Finding the Flaw that Necessitates Renormalization

Here's what Paul Dirac had to say about renormalization:

"Most physicists are very satisfied with the situation. They say: 'Quantum electrodynamics is a good theory and we do not have to worry about it any more.' I must say that I am very dissatisfied with the situation, because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small – not neglecting it just because it is infinitely great and you do not want it!"

So let's see if we can find the flaw that causes infinity to appear in equations and necessitates the ad hoc method of neglecting it in an arbitrary way. First, let's define the variables:

Consider the integral below. It adds up the Coulomb potential energy between two particles. The result is infinity.

If the location of each particle is uncertain and/or there is a superposition of states, we might assume, that at each location there is some energy, and, if we add up each of those energies from zero to infinite r (the distance between the particles) we end up with infinite energy!

Let's assume, arguendo, there is infinite energy. We could get that result if we take the average energy and multiply it by infinity:

Of course, when we measure or observe the two particles, we find the energy is not infinite. So why did the math give us infinity? Well, notice there were no probabilities involved when we solved the integral.

Suppose we assume that, since there is an infinite number of states the particles could be in (due to the distance apart (r) being anywhere from zero to infinity), there must be an infinite number of probabilities. Those probabilities must also add up to one. The average probability is therefore 1/infinity:

If we multiply the average energy by infinity, we get infinity, but if we multiply that by 1/infinity, we get the average energy or expectation value:

This is the same result we would get if we summed each probability and each energy eigenvalue:

When we observe and measure the energy, we get the different eigenvalues. The average energy we will observe is the expectation value. So, it makes perfect sense to multiply the absurd infinity by the average probability. After all, we want our math to agree with nature.

Now, let's consider an example from QED (quantum electrodynamics). We want to calculate the total vacuum energy or ground-state energy. One typical way of doing this is to integrate over k-space. We begin with equation 8 below and work our way to equations 14 and 15 (note: variables including but not limited to Planck's constant are set to one):

At 14 we see the ground-state is infinity. Ridiculous! At 15 we renormalize by subtracting the infinity from the total energy (H). This is exactly the kind of thing Dirac complained of, so let's take what we've learned above and apply it to this situation. We know we can get infinity by multiplying the average observed ground-state energy by infinity:

Even though we are dealing with a field instead of individual particles, let's quantize the field by imagining it is made up of individual particles--each with it's own energy state and finite eigenvalue, and, more importantly, each finite energy has a probability associated with it. Also, the totality of these particles, at any point in time, have an overall state with a probability associated with it. We can imagine an infinite number of possible particle states and overall states with finite energies adding up to infinity, so there must be an infinite number of probabilities that add up to one. The average probability is, once again, 1/infinity:

We get the average ground-state energy if we multiply the infinity by the average probability:

Note that equations 20 and 22 are in agreement. The solution is not infinity, but the expectation value or average ground-state energy? Not quite. The solution is definitely not infinity. Additionally, we are not interested in knowing the average energy. We want to know the total energy, say, in a given volume V.

So the next step is to divide the average energy by a unit volume (Vu):

Now we have an energy density. According to WMAP, the vacuum energy density is approximately what we have at equation 24. At equation 25 we multiply the density by the volume we are interested in to get the total "finite" ground-state energy.

Equation 26 shows the energy above the ground state is no longer the total Hamiltonian (H) minus infinity, but the total energy minus a finite vacuum energy.

Proving the Dirac Delta Function, Etc.

In this post we prove the Dirac Delta function and its sampling property. Here are the variables we need:

So we ask, is equation 1 below true?

To prove equation 1, we first define the Dirac Delta function:

And it helps if we draw a diagram:

In the diagram above, we have a line segment along the x axis ranging from minus infinity to infinity. If we let the value of epsilon go to infinity (or assume an infinite number of points along any epsilon distance) we can make a substitution and create the diagram below:

At line 3 we create a new integral that is equivalent to the one we started with. From there to line 5 we show that the integral does indeed equal 1.

Now let's prove the sampling property. Suppose a particle is located at position 'a' instead of zero. Is the value of the integral f(a)?

We assume the following are true:

We make some changes to equation 6 to get 7:

We draw a new diagram to account for the fact that x equals 'a' instead of zero:

We can now derive equation 8. From there we derive the calculus difference quotient or derivative formula at line 14.

Line 15 above confirms the Dirac Delta function's sampling property.

Why Does Squaring a Wave Amplitude Yield a Probability?

Perhaps you heard the story (that's code for unsubstantiated rumor) where Paul Dirac, a legend in the field of quantum physics, woke up one morning, put on his trousers, put on his shirt, put on his socks and shoes. He then stepped into the shower, forgetting he had already dressed. His mind was elsewhere, but the brisk chill of cascading water drenching his clothes brought him back. "That's it!" he said. "Square the amplitude and you get the probability!"

He was, of course, referring to the fact that you can take the inner product of an eigenvector (or ket) with its complex conjugate (bra) and get a probability of a particle's position, momentum, or whatever. It's a pretty cool trick and it works. The question is why? Why does squaring a wave amplitude yield a probability? Below is a mathematical equation that I thought up while I was taking a shower (with my clothes on):

In the numerator you may recognize Euler's identity. Basically, the numerator is the sum of all possible wave amplitudes. The denominator is a normalization factor: it ensures that when you calculate the inner product of the equation's right-side expression with its complex conjugate you get one. One is a good number to get, since it is the sum of all probabilities. Most importantly, the equation shows the connection between wave amplitudes and probabilities.

Below is a visual aid that I hope will clarify the connection. It is a well-known fact in the subject of trigonometry that sine squared plus cosine squared always equals one. This fact can be used to model not only waves, but probabilities as well, since the sum of all probabilities also equals one. However, to reach a total of one, one must divide the wave amplitude by X1 in the case below where n is equal to one, i.e., where there is only the sum of one Euler's identity multiplied by a coefficient X1.

The lower part of the visual aid shows the inner product between the bra and ket vectors.

As you can see in the diagram below, the sum of squared wave amplitudes equals one, and probabilities P(A), P(B) add up to one.

This system works well if you have only two probabilities: cosine squared can represent one probability and sine squared can represent the other, but what if there are three or more probabilities? Well, that's why I put together the equation we started with. Let's say there are three probabilities, and they must add up to one. In such a case we need to raise n to 2:

There are now three wave amplitudes. When you square them, add them, and divide by the normalization factor (x1^2 + x2^2), you get one. You also get one when you calculate the inner product of the bra-ket vectors:

And, of course, the probabilities also add up to one: