On Vacuum Fluctuations and Particle Masses

Abstract

The idea that the mass m of an elementary particle is explained in the semi-classical approximation by quantum-mechanical zero-point vacuum fluctuations has been applied previously to spin-1/2 fermions to yield a real and positive constant value for m, expressed through the spinorial connection Γ _i in the curved-space Dirac equation for the wave function ψ due to Fock. This conjecture is extended here to bosonic particles of spin 0 and spin 1, starting from the basic assumption that all fundamental fields must be conformally invariant. As a result, in curved space-time there is an effective scalar mass-squared term $m_{0}^{2}=-R/6=2\varLambda_{\mathrm{b}}/3$, where R is the Ricci scalar and Λ _b is the cosmological constant, corresponding to the bosonic zero-point energy-density, which is positive, implying a real and positive constant value for m ₀, through the positive-energy theorem. The Maxwell Lagrangian density $\mathcal{L} =- \sqrt{-g}F_{ij}F^{ij}/4$ for the Abelian vector field F _ij ≡A _j,i −A _i,j is conformally invariant without modification, however, and the equation of motion for the four-vector potential A _i contains no mass-like term in curved space. Therefore, according to our hypothesis, the free photon field A _i must be massless, in agreement with both terrestrial experiment and the notion of gauge invariance.

Appendix: The Fermi Lagrangian of Electromagnetism

The Lagrangian density of the electromagnetic field

$$ \mathcal{L}=-\frac{1}{4}\sqrt{-g}F_{ij}F^{ij} $$

(156)

can be rewritten in Minkowski space-time, where $\sqrt{-g}=1$ and covariant derivatives reduce to partial derivatives, as

$$ \mathcal{L}=\frac{1}{2}(A_{j,i}-A_{i,j})A^{i,j}=\mathcal {L}_\mathrm{F}+ \frac{1}{2}\bigl(A_{j}A^{i,j}\bigr)_{,i}-\frac{1}{2}A_j \bigl(A^{i}_{,i}\bigr)^{,j}, $$

(157)

where the Fermi Lagrangian is defined by

$$ \mathcal{L}_\mathrm{F}=-\frac{1}{2}A_{i,j}A^{i,j}. $$

(158)

Thus, expression (157) differs from $\mathcal{L}_{\mathrm{F}}$ only by a total divergence, if the Lorenz gauge condition (104) is imposed, which is guaranteed by the condition (110). Expression (158) implies a non-vanishing canonical momentum

$$ \pi_0=\partial\mathcal{L}/\partial \bigl(\partial_0 A^0\bigr), $$

(159)

allowing construction of the Hamiltonian.

There is no obvious way of generalizing this procedure to curved space-time, however. For the covariant form of Eq. (158) is

$$ \mathcal{L}_\mathrm{F}=-\frac{1}{2}\sqrt{-g}A_{i;j}A^{i;j}, $$

(160)

and therefore we have to rewrite Eq. (156) as

$$ \mathcal{L}=\frac{1}{2}\sqrt{-g}(A_{j;i}-A_{i;j})A^{i;j} = -\frac{1}{2}\sqrt{-g}A_{i;j}A^{i;j} +\frac{1}{2}\sqrt{-g}A_{j;i}A^{i;j}. $$

(161)

The first term on the right-hand side of Eq. (161) is expression (160), but the second term cannot be expressed as the difference of the total divergence $\frac{1}{2}(\sqrt{-g}A_{j}A^{i;j})_{,i}$ and the product of A _j times the jth derivative of $A^{i}_{;i}$, in particular because $A^{i;j}_{;i}\neq A_{;i}^{i;j}$. Nor is any meaningful simplification achieved by imposing a metric gauge condition, for example, the de Donder–Lanczos condition

$$ \bigl(\sqrt{-g}g^{ij}\bigr)_{,j}=0, $$

(162)

in addition to the electromagnetic gauge condition (104), due to the fact that expression (160) contains a term which is quadratic in both $\varGamma^{i}_{jk}$ and A _i .