algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
In physics, a mass term is, quite generally, the summand in a Lagrangian density, Hamiltonian or equation of motion which expresses the mass of a given (quasi-)particle or field.
Often the terminology is used specifically for masses of spinors/Dirac fields, since here a mass term is subject to subtle constraints which may or may not exclude the mere existence of admissible mass terms.
Schematically, if a massless relativistic Dirac equation (here in Minkowski spacetime) is written in the form
where $k\!\!\!/$ are the Clifford momentum-operator, then for a mass term addition
to lead to the required relativistic dispersion relation
one needs that the new Clifford-generator $\gamma_0$ exists as an operator on the given Hilbert space of spinor $\psi$ such that it skew-commutes with all the Clifford momenta
This is the kind of condition that also appears in Karoubi‘s formlation of topological K-theory-classes as equivalence classes of Clifford modules (see Freed & Hopkins 2021, Thm. 9.63).
In this guise, mass terms for electron-excitations in semi-metals play a role in the K-theory classification of topological phases of matter (see below).
In particle physics the (non-)existence of mass terms for fermionic fundamental particles goes along with their characterization as Dirac spinors, Weyl spinors and Majorana spinors.
This plays a central role notably in discussion of the (still hypothetical) detailed nature of neutrinos.
In solid state physics of crystals, electrons are typically well-approximated by the non-relativistic Dirac equation, but around nodal points in the Brillouin torus, where electron bands cross, the effective dispersion relation is again of form of a relativistic Dirac- or Weyl-equation (whence one speaks of Dirac points or Weyl points).
This phenomenon appears in particular in topological semi-metals, in which case the existence of a mass term, whose addition to the dispersion relation will remove the band crossing by “opening an energy gap”, is thought to signal the decay of the semi-metal-phase into a topological insulator-phase.
See for instance the example of the Haldane model.
A careful discussion of relativistic mass terms is in:
Discussion aimed at the description of topological semi-metal-phases in solid state physics includes:
Takahiro Morimoto and Akira Furusaki, Sec. V of: Topological classification with additional symmetries from Clifford algebras, Phys. Rev. B 88 (2013) 125129 (arXiv:1306.2505, doi:10.1103/PhysRevB.88.125129)
Ching-Kai Chiu, Andreas P. Schnyder, Section A.2 of: Classification of reflection-symmetry-protected topological semimetals and nodal superconductors, Phys. Rev. B 90 205136 (2014) $[$doi:10.1103/PhysRevB.90.205136$]$
Ching-Kai Chiu, Jeffrey C.Y. Teo, Andreas P. Schnyder, Shinsei Ryu, Section III.C of Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88 (2016) 035005 (arXiv:1505.03535, doi:10.1103/RevModPhys.88.035005)
See also discussion in the context of differential KO-theory:
Some of the above material is taken from:
Last revised on February 16, 2023 at 07:41:55. See the history of this page for a list of all contributions to it.