Untangling the Quantum Entanglement Probability

According to the video above, Bell's Inequality theorem shows that two fictional characters, Bob and Alice should get the same results more than 33% of the time. The actual probability of them getting the same outcome is 25%. But why 25%?

Since the particles Bob and Alice possess are entangled, and since Bob and Alice are not, it stands to reason that the probability of Bob and Alice being in sync would be less than expected.

Suppose Bob and Alice each randomly choose one of two polarizers: up or down. They compare notes hoping for a match. Next, they each send one of the entangled particles through their respective polarizers. Each check the spin of his/her particle to see if it is up or down. They compare notes again, hoping for another match.

If the particles weren't entangled the experiment would yield these possible outcomes:

The above outcomes show that Bob and Alice are in sync 50% of the time (which is greater than 33%). Four out of eight outcomes show their particles have the same spin or Alice and Bob picked the same polarizer.

Now here are the possible outcomes if the particles are entangled with opposite spin:

As you can see the probability falls from 50%. Alice's particle is no longer independent of Bob's. If Bob's particle's spin is up, Alice's is down, and vice versa. There are no longer up-up or down-down outcomes.

An Easy Way to Prove Bell's Inequality

Here is a simple way to derive Bell's inequality. Basically you have three random sets A, B, and C. Bell's inequality says if you take A that isn't part of B and add that to all of B that isn't part of C, that value is greater than or equal to all of A that isn't part of C. (See the diagram of sets A, B, C below.)

How Extraterrestrials Compose Music

One of the most famous sci-fi flicks of all time is "Close Encounters of the Third Kind." Below is a clip regarding those musical tones that Richard Dreyfuss kept hearing in his head. But what's up with those tones? What do they mean? Why did the extraterrestrials send them to the lead actor's head? And, most importantly, how do Extraterrestrials compose music? All these questions shall be answered soon after you're done watching the clip:

Put yourself in the extraterrestrial's shoes. You are light-years from your home planet. You and your crew are exhausted after exploring much of the galaxy. Your navigation system is on the fritz due to a meteor shower. You are lost. You pull your flying saucer up to a nearby inhabited planet so you can ask directions.

Hopefully the earthlings can tell you how to get back to the constellation where you are from:

But first you have to let them know which constellation it is. They don't speak your language and the lead actor doesn't have a clue what is happening. He keeps building mounds with his mash potatoes and the the mud in his yard. His wife thinks he's insane and leaves him. That doesn't help you at all. What can you do to get through to these earthlings? You can create a musical language that describes the stars in your constellation. You begin with an alphabet or scale: Do ray me fa so lah te do.

Each note in the scale has its own frequency. When you double the frequency (f) you raise the pitch a whole octave. If you want to go down an octave, cut the frequency in half. If you want to go up or down a half tone, multiply or divide the frequency by 1.06, respectively. There are 12 half-tones per octave, so going up an octave requires 1.06^12 times the frequency (f). Going up a perfect fifth requires (1.06^7)f. Going down one full tone requires f/1.06^2.

Once you get your universal musical language together, you match each star in your constellation with a note:

You can now tell the earthlings where you are from and hopefully they can tell you the way back home. If they don't understand at first, just keep telepathically sending those tones until the whole planet is singing them.

If they still don't understand, try taking Richard Dreyfuss for a ride in your flying saucer and show him your sheet music below:

Extraterrestrials use triangles to communicate their music. The height of the top triangles are the pitch. The base (cosine) of each triangle is the duration. The up-side-down triangles are the loudness. The bigger the sine waves the louder and higher the pitch, and vice versa.

Below I use super-complex numbers to translate the alien music. (If you are not familiar with super-complex numbers, click here.) Hopefully the diagram below will serve as a Rosetta stone. Each note is indexed by the letter u.

To simplify the math further, let's assign first-rank tensor variables (T, P, L) to each cosine and isine term.

We end up with a simple tensor equation that describes the language of the universe: music. Now that we understand what the extraterrestrials want, can someone please give them directions?

Probabilities, Euler's Identity and Super-complex Numbers

Today we are going to examine how super-complex numbers fit in with Euler's identity and probability amplitudes. If you are not familiar with super-complex numbers, read my post entitled: "Introducing Super Complex Numbers."

Euler's identity is as follows:

It has a cosine and one isine or imaginary number. It can be used to model a right triangle:

Or a propagating wave:

Here is the super-complex number version of Euler's identity:

The exponent has i sub-n and theta sub-n. This tells us we will take the cosine of angle theta, and the isine-thetas from 1 to n. In the example below, n = 3.

The super-complex Euler identity is useful if you are working with one cosine and multiple triangles that share the same cosine. Take the exponent i sub-n, theta sub-n; it shows precisely how many isines/triangles are involved. The super-complex Euler identity is a real time saver. Instead of writing a big long string of trig functions we can write one simple exponent.

Since it can represent multiple triangles, it can be used to represent a wing design for a stealth aircraft:

OK, so that wasn't really a stealth aircraft--just more triangles, but you get the idea. Below are multiple waves propagating through space that are modeled by the super-complex Euler ID:

We know that if we multiply Euler's identity with its complex conjugate we get 1. We can also take the inner products of cosine and isine to get probabilities, but notice we are stuck with just two probabilities:

Using the super-complex Euler ID, is it possible to have as many probabilities as we like--and they all add up to one? To answer this question we need to derive the super-complex Euler ID. Start with the normal Euler ID, then split up the isin(theta) into smaller parts. Each part shall have a coefficient of epsilon:

If we examine a unit circle diagram and some trig relations, we discover that each epsilon sub-j isin(theta) equals isin(theta sub-j). We make a substitution:

If we equate i with each i sub-j and i sub-k, we assume the product of any two indexed i's is -1. Unfortunately this will give us non-zero cross product terms.

Let's check to see if the cross product of two arbitrary indexed i's really do give us -1 and not 0. Take the sum of two arbitrary terms and multiply it by its super-complex conjugate. That will give us a real number we label b^2. Like the isine of Euler's ID, we split b into smaller parts. Call those parts c and d.

The last equation above confirms the product of any two indexed i's equals -1. The great thing about Euler's ID is the square of its absolute value yields the same result as a dot product. We want the super-complex Euler's ID to behave the same way--no non-zero cross products! So here's what we do:

At the second equation above, we assume there are no non-zero cross product terms on the right side; however, without the extra cross-product terms, the left side is greater than the right side. We cure this by increasing the value of the isines. We then throw in unit vectors (ej). We now have a multiplication that behaves like an inner or dot product. We make some further refinements below:

Note how each indexed i is converted to an indexed mu, which behaves like a unit vector, giving the result we want:

Each squared sine can represent a probability and the total is 1.

As you can see, the super-complex version of Euler's identity has a great deal more flexibility than the ordinary Euler's identity. It allows us to model complex systems and probability amplitudes with just one simple exponent expression.

Introducing Super Complex Numbers

We know that i and -i are square roots of -1. They are part of the axis of imaginary numbers. Combine them with real numbers and you get complex numbers. Complex numbers are often used to model rotations, spins, oscillations, vibrations. They can even be used to model error margins. 55 +/- 3 can be expressed as 55 +/- 3i.

Complex numbers are useful any time you have a real-number value combined with a number that fluctuates. A good example is a wave. The real number tells you how long the wave is or how far it has moved along the x-axis. The imaginary number tells you the vertical measurement along the i axis. (See diagram below.)

Point A above is the sum of the real part and the imaginary part. Complex numbers work well in the above example, but suppose you want to model, say, an electric wave and a magnetic wave? For that you may want to use super complex numbers. Super complex numbers have an extra imaginary axis so you can model two waves for the price of one:

As you can see, i1 and i2 are both equivalent to i. They are both square roots of -1. Multiplications between them yield -1 or 1 in the same manner as plain old i. Operations of super complex numbers are similar to complex numbers.

Division with super complex numbers is tricky just as it is with complex numbers. You use super complex conjugates ( super complex numbers with the signs reversed) to get a real number solution:

The type of super complex number we've been working with so far is called a type-2 super complex number. It's type-2 because there are two imaginary axes. A complex number is a type-1 super complex number, since it only has one imaginary axis. Real numbers are type-0 for an obvious reason. All this implies we can have type-3 or even type-infinity super complex numbers.

Suppose we have multiple waves propagating through space? We can model the entire system with the following expression:

So far, all our waves have conveniently moved along the x-axis. What about the y and z axes? Or some combination of axes? Suppose we have waves moving along a vector? The imaginary axes would have to become imaginary vectors with the same angular relationship to the real-number vector as they had with the x-axis.

Below is a super complex vector expression:

But why stop at super complex vectors when we can have super complex tensors? The diagram below shows a type-n super complex tensor of rank-2. Below that is a general expression that can fit any super complex number or tensor.

Imagine being able to model a highly complex system filled with fixed values and variations. The weather perhaps? You could also model beams in a building as they vibrate during an earthquake. The beams could make up real-number tensors, and all the different vibrations could make up the imaginary-number tensors. These are but a couple of examples of what you can do with super complex numbers. Their application is only limited by your imagination.

For more information of this topic see "Probabilities, Euler's identity and Super-complex Numbers."

General Relativity's Invariant Tensor Myth

If you took a General Relativity course or read a book on the subject you were probably told the tensors that make up Einstein's field equations are a good thing because they are invariant. What's so special about invariant tensors? Here is a quote from one source:

"As an abstract mathematical entity, tensors have an existence independent of any coordinate system or frame of reference ..."

Wow! Cool! This means I can take a vector (a rank-one tensor), place it in any coordinate system or reference frame (also known as a basis) and it will not be affected by the coordinate system. The only things that will change are the values of the vector's components. The diagrams below show an example of a typical coordinate transformation:

Notice how the vector looks and behaves the same way in the different coordinate systems. However, Einstein's theory of General Relativity would not work if this were really true. If the vector above was a light beam this is what would happen:

The light-beam vector curves if the geometry of the coordinate system is curved. It is not independent of the coordinate system. So technically, it is not a tensor and should not be modeled by tensors. Or, the definition of "tensor" needs to be modified.

The theory of General Relativity claims it is the very geometry of spacetime that causes the light beam to curve. Also, curved spacetime geometry causes other objects that are considered tensors to move along a geodesic or curved trajectory when those objects would move differently in flat spacetime. None of this is consistent with the concept of invariance.

Solving the Dark Energy Mystery

From our point of view, the universe is expanding at an exponential rate. We would have this perception no matter where we are in the universe. It's as if we are a raisin on a raisin cake baking in the oven. Energy from the oven makes the cake expand and the raisins move apart. (See illustration below.)

If raisins could see, each raisin would see the other raisins moving away. In the case of the raisin cake, outside energy is needed to heat the oven so the cake can expand. If the universe is expanding the same way, where is the outside energy coming from? What kind of energy is making the universe expand? One popular hypothesis says the universe is filled with this mysterious stuff called dark energy. Sounds like something out of a sci-fi novel, doesn't it?

If energy in our universe is conserved, it seems inconceivable that a fixed amount of energy could cause the universe to expand at an exponential rate. Perhaps this "dark energy" or vacuum energy is being piped in from from a higher dimension or another universe through a wormhole. Let's do the math and see where it leads us.

We begin our investigation with a metric that includes an exponent scale factor to the power of Hubble's constant (H) times t (time). The x's are dimensions, c is light speed and v is velocity. From here we derive something that resembles the Lorentz factor:

Velocity squared is equivalent to Gm/r. (G is Newton's constant; m is the universe's mass and r is the universe's radius.) We make a substitution below:

Next, calculate the derivative of a sphere's surface area. That gives us 8pi times r. Multiply both sides of the above equation by that figure:

We know that mass is the universe's energy (E) divided by c squared. We make a substitution below and get an expression that resembles the right side of Einstein's field equations.

The 8pi's cancel and we end up with a formula that allows us to calculate the radius of the universe as a function of time:

Notice we can hold energy (E) constant and the universe's radius will increase anyway. The radius increases when time increases. No added outside energy is needed. Say bye bye to the wormhole pipe and the hidden universe or dimension.

So what the heck is going on? This is a real mystery. Time goes by and the universe simply expands at an exponential rate! Let's delve a little deeper and see if we can unravel this conundrum. To make things easier let's set all the constants to one:

Let's take a closer look at energy (E). What is it exactly? It has a mass aspect; it also has a spacetime aspect: the v^2 is a distance squared over a time squared. Let's assume the distance is the universe's radius. Let's assume the time is the proper time or the universe's time rather than the observer's time (t). Set 1/2m to one, and we get the revised equation below:

Now we can solve for the universe's proper time:

Notice we dispensed with r by substituting e^t (e^t*e^t=e^2t). If you plug in some numbers you will discover the universe's proper time changes faster than the observer's time. This gives the observer the impression the universe is expanding at an exponential rate when in fact it is expanding at a steady rate. If the universe's radius grows proportionately with its proper time, energy is conserved.

Since the radius equals the exponent, we can derive the velocity of the expansion:

When the observer's time increases, the velocity of the expansion increases, i.e., the expansion is accelerating from the observer's point of view. Also notice if we substitute a shorter radius, we get a slower velocity. This is consistent with the universe expanding at a slower rate when the observer looks a short distance, and the universe expanding at a faster rate when the observer looks farther. (Another common equation used is v = Hr.)

If we assume relativity is the culprit, our expanding universe no longer seems mysterious.

Update: Here is a mathematical proof showing how the universe's energy is conserved:

Equations 1 and 2 below show the universe has no choice but to expand. Vacuum pressure is never relieved as the universe expands, so it keeps expanding. The remaining equations show how the overall energy is conserved. As volume and entropy increase, temperature decreases.