The n-Category Café

Isham and Doring suggest a general framework for formulating physical theories using topos theory. The framework is very wide: so wide that it can encompass classical theory, quantum theory, (I think) local field theory, and who knows what else. The downside of this flexibility is the small amount of restrictions imposed on such a theory. In particular, it is not enough to specify the topos, the “state object” and the “value object”, one should also explicitely specify which morphisms count as observables.

The daseinization of a projection operator P is the following presheaf on the category of commutative subalgebras of our quantum obervable algerba A: given B a commutative subalgebra, the dasenization delta(P) at B is the smallest projection operator Q in B that is larger than P. Here, one projection operator P1 is said to be larger than another P2 when the image of P1 contains the image of P2.

Maybe I’ll attempt to summarize the paper later, too sleepy and have to get up for work tomorrow :)

Isham and Döring suggest several alternatives for the “value object” $R$ that they denote $R_\geq$, $\R_\leq$ and $R_{\leftrightarrow}$.

$R_\geq$ is the so-called presheaf of order reversing real-valued functions over our category of commutative subalgebras. To any commutative subalgebra $B$ of our noncommutative algebra $A$, $R_\geq$ corresponds to the set of order-reversing real-valued functions on the set of subalgebras of $B$. Given $x$ an element of $A$, we can construct a morphism $$\Sigma \to R_{\geq}$$ as follows.

Consider the spectral family $e_\lambda$ of $x$: $e_\lambda$ is the projector corresponding to $x \lt \lambda$. On each commutative subalgebra $B'$ of $A$, $e_\lambda$ yields the spectral family $$f_\lambda(B'): = \mathrm{inf}_{\mu \gt \lambda} \delta(e_\mu)_{B'}$$

Here $\delta(...)_{B'}$ means daseinization at $B'$. Each such family yields an element $\delta(x)_{B'}$ of $B'$ and hence a function on $\mathrm{Spec} B'$. We have $$\delta(x)_{B'} := \int \lambda df_{\lambda(B')}$$ Now we are ready to construct the morphism. The morphism should assign to each point $p$ in $\mathrm{Spec} B$ an order-reversing real-valued function on the subalgebras $B'$ of $B$. We define this function to be the evaluation of $\delta(x)_{B'}$ at the image of $p$ in $\mathrm{Spec} B'$ (equivalently the evaluation of the image of $\delta(x)_{B'}$ in $B$ at $p$). The $R_{\leq}$ story is similar with the following differences:

1) $R_{\leq}$ is the presheaf of real-valued order-preserving functions.

2) In this case the definition of $f_{\lambda(B')}$ is simpler: $$f_{\lambda(B')} := \delta^i(e_\lambda)_{B'}$$ Here $\delta^i$ is the so-called inner daseinisation. The inner daseinisation of $P$ at $B'$ is the best approximation of $P$ from below by a projector in $B'$ just as the usual (outer) daseinisation is the best approximation of $P$ from above by a projector in $B'$. $R_{\leftrightarrow}$ is the product of the presheaves $R_{\geq}$ and $R_\leq$, the morphism $$\Sigma \to R_{\leftrightarrow}$$ being the product morphism. This allows symmetry between inner and outer daseinisation.

Isham and Döring also discuss so-called $k(R_{\geq})$ as a value object. It is obtained from $R_\geq$ by a procedure that transforms monoid objects into group objects (note that $R_{\geq}$ is merely a monoid object, not a group object).

The reasons I don’t think this approach has great kinship with noncommutative geometry are:

1) Addition and multiplication of noncommuting observables cannot be defined in topos-theoretic terms (as far as I can see). It is true that they are used in the construction, but that just goes to say the construction actually depends on the algebra and not on something less. On the other hand, addition and multiplication are essentially the only structure on the set of “functions” the noncommutative geometry guys use.

2) Each algebra requires its own topos, there is no single topos which allows describing all noncommutative geometries.

In particular, I find point 1 rather disappointing from a physical perspective, and this is why. Once we can formulate statements about the quantum system in some formal language and assign truth value to these statements, the next thing one would like is some sort of an axiom system. I’m not sure how axiom systems work in multi-valued logic but I’m sure Isham and Doring know. :-) In particular, these axioms would represent familiar laws of physics like the second law of Newton for a point particle. However, this law

mx”(t) = F(x(t))

involves a 2nd derivative which is essentially a linear combination and it’s not clear how to describe it topos-theoretically. It appears to me this is an impediment to constructiing a reasonable axiom system and thus to actually understanding physics using the Isham-Doring approach.

A small remark: if we add the additional requirement

a) F(A + B) = F(A) + F(B)

or the weaker requirement

b) F is continuous

Then we get a Kochen-Specker-like theorem form dim H = 2 as well.

But without specifying how V is chosen, you may not satisfy the second condition. For example, say I always set F(A) to be the larger eigenvalue. Then F(f(A))=f(F(A)) will fail whenever f is decreasing.

Squark gave a solution to my Kochen–Specker puzzle. It sounds right… but it doesn’t sound very ‘physical’.

I could be wrong, but I think there’s a way to make the solution sound more like physics: the physics of spin-1/2 particles.

Any self-adjoint operator $A$ on $\mathbb{C}^2$ can be written as

$$A = a_0 1 + a_x \sigma_x + a_y \sigma_y + a_z \sigma_z$$

in a unique way, where the $a_i$ are real and $\sigma_x, \sigma_y, \sigma_z$ are the Pauli matrices. Write $\vec{a} = (a_x,a_y,a_z)$.

So, a self-adjoint operator $A$ on $\mathbb{C}^2$ is just a real number $a$ and a vector $\vec a$. We can think of it as measuring the spin of a spin-1/2 particle in the $\vec a$ direction and adding $a_0$.

I think there’s a way to use $a_0$ and $\vec a$ to write formulas for functions $F$ satisfying the Kochen–Specker conditions:

a) $F(A)$ lies in the spectrum of $A$;

b) For any continuous $f: \mathbb{R} \to \mathbb{R}$, $F(f(A)) = f(F(A))$.

But, I forget how these formulas go, if I ever knew!

Squark also mentions:

if we add the additional requirement

a) $F(A + B) = F(A) + F(B)$

or the weaker requirement

b) $F$ is continuous

then we get a Kochen–Specker-like theorem for $dim H = 2$ as well.

The continuity condition suggests that any formula for $F(A)$ requires an arbitrary choice that we can’t make in a continuous way. I don’t remember hearing about this before, but clearly it’s happening in Squark’s own solution to the puzzle: to define $F(A)$ he needs to pick a 1-dimensional eigenspace of $A$!

Sub-puzzle: what bundle are we finding a section of when we choose a 1-dimensional eigenspace of $A$ for each self-adjoint operator $A$, and why is there no continuous section? What (presumably famous) topological obstruction prevents us?

The other condition Squark mentions,

$$F(A + B) = F(A) + F(B)$$

goes completely against the grain of the Kochen–Specker philosophy. If $A$ and $B$ don’t commute, there’s no way to measure $A$, $B$, and $A+B$ simultaneously, get numbers $F(A)$, $F(B)$ and $F(A+B)$, and check that $F(A + B) = F(A) + F(B)$. The supposedly startling thing about the Kochen–Specker theorem is that the condition

$$F(f(A)) = f(F(A))$$

only involves commuting observables, namely $A$ and $f(A)$ — yet it’s still too strong to get solutions!