To derive Dirac's Lagrangian, we begin with the Dirac field's adjoint spinor (bar-psi) and spinor (psi). Note that each spinor contains right-handed (psi-R) and left-handed (psi-L) fields. (Right-handed means the spin and momentum of the field particles are in the same direction. Left-handed means spin and momentum are opposite.)
Let's put the fields into an algebraic form:
Next, we multiply them together:
On the right side of the equal sign the first two terms are each zero and add to zero. Here's why:
The mixed terms don't equal zero. Here's why:
The first two terms that equal zero won't equal zero if we multiply them both by Dirac's matrix.
Now we can set those first two terms equal to the mixed terms:
On the left side, we take the derivative of the fields and multiply by -i, Planck's constant, and the speed of light (c). That gives us kinetic energy. On the right side we multiply by mass (m) and c^2. That gives us potential energy.
The Lagrangian is kinetic energy minus potential energy, so we subtract the potential energy from both sides to get the Lagrange (L):
There's a new theory that explains every mystery in physics. I call it the magic-dust theory. Like other modern theories it is mathematically consistent. But like other modern theories it is currently not testable. To observe the magic dust requires a particle accelerator the size of our galaxy, maybe bigger.
Here's an example of the mathematics of this promising new theory:
Thanks to this new innovative theory, we now know what caused the Big Bang. There is no empirical evidence, but, as I said before, this theory is mathematically consistent. Here is an example:
All kidding aside, the "magic-dust" theory demonstrates what is wrong with modern theoretical physics. If you stop and think about it, there isn't much difference between magic dust, and strings.
Like magic dust, strings have virtually unlimited power: they can vibrate and make the different particles; they can stretch up to infinity in multiple dimensions to give us d-branes. They can do whatever is needed to explain any mystery. As long as the math is consistent, we can call it science. However, Issac Newton would disagree. Below he discusses gravity and his philosophy:
"I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction."
Note how Newton uses the word "hypothesis" rather than "theory." Technically, string theories and other modern theories aren't theories at all. At best, they are hypotheses, since the entities they invoke (e.g. strings, extra dimensions, etc.) have never been observed. Below, William Whewell has this to add:
"What is requisite is, that the hypotheses should be close to the facts, and not connected with them by other arbitrary and untried facts; and that the philosopher should be ready to resign it as soon as the facts refuse to confirm it."
I have to agree. The challenge is to explain the mysteries without magical thinking; i.e., adding extra entities that have never been observed like extra dimensions, strings, branes, magic dust, etc. The whole idea of science, in my opinion, is to reduce mystery, not add to it. If you have to create a new mysterious object to explain an old mystery, then that is the analogue of digging one hole to fill another.
Using new, unobserved, untested entities to explain a mystery has the disadvantage of increasing your burden of proof. This is why the best hypotheses use known facts as much as possible to explain the unknown.
On the flip side it's true that Democritus dreamed up the atom centuries before it was physically observed. And maybe sometime in the distant future, humans will discover strings, extra dimensions, gods, fairies, pixies, and dragons in their basements ... or ... maybe not. The thing I love about science is we are free to change our minds as soon as the evidence becomes available.
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.
