|San José State University|
& Tornado Alley
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
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
Thh'ρh' ≠ <0
then there must exist a massless particle with the same quantum numbers as Tα.
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
It then follows that
(To be continued.)
Source: T.D. Lee, Particle Physics and Introduction to Field Theory, Harwood Academic Publisher, New York, 1981.
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