Why There is Something Rather Than Nothing

Why is there something rather than nothing? Mathematicians use zero to represent nothing. The concept of zero goes way back to ancient Egypt. The hieroglyph known as 'nfr' indicated "emptiness." Aryabhatta, an Indian mathematician, introduced zero in the 5th century AD.

I was first introduced to zero in the first grade. Our teacher, Mrs. White, taught us that 2 - 2 = 0. Later in life I was introduced to the Lagrange equation, where potential energy is subtracted from kinetic energy. Of course if kinetic energy equals potential energy, you get zero on the right side of the equation.

It's safe to say that all my life I've been brainwashed to believe you can have something, take it away, and end up with nothing: zero. And I'm not alone; philosophers have asked, "Why is there something rather than nothing?" as if nothing were a viable possibility.

The closest thing to nothing measured by the Wilkinson Microwave probe is about 6E-10 joules per cubic meter. The energy density of the vacuum of space is close to zero, but not quite. So the question arises, what is the probability of ever having nothing? In the case of the Lagrange equation, what is the probability that a universe could have the same amount of kinetic energy as potential energy?

Suppose we label kinetic energy "positive," and potential energy "negative." Let's assume all energy is made up of discrete quanta. We take all that quanta (an infinite amount) and put it inside a big cosmic hat. We reach in and pull some out. What is the probability we pulled out the same amount of positive quanta as negative quanta? Or what is the probability we pulled out zero quanta?

The probability we pulled out zero quanta is easy to figure out. Since there is an infinite number of quanta, the probability is 1/infinity--which is virtually zero. So we are virtually guaranteed to have more than zero quanta outside the hat. To have zero net energy, we need an even number of quanta, and equal amounts of negative and positive quanta.

If the quanta number is odd, the probability of netting zero is zero. For example, if we pulled out five quanta from our cosmic hat, the closest combination to zero would be three positives and two negatives--or vice versa. If we pulled out four quanta, there's a possibility we could have two positives and two negatives. The equations below enable us to calculate the probability of netting zero:

Equation 2) takes into account that there's a 0.5 probability that the number of quanta (n) could be even, so the probability that is calculated in equation 1) is cut in half. Thus, in equation 2's denominator, there is 2^(n+1) instead of 2^n.

The above equations show that the larger the number of quanta (n), the less likely we will have zero net energy. But shouldn't the net energy get closer to zero as we add more quanta to the mix? If we flip a coin, we get heads or tails. If we assign a value of minus one to heads and plus one to tails, we get plus one or minus one--never zero. But suppose we toss a trillion coins? We should get an equal amount of heads and tails (or pretty close).

We know if we increase the number of coins, the variance or deviation from zero is reduced.

Equations 3) and 4) clearly show, that as the number of coins (or quanta) increases, the smaller the variance becomes. The sum of coins or quanta converge to zero. This seems to contradict our earlier finding via equations 1) and 2). Let's crunch some numbers and put the data in a couple of tables to see what we get.

The first table above contains data for odd numbers of quanta. We see that the variance (v) decreases as expected. It gets closer to zero as more quanta are added. (This is also true in the second table for even numbers of quanta.) We see in the last two columns the probability of having exactly zero net energy is zero due to an odd number of quanta.

The second table shows an interesting paradox: As the variance decreases, so does the probability of having exactly zero net energy. This paradox is illustrated in the graphs below:

So, the closer we are to zero, the less likely we will have precisely zero--this is why there is something rather than nothing.

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:

How Uncertainty is the Root of All Certainty

If you roll a pair of dice, you never know what you will get. You might get anything from snake eyes to twelve. But what happens when you roll a trillion pairs of dice all at once? Strangely enough, you get seven-trillion or a number very close to that figure when you add up all those dice. The uncertainty has become a certainty.

When you calculate the average dice throw, the number you always get is seven. Seven is the mean number, and two and twelve are the extreme numbers. Even if you were to eliminate all the numbers between two and twelve, you would still have an average of seven.

(2 + 12)/2=7

There is an old theorem in quantum mechanics (Ehrenfest theorem) that says that all our laws of physics that give us certain results are rooted in uncertainty. The “certain result” is called the expectation value. You calculate the expectation value by simply adding up and taking the average of all the uncertain results.

At small scales, matter behaves in very strange and unexpected ways, but at large scales, matter behaves as we expect: it follows the laws of Newtonian physics. It behaves much like a pair of dice on the quantum scale, and behaves like a trillion pairs of dice on our old, familiar scale.