Sean Carroll's everyday equation.

[Alex K]

Let's start with the first term in the square bracket, the letter R.

This is a number which stands for the curvature of spacetime (the 4d- equivalent of the Gaussian curvature called the Ricci scalar). Now, the first path integral [Dg] averages over all possible ways spacetime can wobble between two points in time, and they are all assigned a complex number (think of them as arrows in a plane) whose angle is given by the total curvature of that particular way spacetime could wobble and bend. One can show that most of those ways cancel each other out because for every one there is another with a 180° angle relative to it, and what remains are configurations in which the total curvature is minimal.

Those are the solutions you would also get by solving Einstein's equations. But all the other strange ways in which spacetime could bend don't cancel out *exactly*, and this is where the quantum fuzzyness comes in.

Wow! Okay. I can't remember whether I covered that, it's been so long since I did my studies. I remember doing integration around a path in the complex plane.

I think many mathematicians don't touch path integrals with a ten foot pole because everything diverges and the continuum limit is fishy etc., but despite the difficulties to make them mathematically rigorous in the continuum limit (meaning you really integrate over all possible real functions) they work perfectly.