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Theories of a mathematically real quantum function are discussed. The analysis relies on fundamental elements of quantum theories and proves the existence of a new type of problems in these theories. Particle density plays a ke y role in the discussion. The results show new contradictions in the Majorana neutrino theory and in the Yukawa theory of the nuclear interaction. Experimental data support these conclusions. New problems exist with the theories of the Z and the Higgs bosons.

Mathematically complex wave functions were the first choice of the founders of quantum mechanics. This type of functions is mandatory for the Schroedinger and the Dirac equations, whose form is

Here the Hamiltonian

A different course was taken few years later, when two mathematically real quantum equations were published. The first case was the real version of the Klein-Gordon (KG) equation, which was used in 1935 by Yukawa for a theoretical interpretation of the nuclear force (see e.g. [

The quantum equations of Yukawa and of Majorana have a relativistic covariant structure. The same property holds for other real quantum equations, like those of the Higgs and the

General considerations are described in the second section. The third section contains a derivation of new problematic issues of mathematically real quantum theories. The last section contains concluding remarks. This work uses standard notation of relativistic expressions. Greek indices run from 0 to 3 and Latin indices run from 1 to 3. The metric is diag (1, −1, −1, −1). Units where

The quantum theory is about 90 years old. It aims to explain the dual nature of an elementary particle, namely, a pointlike particle that has wave properties. There is a vast literature that discusses many attributes of this theory. On the other hand, quantum effects provide a basis for modern industry which uses transistors, lasers etc.

Quantum theories can be classified as follows:

1) Nonrelativistic quantum mechanics.

2) Relativistic quantum mechanics (RQM).

3) Quantum field theory (QFT).

These theories apply to an increasing order of their domain of validity. Nonrelativistic quantum mechanics holds for nonrelativistic states and processes. RQM holds for cases where the number of particles can be regarded as a constant of the motion. QFT describes processes containing phenomena of particle creation and annihilation.

For example, the nonrelativistic Schroedinger equation explains some properties of the hydrogen atom. The Dirac equation is the corresponding relativistic equation and it provides a much better explanation for the hydrogen atom properties. Tiny effects of creation and annihilation of virtual particles are explained by QFT. These tiny corrections are confirmed by experiments that measure the hydrogen atom properties. A production of particles, like electron-positron pair production, is demonstrated in high energy experiments. This production is explained by QFT.

Evidently, these theories are connected by appropriate limiting processes. Thus, for small velocity (namely, in cases where

“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 1925-1926, and has been used ever since in atomic, molecular, nuclear and condensed matter physics”.

Later on, this assertion is called Weinberg’s QFT correspondence principle (see also [

The analysis relies on basic properties of a quantum particle which are briefly presented in the following lines. The wave nature of such a particle is considered as a primary attribute of a quantum theory. Here the de Broglie formula for the wave length of a massive particle is related to its momentum (see [

For the simplicity of the discussion, let us examine a free massive quantum particle, like an electron in a re- gion of space where external electromagnetic fields vanish. The phase of its wave takes the form

The de Broglie relationship means that the particle’s energy and momentum appear as elements of its phase where (

It is interesting to note that (3) and (4) prove that the phase

A quantum theory of a given particle must describe the time evolution of its state. This objective takes the form of a differential equation. Relying on the foregoing expressions, one finds that appropriate differential operators are related to the particle's energy and momentum. These operators take the form

These relationships show the connection between dynamical quantities and differential operators.

The following section relies on Weinberg’s QFT correspondence principle and shows how the fundamental quantum issues described above affect the structure of QFT.

As stated above, the quantum Equation (1) proves the well known complex form of the Schroedinger and the Dirac wave function. Let us examine the possibility of describing a massive quantum particle by means of a real wave function. A simple case is that of a free particle moving along the positive x-direction. The form of the factor that describes the undulating properties of its wave function can be written as a linear combination of the following expressions (see [

The last expression of (6) is a complex function which depends on the particle’s energy and momentum. Therefore, it is unsuitable for a real wave function. Evidently, every linear combination of the first and the second functions of (6) can be written in the form

where

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

It follows that for every integer n, the real wave function (8) vanishes identically throughout the entire 3-dimen- sional space at every instant t when

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 the real wave function vanishes at 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 QFT cannot be constructed.

3) Obviously, due to the absence of these spaces, operators used in mathematically real quantum theories be- come meaningless. Another discrepancy that stems from the missing Hilbert and Fock spaces is that a calcula- tion of transition amplitude between quantum states becomes impossible.

These findings prove the existence of inherent contradictions in quantum theories of a mathematically real wave function. Beside the foregoing specific issues, these theories violate the Weinberg’s QFT correspondence principle, because density is a well defined quantity in the nonrelativistic Schrodinger theory. Note that the proof takes a general form which is independent of the specific structure of any given mathematically real quantum theory. Therefore, it applies to all quantum theories of this kind.

Here the following question arises: Why the Noether theorem does not provide a consistent expression for density of a particle whose quantum equation of motion takes a mathematically real form?

In the case of a Majorana particle the answer depends on the absence of an appropriate eigenfunction. Indeed, the Majorana Lagrangian density is (see [

where the four

Another argument that disproves the Majorana theory stems from the definition of the following function of (9)

(see [

is real and so is its solution

The Lagrangian density and its action

are pure imaginary due to the additional

The following argument explains the origin of the problem. The Majorana

A primary attribute of a Majorana particle is that it is identical to its antiparticle (see [

One kind of experimental data is the neutrino (

Another kind of experiment is the search for a neutrinoless double

As far as this work is concerned, the case of a real KG equation is much simpler, because the absence of density and of a conserved 4-current for this quantum field is proved in a textbook (see [

At present there is no experimental support for the real KG particle which is used in the Yukawa theory of nuclear force. Thus, the discovery of quarks in the 1960s proves that a

A search of the literature provides an indirect support for the claim that there is no self-consistent expression for density of a mathematically real quantum theory. Indeed, an expression for density and its associated con- served 4-current can be found for the Schroedinger equation (see e.g. [

The following pure leptonic decay mode of the

Analogous decay modes exist for the

This work analyzes a class of quantum theories whose wave function takes a mathematically real form. It proves that new problems exist with the corresponding parts of presently accepted physical theories. The analysis relies on well documented physical properties of quantum theories that can be found in standard textbooks. The results show contradictions of the Majorana neutrino theory and of the Yukawa theory of the nuclear interactions. Experimental data provide a solid support for these results. Furthermore, serious problems exist with the current theoretical structure of the Z and the Higgs bosons. This outcome calls for a further analysis of the specific results of this work and of their implications.

E. Comay, (2016) Problems with Mathematically Real Quantum Wave Functions. Open Access Library Journal,03,1-6. doi: 10.4236/oalib.1102921