### QFT of Charged n-Particle: Dynamics

#### Posted by Urs Schreiber

Given an $n$-particle (a point, a string, a membrane, etc.) coupled to a $n$-bundle with connection (an electromagnetic field, a Kalb-Ramond field, a 2-gerbe, etc.), what is the corresponding quantum theory?

The answer to this question has a *kinematical* and a *dynamical* aspect and up to now I had concentrated on the kinematics:

Definition, point particle, string on $B G$, open string on $X$.

Here I start talking about **dynamics**. The right abstract point of view on the dynamics of quantum systems, as far as path integrals are concerned, is certainly that adopted for instance in

E. Lupercio, B. Uribe
*Topological Quantum Field Theories, Strings, and Orbifolds*

hep-th/0605255.

A parameter space $\mathrm{par}$, describing the shape of the quantum object (the point, the circle, the sphere, etc.) propagates along “*worldvolumes*” (graphs, surfaces, 3-manifolds, etc.), which are spaces whose boundaries look like $\mathrm{par}$, by pull-push along correspondences of the form
$\array{
& \mathrm{worldvol}
\\
\multiscripts{^{\mathrm{in}}}{\nearrow}{}
&&
\nwarrow^{\mathrm{out}}
\\
\mathrm{par} && \mathrm{par}
}
\,.$

Here I formulate this in an arrow theoretic way (a little more arrow-theoretic than the discussion in the above text, that is) that fits into the context of kinematics that I discussed before.

While everything is categorical, a crucial point is that, as opposed to the kinematics, the dynamics requires the push-forward of a *set* at one point. In the absence of a notion of adjoint morphisms of sets, this requires extra structure on our sets: a *measure*. This is the infamous measure that appears in the path integral.

The following definition is taken from

Recall # that an $n$-particle is something that looks like the ($n-1$)-category $\mathrm{par}$ (the point $\mathrm{par} = \{\bullet\}$, or the interval $\mathrm{par} = \{a \to b\}$, or the sphere, $\mathrm{par} = \Pi_2(S^2)$, etc.) and comes equipped with a choice of maps $(\gamma : \mathrm{par} \to \mathrm{tar}) \in \mathrm{conf} \subset [\mathrm{par},\mathrm{tar}]$ into some $n$-category called “target space” (ordinarily, $\mathrm{tar} = P_n(X)$ are $n$-paths in some “spacetime” $X$).

Each such map is a *configuration* of the $n$-particle: one of many ways for it to sit in spacetime.

If our $n$-particle, however, does not just want to sit around, we get the same picture just slightly lifted in dimension: we look at $n$-categories whose *boundaries* look like $\mathrm{par}$, and think of them as inducing interpolations between different configurations of the $n$-particle.

**Deinition.** *A worldvolume (or diagram) for our $n$-particle is an $n$-category
$\mathrm{worldvol}$
together with a collection of embeddings
$\mathrm{in}_i : \mathrm{par} \to \mathrm{worldvol}$
and
$\mathrm{out}_j : \mathrm{par} \to \mathrm{worldvol}
\,.$
*

*
Given a worldvolume, a space of histories or space of trajectories or space of paths is a choice of subcatgeory
$\mathrm{hist} \subset [\mathrm{worldvol},\mathrm{tar}]$
which is compatible with the above choice of configuration space in that
$\mathrm{in}_i^* \mathrm{hist} \simeq \mathrm{conf}$
and
$\mathrm{out}_j^* \mathrm{hist} \simeq \mathrm{conf}
\,.$
*

For instance, for $n=1$ and $\mathrm{par} = \{\bullet\}$ the single point particle, the simplest worldvolume is the *worldline*, which is nothing but the interval $\{a \to b\}$ with the only two possible injections
$\array{
& \{a \to b\}
\\
\multiscripts{^{\mathrm{in}}}{\nearrow}{}
&&
\nwarrow^{\mathrm{out}}
\\
\bullet && \bullet
}
\,.$
In a simple standard example, we would take target space to be $\mathrm{tar} = \mathrm{Moore}(X)$, the category of Moore paths (paths with a parameter length) in $X$ and would take the category $\mathrm{hist}$ of histories to be the discrete category on the set of paths of parameter length $t \in \mathbb{R}$, say.

Not before long, we will need to equip the space of objects of this category with a measure. The canonical choice here would be the Wiener measure.

The interval $\{a \to b\}$ in the above example is special, in that it is a *cylinder over parameter space.*

**Definition**. *A worldvolume $(\mathrm{worldvol},\mathrm{in},\mathrm{out})$ is a cylinder over parameter space if there is a unique transformation*
$\array{
& \nearrow \searrow^{\mathrm{in}}
\\
\mathrm{par}
&\Downarrow&
\mathrm{worldvol}
\\
& \searrow \nearrow^{\mathrm{out}}
}
\,.$

Whenever we have a worldvolume which is a cylinder (not only then, but let us concentrate on this for the time being), the unique transformation it comes with induces for us a transformation filling the diagram $\array{ & & \mathrm{hist}\times \mathrm{worldvol} \\ & \multiscripts{^{\mathrm{out}^*}}{\swarrow}{} && \searrow^{\mathrm{in}^*} \\ \mathrm{conf}\times\mathrm{par} & &\;\;\Downarrow^{\mathrm{cyl}}& & \mathrm{conf}\times\mathrm{par} \\ & \multiscripts{^{\mathrm{ev}}}{\searrow}{} && \swarrow^{\mathrm{ev}} \\ && \mathrm{tar} } \,.$

This is noteworthy, because this diagram can be regarded as a correspondence through which we may pull-push *states* of the $n$-particle.

Recall that a state of the particle charged under the $n$-bundle with parallel transport $\mathrm{tra} : \mathrm{tar} \to \mathrm{phas}$ is a transformation of the form $\array{ & \nearrow \searrow^{1} \\ \mathrm{conf}\times\mathrm{par} &\;\Downarrow^e& \mathrm{worldvol} \\ & \searrow \nearrow^{\mathrm{ev}^*\mathrm{tra}} } \,.$

Given the above $\mathrm{cyl}$-diagram, we may

- first pull such a state back along $\mathrm{in}^*$, “from the incoming $n$-particle to the space of paths”

- then tranport it by composition with $\mathrm{cyl}$ to the other end of the cylinder

- and finally push it forward along $\mathrm{out}^*$ from the space of paths to the *outgoing* $n$-particle.

The diagrams below illustrate this process, and its appearance in pasting diagrams, in full detail.

Remarkably, while everything in the theory of the charged $n$-particle is completely canonical and based only on abstract arrow theory, here is one point where it *seems* as if the Dao does not yield an answer, and we need to intervene by hand:

in order to reproduce standard quantum theory, we need to interpret the above push-foward as a push-forward of measure spaces.

If we do this, we add to our definition of *space of paths*, $\mathrm{hist}$, a measure on this space. Then the required push-forward is integration over the fiber with respect to this measure.

In our above example, with this measure chosen as the ordinary Wiener measure, the map $e \mapsto e'$ which we get from the worldline of length $t$ is precisely the path integral propagator over time $t$ in standard quantum mechanics.

## Re: QFT of Charged n-Particle: Dynamics

I just wrote a long reply to this post talking in a pretty general way about categories of spans, functors giving configuration spaces and state spaces, and the pull-push procedure for turning spans of configuration spaces into operators on state spaces, and then mistakenly hit the “cancel” button rather than “preview”. Ugh: never edit anything in a web browser. The bottom line can be summed up thus:

You have a span of configuration spaces - one for a cobordism, and one each for its input and output boundaries. Then you have a “state space” which is something like a space of functions on these configuration spaces (or, if the configuration spaces are categories, a category of presheaves). You want to take a function from input to output. You do this by pulling back the function (or presheaf) to the middle of the span (your space of histories) - then pushing forward onto the output. The result is a path integral.

There are a bunch of variations depending on two things: (1) what category $C$ the configuration spaces live in, and (2) what kind of algebraic structure $A$ the functions take values in. In particular, if configuration spaces are literally spaces, and functions are literally complex-valued function, then you find you need a measure to do the pushforward - which is then integration. But there are other cases where no extra structure is needed and the Tao remains un-intervened with - basically, if $A$ has infinitary “sums”. For example, as John has discussed in the “quantization and cohomology” seminar recently, if $A$ is one of these idempotent algebras using “min” instead of summation, you are taking an infemum rather than an integral - so no measure is needed. In a different vein, if we categorify things, and we’re taking presheaves valued in a category $A$ with infinite colimits, then again everything works internal to $A$.

Some questions that occur to me, then: what combinations of $A$ and $C$ work automatically? What sorts of $A$ need what kinds of extra structure on $C$ to make the push-forward part work? How “canonical” are the choices? Etc.