Origin of inertia at rest and the number of generations

Abstract

A new scenario is suggested for the discussion of the old problem of generations. We shall assume that the rest mass $m$ of a particle described by the Lagrangian $L(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x})=\ensuremath{-}m{(1\ensuremath{-}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}}^{2})}^{\frac{1}{2}}$ has its origin in the momentum ${p}_{\ensuremath{\theta}}$, canonical conjugate of a supplementary dimension of space, beyond the usual four dimensions. More precisely, we shall contemplate the possibility that the different generations, or for definiteness, the charged leptons $e, \ensuremath{\mu}, \ensuremath{\tau}, \dots{}$ are states, with different ${p}_{\ensuremath{\theta}}$, of a unique physical system, whose free Lagrangian will depend on $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\theta}}$ in addition to $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}:L(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}, \stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\theta}})$. The requirement that the relativistic relationship between the momentum ${p}_{x}$ and velocity $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}$ is maintained in five dimensions leads to the simplest Lagrangian $L(\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}, \stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\theta}})=\ensuremath{-}\ensuremath{\Lambda}{(1\ensuremath{-}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}}^{2})}^{\frac{1}{2}}\mathrm{exp}[\ensuremath{-}\frac{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\theta}}}{{(1\ensuremath{-}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}}^{2})}^{\frac{1}{2}}}]$, $\ensuremath{\Lambda}$ being a constant with dimension ${[\ensuremath{\theta}]}^{\ensuremath{-}1}$. With this Lagrangian, the function of ${p}_{\ensuremath{\theta}}$ that plays the role of the mass in the usual relativistic relation between ${p}_{x}$ and $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}$ is $m({p}_{\ensuremath{\theta}})={p}_{\ensuremath{\theta}}(1\ensuremath{-}\frac{\ensuremath{\beta}\mathrm{ln}{p}_{\ensuremath{\theta}}}{\ensuremath{\Lambda}})$. The quantization of momentum ${p}_{\ensuremath{\theta}}$ with periodic conditions leads to a mass spectrum compatible with experimental data only if the number of generations is three. In the present work we consider that the mass differences within each generation should be explained in the context of grand unified theories (GUT's). In the last section, though, the complementary information that GUT's might supply in our context is suggested, and a value for the mass of the top quark and a bound for the mass of the $\ensuremath{\tau}$ neutrino are obtained.