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W, Z and Higgs in General Relativity


The Standard Model shows that each of the W, Z and Higgs bosons has the following theoretical properties:
  1. The particle's Lagrangian density has a quadratic mass term of the form
    where m denotes the mass of the corresponding particle and Φ denotes its field function. (Here indices and complex attributes of the field functions are omitted.)

  2. The mass term is free of derivatives.

  3. Therefore, the corresponding Hamiltonian density contains the same mass term but with an opposite sign.
It is well known that the Hamiltonian density is the T00 component of the energy-momentum tensor.

The fundamental equation of General Relativity takes the following form (see [1], p. 297)
Rμν - 12gμνR = κTμν,
where Rμν and R are appropriate contractions of the curvature tensor, gμν is the metric tensor, Tμν is the overall matter and fields energy-momentum tensor and κ is a constant.

It is well known that the classical energy-momentum tensor of matter as well as that of a Dirac particle depends linearly on mass, whereas it is shown above that the Standard Model interpretation of each of the W, Z and Higgs bosons has an energy-momentum tensor that depends quadratically on mass. It follows that if the Standard Model is correct then the W, Z and the Higgs bosons carry a kind of mass which is different from ordinary matter.

The following argument indicates the compatibility of the Dirac equation with the required expression of the energy-momentum tensor. General Relativity is a classical theory. Now, the Schroedinger equation is the fundamental equation of non-relativistic quantum mechanics. Furthermore, this equation is the non-relativistic limit of the Dirac equation. It is also well known that Newtonian mechanics is the classical limit of quantum mechanics. By taking these limits on finds the compatibility of the Dirac theory with General Relativity.

This procedure does not work for the Standard Model interpretation of the W, Z and Higgs bosons. Indeed, the aforementioned limits are continuous processes. Therefore, the second power of the mass term of the Standard Model equations of these bosons cannot jump and become a first power. It follows that these equations are inconsistent with the linear mass dependence of the classical energy-momentum tensor and with General Relativity as well.

Take for example an electron and a positron, which are two pure Dirac particles. It is not clear how and why an electron-positron energetic collision produces successfully an electroweak Z boson and how this process is reconciled with General Relativity.

An analogous problem has been pointed out a long time ago in an analysis of the Proca equation [2].


[1] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Elsevier, Amsterdam, 2005).
[2] E. Comay, Nuove Cimento, B113, 733 (1998).