The Nonlinear Quantum Field Theory as a Generalization of Standard Model

(Geometrical Approach)



The brilliance and poverty

of the Standard Model


  Quantum field theory (QFT), known as the Standard Model (SM), is a contemporary theory of elementary particles and their interactions. This is a very remarkable but strange theory.

On one hand, in the majority of cases, its formulas are confirmed by experiments to a very high accuracy. Since in natural science experimental confirmation is the criterion that is used to establish theories’ validity, the majority of SM formulas, without a doubt, are accurate. Therefore, theoretical physicists and experimenters who work in the area of physics of elementary particles can be proud of SM.

On the other hand, in recent decades, people became aware of the fact that SM is not perfect. Specifically, this otherwise very effective theory has two disappointing disadvantages.

1) There are at least 18 parameters in SM which cannot be calculated, but have to be introduced “by hand” through experimental data. Thus, a need exists to calculate them.

The most serious difficulty is the calculation of particles’ masses. In SM, the particles’ masses are originally equal to zero. However, a special mathematical apparatus - “Higgs’s mechanism” – has been developed to describe the procedure of mass acquisition by particles. The experiments did not yet confirm this “mechanism”, thus making aspects of SM theory questionable.

2) SM, as a theory, does not have an axiomatic structure.

Examples of successful physical axiomatic theories are Newtonian mechanics and classical electrodynamics. In these theories, all formulas that are necessary for the calculation of physical values are derived on the basis of several postulates (or axioms).

At the same time, the structure of quantum theories, in particular SM, is non-axiomatic. One of the creators of SM, Nobel Prize winner Murray Gell-Mann, said briefly and evocatively the following about this property of QFT. (“Questions for the future” in the “The nature of matter”, Wolfson College Lectures 1980 (Oxford, 1981)):

 “Quantum mechanics, this mysterious, confusing discipline, which none of us really understand, but we know how to use… It wonderfully works in the description of physical reality, but, as sociologists would say, this is an anti-intuitive discipline. Quantum mechanics is not theory, but it is a framework, which, as we assume, must contain any correct theory.”

Other well-known physicists characterize QFT as a collection of prescriptions for calculating the necessary parameters of elementary particles.

It is essential to realize that it is perfectly acceptable for the theory to have such a structure if we want to use it for technical applications. However, this structure does not allow us to answer some questions within the framework of SM.

For example, among these questions are the following: what is the origin of particles’ mass; why fundamental particles are points; why the wave function does not have a physical sense, and many others.

The fact that SM is “not theory, but only a framework, which… must contain any correct theory”, makes us want to construct a completely axiomatic theory of elementary particles.

In particular, superstring theory was planned to be such a theory, but the absence of predictions based on superstring theory makes its validity questionable. (You can read about the situation with superstring theory and SM in Peter Woit  “String Theory and the Crisis in Particle Physics”.)

If we assume that there is an axiomatic theory of elementary particles, then what must it be?

Obviously, it must contain a limited number of axioms, like Euclidean geometry, Newtonian mechanics, or Maxwell-Lorentz’s electrodynamics. This will allow us to obtain all necessary results of the theory as theorems.

At the same time, all mathematical results of this theory must completely coincide with, or be equivalent to, the corresponding mathematical results of the Standard Model, which have been experimentally verified.

For example, such a theory must not only confirm the Heisenberg uncertainty principle, but it also must find reasons for its existence. It cannot deny the truthfulness of the statistical interpretation of a wave function, but it must also give its physical interpretation. An axiomatic theory must explain the origin of particles’ mass, but it must allow the mass calculation to be formulated in the form of Higgs’ mechanismxe "Higgs mechanism". Even if Higgs' bosonxe "boson" will never be found, the mathematicians’ description of Higgs’ mechanism can exist in the theory in order to explain the appearance of masses of gauge fieldsxe "gauge fields" in electro-weak theory, and so on. It seems that such a theory can be built.

In order not to violate the principle of superposition, for a long time physicists considered that all fundamental theories must be linear. Specifically, quantum field theory was built on this basis. However, strictly speaking, in nature all processes are nonlinear.

Nonlinearity is the basic feature of the proposed axiomatic theory.  At the same time, SM is shown to be a linear approximation of this nonlinear theoryxe "nonlinear theory". It appears that this nonlinearity is characteristic of physics of elementary particles. It also seems that all difficulties of the existing theory of elementary particles are related to nonlinearity, since we are attempting to describe our nonlinear world using a linear model. By taking nonlinearity into account, the proposed theory answers questions that are not addressed in SM. At the same time, it is not in  conflict with any experimentally verified results of SM.

The proposed theory is based on the fact that wave functions of elementary particles are special nonlinear wave fields. In a normalized form, they can be interpreted statistically as in SM. These fields are non-point fields, but they can be presented in the form of points, thus preserving the ideas of SM for point particles.

The uncertainty principle is valid for these fields in the same way as for any wave packets. However, at the same time, nonlinear “packets” do not have spreading, since they are monochromatic.

This theory describes the appearance of particles’ masses without requiring the presence of Higgs’ vacuum. Instead, the usual physical vacuum is sufficient here, in particular, electromagnetic vacuum. At the same time, the mathematics of spontaneous symmetry breakdown in nonlinear theoryxe "nonlinear theory" can be presented in a way similar to Higgs' mechanism, and so on.

The primary attention of physicists is now focused on the search for the Higgs bosonxe "boson", which is the particle whose existence is necessary for the Standard Model to be complete. Physicists hope that finding this particle will make it possible to begin the study of the structure of our world on a new, deeper level, “beyond the Standard Model”. However, recent results have revealed serious obstacles to achieving these goals.

Early results (2001) obtained at LEP (Europe), and latest results (2009) obtained at Tevatron in Fermilab (Fermixe "Fermi" National Accelerator Laboratory, USA) disproved the existence of the standard Higgs bosonxe "boson" for a wide range of masses (see illustration of these results on

The LHC (Large Hadron Collider, Europe) was built specifically to search for Higgs' boson,xe "boson" at a cost of over ten billion dollars. By the end of 2011, when, as it is assumed, the LHC will provide its first results, Tevatron must almost completely check the remaining range of masses where the existence of the standard Higgs’s boson is possible. If the Tevatron does this, and no Higgs’ boson is  found  (which is very likely), it will mean that Higgs’ mechanismxe "Higgs mechanism" is implemented in nature in some other way. In this case, to construct the “Beyond Standard Model” theory, it is desirable to supplement the program of experiments at LHC by alternative, robust, theoretical research in order not to spend time and money in vain.

The proposed theory makes it possible to complete the construction of the theory of elementary particles, and facilitate the construction of physics “beyond the Standard Model”, since it does not originally contain the Higgs bosonxe "boson" and other difficulties inherent in the Standard Model.

This book presents a concise version of a similar book in published in Russian.  The author is planning to publish a complete version of the theory in subsequent editions, or another book.


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Last modified: 01/10/21