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.

There is some work by Duck and Sudarshan which draws on some earlier work by Sudarshan and which is summarized in a pedagogical manner in Am. J Phys, 66, 4, 284 (1998), that deals with the restrictions that are required for the theorem to be valid. It would seem that Lorentz invariance isn't a requirement and most of the other possible restrictions you mention are probably not required either.

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