Re-normalizing Feynman Diagram Amplitudes in a Non-arbitrary Way

Quantum electrodynamics (QED) is perhaps the most precise and successful theory in all of physics. There is, as I've mentioned in previous posts, a peculiar characteristic within the theory's math: infinities keep cropping up. In this post we deal with the infinities that appear in the math when calculating Feynman-diagram amplitudes.

If you read the previous post, you recall Paul Dirac having a problem with re-normalization. He said, " 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."

Let's see if we can re-normalize Feynman-diagram amplitudes in a non-arbitrary way. First, we define the variables:

Next, let's do a typical textbook calculation and reveal how the infinity arises. Below is the Feynman diagram we will be working with. A and A' are particle and anti-particle, respectively:

The diagram progresses from bottom to top. There are two vertices. The particle (A) and anti-particle (A'), with momenta p1 and p2, meet at the first vertex and annihilate each other. A boson (B) is released. It has an internal momentum q. At the top vertex it creates a new particle (A) and anti-particle (A') with momenta of p3 and p4.

To find the amplitude M, we need a dimensionless coupling constant (-ig) for each vertex. This coupling constant contains the fine structure constant (see equation 1) There are two vertices, so we square the coupling constant (see equation 2):

To conserve momentum we use the Dirac delta function (see 3 and 4). Momenta p1 and p2 are external momenta heading in, and q is the internal momentum heading out (see 3). At 4, q is incoming momentum and p3, p4 are outgoing momenta.

For boson B's internal line we need a propagator, a factor that represents the transfer of propagation of momentum from one particle to another:

We integrate over q using the following normalized measure:

We put all the pieces together to get equation 7. We begin solving the integral at equation 8:

We can solve the integral more easily if we set q equal to p3 and p4. Using some algebraic manipulation, we arrive at equation 11:

Note that at equation 11 we have a red portion and a blue portion. To get the solution at equation 12, we simply throw away the blue portion! We can just imagine Dirac rolling over in his grave. Further, equation 12 is supposed to be the probability of the event illustrated in the Feynman diagram. But probabilities are dimensionless numbers. This probability has dimensions of 1/momentum squared!

Experiments may show that equation 12 is correct within a tiny margin of error, but can the math that leads to it be more sloppy and arbitrary? Sure it can. But let's try to make it less sloppy and arbitrary. We can start by changing the normalized measure:

Next, we can recognize that momentum is conserved, so the Dirac delta functions will equal 1:

As a result, a lot of the stuff we arbitrarily threw away is now properly cancelled. We end up with equation 19:

If we evaluate the integral, we get an infinity (see 20). The good news is we can convert that infinity to the expression at 21. If we introduce a gamma probability amplitude factor, the infinity becomes a finite number at 21b.

We make a substitution at equation 22:

If we throw away the blue section at equation 22, it makes logical sense when you treat that section as all the probable outcomes that could have happened but didn't happen when the observation was made. The observer saw the expression outlined in red--the eigenvalue. That eigenvalue is paired with what is supposed to be its probability amplitude. Notice if we multiply this amplitude by the gammas in the summation, we get the probability amplitudes for all the eigenvalues that add up to infinity. As a result, the right side of equation 22 is no longer infinite. If we take the sum of squared probability amplitudes multiplied by their respective eigenvalues we get the expectation value.

The expectation value is not what we want, however. We want the actual observed value outlined in red, so we ignore "what could have happened but wasn't observed" outlined in blue. This approach is logical instead of arbitrary.

Now, let's see what we can do to fix the dimension problem. At 23 we pull out a momentum unit and set it to one. This leads us to a new solution at 24:

At 24 we end up with an eigenvalue multiplied by a probability amplitude--and the dimensions come out right. The eigenvalue fits nicely into Einstein's energy equation:

So we have a solution for four-dimensional spacetime. For three-dimensional space, we get equation 27:

At 27, the eigenvalue is just q, the internal momentum of the Feynman diagram. The probability of q is the same as the Feynman-diagram event. We obtain the probability by squaring the phi amplitude:

In conclusion, if you encounter an infinity in QED math, it is OK to discard it. It's not really arbitrary to do so, because you are only interested in what you observed. You are not interested in an infinite number of probable events you didn't observe in your experiment.

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.

Adding Gravity to the Standard Model Lagrangian

Here is one version of the Standard Model Lagrangian. Click on the image below to learn more details. Lagrangian L =

To read a pretty good outline re: the Standard Model Lagrangian, click here.

Now you have probably been told a gazillion times that the Standard Model does not include gravity, that Einstein's field equations are incompatible with the big messy equation you see above. Let's have a look at the field equations:

The energy-stress tensor (Tij) provides a clue to how we can unify gravity with the Standard Model. Its units are energy density or energy over a volume (V). The Lagrangian has units of energy. Hmmmm ... if we contract the tensor indices and do a little algebra, we get equation 5 below:

Equation 5 shows that the scalar or zero-order tensor T is equivalent to the Lagrangian (L) divided by a volume (V). That leads us to equation 6 below:

We can now see the relationship between gravity and the other forces. If we do a little more algebra, we get an interesting result:

At equation 9 we add the newly-formed gravity Lagrangian to the Standard Model Lagrangian. Doing this yields the ground-state vacuum energy--and this quantity is consistent with the Wilkinson Microwave Anisotropy Probe measurement. So if we add gravity to the Standard Model as illustrated above, we not only get a result that is mathematically consistent, but also consistent with observations.