Nonperturbative quantization approach for QED on the Hopf bundle

We consider the Dirac equation and Maxwell's electrodynamics in $\mathbb{R} \times S^3$ spacetime, where a three-dimensional sphere is the Hopf bundle $S^3 \rightarrow S^2$. In both cases, discrete spectra of classical solutions are obtained. Based on the solutions obtained, the quantization of free, noninteracting Dirac and Maxwell fields is carried out. The method of nonperturbative quantization of interacting Dirac and Maxwell fields is suggested. The corresponding operator equations and the infinite set of the Schwinger-Dyson equations for Green's functions is written down. To illustrate the suggested scheme of nonperturbative quantization, we write a simplified set of equations describing some physical situation. Also, we discuss the properties of quantum states and operators of interacting fields.


I. INTRODUCTION
Quantum electrodynamics (QED) and the electroweak theory are very successful in explaining quantum phenomena for electromagnetic and weak interactions. Their predictions agree with experimental data to great precision. This progress was achieved despite the fact that the calculations are perturbative and one needs to involve, for example, a renormalization procedure. R. Feynman called this procedure "sweeping the garbage under the rug." L. Landau et al. [1] wrote on this subject: "Although at present there exist methods to remove these singularities (regularization), which clearly lead to correct results, such method of action has the artificial nature. The singularities arise in the theory due to the pointlike interaction described by delta functions (operators of interacting fields are taken at one point)." In Refs. [2][3][4], he and his co-authors study this question trying to remove such singularities in QED.
Attempts to use this technique for strong interactions do not lead to a full success, and in gravity they are unsuccessful. This suggests that at present we do not clearly understand the nature of quantization. It is reasonable to assume that there should exist some well-defined mathematical procedure of quantization which can be applied to any field theory. This cannot be a perturbative technique used, for instance, in QED, since it leads to nonrenormalizable theories. This suggests that the aforementioned procedure of quantization, applicable to any field theory, must be nonperturbative. It should be pointed out here that the nonperturbative quantization technique was perhaps first suggested by W. Heisenberg in Ref. [5] where he proposed an idea that an electron can be described by a nonperturbatively quantized nonlinear spinor field.
Taking all this into account, it is of great interest to construct nonperturbative QED for some simple case in order to compare perturbative and nonperturbative QED. It would allow one to understand the physical essence of such phenomena like the renormalization, convergence of the Feynman integral, etc. One can assume that such a possibility may arise in constructing QED on some compact manifold, since in this case spectra of eigenvalues of the Dirac and Maxwell equations will presumably be discrete. This would permit one to replace the Fourier integral in a general solution by a summation over quantum numbers that give the number of the eigenvalues. In doing so, Dirac delta functions would be replaced by the Kronecker symbols, and all calculations would be simplified; this presumably would permit one to construct nonperturbative QED. Simultaneously, it would be possible to construct perturbative QED according to conventional methods, but taking into account the compactness of three-dimensional space. After that, there would appear the possibility of comparison of perturbative and nonperturbative quantum theories.
In the present paper, we find a discrete spectrum of classical solutions describing noninteracting Dirac and Maxwell fields in a spacetime with a spatial cross-section in the form of the Hopf bundle. Then we use the spectra obtained to quantize the Dirac equation and Maxwell's electrodynamics. Finally, we suggest a scheme of nonperturbative quantization of coupled Dirac and Maxwell fields.
The paper is organized as follows. In Sec. II, we give the Lagrangian, field equations, and Ansatze for the Dirac and Maxwell equations on the Hopf bundle. Using them, we obtain classical solutions to the Dirac and Maxwell equations separately. In Sec. III, we quantize free, noninteracting fields and obtain expressions for the corresponding propagators. In Sec. IV, we carry out the nonperturbative quantization on the Hopf bundle. Finally, in Sec. V, we discuss the results obtained and list the important problems in nonperturbative quantum field theory.

II. CLASSICAL ELECTRODYNAMICS PLUS THE DIRAC EQUATION
In this section we consider classical electrodynamics coupled to spinors obeying the Dirac equation in R × S 3 spacetime with a spatial cross-section in the form of the Hopf bundle S 3 → S 2 . One can say that a relativistic quantum theory of an electron interacting with an electromagnetic field and living on the the Hopf bundle is under consideration.

A. Ansatze and field equations
In this subsection we closely follow Ref. [6]. Consider Dirac-Maxwell theory with the source of electromagnetic field taken in the form of a massless Dirac field. The corresponding Lagrangian can be chosen in the form (hereafter, we work in units where = c = 1) with the covariant derivative ψ ;µ ≡ ∂ µ + 1/8 ω abµ γ a γ b − γ b γ a − ieA µ ψ, where γ a are the Dirac matrices in flat space (below we use the spinor representation of the matrices); a, b and µ, ν are tetrad and spacetime indices, respectively; F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic field tensor; A µ are four-potentials of the electromagnetic field; e is a charge in Maxwell theory. In turn, the Dirac matrices in curved space, γ µ = e µ a γ a , are derived using the tetrad e µ a , and ω abµ is the spin connection [for its definition, see Ref. [7], formula (7.135)]. Varying the corresponding action with the Lagrangian (1), one can derive the following set of equations: where g is the determinant of the metric tensor and j µ = eψγ µ ψ is the four-current. The above equations will be solved in R × S 3 spacetime with the Hopf coordinates χ, θ, ϕ on a sphere S 3 with the metric where dS 2 3 is the Hopf metric on the unit S 3 sphere; r is a constant; 0 ≤ χ, ϕ ≤ 2π and 0 ≤ θ ≤ π. A detailed description of the Hopf bundle can be found in Appendix A.
Then, because of the presence of the factors e inχ and e imϕ in Eq. (5), the spinors ψ mn and ψ pq with different pairs of indices (m, n) and (p, q) are orthogonal.
To solve the equations, we use the tetrad coming from the metric (4). On substitution of the Ansatze (5) and (6) in Eqs. (2) and (3), we have where the prime denotes differentiation with respect to θ. Notice that the set of equations (7)-(11) must be solved as an eigenvalue problem for Ω with the eigenfunctions Θ 1,2 .
We seek a solution of the above equations in the following particular form: and the eigenparameter Ω is given by the expression It is seen from this formula that the energy of the system Ω depends on n only, that is, the energy spectrum is degenerate with respect to m. For the above particular solution, Eq. (11) is identically satisfied, and the remaining equations give the set of equations for the functions Θ, φ, ρ, and λ, where new dimensionless variablesρ,λ,φ = er (ρ, λ, φ) andΘ = er 3/2 Θ have been introduced. From the Maxwell equations (14) and (15), it then follows thatφ = ∓2ρ.

B. Classical vacuum solutions to the Dirac equation
In this subsection we consider the case of "frozen" electric and magnetic fields with zero scalar and vector potentials, φ = ρ = λ = 0. The solution of Eq. (13) is then [8,9] (Θ 2 ) nm = ± (Θ 1 ) nm = ±Θ nm = ± C r 3/2 sin α θ 2 cos β θ 2 with Here, we explicitly choose the integration constant in the form C/r 3/2 so that the constant C would be dimensionless. The spinors ψ mn and ψ pq are orthogonal for (m, n) = (p, q). Consider the normalization condition for the upper sign in (17), from which one can find the value of the normalization constant The condition of the positiveness of the Γ functions leads to the following inequalities for the quantum numbers m and n and the energy spectrum: For the lower sign in the solution (17), we have the following spectrum with the corresponding restrictions on the quantum numbers m, n: Similar Ansatz for the Weyl spinor yields the solution For the upper sign, we have the following possible values of the quantum numbers m and n: Correspondingly, for the lower sign n 0, |m| n, In this subsection we consider classical solutions with the "frozen" spinor field ψ = 0. The four-potential A µ for the Maxwell equations (3) can be written in the form where we have chosen the gauge A θ = 0, and p, q are integers. Using this potential, the Maxwell equations yield This set of equations must be regarded as an eigenvalue problem with the eigenfunctions φ pqn , ρ pqn , λ pqn and the eigenvalue ω pqn , where the integer n numbers the eigenvalues of ω for fixed values of the integers p and q. This set of equations has the following discrete symmetries: Below we consider regular solutions, as well as singular solutions supported by a pointlike charge and a current located at the points θ = 0, π.

Regular solutions
Deriving an analytic solution to the set of equations (27)-(30) runs into great difficulty. Therefore, in this subsection, we first find numerical solutions and then show that there are analytic solutions for some particular values of the numbers p, q and of the functions φ, ρ, and λ.
a. Particular solutions. To perform numerical computations, it is necessary to assign the values of the functions and their derivatives at the point θ ≪ 1. To do this, let us seek a solution in the form Then Eq. (29) yields the following restriction on the parameters φ 0 , λ 0 , and ρ 0 : with α = p + q and β = 2 + α. . For all graphs Here, it is possible to find an analytic solution. In this case the Maxwell equations (27)-(30) reduce to one equation where 2 F 1 is the hypergeometric function. This solution is regular for the following eigenvalues of ω: where n is integer.

Singular solutions
Singular solutions are supported by a pointlike charge and a pointlike current, and for them p = q = ω = 0. In this case the Maxwell equations (27)-(30) take the form where a i and c j are integration constants. In the above expressions, the terms with ln sin θ 2 describe the electric and magnetic fields created by the pointlike charge and current located at the point θ = 0 where the potentials diverge. Similarly, one can regard the terms with ln cos θ 2 as describing the electric and magnetic fields created by the pointlike charge and current located at the point θ = π.
The components of the electric, E θ = −∂ θ φ, and magnetic, H χ,ϕ , fields are The first term in the right-hand side of Eq. (37) describes the electric field created by the pointlike charge q located at the point θ = π, and the second term -the field created by the pointlike charge q located at the point θ = 0. Thus the solutions (34)-(36) and (37)-(39) describe the scalar/vector potentials and electric/magnetic fields created by two pointlike charges and currents located at the points θ = 0, π.

III. QUANTIZATION OF LINEAR FIELDS
In this section we consider the quantization of the Dirac field and Maxwell's electrodynamics in R × S 3 spacetime where a spatial cross-section S 3 is the Hopf bundle S 3 → S 2 .
The distinctive feature of the field theories under consideration on the Hopf bundle is that a three-dimensional sphere S 3 is a compact space. This results in the fact that the noninteracting field systems in question (spinor Dirac field and Maxwell's electrodynamics) have discrete spectra of solutions; in both cases, this enables us to write a general solution as a discrete sum over the corresponding eigenvalues. This is the principle difference compared to Minkowski space where a general solution is given by the Fourier integral. The replacement of the integral by the sum should lead to a considerable simplification of the quantization process.

A. Quantization of the Dirac field
Consistent with the discrete spectrum of the solutions (17), a general solution to the Dirac equation in the classical case can be represented as a sum of these solutions numbered by the integers m, n. Then the field operatorsψ and ψ † can be written in the form where the spinors u s and v s describe the states with the spin ±1/2, respectively, The function Θ nm is defined according to Eqs. (17), (18), and (20) as Θ nm = C nm r 3/2 sin |n|±m−1/2 θ 2 cos |n|∓m−1/2 θ 2 , |m| |n| , and the energy given by Eqs. (19) and (21) for the first term in (40) is and for the second term in (40) [given by Eqs. (24) and (26)] is The operatorb s nm describes the annihilation of a particle with the spin s, the energy rΩ n = 1/2 + 2 |n|, and the quantum numbers n, m. Correspondingly, the operator bs nm † describes the creation of such a particle. Similarly, the operatorĉ s nm describes the annihilation of a particle with the spin s, the energy rΩ n = −1/2 − 2 |n|, and the quantum numbers n, m, and the operator (ĉ s nm ) † describes the creation of such a particle. We impose the following anticommutation relations on these operators: where f nm is a numerical factor depending on n and m, and this factor is chosen such that the infinite sum over n in the propagator (45) would be convergent. The fermion propagator is defined as usual by the expression whereS αβ = s u s αū s β = s v s αv s β is a constant matrix. Note that this propagator is not translationally invariant, since it contains the product Θ nm (θ)Θ nm (θ ′ ).
Consider the convergence of the series (45) on summation over n and m. The analytic study of the convergence is an extremely complex problem; for this reason, we examine this question numerically. For the simplest case of x µ = x ′µ (where x µ = t, χ, θ, ϕ) and with θ = θ ′ , we have studied the behavior of the function S αβ (0, 0, θ, θ, 0) for the finite sum To calculate the Hamilton operator, let us write out its density Then, after standard calculations, we arrive at the following expression: The question of whether the last term in the square brackets leads to the divergence or not requires a special study, since Ω n can be both positive and negative.

B. Quantization of Maxwell's electrodynamics
In Sec. II C 1, the numerical study of solutions within classical electrodynamics has been carried out, and we gave arguments to claim that these solutions form a discrete spectrum. The same situation takes place for the Dirac equation as well. The reason for that is that a three-dimensional sphere on the Hopf bundle S 3 → S 2 is a compact space; as a result, the spectra of solutions of the Dirac and Maxwell equations are discrete.
For such a case, the operator of the electromagnetic field four-potential can be written in the form As for the anticommutation relations (42) and (43), we have introduced here the numerical factor f pq , which will possibly be needed to ensure the finiteness of the sum over the quantum states pq. The momentum operators conjugate to the potential A µ are defined aŝ π 0 = 0, × ω (λ pqn cos θ + ρ pqn ) − φ pqn (p + q cos θ) sin 2 θ , × ω (λ pqn + ρ pqn cos θ) − φ pqn (p + q cos θ) sin 2 θ .
Let us now calculate the commutators: It must be mentioned here that the Green functions G µν are not translationally invariant, since they contain the product Ã µ pqn (θ) Ã ν pqn (θ ′ ). These expressions permit us to calculate the Feynman Green's function

IV. NONPERTURBATIVE QUANTIZATION OF MAXWELL'S ELECTRODYNAMICS COUPLED TO A SPINOR FIELD
In this section we suggest a method of nonperturbative quantization of Maxwell-Dirac theory on the Hopf bundle. As we saw in Sec. III, a fact of compactness of a three-dimensional sphere S 3 very much simplifies the procedure of quantization of free fields: when quantizing, no Dirac delta functions appear, since they are replaced by the Kronecker symbols. This would lead us to expect that the quantization of interacting fields on a sphere S 3 will be considerably simplified as well. Notice also that there appears a considerable difference in the behavior of quantum fields on a sphere S 3 and in Minkowski space: the propagators of free fields on the compact space are not translationally invariant.
According to Heisenberg [5], the process of nonperturbative quantization consists in that a set of equations describing interacting fields [in our case, these are Eqs. (2) and (3)] is written in the operator form where the covariant derivative of the spinor field, the operators of the field strength and of the current are defined aŝ It is worth to point out that the above equations involve the operators of the interacting fieldsψ andÂ µ , whose properties differ from those of free, noninteracting fields considered in the previous section. Notice also the presence of the nonlinear quantitiesÂ µψ andψγ µψ . The former describes the interaction between the fields, and the latter is the source of the electromagnetic field. These nonlinear quantities do not allow to quantize the fields ψ and A µ , as was done in Sec. III. For free fields, we have discrete spectra of solutions, a linear combination of which permits one to find any solution to the Maxwell or Dirac equations. With such spectra in hand, one can quantize the fields by writing the operatorsψ andÂ µ as a superposition of solutions of the discrete spectrum and by introducing the creation and annihilation operators of the corresponding quantum states which are the coefficients before each such solution. In Minkowski space, such operators are called the creation/annihilation operators for particles. But in our case we cannot speak of particles, since the functions (22) and (33) do not correspond to plane waves.
When quantizing free fields, we saw that the propagators (45) and (46) are ordinary (not distribution) functions. These propagators do not involve Dirac delta functions, and therefore the product of two operators at one point is well defined. This was demonstrated for the propagator S αβ (0, 0, θ, θ, 0) in Sec. III A. It is reasonable to expect that this property also persists for the operators of interacting fields. Hence, the products of the interacting operatorŝ A µψ andψγ µψ for the discrete spectrum on the Hopf bundle are well defined, in contrast to the same product of operators of free fields in the case of perturbative quantization in Minkowski space.
According to Ref. [5], the main idea of nonperturbative quantization consists in that the operator equations (47) and (48) are replaced by an infinite system for all Green's functions. The first equations are quantum average of Eqs. (47) and (48). They contain the Green functions G(A µ (x), ψ(y)) = Â µ (x)ψ(y) and G(ψ(x), ψ(y)) = ψ (x)ψ(y) . In order to close the set of equations, it is necessary to derive equations for these Green functions. To do this, the operator equations (47) and (48) are multiplied by the corresponding operators and are averaged; this is done an infinite number of times. As a result, one arrives at the following infinite set of equations: . . .
Note that the Green function G(A µ (x), ψ(y)) is a function of two variables, and for its definition we need two equa- It is evident that this infinite set of equations cannot be solved explicitly and analytically. Therefore, the question arises as to whether it is possible to find its approximate solution. The problem of how to cut off an infinite set of equations is called the closure problem, and it is well known in turbulence modeling (see, e.g., the textbook [10]). In that case, the Navier-Stokes equation is averaged, and it is known as the Reynolds-averaged Navier-Stokes equation. But this equation contains an unknown quantity -the Reynolds-stress tensor, for which one has to have an extra equation called the Reynolds-stress equation, which in turn contains more unknown functions, and so on.
For better understanding of the situation, it is useful to consider some simple example illustrating this process. Consider the case where, to solve the set of equations (49)-(54), one can employ the Ansatze (5) for the spinor field and (6) for the potential of the electromagnetic field. Replacing the functions by operators, we then havê In this case, similarly to the situation with the classical fields (13)-(16), we obtain the following equations coming from the quantum equations (49)-(54): After quantum averaging, the first equation (55) will be the equation for Θ , the second one -for φ , the third one -for ρ , and the fourth one -for λ . But these equations contain the following new Green's functions: for which one must have their own equations. The equation for the Green function G Θρ (x, y) can be obtained by multiplying the operator equation (55) on the right byρ and by performing the quantum averaging, where we have introduced the following notation: HereΘ denotes that the derivative is taken with respect to the function Θ with the bar. The Green function G Θρ (θ 1 , θ 2 ) is a function of two variables θ 1 and θ 2 ; hence, one has to have one more differential equation for the variable θ 2 . This equation can be obtained by multiplying the equation (57) on the left byΘ and by performing the quantum averaging, Similarly, one can obtain equations for the Green functions G Θλ (θ 1 , θ 2 ) and G ΘΘ (θ 1 , θ 2 ). As a result, we arrive at an infinite set of equations for an infinite number of Green's functions . . . = . . .
As expected, Eqs. (63), (65), and (67) contain new 3-point Green's functions, for which in turn one has to have differential equations determining these Green's functions. As a result of this process, we finally get an infinite set of differential equations describing all Green's functions of the operatorsΘ,ρ,φ, andλ appearing in the operator equations (55)-(58). As was mentioned above, the infinite set of equations obtained can scarcely be solved explicitly, and hence the question of its approximate solving arises. Using the experience accumulated in turbulence modeling, one can assume that this can be done by cutting off the infinite set of equations to a finite one using some physical assumptions concerning higher-order Green's functions. In doing so, one can involve the following suppositions: • One can neglect n−th order Green's functions compared with (n − 1)−th order Green's functions.
• n−th order Green's functions are polylinear combinations of lower-order Green's functions.
• One can use the energy conservation law together with some physically reasonable propositions concerning its separate components.

V. DISCUSSION AND CONCLUSIONS
We have considered the Dirac equation and Maxwell's electrodynamics in R × S 3 spacetime. The distinctive feature of these theories on the Hopf bundle is that they have discrete spectra of solutions both for the Dirac equation and for Maxwell's electrodynamics. This is a consequence of the fact that a three-dimensional sphere S 3 is a compact space. For the Dirac equation, this was shown explicitly by finding discrete solutions in analytic form. For the Maxwell equations, we have obtained numerical solutions, and the numerical analysis permits us to assume that a discrete spectrum of solutions does exist.
The presence of the discrete spectrum allows one to quantize the free, noninteracting Dirac and Maxwell fields on the Hopf bundle. For the Dirac equation, the quantization is suggested by introducing the creation and annihilation operators for the corresponding quantum states. The standard anticommutation relations are imposed on these operators, and an additional numerical factor in the right-hand side of the anticommutator is introduced. Using these relations, the propagator for the spinor field is calculated. The calculations indicate that this propagator is a sum over the discrete spectrum numbered by the quantum numbers m and n. In order to ensure the convergence of the sum, it is necessary that the aforementioned factor would possess a perfectly definite dependence on the quantum number n.
For the free electromagnetic field, a similar scheme of quantization has been suggested.
The most important part of the present study is the procedure of nonperturbative quantization suggested for the interacting Dirac and Maxwell fields. Following Heisenberg [5], we have replaced the classical equations by equations for operators of the corresponding fields. Since the operator equation can scarcely be solved somehow, it is replaced by an infinite set of equations for Green's functions. Such a set is known as the Schwinger-Dyson equations, but they are usually employed in perturbative quantum field theory.
To illustrate the suggested scheme of nonperturbative quantization, we have considered some physical system possessing perfectly definite Ansatze for the spinor and electromagnetic fields. This gives the much more simple set of equations, for which we have written out the first few equations for 1-and 2-point Green's functions. For the equations describing 2-point Green's functions, we have explicitly written out 3-point Green's functions appearing in these equations.
The significance of examination of the scheme of nonperturbative quantization is that if, in nature, some fields are quantized, then apparently there should exist a mathematically well-defined quantization procedure for any fields, including those that are not quantized due to the perturbative nonrenormalizibilty of the theory.
Let us note some features of the nonperturbative quantization.
• The properties of the operators of interacting fields can differ drastically from those of free fields. For example, in the quantum theory of strongly interacting fields, there can exist static field configurations similar to those described by soliton, monopole, instanton, etc. solutions in classical field theory.
• The properties of the operators of interacting fields cannot be assigned by their commutators/anticommutators. The algebra of these fields is much more complicated compared with the algebra given only by commutators/anticommutators. These properties are determined by the infinite set of the Schwinger-Dyson equations as a whole.
• For strongly interacting quantum fields, it is impossible to introduce creation and annihilation operators, since the field operators cannot be represented as a superposition of plane waves with the coefficients that are creation and annihilation operators.
• Separate consideration of the notion of quantum state is required, since in perturbative quantum field theory quantum states are defined using creation and annihilation operators. According to what has been said in the previous item, such operators cannot be defined for strongly interacting fields, and hence the definition of quantum states requires special consideration.
• It is possible that the properties of the operators of interacting fields and of quantum states are related to the properties of the complete set of Green's functions defined by the Schwinger-Dyson equations.