The n-Category Café

The formalization of the notion quantum mechanical system 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 $\mathbb{C}$-linear map $\rho : A \to \mathbb{C}$ (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 $(a, \rho) \mapsto \rho(a) \in \mathbb{C}$ 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 $\mathbb{R} \to Aut(A)$ (inner automorphisms for “localized” systems) is thought of as as being an evolution of the system, for instance in time or more generally 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 $C^\ast$-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 $Spec A \in C^\ast Alg^{op}$ that is the incarnation of $A \in C^\ast Alg$ 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 $Spec A \to \mathbb{R}$ in $C^\ast Alg^{op}$, which dually is a $C^\ast$-algebra homomorphism $C(\mathbb{R})_0 \to A$. That such correspond to self-adjoint operators of $A$ is the statement of functional calculus for $C^\ast$-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-$C^\ast$-algebra $\langle \{a_i\}_i\rangle \subset A$ generated by them is commutative. Therefore for the phenomenological interpretation of the axioms it seems to make no sense to demand that a state $\rho : A \to \mathbb{C}$ 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 $\rho : A \to \mathbb{C}$ which is required to be $\mathbb{C}$-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 – nontrivially – 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 $dim H \gt 2$ and $A = B(H)$ the $C^\ast$-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 $\rho : A \to \mathbb{C}$ which is not just linear but also an algebra homomorphism on each commuting subalgebra. Notice that, by Gelfand duality , every commutative $C^\ast$-algebra $C$ is the algebra of continuous functions on some topological space $sp(C)$. Under this duality a state on $C$ is a probability distribution on $sp(C)$, while an algebra homomorphism $C \to \mathbb{C}$ is a point of $X_C$. Therefore a quasi-state which is commutative-subalgebra-wise even 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 variables of the quantum mechanical system: it’s 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:

Foundational context.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 therfore 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 interst, that of functions defined on commutative subalgebras:

for $A$ any C-star algebra, write $\mathcal{C}(A)$ for the set of all its commutative $C^\ast$-subalgebras. This is naturally a poset under inclusion of subalgebras. A (co)presheaf of this set is a functor $\mathcal{C}(A) \to Set$. 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 $[\mathcal{C}(A), Set]$ – 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 $C^\ast$-algebra $A$ naturally induces a commutative C-star algebra object $\underline{A}$ internalization|internal to $[\mathcal{C}(A), Set]$: namely the copresheaf defined by the tautological assignment

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

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

Elsewhere we shall discuss 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

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

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 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 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 $Bohr(A)$.

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

In particular (Gleason’s theorem): if $A = B(H)$ for $dim H \gt 2$ 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 $sp(\underline{A})$ 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 $Bohr(A)$ – somewhat more naturally than as a $C^\ast$-algebra $A$. (The second, in a slightly different setup, was emphasized in (Isham-Hamilton-Butterfield98), which inspired all of the following discussion, the first in (Spitters)). In fact, thinking of ringed toposes as generalized spaces in higher geometry, it suggestes that the Bohr topos $Bohr(A)$ itself is the quantum phase space of the quantum mechnanical system in question.

To which extent this perspective is genuinely useful is maybe still to be established. 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 (Nuiten11) 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

$$A : Op(X) \to C^\ast Alg$$

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 $Bohr(A_U)$. This yields a presheaf

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

of ringed toposes whose internal ring object has the structure of a commutative $C^\ast$-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 (Nuiten11) 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 of the original copresheaf of observables $A : Op(X) \to C^\ast Alg$ 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 perhaps 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.