nLab mass term

Contents

Context

Algebraic Quantum Field Theory

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

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

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).

Examples

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.

References

A careful discussion of relativistic mass terms is in:

Discussion aimed at the description of topological semi-metal-phases in solid state physics includes:

Some of the above material is taken from:

Last revised on May 18, 2022 at 14:10:11. See the history of this page for a list of all contributions to it.