Confusion then arises in the quantum many-body theory of condensed matter because there are theories [and materials!] which involve quasi-particles which appear to violate this theorem. Here are some examples:
- Spinless fermions. These arise in one dimensional models. For example, the transverse field Ising model.
- Schwinger bosons which are spinors (bosonic spinons). These arise in Sp(N) representations of frustrated quantum antiferromagnets. They were introduced by Read and Sachdev.
- Anyons. In two dimensions one can have quasi-particles which obey neither bose nor fermi statistics.
- The theory has a Lorentz invariant Lagrangian.
- The vacuum is Lorentz invariant.
- The particle is a localized excitation. Microscopically, it is not attached to a string or domain wall.
- The particle is propagating, meaning that it has a finite, not infinite, mass.
- The particle is a real excitation, meaning that states containing this particle have a positive definite norm.
I think three dimensions [and a non-interacting, i.e. quadratic Hamiltonian] may be other assumptions.
In condensed matter, one or more of the above assumptions may not hold. For example,
- inclusion of a discrete lattice breaks Galilean invariance
- spontaneous symmetry breaking
- topological order can lead to non-local excitations
- in one dimension spinless fermions may be non-local [e.g. associated with a domain wall]
Thanks to Ben Powell for reminding me of this question.