## Abstract

For each quantum system described by an operator algebra of physical quantities, we provide a (generalized) state space, notwithstanding the Kochen–Specker theorem. This quantum state space is the spectral presheaf . 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 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

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 , formed by the physical quantities (or observables) of a quantum system. This is an algebra over the complex numbers . We assume throughout that our algebra is unital, i.e. it contains an element 1 that serves as a multiplicative unit. In order to define self-adjoint and unitary operators, is equipped with an involution . The subset of self-adjoint operators is denoted as . These operators represent physical quantities. is closed under taking real linear combinations, so it is a real vector space. The unitary operators form a group with 1 as its neutral element.

Usually, completeness of the algebra with respect to the topology induced by the operator norm is assumed, that is, is a Banach algebra. The interplay between the norm and the involution is given by the *C**-rule
1.1for all , which makes 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 (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 of all bounded operators on a given Hilbert space . Every represented *C**-algebra (respectively, *W**-algebra) is a norm-closed (respectively, weakly closed) subalgebra of some .

In the algebraic picture, physical states are described by states of the algebra, that is, linear functionals
1.2that 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 in the state *ρ*.

Two more remarks:

(a) For a finite-dimensional Hilbert space , the algebra is simply the algebra of complex

*n*×*n*-matrices.(b) The canonical observables position and momentum generate the Heisenberg algebra. If one considers its weak closure , then one obtains the von Neumann algebra , where is infinite-dimensional.

### (b) Gelfand spectrum and Gelfand representation

Every *Abelian* *C**-algebra or von Neumann algebra has a Gelfand spectrum , which is a compact Hausdorff space.^{1} The elements of are the algebra homomorphisms from to , also called characters,
1.3Equivalently, the elements of the Gelfand spectrum are the multiplicative states of . The set , being a subset of the state space of , 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, complex-valued functions on a compact Hausdorff space *X*. The norm on *C*(*X*) is the supremum norm,
1.4and the involution is given by taking the complex conjugate in the image,
1.5where *f**(*x*):=*f*(*x*)* for all *x*∈*X*. The Gelfand representation theorem shows that, for every Abelian *C**-algebra , there is an isomorphism
1.6of *C**-algebras, that is, *G* is an algebra isomorphism that preserves the norm and the involution. Concretely, the Gelfand transform of is given by
1.7Note that self-adjoint operators are mapped to real-valued functions.

Physically, each is a pure state of , and λ(*a*) is the value that the physical quantity 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 have joint eigenstates, because is Abelian.

Hence, physically, the Gelfand spectrum consists of the pure states of the system described by the Abelian algebra , and by the Gelfand representation theorem, we can think of concretely as the algebra of continuous functions on . The Gelfand spectrum plays the role of a state space of the system, and the physical quantities, given by the self-adjoint operators in , are represented as real-valued functions on the state space .

### (c) The main question

The situation for systems described by Abelian *C**- or *W**-algebras is quite straightforward and appealing: to each algebra of physical quantities, there corresponds a compact Hausdorff space such that is isomorphic to the concrete algebra of functions , and, physically, plays the role of the state space for the system. All this hinges on being Abelian; for non-Abelian algebras, there is no direct generalization of the Gelfand spectrum 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 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 plays the role of a quantum state space and hence gives a geometric counterpart to the non-Abelian algebra 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.

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 non-commutative 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.

## 2. The spectral presheaf of a quantum system

We introduce the spectral presheaf 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 . Because is non-Abelian, it has no Gelfand spectrum, but each Abelian *C**- or *W**-subalgebra has a Gelfand spectrum *Σ*(*C*). Moreover, if *C*_{1},*C*_{2} are two Abelian subalgebras of such that *C*_{2}⊂*C*_{1}, then there is a canonical map
2.1between their Gelfand spectra, where simply denotes the restriction of to the smaller subalgebra *C*_{2}. It is well known that the restriction map *r*_{C1C2} is continuous, closed and surjective.

The basic idea in the construction of the spectral presheaf of is very simple: we collect the Gelfand spectra *Σ*(*C*) of all the Abelian subalgebras of and all the functions *r*_{C1C2} between their spectra (where *C*_{2}⊂*C*_{1}) into one object. In order to do this systematically, we first define the *context category* : its objects are the unital Abelian *C**-subalgebras *C* of the non-Abelian algebra . Following tradition in physics, these Abelian *C**-subalgebras are also called *contexts*. We consider only contexts such that holds. The arrows in the category are the inclusions of smaller contexts into larger ones, *i*_{C2C1}:*C*_{2}↪*C*_{1}. Hence, the context category is simply the partially ordered set of contexts.

The *spectral presheaf * then arises as a contravariant functor from the context category to **Set**, the category of sets and functions:

(a) On objects , we define , the Gelfand spectrum of

*C*.(b) On arrows

*i*_{C2C1}:*C*_{2}↪*C*_{1}, we define , λ↦λ|_{C2}.

Contravariant, **Set**-valued functors are traditionally called *presheaves*, hence the name. As we saw above, each Gelfand spectrum is, in fact, a compact Hausdorff space, and each restriction map 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 for a *W**- or von Neumann algebra from now on, and for the context category, i.e. the poset of Abelian *W**- or von Neumann subalgebras of . A generic element of will be denoted as *V* .

The spectral presheaf was first defined in similar form by Isham & Butterfield [5] and for the algebra by Isham *et al.* [6]. Physically, the spectral presheaf 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 , the contexts of , 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 be a presheaf over the context category , i.e. a functor . The analogue of an element *x* of a set *X* is a global element of a presheaf , that is, a natural transformation (in the sense of category theory [32])
2.2from the constant presheaf to . Such a natural transformation picks one element at each context such that, whenever *C*_{1}⊂*C*_{2}, we have *p*_{C1}|_{C2}=*p*_{C2}: the elements picked out by *p* must fit together under restriction.

If we consider the spectral presheaf , a global element would pick one for each context . As we saw above, λ_{C} is a pure state of *C*, and, for each physical quantity *a*∈*C*_{sa}, λ(*a*) is the value that *a* has in the state λ. Moreover, if *C*_{2}⊂*C*, then λ_{C}|_{C2}=λ_{C2}=*p*_{C2}, because we assume that *p* is a global element of . This implies that the value λ_{C}(*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 *a*^{2} is the square of the value assigned to *a*.

But, at least for the case that is a *W**-algebra with no summand of type *I*_{2}, 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 , by Isham *et al.* [6].

Mathematically, the fact that has no points is an expression of its character as a non-commutative 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 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 , 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 , called*clopen subobjects*.^{2}These subobjects form a complete Heyting algebra , 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 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 or, equivalently, a Dirac measure concentrated at*s*. As mentioned, the quantum state space 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 logic of the topos of presheaves over the context category , 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 , the energy has a certain value (in a suitable, fixed system of units), so energy is described by a function . 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 to a space 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*μ*_{ρ}on . Conversely, every probability measure*μ*on determines a unique quantum state*ρ*_{μ}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].

These aspects of the topos formalism for quantum theory show that the spectral presheaf indeed, plays the role of a quantum state space, in strong analogy to the state space 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 , 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 of physical quantities. But, if is a von Neumann algebra, we have that *h* is affiliated with , that is, all spectral projections of *h* lie in . We consider the one-parameter group of unitaries generated by *h*:
3.1By 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*):=*u*_{t}*ψ*_{0} for all , where *ψ*_{0} is the initial state at *t*=0. Time evolution in the Heisenberg picture is given by *a*(*t*):=*u*_{−t}*a*_{0}*u*_{t} for all , where *a*_{0} 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 *ϕ*_{H}. The latter is a one-parameter group 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 . A number of mathematical aspects have been developed in [35,36], but we will take a more down-to-earth physics approach here.

Let , and let be a context. We first observe that the unitary *u*_{t} acts on *V* by conjugation, sending it to . In general, *u*_{t}*V* *u*_{−t} will not be the same context as *V* , so *u*_{t} moves the elements of the context category around. Of course, *u*_{−t} induces the opposite transformation. Importantly, if *V* _{2}⊂*V* _{1}, then *u*_{t}*V* _{2}*u*_{−t}⊂*u*_{t}*V* _{1}*u*_{−t}, so *u*_{t} preserves the order on the context category , and we obtain an order-isomorphism
3.2for each . In the quantum state space , the Gelfand spectrum 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 in our case, to the topos of presheaves over , where the spectral presheaf lives, is by using a so-called geometric morphism [37,38]. We do not go into the details here, but just state that each acts on the spectral presheaf by the inverse image part of the geometrical morphism to give
3.3
3.4
That is, the action of on results in a ‘twisted’ version of the spectral presheaf, denoted . The component of at the context *V* is the Gelfand spectrum of *u*_{t}*V* *u*_{−t}.

Hence, if a context *V* is moved by *u*_{t}, then the corresponding Gelfand spectrum will also be moved, which means that we do not map the quantum state space into itself. We need a second step mapping the spectrum back into , and, globally, mapping back into . Acting by on would achieve this, but only in a trivial manner, because the composite is simply the identity on , independent of *t*. Clearly, we need a different construction.

We make use of the fact that the context *V* is isomorphic to the context *u*_{t}*V* *u*_{−t} as a commutative von Neumann algebra. Gelfand duality [3] implies that is isomorphic to . Concretely, let us write
3.5for the isomorphism between the unital, Abelian von Neumann algebras *V* and *u*_{t}*V* *u*_{−t}. Then
3.6is the isomorphim between the Gelfand spectra that we are looking for. It is easy to see that the maps *g*_{ut;V}, for *V* varying over , form a natural isomorphism (in the sense of category theory [32]),
3.7Then by composing
3.8we indeed map the quantum state space into itself in a non-trivial and invertible way. We denote the composite map by
3.9the inverse being *F*_{−t}.^{4} We interpret *F*_{t} as the analogue of a diffeomorphism from to itself. Because we do not have the analogue of a differential structure on yet, we more modestly (and correctly) call *F*_{t} an automorphism of . The automorphisms of the spectral presheaf form a group , and, by letting *t* vary, we obtain a one-parameter group of automorphisms,
3.10which we call a *Hamiltonian flow on the quantum state space *. By construction, *F*_{0}=*F*(0) is the identity on . In §4c, we consider flows on the spectral presheaf in some more detail. For more details on the automorphism group , 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 has no points, as discussed in §2. Instead, pure quantum states are described by subobjects of that are ‘as close to being a point’ as possible. We describe this representation now. We assume that the algebra of observables is a von Neumann algebra, because we need sufficiently many projections for some constructions to work. Contexts are Abelian von Neumann subalgebras of .

#### (i) Definition of pseudo-states

Let be a pure state, and let *p*_{ψ} be the corresponding projection onto the ray . For every context , let denote the lattice of projections in *V* and define
3.11Geometrically, this means that we pick the smallest projection *in V* that projects onto a closed subspace of that contains the ray . The projection clearly depends on the context

*V*, because

*V*may contain many or few projections. If , then ; otherwise, .

For each context , consider
3.12If *V* ′,*V* are two contexts such that *V* ′⊂*V* , then it holds for all that
3.13so . This means that the sets form a *subobject* of the spectral presheaf . This subobject is called a *pseudo-state* and represents the pure state *ψ*.

The pseudo-state is, intuitively speaking, as close to being a point of as possible: by construction, it is the smallest subobject that ‘covers’ the ray .

#### (ii) Time evolution of pseudo-states

In order to describe the time evolution of , we use the Hamiltonian flow on the quantum state space as defined in §3a. Let be the strongly continuous one-parameter group of unitaries generated by the Hamiltonian operator *h*, and let be the corresponding flow on (see (3.10)).

For each time , is an automorphism of the quantum state space . In order to define a map between subobjects of , it is natural to mimic the inverse image of a function, so we use the inverse image of the map *F*_{t},
3.14We now have to describe how this acts on subobjects of , and in particular on a pseudo-state .

Let be a clopen subobject. Then, *F*_{−t} acts on by
3.15Here, *G*_{−t} is the natural transformation from to with components
compare equation (3.6). Finally, denotes the image of under *G*_{−t}. It is straightforward to check that is a clopen subobject of .

If is a pseudo-state, which is a small subobject of representing a vector state, and *t*↦ e^{ith} is a strongly continuous one-parameter group of unitaries, we define the time evolution of in the Schrödinger picture by
3.16Alternatively, we could simply have defined
3.17but this makes use of the Hilbert space structure directly, because *u*_{t} acts on *ψ*_{0}. 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.

It is still sensible to check whether both definitions of time evolution of pseudo-states coincide. First note that, for all and all ,
3.18
3.19
3.20
3.21It is easy to show that . Define a map *α*_{V} on projections in *V* by for all . Then, we obtain
3.22
3.23
3.24
3.25
3.26
3.27
3.28
3.29
This shows that time evolution of quantum states, when described by flows on the quantum state space (acting on subobjects such as ), 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 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 . 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) physical quantities. The prime example are the self-adjoint operators on a Hilbert space, equipped with the Jordan product
4.1The 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 of projections such that

(i) for all , the operator commutes with

*a*,(ii) for all

*r*<−|*a*| and for all*r*>|*a*|,(iii) for

*r*<*s*,(iv) , and

(v) 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 non-commutative 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 has an associated *JBW*-algebra with the symmetrized product given by (4.1). Often, it is enough to consider the real *JBW*-algebra 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 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*,
4.2Usually, the antiunitary option is disregarded. Yet, it is known that every Jordan *-automorphism is implemented by either a unitary or an antiunitary operator; see [42]. Hence, Wigner's theorem actually says that every automorphism of the projection lattice lifts to a Jordan *-automorphism . 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 in a von Neumann algebra generates a one-parameter group e^{ita} of unitaries in . Moreover, if is some other self-adjoint operator, then
4.3so commutators are the infinitesimal aspect of such one-parameter groups acting on other self-adjoint operators. The self-adjoint operators in a von Neumann algebra 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*↦e^{ith}, where *h* is the Hamiltonian of the system^{5} and the quantum state changes as *ψ*_{0}↦*ψ*_{t}=e^{ith}*ψ*_{0} (Schrödinger picture) or the physical quantities change as *a*↦e^{−ith}*a* *e*^{ith} (Heisenberg picture). The infinitesimal change of *a* under time evolution is given by (d/d*t*)(e^{−ith}*a* e^{ith})|_{t=0}=i[*h*,*a*].

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 can be decomposed into its symmetrized and antisymmetrized parts,
4.4
4.5
which gives the Jordan product and the commutator. (The factor is absorbed into the Jordan product by convention; otherwise, the formula would look entirely symmetrical.) The non-commutative, 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 be von Neumann algebras not isomorphic to and with no type *I*_{2} summands. and are isomorphic as complex Jordan algebras if and only if and are isomorphic as posets. Concretely, for every order-isomorphism , there exists a unique Jordan *-isomorphism such that
4.6Conversely, every Jordan *-isomorphism induces an order-isomorphism .

It is easy to see that this can be strengthened to include topology: for every order-isomorphism , there exists a unique *normal* Jordan *-isomorphism and vice versa [45]. Restricting to self-adjoint operators, one obtains a one-to-one correspondence between isomorphisms of real *JBW*-algebras and order-isomorphisms of context categories.

Hence, the context category determines a von Neumann algebra 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 is entirely encoded by the context category , that is, by the order structure on the Abelian von Neumann subalgebras of .

It turns out that there is also a one-to-one correspondence between isomorphisms of spectral presheaves and order-isomorphisms , and, hence, a one-to-one correspondence between isomorphisms and normal Jordan *-homomorphisms (see [35]). This means that the spectral presheaf determines a von Neumann algebra 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 (which also are states of the real *JBW*-algebra ) and probability measures on . 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.

Hence, our quantum state space, the spectral presheaf , can be seen as the measurable space underlying the quantum theory of the system described by , and quantum states are represented by probability measures on this space.

#### (ii) Flows and Lie algebra structure

We now proceed to describe the Lie algebra structure on , the set of self-adjoint operators in the algebra of physical quantities , in geometrical terms relating to the spectral presheaf . 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 between von Neumann algebras is a homomorphism of von Neumann algebras if and only if *f* preserves commutators, i.e. if and only if
4.7If this holds, then
4.8
4.9
4.10
Note that for this argument we make use of the non-commutative products in and , 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 such that e^{tδ}(*A*^{+})⊆*A*^{+} for all , that is, *t*↦e^{tδ} is a one-parameter group of order-automorphisms. An order derivation *δ* is *skew* if *δ*(1)=0. We write *OD*_{s}(*A*) for the set of skew order derivations on *A*.

Alfsen and Shultz showed that if , the self-adjoint part of a von Neumann algebra, then the skew order derivations on correspond bijectively to the elements of by
4.11That is, skew order derivations exactly encode commutators. We can reformulate condition (4.7) as
4.12Note that we can restrict to self-adjoint operators and that universal quantification over *b* is implicit, because is a function. Similarly, 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
4.13This 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*↦e^{ita}, hence we can pass from the Lie algebra to (one-parameter subgroups of) the Lie group 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.

For every *t*, the exponentiated map e^{tδia} is a linear map from to itself. In fact, Alfsen and Shultz showed that is a one-parameter group of *Jordan* automorphisms, and—as one might expect—it acts by unitaries: for all ,
4.14

By assumption, is a normal Jordan homomorphism, so both and are one-parameter groups of normal Jordan homomorphisms. They can be extended to one-parameter groups of Jordan *-homomorphisms from to 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 to and isomorphisms of spectral presheaves from to . (At this point, we have to specialize to isomorphisms. Both and are normal Jordan *-automorphisms, so it is enough to assume that is a Jordan *-isomorphism in order to guarantee that *f*°e^{tδia} and e^{tδif(a)}°*f* are Jordan *-isomorphisms.)

Concretely, to the one-parameter group of Jordan *-isomorphisms there corresponds the one-parameter group of isomorphisms
4.15of spectral presheaves, and to the one-parameter group of Jordan *-isomorphisms *t*↦e^{tδif(a)}°*f* there corresponds the one-parameter group of isomorphisms
4.16These one-parameter groups are constructed from *t*↦*f*°e^{tδia} and from *t*↦e^{tδif(a)}°*f*, respectively, in the same way as the Hamiltonian flow was constructed from the one-parameter group of automorphisms *t*↦*u*_{t}=e^{ith} of in §3a. For details, see [45].

If and only if the two one-parameter groups are equal, the map is an isomorphism of von Neumann algebras. In other words, if and only if for all and all the diagram 4.17commutes, is an isomorphism of von Neumann algebras.

We call a representation
4.18of the additive group of real numbers by automorphisms of the spectral presheaf a *flow on the spectral presheaf* . Let be the one-parameter group of normal Jordan automorphisms of corresponding to *F*. We call a flow *F* *inner* if is an inner automorphism for every .

The Hamiltonian flows of §3a are examples of inner flows on the spectral presheaf. More generally, we saw above that every self-adjoint operator induces a flow
4.19on , the spectral presheaf of . Given a Jordan *-isomorphism , *a* also induces a flow
4.20on , the spectral presheaf of . Thus, diagram (4.17) shows that is an isomorphism of von Neumann algebras if and only if, for the induced isomorphism between the spectral presheaves, it holds that
4.21that is, *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 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 can be seen as a ‘local’ state space for the physical quantities in the context *V* . The seemingly simple-minded construction of the spectral presheaf 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 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 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 one-parameter groups of automorphisms, called inner flows, of the quantum state space . We saw that a Jordan isomorphism 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 between the spectral presheaves induced by *f* preserves the inner flows induced by the self-adjoint elements of resp. .

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.

## Acknowledgements

I thank Chris Isham for many discussions, for his friendship and support, and for 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’.

## Footnotes

One contribution of 13 to a theme issue ‘New geometric concepts in the foundations of physics’.

↵1 If is a non-unital

*C**-algebra, is only locally compact. We assume throughout that is unital.↵2 A subobject of is called clopen if every component , , is a clopen subset of .

↵3 Whether vector states are pure states depends on the algebra at hand. For , vector states are pure.

↵4 Technically, 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 .

↵5

*h*need not be an element of the algebra of physical quantities, but it is affiliated with it, i.e. all spectral projections of*h*are in .

- Accepted March 9, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.