# Relativistic fermionic spinor and emergent Lorentz symmetry from non-relativistic fermion band [closed]

It is commonly stated that the low energy theory of some non-relativistic fermion band structure with a linear dispersion $$E = vk$$ implies the

1. low energy effective theory is a relativistic fermion theory.

2. there is an emergent Lorentz symmetry at or near the node

Of course $$E = vk$$ is necessary condition, but it does not seem to be sufficient to show 1 and 2. Also the Lorentz spinor degrees of freedom is unclear.

## My questions

• why 1 and 2 are true? What conditions guarantee those? (Sketch a physics proof?)

• How to determine which relativistic fermions they have at low energy: Dirac, Majorana, Weyl or Majorana-Weyl spinors? as Lorentz spinors?

• why are they Lorentz spinors? (they can be approximate spatial rotational symmetries, but not the Lorentz spacetime boost + rotational symmetries ?)