Gauge and Lorentz Transformation Placed on the Same Foundation

Abstract

We show that a “dynamical” interaction for arbitrary spin can be constructed in a straightforward way if gauge and Lorentz transformations are placed on the same foundation. As Lorentz transformations act on space-time coordinates, gauge transformations are applied to the gauge field. Placing these two transformations on the same ground means that all quantized field like spin-1/2 and spin-3/2 spinors are functions not only of the coordinates but also of the gauge field components. As a consequence, on this stage the (electromagnetic) gauge field has to be considered as classical field. Therefore, standard quantum field theory cannot be applied. Despite this inconvenience, such a common ground is consistent with an old dream of physicists almost a century ago. Our approach, therefore, indicates a straightforward way to realize this dream.

1. Introduction

After the formulation of general relativity which explained fources on a geometric ground, physicists and mathematicians tried to incorporate the electromagnetic interaction into this geometric picture. Weyl claimed that the action integral of general relativity is invariant not only under space-time Lorentz transformations but also under the gauge transformation, if this is incorporated consistently [1]. However, the theories at that time were not ready to incorporate this view. Nowadays, we see more clearly that all physical variables (like position, momentum, etc.), quantum wave functions, and fields transform as finite-dimensional representations of the Lorentz group. The reason is that interactions between fundamental particles (as irreducible representations of the Poincaré group) are most conveniently formulated in terms of field operators (i.e., finite-dimensional representations of the Lorentz group) if the general requirements like covariance, causality, and so forth are to be incorporated in a consistent way. The relation between these two groups and their representations is given by the Lorentz-Poincaré connection [2]. In this paper we show that if gauge transformation is put on the same foundation, the resulting nonminimal “dynamical” interaction obeys all necessary symmetries which for higher spins are broken if the interaction is introduced by the usual minimal coupling.

In Section 2 we explain details of the Poincaré group which are necessary in the following. In Section 3 we deal with linear wave equations as objects to the Lorentz transformation. In Section 4 we introduce the external electromagnetic field by a nonsingular transformation. In Section 5 we specify the nonlinear transformation by the claim of gauge invariance of the Poincaré algebra. Finally, in Section 6 we give our conclusions.

2. The Poincaré Group

Relativistic field theories are based on the invariance under the Poincaré group 𝒫_1,3 (known also as inhomogeneous Lorentz group ℐℒ [2–11]). This group is obtained by combining Lorentz transformations Λ and space-time translations 𝑎_𝑇, (𝑎,Λ)≡𝑎_𝑇Λ∶𝔼_1,3∋𝑥^𝜇⟶Λ^𝜇𝜈𝑥^𝜈+𝑎^𝜇∈𝔼_1,3.(2.1) The group’s composition law (𝑎₁,Λ₁)(𝑎₂,Λ₂)=(𝑎₁+Λ₁𝑎₂,Λ₁Λ₂) generates the semidirect structure of 𝒫_1,3, 𝒫_1,3=𝒯_1,3⊙ℒ,(2.2) where 𝒯_1,3 is the abelian group of space-time translations (i.e., the additive group ℝ⁴) and ℒ={Λ∶detΛ=+1,Λ⁰⁰≥1} is the proper orthochronous Lorentz group acting on the Minkowski space 𝔼_1,3 with metric 𝜂_𝜇𝜈=diag(1,−1,−1,−1).(2.3) The condition of the metric to be invariant under Lorentz transformations Λ takes the form Λ^𝜇𝜌𝜂_𝜇𝜈Λ^𝜈𝜎=𝜂_𝜌𝜎.(2.4) Under the Lorentz transformation Λ∈ℒ the transformation of the covariant functions 𝜓 according to a representation 𝜏(Λ) of the Lorentz group [3–16] is determined by the commutative diagram (2.5) That is,𝑇(Λ)𝜓(𝑥)=(𝜏(Λ)𝜓)(Λ𝑥)≡𝜓^Λ(Λ𝑥).(2.6) The map 𝑇∶Λ→𝑇(Λ) is a finite-dimensional representation of ℒ. If we parametrize the element Λ∈ℒ by Λ(𝜔)=exp(−(1/2)𝜔_𝜇𝜈𝑒^𝜇𝜈) where the Lorentz generators are given by 𝑒_𝜇𝜈^𝜌𝜎=−𝜂^𝜌𝜇𝜂_𝜈𝜎+𝜂_𝜇𝜎𝜂^𝜌𝜈(2.7) and 𝜔^𝜇𝜈=−𝜔^𝜈𝜇 are six independent parameters, the parametrization of 𝑇 reads 𝑇(Λ(𝜔))=exp−𝑖2𝜔^𝜇𝜈𝑠^𝜇𝜈.(2.8) The Lorentz group ℒ is noncompact. As a consequence, all unitary representations are infinite dimensional. In order to avoid this, we introduce the concept of 𝐻-unitarity (see, e.g., [9] and references therein). A finite representation 𝑇 is called 𝐻-unitary if there exists a nonsingular Hermitian matrix 𝐻=𝐻^† so that 𝑇^†(Λ)𝐻=𝐻𝑇⁻¹(Λ)⟺𝑠^†𝜇𝜈𝐻=𝐻𝑠_𝜇𝜈.(2.9) Notice that an 𝐻-unitary metric is always indefinite, so that the inner product ⟨,⟩ generated by 𝐻 is sesquilinear sharing the hermiticity condition ⟨𝜓,𝜑⟩=⟨𝜑,𝜓⟩^∗. The most famous case of 𝐻-unitarity is given in the Dirac theory of spin-1/2 particles where 𝐻=𝛾⁰.

For an operator 𝒪 [17, 18] acting on the 𝜓-space of covariant functions (we have to impose the action on covariant functions because in case of higher spins the relations between operators we obtain are valid only as weak conditions) the transformation 𝜏(Λ) in (2.6) is a covariant transformation if the diagram(2.10)

is commutative, that is,𝜏(Λ)𝒪𝜏⁻¹(Λ)(Λ𝑥)(𝜏(Λ)𝜓)(Λ𝑥)=𝑇(Λ)𝒪(𝑥)𝜓(𝑥).(2.11) Using (2.6) we obtain 𝜏(Λ)𝒪𝜏⁻¹(Λ)(Λ𝑥)𝑇(Λ)𝜓(𝑥)=𝑇(Λ)𝒪(𝑥)𝜓(𝑥).(2.12) Notice that the covariance of the transformation embodies only the property of equivalence of reference systems. The covariant operator 𝒪 is invariant under transformation (2.6) if in addition 𝜏(Λ)𝒪𝜏⁻¹(Λ)=𝑂. As a consequence we obtain the commutative diagram (2.13) or 𝒪(Λ𝑥)𝑇(Λ)𝜓(𝑥)=𝑇(Λ)𝒪(𝑥)𝜓(𝑥) which means 𝒪(Λ𝑥)𝑇(Λ)=𝑇(Λ)𝒪(𝑥)(2.14) on the 𝜓-space. The invariance is a symmetry of the physical system and implies the conservation of currents. In particular, the symmetry transformations leave the equations of motion form-invariant.

While the Lorentz transformation 𝑇(Λ) changes the wave function 𝜓 itself as well as the argument of this function (cf. (2.6)), the proper Lorentz transformation 𝜏(Λ) causes a change of the wave function only. On the ground of infinitesimal transformations, this change is performed by the substantial variation. Starting from an arbitrary infinitesimal coordinate transformation Λ(𝛿𝜔)∶𝑥^𝜇→𝑥^𝜇+𝛿𝜔^𝜇𝜈𝑥_𝜈, the substantial variation is given by [13] 𝛿₀𝜓(𝑥)≡𝜓^(𝑥)−𝜓(𝑥)=−𝑖2𝛿𝜔^𝜌𝜎𝑀_𝜌𝜎𝜓(𝑥),(2.15) where 𝑀_𝜌𝜎=ℓ_𝜌𝜎+𝑠_𝜌𝜎, ℓ_𝜌𝜎=𝑖(𝑥_𝜌𝜕_𝜎−𝑥_𝜎𝜕_𝜌). The corresponding finite proper Lorentz transformation can be written as 𝜏(Λ(𝜔))=exp−𝑖2𝜔^𝜇𝜈𝑀^𝜇𝜈,(2.16) and the multiplicative structure of the group generates the adjoint action Ad_𝜏(Λ)∶𝑀_𝜇𝜈⟶𝜏⁻¹(Λ)𝑀_𝜇𝜈𝜏(Λ)=Λ^𝜌𝜇Λ^𝜎𝜈𝑀_𝜌𝜎.(2.17) Due to (2.9) the generators 𝑠_𝜌𝜎 fulfill 𝑠^†𝜌𝜎𝐻=𝐻𝑠_𝜌𝜎. They depend on the spin of the field but not on the coordinates 𝑥_𝜇. Therefore, we have [ℓ_𝜇𝜈,𝑠_𝜌𝜎]=0. If a generic element of the translation group is written as exp+𝑖𝑎_𝜇𝑃^𝜇,(2.18) the commutator relations of the Lie algebra are given by 𝑀_𝜇𝜈,𝑀_𝜌𝜎=𝑀𝑖𝜂_{𝜇𝜎𝜇𝜈}𝑀_𝜈𝜌+𝜂_𝜈𝜌𝑀_𝜇𝜎−𝜂_𝜇𝜌𝑀_𝜈𝜎−𝜂_𝜈𝜎𝑀_𝜇𝜌,,𝑃_𝜌𝑃=_𝜇𝑖𝜂_𝜈𝜌𝑃_𝜇−𝜂_𝜇𝜌𝑃_𝜈,,𝑃_𝜈=0.(2.19) The Casimir operators of the algebra are 𝑃²=𝑃_𝜇𝑃^𝜇 and 𝑊²=𝑊_𝜇𝑊^𝜇, where 𝑊^𝜇=+12𝜖^{𝜇𝜈𝜌𝜎}𝑀_𝜈𝜌𝑃_𝜎(2.20) is the Pauli-Lubanski pseudovector, [𝑃_𝜇,𝑊_𝜈]=0. In coordinate representation we have 𝑃_𝜇=𝑖𝜕_𝜇, and the finite Poincaré transformation has the form 𝜏(𝑎,Λ)∶𝜓(𝑥)⟶(𝜏(𝑎,Λ)𝜓)(𝑥)=𝑇(Λ)𝜓Λ⁻¹(𝑥−𝑎).(2.21) This relation constitutes the Lorentz–Poincaré connection [2]. While the representation 𝑇 generally generates a reducible representation of 𝒫_1,3, the spectra of the Casimir operators 𝑃² and 𝑊² determine the mass and spin content of the system.

3. The Wave Equations

As an operator 𝒪 in the above sense we consider the operator of the wave equation. The Dirac-type wave equation we will consider has the form 𝒟(𝜕)𝜓(𝑥)≡𝑖𝛽^𝜇𝜕_𝜇−𝜌𝜓(𝑥)=0,(3.1) where 𝜓 is an 𝑁-component function, 𝛽^𝜇(𝜇=0,1,2,3), and 𝜌 are 𝑁×𝑁 matrices independent of 𝑥. Following Bhabha's conception [19], it is “…logical to assume that the fundamental equations of the elementary particles must be first-order equations of the form (3.1) and that all properties of the particles must be derivable from these without the use of any further subsidiary conditions.”

The principle of relativity states that a change of the reference frame cannot have implications for the motion of the system. This means that (3.1) is invariant under Lorentz transformations. Equivalently, the Lorentz symmetry of the system means the covariance and form-invariance of (3.1) under the transformation in (2.6), that is, the transformed wave equation is equivalent to the old one. Therefore, we require that every solution 𝜓^Λ(Λ𝑥) of the transformed equation 𝒟^Λ(Λ𝜕)𝜓^Λ(Λ𝑥)=0(3.2) can be obtained as Lorentz transformation of the solution 𝜓(𝑥) of (3.1) in the original system and that the solutions in the original and transformed systems are in one-to-one correspondence. The explicit form of the covariance follows from (2.11), 𝜏(Λ)𝒟𝜏⁻¹(Λ)(Λ𝜕)(𝜏(Λ)𝜓)(Λ𝑥)=𝑇(Λ)𝒟(𝜕)𝜓(𝑥)=0,(3.3) and leads to the explicit Lorentz transformations 𝛽^Λ𝜇=Λ^𝜇𝜌𝑇(Λ)𝛽^𝜌𝑇⁻¹(Λ),𝜌^Λ=𝑇(Λ)𝜌𝑇⁻¹(Λ).(3.4) The Lorentz invariance is given by the substitution 𝒟(𝜕)𝜓(𝑥)=0⟶^(2.6)𝒟(𝜕)𝜓^Λ(𝑥)=0.(3.5) or 𝑇⁻¹(Λ)𝛽^𝜇𝑇(Λ)=Λ^𝜇𝜌𝛽^𝜌,𝑇⁻¹(Λ)𝜌𝑇(Λ)=𝜌.(3.6) The difference of the original and transformed wave equation is given by the wave equation where the wave function 𝜓 is replaced by the substantial variation 𝛿₀𝜓, 𝐷(𝜕)𝛿₀𝜓(𝑥)=0. As a consequence we obtain [𝐷,𝑀^𝜌𝜎]=0 or 𝛽^𝜇,𝑠^𝜌𝜎=𝑖(𝜂^𝜇𝜌𝛽^𝜎−𝜂^𝜇𝜎𝛽^𝜌),𝜌,𝑠^𝜌𝜎=0.(3.7) An excellent discussion of such matrices 𝛽 can be found in [13, 19–23]. The hermiticity of the representation 𝑇 in (2.9) implies the hermiticity of (3.1). Including a still unspecified Hermitian matrix 𝐻 the hermiticity condition reads 𝒟(𝜕)^†𝐻=^!(𝒟(𝜕)𝐻)^†=𝐻𝒟(−𝜕) or 𝛽^𝜇†𝐻=𝐻𝛽^𝜇,𝜌𝐻=𝐻𝜌.(3.8) Writing 𝜓=𝜓^†𝐻, one obtains the adjoint equation 𝜓𝒟−^←𝜕=𝜓−𝑖𝛽^𝜇←𝜕_𝜇−𝜌=(𝐻𝒟(𝜕)𝜓)^†=0.(3.9)

4. Introduction of the External Field

It may be reasonable to introduce an external field directly into the Poincaré algebra which can be applied to classically understand the elementary particle. To do so one has to transform the generators of the Poincaré group to be dependent on the external field in such a way that the new, field-dependent generators obey the commutation relations (2.19). As it was proposed by Charkrabarti [24] and Beers and Nickle [25], the simplest way to build such a field-dependent algebra is to introduce the external field 𝐴 by a nonsingular transformation Ad_𝒱(𝐴)∶𝑝_1,3⟶𝑝^𝑑1,3(𝐴)=𝒱(𝐴)𝑝_1,3𝒱⁻¹(𝐴).(4.1) In case of a particular external electromagnetic field 𝐴, the external field can be introduced by using an evolution operator 𝒱(𝐴), called the “dynamical” representation [26, 27]. By analogy with the free-particle case one can realize this representation on the solution space of relativistically invariant equations. Expressing the operators explicitly in terms of free-field operators, one obtains the “dynamical” interaction. Applying, for instance, the operator 𝒱(𝐴) to (3.1) one obtains 𝒱(𝐴)∶𝒟(𝜕)𝜓(𝑥)=0⟶𝒟^𝑑(𝜕,𝐴)Ψ(𝑥,𝐴)=0,(4.2) where 𝒟^𝑑(𝜕,𝐴)=𝒱(𝐴)𝒟(𝜕)𝒱⁻¹(𝐴) and Ψ(𝑥,𝐴)=𝒱(𝐴)𝜓(𝑥)(4.3) (here and in the following we will skip the argument 𝑥 for Ψ and the argument 𝜕 for 𝐷^𝑑). Having introduced the external gauge field 𝐴, we introduce gauge covariance on the same foundation as Lorentz covariance in (2.6), that is, by claiming that the diagram (4.4) is commutative, that is,Ψ^𝜆(𝐴+𝜕𝜆)=𝐺(𝜆)Ψ(𝐴).(4.5) According to (2.13), the “dynamical” interaction 𝒟^𝑑 is gauge invariant under the gauge transformation 𝐴→𝐴^𝜆≡𝐴+𝜕𝜆 if the diagram (4.6)

is commutative, that is,𝒟^𝑑(𝐴+𝜕𝜆)Ψ^𝜆(𝐴+𝜕𝜆)=𝐺(𝜆)𝒟^𝑑(𝐴)Ψ(𝐴).(4.7) Together with (4.5) we obtain 𝒟^𝑑(𝐴+𝜕𝜆)𝐺(𝜆)Ψ(𝐴)=𝐺(𝜆)𝒟^𝑑(𝐴)Ψ(𝐴) or 𝒟^𝑑(𝐴+𝜕𝜆)𝐺(𝜆)=𝐺(𝜆)𝒟^𝑑(𝐴)(4.8) on the 𝜓-space. Note that up to now we have not specified the explicit shape of the finite-dimensional representation 𝐺∶𝜆→𝐺(𝜆) of the gauge group.

5. Specifying 𝒱(𝐴) by Gauge Invariance

At this point we specify 𝒱(𝐴) by two claims. Due to gauge symmetry as a fundamental principle the dynamical transformation 𝒱 has to be compatible with the gauge transformation. Therefore, we first claim the gauge invariance in (4.8) not only for the operator 𝒟^𝑑 but for the whole dynamical Poincaré algebra 𝑝^𝑑1,3(𝐴), 𝑝^𝑑1,3(𝐴+𝜕𝜆)𝐺(𝜆)=𝐺(𝜆)𝑝^𝑑1,3(𝐴).(5.1) By using (4.1) and multiplying by 𝐺(𝜆)⁻¹ from the right we obtain 𝒱(𝐴+𝜕𝜆)𝑝_1,3𝒱⁻¹(𝐴+𝜕𝜆)=𝐺(𝜆)𝒱(𝐴)𝑝_1,3(𝐺(𝜆)𝒱(𝐴))⁻¹.(5.2) This means that the first claim is fulfilled if 𝑉(𝐴+𝜕𝜆)=𝐺(𝜆)𝑉(𝐴).(5.3) On the other hand, with (4.3) and (4.5) we obtain 𝒱^𝜆(𝐴+𝜕𝜆)𝜓(𝑥)=𝐺(𝜆)𝒱(𝐴)𝜓(𝑥)(5.4) and, therefore, 𝒱^𝜆=𝒱 on the 𝜓-space. To summarize, by the first claim the gauge symmetry determines the gauge properties of 𝒱(𝐴) and, therefore, of the interacting field Ψ(𝐴).

The second claim is that the dynamical transformation operator 𝒱(𝐴) should be of Lorentz type, that is, for the generators 𝑠_𝜇𝜈 of the Poincaré algebra 𝑝_1,3 one has 𝒱(𝐴)𝑠^𝜇𝜈𝒱⁻¹(𝐴)=𝑉^𝜇𝜌(𝐴)𝒱^𝜈𝜎(𝐴)𝑠^𝜌𝜎(5.5) which is a local extension of (2.17). 𝑉(𝐴)=𝑉(𝑥,𝐴) is the local Lorentz transformation generated by the external field 𝐴 and obeying 𝑉_𝜇𝜌(𝐴)𝑉^𝜇𝜎(𝐴)=𝑉_𝜌𝜇(𝐴)𝑉^𝜇𝜎(𝐴)=𝜂_𝜌𝜎.(5.6) If such a local Lorentz transformation exists, the problem is solved. Therefore, in the following we make the attempt to find explicit realizations of the local Lorentz transformation 𝑉_𝜇𝜈(𝐴). It is hard to find the Lorentz transformation 𝑉_𝜇𝜈(𝐴) in general. However, as first shown by Taub [28], in the case of a plane-wave field we obtain 𝑉_𝜇𝜈(𝐴)=𝜂_𝜇𝜈−𝑞𝑘_𝑃𝐺_𝜇𝜈−𝑞²2𝑘^2𝑃𝐴²𝑘_𝜇𝑘_𝜈,(5.7) where 𝑞 is the electric charge of the particle and 𝐺_𝜇𝜈=𝑘_𝜇𝐴_𝜈−𝑘_𝜈𝐴_𝜇. The plane-wave field 𝐴_𝜇=𝐴_𝜇(𝜉), 𝜉=𝑘𝑥 is characterized by its lightlike propagation vector 𝑘_𝜇, 𝑘²=0, and its polarization vector 𝑎^𝜇 such that 𝑎²=−1 and 𝑘𝑎=0. The operator 𝑘_𝑃≡𝑘_𝜇𝑃^𝜇 commutes with any other and has a special role in the theory. For particles with nonzero mass one has 𝑘_𝜇𝑃^𝜇≠0. Therefore, for the plane wave the differential operator 1/𝑘_𝑃 is local and well defined for the plane-wave solution 𝜓_𝑃 of the Klein-Gordon equation. In all other cases, 1/𝑘_𝑃 is assumed to exist.

Note that the plane-wave solution of the Dirac equation was found more than 70 years ago by Wolkow [29] and extended later on to a field of two beams of electromagnetic radiation [30, 31]. However, these approaches did not make use of the nonsingular transformation 𝒱(𝐴). The realization of 𝒱(𝐴) can be achieved by the nonsingular transformation 𝒱(𝐴)=𝒱₀(𝐴)𝒱_𝑠(𝐴), where 𝒱₀(𝐴)=exp−𝑖𝑑𝜉2𝑘_𝑃2𝑞(𝐴𝑃)−𝑞²𝒱_𝑠𝐴²,(𝐴)=exp−𝑖𝑞2𝑘_𝑃𝐺_𝜇𝜈𝑠^𝜇𝜈.(5.8) It has to be mentioned that the evolution operator 𝒱(𝐴) may be chosen to be 𝐻-unitary according to the representation 𝑇 in (2.9), that is,𝒱^†(𝐴)𝐻=𝐻𝒱⁻¹(𝐴).(5.9) Considering the nonsingular transformation of Dirac-type wave equation 𝒱(𝐴)∶𝛽^𝜇𝑃_𝜇−𝑚𝜓=0⟶Γ^𝜇(𝐴)Π_𝜇(𝐴)−𝑚Ψ(𝐴)=0,(5.10) with the help of (5.8) the “dynamical” counterparts to the operator 𝑃_𝜇=𝑖𝜕_𝜇 can be calculated to be Π_𝜇(𝐴)=𝒱(𝐴)𝑃_𝜇𝒱⁻¹(𝐴), ^𝑃_𝜇^⟶Π_𝜇^(𝐴)=𝑃_𝜇^+𝑘_𝜇^𝑞_{2𝑘𝑃𝑞𝐴}^{2−2𝐴𝑃−𝐹,𝑃(5.11)}_{2⟶Π2(𝐴)=(}/_{𝑃−𝑞𝐴)2−𝑞}^𝐹/^(5.12) (^𝐹/^{≡𝑠𝜇𝜈𝐹}_𝜇𝜈) while the “dynamical” counterpart to 𝛽^𝜇 is given by Γ^𝜇(𝐴)=𝒱(𝐴)𝛽^𝜇𝒱⁻¹(𝐴), Γ^𝜇(𝐴)=𝑉^𝜇𝜈(𝐴)𝛽^𝜈=𝛽_𝜇−𝑞𝑘_𝑃2𝑘𝑞_𝑃𝐴²𝑘_𝜇𝑘_𝜈+𝐺_𝜇𝜈𝛽^𝜈.(5.13) In terms of Π_𝜇(𝐴) and Γ^𝜇(𝐴) we have 𝒟^𝑑(𝐴)Ψ(𝐴)=Γ^𝜇(𝐴)Π_𝜇(𝐴)−𝑚Ψ(𝐴)=0.(5.14) However, expressed in terms of 𝐷_𝜇=𝑃_𝜇−𝑞𝐴_𝜇 and 𝛽^𝜇, we obtain ^{𝒟𝑑𝛽(𝐴)Ψ(𝐴)≡𝜇𝐷}_𝜇^−𝑞_2𝑘𝑃^𝑘//^{𝐹−𝑚Ψ(𝐴)=0,(5.15)} where ^𝑘/^{≡𝛽𝜇𝑘}_𝜇. This interaction is nonminimal. However, as we have shown before, it is determined completely by the claim of gauge invariance.

Note that due to the antimutation of the 𝛾-matrices, in the spin-1/2 case the dynamical interaction in (5.15) reduces to the minimal coupling. However, in order to obtain the correct values of the gyromagnetic factor, in some cases the (phenomenological) Pauli term 𝛾_𝜇𝛾_𝜈𝐹^𝜇𝜈 has to be added by hand to the minimal coupling of the Dirac equation (see also [32, page 109]). In case of plane waves the exact solution of this (supplemented) Dirac equation as given by Charkrabarti [24] obeys the same gauge invariance condition Ψ(𝐴+𝜕𝜆)=𝐺(𝜆)Ψ(𝐴). This property is found also in the book by Fried [33].

Finally, as a consequence of the explicit form (5.8), the associated transformation of the evolution operator 𝒱(𝐴) under the local gauge transformation for the plane wave field, 𝐴_𝜇(𝜉)⟶𝐴_𝜇(𝜉)+𝜕_𝜇𝜆(𝜉),(5.16) becomes 𝒱(𝐴)⟶𝒱(𝐴+𝜕𝜆)=𝑒^{−𝑖𝑞𝜆}𝒱(𝐴).(5.17) As an example of higher spin, the spin-3/2 case is considered in detail in [34]. As it turns out, the Rarita-Schwinger spin-3/2 equation on the presence of a “dynamical” interaction is algebraically consistent and causal.

6. Conclusions

As a consequence of gauge invariance and Lorentz type of 𝒱(𝐴) we obtain (1)the invariance of the wave function under gauge transformations, Ψ^𝜆(𝐴+𝜕𝜆)=𝒱^𝜆(𝐴+𝜕𝜆)𝜓=𝒱(𝐴+𝜕𝜆)𝜓=Ψ(𝐴+𝜕𝜆),(6.1) that is, Ψ^𝜆=Ψ, (2)the explicit shape of 𝐺(𝜆) in (4.5), Ψ^𝜆(𝐴+𝜕𝜆)=𝒱(𝐴+𝜕𝜆)𝜓=𝑒^{−𝑖𝑞𝜆}𝒱(𝐴)𝜓=𝑒^{−𝑖𝑞𝜆}Ψ(𝐴),(6.2) that is, 𝐺(𝜆)=𝑒^{−𝑖𝑞𝜆}, (3)the invariance of 𝒟^𝑑 under gauge transformations from (4.7) and 𝒟^𝑑(𝐴+𝜕𝜆)Ψ^𝜆(𝐴+𝜕𝜆)=𝒟^𝑑(𝐴+𝜕𝜆)𝑒^{−𝑖𝑞𝜆}Ψ(𝐴),(6.3) that is, 𝒟^𝑑(𝐴+𝜕𝜆)𝐺(𝜆)=𝐺(𝜆)𝒟^𝑑(𝐴) on the 𝜓-space, (4)the “dynamical” interaction for any spin as given by ^{𝒟𝑑𝛽(𝐴)Ψ(𝐴)=𝜇𝐷}_𝜇^−𝑞_2𝑘𝑃^𝑘//^{𝐹−𝑚Ψ(𝐴)=0(6.4)} being nonminimal but completely determined by gauge invariance, thereby causing Poincaré symmetry,(5)as a consequence of (5.12), the gyromagnetic factor in the presence of a “dynamical” interaction as being 𝑔=2 for any spin [27].

Let us close again with Weyl In [1] he honestly confessed: “Die entscheidenden Folgerungen in dieser Hinsicht verschanzen sich aber noch hinter einem Wall mathematischer Schwierigkeiten, den ich bislang nicht zu durchbrechen vermag.” (“However, the crucial consequences in this respect entrench oneself still behind a bank of mathematical difficulties which up to now I am not able to penetrate.”) We hope that our work breaks a small bay into this mathematical bank.

Acknowledgments

The work is supported by the Estonian target financed Project no. 0180056s09 and by the Estonian Science Foundation under grant no. 8769. S. Groote acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) under Grant 436 EST 17/1/06.