Canonical Quantization of Scalar Field in Terms of World Lines

Yuri A. Rylov

Institute for Problems in Mechanics, Russian Academy of Sciences,
101, bld.1 Vernadskii Ave. , Moscow, 117526, Russia.


It is not a usual practice to write prefaces for articles. But the present article is dissidentic, although nobody recognizes this, and that is why I would like to discuss the problems connected with this situation.

This paper states essentially that the conventional quantum field theory (QFT) is inconsistent. This inconsistency of QFT is a consequence of some additional constraint (identification of the energy and the Hamiltonian) which is incompatible with the dynamic equations for interacting fields.

The inconsistency of the conventional QFT manifests itself primarily in the non-stationarity of the vacuum state. Let me demonstrate the purpose of the present paper with a simple example.

Let us imagine some theory FT based on a set of base statements SBS. Let this theory FT be invalid, because SBS contains some incorrect statement F. Let, for instance, theory FT results in an invalid conclusion Y=a, whereas the experimentally correct conclusion is Y=b.

Let us consider another theory INCT which contains a set of basic statements SBS+INC, where INC is some statement incompatible with SBS. Let for concreteness the statement INC be 2+2=5, incompatible with the standard arithmetic rules contained in SBS. Let us show that using the theory INCT one can derive the correct conclusion Y=b from the incorrect conclusion Y=a by using valid logical and mathematical operations. Indeed, adding to Y=a the relation INC multiplied by b-a and adding 4(a-b) to both sides of the equality, one obtains the experimentally correct relation Y=b. The derivation of the experimentally correct relation gives a possibility to declare that the theory INCT is a true theory, because it is substantiated experimentally, although INCT is invalid twice: due to F and due to INC. But one mistake compensates another, and one has a "true" but fruitless theory.

Let somebody notice that the statement INC is incompatible with SBS and suggest to consider a theory without INC. Using FT, one obtains the experimentally incorrect conclusion Y=a. But investigating the incorrect theory FT, one has a chance to discover the incorrect statement F, whereas dealing with the inconsistent , but "experimentally substantiated" theory INCT, one does not have such a chance. At least, a discovery of F inside INCT is much more difficult, than inside FT.

Of course, this example is grotesque. The energy-Hamiltonian identification is incompatible only with the dynamic equations (but not with the rules of arithmetics as INC). Besides, the energy-Hamiltonian identification is incompatible with the dynamic equations only in the case when pair production effect is essential.

Let us apply this to the conventional QFT. Then FT is QFT, INC is the energy-Hamiltonian identification. An what is F? Apparently , F is an essential usage of the quantum axiomatics, in particular, application of the Heisenberg representation and the second quantization. This became clear quite recently, although the incompatibility of the energy-Hamiltonian identification with the dynamic equations of the QFT is known for as long as twenty five years (see. ref. [2,5] of the present paper).

The considered example illustrates the well known thesis that consistency, or inconsistency of a mathematical theory is a logical category which cannot be tested by a direct experiment, because it has nothing to do with the trueness of the theory. A consistent theory may be not valid, it may disagree with experimental data, but nevertheless it is a consistent theory. At the same time an inconsistent theory may agree with the experimental data, but in reality it is not a mathematical theory. It is rather a kind of intellectual game, because in the inconsistent theory one can derive practically any desirable result, provided enough ingenuity is shown. Such a capacity of an incosistent theory is its principal defect. If a theory is inconsistent, the question of whether or not this theory agrees with experimental data is incorrect.

I have provided this example because the reaction of the scientific community to the first publication (Ref.[2]) (I mean discussions with my colleagues) was unexpected and incomprehensible for me. This reaction looked approximately as follows:

"Please, develop your conception and demonstrate that the QFT without the energy-Hamiltonian identification is more effective than the conventional QFT. Then your conception will be considered and, maybe , accepted".

In other words my paper was considered as an improvement of the QFT (not as a criticism).

When the present paper was submitted to one of the most liberal physical journals, the paper was declined on the base of the referee's report which was very short and looked as follows:

As the author himself admits, the proposed quantization scheme runs into many difficulties, even if the classical picture looks sensible as a starting point. Essentially the author disproves the viability of his own theory. The paper is not worth publishing.

Submitting the paper to another physical journal, I have received the following answer:

We are sorry to say that after a careful review of your manuscript we find that our current publication program is not suited for it. We thank you for letting your paper for publication in ****.

We hope that you will not be discouraged by this decision. While physics is an exact science, physics publishing is not. Often a paper declined by one journal is published by another with different requirements. We wish you success in locating an appropriate medium for your research.

Due to our limited resources we cannot provide you with a review of the paper. We will keep your manuscript on file for your convenience for at least four years unless you request us to return them.

After comparing all responses to my papers concerning the energy-Hamiltonian identification, I recognized that something is wrong. But what? This could not be a mathematical or a logical mistake. It could not be unconscientiousness or incompetence of the referees. Finally I realized that the reason of such responses is in different ways of thinking. I use a good old classical way of thinking, whereas my opponents use another way of thinking which is characteristic to the researchers dealing with the quantum phenomena theory.

The old classical way of thinking clearly distinguishes between the incompatibility of the basic statements of a theory and incompatibility of the predictions of a theory with experimental data. Mutual compatibility of the basic statements (consistency of a theory) is more important than agreement of the predictions with experimental data, because predictions of a theory can be corrected within the same set of basic statements. For instance, it can be done by changing Lagrangian terms or using a different statement of a problem.

The way of thinking used in QFT does not distinguish logical incompatibility of the basic QFT statements from the discrepancy between theoretical predictions and experimental data. All basic statements are treated as equally possible, and agreement with experments is a sole criterion of the theory. Such approach is probably connected with a fact that many quantum axioms have no obvious meaning making it very difficult to understand whether or not they are mutually compatible. Researchers dealing with QFT are often working with quantum techniques without understanding what their manipulations mean. The only possible criterion of such activity is (quite obviously) agreement with experiments.

Now, when there are more arguments suggesting that the QFT should be constructed on the dynamic base without any reference to the quantum axiomatics and second quantization, the present paper can be considered as an additional argument in favour of a revision of the QFT.

Yuri Rylov
January, 1997


A scheme of quantization in terms of world lines (E-scheme) is proposed. Being a modification of the conventional quantization scheme (PA-scheme), the E-scheme does not need a use of perturbation theory techniques. Vacuum state is stationary in E-scheme. The total quantization problem is separated into problems which contain one, two,... etc. world lines. The scattering problem and bound states are investigated.

(There is the paper in the form of a postscript file. The figure for the paper is here: