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

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.

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.