nLab
Bohr topos

Idea

Brief

To every quantum mechanical system is associated its Bohr topos : a ringed topos which plays the role of the quantum phase space. The idea of this construction – Bohrification – is that it naturally captures the geometric and logical aspects of quantum physics in terms of higher geometry/topos theory.

One way to understand Bohrification is as a generalization of the construction of the Gelfand spectrum of a commutative C-star algebras to a context of noncommutative $C^{*}$-algebras. It assigns to a noncommutative $C^{*}$-algebra $A$ essentially a system of Gelfand dual spaces to each of its commutative subalgebras. Together, this system yields a generalized Gelfand spectrum in the form of a locale ${\underline{\Sigma }}_A$ internal to the sheaf topos ${𝒯}_A$ over the poset of commutative subalgebras of $A$, or equivalently its externalization ${\Sigma }_A\to 𝒞(A)$ regarded as a locale over the locale of open subalgebras.

As a topos, the Bohr topos is just a presheaf topos, the topos of presheaves on this poset of commutative subalgebras of $A$, but the point is that it is naturally a ringed topos with the original non-commutative algebra $A$ appearing as a commutative $C^{*}$-algebra internal to the Bohr topos. This allows to talk about quantum states on $A$ much like classical states, but internal to the Bohr topos.

In fact, under mild assumptions on $A$, its poset of commutative subalgebras, and hence the Bohr topos over it, encodes precisely the Jordan algebra underlying $A$. As discussed at Jordan algebra, this is precisely that part of $A$ which knows about the quantum observables themselves. In order to have the Bohr topos remember the full non-commutative algebra structure of $A$, it needs to be equipped with the information about Hamiltonian flows induced on $A$ by automorphisms of the form $\exp (i[H,-])$, where $[-,-]$ is the commutator (that gets discarded as one passes to the Jordan algebra). According to Andreas Döring (private communication at MPI Bonn, April 2013), this can be formulated nicely in topos theory.

More detailed

The formalization of the notion quantum mechanical system (see there for details) with its states and observables is the following.

• The system as such is encoded by a C-star algebra $A$;

• a self-adjoint operator $a\in A$ is thought of as being an observable of the system: a kind of observation that one can make about it;

• a $ℂ$-linear map $\rho :A\to ℂ$ (which is “positive” and “normalized”) is thought of as being a state that the system can be in – a physical configuration (or rather: a probability distribution of such);

• the pairing $(a,\rho )\mapsto \rho (a)\in ℂ$ is thought of as being the value of the observable $a$ made on the system in state $\rho$: for instance the total energy of the system as measured in some chosen unit;

• a one-parameter group of automorphisms $ℝ\to \mathrm{Aut}(A)$ (inner automorphisms for “localized” systems, see below) is thought of as being an evolution of the system, for instance in time? or more general in spacetime.

(Often in the literature quantum mechanical systems are instead dually conceived of in terms of Hilbert spaces of pure states. The relation between these two descriptions is established by the GNS-construction which allows to pass from one to the other.)

Notice that these axioms are naturally thought of as exhibiting the $C^{*}$-algebra $A$ as a formal dual of the would-be quantum phase space of the physical system – not quite an ordinary topological space (which by Gelfand duality it would be if $A$ were commutative) but still a kind of generalized space. Traditionally it is common to regard this as a space in noncommutative geometry. Notice that if we do so – hence if we regard the object $\mathrm{Spec}A\in C^{*}{\mathrm{Alg}}^{\mathrm{op}}$ that is the incarnation of $A\in C^{*}\mathrm{Alg}$ in the opposite category – then the definition of observable above reduces to the evident notion of real-valued functions on quantum phase space: such as function is a morphism $\mathrm{Spec}A\to ℝ$ in $C^{*}{\mathrm{Alg}}^{\mathrm{op}}$, which dually is a $C^{*}$-algebra homomorphism $C(ℝ{)}_0\to A$ (where on the left we have the unitization of functions with compact support). That such correspond to self-adjoint operators of $A$ is the statement of functional calculus for $C^{*}$-algebras.

More subtle is the interpretation of the axiom for states. Historically this had been been subject to some discussion: by the spectral theorem two different observables $a_1,a_2\in A$ have a compatible set of observable values if and only if these elements commute with each other in $A$. Generally, a set $\{a_i{\}}_i$ of observables has a jointly consistent set of observable values if and only if the sub-$C^{*}$-algebra $⟨\{a_i{\}}_i⟩\subset A$ generated by them is commutative. Therefore for the phenomenological interpretation of the axioms it seems to make no sense to demand that a state $\rho :A\to ℂ$ be linear on non-commuting observables: if $a_1$ and $a_2$ do not commute, it is not a-priori clear that it makes sense to require that $\rho (a_1+a_2)=\rho (a_1)+\rho (a_2)$. This might experimentally fail, and hold only for commuting $a_1,a_2$.

Therefore the notion of quasi-state was introduced: a quasi-state on $A$ is defined to be a (positive and normalized) function $\rho :A\to ℂ$ which is required to be $ℂ$-linear only on all commutative subalgebras of $A$. Operationally, quasi-states should be the genuine states!

One would therefore tend to think that the terminology has been chosen in an unfortunate way. While maybe true, it turns out – non-trivially – that in a major class of cases of interest the distinction does not matter: namely Gleason's theorem states that for $H$ a separable complex Hilbert space with $\dim H>2$ and $A=B(H)$ the $C^{*}$-algebra of bounded operators on $H$, all quasi-states on $A$ are automatically states: a function that is linear on all commutative subalgebras is automatically also linear on all of $A$.

While this means that the distinction between states and quasi-states disappears in a major case of interest, it does not disappear in all cases of interest. In particular, other foundational theorems about quantum mechanics concern the collection of commutative subalgebras, too.

Notably, one may wonder about the evident strengthening of the notion of quasi-states to that of a map $\rho :A\to ℂ$ which is not just linear but also an algebra homomorphism on each commuting subalgebra. Notice that by Gelfand duality every commutative $C^{*}$-algebra $C$ is the algebra of continuous functions on some topological space $\mathrm{sp}(C)$. Under this duality a state on $C$ is a probability distribution on $\mathrm{sp}(C)$, while an algebra homomorphism $C\to ℂ$ is a point of $X_C$. Therefore a quasi-state which is commutative-subalgebra-wise an algebra homomorphism may be thought of as encoding a collection of precise numerical values (as opposed to just expectation values) of all possible observables. Such a hypothetical quasi-state is sometimes called a collection of hidden variable?s of the quantum mechanical system: its existence would mean that despite the apparent probabilistic nature of quantum mechanics, there are “hidden” non-probabilistic states. But there are not. This is the statement of the Kochen-Specker theorem: under precisely the assumptions that make Gleason's theorem work, there is no quasi-state which is commutative-subalgebra-wise an algebra homomorphism.

In summary, this means:

The foundational characteristics of quantum physics are encoded in notions of functions on the algebra of observables $A$ which are homomorphisms only commutative-subalgebra-wise .

Since the notion of commutative-subalgebra-wise homomorphism is at the heart of quantum physics, it seems worthwhile to consider natural formalizations of this notion. There is indeed a very natural and general abstract one: whenever any notion of function is defined only locally it is natural to consider the sheaf of such functions over all possible local patches.

The historically motivating example, and possibly still the most widely familiar one, is that of holomorphic functions on a complex manifold: there are in general very few holomorphic functions defined over all of a complex manifold, but plenty of them defined over any small enough subset. And it is of fundamental interest to consider the collections of holomorphic functions over each such subset, and how these restrict to each other under restriction of subsets. This collection of local data is a sheaf of functions on the complex manifold.

There is an evident analog setup of this situation that applies in the present case of interest, that of functions defined on commutative subalgebras:

for $A$ any C-star algebra, write $𝒞(A)$ for the set of all its commutative $C^{*}$-subalgebras. This is naturally a poset under inclusion of subalgebras. A (co)presheaf of this set is a functor $𝒞(A)\to \mathrm{Set}$. Any such functor we may think of as a collection of commutative-subalgebra-wise data on $A$, consistent with restriction of subalgebras. The collection of all such functors – which we write $[𝒞(A),\mathrm{Set}]$ – is a category called a presheaf topos.

Inside this topos, all the above discussion of foundations of quantum mechanics finds a natural simple equivalent reformulation:

first of all, the non-commutative $C^{*}$-algebra $A$ naturally induces a commutative C-star algebra object $\underline{A}$ internal to $[𝒞(A),\mathrm{Set}]$: namely the copresheaf defined by the tautological assignment

$$\underline{A}:(C\in 𝒞(A))\mapsto C.$$\underline{A} : (C \in \mathcal{C}(A)) \mapsto C \,.

In words this means nothing but that the collection of all commutative subalgebras of $A$ may naturally be regarded as a single commutative $C^{*}$-algebra internal to the topos $[𝒞(A),\mathrm{Set}]$.

Below we shall discuss (here) that in a precise sense this commutative internal $\underline{A}$ captures precisely all the kinematical information encoded in the quantum mechanical system of $A$ – everything related to states and observables but not information about (time) evolution. So everything we have discussed so far.

The pair of these two ingredients

$$\mathrm{Bohr}(A):=([𝒞(A),\mathrm{Set}],\underline{A})$$Bohr(A) := ([\mathcal{C}(A), Set], \underline{A})

constitutes what is called a ringed topos – a special case of the notion of a locally ringed topos. This notion is a fundamental notion for generalized spaces in higher geometry. The most advanced general theory of higher geometry (Lurie09) is based on modelling spaces as ringed toposes.

We shall call this ringed topos the Bohr topos of $A$.

This terminology is meant to indicate that one may think of this construction as formalizing faithfully and usefully a heuristic that has been emphasized by Nils Bohr – one of the founding fathers of quantum mechanics – and is known as the doctrine of classical concepts (Scheibe) in quantum mechanics. This states that nonclassical/noncommutative as the logic/geometry of quantum mechanics may be, it is to be probed and detected by classical/commutative logic/geometry.

Namely in terms of the Bohr topos we have the following equivalent reformulations of the foundational facts about quantum physics discussed above, now internally in $\mathrm{Bohr}(A)$.

Consistent quantum mechanical states. A quasi-state on $A$ is precisely an ordinary classical state on $\underline{A}$, internal to $\mathrm{Bohr}(A)$.

In particular (Gleason's theorem): if $A=B(H)$ for $\dim H>2$ then a quantum state on the external $A$ is precisely a classical state on the internal $\underline{A}$.

and

Non-existence of hidden quantum variables. The Gelfand spectrum $\mathrm{sp}(\underline{A})$ of $\underline{A}$ internal to the Bohr topos has no global point. (Kochen-Specker).

These two statements might be taken as suggesting that a quantum mechanical system $A$ is naturally regarded in terms of its Bohr topos $\mathrm{Bohr}(A)$ – somewhat more naturally than as a $C^{*}$-algebra $A$. (The second, in a slightly different setup, was emphasized in IshamHamiltonButterfield, which inspired all of the following discussion, the first in Spitters). In fact, thinking of ringed toposes as generalized spaces in higher geometry, it suggests that the Bohr topos $\mathrm{Bohr}(A)$ itself is the quantum phase space of the quantum mechanical system in question.

To which extent this perspective is genuinely useful is maybe still to be established. For pointers to the literature see the references below. Discussion along the above lines may suggest that this perspective is indeed useful, but what is probably still missing is a statement about quantum physics that can be formulated and proven in terms of Bohr toposes, while being hardly conceivable or at least more unnatural without. It is probably currently not clear if such statements have been found.

One potential such statement has been suggested in (Nuiten) after discussion with Spitters:

in the formalization of quantum field theory by the Haag-Kastler axioms – called AQFT – every quantum field theory is entirely encoded in terms of its local net of observables over spacetime $X$. This is a copresheaf of C-star algebras

$$A:\mathrm{Op}(X)\to C^{*}\mathrm{Alg}$$A : Op(X) \to C^\ast Alg

which assigns to every open subset $U\subset X$ of spacetime the quantum subsystem $A_U$ of quantum fields supported in that region. By the above, we may consider for each of these quantum systems their associated quantum phase spaces given by the correspondong Bohr toposes $\mathrm{Bohr}(A_U)$. This yields a presheaf

$$\mathrm{Bohr}(A):\mathrm{Op}(X{)}^{\mathrm{op}}\to C_{\mathrm{com}}^{*}\mathrm{TopSpace}\hookrightarrow C_{\mathrm{com}}^{*}\mathrm{Topos}$$Bohr(A) : Op(X)^{op} \to C^\ast_{com} TopSpace \hookrightarrow C^\ast_{com}Topos

of ringed toposes whose internal ring object has the structure of a commutative $C^{*}$-algebra. With the copresheaf thus turned into a presheaf it is natural to ask under which conditions this is a sheaf: under which conditions this presheaf satisfies descent.

In (Nuiten) the following is observed: if $A$ satisfies what is called the split property (a strong form of the time slice axiom) then the Bohr-presheaf of quantum phase spaces satisfies spatial descent by local geometric morphisms precisely if the original copresheaf of observables $A:\mathrm{Op}(X)\to C^{*}\mathrm{Alg}$ is indeed local – spatially and causally. So this means that a natural property of quantum physics – spatial and causal locality – corresponds from the perspective of Bohr toposes to a natural property of presheaves of quantum phase spaces: descent.

One can probably view this as further suggestive evidence that indeed quantum physics is naturally regarded from the point of view of the Bohr topos. But for seeing where this perspective is headed, it seems that more insights along these lines would be useful.

Outline

The discussion below proceeds in the following steps (following (Nuiten11))

1. Bohr topos of a quantum mechanics system

   This discusses the Bohr topos incarnation of a quantum mechanical system – the topos-theoretic quantum phase space – and its functoriality.

2. Kinematics in a Bohr topos

   This discusses how the classical kinematics internal to a Bohr topos is the external quantum kinematics of the underlying quantum mechanical system.

3. (Pre-)Sheaf of Bohr toposes of a quantum field theory

   This discusses how the presheaves of Bohr toposes obtained by applying Bohrification to a local net of observables of a quantum field theory.

Bohr topos of a quantum mechanical system

We discuss the definitions and some basic properties of Bohr toposes: certain ringed toposes – in fact ringed spaces – associated with any (possibly non-commutative) algebra. We formulate the construction for C-star algebras, since this is the standard model for quantum mechanical systems, but actually much of it does not depend on either the topology or the star-algebra structure until we come to the discussion of the kinematics in a Bohr topos below.

$C^{*}$-algebras

Definition

A C-star algebra is (…)

Definition

For $A$ a star-algebra, an element $a\in A$ is called a normal element if $a^{*}a=a{a}^{*}$.

Remark

Every element of a $C^{*}$-algebra is the sum of two normal elements, because

$$a=\frac{1}{2}((a+a^{*})+(a-a^{*})).$$a = \frac{1}{2} \left( \left( a + a^* \right) + \left( a - a^* \right) \right) \,.

This means whenever a linear morphism between the vector spaces underlying two $C^{*}$-algebras is defined on normal elements, it is already defined on all elements. This will be used in several of the arguments below.

Definition

A partial $C^{*}$-algebra is a set $A$ equipped with

• a symmetric and reflexive binary relation $C\subset A\times A$;

• elements $0,1\in A$;

• an involution $*:A\to A$;

• a function $(-)\cdot (-):ℂ\times A\to A$;

• a function $\parallel -\parallel :A\to ℝ$;

• (partial) binary operations $+,\times :C\to A$

such that every set $S\subset A$ of elements that are pairwise in $C$ is contained in a set $T\subset A$ whose elements are also pairwise in $C$ and on which the above operations yield the structure of a commutative C-star algebra.

A homomorphism of partial $C^{*}$-algebra is a function preserving this structure. This defines a category $\mathrm{PCstar}$ of partial $C^{*}$ algebras.

This appears as (vdBergHeunen, def. 11,12).

Definition

For $A$ a C-star algebra, write

$$N(A):=\{a\in A\mid a{a}^{*}=a^{*}a\}$$N(A) := \{a \in A | a a^* = a^* a\}

for its set of normal operators. This is naturally a partial C-star algebra with $C\subset N(A)\times N(A)$ the set of pairs of elements that commute in $A$.

The Bohr site

Definition

For $A$ a partial C-star algebra write $𝒞(A)$ for the poset of total (not partial) commutative sub C-star algebras. We call this the poset of commutative subalgebras.

Proposition

This construction extends to a functor

$$𝒞:C^{*}\mathrm{Alg}\to \mathrm{Poset}.$$\mathcal{C} : C^* Alg \to Poset \,.

to the category Poset of posets: for $f:A\to B$ a homomorphism we let $𝒞(f)$ be over any $C\in 𝒞(A)$ be the image $\mathrm{im}(f{\mid }_C)\in C^{*}\mathrm{Alg}$ of the restriction of $f$ to $C$.

Remark

By standard properties of C-star algebras (see here), this image $\mathrm{im}(f{\mid }_C)$ is simply the set-theoretic image $f(C)$.

Notice the following fact about Alexandroff spaces:

Proposition

The functor

$$\mathrm{Alex}:\mathrm{Poset}\to \mathrm{Top}$$Alex : Poset \to Top

from posets to topological spaces that sends a poset $P$ to the topological space whose underlying set is the underlying set of $P$ and whose open subsets are the upward closed subsets $\mathrm{Up}(P)$ exhibits an equivalence of categories

$$\mathrm{Poset}\underset{\simeq }{\overset{\mathrm{Alex}}{\to }}\mathrm{AlexTop}\hookrightarrow \mathrm{Top}$$Poset \underoverset{\simeq}{Alex}{\to} AlexTop \hookrightarrow Top

of Poset with the full subcategory of Alexandroff spaces.

Definition

For $A\in C^{*}\mathrm{Alg}$ we call $\mathrm{Alex}𝒞(A)$ the Bohr site of $A$.

Definition

A morphism $f:A\to B$ in $C^{*}\mathrm{Alg}$ is called commutativity reflecting if for all $a_1,a_2\in A$ we have that if $f(a_1)$ commutes with $f(a_2)$ in $B$ then already $a_1$ commutes with $a_2$ in $A$.

Write

$$C^{*}{\mathrm{Alg}}_{\mathrm{cr}}\subset C^{*}\mathrm{Alg}$$C^* Alg_{cr} \subset C^* Alg

for the subcategory of C-star algebras on the commutativity-reflecting morphisms.

Remark

Every monomorphism $A\hookrightarrow B$ in $C^{*}\mathrm{Alg}$ is commutativity reflecting.

Proposition

A morphism $f:A\to B$ in $C^{*}\mathrm{Alg}$ is commutativity reflecting precisely if the morphism $𝒞(f)$ has a right adjoint

$$(𝒞(f)⊣R_f)𝒞(A)\stackrel{\stackrel{R_f}{\leftarrow }}{\underset{𝒞(f)}{\to }}𝒞(B).$$(\mathcal{C}(f) \dashv R_f) \mathcal{C}(A) \stackrel{\overset{R_f}{\leftarrow}}{\underset{\mathcal{C}(f)}{\to}} \mathcal{C}(B) \,.

This appears as (Nuiten, lemma 2.6).

Definition

Write

$$C^{*}{\mathrm{Alg}}_{\mathrm{inc}}\subset C^{*}{\mathrm{Alg}}_{\mathrm{cr}}\subset C^{*}\mathrm{Alg}$$C^* Alg_{inc} \subset C^* Alg_{cr} \subset C^* Alg

for the subcategories of $C^{*}\mathrm{Alg}$ on the monomorphisms and on the commutativity-reflecting morphisms, respectively.

The Bohr topos

Definition

Write

$$C_{\mathrm{com}}^{*}\mathrm{Top}\hookrightarrow C_{\mathrm{com}}^{*}\mathrm{Topos}$$C^*_{com} Top \hookrightarrow C^*_{com} Topos

for the categories of ringed spaces and ringed toposes, where the internal ring object is equipped with the structure of an internal commutative C-star algebra and the morphisms respect the $C^{*}$-algebra structure.

Hence a morphism $(ℰ,\underline{A})\to (ℱ,\underline{B})$ in $C_{\mathrm{com}}^{*}\mathrm{Topos}$ is

1. a geometric morphism $f:ℰ\to ℱ$ such that the inverse image $f^{*}\underline{B}$ is still an internal $C^{*}$-algebra in $ℰ$;

2. and a morphism of internal $C^{*}$-algebras

   $$f^{*}\underline{B}\to \underline{A}$$f^\ast \underline{B} \to \underline{A}

   in $ℰ$.

Definition

For $A\in C^{*}\mathrm{Alg}$ the Bohr topos of $A$ is the $C^{*}$-space/topos

$$\mathrm{Bohr}(A):=(\mathrm{Sh}(\mathrm{Alex}(𝒞(A))),\underline{A})\in C_{\mathrm{com}}^{*}\mathrm{Top}\hookrightarrow C_{\mathrm{com}}^{*}\mathrm{Topos}$$Bohr(A) := ( Sh(Alex(\mathcal{C}(A))), \underline{A}) \in C^\ast_{com} Top \hookrightarrow C^\ast_{com} Topos

whose underlying topological space is (that corresponding to) the Bohr site, and whose internal $C^{*}$-algebra is the tautological copresheaf

$$\underline{A}:(C\in 𝒞(A))\mapsto C$$\underline{A} : (C \in \mathcal{C}(A)) \mapsto C

in $[𝒞(A),\mathrm{Set}]\simeq \mathrm{Sh}(\mathrm{Alex}(𝒞(A)))$ equipped with the evident objectwise commutative $C^{*}$-algebra structure.

Moreover, write

$${\mathrm{Bohr}}_{\neg \neg }(A):=({\mathrm{Sh}}_{\neg \neg }(\mathrm{Alex}𝒞(A)),\underline{A})$$Bohr_{\not \not}(A) := (Sh_{\not \not}(Alex \mathcal{C}(A)), \underline{A})

for the $C^{*}$-topos whose underlying sheaf topos is that for the double negation topology on the plain Bohr topos.

The general notion of morphisms between toposes are geometric morphisms. But those that remember the morphisms of Bohr sites are essential geometric morphisms.

Remark

Every functor $f:𝒞(A)\to 𝒞(B)$ induces an essential geometric morphism

$$(f_{!}dasghv{f}^{*}⊣f_{*}):[𝒞(A),\mathrm{Set}]\stackrel{\stackrel{f_{!}:={\mathrm{Lan}}_f}{\to }}{\stackrel{\stackrel{f^{*}:=(-)\circ f}{\leftarrow }}{\underset{f_{*}:={\mathrm{Ran}}_f}{\to }}}[𝒞(B),\mathrm{Set}]$$(f_! \dasghv f^* \dashv f_*) : [\mathcal{C}(A), Set] \stackrel{\overset{f_! := Lan_f}{\to}}{\stackrel{\overset{f^* := (-) \circ f}{\leftarrow}}{\underset{f_* := Ran_f }{\to}}} [\mathcal{C}(B), Set]

where ${\mathrm{Lan}}_f$ and ${\mathrm{Ran}}_f$ are left and right Kan extension along $f$, respectively.

We also write $[f,\mathrm{Set}]:[𝒞(A),\mathrm{Set}]\to [𝒞(B),\mathrm{Set}]$ for this. Notice that by the equivalence of copresheaves on posets and sheaves on the corresponding Alexandroff locales (see there for details) this is equivalently

$$\mathrm{Sh}(\mathrm{Alex}(f))):\mathrm{Sh}(\mathrm{Alex}𝒞(A))\to \mathrm{Sh}(\mathrm{Alex}𝒞(B)).$$Sh (Alex(f))) : Sh(Alex \mathcal{C}(A)) \to Sh(Alex \mathcal{C}(B)) \,.

The next proposition asserts that all essential geometric morphisms between Bohr toposes arise this way:

Proposition

The 2-functor

$$[-,\mathrm{Set}]:\mathrm{Poset}\hookrightarrow {\mathrm{Topos}}_{\mathrm{ess}}$$[-,Set] : Poset \hookrightarrow Topos_{ess}

is a full and faithful 2-functor.

Analogously, essential geometric morphisms of the underlying toposes $\mathrm{Bohr}(A)\to \mathrm{Bohr}(B)$ are precisely those in image under the functor $\mathrm{Sh}\circ \mathrm{Alex}$ of functors $𝒞(A)\to 𝒞(B)$.

Proof

By the discussion in the section In terms of essential geometric morphisms at Cauchy complete category we have a full and faithful embedding of Cauchy-complete catgeories $[-,\mathrm{Set}]:{\mathrm{Cat}}_{\mathrm{Cauchy}}\hookrightarrow {\mathrm{Topos}}_{\mathrm{ess}}$. But posets are trivially Cauchy, complete, hence this restricts to an embedding $[-,\mathrm{Set}]:\mathrm{Poset}\hookrightarrow {\mathrm{Cat}}_{\mathrm{Cauchy}}\hookrightarrow {\mathrm{Topos}}_{\mathrm{ess}}$.

In terms of Alexandroff topologies: by the discussion of Alexandroff locales (in the entry Alexandroff space) we have that the functor $\mathrm{Alex}:\mathrm{Poset}\to \mathrm{Locale}$ takes values precisely on those morphisms of locales whose inverse image has a left adjoint. The statement then follows using the properties of localic reflection, which says that the 2-functor $\mathrm{Sh}:\mathrm{Locale}\to \mathrm{Topos}$ is a full and faithful 2-functor.

For such essential geometric morphisms to be parts of morphisms in $C^{*}\mathrm{Topos}$, def. 9 we need that their inverse images respect internal $C^{*}$-algebras:

Proposition

For $f:\mathrm{Bohr}(A)\to \mathrm{Bohr}(B)$ an essential geometric morphism, the inverse image $f^{*}\underline{B}$ is naturally a $C^{*}$-algebra in $[𝒞(A),\mathrm{Set}]$.

Proof

According to (HLS09, 4.8) every functor $𝒞(A)\to C^{*}\mathrm{Alg}$ is an $C^{*}$-algebra internal to $[𝒞(A),\mathrm{Set}]$. Here $f^{*}\underline{B}$ is such a functor, sending $(C\in 𝒞(A))\mapsto {\mathrm{im}}_f(C)$.

Using this we now discuss morphisms of Bohr toposes in $C^{*}\mathrm{Topos}$.

Proposition

The construction of Bohr toposes from def. 10 extends to a functor of the form

$$\mathrm{Bohr}:C^{*}{\mathrm{Alg}}_{\mathrm{cr}}^{\mathrm{op}}\to C_{\mathrm{com}}^{*}\mathrm{TopSpace}\hookrightarrow C_{\mathrm{com}}^{*}\mathrm{Topos}$$Bohr : C^\ast Alg_{cr}^{op} \to C^\ast_{com} TopSpace \hookrightarrow C^\ast_{com} Topos

with the special property that any $f:A\to B$ is sent to

• an essential geometric morphisms $(f^{*}⊣f_{*}):\mathrm{Bohr}(B)\to \mathrm{Bohr}(A)$ with an extra right adjoint

  $$\mathrm{Bohr}(B)\stackrel{\stackrel{f_{!}}{\to }}{\stackrel{\stackrel{f^{*}}{\leftarrow }}{\stackrel{\stackrel{f_{*}}{\to }}{\underset{f^{!}}{\leftarrow }}}}\mathrm{Bohr}(A)$$Bohr(B) \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\stackrel{\overset{f_*}{\to}}{\underset{f^!}{\leftarrow}}}} Bohr(A)

• such that the corresponding internal homomorphism of internal algebras $\underline{A}\to f_{*}\underline{B}$ is an epimorphism.

This is (Nuiten, lemma 2.7). (Essentially this argument also appears as (vdBergHeunen, prop. 33), where however the extra right adjoint is not made use of and instead the variances of the morphisms involved in the definition of $C^{*}\mathrm{Topos}$ are redefined in order to make the statement come out.)

Proof

To a morphism $f:A\to B$ in $C^{*}{\mathrm{Alg}}_{\mathrm{cr}}$ which by prop. 3 corresponds to an adjoint pair

$$(𝒞(f)⊣R_f):𝒞(A)\stackrel{\stackrel{R_f}{\leftarrow }}{\underset{𝒞(f)}{\to }}𝒞(B)$$( \mathcal{C}(f) \dashv R_f) : \mathcal{C}(A) \stackrel{\overset{R_f}{\leftarrow}}{\underset{\mathcal{C}(f)}{\to}} \mathcal{C}(B)

we assign the essential geometric morphism

$$(f_{!}⊣f^{*}⊣f_{*}⊣f^{!}):=[𝒞(B),\mathrm{Set}]\stackrel{\stackrel{f_{!}:={\mathrm{Lan}}_{R_f}}{\to }}{\stackrel{\stackrel{f^{*}:=(-)\circ R_f}{\leftarrow }}{\stackrel{\stackrel{f_{*}:=(-)\circ 𝒞(f)}{\to }}{\underset{f^{!}:={\mathrm{Ran}}_{𝒞(f)}}{\leftarrow }}}}[𝒞(A),\mathrm{Set}]$$(f_! \dashv f^* \dashv f_* \dashv f^!) := [\mathcal{C}(B), Set] \stackrel{\overset{f_! := Lan_{R_f}}{\to}}{\stackrel{\overset{f^* := (-)\circ R_f}{\leftarrow}}{\stackrel{\overset{f_* := (-)\circ \mathcal{C}(f)}{\to}}{\underset{f^! := Ran_{\mathcal{C}(f)}}{\leftarrow}}}} [\mathcal{C}(A), Set]

(wher $\mathrm{Lan}$ and $\mathrm{Ran}$ denote left and right Kan extension, respectively) equipped with the morphism of internal $C^{*}$-algebras

$${\eta }_f:\underline{A}\to f_{*}\underline{B}$$\eta_f : \underline{A} \to f_* \underline{B}

which over $C\in 𝒞(A)$ is the restriction of $f$ to $C$ and corestriction to $𝒞(f)(C)$

$$f{\mid }_C:C\to 𝒞(f)(C)$$f|_C : C \to \mathcal{C}(f)(C)

(to the $C^{*}$-completion of the algebraic image of $f{\mid }_C$).

Using prop. 4 the above prop. 6 has the following partial converse.

Proposition

For $A,B\in C^{*}\mathrm{Alg}$, morphisms $f:\mathrm{Bohr}(B)\to \mathrm{Bohr}(A)$ in $C^{*}\mathrm{Top}$ for which

1. the underlying geometric morphism has an extra right adjoint

2. the morphism of internal algebras $\underline{A}\to f_{*}\underline{B}$ is an epimorphism

are in bijection with functions

$$f:A\to B$$f : A \to B

that are homomorphisms on all commutative subalgebras and reflect commutativity.

In particular when $A$ is already commutative, morphisms $\mathrm{Bohr}(B)\to \mathrm{Bohr}(A)$ with an extra right adjoint and epi ring homomorphism are in bijection with algebra homomorphisms $A\to B$.

Proof

By prop 4 every essential geometric morphism $\mathrm{Sh}(\mathrm{Alex}𝒞(A))\to \mathrm{Sh}(\mathrm{Alex}𝒞(B))$ comes from a morphism of locales $\mathrm{Alex}𝒞A\to \mathrm{Alex}𝒞A$, which by the discussion at Alexandroff space is equivalently a morphism of posets $𝒞(f):𝒞(A)\to 𝒞(B)$. By the assumption of the extra right adjoint we also have a geometric morphism the other way round, and hence, again by prop. 4, an adjoint pair

$$(𝒞(f)⊣R_f):𝒞(A)\leftrightarrow 𝒞(B)$$(\mathcal{C}(f) \dashv R_f) : \mathcal{C}(A) \leftrightarrow \mathcal{C}(B)

that induces functors between toposes as in prop. 6. Then the fact that $f$ is a morphisms of $C^{*}$-toposes implies algebra homomorphisms

$$f_C:C\to f(C)$$f_C : C \to f(C)

natural in $C\in 𝒞(A)$.

By the assumption that this are the components of an epimorphism of copresheaves all these component morphisms are themselves epimorphisms and hence we have that indeed $f(C)={\mathrm{image}}_{f_C}(C)$.

Kinematics in a Bohr topos

A key aspect about the Bohr topos $\mathrm{Bohr}(A)$ is that that classical kinematics of the commutative internal $C^{*}$-algebra $\underline{A}\in \mathrm{Bohr}(A)$ is the quantum kinematics of $A$. In fact, the very definition of $\mathrm{Bohr}(A)$ provides a formal context in which Gleason's theorem has a natural formulation:

Theorem

(Gleason’s theorem)

For $H$ a Hilbert space of dimension $\dim H>2$, and $A=B(H)\in C^{*}\mathrm{Alg}$ its algebra of bounded operators, a state on $A$ is a function

$$\rho :A\to ℂ$$\rho : A \to \mathbb{C}

which is a $ℂ$-linear map when restricted to each commutative subalgebra $C\subset A$.

A function that preserves certain structure locally – here: over each commutative subalgebra – is precisely an internal fully structure preserving homomorphism in the presheaf topos over these local objects – here: over commutative subalgebras. Hence we have the following direct topos-theoretic equivalent reformulation of Gleason’s theorem.

Corollary

For $A=B(H)\in C^{*}\mathrm{Alg}$ as above, we have a natural bijection between the quantum states on $A$ and the (classical) states of $\underline{A}$ internal to $\mathrm{Bohr}(H)$.

The phase space

The idea is is that for $A\in C^{*}\mathrm{Alg}$, the Bohr topos $\mathrm{Bohr}(A)=(\mathrm{Sh}(\mathrm{Alex}(𝒞(A))),\underline{A})\in C^{*}\mathrm{TopSpace}\subset C^{*}\mathrm{Topos}$ is the corresponding quantum phase space. More precisely, we may think of the internal commutative $C^{*}$-algebra $\underline{A}\in \mathrm{Bohr}(A)$ as the formal dual to the quantum phase space.

The following discussion makes this precise by exhibiting this formal dual as an internal locale. Since $\mathrm{Bohr}(A)$ is a spatial topos, this internal locale is in fact an ordinary topological space bundle $\Sigma \to \mathrm{Alex}𝒞(A)$ over the Alexandroff space $\mathrm{Alex}𝒞(A)$.

Definition

Write ${\underline{\Sigma }}_A$ and ${\Sigma }_A^{\neg \neg }$, respectively for the corresponding internal locales associated to $\underline{A}$ by internal constructive Gelfand duality. Write

$${\Sigma }_A\to 𝒞(A)$$\Sigma_A \to \mathcal{C}(A)

and

$${\Sigma }_A^{\neg \neg }\to 𝒞(A)$$\Sigma^{\not \not}_A \to \mathcal{C}(A)

for the corresponding external locale, given under the equivalence of categories

$$\mathrm{Loc}(\mathrm{Sh}(𝒞(A)))\simeq \mathrm{Loc}/𝒞(A)$$Loc(Sh(\mathcal{C}(A))) \simeq Loc/\mathcal{C}(A)

discussed at internal locale.

Definition

For $A$ a (noncommutative) C-star algebra, the assignment

$$A\mapsto {\Sigma }_A$$A \mapsto \Sigma_{A}

or

$$A\mapsto {\Sigma }_A^{\neg \neg }$$A \mapsto \Sigma^{\not \not}_{A}

is called the Bohrification of $A$.

Proposition

Let $\underline{\Sigma }(\underline{(}A))$ be the internal locale from def. 11.

Regarded as an object

$$({\Sigma }_A\to 𝒞(A))\in \mathrm{Loc}/𝒞(A)$$(\Sigma_A \to \mathcal{C}(A)) \in Loc/\mathcal{C}(A)

of external locales over $𝒞(A)$, this is the topological space whose underlying set is given by the disjoint union

$${\Sigma }_A=\coprod _{C\in 𝒞(A)}\Sigma (C)$$\Sigma_A = \coprod_{C \in \mathcal{C}(A)} \Sigma(C)

over all commutative $C^{*}$-subalgebras of $A$ of the ordinary Gelfand spectra $\Sigma (C)$ of these commutative $C^{*}$-algebras, and whose open subsets are defined to be those $𝒰\subset {\Sigma }_A$ for which

1. $𝒰{\mid }_C\in 𝒪(\Sigma (C))$ for all commutative subalgebras $C$;

2. For all $C\hookrightarrow D$, if $\lambda \in 𝒰{\mid }_C$ and $\lambda \prime \in \Sigma (D)$ such that $\lambda \prime {\mid }_C=\lambda$, then $\lambda \prime \in 𝒰{\mid }_D$.

Regarded equivalently as an internal locale in $\mathrm{Sh}(𝒞(A))$ this

As a presheaf on the poset $𝒞(A)$ this is given by

$$\underline{\Sigma }(\underline{A}):U\mapsto {\Sigma }_U,$$\underline{\Sigma}(\underline{A}) : U \mapsto \Sigma_U \,,

where for $U\in 𝒪(𝒞(A))$ we set

$${\Sigma }_U:=\coprod _{C\in U}\Sigma (C)$$\Sigma_U := \coprod_{C \in U} \Sigma(C)

with the relative topology inherited from ${\Sigma }_A$.

This appears as (HLSW, theorem 1).

Proposition

The Bohrification of $A\in \mathrm{ncCStar}$ only depends on its partial C-star algebra $N(A)$ of normal elements

$${\Sigma }_A\simeq {\Sigma }_{N(A)}.$$\Sigma_A \simeq \Sigma_{N(A)} \,.

This is highlighted in (vdBergHeunen).

Proposition

For $A$ a commutative $C^{*}$-algebra and ${\Sigma }_A^{\mathrm{Gelf}}\in$ Loc its ordinary Gelfand spectrum, we have that Bohrification in the double negation topology reproduces this ordinary Gelfand spectrum:

$${\Sigma }_A^{\neg \neg }\simeq {\Sigma }_A^{\mathrm{Gel}}.$$\Sigma^{\not \not}_A \simeq \Sigma_A^{Gel} \,.

This is (Spitters06, theorem 9, corollary 10).

Proposition

Then the construction of the ringed topos over the poset of commutative subalgebras

$$A\mapsto ([𝒞(A),\mathrm{Set}],\underline{A})$$A \mapsto ([\mathcal{C}(A), Set], \underline{A})

extends to a functor

$$C^{*}{\mathrm{Alg}}_{\mathrm{incl}}\to C^{*}\mathrm{TopSpace}\hookrightarrow C^{*}\mathrm{Topos},$$C^* Alg_{incl} \to C^*TopSpace \hookrightarrow C^* Topos \,,

where on the right the morphisms of internal rings are even morphisms of internal C-star algebras.

Proof

With the components of the morphism of internal rings the evident objectwise inclusions, this is directly checked.

Proposition

Let ${\mathrm{Cstar}}_{\mathrm{inc}}$ be the category of C-star algebras and inclusions. Then Bohrification extends to a functor

$${\Sigma }_{(-)}:{\mathrm{CStar}}_{\mathrm{inc}}^{\mathrm{op}}\to \mathrm{Loc}.$$\Sigma_{(-)} : CStar_{inc}^{op} \to Loc \,.

This is effectively the functoriality of the internal constructive Gelfand duality applied to the above observation. The statement appears as (vdBergHeunen, theorem 35).

The observables

For $A\in C^{*}\mathrm{Alg}$ we call a self-adjoint operator $a\in A$ an quantum observable. The following statement asserts that quantum observables on $A$ are in a precise sense the $ℝ$-valued “functions” on the Bohr topos of $A$.

Write $C(ℝ)$ for the $C^{*}$-algebra of continuous complex functions on the real line. We think of $\mathrm{Bohr}(C(ℝ))$ as the incarnation of $ℝ$ in the context of Bohr toposes.

Proposition

Morphisms $f:\mathrm{Bohr}(A)\to \mathrm{Bohr}(C(ℝ{)}_0)$ with an extra right adjoint and $C(ℝ{)}_0\to f_{*}\underline{A}$ epi are in bijection to the observables on $A$.

Proof

By prop. 7 such morphisms are in bijection to algebra homomorphisms

$$C(ℝ{)}_0\to A.$$C(\mathbb{R})_0 \to A \,.

By functional calculus: every self-adjoint operator $a\in A$ provides such a homomorphism by $f\mapsto f(a)$. Conversely, given such an algebra homomorphism, its image of $i:x\mapsto x$ is a self-adjoint operator in $A$, and these two constructions are clearly inverses of each other.

Let $x_{t_1}\in A$ be an observable and write $⟨x_{t_1}⟩$ for the subalgebra generated by it. Then (by general properties of presheaf over-toposes) the over topos $\mathrm{Bohr}(A)/⟨x_{t_1}⟩$ is equivalent to

$$\mathrm{Bohr}(A)/⟨x_{t_1}⟩\simeq [⟨x_{t_1}⟩𝒞(A),\mathrm{Set}],$$Bohr(A)/\langle x_{t_1}\rangle \simeq [\langle x_{t_1}\rangle \mathcal{C}(A), Set] \,,

where on the right we have the copresheaves over the under-category $⟨x_{t_1}⟩/𝒞(A)$. This is precisely the sub-poset of commutative subalgebras on those commutative subalgebras that contain $x_t$. This means that a classically consistent observation in the slice topos is one that is consistent with the observation of $x_{t_1}$.

(…)

The states

Definition

For $A\in C^{*}\mathrm{Alg}$ write $(\mathrm{Sh}(\mathrm{Alex}𝒞(A)),ℝ)$ for the ringed topos as indicated, where $ℝ$ denotes the copresheaf constant on $ℝ$.

The internal $C^{*}$-algebra $\underline{A}\in \mathrm{Bohr}(A)$ is an internal $ℝ$-module. Forgetting the algebra structure and only remembering the $ℝ$-module structure, we get a category of ”$ℝ$-module toposes”.

Observation

There is a canonical morphism of ringed toposes

$$\pi :\mathrm{Bohr}(A)\to (\mathrm{Sh}(\mathrm{Alex}𝒞(A)),ℝ)$$\pi : Bohr(A) \to (Sh(Alex \mathcal{C}(A)), \mathbb{R})

whose underlying geometric morphism is the identity (and whose morphism of internal ring objects is the unique).

Remark

This bundle is the $C^{*}$-topos incarnation of the morphism $\Sigma \to \mathrm{Alex}𝒞(A)$ of locales discussed above.

Observation

A state $\rho$ on $A$ is a section of $\pi$ in the category of $ℝ$-moduled toposes that is positive and normalized.

(…)

(Pre-)Sheaf of Bohr toposes of a quantum field theory

Notice that in the context of AQFT a quantum field theory is encoded by a local net of C-star algebras on spacetime.

Note

Let $X$ be a Lorentzian manifold and

$$A:𝒪(X)\to C^{*}{\mathrm{Alg}}_{\mathrm{inc}}$$A : \mathcal{O}(X) \to C^* Alg_{inc}

be a local net of algebras. Notice that by definition this indeed takes values in $C^{*}$-algebras and inclusions . Then postcomposition with the Bohr topos-functor yields a presheaf of ringed spaces

$$\mathrm{Bohr}(A):𝒪(X{)}^{\mathrm{op}}\to C^{*}{\mathrm{Alg}}_{\mathrm{inc}}^{\mathrm{op}}\hookrightarrow C^{*}{\mathrm{Alg}}_{\mathrm{cr}}^{\mathrm{op}}\stackrel{\mathrm{Bohr}}{\to }{C}^{*}\mathrm{TopSpace}\hookrightarrow C^{*}\mathrm{Topos}.$$Bohr(A) : \mathcal{O}(X)^{op} \to C^* Alg_{inc}^{op} \hookrightarrow C^* Alg_{cr}^{op} \stackrel{Bohr}{\to} C^* TopSpace \hookrightarrow C^* Topos \,.

This appears as (Nuiten, def. 17).

Locality and descent

Assume that a net of observables $A:\mathrm{Op}(X)\to C^{*}{\mathrm{Alg}}_{\mathrm{inc}}$ satisfies the split property?. Then it is local precisely if the corresponding presheaf of Bohr toposes $\mathrm{Bohr}(A):\mathrm{Op}(X{)}^{\mathrm{op}}\stackrel{A}{\to }{C}^{*}{\mathrm{Alg}}_{\mathrm{inc}}\stackrel{\mathrm{Bohr}}{\to }{C}^{*}\mathrm{Top}$ satisfies spatial descent by local geometric morphisms (meaning that for every spatial hyperslice $\Sigma \subset X$ the induced presheaf $\mathrm{Bohr}(A){\mid }_{\Sigma }:\mathrm{Op}(\Sigma {)}^{\mathrm{op}}\to C^{*}\mathrm{Top}$ satisfies descent by local geometric morphisms.)

(…)

This appears as (Nuiten, theorem 4.2).

(…)

Boundary field theories and descent

(Nuiten, section 4.3)

References

Niels Bohr’s views on quantum mechanics that give the construction of Bohrification its name are reviewed in

• Erhard Scheibe, The logical analysis of quantum mechanics . Oxford: Pergamon Press, 1973.

The term Bohrification and the investigations associated with it are initiated in

• Chris Heunen, Klaas Landsman, Bas Spitters, Bohrification of operator algebras and quantum logic (arXiv:0905.2275)

See also

• Chris Heunen, Klaas Landsman, Bas Spitters, Bohrification in Deep Beauty Cambridge University Press(2009) (arXiv:0909.3468)

The computation of the internal Gelfand spectrum $\underline{\Sigma }$ was initiated in

• Chris Heunen, Klaas Landsman, Bas Spitters A topos for algebraic quantum theory, Comm. Math. Phys. 291:63–110, 2009, (arXiv:0709.4364, doi)

with some results in section 5 and 6 of

• Martijn Caspers, Gelfand spectra of $C^{*}$-algebras in topos theory (pdf)

and completed in

• Sander Wolters, Contravariant vs covariant quantum logic: A comparison of two topos-theoretic approaches to quantum theory (arXiv:1010.2031)

An complete outline of the full proof is given in

• Chris Heunen, Klaas Landsman, Bas Spitters, Sander Wolters, The Gelfand spectrum of a noncommutative $C^{*}$-algebra: a topos-theoretic approach, arxiv:1010.2050

Applications and examples for $A$ a matrix algebra are discussed in

• Martijn Caspers, Chris Heunen, Klaas Landsman, Bas Spitters, Intuitionistic Quantum Logic of an $n$-level System (pdf)

The functoriality of Bohrification is observed in

• Benno van den Berg, Chris Heunen, Noncommutativity as a colimit (arXiv:1003.3618)

The application of the double negation topology to make Bohrification coinicide with ordinary Gelfand duality on commutative $C^{*}$-algebras is discussed in

• Bas Spitters, The space of measurement outcomes as a spectrum for non-commutative algebras in Cooper, Kashefi, Panangaden (eds.) Developments in computational models (DCM 2010)(arXiv:1006.1432)

The generalization of Bohrification from quantum mechanics to quantum field theory (AQFT) is discussed in

• Joost Nuiten, Bohrification of local nets

The original suggestion to interpret the Kochen-Specker theorem in the topos over the poset of commutative subalgebras (there taken to be presheaves instead of copresheaves) is due to

• Jeremy Butterfield, John Hamilton, Chris Isham, A topos perspective on the Kochen-Specker theorem, I. quantum states as generalized valuations, Internat. J. Theoret. Phys. 37(11):2669–2733, 1998, MR2000c:81027, doi; II. conceptual aspects and classical analogues Int. J. of Theor. Phys. 38(3):827–859, 1999, MR2000f:81012, doi; III. Von Neumann algebras as the base category, Int. J. of Theor. Phys. 39(6):1413–1436, 2000, arXiv:quant-ph/9911020, MR2001k:81016,doi; IV. Interval valuations, Internat. J. Theoret. Phys. 41 (2002), no. 4, 613–639, MR2003g:81009, doi