The Dirac equation is invariant, but there are a lot of problems with the concept of locality. For example, if you want to create localised one-particle states that remain local in any reference frame and form an orthonormal eigenbasis of the (one-particle subspace of) the Hilbert space, you will find it impossible.
The canonical solution in axiomatic QFT is to begin with local operations instead of localised particles. However, to see the problem, one has to question the notion of measurement and define it in a covariant manner, which may be a mess. See e.g.
I am sympathetic to the approach which is used by quantum gravitists, which uses extended Hamiltonian formalism and the Wheeler—DeWitt equation instead of the Schrödinger one. This approach doesn’t use time as special, however the phase space isn’t isomorphic to the state space on the classical level and similar thing holds on the quantum level for the Hilbert space, which makes the interpretation less obvious.
The Dirac equation is invariant, but there are a lot of problems with the concept of locality. For example, if you want to create localised one-particle states that remain local in any reference frame and form an orthonormal eigenbasis of the (one-particle subspace of) the Hilbert space, you will find it impossible.
The canonical solution in axiomatic QFT is to begin with local operations instead of localised particles. However, to see the problem, one has to question the notion of measurement and define it in a covariant manner, which may be a mess. See e.g.
http://prd.aps.org/abstract/PRD/v66/i2/e023510
I am sympathetic to the approach which is used by quantum gravitists, which uses extended Hamiltonian formalism and the Wheeler—DeWitt equation instead of the Schrödinger one. This approach doesn’t use time as special, however the phase space isn’t isomorphic to the state space on the classical level and similar thing holds on the quantum level for the Hilbert space, which makes the interpretation less obvious.