recall the coefficient of Dirac plane wave squrae-root-of-spacetime-momentum-spinor-representation
positive definite inner product
Hermitian symmetric tensor
It doesn't look special, but if we consider the sum of the results for and , using , their sum is
Conjugate phase plane wave case
[square-root-of-harmonic-oscillator]
Inspired by the treatment of KG field quantization
We can also define the square root of a point particle complex harmonic oscillator
Dirac plane wave with and
If we ignore the plane wave, even if there is a loss of precision, we get an ODE
The solution can be written in the form of the square root of the spacetime momentum spinor representation . Using
Dirac conjugate phase plane wave with and
gives the ODE
Solution . Using
If we ignore the plane wave restriction of Dirac eq
-
Solution of the ODE given by
-
Solution of the ODE given by
Using general ODE theory, for constant coefficient linear ODEs
perform ification. Use The series rule is
The result is
Written as complex exponential
Lagrangian of the ODE
Energy
Momentum (if it makes sense)