7.2. Vacuum polarization and the value of
Zeldovich (1968), having demonstrated that the energy density of the vacuum was infinite at the one-loop level, suggested that after the removal of divergences, the `regularized' vacuum polarization contributed by a fundamental particle of mass m would be described by the expression
(81) |
One can arrive at this result by means of the following argument: the vacuum consists of virtual particle-antiparticle pairs of mass m and separation = /mc. Although the regularized self-energy of these pairs is zero, their gravitational interaction is finite and results in the vacuum energy density _{vac} _{vac} c^{2} ~ (G m^{2} / ) / ^{3} = Gm^{6} c^{4} / ^{4} corresponding to (81). (In terms of Feynman diagrams this corresponds to the energy associated with the two-loop vacuum graph shown in figure 13.) Substituting m m_{e}(m_{p}) we find that the electron (proton) mass gives too small (large) a value for _{}. On the other hand, the pion mass gives just the right value [106] ^{(13)}
(83) |
Finally, a small value of can be derived from dimensionless fundamental constants of nature using purely numerological arguments. For instance, the fine structure constant e^{2} / c 1/137 when combined with the Planck scale _{P}, suggests the relation [187]
(84) |
Or, when expressed in terms of _{} = (8 G _{} / 3 H_{0}^{2}) we get _{} h^{2} = 0.335, in excellent agreement with observations. In principle, could be some other fundamental constant, such as the `string constant' associated with superstring theory, which might enter into exponentially small expressions for of this type.
Figure 13. This figure shows the one-loop (a) and two-loop (b) vacuum diagrams which contribute towards the vacuum energy density discussed in sections 5 & 7.2 respectively. |
^{13} The large difference between _{} obtained using (81) for the proton and its observed value prompted Zeldovich to suggest that Fermi's weak interaction constant G_{F} might play a role in determining the vacuum energy, so that
(82) |
Although this leads to some improvement, _{} for the proton is still several orders of magnnitude larger than its observed value. Back.