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.

Warp Drive Mathematics and Physics

"Scotty!" barked Captain Kirk, "we need more power!"

"I don' know, Cap'n!" replied Scotty, "we're on impulse engines alone!"

This classic exchange comes from the Star Trek series. It takes place in the 23'rd century, a time when there is warp-drive technology. In this post we work out the mathematics and describe the physics behind warp drive.

What exactly is warp drive? According to the series, it is powered by dilithium crystals. Warp drive pulls the starship's destination closer and pushes the ship's starting coordinates further back. Essentially, the spacetime shrinks in front of the starship and stretches out behind it. This implies shorter spacetime wavelengths in front and longer spacetime wavelengths in back. It is possible to derive an equation that models this. Let's begin with the classic Hamiltonian:

Why the Hamiltonian? It is the sum of kinetic and potential energy. We can think of kinetic energy as energy needed to move a particle through space. Potential energy is, of course, stored energy, or, time energy, since a particle moves through time when it is at rest.

Energy conservation suggests that when there is more kinetic energy (more movement through space), there is less potential energy (less movement through time), and vice versa. At equations 4 and 5 below, we show the equivalency of time and potential energy; and, space and kinetic energy:

We can also create a Minkowski diagram:

From the Minkowski diagram we can derive the Lorentz factor (see equation 11 below):

If we start with the Planck mass squared, we can derive and define the spacetime wavelength (lambda) as well as proper time (lambda/c). (See equations 16 and 17):

Using a scale factor (alpha) we can build a second energy equation equal to the one we derived from the Minkowski diagram.

At equation 20 we set the kinetic energy equal to the gravitational energy. Gravitational energy is the warped spacetime that allows the starship to stay at rest, yet, seemingly move through space. It actually moves with space rather than through it. This enables the starship to reach destinations at super-light speeds.

At 21 and 22 we equate the classical Hamiltonian with the energy's quantum representation. The alpha scale factor makes this possible. Also, notice energy would not be conserved without it. When gravitational energy increases, the wavelength (lambda) decreases. This conserves energy on the right side of equation 22, but the left side can become infinite. Dividing the left side by alpha fixes this problem.

Using a bit of algebra we derive equation 29 below:

Equation 29 is the warp-drive equation. We know that massive galaxies move away from us faster than light if they are far enough away. Equation 29's first term contains Hubble's constant and bar-lambda. This is a velocity with long wavelengths or vast distance. The second term contains a velocity with short wavelengths or distance. The greater the difference, the faster the starship moves with space. It's like dark energy pushing from behind and gravity pulling in front. We can use an integral to sum every point in space along the path between the longest wavelength to the shortest:

At 30 and 31 we show how energy is conserved in spite of the fact that gravitational energy seems to have no upper limit. Shorter wavelengths (lambda) offset the longer wavelengths (bar-lambda):

Below we restate equation 26 at 32. From there we show how Einstein's field equations can be derived.

The fact we can derive the field equations confirms that the warp-drive equation is a solution. Caveat: Unfortunately there is still that pesky second postulate of special relativity and the apparent fact that the photons within any system can't ever be observed going faster than light.

Update: "Alcubierre drive shifts space around an object so that the object would arrive at its destination faster than light would in normal space without breaking any physical laws."--Wikipedia.

OK, so how long does it take the Alcubierre drive to shift space around? Let's say the goal is to bring point B closer to point A. If no physical laws are broken, then the minimum time (t) needed is t = (B-A)/c, where c is light speed. Distance B-A = ct, the shortest distance possible. So if a spaceship goes light speed, it will cover the distance just as fast or faster than if you take the time to shift point B closer to point A and then pretend you covered the distance faster than light.

How to Whip Time Dilation During High-Speed Interstellar Space Flight.

In our previous post we showed that infinite energy is not necessary to accelerate a mass to light speed. Click here and read all about it. From the Einstein Field equations we were able to derive equation 10 below. Equation 10 shows that faster-than-light speed is possible in a gravitational field. (Equations 11 and 12 show how energy is conserved.)

Of course the problem with going at, near, or above light speed is the time dilation problem. If you travel at light speed, theoretically time does not pass for you. It's as though you reach your destination instantaneously. The only problem is, the exoplanet you were planning to visit and colonize is long gone, its sun was a supernova eons ago. That's because time passed normally for the rest of the universe. If we are to explore the cosmos, we need to solve this time dilation problem.

To find a solution we need to take a closer look at the Lorentz factor and its limitations. If we create a time unit using Planck's constant, we can derive equation 18:

With another step we derive equation 19. At equation 20 we assume a particle is traveling at light speed. We get eye-opening results at 23 and 24:

Equation 23 shows the particle has infinite energy! This is expected if the particle has mass. But what if the particle is a photon? Most photons don't have or require infinite energy to go light speed. Assuming the particle is a photon and said photon has finite energy, then equation 24 shows that proper time for said photon is greater than zero!

We can spot another problem if we assume the particle is going faster than light. Equations 25 through 29 demonstrate that faster-than-light speed requires less energy than light speed!

To make matters worse, the energy for faster-than-light speed is an imaginary number. Surely we want a real number.

Below are some more problems. First the value of proper time (t') depends on the value of t. The Lorentz equation fails to give us an exact proper time. It just gives a relative proper time. Further, the energy needed to go velocity v is the same for all masses! The Lorentz equation fails to take into account how much mass a particle has.

But that's not all! If you change your energy units from electron volts to Joules, the energy is increased! Just compare equations 30 and 31 above. And, more significantly, the value for proper time (t') changes! Yet, we are talking about the same particle, going the same velocity, with the same energy.

What we want is better precision. We want a true value of proper time (t'). Perhaps we can get that by dividing Plank's constant by the energy (E):

At 33 and 34 we realize we can convert the non-specific proper time into a specific number by using a unit-less conversion factor alpha. Using a Minkowski diagram we can graphically show the conversion factor works:

We derived the Lorentz equation at 36. Equation 37 demonstrates that time (t) does not know how big or small it should be, so why not set it to t/a? That way the precise proper time we get from equation 34 agrees with the Lorentz equation.

OK, let's take what we discovered and derive a precise way of finding proper time in a gravitational field with energy GMm/r. We derive equation 51 below:

At equations 52 and 53 we steal an idea from quantum physics and apply it to conserving the energy of a star:

At 53, even if the gravitational energy is infinite (a black hole?), time (t') is zero and the finite energy of the the original star is conserved. So far, so good. But notice the proper time will never go to zero unless energy is infinite. Is this really true? Photons allegedly experience zero proper time with their finite energy. To resolve this contradiction, we need to understand the nature of time better. We can do this by building a quantum clock. First, here are the variables:

At equation 54 we put the gravitational energy into one mass variable. At equation 55, momentum (rho) is conserved--an increase in mass (m) causes a decrease in velocity (v). At 56, momentum (p) is not conserved--an increase in mass does not change velocity (c). At 57 we give time (t) and frequency (omega) definite values by dividing Planck's constant by ground-state energy. At 58 we take the ratio of conserved momentum to non-conserved momentum. (Note when mass increases, velocity v decreases and proper time decreases.) At 59 we cancel the masses. At 60 and 61, we convert the velocities into oscillators. Because oscillators are cyclical, they make excellent little clocks.

At equation 62 we see that when velocity v slows, the wavelength lambda shortens. So we discover a correlation between more mass, shorter wavelengths, and slower time. At 63 through 65 we prove that the wavelength is proper time t' multiplied by light speed c. At 66 through 68, we set the final parameters for our quantum clock.

Below is a diagram of the quantum clock. The clock's imaginary hand is radius mu. When there is more mass or energy, it goes around the clock slower. To stop the clock requires infinite energy. Thus, it appears to be true! Photons with finite energy experience non-zero proper time.

To be sure we are right about proper time, let's take another look at the Lorentz factor equation for mass. Photons have zero mass and have velocity c (see equations 69 and 70). At 73 we get a relative mass (m') that is between zero and infinity. At 74 we convert the relative mass into photon energy divided by c^2. Thus we confirm the photon's energy can be less than infinite.

At 75 we convert the photon's frequency into the reciprocal of its proper time (t'). If we convert the photon's zero mass into photon energy, variable t would need to be infinite (see equations 76 and 77). At 78 we make some substitutions and with a few more steps we get 81 and 82.

Looking at 81 and 82 we see that a photon's proper time does not have to be zero. And, equations 83 to 88 confirm that infinite energy is required to have zero proper time.

Thus our formula for calculating a precise proper time is correct. Our final formula for a gravitational field is at equation 90:

What does equation 90 tell us. It tells us that proper time never goes below zero even if velocity squared Gm/r goes to infinity! This means if you have a twin, if he/she stays on earth, and you traveled to the nearest star at, say, 24 times the speed of light (using a gravitational warp drive), your round trip would take about four months (instead of years at light speed). Your twin will only be four months older than you. The age gap decreases if you travel at even faster speeds. Plus, you can reach that exoplanet mentioned earlier in a timely fashion. Of course this is more science fiction than science. It is not clear how one can travel faster than light without violating the second postulate of relativity which states that all observers must see the photons in your spaceship going light speed in a vacuum.