Goldstone's theorem states that a spontaneously broken continuous internal global symmetry in field theory of a local relativistic quantum field theory produces massless Goldstone bosons. For a Lorentz invariance vacuum breaking

G

to its stabilizer

H

, there are

dim G - dim H

independent symmetry-required massless directions. Additional accidentally massless modes are possible. A gauge redundancy does not obey this physical-state counting, and nonrelativistic systems need not have one independent mode per broken generator.

For the classical proof, consider real scalar fields

ϕ_{i}

with a positive canonical kinetic term and an invariant smooth scalar potential

V

. An infinitesimal generator

T^{a}

satisfies

\frac{\partial V}{\partial ϕ _{i}} (T^{a} ϕ)_{i} = 0.

(1)

Let

v

be a stationary vacuum and differentiate this identity with respect to

ϕ_{j}

. At

v

, the term proportional to

\partial_{i} V

vanishes, leaving

M_{ji}^{2} (T^{a} v)_{i} = 0, M_{ji}^{2} = \frac{\partial ^{2} V}{\partial ϕ _{j} \partial ϕ _{i}}_{v} .

(2)

The Hessian matrix

M^{2}

is the scalar mass matrix. An unbroken generator has

T^{a} v = 0

; broken generators span the tangent space of the vacuum manifold

G / H

. Each independent nonzero tangent vector is a zero eigenvector of

M^{2}

. The quadratic Euler-Lagrange equations in these directions are

□ π^{a} = 0

, giving the massless dispersion relation

ω^{2} = p^{2}

. This proves the classical result and explains why the Goldstone bosons describe motion along the degenerate minima rather than against a restoring curvature.