filter can be extended to a maximal one.
We denote by OML the variety of orthomodular lattices. Let L =
hL, ∨, ∧, ¬, 0, 1i be an orthomodular lattice. Given a, b, c in L, we write:
(a, b, c)D iff (a∨b)∧c = (a∧c)∨(b∧c); (a, b, c)D
∗
iff (a∧b)∨c = (a∨c)∧(b∨c)
and (a, b, c)T iff (a, b, c)D, (a,b,c)D
∗
hold for all permutations of a, b, c. An
element z of a lattice L is called central iff for all elements a, b ∈ L we have
(a, b, z)T . We denote by Z(L) the set of all central elements of L and it is
called the center of L. Z(L) is a Boolean sublattice of L (Maeda and Maeda,
1970; Theorem 4.15).
3 Sheaf-theoretic view of contextuality
Let H be the Hilbert space associated to the physical system and L(H) be
the set of closed sub s paces on H. If we consider th e set of these subspaces
ordered by inclusion, then L(H) is a complete orthomodular lattice (Maeda
and Maeda, 1970). It is well known that each self-adjoint operator A that
represents a physical magnitude A may be associated with a Boolean sub-
lattice W
A
of L(H). More precisely, W
A
is the Boolean algebra of projectors
P
i
of the spectral decomposition A =
P
i
a
i
P
i
. We will refer to W
A
as the
spectral algebra of the operator A. Any proposition about the s y s tem is
represented by an element of L(H) which is the algebra of quantum logic
introduced by G. Birkhoff and J. von Neumann (Birkhoff and von Neumann,
1936).
Assigning values to a physical quantity A is equivalent to establishing
a Boolean homomorph ism v : W
A
→ 2 (Isham, 1998). Thus, it is natural
to consider the following definition which provides u s with a compatibility
3
condition:
Definition 3.1 Let (W
i
)
i∈I
be the family of Boolean sublattices of L (H).
A global valuation over L(H) is a family of Boolean homomorphisms (v
i
:
W
i
→ 2)
i∈I
such that v
i
| W
i
∩ W
j
= v
j
| W
i
∩ W
j
for each i, j ∈ I.
Kochen-Specker theorem (KS) precludes the possibility of assigning def-
inite p roperties to the physical system in a non-contextual fashion (Kochen
and Specker, 1967). An algebraic version of KS theorem is given by (Domenech
and Freytes, 2005; Theorem 3.2):
Theorem 3.2 If H be a Hilbert space such that dim(H) > 2, then a global
valuation over L(H) is not possible. 2
It is also possible to give a topological version of th is theorem in the frame
of local sections of sheaves. In fact, let L be an orthomo dular lattice. We
consider the family W
L
of all Boolean subalgebras of L ordered by inclusion
and the topological space hW
L
, W
+
L
i. On the set
E
L
= {(W, f ) : W ∈ W, f : W → 2 f is a Boolean homomorphism}
we define a partial ordering given by (W
1
, f
1
) ≤ (W
2
, f
2
) iff W
1
⊆ W
2
and
f
1
= f
2
| W
1
. Thus we can consider the topological space hE
L
, E
+
L
i whose
canonical base is given by the principal decreasing sets ((W, f)] = {(G, f |
G) : G ⊆ W }. By simplicity ((W, f )] is noted as (W, f].
Definition 3.3 The map p
L
: E
L
→ W
L
such that (W, f) 7→ W is a sheaf
over W
L
called spectral sheaf associated to the orthomodular lattice L.
Let ν : U → E
L
be a local section of p
L
. By (Domenech and Freytes,
2005; Proposition 4.2), f or each W ∈ U we have that ν(W ) = (W, f) f or some
Bo olean homomorphism f : W → 2 and if W
0
⊆ W , then ν(W
0
) = (W
0
, f |
W
0
). From a physical perspective, we may say that the s pectral s heaf takes
into account the whole set of possible ways of assigning truth values to the
propositions associated with the projectors of the spectral decomposition
A =
P
i
a
i
P
i
. The continuity of a local section of p guarantees that the
truth value of a proposition is maintained when considering the inclusion
of subalgebras. In this way, the compatibility condition 3.1 of the Boolean
valuation with respect to the intersection of pairs of Boolean sublattices of
4
L(H) is m aintained. Thus, continuous local sections of p
L
are identifiable
to compatible contextual valuations.
We us e ν(a) = 1 to note that there exists W ∈ U such that a ∈ W ,
ν(W ) = (W, f ] and f (a) = 1. On the other hand, if f : W → 2 is a
Bo olean homomorphism, ν : (W ] → E
L
is such that for each W
i
∈ (W ],
ν(W
i
) = (W
i
, f /W
i
) is a local section of p
L
. We call this a principal local
section.
A global section τ : W
L
→ E
L
of p
L
is interpreted as follows: the map
assigns to every W ∈ W
L
a fixed Boolean valuation τ
w
: W → 2 obviously
satisfying th e compatibility condition. Thus, KS theorem in terms of the
spectral sheaf r eads (Domenech and Freytes, 2005; Theorem 4.3):
Theorem 3.4 If H is a Hilbert space such that dim(H) > 2 then the spectral
sheaf p
L(H)
has no global sections. 2
4 An algebraic study of modality
With these tools, we are now able to build up a framework to include modal
propositions in the same structure as actual ones. To do so we en rich the or-
thomodular lattice with a modal operator taking into account the following
considerations: 1) Propositions about the properties of the physical system
are interpreted in the orthomodular lattice of closed subspaces of the Hilbert
space of the (pure) states of the s ys tem. 2) Given a proposition about
the system, it is possible to define a context from which one can p redicate
with certainty about it together with a set of propositions that are compat-
ible with it and, at the same time, predicate probabilities about the other
ones. In other words, one may predicate truth or falsity of all possibilities
at the same time, i.e. possibilities allow an interpretation in a Boolean al-
gebra. In rigorous terms, for each proposition P , if we refer with 3P to
the possibility of P , then 3P will be a central element of the orth omodular
structure. 3) If P is a proposition about the system and P occurs, then
it is trivially possible that P occurs. This is expressed as P ≤ 3P . 4)
Assuming an actual property and a complete set of properties that are com-
patible with it determines a context in which the classical discourse holds.
Classical consequences that are compatible with it, for example probability
assignments to the actuality of other pr opositions, shear the classical frame.
These consequences are the same ones as those which would be obtained by
considering the original actual property as a possible one. This is interpr eted
in the following way: if P is a property of the system, 3P is the smallest
5