nLab path integral as a pull-push transform



Functorial quantum field theory


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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 quantum particle

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=X = \mathbb{R} – the line.

In a finite approximation, one considers for NN \in \mathbb{N} paths consisting on NN discrete steps: let I N:={0,1,,N}I_N := \{0, 1, \cdots, N\} be the set of N+1N+1 elements. Regarding this as an abstract discrete cobordism of NN steps, we consider the cospan

I N in out * *, \array{ && I_N \\ & {}^{\mathllap{in}}\nearrow && \nwarrow^{\mathrlap{out}} \\ * &&&& * } \,,

where in:*I Nin : * \to I_N is the inclusion of the first elements 0, and out:*I Nout : * \to I_N the inclusion of the last element, NN.

The mapping space X I NX×X××XX^{I_N} \simeq X \times X \times \cdots \times X is the space of paths – the space of paths in X=X = \mathbb{R} consisting of nn linear steps. Homming the above cospan into XX produces the span

X I N X in X out XX * X *X. \array{ && X^{I_N} \\ & {}^{\mathllap{X^{in}}}\swarrow && \searrow^{\mathrlap{X^{out}}} \\ X \simeq X^* &&&& X^* \simeq X } \,.

In the canonical coordinates on X I NX^{I_N} a path γX I N\gamma \in X^{I_N} is parameterized as

γ=(x 0,x 1,,x N). \gamma = (x_0, x_1, \cdots, x_N) \,.

The action functional that encodes the dynamics of the free particle on XX is the differential form on X I NX^{I_N} given by

exp(iS):=exp(i i=1 Nm2(x ix i11/N) 2)dx 0dx 1dx N1Ω N(X I n). \exp(i S) := \exp(i \sum_{i = 1}^N \frac{m}{2}(\frac{x_i - x_{i-1}}{1/N})^2) d x_0 \wedge d x_1 \wedge \cdots \wedge d x_{N-1} \in \Omega^{N}(X^{I_n}) \,.

Denote by

(X in) *:Ω (X)Ω (X I N) (X^{in})^* : \Omega^\bullet(X) \to \Omega^\bullet(X^{I_N})

the pullback of differential forms along X inX^{in}, and by

(X out) *:= X I N/X:Ω (X I N)Ω N(X) (X^{out})_* := \int_{X^{I_N}/X} : \Omega^\bullet(X^{I_N}) \to \Omega^{\bullet - N}(X)

the “pushforward”: the fiber integration of differential forms.

Then then standard path integral for the particle is given by

  1. pulling back functions along X inX^{in};

  2. taking their wedge product with exp(iS)\exp(i S);

  3. pushing the result forward along X outX^{out}.

This gives the map

(X out) *exp(iS)()(X in) *:Ω 0(X)(X in) *C (X I n)exp(iS)Ω N(X I N)(X out) *Ω 0(X) (X^{out})_* \circ exp(i S) \wedge (-) \circ (X^{in})^* : \Omega^0(X) \stackrel{(X^{in})^*}{\to} C^\infty(X^{I_n}) \stackrel{\exp(i S)\wedge}{\to} \Omega^{N}(X^{I_N}) \stackrel{(X^{out})_*}{\to} \Omega^0(X)

that acts by

(ψC (X))(x X I N/Xψ(x 0)exp(im2 i=1 N(x ix i11/N) 2)dx 0dx N1). (\psi \in C^\infty(X)) \mapsto \left( x \mapsto \int_{X^{I_N}/X} \psi(x_0) \exp(i \frac{m}{2} \sum_{i = 1}^N (\frac{x_i - x_{i-1}}{1/N})^2 ) d x_0 \wedge \cdots \wedge d x_{N-1} \right) \,.

This is the standard expression for the path integral of the particle on the line, at the approximation of NN discrete steps.

Gromov-Witten theory

String topology

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.

Geometric \infty-function theory

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

and fully explicitly in

The description of string topology operations as an HQFT defined by pull-push transforms was originally realized in

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.

Last revised on August 8, 2013 at 11:36:43. See the history of this page for a list of all contributions to it.