Notes

(1) To obtain a theory with only one universal (bare) coupling constant for interactions of gauge bosons, the group G must be a simple Lie group (or perhaps a product of simple groups with a suitable discrete symmetry imposed).

(2) Unless they have become spatially separated. This possibility is difficult to implement because of the small volume of the universe in causal contact at (according to the standard cosmological model).

(3) In the literature the terms ``hard'' and ``soft'' have been used for ``intrinsic'' and ``spontaneous'' violation. We shall reserve ``hard'' and ``soft'' to describe lagrangian terms of dimensions four and lower, respectively.

(4) For a cubic lattice, the fracton of cells must be for infinite domains to exist. Other lattice geometries give slightly different percolation thresholds; all are below 0.5.

(5) We use throughout units such that , where is Boltzmann's constant. Since we do not set the gravitational constant , the Planck mass appears explicitly.

(6) Note that the effective Planck mass given here assumes that all particles obey Maxwell-Boltzmann statistics. If instead, all were to obey Bose-Einstein statistics, would decrease by a factor 0.96, and if Fermi-Dirac, increase by a factor of 1.03.

(7) This discussion has a potential application to thermodynamic models for hadron production in high-energy collisions. If the excited hadron material does not expand rapidly enough, the effective lifetime may be increased by copious production in inverse decay processes, thereby modifying correlations between final pions.

(8) Formal approaches based on path integrals periodic in imaginary time are suitable for calculating static correlation functions. These may be used to deduce the V propagator, but may not be used directly in calculations on the complete time-dependent system.

(9) Methods used to treat gravitational clumping are even inadequate in this case; baryon number may be modified by gauge interactions involving arbitrarily small momentum transfers, so that their long-range effects are still more prevalent.

(10) Note that if is stable, and can be produced or destroyed only in pairs, then the first term in (4.5.1) is absent, and the second term gives the complete Boltzmann equation for the evolution of its number density. This is approximately the case for the heavy right-handed neutrino of the SO(10) model, discussed in sect. 7 and in ref. [35]. The solution of eq. (4.5.1) for stable particles in such a case was discussed in ref. [36].

(11) This is the case whenever decays into several light fermion states with different baryon numbers, as discussed in sect. 2.

(12) These vertices may be represented schematically by the interaction lagrangian

where all Lorentz structure has been suppressed.

(13) Some of the results of this section are summarized in ref. [41].

(14) Assuming that no families with vanishing mixing angles exist.

(15) Many of the results in this section are also presented in refs. [47,48].

(16) This is a consequence of the fact that the symmetric product of the adjoint representation with itself does not contain the adjoint, and thus that the coefficients to which anomalies would be proportional, must vanish.

(17) Since the couples antisymmetrically to fermions, it must yield an antisymmetric fermion mass matrix with a zero eigenvalue for at least one out of three families.

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