Can the Laplacian be Factorized into Two First Order Differential Operators?

[zetafunction]

given the Laplacian for a certain metric 'g'

$$\Delta f = \partial _i \partial ^{i}f + (\partial ^{i}f) \partial _ i log |g|^{1/2}$$

where a sum over 'i' dummy variable is assumed

the idea is , could we factorize this Hamiltonian a second order differential operator into two first order differential operator as

$$1/2 + iL$$ and $$1/2 - iL$$ so $$\Delta f= (1+2+iL)(1/2-iL)f$$ ?

the idea taken from mathematics is the following:

accroding to selberg trace $$\lambda =s(1-s)$$ where s is a complex zero and lambda is the eigenvalue of the Laplacian of a surface.

in terms of Riemann Hypothesis : if we can find such factorization eigenvalues of $$1/2 + iL$$ would be precisely the zeros of Riemann zeta function , providing that $$1/4+ \gamma ^2$$ are eigenvalues of a certain surface.

 

[gtoguy97]

In general this factorization is possible in other cases too, but it depends on the particular metric.