How Entangled Particles Correlate (or Anti-correlate)

When we last checked in on Alice and Bob they were observing the spins of two entangled particles. According to their findings, the spins of particles A and B always seem to correlate (or anti-correlate)--no matter how far apart they are.

Alice noticed when A's spin changed, Bob reported that B's spin also changed in consistent manner with A. Particles A and B behave as if they are joined at the hip rather than as if they are two distinct and separate particles.

We know that space and time are relative. In the case of particles A and B we are interested in the space or distance between them (x' and x). Perhaps the Lorentz factor below can model the relationship between the observed distance between the particles (x) and the distance the particles experience (x').

If A and B moved at light speed (c), then v^2 in the above equation would equal c^2. Distance x' would be zero. It would be as though A and B never separated. If they never separated, then it makes sense they would behave as if they were a single particle. All their information would be contained in particle A-B, so to speak.

According to Alice and Bob, though, A and B are at rest, so their distance would be x, the distance Alice and Bob observe. Then again, a mass (m) can be at rest and, along with its radius (r), can produce a velocity-squared equivalent (G=Newton's constant):

If mass (m) is big enough, distance x' would drop to zero. Unfortunately, particles A and B have tiny masses. The question then arises: Do entangled particles have a velocity-squared equivalent of some kind that is equal to c^2? And what of particles that aren't entangled (separable)? If a velocity-squared equivalent exists, what would it be for them?

We know that all particles have mass or a mass equivalent (m), and said mass is coupled with c^2. Particles A and B have the following parameters (E=energy):

Separable particles also have an mc^2. If we use c^2 as our v^2 equivalent, there seems to be no distinction between entangled and separable particles. Entangled particles are like particles that have never separated and they are statistically dependent on each other. Separable particles, by contrast, behave like they are far apart and are statistically independent; i.e., are orthogonal.

Dot products (inner products) could be used to model orthogonal separable particles and statistically dependent entangled particles. Below i and i are not orthogonal and yield a scalar value of one; whereas, i and j are orthogonal and yield zero.:

Using the entangled particles' parameters and the dot product (inner product) concept we formulate the following equation:

Let's put the above equation to work and see what happens. We shall first calculate the distance x' between two separable particles:

The distance x' between two separable particles is simply x--what Alice and Bob perceive and what the two particles experience--since none of the aforementioned are part of an entangled system. Note that the separable particles' v^2 equivalent is zero. How can that be when they each have an mc^2? Answer: The v^2 does not represent the individual particles; it is the v^2 of the entangled system. Since separable particles are not entangled, the entangled system is nonexistent and the v^2 equivalent is zero.

Now we determine the distance x' between entangled particles:

The distance x' between two entangled particles is zero. From A and B's point of view, they never separated. They don't send information back and forth like Alice and Bob imagine. Poor Alice and Bob are still not part of the entanglement and only see distance x and believe the particles are far apart. As a result, they scratch their heads and wonder how A and B can send information to each other faster than the speed of light. But A and B don't need to send information back and forth--they are particle A-B--they are entangled.