Within quantum field theory we find two equations based on Einstein's equation for energy (E = energy; p = momentum; c = light speed; m = mass):
The first is Dirac's equation. The second, the Klein Gordon equation (i = imaginary number; h-bar = Planck's constant; bold y with arrow = Dirac-Pauli matrices; a-sub-u = first derivative; psi = quantum field; t = time; up-side-down triangle = Laplacian).
The goal is to take elements and ideas from the above equations and combine them with Einstein's field equation below (Guv = Einstein's tensor; Gn = Newton's constant; Tuv = the energy-stress tensor; and then there's pi):
Let's first write Guv in quantum terms (psi-sub-r = relativity-field-wave function; Aj = a coefficient element in series j). Note there's a double derivative of psi with respect to an x-vector, and tensor indices (uv) after the closing parenthesis. There is also psi's complex conjugate, which, when multiplied my psi and the rest, gives the expectation value; i.e., a result we expect on the classical scale.
Next, we write Tuv in quantum terms (V = volume). Again we get an expectation value, and the entire term is placed in parenthesis and made a tensor with indices uv.
Now that we have Einstein's equation in quantum terms, let's define each element in greater detail. Below is the definition of psi:
Next we define k. The k represents wave number(s) for one or more particles summed (j=1 to n) by the summation sign. The arrow over k indicates it is a 4-vector. The "a" with a dagger is the creation operator. The zero in the ket represents the field in the ground state.
The variable x is also a 4-vector:
The bold b with an arrow can be any one of three types of spin matrices: Dirac-Pauli matrices (y) for half-spin particles; S-matrices for integer-spin particles; and the number one for zero-spin particles.
Below is a full demo of the Dirac-Pauli matrices:
Here is an example of one Dirac matrix acting on the k 4-vector:
The following are 4X4 matrices I designed for spin-1 particles. I always admired Dirac's equation and matrices, but they are limited to half-spin particles. Now it's possible to include bosons in the field.
We now have a quantized version of Einstein's field equations where we can work with both matter and anti-matter. We can work with zero-spin particles (Higgs field), spin-1 bosons, fermions, atoms, molecules and beyond ... and see how they curve spacetime in terms of the wave number(s) k.
The interesting thing about k^2 is it has the same units as Einstein's field equations: 1/L^2. Thus if we take the double derivative of psi with respect to x, we get k^2. Multiply that by the unit-less coefficient A; multiply psi by its complex conjugate--and we get the equivalent of an Einstein tensor element. Put 16 elements together and we get the complete Einstein tensor.
