Today's buzzword: nonrenormalizable

 kw: science, philosophy of science, explanations

In a post last week I reviewed The Grand Biocentric Design: How Life Creates Reality, by Robert Lanza, MD and Matej Pavšič, PhD, with Bob Berman. Much of the post expressed my objections to the theory that "life creates reality", a theory based on a dramatic over-extension of the Copenhagen Interpretation of quantum theory (I disagree also with Neils Bohr, who concocted CI). 

I decided now to add a brief discussion of a point made several times in the book, that the version of quantum gravity theory these authors espouse is "non-renormalizable". The word is never explained. Perhaps I can at least give the public some understanding of "normalization" and "renormalization".

A problem arose in the early 1900's when scientists began to solidify the mathematics of quantum electrodynamics (i.e., the dynamics of charged particles in the quantum realm). If equations for the behavior of an electron (for example) were to be solved, partway through the solution one had to contend with expressions that divided by zero, leading to "infinities". Eventually, mathematical methods were developed that "renormalized" these equations, so that the term(s) leading to division by zero could be removed before one actually had to calculate the result.

To understand renormalization, it helps to first understand normalization, which is used to derive the basic equations of differential calculus. The term "derivative" means a function, derived from another function, that expresses the slope at any point. It is the basis for a large family of functions that are needed to optimize functions and, used "in reverse", to calculate areas and volumes (among many other useful summation operations). 

I will illustrate by deriving the "first derivative" of two functions. The first is a basic parabolic function, y = 5x². This states that, in an x-y coordinate system, the value of y is found by squaring x and multiplying by 5. Thus, for x = 3, y = 5*3*3 = 45. The second is a simple quartic, y = x⁴. For this one, when x = 3, y = 3*3*3*3 = 81. Both functions are very useful in mechanics: the position of an object falling in a uniform gravitational field is expressed by a parabolic function, and the energy radiated by a heated body is related to the temperature by a quartic function.

This illustration shows both functions graphed in the domain x = [-1,2].

NOTE: There are several notations used in calculus, because of the complicated history of its discovery three-plus centuries ago. Here I use notation based on that of Gottfried Liebniz (d. 1716).

The mathematical slope of either of these curves is dy/dx, and one can approximate it by calculating the functions at x and x+dx, for very small dx. But the exact value can only be found when dx = 0!

Below is the derivation for the parabolic equation as I learned it in high school.

"Lim" means the limit of the expression in brackets, as the term in parentheses is satisfied. We do the derivation using dx and dy as algebraic variables.

The three lines in the middle expand the function. The following line is the "normalization": dividing by dx removes dx from one of the terms in the equation.

Then, as shown in the last line, when we set dx to zero, the term that contains it vanishes, leaving us with a function in x only. This function is the first derivative of function y1.

Let's drive the point home by performing the same operation on the quartic function:

This time, multiplying out the function results in a larger number of terms. As before, the calculations are shown in the three-line cluster in the middle. The next step, normalization, divides by dx to yield one term that doesn't include dx, and three others that include it. Setting dx to zero yields the first derivative of function y2.

To summarize: Normalization is a mathematical process of deriving expressions that would contain a division by zero if carried out with numbers, but then dividing out the term that will become zero, so that a perfectly calculable function remains.

We can do a simple verification to see how these derivatives perform. Here is a table of the two functions, calculated from the values of x as given. The graph above was produced from these numbers.

In the graph above the green lines are anchored to x = 1.5. Let's start with dx = 0.1 and calculate y1 and y2 at x = 1.6, subtracting values found in this table at 1.5:

y1(1.6) = 5*1.6*1.6 = 12.8; dy1 = 12.8 - 11.25 = 1.55; dy1/dx = 15.5.

y2(1.6) = (1.6)⁴ = 6.5536; dy2 = 6.5536 - 5.0625 = 1.4911; dy2/dx = 14.911.

We can shrink dx to 0.01, so we calculate at 1.51. The results are then 

dy1 = 11.4005 - 11.25 = 0.1505; dy1/dx = 15.05.

dy2 = 5.19885601 - 5.0625 = 0.13635601; dy2/dx = 13.635601.

These numbers seem to be closing in on 15 and something more than 13.

Now let's find exact slopes using the two derivatives:

Slope #1 = 10*1.5 = 15.0

Slope #2 = 4*(1.5)³ = 13.5

It's possible to use smaller and smaller values of dx to get a converging series, from which we can get pretty accurate results. However, the mathematics of the derivative ensure that the values calculated by them are exact.

Based on all that, what is Renormalization? Here I must get conceptual because the functional expressions are enormous, and people got Nobel prizes for figuring them out. Calculus, which is already based on Normalization, was used to determine the functions needed by quantum electrodynamics. The equations were labored over until a version of each derivation could be performed that "divided out" the expressions (analogous to dx but much more elaborate) that caused trouble. This second dividing-out process is renormalization.

In one of his books, Richard Feynman tells of being asked, at the Nobel Prize ceremony, "What did you do?" He answered, "Buddy, if I could tell you that in one minute, it wouldn't be worth a Nobel Prize." Renormalization was part of it.

So what is Non-renormalizable? The authors of The Grand Biocentric Design make much of their contention that a quantum theory of gravity cannot be renormalized. No such theory has yet been fully developed, as the authors admit. Since we don't know what form such a theory will take, we (and they) don't have a way to determine if their contention is anywhere close to being correct.

On a plane ride several years ago I sat with a young man who, it developed, is a cosmologist. I said, "I have a question I've been hoping to ask a cosmologist, if you're willing to give it a crack." He agreed. I asked, "If gravity is a quantum phenomenon, that means there is a gravitational quantum, a 'graviton'. The most powerful sources of these gravitons are black holes. What is it, for a black hole or for any other massive body, that emits the gravitons? If they start 'inside', where the mass is, how do they get out?" He said, "That's two questions, but it's worth some thought." After a half hour of silence he said, "By analogy to photons, which are emitted when charged particles such as electrons are accelerated or make a quantum jump from one orbital to another: The motion of an electron may cover at most a nanometer or two, particularly in an orbital transition. The photon that is emitted has a wavelength of a few hundred nanometers or more. We don't yet know what is the wavelength of a typical graviton. But I think gravitons are emitted over an area larger than the typical black hole, not from inside it. That's the best I can say on short notice." That was pretty good! I reckon it will be a while before a better answer is in the offing.