Something vs. Nothing vs. Mixed States

Yes, we could ask, "Why is there something rather then nothing?" However, if we take quantum physics into account, we shouldn't limit ourselves to pondering something or nothing--we also need to ponder mixed states. Before we do, let's work out the probability of there being something or nothing using a kind of information theory. We can let '1' represent something, and '0' represent nothing.

Imagine a coin with a '1' on one side and a '0' on the other. We assume the probability of either side being face up is 0.5. However, the coin might favor one side over the other. How do we fix this? We get a trillion coins where one side might be more or less probable than the other. The probabilities could range from 0 to 1, but the average probability will be at or near 0.5 for each side. So it's safe to assume, if we perform the coin toss many many times (and switch coins), on average, we will get each side 50 percent of the time.

To make our analysis easier, let's assume we are using a coin that has the expected (average) value of 0.5 for each side. Half the time we are going to get a universe with something, and half the time we will get nothing. If we define "nothing" as absolutely nothing, we can improve our odds of getting "something" by adding additional coins. Let "n" in the last two equations below be the number of coins:

Using two coins yields the following results:

The [0 0] above is nothing. Those pairs above it have a '1'--so they are something. By only using a pair of coins we have improved the chance of getting something to 0.75. What happens if we use three coins?

The probability of getting something jumps to 0.875. If we use an infinite number of coins, the probability of getting something is one--a sure thing. The probability of getting nothing is zero.

Thus we have a reason why there is something rather than nothing. But what if what we call "something" is really something else: a mixed state? What exactly is a mixed state? You could think of it as being both something and nothing--an undecided state, a state with both ones and zeros. By contrast, a pure state has all ones or all zeros:

If we dare to ask where something or nothing came from, one answer is a mixed state. The mixed state evolves into a pure state. When we flip a coin, we don't know if it's heads or tails when it is in the air. We can say it's both heads and tails--a mixed sate. When it hits the ground and comes to rest, it's in a pure state.

Another example analogous to a mixed state is a ball sitting on a hill. It can roll down one side or the other. It stays in a mixed state until something disturbs it:

In the above diagram, the ball's state can evolve into the pure state of '1' or '0.'

So instead of imagining a universe with just something or nothing, imagine a universe filled with ones and zeros, and, to make things more interesting, let's add an additional state of '-1.'

Instead of coin tosses we have binary number sequences that continuously change over time (t). The above diagram is an example of a two-digit binary sequence. The diagram below models the evolution of these binary numbers:

The diagram above starts with the pure state of [1 1]. It then moves through mixed states to get to [0 0], then moves through more mixed states to get to [-1 -1], and so on. Basically we have something that resembles a sine wave. There is a problem, however: it is too predictable for our probabilistic universe. The evolution of states should be random:

The diagrams above are more realistic. The different states should have varied time duration(t00,t11,etc.). If, say, state [00] jumps to [11] without passing through [01] and [10], We say that these mixed states each have a zero duration. It is also possible for any state to have an exceptionally long duration.

So how do we get the predictable sine wave we started with? We take a trillion random waves and calculate the average time duration for each state. We can then add those average times together to get the total time and average wavelength:

Our predictable binary sine wave is really the expectation value of a bunch of random binary waves:

Now, our binary sine wave isn't very smooth nor continuous. It has a clunky, stair-step quality. We can make it smoother if we add more digits to the states:

In the diagram above, notice the wave with only one digit is the clunkiest. The smooth wave at the bottom has an infinite number of digits or at least a really big number of digits. So let's take the bottom wave and make it our official sine wave:

Sooner or later, our sine wave will interact with another wave (superposition). At one extreme the phase difference between the waves could be 180 degrees, resulting in destructive interference:

At the other extreme the phase difference could be zero degrees, resulting in constructive interference:

If we have a bunch of waves, the average or median phase between each pair will be 90 degrees. Thus, we end up with both a sine and cosine wave.

Having sine and cosine waves lead to wave functions and matter--the stuff we call "something."

However, as we have witnessed, the stuff we call "something" is really a mixture of both something and nothing: zeros and ones. The stuff we call "something" also oscillates between pure and mixed states.

You can find more on this topic if you click here.

The Quantum Differences Between Gravity and Electromagnetism

Welcome to part five of the gravity series. To read the other parts, click here and here. Are gravity and electromagnetism (EM) the same? There are some who think they are the same force. There may have been a time in the early universe when all the forces were one force, and, as time passed, the one force evolved into the ones we are familiar with. If gravity and EM are the same, they have some distinct differences we will examine at the quantum and cosmic levels.

In previous posts we discovered that gravity is a field with ever shortening spacetime wavelengths, or, increasing energy as a falling body moves closer to where there is greater mass or energy density. In addition, particle-waves' wave numbers increase. These wave numbers have the same units as curved spacetime. Gravitational acceleration is due to the difference in energy or wave number between an upper surface layer of spacetime and a lower one.

Imagine a particle falling in a gravitational field. We can model this using Schrodinger's equation with a twist: we take the difference between the Hamiltonian in the upper layer and the Hamiltonian in the lower layer:

We want to find the change in wave number (k):

We perform a couple of more steps to get the particle's change in kinetic energy:

If we take the wave number (k) and multiply it by Planck's constant (h-bar) and divide it by the particle's mass, we get the particle's change in velocity (v).

At equation 12) notice that the first term has momentum (p). Here are two kinds of momentum: mv (mass X velocity) and fh/c (frequency X Planck's constant/light speed). Velocity (v) could be a function of one or both of these momenta--or we could have two types of velocity: equations 13 and 14 below are derived from the first term of equation 12).

Equation 13) isn't fully simplified. We want to emphasize that the particle's mass cancels itself. The velocity is the same whether the mass is big or small. We can label this velocity the change in velocity due to gravity (Vg). Take note that momentum is not conserved: a big falling mass has more momentum than a smaller falling mass.

Equation 14) tells a different tale. A change in mass does change the velocity (Ve). Momentum is conserved. This equation fits the EM force.

Equations 13) and 14) reveal that when mass (m) is very large, gravity's influence stays the same; whereas, EM's impact diminishes. When mass (m) is small, EM becomes the dominant force--gravity becomes less significant.

Another key difference between gravity and EM is EM is a function of charge; whereas, gravity is a function of mass or energy density. Gravity attracts but EM obeys Column's law (like charges repel, opposite charges attract).

The crude diagrams below demonstrate EM interactions. When the field arrows are pointing towards each other, particles A and B push apart. When the arrows point away from each other, A and B separate. Particles A and B attract each other when the field lines (arrows) flow in the same direction, as if B is flowing towards A.

The following diagrams show how A and B share gravitational field lines. This sharing causes the energy field between A and B to become more intense than the fields at the far right and left of A and B. A and B want to move away from each other and move closer together. The shared energy between them makes the latter more probable. To see how this works in more detail, click here. As the distance between A and B decreases, the shared energy between them becomes stronger and the gravitational acceleration increases (the inverse square law).

One thing EM and gravity have in common are mass-less bosons. Since they are mass-less, these bosons have unlimited range and they travel at light speed. This raises a troublesome paradox, for we know EM is much much stronger than gravity. How can atoms and molecules ever get together via gravity when their outer-shell electrons have a repulsive force far greater than gravity's attractive force?

Consider two hydrogen atoms that are close together. We fully expect the electrons to repel each other, same goes for the protons. But could the proton in one atom be attracted to the electron in the other? If so, that attraction could, to some extent, cancel the repulsive force. We can use the following equations to see if gravity is stronger or weaker than the net EM repulsive force.

When we crunch the numbers we find that as distance (d) between the atoms increases, the EM force (Fe) drops more quickly than gravity (Fg). In the diagram below, where the atoms are close together, the distance between the electrons is very short compared to the distances between opposite charges, so the repulsive force is strong. This is good news! It means the atoms will remain distinct and separate. It means the ground will be solid beneath your feet.

By contrast, when the atoms are far apart, the different distances between opposites and same-charge particles are less dramatic. The charges cancel each other and gravity dominates.

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.

Deriving Maxwell's Equations From Heisenberg and Einstein

What is the node that connects Heisenberg's uncertainty principle, Einstein's field equations, and Maxwell's equations? To find out, let's start with Heisenberg's uncertainty principle and see what we can derive:

We derive a distance (r). We can do the same if we start with Einstein's field equations:

We can now equate the right side of equation 6) with equation 11). If we do this, we can discover the above-mentioned node.

Equation 15) reveals the "node" to be energy (E). Equation 16) shows that energy not only connects the uncertainty principle and field equations, it shows the electric field is linked as well. From 16) we can derive the integral form of Gauss's law for electric fields:

Equation 20) basically says electric charge produces an electric field. The field flux passing through a closed surface is proportionate to the charge contained within the surface.

If we take equation 20's integral and divide it by a volume (V), we can derive the differential form of Gauss's law for electric fields:

Equations 22) and 23) are just different ways of saying the electric field produced by electric charge diverges from a positive charge, or it converges upon a negative charge.

Going back to energy (E), we can begin again and derive Gauss's law for magnetic fields:

Equation 35) is the integral form. It says the magnetic flux passing through a closed surface is zero. According to equation 36), the magnetic field's divergence at any point is zero.

Given what we have derived so far, we can also derive the integral form of Faraday's law:

Equation 44) describes the electric generator. As the magnetic flux through a surface changes, an electric field is induced.

The differential form of Faraday's law is derived as follows:

According to equation 51), a magnetic field that changes with time produces a circulating electric field.

Next, we shall derive the integral form of the Ampere-Maxwell law:

Equation 63) says an electric current (I) produces a circulating magnetic field. The differential form is derived as follows:

According to equation 67), an electric field that changes with time produces a circulating magnetic field.

To put a cherry on top of the work we've done so far, we use Heisenberg's uncertainty principal and Einstein's field equations to derive the electromagnetic tensor (of equation 82) below):

Notice in equation 82) the Schur product was used to multiply the matrices to get the final result: the electromagnetic tensor.

To learn more about Maxwell's equations, there is an excellent book entitled "A Student's Guide to Maxwell's Equations" by Daniel Fleisch who explains every detail, symbol, and nuance. There's also this video which is also excellent:

Which Way Will the Pencil Fall?--Playing with Perturbation Theory

On a smooth, flat table you have a pencil balancing on its tip. Which way will it fall? Or will it just stay balanced on its tip? Variable L is the displacement. If the pencil stays balanced, L equals zero. If the cat jumps on the table ... L is greater than zero. In fact, L is the length of the pencil laying on its side, thanks to Felix the feline pencil flopper.

We can mathematically represent this feline faux pas with the following equation straight out of perturbation theory:

Whether the displacement (L) is zero or greater depends on the value of epsilon. If epsilon is zero, then L just equals Lo which is zero. In that instance, the pencil is perfectly balanced on that smooth, shiny tabletop. Epsilon represents a small disturbance (or a big one). When Felix jumps on the table, epsilon is greater than zero, and that causes L1 and L2 to come into play, which causes L to be greater than zero--so the pencil falls on its side.

The above equation, however, does not tell us which way the pencil fell (or the color of Felix's fur). It does not tell us much about the forces that make up epsilon. We want to know which way the pencil fell and the magnitude and direction of the forces involved. We could represent the direction and angle of displacement (L) if we use a complex number: a + bi. We can also use angle phi:

The maximum displacement (L) (i.e., the length of the pencil laying on its side) is equal to the square root of the complex number times its complex conjugate:

Epsilon can be divided into three dimensions of force: epsilon(a) is the force(s) that causes the pencil to fall in the "a" direction or "x" direction. Epsilon(b) is the force(s) pertaining to the "bi" direction or "y" direction. Since the pencil does not rise vertically along the "z" axis, or drill into the table, we only need to consider two dimensions.

The equation below tells us what epsilon is along directions "a" and "b." You will note that the complex number is divided by itself. This yields a one or a zero. The value of N equals one if the pencil falls, and zero if it does not.

We can find the values of a and b as follows:

And let's not forget the values of epsilon(a) and epsilon(b):

Last, but not least, we can find angle phi:

Click here for a totally excellent tutorial on perturbation theory.