applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA

The Goldstone Theorem: Its Nature and Proof

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'}^{α}φ_{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}^{α}n,k><n,k|φ_{h}(0)|vac> = 0 if j_{n}≠0
and
<vac|J_{μ}^{α}n,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.