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Problems with the Electroweak  Factor

The following discussion relies on these principles:

  1. The primary objective of a physical theory is to provide an acceptable prediction and interpretation of experimental results that are carried out within its domain of validity.
  2. Quantum mechanics uses a wave-function and operators. An application of an appropriate operator to the wave function that represents a quantum system, yields the required quantity.
  3. There is a correspondence between quantum mechanics (QM) and quantum field theory (QFT) (see [1], p. 49).
  4. The Hilbert space is a crucial element of QM (see [1], pp. 49, 50). The Fock space of QFT is an extension of a Hilbert space of QM [2].
  5. A quantum theory of a specific interaction relies on an appropriate Lagrangian density. The electroweak Lagrangian density contains the factor which operates on a Dirac spinor (see [4], p. 308). The following analysis uses the principles 1-5 for an examination of the validity of a more general factor , where  is a pure number.

Let us see what happens after applying to the function of a free motionless spin-up Dirac spinor

Here the notation of the matrices is that of [5], p. 17.

Let us examine the three cases where .

  • If  then the right-hand side of (1) is a negative-energy Dirac spinor (see [5], pp. 28-30). It means that in this case the operator  projects a massive motionless Dirac particle into an unphysical state. This state is not included in the Hilbert space and expectation values of observables cannot be calculated.
  • If  then the right-hand side of (1) is a Dirac spinor that has an infinite energy-momentum (see [5], p. 30). It means that the operator projects a massive motionless Dirac particle into an unphysical state and the previous result holds.
  • then the right-hand side of (1) is a Dirac spinor that moves in the z-direction. The energy of this spinor is greater than that of the original motionless spinor. It follows that in this case the operator   violates energy conservation.

Conclusion: The electroweak factor is unacceptable.

Research topic: Can the above mentioned discrepancy of the electroweak theory be resolved?

References

[1] S.Weinberg, The Quantum Theory of Fields, Vol. I (Cambridge University Press, Cambridge, 1995).

[2] https://en.wikipedia.org/wiki/Fock_space

[3] R. P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958).

[4] S. Weinberg, The Quantum Theory of Fields, Vol. II (Cambridge University Press, Cambridge, 1995).

[5] J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964).