### The n-Café Quantum Conjecture

#### Posted by Urs Schreiber

**Conjecture** (roughly): *Quantum mechanics, which has historically been considered as a structure internal to $0\mathrm{Cat}$, has a more natural formulation as a structure internal to $1\mathrm{Cat}$. This involves refining functions to “bundles of numbers” (namely fibered categories, or, dually, refining (action) functors to pseudofunctors) and it involves refining linear maps by spans of groupoids.*

*In this natural formulation, quantization will no longer be a mystery – but a pushforward: the quantum propagator $(t \mapsto U(t))$ is the pushforward to a point of the classical action (pseudo)-functor.*

*$n$-Dimensional quantum field theory works the same way, but internal to $n\mathrm{Cat}$.*

**Evidence.**

1) The **Tale of groupoidification** (or “Tale of spanification”): The shift in degree is real.

Linear algebra and linear representation theory is in many respects better thought of as an unrefined version of what should really be the study of spans in groupoids. Taking this seriously and applying it to the linear algebra appearing in quantum mechanics might be responsible for the categorical refinement in the above conjecture.

John Baez has been reporting on joint work on this with Jim Dolan and Todd Trimble in the last couple of installments of This Week’s Finds: 247, 248, 249, 250, 251, 252.

2) **It does make sense** when applied to quantum mechanics.

Before the *Narn i Groupoidification* appeared, Jeffrey Morton had already applied the central ideas appearing there to the quantum harmonic oscillator. While there remain lots of open questions, the main insight is that it does make good sense. Indeed, many concepts in quantum mechanics become *simpler* (conceptually) this way.

Jeffrey Morton
*Categorified Algebra and Quantum Mechanics*

arXiv:math/0601458v1

(see also Jamie Vicary’s work, discussed in our recent thread Categorifying Quantum Mechanics)

The example John emphasizes a lot is: the canonical commutation relations for Fock operators which play such a crucial role throughout quantum theory $[a,a^\dagger] = \mathrm{Id}$ becomes a mere combinatorial statement: it says that there is one more way to first insert one item into a box of similar items and then remove one, than there is to first remove one and then put it back in.

This is the kind of “simplification of concepts” that is to be expected. But it should only be a start.

3) The **process of quantization becomes simpler**, conceptually.

As John Baez has explained in great detail in his lectures on *Quantization and Cohomology (* week 18 21, and others) a classical action functional is really a functor.

In particular, the non-kinetic part of the action is a parallel transport functor. The kinetic part is, I think, better thought of as being part of the measure which appears once we realize the shift in categorical degree and then push-forward this parallel transport functor to a point.

This should be what quantization is: $\array{ \text{classical (non-kinetic) action} &\stackrel{\text{quantization}}{\rightarrow}& \text{quantum theory} \\ \text{parallel transport functor} &\stackrel{\text{push-forward to a point}}{\rightarrow}& \text{quantum propagation functor} } \,.$

One can check that with everything conceived in its usual un-refined formulation, this push-forward does produce the right answers on $(k \lt n)$-morphisms.

For instance, for $n=1$ it is easy to see that the push-forward computes the space of states of the quantum particle: The First Part of the Story of Quantizing by Pushing to a Point….

Then, for $n=2$ it does, apparently, produce the right kinematics for the string: the D-branes over the ends of the open string appear as 2-spaces of states of the 2-particle : Quantum 2-States: Sections of 2-vector bundles, QFT of Charged n-particle: Chan-Paton Bundles.

(For $n=3$ one can see states of the 3-particle in the context of the Chern-Simons membrane reproduce aspects of 2-dimensional conformal field theory. I will summarize some of the aspects of this in a future entry.)

But in the unrefined version of quantum mechanics, the canonical push-forward to a point does not even exist for top level $n$-morphisms. One has to fake it by introducing measures and intervening by hand.

However, once we do refine the parallel transport from functors to pseudofunctors, it seems that also the usual kinetical part of the quantum theory is produced by a canonical god-given push-forward to a point: The Canonical 1-Particle.

And it turns out that in fact even without any intent to quantize it, a parallel transport functor is very naturally thought of in terms of its *curvature pseudo-functor*. As described in $n$-Curvature In fact, this does clarify a couple of things in gauge and higher gauge theory.

I expect that this classical passage from parallel transport functors to pseudofunctors roots in the same general abstract nonsense which the shift appearing in the “Tale of Groupoidification” does: judging from what I understood, it seems that in both cases the deal is the *Grothedieck construction*, or a variation of that theme, which relates fibered (1-)categories
$p : P \to X$
to pseudofunctors
$X \to \mathrm{Cat}
\,.$

## Re: The n-Café Quantum Conjecture

Urs, what’s your position on the point behind the question we finally got articulated here:

The idea might be expressed as that classical mechanics should be cast at the same category theoretic level as quantum mechanics. It’s just a rig change away.