Functional Analytic Square Root of Hamiltonian Alternative to Dirac

Question

I was thinking about the history of the Dirac equation and asked myself, what happens if one simply considers the Schrödinger equation $$i\hbar\frac{\partial\phi}{\partial t}=\sqrt{-c^2\hbar^2\Delta+m^2c^4}\phi~?$$ The literature seems to suggest that the square root is troublesome. However, in spectral theory it is well known how to take the square root of positive self-adjoint operators. So, what goes wrong with this?

Answer 1

I think that the problem is that the square root of the Laplacian is a non-local operator and non-locality is usully regarded as a bad thing in physics. The long range nature shows up in the general expression
$$ (-\nabla^2)^s f(x)\equiv \frac{4^s}{\pi^{n/2}}\frac{\Gamma(s+n/2)}{\Gamma(-s)} \int_{{\mathbb R}^n} d^ny \frac{\left\{f(x)-f(y)\right\}}{|x-y|^{2s+n}}, \quad 0<s<1, $$ for fractional powers of the Laplacian as a convolution integral. (The limit on the range of $s$ is to ensure that no further subtractions are required to define $(-\nabla^2)^s$ as a distribution.)