Comment: look here for an analysis of the new particle which was found in LHC.

The Higgs Boson belongs to a family of particles which, by definition, are massive, elementary and spinless. So far it has been assumed that such particles really exist, and scientists even won Nobel prizes for allegedly finding a particle from this family.

Based on the knowledge existing today, this kind of particle has clearly not yet been found. This fact has now become common knowledge, as illustrated, for example, by the Klein–Gordon equation or Higgs Boson entry in Wikipedia. Are we about to witness the discovery of the first particle belonging to this family, as predicted by the Standard Model?

Can spinless, massive elementary particles exist at all?

Some definitions

There are particles which always travel at the speed of light, such as the photon, which have no self-mass because they are never at rest. Another kind of particles is called “massive”, they rest or move at a speed lower than the speed of light, and their mass is defined as their resting mass.

Experiments show that there are some particles in nature, with a volume smaller than the experimental detection threshold, and they are assumed to be point-like particles. These particles are called “elementary”, and the most famous of them is the electron. Various measurements show that the quark, residing within the proton, is a point-like particle as well. Today, there are 24 known point-like particles that have a mass: six quarks, three types of neutrinos, three types of electron-like particles, and their anti-particles. All of these particles have spin 1/2.

Some history

In the middle of the 1920s, Schrödinger tried to find an equation that would describe the electron’s quantum behavior. He used the experimentally well-known energy levels of the hydrogen atom to test the validity of his equation. He first tried to use an equation later called “The Klein-Gordon Equation” but realized that it did not yield the desired results. He thus developed the Schrödinger Equation, describing a massive elementary particle.

This equation, published in 1926, was considered as a breakthrough, but had a few drawbacks: it did not take the spin into account, and was not coherent with the theory of Special Relativity. In 1927, Pauli published the Pauli Matrices, which extend the scope of the Schrödinger equation and treat the electron as a particle with spin.

A year later, Dirac published the Dirac Equation, describing an elementary particle, while being coherent with Special Relativity. The Dirac Equation gave an excellent description of the electron, which was the only massive elementary particle known at the time. The Dirac Equation explains the spin of the electron, as well as the magnetic moment associated to it. Another outcome of this equation is the existence of an anti-particle for each of the particles the equation applies to. This result is considered today as an inalienable asset of particle physics.

The Dirac Equation minimized Pauli’s achievement. It is, therefore, easy to understand why Pauli used to tease Dirac in different occasions. He said about Dirac, who was atheist: “There is no God and Dirac is his prophet” ([1], page 138), and following the publication by Dirac of an article about magnetic monopoles, Pauli named him “Monopoleon” ([1], page 343).

In 1934, Pauli and Weiskopf revived the Klein-Gordon Equation, describing spinless particles. However, spinless particles were not known at that time. When publishing the article, Pauli himself admitted that he didn’t find any use for the equation, but that he was happy that he “can again cast aspersion on my old enemy – the Dirac theory of the spinning electron” ([2], page 70).

Dirac contested this equation, and published throughout his life several arguments explaining why it was invalid. But the scientific community ignored Dirac, whether for relevant reasons, or because they attributed his position to his rivalry with Pauli. Since the revival of the Klein-Gordon Equation by Pauli and Weiskopf, it has often been used to predict new elementary particles.

The Yukawa Particle

In his attempt to explain the “Strong Nuclear Interaction”, which is the force holding the protons together in the nucleus, the Japanese physicist Yukawa invented the idea of a particle carrying the force. According to Yukawa’s Theory, published in 1935, this force-carrying particle was elementary, massive and spinless, and therefore could be described by the Klein-Gordon Equation. Yukawa also gave an evaluation of the particle’s mass.

In 1947, a spinless particle was found, whose mass corresponded to Yukawa’s evaluation, and Yukawa won the Nobel Prize in 1949. But after the discovery of the quarks, it turned out that this particle, the pi-meson, is composed of a pair of quark and anti-quark, was not elementary, and could not carry the Strong Interaction.

Why can there be no Yukawa particle?

When Yukawa constructed the quantum function for his particle, he used a “real” function as a solution to the Klein-Gordon Equation. It is important to note here, that the notion of a “real” function was different from the admitted structure of Quantum Mechanics. And in fact, Quantum Mechanics uses what mathematicians call “complex functions”. These functions are used as solutions of the quantum equations and describe the physical properties of massive particles.

There is a profound reason for using complex functions in describing massive particles. A massive particle can be at rest, so that a real function describing its state should be time independent. On the other hand, in Quantum Mechanics, the particle’s variation with time is related to its energy. But for a particle which doesn’t change in time, a quantum function of real numbers implies that the particle’s energy (and mass) is zero. For this reason, a massive particle cannot be described using a real function, which means that the mathematical structure of Yukawa’s particle is wrong from the start.

This error may seem fundamental, and people may find it hard to believe that a major physicist such as Yukawa could have gone wrong. It turns out, however, that not only great physicists can and do make mistakes, but that this specific error is taught as a valid theory up to this very day and it features in the textbooks. A more precise and formal demonstration of this contradiction is brought in the first link on the page “A challenge to physicists” in the present website.

It would be plausible to assume that the source of the error was the application of calculation methods corresponding to the mass-less photon, to calculate a massive particle. But this shortsightedness, discovered by Eliyahu Comay over 60 years later, proves that major errors may find their ways into basic textbooks and remain in the foundations of entire theories. Comay admits that he himself took part in this shortsightedness for decades, and only after discovering additional contradictions in Yukawa’s theories, he decided to thoroughly examine the theoretical foundations of Yukawa’s particle.

The Klein-Gordon Equation in the test of experiment

In 1934, when Pauli and Weiskopf recycled the Klein-Gordon Equation, very little was known about elementary particles, and in this respect, theoretical physicists were groping in the dark. Even Pauli admitted that his revived Klein-Gordon equation “had little to do with reality” ([2], page 70).

In the 1940s, several such particles have been discovered, the most famous of which were a group of 3 particles called pi-mesons, or in short – Pions. The discovery of these particles was followed by a major rush of physicists toward the work of Pauli and Weiskopf.

In the 1960s, the existence of quarks was recognized and established. It turned out that these quarks actually compose the pions, among others, and that they were actually the building blocks of all the spinless particles discovered in experiments. Hence, no spinless elementary particle is known today. We are therefore back to point zero, when Pauli complained in 1934 that there are no known particles for which the Klein-Gordon equation can apply. On the other hand – the point-like massive elementary particle family grew substantially, and it turns out that they all have spin 1/2, as implied by Pauli’s “enemy”- the Dirac equation.

The errors of the Klein-Gordon Equation

The problematic nature of the Klein-Gordon’s Equation is not limited to the real functions used by Yukawa. There are fundamental flaws in the complex solutions of the equation as well. Some of these flaws had already been mentioned by Dirac.

Comay’s article ([3], chapter 4) mentions at least 3 additional contradictions. The most interesting of them shows how physicists, while being fully aware of the ensemble of Physics’ laws, fail to understand the deep connections between these laws. The point is this: the Lagrangian density and the Variational Principle yield a differential equation which constitutes a particle’s equation of motion. Another differential equation can be obtained from a quantity called “Hamiltonian”, which yields a first order differential equation in time.

A non-physicist reader should realize that Lagrangian and Hamiltonian are not simply the names of famous Armenian physicists. These are physical and mathematical notions, which yield two differential equations describing the same particle. Both are derived from the same original equation – but each of them is obtained from a different mathematical development. Fortunately, it turns out, that with the Dirac equation, these two equations miraculously unite !!

The Klein-Gordon Equation wasn’t as lucky. The two equations deriving from it do not unite, and even end up contradicting each other. Therefore, the Klein-Gordon Equation cannot be correct…

Another flaw specified in this article (understandable only to physicists) is that it is not possible to build a consistent inner product for Hilbert Space, because the density function of a complex Klein-Gordon particle depends not only on its wave function, but also on the potential defined by an external source. This potential can vary with time and disrupt the definition of the Hilbert Space inner product.

One of the common features of natural laws, is that if a theory is wrong – nature has many ways to show it.

Why can there be no Higgs

The Higgs Boson is a massive, elementary, spinless particle as well, predicted some 50 years ago and not yet found. Its existence is crucial for the Standard Model Theory. Many billions of dollars have been invested until today in order to find it, and many billions are still expected to be spent for the same purpose in the few coming years.

The Higgs Boson Equation is a kind of an extension of the Klein-Gordon Equation. Both equations describe a spinless particle. And the flaws characterizing the Klein- Gordon equation apply to the Higgs equation as well. The reliance of Physics on the Mathematical sciences is deep and fundamental (cf. discussion of this issue in Mario Livio’s book: “Is God a Mathematician“?). Therefore, the Higgs boson, predicted by a flawed mathematical instrument, does not exist.

The W and Z Bosons

The W and Z bosons are two spin-1 massive particles. The Standard Model assumes these two are elementary particles that carry the weak force. The Standard Model equations of the two particles suffer from similar problems as the Higgs boson equations. The issue can be noticed once one considers the classical counterpart of these equations, namely the Proca Lagrangian. The theory associated with this Lagrangian describes a massive photon. The matter described by the Proca Lagrangian is fundamentally different than matter as is known to science [4]. This casts doubt on its validity, and may be one theoretical reason behind the failure to find such matter. Indeed, the experimental upper limit of the photon mass is 23 orders of magnitude lower than that of the electron.

Electrodynamics and the strong force are the domains of Comay’s main scientific work, leaving the weak force domain to others. However, as you may have noticed by now, he likes asking questions when he notices theoretical difficulties. Comay recommends that the scientific community keeps an open mind when interpreting the W and Z bosons findings. As an example, the theoretical issues described above stem from the assumption that W and Z are elementary particles. Dropping this assumption opens the door to other interpretations, such as that W and Z particles are not elementary Bosons, but Mesons of the Top quark. The fact that the Top quark is more massive than the W and Z particles serves as a positive sign regarding such a direction, and so is the absence of any Top quark meson from the PDG table of Mesons. Comay believes that the last word has not yet been said regarding these questions, and it is worthwhile to leave it open for a further consideration in the future.

[1] G. Farmelo, “The Strangest Man” (Basic Books, New York, 2009)

[2] A. I. Miller “Early Quantum Electrodynamics” (University Press, Cambridge, 1994)

[3] E. Comay, “Physical Consequences of Mathematical Principles“, (Progress in Physics, October 2009 Vol 4)

[4] E. Comay, Nuovo Cimento, B113, 733 (1998)