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The Goldstone Theorem:
Its Nature and Proof

Consider a set of H spin-0 Hermitian fields φh(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 Tα where α=1,2,…,N.

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

φh(x) → φh(x) + δφh(x)
where
δφh(x) = i(Thh'αφh'(x))dθα

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

It is assumed that for each generator Tα there is a conserved current Jμα. 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

[Tα, φh(x)] = iTh,h'φh'(x)


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

Thh'ρh' ≠ <0

then there must exist a massless particle with the same quantum numbers as Tα.


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 mn, its electric charge qn, its spin jn and its helicity.

For states with zero momenta in all three space directions

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

It then follows that

<vac|Jμαn,k><n,kh(0)|vac> = 0 if jn≠0
and
<vac|Jμαn,k><n,kh(0)|vac> ≠ 0 if jn=0

(To be continued.)


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


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