Born to do Math 77 — Renormalization in Quantum Theory and Infinities
Scott Douglas Jacobsen: I was watching a short Business Insider clip with Brian Greene from Columbia University and in it he was talking about renormalization.
Rick Rosner: Physics professor?
Jacobsen: A physics professor, specializes in string theory and some fundamental work alongside Witten and Kaku, who are some of the founders in string theory. Witten is known for being something akin to Einstein within that field, where he really blazes new trails to use that cliché.
And one of the points in that Business Insider clip that I was noting is his discussion of infinities. When he was talking about those infinities, he was looking into renormalization in quantum field theory.
I see there are a few types of infinities that are different than that. So, let’s cover two types of things first: one on how renormalization in quantum field theory deals with one type infinity, but how I see the other type infinities having different types and forms and consequences.
Rosner: I haven’t looked at the math of renormalization theory in a while, but basically the equations generate infinities at some points. You need to do tricks that aren’t precisely allowed by the rules of math to cancel out infinities.
And once you do that, you end up with numbers that really accurately predict the values, the things that are being described by the equations in the real world. The equations, once you’ve done these forbidden tricks to them, accurately describe real physics, but you don’t have to assume that this means that the universe itself is cancelling out infinities.
It’s a better way of thinking to think that it doesn’t quite have the right math; it’s good math, but it doesn’t quite encompass all the actual processes that are happening in the world down to the nth degree.
There are all sorts of things that have hidden infinities, but not the world itself. When we’ve talked one of the principles, we talk about that we live in a world that has vast numbers in it, but none of those numbers reach infinity.
And the world is approximated by things that include infinity, for instance, when an object goes from point A to point B in our geometric model, our mental model of traveling from point A to point B has it hitting every single infinite point.
We’ve been taught in school that a number line has an infinity of points along it and not just a countable infinity, but the trans-countable infinity; not just the rational numbers on a number line, but also the irrational numbers, which are uncountable.
There’s so many of them. You can’t even count them using the lowest level of infinity and so you think of things moving along a line and you think they’re hitting an infinite number of points. But we live in a quantum world where position in space isn’t precisely defined.
Things that are happening in a physical framework that’s established by quantum rules; you can’t pin down an object with such precision that you can say that it travels through an infinity of points to get from one point to another. Space isn’t defined that precisely.
There is another set of hidden infinities with counting numbers. The counting numbers seem as finite as you can get; 1, 2, 22, 104… those are finite numbers. But every one of those numbers has an infinite number of digits beyond the decimal place. 223.00000… and the zeros go out to infinity.
One is precisely one to an infinite degree; it’s precisely defined. We just deal with objects in the world as if they are infinitely precise in their unit-ness. If you have two eggs, you have two eggs. 2.000… all the way out to infinity and there are other hidden infinities just in counting numbers, where their infinite precision is actually defined by an infinite series of relationships among each other.
That the prime numbers are distributed along the number line in such a way that they determine the infinite precision of counting numbers. But the deal is those infinities in numbers don’t necessarily reflect actual infinities in the world.
You have one apple. You have 12 eggs. But the oneness of the apple and the twelve-ness of your dozen eggs don’t reflect an infinite precision in the number of things that you have. The world itself is defined by the relationships among the less than infinite particles in the world.
So, objects in the world are highly precisely defined, but not infinitely precisely defined and the oneness of one apple of the dozen-ness of a dozen eggs are abstract characteristics with hidden infinities assigned to the objects that are not infinitely precisely defined because they’re real and they’re in a finite world.
You mentioned off tape of the infinity the ratio of the circumference of a circle, or a wheel, or a tire to its diameter because pi just keeps going for an infinite number of random feeling digits. Its pie is infinitely precise, but when dealing with real objects you can’t infinitely precisely measure or define that ratio.
That ratio is an abstract thing you are assigning to this wheel or tire you’re dealing with; and the wheel or tire is made of atoms and molecules that are held together by Van der Waal’s forces and other electromagnetic intermolecular forces, plus their atoms are held together with nuclear forces and the more electromagnetic forces between the atom, the electrons, and the protons.
But all those particles are imprecisely defined in space. There are probability waves and because they’re imprecisely defined, your tire and the ratios that you’re assigning to it, the ratios can be infinitely precise, but they don’t reflect an infinite precision in the position in space and the shape of the tire and the relationships among its constituent particles.
Everything’s a little fuzzy and the fuzziness reflects a lack of infinity and a lack of infinite precision.
Jacobsen: And then I see that resolves the distinctions of some infinities. In the description of both, renormalizations in quantum field theory as well as infinities of things around infinite digit spans in numbers as well, but in the end that resolves an issue to deal with…
Rosner: We use the tools we have and our tools are symbols. Our mathematical systems are abstract. They contain all sorts of hidden infinities and they work really well when describing a world that is very well defined, but not infinitely precisely defined.
Our tools are not perfectly accurate, if you wanted to perfectly accurately define a tire in space you could do it using quantum mechanical description. For instance, there’s a well-known principle from the beginning physics of the De Broglie matter wave as a wavelength that is inversely proportional to its math.
So, an electron is not weighing much at all. It’s very fuzzy in space. You can’t really pin down an electron very well.
You can pin it down, but only to a limit and the usual example that I’ve seen in physics textbooks is that you compare the matter wave of a baseball and the uncertainty in the space of a baseball to the uncertainty in space of an electron since a baseball weighs like 10 to the 28thor 29th, 10 to the 30thtimes more than electron. A baseball is 10 to the 30thtimes more well-defined in space.
Originally published at borntodomath.blogspot.com on May 4, 2018.