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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
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 [1], p. 18)
|
cos(kx - ωt),
sin(kx - ωt),
exp(±(kx - ωt)).
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(2)
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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
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ψ(t,x) = Asin(kx - ωt - δ),
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(3)
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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),
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(4)
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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:
- A conserved density cannot be consistently defined for a
particle described by
a mathematically real wave function.
- 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.
- 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.
For reading a full article that discusses this issue, click
here
.
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 [2], p. 49). One can also
see on the same page of [2] 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
)
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 [3], 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:
[1] L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955).
[2] S. Weinberg, The Quantum Theory of Fields, Vol. I, (Cambridge
University Press, Cambridge, 1995). (See the following link
here
.)
[3] F. Halzen and A. D. Martin,
Quarks and Leptons, An Introductory Course in Modern Particle Physics
(John Wiley, New York,1984).
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