Relativistic canonical quantization:
Questions and answers

D. A. Arbatsky

May, 2005
(revised in May, 2006)

Abstract

What is “Relativistic Canonical Quantization”?

Relativistic canonical quantization (RCQ) is a modern method of construction (“quantization”) of relativistic quantum fields from classical fields described by the language of invariant Hamiltonian (“canonical”) formalism.

This method has the following features:

• Quantization is performed constructively. Properties of a quantized field (for example, commutation relations) are not postulated, but follow from the construction. Consequently, we can say that in this case we deal with “representation” of relativistic canonical quantization.
• The construction is explicitly relativistic-invariant.
• The method makes possible to see, how quantum system acquires integrals of motion connected with symmetry group, like initial classical system. So, it becomes possible to formulate the mathematically rigorous analog of the Noether theorem.
• The method also allows to see anew the origin of discrete symmetries, including anti-unitary transformation of time reversal.
• The method naturally covers the possibility of quantization in a space with indefinite scalar product.

A more concrete answer to this question is given in the review What is “Relativistic Canonical Quantization”?.

The detailed description is given in the papers On quantization of electromagnetic field.

Is not RCQ a re-account of second quantization in Fock space in other terms?

For the aims of modern quantum field theory the method of second quantization in Fock space must be considered unsatisfactory.

At the outset of the quantum field theory it seemed (and Fock proceeded from this) that wave function is a mathematical object organically inherent in one quantum particle. Dirac even suggested the equation for electron proceeding from these ideas.

But later it became obvious that this is incorrect.

A quantum field can not be defined on the base of the theory of one quantum particle. As far as there is not any formal method (in the frame of Wigner-Mackey theory) for introduction of wave function, then (according to Fock) it is impossible to formally introduce local operators of quantized field. So, although some concrete quantum fields have been already quite correctly constructed, it should be understood that these particular constructions did not imply any general abstract mathematical scheme.

The method of RCQ have solved this problem completely. It turned out that, from abstract-algebraic point of view, quantum fields are objects derivative of classical fields (described by the language of the invariant Hamiltonian formalism), and not of quantum particles.

Does this mean that the Fock construction is incorrect?

No. It is correct to the extent in which it follows from the RCQ construction.

RCQ constructs quantum fields from classical, i. e. it derives properties of quantum objects from properties of classical objects. Can such a situation be considered satisfactory from methodological point of view?

Whether we like it or not, but with the appearance of the method of RCQ we are forced to recognize it as necessary that the whole quantum field theory turned out to be not so “quantum” as we would like it to think. The mankind just had not enough fantasy yet to deviate far enough from old ideas.

It seems appropriate to recall Dirac's words about quantum field theory:

„Нам нужна какая-то новая математика, столь же поразительная и непохожая на то, к чему мы привыкли, как некоммутативная алгебра Гейзенберга во времена, когда физики всё ещё работали с боровскими орбитами.“

[Taken from the Russian edition of: P. Dirac “The origin of quantum field theory”, in The impact of modern scientific ideas on society, Dordrecht, Holland: D. Reidel (1981). Sorry, I do not want to make my own reverse translation.]

Why is Gupta-Bleuler formalism unsatisfactory? What new has the method of RCQ given to quantization of electromagnetic field?

Gupta-Bleuler “formalism” can be called by this word only for the reason of its formality. It gives some formal relations for “operators” of quantum electromagnetic field, but it does not give any concrete answer to the question, in what functional space and how the symbols under consideration act. In other words, it does not define representation.

Of course, before the method of RCQ not everything was so bad. Using the analogy with the scalar field physicists have defined how “operators” act in the “space of wave functions”. But, in fact, the “space of wave functions” remained undefined as a functional space.

With appearance of the method of RCQ it became obvious that working only with Hilbert space is just a prejudice, which physicists clung to only from force of habit. It turned out that in some cases it is possible (though, not necessary) to manage with Hilbert space. But in the case of electromagnetic field everything is completely different.

It should be noticed also that the method of RCQ allows to look completely differently at the questions of gauge invariance and at the problem of arbitrariness in the choice of Lagrangian.

Why in the papers On quantization of electromagnetic field is so little attention devoted to distinction of usual and generalized functions? It is well known that operator amplitudes of a quantized field in a fixed point of space-time are not well-defined operators.

Once Bohr and Rosenfeld have used very delicate physical arguments to prove this fact. But from the point of view of the method of RCQ this is obvious and does not influence mathematical strictness of the construction.

It seems appropriate to notice also that the paper of Bohr and Rosenfeld have not become antiquated in any way with appearance of RCQ. Because the method of RCQ uses the concrete abstract scheme; and the indefiniteness of local operators (without results of Bohr and Rosenfeld) would arise bewilderment.

Is the method of RCQ applicable to non-linear fields and to interacting fields?

In the “Introductory remarks” in the papers On quantization of electromagnetic field I said: “we cannot quantize non-linear fields”.

As regards application of the method of RCQ to formal rows of perturbation theory, it is not only possible, but necessary.

How is the method of RCQ connected with the method of functional (path) integral?

RCQ is a tree. The method of functional integral is a branch of that tree (less rigorous yet). The fact that historically the branch was found first, and the tree was found later, should not puzzle too much: in dense mist everything is possible.

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