Bounding the Feynman Integral

Feynman's Integrals are now more-tractable. The Turtle relates them to high-energy particle physics, where C=1 such that velocity-values are relativistic. Mostly he's talking quantum. Might we generalise this past quantum?

Stefan Weinzierl wrote the book on these integrals, which book is now free and - well, might be worth it at that price, at least for the later chapters on how to solve them. Up to that point it is almost comprehensible!

Chapter 2 gets everyone up to speed on the usual 4-dimensional spacetime, rather timespace with the spatial dimensions not necessarily numbering three.

There's some metric here called the Minkowski where the first dimension is +1 with the others all -1 (or -1, 1,1,1&c). This reminded me of vectors on a computer where y increments downward. Also of quaternions where three dimensions are spatial and the last dimension some function of an angle. That last dimension as a function of change is a delta-V so, I say, be a function of t. Anyway whenever needed, you can transform the Minkowski to Euclid. Maybe with a quaternion.

Mostly chapter 2 teaches graph theory. You are at point a in space and time. After the collider-experient, you get b. What's b? This book concerns the connected-graph, so that's the "simplification". Feynman adds: an orientation (thus our graph becomes an oriented graph), a D-dimensional vector p (the momentum) and a number m (the mass). Turn that graph into an equation, says Feynman. That means a horrendous integral. We can't even solve an integral for a Gaussian or an ellipse. So, turn it into something a computer can solve. Here this means: turn it all into differential-equations. Profit!! So, back to the integral: if you want a single number at the end, you need boundaries on that wiggly line on the left. Apparently this is the step where the gnomes stole everyone's underwear.

Yan-Qing Ma and Zhi-Feng Liu have solved the boundary problem. It's linear algebra - and a lot of it. Physicists will need a computer for this too. FASTER QUIETER 4/1/23 Analog for diffy-Q.