nLab mass term



Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



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

0ψ=k/ψ, \partial_0 \psi \;=\; k\!\!\!/ \cdot \psi \,,

where k/k\!\!\!/ are the Clifford momentum-operator, then for a mass term addition

0ψ=k/ψ+mγ 0, \partial_0 \psi \;=\; k\!\!\!/ \cdot \psi \;+\; m \gamma_0 \,,

to lead to the required relativistic dispersion relation

E=k 2+m 2 E \,=\, \sqrt{k^2 + m^2}

one needs that the new Clifford-generator γ 0\gamma_0 exists as an operator on the given Hilbert space of spinor ψ\psi such that it skew-commutes with all the Clifford momenta

γ 0k/+k/γ 0=0. \gamma_0 \cdot k\!\!\!/ \;+\; k\!\!\!/\cdot \gamma_0 \;=\; 0 \,.

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

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 topological phases of matter

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).

(graphics from SS22)

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:

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.