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7.2. Vacuum polarization and the value of Lambda

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

Equation 81 (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 lambda = hbar/mc. Although the regularized self-energy of these pairs is zero, their gravitational interaction is finite and results in the vacuum energy density epsilonvac ident rhovac c2 ~ (G m2 / lambda) / lambda3 = Gm6 c4 / hbar4 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 rightarrow me(mp) we find that the electron (proton) mass gives too small (large) a value for rhoLambda. On the other hand, the pion mass gives just the right value [106] (13)

Equation 83 (83)

Finally, a small value of Lambda can be derived from dimensionless fundamental constants of nature using purely numerological arguments. For instance, the fine structure constant alpha ident e2 / hbar c appeq 1/137 when combined with the Planck scale rhoP, suggests the relation [187]

Equation 84 (84)

Or, when expressed in terms of OmegaLambda = (8pi G epsilonLambda / 3 H02) we get OmegaLambda h2 = 0.335, in excellent agreement with observations. In principle, alpha could be some other fundamental constant, such as the `string constant' associated with superstring theory, which might enter into exponentially small expressions for Lambda of this type.

Figure 13

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 rhoLambda obtained using (81) for the proton and its observed value prompted Zeldovich to suggest that Fermi's weak interaction constant GF might play a role in determining the vacuum energy, so that

Equation 82 (82)

Although this leads to some improvement, rhoLambda for the proton is still several orders of magnnitude larger than its observed value. Back.

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