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The path integral in quantization can be understood – to the extent and in the cases that it can be understood at all – as an integral transform induced from a span that is given by hom-spaces. According to the general reasoning of integral transforms on sheaves this means that it is given by a pull-push operation through spans.
The archetypical example of the path integral is that for the sigma model that describes the quantum mechanics of the particle propagating on the target space $X = \mathbb{R}$ – the line.
In a finite approximation, one considers for $N \in \mathbb{N}$ paths consisting on $N$ discrete steps: let $I_N := \{0, 1, \cdots, N\}$ be the set of $N+1$ elements. Regarding this as an abstract discrete cobordism of $N$ steps, we consider the cospan
where $in : * \to I_N$ is the inclusion of the first elements 0, and $out : * \to I_N$ the inclusion of the last element, $N$.
The mapping space $X^{I_N} \simeq X \times X \times \cdots \times X$ is the space of paths – the space of paths in $X = \mathbb{R}$ consisting of $n$ linear steps. Homming the above cospan into $X$ produces the span
In the canonical coordinates on $X^{I_N}$ a path $\gamma \in X^{I_N}$ is parameterized as
The action functional that encodes the dynamics of the free particle on $X$ is the differential form on $X^{I_N}$ given by
Denote by
the pullback of differential forms along $X^{in}$, and by
the “pushforward”: the fiber integration of differential forms.
Then then standard path integral for the particle is given by
pulling back functions along $X^{in}$;
taking their wedge product with $\exp(i S)$;
pushing the result forward along $X^{out}$.
This gives the map
that acts by
This is the standard expression for the path integral of the particle on the line, at the approximation of $N$ discrete steps.
On the singular homology of smooth manifolds and other topological spaces, pullback operations can be defined by Thom isomorphisms and fiber integration (“Umkehr maps”). Together with the canonically defined push-forward of singular cycles, this yields a definition of pull-push transformations on singular homology.
It was realized in (CohenGodin) and (Godin) that such pull-push operations define a 2-dimensional HQFT whose space of states is the singular complex, and that the correlators of the thus defined FQFT are the Chas-Sullivan string topology operations. See there for more details.
Every perfect derived stack in dg-geometry forms the target space for a pull-push transform on the stable (infinity,1)-category of quasicoherent sheaves and yields a 2-dimensional TQFT. For details on this see geometric infinity-function theory .
The pull-push nature of the path integral was originally amplified somewhat implicitly in
Quantum groups from path integrals (arXiv:q-alg/9501025)
Higher algebraic structures and quantization (arXiv:hep-th/9212115)
and fully explicitly in
The description of string topology operations as an HQFT defined by pull-push transforms was originally realized in
Ralph Cohen, Veronique Godin, A Polarized View of String Topology
Hirotaka Tamanoi, Loop coproducts in string topology and triviality of higher genus TQFT operations (2007) (arXiv:0706.1276)
A detailed discussion and generalization to the open-closed HQFT in the presence of a single space-filling brane is in
For more see the references at motivic quantization.