The classical Goldstone theorem concisely states that for every spontaneously broken continuous symmetry, there exists a corresponding massless field, famously known as a Goldstone boson.
Understanding Spontaneous Symmetry Breaking
In physics, a continuous symmetry is considered spontaneously broken when the ground state, or vacuum, of a system does not exhibit the full symmetry of the underlying physical laws or Lagrangian. This means that while the laws governing the system possess a certain symmetry, the lowest energy configuration (the vacuum) does not.
A crucial aspect of this phenomenon is the concept of a broken generator. A generator (often denoted as Ta) is an operator that, when applied to a field or the vacuum, would normally transform it while preserving the symmetry. When a symmetry is spontaneously broken, the generator associated with that symmetry acts non-trivially on the vacuum or the ground state fields. This implies that the action of the broken generator, Ta, on the field φ results in a non-zero outcome (i.e., Taφ ≠ 0). This non-zero action is the direct indicator of the symmetry's spontaneous breaking.
The Emergence of Massless Fields
The classical Goldstone theorem provides a direct link between this spontaneous symmetry breaking and the appearance of massless particles. The mechanism unfolds as follows:
- Action of a Broken Generator: When a generator Ta is broken, its action on the ground state or field (represented as φ) is non-zero (e.g., Taφ ≠ 0). This non-zero action signifies that the ground state is not invariant under the symmetry transformation.
- Implication for the Mass Matrix: This non-zero action on the vacuum or field directly leads to a fundamental property of the system's mass matrix (often denoted as Mik). Specifically, the mass matrix is guaranteed to possess a null eigenvector.
- Resulting Null Eigenvalue: By definition, a null eigenvector corresponds to a null eigenvalue.
- Massless Particle: Since the eigenvalues of the mass matrix represent the masses of the particles within the system, a null eigenvalue unambiguously indicates the existence of a massless particle. This massless particle is the Goldstone boson.
In essence, the classical Goldstone theorem provides a powerful statement: for each continuous symmetry that is spontaneously broken, a distinct massless scalar boson must appear in the spectrum of the theory.
