Advances in Mathematical Physics

Advances in Mathematical Physics / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 652126 |

Rein Saar, Stefan Groote, Hannes Liivat, Ilmar Ots, "Gauge and Lorentz Transformation Placed on the Same Foundation", Advances in Mathematical Physics, vol. 2011, Article ID 652126, 12 pages, 2011.

Gauge and Lorentz Transformation Placed on the Same Foundation

Academic Editor: H. Neidhardt
Received25 Mar 2011
Accepted14 Apr 2011
Published13 Jul 2011


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 𝒫1,3 (known also as inhomogeneous Lorentz group [211]). This group is obtained by combining Lorentz transformations Λ and space-time translations 𝑎𝑇, (𝑎,Λ)𝑎𝑇Λ𝔼1,3𝑥𝜇Λ𝜇𝜈𝑥𝜈+𝑎𝜇𝔼1,3.(2.1) The group’s composition law (𝑎1,Λ1)(𝑎2,Λ2)=(𝑎1+Λ1𝑎2,Λ1Λ2) generates the semidirect structure of 𝒫1,3, 𝒫1,3=𝒯1,3,(2.2) where 𝒯1,3 is the abelian group of space-time translations (i.e., the additive group 4) and ={ΛdetΛ=+1,Λ001} is the proper orthochronous Lorentz group acting on the Minkowski space 𝔼1,3 with metric 𝜂𝜇𝜈=diag(1,1,1,1).(2.3) The condition of the metric to be invariant under Lorentz transformations Λ takes the form Λ𝜇𝜌𝜂𝜇𝜈Λ𝜈𝜎=𝜂𝜌𝜎.(2.4) Under the Lorentz transformation Λ the transformation of the covariant functions 𝜓 according to a representation 𝜏(Λ) of the Lorentz group [316] is determined by the commutative diagram 652126.fig.00a(2.5) That is,𝑇(Λ)𝜓(𝑥)=(𝜏(Λ)𝜓)(Λ𝑥)𝜓Λ(Λ𝑥).(2.6) The map 𝑇Λ𝑇(Λ) is a finite-dimensional representation of . If we parametrize the element Λ by Λ(𝜔)=exp((1/2)𝜔𝜇𝜈𝑒𝜇𝜈) where the Lorentz generators are given by 𝑒𝜇𝜈𝜌𝜎=𝜂𝜌𝜇𝜂𝜈𝜎+𝜂𝜇𝜎𝜂𝜌𝜈(2.7) and 𝜔𝜇𝜈=𝜔𝜈𝜇 are six independent parameters, the parametrization of 𝑇 reads 𝑖𝑇(Λ(𝜔))=exp2𝜔𝜇𝜈𝑠𝜇𝜈.(2.8) The Lorentz group 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=𝐻𝑇1(Λ)𝑠𝜇𝜈𝐻=𝐻𝑠𝜇𝜈.(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 diagram652126.fig.00b(2.10)

is commutative, that is,𝜏(Λ)𝒪𝜏1(Λ)(Λ𝑥)(𝜏(Λ)𝜓)(Λ𝑥)=𝑇(Λ)𝒪(𝑥)𝜓(𝑥).(2.11) Using (2.6) we obtain 𝜏(Λ)𝒪𝜏1(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)𝒪𝜏1(Λ)=𝑂. As a consequence we obtain the commutative diagram 652126.fig.00c(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)𝑥𝜇𝑥𝜇+𝛿𝜔𝜇𝜈𝑥𝜈, the substantial variation is given by [13] 𝛿0𝜓(𝑥)𝜓𝑖(𝑥)𝜓(𝑥)=2𝛿𝜔𝜌𝜎𝑀𝜌𝜎𝜓(𝑥),(2.15) where 𝑀𝜌𝜎=𝜌𝜎+𝑠𝜌𝜎, 𝜌𝜎=𝑖(𝑥𝜌𝜕𝜎𝑥𝜎𝜕𝜌). The corresponding finite proper Lorentz transformation can be written as 𝑖𝜏(Λ(𝜔))=exp2𝜔𝜇𝜈𝑀𝜇𝜈,(2.16) and the multiplicative structure of the group generates the adjoint action Ad𝜏(Λ)𝑀𝜇𝜈𝜏1(Λ)𝑀𝜇𝜈𝜏(Λ)=Λ𝜌𝜇Λ𝜎𝜈𝑀𝜌𝜎.(2.17) Due to (2.9) the generators 𝑠𝜌𝜎 fulfill 𝑠𝜌𝜎𝐻=𝐻𝑠𝜌𝜎. They depend on the spin of the field but not on the coordinates 𝑥𝜇. Therefore, we have [𝜇𝜈,𝑠𝜌𝜎]=0. If a generic element of the translation group is written as exp+𝑖𝑎𝜇𝑃𝜇,(2.18) the commutator relations of the Lie algebra are given by 𝑀𝜇𝜈,𝑀𝜌𝜎𝜂=𝑖𝜇𝜎𝑀𝜈𝜌+𝜂𝜈𝜌𝑀𝜇𝜎𝜂𝜇𝜌𝑀𝜈𝜎𝜂𝜈𝜎𝑀𝜇𝜌,𝑀𝜇𝜈,𝑃𝜌𝜂=𝑖𝜈𝜌𝑃𝜇𝜂𝜇𝜌𝑃𝜈,𝑃𝜇,𝑃𝜈=0.(2.19) The Casimir operators of the algebra are 𝑃2=𝑃𝜇𝑃𝜇 and 𝑊2=𝑊𝜇𝑊𝜇, where 𝑊𝜇1=+2𝜖𝜇𝜈𝜌𝜎𝑀𝜈𝜌𝑃𝜎(2.20) is the Pauli-Lubanski pseudovector, [𝑃𝜇,𝑊𝜈]=0. In coordinate representation we have 𝑃𝜇=𝑖𝜕𝜇, and the finite Poincaré transformation has the form Λ𝜏(𝑎,Λ)𝜓(𝑥)(𝜏(𝑎,Λ)𝜓)(𝑥)=𝑇(Λ)𝜓1(𝑥𝑎).(2.21) This relation constitutes the Lorentz–Poincaré connection [2]. While the representation 𝑇 generally generates a reducible representation of 𝒫1,3, 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)𝑖𝛽𝜇𝜕𝜇𝜓𝜌(𝑥)=0,(3.1) where 𝜓 is an 𝑁-component function, 𝛽𝜇(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 𝜓Λ(Λ𝑥) of the transformed equation 𝒟Λ(Λ𝜕)𝜓Λ(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)𝒟𝜏1(Λ)(Λ𝜕)(𝜏(Λ)𝜓)(Λ𝑥)=𝑇(Λ)𝒟(𝜕)𝜓(𝑥)=0,(3.3) and leads to the explicit Lorentz transformations 𝛽Λ𝜇=Λ𝜇𝜌𝑇(Λ)𝛽𝜌𝑇1(Λ),𝜌Λ=𝑇(Λ)𝜌𝑇1(Λ).(3.4) The Lorentz invariance is given by the substitution 𝒟(𝜕)𝜓(𝑥)=0(2.6)𝒟(𝜕)𝜓Λ(𝑥)=0.(3.5) or 𝑇1(Λ)𝛽𝜇𝑇(Λ)=Λ𝜇𝜌𝛽𝜌,𝑇1(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,𝑀𝜌𝜎]=0 or 𝛽𝜇,𝑠𝜌𝜎=𝑖(𝜂𝜇𝜌𝛽𝜎𝜂𝜇𝜎𝛽𝜌),𝜌,𝑠𝜌𝜎=0.(3.7) An excellent discussion of such matrices 𝛽 can be found in [13, 1923]. 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 𝛽𝜇𝐻=𝐻𝛽𝜇,𝜌𝐻=𝐻𝜌.(3.8) Writing 𝜓=𝜓𝐻, one obtains the adjoint equation 𝜓𝒟𝜕=𝜓𝑖𝛽𝜇𝜕𝜇𝜌=(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 Ad𝒱(𝐴)𝑝1,3𝑝𝑑1,3(𝐴)=𝒱(𝐴)𝑝1,3𝒱1(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)=0𝒟𝑑(𝜕,𝐴)Ψ(𝑥,𝐴)=0,(4.2) where 𝒟𝑑(𝜕,𝐴)=𝒱(𝐴)𝒟(𝜕)𝒱1(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 𝐷𝑑). 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 652126.fig.00d(4.4) is commutative, that is,Ψ𝜆(𝐴+𝜕𝜆)=𝐺(𝜆)Ψ(𝐴).(4.5) According to (2.13), the “dynamical” interaction 𝒟𝑑 is gauge invariant under the gauge transformation 𝐴𝐴𝜆𝐴+𝜕𝜆 if the diagram 652126.fig.00e(4.6)

is commutative, that is,𝒟𝑑(𝐴+𝜕𝜆)Ψ𝜆(𝐴+𝜕𝜆)=𝐺(𝜆)𝒟𝑑(𝐴)Ψ(𝐴).(4.7) Together with (4.5) we obtain 𝒟𝑑(𝐴+𝜕𝜆)𝐺(𝜆)Ψ(𝐴)=𝐺(𝜆)𝒟𝑑(𝐴)Ψ(𝐴) or 𝒟𝑑(𝐴+𝜕𝜆)𝐺(𝜆)=𝐺(𝜆)𝒟𝑑(𝐴)(4.8) on the 𝜓-space. Note that up to now we have not specified the explicit shape of the finite-dimensional representation 𝐺𝜆𝐺(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 𝒟𝑑 but for the whole dynamical Poincaré algebra 𝑝𝑑1,3(𝐴), 𝑝𝑑1,3(𝐴+𝜕𝜆)𝐺(𝜆)=𝐺(𝜆)𝑝𝑑1,3(𝐴).(5.1) By using (4.1) and multiplying by 𝐺(𝜆)1 from the right we obtain 𝒱(𝐴+𝜕𝜆)𝑝1,3𝒱1(𝐴+𝜕𝜆)=𝐺(𝜆)𝒱(𝐴)𝑝1,3(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 𝒱𝜆(𝐴+𝜕𝜆)𝜓(𝑥)=𝐺(𝜆)𝒱(𝐴)𝜓(𝑥)(5.4) and, therefore, 𝒱𝜆=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 𝑠𝜇𝜈 of the Poincaré algebra 𝑝1,3 one has 𝒱(𝐴)𝑠𝜇𝜈𝒱1(𝐴)=𝑉𝜇𝜌(𝐴)𝒱𝜈𝜎(𝐴)𝑠𝜌𝜎(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 𝑉𝜇𝜌(𝐴)𝑉𝜇𝜎(𝐴)=𝑉𝜌𝜇(𝐴)𝑉𝜇𝜎(𝐴)=𝜂𝜌𝜎.(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 𝑉𝜇𝜈(𝐴). It is hard to find the Lorentz transformation 𝑉𝜇𝜈(𝐴) in general. However, as first shown by Taub [28], in the case of a plane-wave field we obtain 𝑉𝜇𝜈(𝐴)=𝜂𝜇𝜈𝑞𝑘𝑃𝐺𝜇𝜈𝑞22𝑘2𝑃𝐴2𝑘𝜇𝑘𝜈,(5.7) where 𝑞 is the electric charge of the particle and 𝐺𝜇𝜈=𝑘𝜇𝐴𝜈𝑘𝜈𝐴𝜇. The plane-wave field 𝐴𝜇=𝐴𝜇(𝜉), 𝜉=𝑘𝑥 is characterized by its lightlike propagation vector 𝑘𝜇, 𝑘2=0, and its polarization vector 𝑎𝜇 such that 𝑎2=1 and 𝑘𝑎=0. The operator 𝑘𝑃𝑘𝜇𝑃𝜇 commutes with any other and has a special role in the theory. For particles with nonzero mass one has 𝑘𝜇𝑃𝜇0. Therefore, for the plane wave the differential operator 1/𝑘𝑃 is local and well defined for the plane-wave solution 𝜓𝑃 of the Klein-Gordon equation. In all other cases, 1/𝑘𝑃 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)𝒱𝑠(𝐴), where 𝒱0(𝐴)=exp𝑖𝑑𝜉2𝑘𝑃2𝑞(𝐴𝑃)𝑞2𝐴2,𝒱𝑠(𝐴)=exp𝑖𝑞2𝑘𝑃𝐺𝜇𝜈𝑠𝜇𝜈.(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=𝐻𝒱1(𝐴).(5.9) Considering the nonsingular transformation of Dirac-type wave equation 𝒱𝛽(𝐴)𝜇𝑃𝜇Γ𝑚𝜓=0𝜇(𝐴)Π𝜇Ψ(𝐴)𝑚(𝐴)=0,(5.10) with the help of (5.8) the “dynamical” counterparts to the operator 𝑃𝜇=𝑖𝜕𝜇 can be calculated to be Π𝜇(𝐴)=𝒱(𝐴)𝑃𝜇𝒱1(𝐴), 𝑃𝜇Π𝜇(𝐴)=𝑃𝜇+𝑘𝜇𝑞2𝑘𝑃𝑞𝐴22𝐴𝑃/𝐹,𝑃(5.11)2Π2(𝐴)=(𝑃𝑞𝐴)2𝑞/𝐹(5.12) (/𝐹𝑠𝜇𝜈𝐹𝜇𝜈) while the “dynamical” counterpart to 𝛽𝜇 is given by Γ𝜇(𝐴)=𝒱(𝐴)𝛽𝜇𝒱1(𝐴), Γ𝜇(𝐴)=𝑉𝜇𝜈(𝐴)𝛽𝜈=𝛽𝜇𝑞𝑘𝑃𝑞2𝑘𝑃𝐴2𝑘𝜇𝑘𝜈+𝐺𝜇𝜈𝛽𝜈.(5.13) In terms of Π𝜇(𝐴) and Γ𝜇(𝐴) we have 𝒟𝑑Γ(𝐴)Ψ(𝐴)=𝜇(𝐴)Π𝜇(𝐴)𝑚Ψ(𝐴)=0.(5.14) However, expressed in terms of 𝐷𝜇=𝑃𝜇𝑞𝐴𝜇 and 𝛽𝜇, we obtain 𝒟𝑑𝛽(𝐴)Ψ(𝐴)𝜇𝐷𝜇𝑞2𝑘𝑃/𝑘/𝐹𝑚Ψ(𝐴)=0,(5.15) where /𝑘𝛽𝜇𝑘𝜇. 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 𝛾𝜇𝛾𝜈𝐹𝜇𝜈 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, 𝐴𝜇(𝜉)𝐴𝜇(𝜉)+𝜕𝜇𝜆(𝜉),(5.16) becomes 𝒱(𝐴)𝒱(𝐴+𝜕𝜆)=𝑒𝑖𝑞𝜆𝒱(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, Ψ𝜆(𝐴+𝜕𝜆)=𝒱𝜆(𝐴+𝜕𝜆)𝜓=𝒱(𝐴+𝜕𝜆)𝜓=Ψ(𝐴+𝜕𝜆),(6.1) that is, Ψ𝜆=Ψ, (2)the explicit shape of 𝐺(𝜆) in (4.5), Ψ𝜆(𝐴+𝜕𝜆)=𝒱(𝐴+𝜕𝜆)𝜓=𝑒𝑖𝑞𝜆𝒱(𝐴)𝜓=𝑒𝑖𝑞𝜆Ψ(𝐴),(6.2) that is, 𝐺(𝜆)=𝑒𝑖𝑞𝜆, (3)the invariance of 𝒟𝑑 under gauge transformations from (4.7) and 𝒟𝑑(𝐴+𝜕𝜆)Ψ𝜆(𝐴+𝜕𝜆)=𝒟𝑑(𝐴+𝜕𝜆)𝑒𝑖𝑞𝜆Ψ(𝐴),(6.3) that is, 𝒟𝑑(𝐴+𝜕𝜆)𝐺(𝜆)=𝐺(𝜆)𝒟𝑑(𝐴) on the 𝜓-space, (4)the “dynamical” interaction for any spin as given by 𝒟𝑑𝛽(𝐴)Ψ(𝐴)=𝜇𝐷𝜇𝑞2𝑘𝑃/𝑘/𝐹𝑚Ψ(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.


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.


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