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Spectral presheaves as
quantum state spaces
rsta.royalsocietypublishing.org
Research
Cite this article: Döring A. 2015 Spectral
presheaves as quantum state spaces. Phil.
Trans. R. Soc. A 373: 20140247.
http://dx.doi.org/10.1098/rsta.2014.0247
Accepted: 9 March 2015
One contribution of 13 to a theme issue
‘New geometric concepts in the foundations
of physics’.
Subject Areas:
quantum physics, category theory,
mathematical physics
Keywords:
state space, operator algebra,
category, topos, flow
Andreas Döring
Institute of Theoretical Physics I, Department of Physics,
Friedrich-Alexander-Universität Erlangen-Nürnberg,
Staudtstraße 7, 91058 Erlangen, Germany
For each quantum system described by an operator
algebra A of physical quantities, we provide a
(generalized) state space, notwithstanding the
Kochen–Specker theorem. This quantum state
space is the spectral presheaf Σ. We formulate the
time evolution of quantum systems in terms of
Hamiltonian flows on this generalized space and
explain how the structure of the spectral presheaf
Σ geometrically mirrors the double role played by
self-adjoint operators in quantum theory, as quantum
random variables and as generators of time evolution.
1. Introduction
We introduce the mathematical setting and the key
physical question: what is a physically sensible and
useful definition of a quantum state space?
(a) Algebraic quantum theory
Author for correspondence:
Andreas Döring
e-mail: [email protected]
Throughout this paper, we emphasize an algebraic
approach to the description of quantum systems. The
object that we start from is the algebra, denoted as A,
formed by the physical quantities (or observables) of a
quantum system. This is an algebra over the complex
numbers C. We assume throughout that our algebra A
is unital, i.e. it contains an element 1 that serves as a
multiplicative unit. In order to define self-adjoint and
unitary operators, A is equipped with an involution
∗ : A → A. The subset of self-adjoint operators is denoted
as Asa . These operators represent physical quantities. Asa
is closed under taking real linear combinations, so it is
a real vector space. The unitary operators form a group
U(A) with 1 as its neutral element.
Usually, completeness of the algebra A with respect
to the topology induced by the operator norm |_| : A → R
2015 The Author(s) Published by the Royal Society. All rights reserved.
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is assumed, that is, A is a Banach algebra. The interplay between the norm and the involution is
given by the C∗ -rule
for all a ∈ A, which makes A into a C∗ -algebra.
For many applications, it is useful to consider W ∗ -algebras. These are C∗ -algebras that are the
dual of Banach spaces. There is a very rich literature on C∗ -algebras and W ∗ -algebras and their
representation theory [1–4]. We just need some basics. Importantly, every C∗ - and W ∗ -algebra can
be faithfully represented on a suitable Hilbert space H (represented W ∗ -algebras are called von
Neumann algebras), and we usually think of a concrete algebra of operators on a Hilbert space.
An important, if special, example is the algebra B(H) of all bounded operators on a given Hilbert
space H. Every represented C∗ -algebra (respectively, W ∗ -algebra) is a norm-closed (respectively,
weakly closed) subalgebra of some B(H).
In the algebraic picture, physical states are described by states of the algebra, that is, linear
functionals
ρ : A −→ C
(1.2)
a −→ ρ(a)
that are positive, i.e. a > 0 implies ρ(a) ≥ 0, and normalized, i.e. ρ(1) = 1, the identity operator 1 is
mapped to the number 1. Physically, ρ(a) is interpreted as the expectation value of the physical
quantity a ∈ Asa in the state ρ.
Two more remarks:
(a) For a finite-dimensional Hilbert space H = Cn , the algebra B(H) is simply the algebra
Mn (C) of complex n × n-matrices.
(b) The canonical observables position and momentum generate the Heisenberg algebra. If
one considers its weak closure A, then one obtains the von Neumann algebra B(H), where
H is infinite-dimensional.
(b) Gelfand spectrum and Gelfand representation
Every Abelian C∗ -algebra or von Neumann algebra A has a Gelfand spectrum Σ(A), which is a
compact Hausdorff space.1 The elements of Σ(A) are the algebra homomorphisms from A to C,
also called characters,
λ : A −→ C.
(1.3)
Equivalently, the elements of the Gelfand spectrum are the multiplicative states of A. The set
Σ(A), being a subset of the state space of A, is equipped with the relative weak∗ -topology, which
makes it into a compact Hausdorff space.
The prototypical example of an Abelian C∗ -algebra is given by C(X), the continuous, complexvalued functions on a compact Hausdorff space X. The norm on C(X) is the supremum norm,
⎫
|_| : A −→ R
⎬
(1.4)
f −→ |f | := | sup f (x)|,⎭
x∈X
and the involution is given by taking the complex conjugate in the image,
∗
: A −→ A
f −→ f ∗ ,
1
If A is a non-unital C∗ -algebra, Σ(A) is only locally compact. We assume throughout that A is unital.
(1.5)
.........................................................
(1.1)
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|a∗ a| = |a|2
2
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where f ∗ (x) := f (x)∗ for all x ∈ X. The Gelfand representation theorem shows that, for every
Abelian C∗ -algebra A, there is an isomorphism
of C∗ -algebras, that is, G is an algebra isomorphism that preserves the norm and the involution.
Concretely, the Gelfand transform ā of a ∈ A is given by
ā : Σ(A) −→ C
λ −→ ā(λ) := λ(a).
(1.7)
Note that self-adjoint operators are mapped to real-valued functions.
Physically, each λ ∈ Σ(A) is a pure state of A, and λ(a) is the value that the physical quantity
a ∈ Asa has in the state λ. The value λ(a) can be regarded as an actual value, not just an expectation
value, because λ is a pure state and an eigenstate of a, and all the operators in A have joint
eigenstates, because A is Abelian.
Hence, physically, the Gelfand spectrum Σ(A) consists of the pure states of the system
described by the Abelian algebra A, and by the Gelfand representation theorem, we can think
of A concretely as the algebra C(Σ(A)) of continuous functions on Σ(A). The Gelfand spectrum
plays the role of a state space of the system, and the physical quantities, given by the self-adjoint
operators in A, are represented as real-valued functions on the state space Σ(A).
(c) The main question
The situation for systems described by Abelian C∗ - or W ∗ -algebras is quite straightforward and
appealing: to each algebra A of physical quantities, there corresponds a compact Hausdorff space
Σ(A) such that A is isomorphic to the concrete algebra of functions C(Σ(A)), and, physically,
Σ(A) plays the role of the state space for the system. All this hinges on A being Abelian; for nonAbelian algebras, there is no direct generalization of the Gelfand spectrum Σ(A) available. But
of course, in quantum theory, we have to deal with non-Abelian algebras of physical quantities
from the start, because certain physical quantities such as position and momentum, or spin-x
and spin-z, are incompatible and are described by pairs of non-commuting self-adjoint operators.
A state space of a quantum system would have to be a space corresponding to a non-Abelian
algebra A of physical quantities, i.e. a non-commutative space. The following question arises
naturally:
What is a physically sensible and useful definition of a quantum state space?
Having found such a quantum state space, how do we implement various physical aspects such
as representation of physical quantities, states, time evolution, etc. with respect to it? We present
a solution to these questions in this paper, building on previous work in the topos approach to
quantum theory, which was initiated by Isham, Butterfield and Hamilton [5,6] and substantially
developed mainly by Isham, Döring and others [7–20]. For some closely related work by Heunen
et al. and by Wolters; see [21–24].
We show that the spectral presheaf Σ plays the role of a quantum state space and hence gives
a geometric counterpart to the non-Abelian algebra A of physical quantities. The key step is the
generalization from a set (of points) with structure, such as a compact Hausdorff space, to a
presheaf (with ‘no points’, in a sense to be explained) with structure. This allows us to define
a generalization of the Gelfand spectrum to non-Abelian C∗ - and W ∗ -algebras, which is just the
technical tool we need. We emphasize physical interpretation rather than mathematical details,
for which we provide sufficient references. No knowledge of topos theory is required, and, in fact,
topos theory plays no major role in this paper.
.........................................................
(1.6)
a −→ ā
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373: 20140247
G : A −→ C(Σ(A))
3
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We introduce the spectral presheaf Σ and briefly review previous results supporting its
interpretation as a quantum state space.
(a) Definition and basic properties of the spectral presheaf
Let S be a quantum system whose physical quantities are described by the self-adjoint elements
of a non-Abelian C∗ - or W ∗ -algebra A. Because A is non-Abelian, it has no Gelfand spectrum, but
each Abelian C∗ - or W ∗ -subalgebra C ⊂ A has a Gelfand spectrum Σ(C). Moreover, if C1 , C2 are
two Abelian subalgebras of A such that C2 ⊂ C1 , then there is a canonical map
rC1 C2 : Σ(C1 ) −→ Σ(C2 )
(2.1)
λ −→ λ|C2
between their Gelfand spectra, where λ|C2 : Σ(C2 ) → C simply denotes the restriction of λ :
Σ(C1 ) → C to the smaller subalgebra C2 . It is well known that the restriction map rC1 C2 is
continuous, closed and surjective.
The basic idea in the construction of the spectral presheaf of A is very simple: we collect the
Gelfand spectra Σ(C) of all the Abelian subalgebras of A and all the functions rC1 C2 between
their spectra (where C2 ⊂ C1 ) into one object. In order to do this systematically, we first define
the context category C(A): its objects are the unital Abelian C∗ -subalgebras C of the non-Abelian
algebra A. Following tradition in physics, these Abelian C∗ -subalgebras are also called contexts.
We consider only contexts C ⊂ A such that 1C = 1A holds. The arrows in the category C(A) are the
inclusions of smaller contexts into larger ones, iC2 C1 : C2 → C1 . Hence, the context category C(A)
is simply the partially ordered set of contexts.
The spectral presheaf Σ then arises as a contravariant functor from the context category C(A) to
Set, the category of sets and functions:
(a) On objects C ∈ C(A), we define Σ C := Σ(C), the Gelfand spectrum of C.
(b) On arrows iC2 C1 : C2 → C1 , we define Σ(iC2 C1 ) := rC1 C2 : Σ C1 → Σ C2 , λ → λ|C2 .
Contravariant, Set-valued functors are traditionally called presheaves, hence the name. As we
saw above, each Gelfand spectrum Σ C = Σ(C) is, in fact, a compact Hausdorff space, and each
restriction map Σ(iC2 C1 ) = rC2 C1 is a continuous map, so the spectral presheaf takes values in
KHaus, the category of compact Hausdorff spaces and continuous maps.
There is an obvious W ∗ -analogue to all these constructions, which gives the spectral presheaf
of a non-Abelian W ∗ -algebra (or von Neumann algebra, if we implicitly use a representation on
a Hilbert space, as we usually do). For clarity, we use the notation N for a W ∗ - or von Neumann
algebra from now on, and V(N ) for the context category, i.e. the poset of Abelian W ∗ - or von
Neumann subalgebras of N . A generic element of V(N ) will be denoted as V.
.........................................................
2. The spectral presheaf of a quantum system
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There are various other sophisticated and highly developed approaches to non-commutative
geometry, see, for example, Connes and Marcolli from a differential geometric perspective [25,26];
Manin and Majid from deformation and quantum groups [27,28]; Kontsevich and Rosenberg
from algebraic geometry [29,30]; also Hrushovski & Zilber from geometric model theory [31].
Our approach differs from all these in various respects. The topos approach provides the most
concrete geometric examples of non-commutative spaces in the form of spectral presheaves,
whereas the other approaches are more algebraic in nature. Yet, the other approaches to noncommutative spaces and their geometry are developed more fully, and we hope and expect to
implement some of their structures and results in the topos picture in the future. For example,
non-commutative geometry à la Connes, but also à la Manin, contains beautiful generalizations
of differential geometry to the non-commutative case—something that would clearly be desirable
to also have in our picture.
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(2.2)
from the constant presheaf 1 to P. Such a natural transformation picks one element pC ∈ PC at each
context C ∈ C(A) such that, whenever C1 ⊂ C2 , we have pC1 |C2 = pC2 : the elements picked out by p
must fit together under restriction.
If we consider the spectral presheaf Σ, a global element p : 1 → Σ would pick one λC :=
pC ∈ Σ C for each context C ∈ C(A). As we saw above, λC is a pure state of C, and, for each
physical quantity a ∈ Csa , λ(a) is the value that a has in the state λ. Moreover, if C2 ⊂ C, then
λC |C2 = λC2 = pC2 , because we assume that p is a global element of Σ. This implies that the value
λC (a) assigned to a physical quantity a does not depend on the context C we are considering.
Furthermore, it is easy to see that the value assignment is compatible with taking (continuous)
functions of operators: if f (a) is another self-adjoint operator, then λ(f (a)) = f (λ(a)). As a simple
example, consider taking squares. The value assigned to a2 is the square of the value assigned
to a.
But, at least for the case that N is a W ∗ -algebra with no summand of type I2 , we know that
such an assignment of values to all physical quantities, preserving functional relations between
them, does not exist. This is exactly the content of the famous Kochen–Specker theorem [33]
and its generalization to von Neumann algebras [34]. Hence, the Kochen–Specker theorem is
equivalent to the fact that the spectral presheaf, our quantum state space, has no points. This
was first observed by Isham & Butterfield [5] and, for the case of the von Neumann algebra B(H),
by Isham et al. [6].
Mathematically, the fact that Σ has no points is an expression of its character as a noncommutative space. Physically, we see that the lack of points is equivalent to one of the key
theorems in foundations of quantum theory, the Kochen–Specker theorem.
(b) Previous results on the spectral presheaf as a quantum state space
The interpretation of the spectral presheaf Σ as a quantum state space is supported by a number
of further results:
— In classical physics, propositions about the physical world are represented by (Borel)
subsets of the state space S, and the Borel subsets form a σ -complete Boolean algebra.
In [8], it was shown that there is a distinguished family of subpresheaves (the presheaf
analogues of subsets of a set) of the spectral presheaf Σ, called clopen subobjects.2 These
subobjects form a complete Heyting algebra Subcl Σ, and there is a systematic way, called
daseinization, of mapping propositions about the physical world to clopen subobjects. The
fact that we arrive at a Heyting algebra rather than at a Boolean algebra signifies the shift
from classical, two-valued Boolean logic to a form of intuitionistic logic, which turns out
to be multi-valued. The logical aspects of the topos formalism for quantum theory arise
2
A subobject S of Σ is called clopen if every component SV , V ∈ V(N ), is a clopen subset of Σ V .
.........................................................
p : 1 −→ P
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The spectral presheaf was first defined in similar form by Isham & Butterfield [5] and for the
algebra B(H) by Isham et al. [6]. Physically, the spectral presheaf Σ associated with a quantum
system described by a non-Abelian C∗ - or W ∗ -algebra serves as a (generalized) state space.
It collects all the state spaces of the Abelian parts C ∈ C(A), the contexts of A, into a whole.
Importantly, the ‘local’ state spaces (where ‘local’ means associated with an Abelian part of the
‘global’, non-Abelian algebra) are not independent, but are glued together by the restriction maps.
Presheaves generalize sets, they are also called varying sets. A presheaf can simply be regarded
as a collection of sets, one for each object of the base category, and with functions between these
sets in the opposite direction of the arrows in the base category. Let P be a presheaf over the
context category C(A), i.e. a functor P : C(A) → Setop . The analogue of an element x of a set X
is a global element of a presheaf P, that is, a natural transformation (in the sense of category
theory [32])
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—
—
These aspects of the topos formalism for quantum theory show that the spectral presheaf
Σ, indeed, plays the role of a quantum state space, in strong analogy to the state space S of a
classical system.
Yet, we have considered only kinematic aspects so far, whereas dynamics has not been treated.
We turn to this in §3.
3. Time evolution and Hamiltonian flows on the quantum state space
We introduce Hamiltonian flows on the quantum state space Σ, in analogy to classical
Hamiltonian mechanics, and show that the Schrödinger time evolution of (representatives of)
vector states can naturally be phrased in terms of these flows.
(a) Hamiltonian operators and flows
Let h be the Hamiltonian operator of the quantum system under consideration. The operator h
need not be bounded from above and hence need not be an element of the algebra A of physical
quantities. But, if A = N is a von Neumann algebra, we have that h is affiliated with N , that is,
3
Whether vector states are pure states depends on the algebra N at hand. For B(H), vector states are pure.
.........................................................
—
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—
from the internal logic of the topos of presheaves over the context category; for details
see [8,13,14,16,18]. We will not be concerned with this part of the theory here.
When representing pure states, a difference between classical physics and quantum
physics in the topos formulation shows up. In classical physics, a pure state is simply
a point s of the state space S or, equivalently, a Dirac measure concentrated at s. As
mentioned, the quantum state space Σ has no points by the Kochen–Specker theorem.
Pure quantum states, given by vector states,3 are represented by certain subobjects
that are ‘as close to being a point as possible’. This is shown explicitly below in §3b;
see also [8,15].
Combining (representatives of) propositions and pure states, one can use the internal
logic of the topos to assign truth values to all propositions at once, without any reference
to instrumentalist concepts such at measurements and observers. This becomes possible,
because the topos provides a richer logical system than standard Boolean logic (and
quantum logic) which is multi-valued and intuitionistic. Because we employ the internal
op
logic of the topos SetV(N ) of presheaves over the context category V(N ), the new logic
for quantum systems is contextual by construction; see [8,14,18].
In classical physics, physical quantities are represented by real-valued functions on the
state space. For example, in any state s ∈ S, the energy has a certain value (in a suitable,
fixed system of units), so energy is described by a function E : S → R. In a topos, the
analogue of a function is an arrow between two objects. Hence, in the topos formulation
of quantum theory, a physical quantity is represented by an arrow in the topos of
presheaves from the quantum state space Σ to a space R↔ of values. This space of values
is a presheaf itself. It is related to the real numbers, but in order not to run into difficulty
with the Kochen–Specker theorem, one has to include not just real numbers but also finite
real intervals as generalized, ‘unsharp’ values. Details can be found in [9,13,15].
Mixed states are represented classically by probability measures on the state space. In
the topos formalism, mixed states are given by probability measures on the quantum
state space. Each quantum state ρ induces a unique probability measure μρ on Σ.
Conversely, every probability measure μ on Σ determines a unique quantum state ρμ by
Gleason’s theorem (and its generalization to von Neumann algebras). The usual Born rule
is captured by the topos formalism, and one can calculate expectation values of physical
quantities [11]. Moreover, quantum probabilities can be absorbed into the logic of a larger
topos [13].
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for each t ∈ R. In the quantum state space Σ, the Gelfand spectrum Σ V is attached to a context V.
It is known from topos theory that the appropriate way to lift a map from the base category, which
is the context category V(N ) in our case, to the topos of presheaves over V(N ), where the spectral
presheaf Σ lives, is by using a so-called geometric morphism [37,38]. We do not go into the details
here, but just state that each ut : V(N ) → V(N ) acts on the spectral presheaf by the inverse image
part of the geometrical morphism to give
ũt : Σ −→ Σ t ,
(3.3)
∀V ∈ V(N ) : (Σ t )V = (ũt (Σ))V := (Σ ◦ ut )V = Σ ut Vu−t .
(3.4)
That is, the action of ũt on Σ results in a ‘twisted’ version of the spectral presheaf, denoted Σ t .
The component of Σ t at the context V is the Gelfand spectrum of ut Vu−t .
Hence, if a context V is moved by ut , then the corresponding Gelfand spectrum Σ V will also be
moved, which means that we do not map the quantum state space Σ into itself. We need a second
step mapping the spectrum Σ ut Vu−t back into Σ V , and, globally, mapping Σ t back into Σ. Acting
by ũ−t on Σ t would achieve this, but only in a trivial manner, because the composite ũ−t ◦ ũt is
simply the identity on Σ, independent of t. Clearly, we need a different construction.
We make use of the fact that the context V is isomorphic to the context ut Vu−t as a commutative
von Neumann algebra. Gelfand duality [3] implies that Σ ut Vu−t is isomorphic to Σ V . Concretely,
let us write
ut;V : V −→ ut Vu−t
(3.5)
a −→ ut au−t
for the isomorphism between the unital, Abelian von Neumann algebras V and ut Vu−t . Then
⎫
⎬
gut ;V : Σ ut Vu−t −→ Σ V
(3.6)
λ −→ λ ◦ ut;V ⎭
.........................................................
By a theorem by Stone and von Neumann, this one-parameter group is continuous in the strong
operator topology. In standard quantum theory, time evolution of vector states in the Schrödinger
picture is given by ψ(t) := ut ψ0 for all t ∈ R, where ψ0 is the initial state at t = 0. Time evolution
in the Heisenberg picture is given by a(t) := u−t a0 ut for all t ∈ R, where a0 is the initial physical
quantity at t = 0.
It is not immediately obvious how time evolution translates into the topos formulation. Again,
the situation in classical physics can serve as a guide. In Hamiltonian mechanics, there is a
Hamiltonian function H and the associated Hamiltonian flow φH . The latter is a one-parameter
group φH : R → Diff(S) of diffeomorphisms (more precisely, symplectomorphisms) that maps the
state space of the system into itself. We will show that, in analogy, the Hamiltonian h induces a
Hamiltonian flow on the quantum state space Σ. A number of mathematical aspects have been
developed in [35,36], but we will take a more down-to-earth physics approach here.
Let t ∈ R, and let V ∈ V(N ) be a context. We first observe that the unitary ut acts on V by
conjugation, sending it to ut Vu−t = {ut au−t ∈ N | a ∈ V}. In general, ut Vu−t will not be the same
context as V, so ut moves the elements of the context category around. Of course, u−t induces
the opposite transformation. Importantly, if V2 ⊂ V1 , then ut V2 u−t ⊂ ut V1 u−t , so ut preserves the
order on the context category V(N ), and we obtain an order-isomorphism
ut : V(N ) −→ V(N )
(3.2)
V −→ ut Vu−t
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all spectral projections of h lie in N . We consider the one-parameter group of unitaries generated
by h:
⎫
⎬
u : R −→ U(A)
(3.1)
t −→ ut := eith .⎭
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(3.7)
Then by composing
ũt
Gt
Σ −→ Σ ◦ ut −→ Σ,
(3.8)
we indeed map the quantum state space Σ into itself in a non-trivial and invertible way. We
denote the composite map by
Ft := ũt , Gt = Gt ◦ ũt : Σ −→ Σ,
(3.9)
the inverse being F−t .4 We interpret Ft as the analogue of a diffeomorphism from Σ to itself.
Because we do not have the analogue of a differential structure on Σ yet, we more modestly
(and correctly) call Ft an automorphism of Σ. The automorphisms of the spectral presheaf form a
group Aut(Σ), and, by letting t vary, we obtain a one-parameter group of automorphisms,
F : R −→ Aut(Σ)
(3.10)
t −→ Ft = ũt , Gt ,
which we call a Hamiltonian flow on the quantum state space Σ. By construction, F0 = F(0) is the
identity on Σ. In §4c, we consider flows on the spectral presheaf in some more detail. For more
details on the automorphism group Aut(Σ), see [35].
This shows that time evolution in the topos formulation of quantum theory looks structurally
very similar to time evolution in classical Hamiltonian mechanics. In both cases, there is a state
space and a one-parameter group of automorphisms, called a Hamiltonian flow, that describes
time evolution. Such a geometrical picture of time evolution based on state spaces is missing
from the standard Hilbert space formulation of quantum theory.
(b) Schrödinger evolution of pure states in the topos picture
In [36], it was shown how Heisenberg and Schrödinger evolution can be described
mathematically in the topos picture, but pure states were not treated. Here, we consider pure
states and their time evolution in the topos picture, emphasizing a more physical perspective.
A (pure) state in classical physics is given by a point s of the state space. Under time evolution,
the state s will move with the Hamiltonian flow and will evolve into other states (in general). Our
goal is to show that the topos picture provides a similar way of describing the time evolution of
quantum systems.
The immediate problem is that the quantum state space Σ has no points, as discussed in §2.
Instead, pure quantum states are described by subobjects of Σ that are ‘as close to being a point’
as possible. We describe this representation now. We assume that the algebra N of observables is
a von Neumann algebra, because we need sufficiently many projections for some constructions
to work. Contexts V ∈ V(N ) are Abelian von Neumann subalgebras of N .
(i) Definition of pseudo-states
Let ψ ∈ H be a pure state, and let pψ be the corresponding projection onto the ray Cψ. For every
context V ∈ V(N ), let P(V) denote the lattice of projections in V and define
o
(pψ ) := {q ∈ P(V) | q ≥ pψ }.
(3.11)
δV
Technically, Ft = Gt ◦ ũt : Σ → Σ is the composite of a natural isomorphism (an arrow in the topos) with the inverse image
part of a geometric morphism (a map from the topos to itself); hence, it is not the composite of two arrows in a single category.
For this reason, we prefer the notation ũt , Gt .
4
.........................................................
Gt : Σ t −→ Σ.
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is the isomorphim between the Gelfand spectra that we are looking for. It is easy to see that
the maps gut ;V , for V varying over V(N ), form a natural isomorphism (in the sense of category
theory [32]),
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(3.12)
If V , V are two contexts such that V ⊂ V, then it holds for all λ ∈ wψ V that
o
o
o
λ|V (δV
(pψ )) = λ(δV (pψ )) ≥ λ(δV (pψ )) = 1,
(3.13)
so λ|V ∈ wψ V . This means that the sets (wψ V )V∈V(N ) form a subobject wψ of the spectral presheaf
Σ. This subobject is called a pseudo-state and represents the pure state ψ.
The pseudo-state wψ is, intuitively speaking, as close to being a point of Σ as possible: by
construction, it is the smallest subobject that ‘covers’ the ray Cψ.
(ii) Time evolution of pseudo-states
In order to describe the time evolution of wψ , we use the Hamiltonian flow on the quantum state
space Σ as defined in §3a. Let (ut )t∈R = (eith )t∈R be the strongly continuous one-parameter group
of unitaries generated by the Hamiltonian operator h, and let F : R → Aut(Σ) be the corresponding
flow on Σ (see (3.10)).
For each time t ∈ R, Ft = ũt , Gt : Σ → Σ is an automorphism of the quantum state space Σ.
In order to define a map between subobjects of Σ, it is natural to mimic the inverse image of a
function, so we use the inverse image of the map Ft ,
−1
F−1
= ũ−t , G−t .
t = ũt , Gt (3.14)
We now have to describe how this acts on subobjects of Σ, and in particular on a pseudo-state wψ .
Let S ∈ Subcl Σ be a clopen subobject. Then, F−t acts on S by
ũ−t
G−t
S −→ S ◦ u−t −→ F−t (S).
(3.15)
Here, G−t is the natural transformation from S ◦ u−t to Σ with components
∀V ∈ V(N ) : gu−t ;V : (S ◦ u−t )V −→ Σ V
λ −→ λ ◦ u−t;V ,
compare equation (3.6). Finally, F−t (S) denotes the image of S ◦ ũ−t under G−t . It is
straightforward to check that F−t (S) is a clopen subobject of Σ.
If wψ = wψ (0) is a pseudo-state, which is a small subobject of Σ representing a vector state, and
t → eith is a strongly continuous one-parameter group of unitaries, we define the time evolution
of wψ (0) in the Schrödinger picture by
∀t ∈ R : wψ (t) := F−t (wψ (0)).
(3.16)
Alternatively, we could simply have defined
wψ (t) = wut ψ0 ,
(3.17)
but this makes use of the Hilbert space structure directly, because ut acts on ψ0 . Our goal was to
refer to the Hilbert space structure as little as possible, and to use flows on the spectral presheaf
instead to describe time evolution.
.........................................................
o
(pψ )) = 1}.
wψ V := {λ ∈ Σ V | λ(δV
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Geometrically, this means that we pick the smallest projection in V that projects onto a closed
o (p ) clearly depends on the context
subspace of H that contains the ray Cψ. The projection δV
ψ
o (p ) = p ; otherwise,
V, because V may contain many or few projections. If pψ ∈ P(V), then δV
ψ
ψ
o
δV (pψ ) > pψ .
For each context V ∈ V(N ), consider
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It is still sensible to check whether both definitions of time evolution of pseudo-states coincide.
First note that, for all V ∈ V(N ) and all t ∈ R,
= gu−t ,V ((wψ ◦ ũ−t )V )
(3.19)
= gu−t ,V (wψ u−t Vut )
(3.20)
= wψ u−t Vut ◦ u−t;V .
(3.21)
o (u p u )u . Define a map α on projections in V by
It is easy to show that δuo −t Vut (pψ ) = u−t δV
t ψ −t t
V
αV (p) = {λ ∈ Σ V | λ(p) = 1} for all p ∈ P(V). Then, we obtain
λ ∈ wψ (t)V
(3.22)
⇐⇒ λ ∈ wψ u−t Vut ◦ u−t;V
(3.23)
⇐⇒ λ ◦ ut ∈ wψ u−t Vut ◦ u−t;V ◦ ut;V
(3.24)
⇐⇒ (λ ◦ ut )(δuo −t Vut (pψ0 )) = 1
(3.25)
o
⇐⇒ (λ ◦ ut )(u−t δV
(ut pψ0 u−t )ut ) = 1
(3.26)
o
(put ψ0 )ut u−t ) = 1
⇐⇒ λ(ut u−t δV
(3.27)
o
(put ψ0 ))
⇐⇒ λ ∈ αV (δV
(3.28)
⇐⇒ λ ∈ (wut ψ0 )V .
(3.29)
This shows that time evolution of quantum states, when described by flows on the quantum state
space Σ (acting on subobjects such as wψ ), indeed, gives the answer one would expect from
standard quantum theory.
4. Why the spectral presheaf?
Here, we give a new explanation for the structure of the spectral presheaf. We show that the
various algebraic structures on operators (non-commutative, Lie and Jordan algebras) have
corresponding geometric counterparts within the spectral presheaf.
The starting point is the well-known observation that the self-adjoint operators of a quantum
system play a dual role: on the one hand, they serve as (quantum) random variables, and, together
with quantum states, they provide the probabilistic predictions of quantum theory. On the other
hand, self-adjoint operators are generators of one-parameter groups of unitaries, and thus provide
the dynamics of quantum theory.
These two distinct roles have algebraic counterparts, as we will explain below: the probabilistic
aspects relate to the Jordan algebra formed by the self-adjoint operators, and the dynamical
aspects relate to the Lie algebra formed by them.
The spectral presheaf of a von Neumann algebra N combines these two distinct aspects in a
geometric way. As we will show, the Jordan algebra aspect is fully incorporated by the partial
order on contexts, that is, by the context category V(N ). The Lie algebra aspect is implemented
geometrically by (generators of) inner flows on the spectral presheaf.
(a) Jordan algebra structure and quantum probability
Jordan and co-workers introduced Jordan algebras in [39,40] in order to describe the probabilistic
aspects of quantum theory. Jordan algebras are real algebras, formed by the (representatives of)
.........................................................
(3.18)
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wψ (t)V = (F−t (wψ (0)))V
10
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(i)
(ii)
(iii)
(iv)
(v)
for all r ∈ R, the ear operator commutes with a,
ear = 0 for all r < −|a| and ear = 1 for all r > |a|,
ear ≤ eas for r < s,
∀r ∈ R : ear = s>r eas , and
|a|
a = −|a| r dear in the sense of norm convergence of approximating Riemann sums.
This implies that we can define the usual spectral measures in the setting of JBW-algebras; a noncommutative von Neumann algebra is not required. Moreover, quantum states can be interpreted
as states of JBW-algebras, and the Born rule can be formulated, so all the probabilistic aspects
of standard quantum theory are available at the level of JBW-algebras. Note that every von
Neumann algebra N has an associated JBW-algebra (N , ·) with the symmetrized product given
by (4.1). Often, it is enough to consider the real JBW-algebra (Nsa , ·) formed by the self-adjoint
elements.
Even Wigner’s theorem, a standard result of basic quantum theory, can best be understood in
terms of Jordan algebras: in modern language, Wigner’s theorem states that each automorphism
φ : P(H) → P(H) of the orthomodular lattice of projections on a Hilbert space of dimension 3 or
greater is induced by either a unitary or an antiunitary operator u,
∀p ∈ P(H) : φ(p) = upu∗ .
(4.2)
Usually, the antiunitary option is disregarded. Yet, it is known that every Jordan ∗-automorphism
Φ : B(H) → B(H) is implemented by either a unitary or an antiunitary operator; see [42]. Hence,
Wigner’s theorem actually says that every automorphism φ : P(H) → P(H) of the projection
lattice lifts to a Jordan ∗-automorphism Φ : B(H) → B(H). The generalization of Wigner’s theorem
to von Neumann algebras is known as Dye’s theorem [43].
(b) Lie algebra structure and dynamics
It is well known that every self-adjoint operator a ∈ N in a von Neumann algebra N generates
a one-parameter group eita of unitaries in N . Moreover, if b ∈ Nsa is some other self-adjoint
operator, then
d ita −ita
(e b e )|t=0 = i[a, b],
(4.3)
dt
so commutators are the infinitesimal aspect of such one-parameter groups acting on other selfadjoint operators. The self-adjoint operators Nsa in a von Neumann algebra N form a Lie algebra
with the Lie bracket (a, b) → i[a, b].
Of course, the time evolution of a quantum system is described in terms of the one-parameter
group of unitaries t → eith , where h is the Hamiltonian of the system5 and the quantum
state changes as ψ0 → ψt = eith ψ0 (Schrödinger picture) or the physical quantities change as
a → e−ith a eith (Heisenberg picture). The infinitesimal change of a under time evolution is given
by (d/dt)(e−ith a eith )|t=0 = i[h, a].
h need not be an element of the algebra N of physical quantities, but it is affiliated with it, i.e. all spectral projections of h are
in N .
5
.........................................................
The work by Jordan, von Neumann and Wigner was later extended quite massively into the
theory of Jordan operator algebras (see [41] and references therein), with norm-closed JB-algebras
and weakly closed JBW-algebras. Even if these algebras are not very well known in physics,
they provide all the structure needed for quantum probability. In particular, in JBW-algebras,
the spectral theorem holds: let M be a JBW-algebra, and let a ∈ M. Then, there is a unique family
ea = (ear )r∈R of projections such that
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physical quantities. The prime example are the self-adjoint operators on a Hilbert space, equipped
with the Jordan product
(4.1)
a · b = 12 (ab + ba) = 12 {a, b}.
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= a · b + 12 [a, b],
(4.4)
(4.5)
which gives the Jordan product and the commutator. (The factor 12 is absorbed into the Jordan
product by convention; otherwise, the formula would look entirely symmetrical.) The noncommutative, associative product (a, b) → ab has no direct physical meaning, but both the Jordan
product and the commutator have a good physical interpretation, as we argued above.
Note also that we can restrict attention to self-adjoint operators a, b. If we do so, we see that the
Jordan product a · b is the real part of ab, whereas i[a, b] is −1 times the imaginary part of ab.
(c) The spectral presheaf in this light
(i) Contexts and Jordan algebra structure
In [44], the following was shown: let M, N be von Neumann algebras not isomorphic to C ⊕ C
and with no type I2 summands. M and N are isomorphic as complex Jordan algebras if and only
if V(M) and V(N ) are isomorphic as posets. Concretely, for every order-isomorphism φ : V(M) →
V(N ), there exists a unique Jordan ∗-isomorphism Φ : M → N such that
∀V ∈ V(N ) : φ(V) = Φ[V].
(4.6)
Conversely, every Jordan ∗-isomorphism Φ : M → N induces an order-isomorphism φ : V(M) →
V(N ).
It is easy to see that this can be strengthened to include topology: for every order-isomorphism
φ : V(M) → V(N ), there exists a unique normal Jordan ∗-isomorphism Φ : M → N and vice versa
[45]. Restricting to self-adjoint operators, one obtains a one-to-one correspondence between
isomorphisms Φ : Msa → Nsa of real JBW-algebras and order-isomorphisms φ : V(M) → V(N ) of
context categories.
Hence, the context category V(N ) determines a von Neumann algebra N up to normal Jordan
∗-isomorphism, or briefly:
Contextuality is Jordan structure.
This means that, remarkably, the rather sophisticated structure of the weakly closed, complex
Jordan ∗-algebra (N , ·) is entirely encoded by the context category V(N ), that is, by the order
structure on the Abelian von Neumann subalgebras of N .
It turns out that there is also a one-to-one correspondence between isomorphisms of spectral
presheaves Σ N → Σ M and order-isomorphisms φ : V(M) → V(N ), and, hence, a one-to-one
correspondence between isomorphisms Σ N → Σ M and normal Jordan ∗-homomorphisms Φ :
M → N (see [35]). This means that the spectral presheaf Σ N determines a von Neumann algebra
N up to normal Jordan ∗-isomorphism.
As we saw, the Jordan structure on self-adjoint operators is all that is needed to formulate the
Born rule and the probabilistic part of quantum theory. Indeed, it was shown in [11] that there
is a one-to-one correspondence between states of the von Neumann algebra N (which also are
states of the real JBW-algebra (Nsa , ·)) and probability measures on Σ N . It was also shown in [11]
how to calculate the usual quantum-mechanical probabilities and expectation values using these
measures, so the Born rule is captured by the topos formalism.
.........................................................
∀a, b ∈ N : ab = 12 (ab + ba) + 12 (ab − ba)
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In this way, time evolution relates to the Lie algebra structure on self-adjoint operators. This is
in analogy with classical physics, where the physical quantities also form a Lie algebra under the
Poisson bracket.
The non-commutative product in the von Neumann algebra N can be decomposed into its
symmetrized and antisymmetrized parts,
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We now proceed to describe the Lie algebra structure on Nsa , the set of self-adjoint operators in
the algebra of physical quantities N , in geometrical terms relating to the spectral presheaf Σ N .
We draw on [45], which in turn uses the theory of orientations on Jordan algebras by Connes [46]
and Alfsen & Shultz [42,47,48]. This is the most technical part of the paper, but we hope to convey
the geometric intuition behind the constructions.
First, we observe that a normal unital Jordan ∗-homomorphism f : M → N between von
Neumann algebras is a homomorphism of von Neumann algebras if and only if f preserves
commutators, i.e. if and only if
∀a, b ∈ M : f ([a, b]) = [f (a), f (b)].
(4.7)
If this holds, then
∀a, b ∈ M : f (ab) = f (a · b + 12 [a, b])
= f (a) · f (b) + 12 [f (a), f (b)]
= f (a)f (b).
(4.8)
(4.9)
(4.10)
Note that for this argument we make use of the non-commutative products in M and N ,
respectively. At the level of Jordan algebras, these products are not available, of course,
and commutators with respect to the Jordan product are trivial (because a Jordan algebra is
commutative). We need to encode commutators in some different way. It turns out that skew
order derivations are the appropriate tool; see [47,48]. An order derivation on a JBW-algebra A
is a bounded linear operator δ : A → A such that etδ (A+ ) ⊆ A+ for all t ∈ R, that is, t → etδ is a
one-parameter group of order-automorphisms. An order derivation δ is skew if δ(1) = 0. We write
ODs (A) for the set of skew order derivations on A.
Alfsen and Shultz showed that if A = Nsa , the self-adjoint part of a von Neumann algebra, then
the skew order derivations on Nsa correspond bijectively to the elements of Nsa by
⎫
⎪
ψ : Nsa −→ ODs (Nsa )
⎬
(4.11)
i
⎭
a −→ δia := [a, −].⎪
2
That is, skew order derivations exactly encode commutators. We can reformulate condition
(4.7) as
∀a ∈ Msa : f ◦ δia = δif (a) ◦ f .
(4.12)
Note that we can restrict to self-adjoint operators and that universal quantification over b is
implicit, because δia : Msa → Msa is a function. Similarly, δif (a) : Nsa → Nsa is a function. Using
the series expansion of the exponential series, it is easy to check that condition (4.12) holds if and
only if
∀a ∈ Msa ∀t ∈ R : f ◦ etδia = etδif (a) ◦ f .
(4.13)
This holds due to the theorem by Stone and the von Neumann theorem already cited above,
because there is a bijection between (bounded) self-adjoint operators a and (norm-continuous)
one-parameter groups of unitaries t → eita , hence we can pass from the Lie algebra Msa to (oneparameter subgroups of) the Lie group U(M) uniquely. In future generalizations of our formalism
beyond the case of von Neumann algebras, where the Stone–von Neumann theorem may not
apply anymore, it will be interesting to distinguish two Lie algebras which have the same Lie
group.
.........................................................
(ii) Flows and Lie algebra structure
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Hence, our quantum state space, the spectral presheaf Σ N , can be seen as the measurable
space underlying the quantum theory of the system described by N , and quantum states are
represented by probability measures on this space.
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b −→ e(i/2)ta b e−(i/2)ta .⎭
(4.14)
By assumption, f : Msa → Nsa is a normal Jordan homomorphism, so both f ◦ etδia : Msa → Nsa
and etδif (a) ◦ f : Msa → Nsa are one-parameter groups of normal Jordan homomorphisms. They can
be extended to one-parameter groups of Jordan ∗-homomorphisms from M to N canonically. f is
a morphism of von Neumann algebras if and only if these two one-parameter groups are equal.
As we discussed at the beginning of this subsection, there is a one-to-one correspondence
between normal Jordan ∗-isomorphisms from M to N and isomorphisms of spectral presheaves
from Σ N to Σ M . (At this point, we have to specialize to isomorphisms. Both etδia : M → M and
etδif (a) : N → N are normal Jordan ∗-automorphisms, so it is enough to assume that f : M → N is a
Jordan ∗-isomorphism in order to guarantee that f ◦ etδia and etδif (a) ◦ f are Jordan ∗-isomorphisms.)
Concretely, to the one-parameter group of Jordan ∗-isomorphisms t → f ◦ etδia : M → N there
corresponds the one-parameter group of isomorphisms
tδia , G tδ ◦ f˜, G t → f
◦ etδia , Gf ◦ etδia = e
f
e ia
(4.15)
of spectral presheaves, and to the one-parameter group of Jordan ∗-isomorphisms t → etδif (a) ◦ f
there corresponds the one-parameter group of isomorphisms
tδif (a) , G tδ
if (a) ◦ f , G tδ
t → etδ
= f˜, Gf ◦ e
.
e if (a) ◦f
e if (a)
(4.16)
These one-parameter groups are constructed from t → f ◦ etδia and from t → etδif (a) ◦ f , respectively,
in the same way as the Hamiltonian flow Ft = ũt , Gt was constructed from the one-parameter
group of automorphisms t → ut = eith of N in §3a. For details, see [45].
If and only if the two one-parameter groups are equal, the map f : M → N is an isomorphism
of von Neumann algebras. In other words, if and only if for all a ∈ Msa and all t ∈ R the diagram
ΣN
f̃ ,Gf tδ
e if (a) ,G
/ ΣM
(4.17)
tδia
,Getδia e
tδ
e if (a)
ΣN
f̃ ,Gf / ΣM
commutes, f : M → N is an isomorphism of von Neumann algebras.
We call a representation
F : R −→ Aut(Σ M )
t −→ φ̃t , ηt (4.18)
of the additive group of real numbers by automorphisms of the spectral presheaf a flow on
the spectral presheaf Σ. Let F̃ : R → AutJBW (Msa ) be the one-parameter group of normal Jordan
automorphisms of (Msa , ·) corresponding to F. We call a flow F inner if F̃(t) is an inner
automorphism for every t ∈ R.
.........................................................
⎫
⎬
etδia : Msa −→ Msa
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For every t, the exponentiated map etδia is a linear map from Msa to itself. In fact, Alfsen and
Shultz showed that etδia : Msa → Msa is a one-parameter group of Jordan automorphisms, and—as
one might expect—it acts by unitaries: for all t ∈ R,
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on Σ N , the spectral presheaf of N . Thus, diagram (4.17) shows that f : M → N is an isomorphism
of von Neumann algebras if and only if, for the induced isomorphism f˜, Gf : Σ N → Σ M
between the spectral presheaves, it holds that
∀a ∈ Msa : Fa ◦ f˜, Gf = f˜, Gf ◦ Ff (a) ,
(4.21)
that is, f˜, Gf preserves flows. We recall that this is equivalent to f preserving commutators,
which was our starting point. Hence, the Lie algebra structure on the self-adjoint operators (or
rather, its exponentiated version) is faithfully represented geometrically by inner flows on the
spectral presheaf.
This ties in neatly with the description of time evolution in the topos approach, given in terms
of Hamiltonian flows, which are inner flows on the spectral presheaf as discussed in §3a.
5. Conclusion
Standard quantum theory lacks a geometric underpinning in the form of a state space picture
in analogy to classical (Hamiltonian) mechanics. The Kochen–Specker theorem seems to pose
an insurmountable obstacle to the construction of any such quantum state space. Yet, as we
showed here, by relaxing the assumptions needed to prove the Kochen–Specker theorem, we can
construct a quantum state space after all and find geometric counterparts to many key aspects of
quantum theory.
Concretely, we did not demand that our quantum state space must be a set but considered
a more general varying set or presheaf. The spectral presheaf Σ is constructed in the simplest
possible manner by gluing together the Gelfand spectra of all the Abelian C∗ - or von Neumann
subalgebras of the non-Abelian algebra of physical quantities. Each Gelfand spectrum Σ V can be
seen as a ‘local’ state space for the physical quantities in the context V. The seemingly simpleminded construction of the spectral presheaf Σ proves to be quite powerful: in previous work,
we had already shown that the kinematic and probabilistic aspects of (non-relativistic) quantum
theory can be reformulated in a more geometric way with respect to the spectral presheaf. For
example, quantum states correspond to probability measures on Σ and vice versa.
In this paper, we mostly considered time evolution. We introduced Hamiltonian flows, which
are one-parameter groups of automorphisms of the quantum state space, and showed that the
Schrödinger picture of time evolution of vector states can naturally be reformulated using these
flows. This provides a geometrical picture of time evolution of quantum systems in striking
analogy to classical, Hamiltonian mechanics. We then showed that the structure of the spectral
presheaf Σ geometrically mirrors the double role that self-adjoint operators play in quantum
theory: as quantum random variables, for which the operators can be organized into a Jordan
algebra, and as generators of time evolution, for which they can be organized into a Lie algebra.
If we picture the poset of contexts in a vertical fashion, with the maximal Abelian subalgebras
on top and smaller ones below, we see that the Jordan algebra structure is determined by this
vertical (order) structure alone. Unitary operators act horizontally in this picture, moving contexts
around, but preserving the order while doing so. One-parameter groups of unitaries lift to oneparameter groups of automorphisms, called inner flows, of the quantum state space Σ. We saw
.........................................................
on Σ M , the spectral presheaf of M. Given a Jordan ∗-isomorphism f : M → N , a also induces a
flow
⎫
Ff (a) : R −→ Aut(Σ N )
⎬
(4.20)
⎭
tδif (a) , G tδ
t −→ e
if
(a)
e
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The Hamiltonian flows of §3a are examples of inner flows on the spectral presheaf. More
generally, we saw above that every self-adjoint operator a ∈ Msa induces a flow
⎫
Fa : R −→ Aut(Σ M ) ⎬
(4.19)
tδia , G tδ ⎭
t −→ e
e ia
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introducing the spectral presheaf in the first place. I thank Carmen Constantin, Daniel Marsden, Rui Soares
Barbosa, Pedro Resende and Jonathon Funk for valuable discussions. Moreover, I am grateful to the Royal
Society for providing me with the opportunity to be a guest editor for the theme issue of Philosophical
Transactions of the Royal Society A with the title ‘New geometric concepts in the foundations of physics’.
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.........................................................
Acknowledgements. I thank Chris Isham for many discussions, for his friendship and support, and for
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that a Jordan isomorphism f : M → N between von Neumann algebras is an isomorphism of von
Neumann algebras, i.e. it preserves the Lie algebra structure and hence the non-commutative
product, if and only if the isomorphism f˜, Gf between the spectral presheaves induced by f
preserves the inner flows induced by the self-adjoint elements of M resp. N .
Summing up, the spectral presheaf is a new kind of quantum state space that allows us to
formulate the kinematics and dynamics of standard algebraic quantum theory in a geometric
manner closely analogous to classical mechanics. For the future, several extensions of the
formalism will be of interest: for example, to composite systems, to relativistic systems and to
field theories.
In future work, we will also consider the possibility of a ‘multi-fingered’ or contextual time
variable. In the current formulation, as in standard non-relativistic quantum theory, time is an
external parameter. The topos formalism, which is based on the poset of contexts, suggests a
potential dependence of time on the context, at least from an internal perspective (which is always
possible in a topos, but which we did not develop here). Mathematically, this could be realized
as a global section of a more general presheaf than the real number object in the topos. Such
generalized real values were already considered in the topos approach as generalized values of
physical quantities [9]. The physical interpretation of contextual time(s) has yet to be explored.
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