The New Relativity Particle Wave Function

General Relativity and quantum mechanics don't mix--so we are told by an echo chamber that has lasted a century. The Schrodinger time-dependent and time-independent equations don't fit with the 4D tensor format of Einstein's field equations. I decided to take on the challenge.

The root of the problem is the wave function (represented by the Greek letter psi). It depends on three space dimensions multiplied by a factor of k (momentum[p]/h-bar) and one time dimension multiplied by frequency (f).

To be compatible with relativity, we need to express the wave function in four spacial dimensions, where light speed times time (ct) makes up the fourth dimension. We multiply that by k4 to make it equivalent to frequency times time (ft).

Below are the mathematical steps needed to make an incompatible wave function into a compatible one. The variable A can be any coefficient for the exponent (exp()).

Notice the variable k has become four variables (k1, k2, k3, k4). The logic behind this will become clear when we test the new wave function later. For now, more variables offer more flexibility. They can all be equal to k or their values can vary depending on the situation.

We would very much like the the new wave function to do all the tricks the old one can do and then some. Let's do the time-dependent first-derivative test and see if the new wave function yields the same result as the old one. We set the variable A to 1.

Looks like a winner! Let's try the time-independent second-derivative test for the x-axis:

We do indeed get the same result for both wave functions. The only difference is the syntax. k becomes k1. Variable p^2 (momentum) becomes p1p1. It should be obvious that what works for the x-axis also works for y and z. So we can use this new wave function when working with Schrodinger's equation.

Now let's convert variables x, y, z, and ct to x1, x2, x3, x4. Let's see how well the new wave function works with a quantized version of the field equations. Uij is potential energy; h-bar is Planck's constant; m is mass; G is Newton's constant; c is light speed; V is volume; N is the large number of particles needed to make a little gravity; gijGij are scalar coefficients for each element of the rank-2 tensor created by the double partial derivative of the wave function psi along xi, xj.

With a little differential calculus and a pinch of algebra we can derive the field equations:

Notice how the wave function's complex conjugate is used to get the expectation value and eliminate the wave function psi.

Now I will show you the advantage of having more than one k variable. If we take double-partial derivatives of psi along k1, k2, k3 and k4, we can derive the spacetime metric and Lorentz factor. We can also derive yet again the field equations. You can see the details in my blog post entitled "Einstein's Field Equations Simplified."

OK, so we can derive a bunch of stuff with this new wave function. What's the big deal? Notice how we've unified General Relativity with quantum mechanics without invoking strings, branes or extra dimensions. Heck, we didn't even invoke the graviton. These things may or may not exist, but if they are never discovered we have a way to work with quantum gravity.

Since we don't need strings or extra dimensions to do this work, we have far less to prove than string theorists. We can make it work with established, empirically verified physics--and they can't. That puts us way ahead of the curve.