A free parameter when switching from $\phi$ to $a$

Question

In the unnumbered equation above (8) the authors of https://arxiv.org/abs/2109.05547 introduce a free parameter $m_x$ (presumably of mass dimension), when switching from the real scalar field $\phi$ (discretized on a spatial lattice) to creation operators:

$$ \phi(x)= \dfrac{1}{\sqrt{2m_x}}(a_x+a^\dagger_x). $$

They say then that the commutation relation between the creation/annihilation operators are $$[a_x,a^\dagger_y]=a^{-1}\delta_{x,y}.$$ How can this commutation relation not be affected by $m_x$, given the fixed commutation relation between the fields from the unnumbered equation above (1), $$[\phi(x),\pi(x)]=ia^{-1}\delta_{x,y}~?$$ (This seems correct from the dimensional analysis perspective.)

I'm also wondering whether $m_x$ can actually explicitly depend on $x$, similarly to $\omega_{\bf k}$ in the momentum space.

Answer 1

It should be fine as long as you put the right normalization in $\pi$. If one takes $$\pi(x) = i\sqrt{\frac{m_x}{2}} (a_x^\dagger - a_x)$$ then everything works out.

In general, you can always add a scaling factor like this in front of your creation and annihilation operators without changing the canonical commutation relations. What it does change is a prefactor in your Hamiltonian which depends on the scale.