Sean Carroll's everyday equation.

[Jehanne]

Can someone explain this, or know of a resource that does explain it:



It's from Sean Carroll's blog:

http://www.preposterousuniverse.com/blog...-equation/

[Alex K]

I can tell you all about it and then some. It's a mathematical expression called the "Partition Function" or in quantum field lingo the "generating functional" (written in very short form) that describes the dynamics of the particles and fields of the standard model, as well as spacetime via Einstein's General Relativity.
The Integrals in the front (the big S signs) describe taking the average over all possible ways the fields can change between two moments in time (as quantum things do...) . The <Lambda indicates that we don't have the complete quantum gravity

Edit:it's not quite the generating functional, but almost

[robvalue]

Wow!

Well, I recognise a lot of the symbols, and I see the general idea, but not enough to understand how it all goes together. I can see it's an integral performed over four variables. I can explain how simple integrals work, but I can't explain all of this bad boy I'll leave that to Alex.

Integrals are a way of summing continuous data, as opposed to simple addition which sums discrete data.

[ignoramus]

That's weird! Even though it's nearly all Greek. it's still nearly all Greek to me.

[Alex K]

So the idea of this description of quantum physics called path integrals is that - simply speaking - nature takes all kinds of paths between two points in time at once, but they are weighted with different probabilities (for the experts: every possible way the fields can evolve in that period of time gets assigned a complex number called the amplitude)
They all get added together via the functional integrals(which, rob, aren't merely integrals over a real variable, but over entire functions, a problematic thing to define properly)

[robvalue]

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.

[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.

[Alex K]

Now, this was the description of quantum gravity. The remaining terms in the square bracket describe the standard model of particle physics.

It turns out that one can give a mathematical expression for all the particle fields of the standard model called the "action", which plays a similar role as the curvature R, but for the other fields rather than spacetime. For example, (part of) the second expression F_mu nu F^mu nu , in the case of electromagnetism, is simply the electrical field squared minus the magnetic field squared once you disentangle the formalism. It turns out that if you average over all possible field values but weighted with a complex number whose angle is given by that number El.Field^2-Mag.Field^2, what remains after all else cancels out, are none other than the solutions to Maxwell's equations. Again, the other stuff doesn't cancel out exactly, and that's why photons have quantum superpositions and fuzzyness.

The remainder of the square bracket does the same, but for all the other fields, the fermions (yielding the Dirac equation) and the Higgs (yielding the Klein Gordon equation)

[Excited Penguin]

It boggles the mind the stuff you know.

[Alex K]

The constraint k<Lambda on the path integral is an interesting issue. It says that one should not average over *all* ways the fields and spacetime can bend and change, but only those above a certain length scale, leaving out those below a certain scale (e.g. the planck scale). This is why this number Lambda is called a "cut off". This takes care of the fact that we don't know how quantum gravity works near the planck scale - what is ordinarily summarized too simplistically as "we don't have a theory of quantum gravity". The above Functional therefore only yields a consistent description of nature at energies below the planck energy.