# Integrating distributions over submanifolds

Distributions may be "integrated against" (i.e. evaluated on) test functions, the following notation (for $$T$$ a distribution) is fairly common: $$T(f) = \int T(x) f(x)\, \mathrm{d}x$$ I'm wondering if anything is known about distributions which may be "integrated" in some sense over, say, smooth hypersurfaces (or other submanifolds) in the ambient space, so as to give a meaning to expressions of the form $$\int_\Sigma T(x)\,\mathrm{d}x$$ for $$\Sigma$$ a smooth hypersurface. These objects would presumably exist "between" functions and distributions: not necessarily regular enough to have values at points but still regular enough that the smearing provided by test functions is not required in all directions.

I can only motivate this question by appealing to QFT: in the Wightman mindset, one would expect the energy-momentum tensor $$T_{\mu\nu}(x)$$ to be a distribution valued in some locally convex algebra of unbounded operators. Then there is a general principle that this should "generate" the momenta $$P_\mu$$ (the generators of the translations in the representation of the Poincaré group) in the sense that $$P_\mu = \int_{x^0=0}T_{\mu 0}(0,{\bf x})\,\mathrm{d}^3{\bf x}.$$ Of course the formal expression on the right isn't immediately defined since the would-be integrand is a distribution. In fact this same problem threatens to arise whenever we have a conserved charge and try to write it as the integral of a current.

Moving to the simpler case where our distributions are scalar-valued (i.e. just standard distributions), we come back to the beginning of this question, wondering about integrals of distributions over hypersurfaces. Even though this seems like a fairly obvious issue, I have only seen it discussed in a paper by Verch, precisely in a QFT context.

• Thanks for the suggestion, but I'm not sure that this is what I'm after since as far as I can tell currents still need a form defined on the whole manifold $M$ as their argument. I'm looking for something closer to ordinary distributions which, however, can be evaluated also on smooth functions (perhaps with compact support) $f\in C^\infty(\Sigma)$ where $\Sigma\subset M$ is any smooth hypersurface. Note that since $\Sigma$ is not fixed, this does not just correspond to distributions on $\Sigma$.