   The de Broglie Principle Vs. the Standard Model Home Contact Me

The analysis is based on an undergraduate Quantum Mechanics course. Therefore, it is understandable by all physicists, chemists and many engineers. It examines properties of quantum theories that use a mathematically real wave function. The de Broglie hypothesis of the wave nature of massive particles "forms a central part of the theory of quantum mechanics" (see here ). This hypothesis says that the following relation holds between the particle's wave length and its linear momentum

 λ = h/p. (1)

The form of the factor that describes the undulating properties of the particle's wave function can be written as a linear combination of the following expressions (see , p. 18)

 cos(kx - ωt),    sin(kx - ωt),    exp(±(kx - ωt)). (2)

Mathematically real functions can be written as a linear combination of the first and the second functions of (2). Hence, a real wave function of a free massive particle moving along the positive x-direction takes the form

 ψ(t,x) = Asin(kx - ωt - δ), (3)

where A is a real normalization factor and δ is a real constant.

The free quantum particle that is analyzed here is massive and it has a rest frame. (It should be noted that the following analysis does not apply to the photon because this particle has no rest frame.) In this frame the particle's linear momentum is k = p = 0 and its wave function (3) reduces to the form

 ψRest(t,x) = Asin(- ωt - δ), (4)

It follows that for every integer n, the real wave function (4) vanishes identically throughout the entire 3-dimensional space at the instant t when ωt + δ = nπ.

Evidently, if the wave function vanishes at a certain point then the particle's probability density vanishes there. This is the basis for the quantum interpretation of an interference pattern. Therefore, the fact that a real wave function vanishes throughout the entire 3-dimensional space means that at the corresponding instant the particle disappears. Hence, the following results are obtained:
1. A conserved density cannot be consistently defined for a particle described by a mathematically real wave function.

2. The lack of a consistent expression for density means that a Hilbert space of quantum mechanics and its associated Fock space of quantum field theory (QFT) cannot be constructed.

3. Obviously, due to the absence of these spaces, operators used in mathematically real quantum theories become meaningless. Another discrepancy that stems from the missing Hilbert and Fock spaces is that a calculation of a transition amplitude between quantum states becomes impossible.
The foregoing analysis applies to all quantum theories of a massive particle that use a mathematically real wave function, like the Majorana neutrino theory, the Yukawa theory of the nuclear force, the Z boson of the electroweak theory and the 125 GeV Higgs boson. In particular, the Z and the Higgs bosons are fundamental elements of the Standard Model of particle physics. Hence, the foregoing analysis shows that the de Broglie Principle disproves the validity of the Standard Model.

It should be pointed out that this conclusion applies to quantum mechanics and to QFT as well. Indeed, S. Weinberg states clearly in his QFT textbook: "First, some good news: quantum field theory is based on the same quantum mechanics that was invented by Schroedinger, Heisenberg, Pauli, Born, and others in 1925-26, and has been used ever since in atomic, molecular, nuclear and condensed matter physics" (see , p. 49). One can also see on the same page of  that the construction of a Hilbert space is a vital element of quantum mechanics.

The following process illustrates this issue. Consider the electron-positron decay mode of the Z boson (see here )

 e+ ← Z → e-

This is a pair production event that belongs to the QFT validity domain. The final result comprises a free electron and a free positron that can be described by quantum mechanics. The decay is a continuous time-dependent process. Hence, QFT continuously corresponds to quantum mechanics.

The following argument shows that the experimental side supports this result. The assignment of the outgoing e+e- to the Z boson is based on a very small space-time distance between certain points of the trajectories of these particles. This evidence indicates locality of the experimental particle Z. It is proved above that a real wave function cannot describe this effect.

These claims can simply be refuted by showing the page number of a textbook where density is consistently defined for even one particle of this kind. As of today, few Standard Model supporters have already failed to show such a reference.

P.S. The standard expression of a quantum field's 4-current is obtained from the Noether theorem. Here the invariance of the Lagrangian density with respect to the complex factor exp(iα) is applied (see , p. 314). This transformation does not apply to a mathematically real function. This argument shows the compatibility of the discussion presented above with general principles of physics.

References:

 L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955).

 S. Weinberg, The Quantum Theory of Fields, Vol. I, (Cambridge University Press, Cambridge, 1995). (See the following link here .)

 F. Halzen and A. D. Martin, Quarks and Leptons, An Introductory Course in Modern Particle Physics (John Wiley, New York,1984).