Consider a set of H spin-0 Hermitian fields φ_h(x) defined for all space-time points x over 4 dimensional space-time. For the system there is a mathematical group G represented by a set of H×H complex matrices. This group is the symmetry group for the system. Furthermore the elements of the group G can be generated by a subset of N generators. These are elements of G such that any element of G is equal to a product of powers of these generators. Let these generators be denoted as matrices T^α where α=1,2,…,N.

The system is assumed to be invariant for continuous transformation of the form

φ_h(x) → φ_h(x) + δφ_h(x)
where
δφ_h(x) = i(T_hh'^αφ_h'(x))dθ^α

where T_hh'^α is one of the H×H generator matrices of the group representation.

It is assumed that for each generator T^α there is a conserved current J_μ^α. This assumption implies that the time rate of change of the generator is zero.

The commutator relations for the generators and the fields is such that

[T^α, φ_h(x)] = iT_h,h'φ_h'(x)

The Goldstone Theorem: If the vacuum expected value <vac|φ_h|vac>
is denoted as ρ_h and for at least some α and h

T_hh'ρ_h' ≠ <0

then there must exist a massless particle with the same quantum numbers as T^α.

Proof:

The state is specified by two sets of number (n, k), where k is the momentum vector and n is the all of the other numbers required to specify a state, such as its invariant mass m_n, its electric charge q_n, its spin j_n and its helicity.

For states with zero momenta in all three space directions

<n, k=0| φ_h(0) | vac> ≠ 0
only if
the spin j_n=0

It then follows that

<vac|J_μ^αn,k><n,k|φ_h(0)|vac> = 0 if j_n≠0
and
<vac|J_μ^αn,k><n,k|φ_h(0)|vac> ≠ 0 if j_n=0

(To be continued.)

Source: T.D. Lee, Particle Physics and Introduction to Field Theory, Harwood Academic Publisher, New York, 1981.