# Gauge and Lorentz Transformation Placed on the Same Foundation

Authors: Rein Saar; Stefan Groote; Hannes Liivat; Ilmar Ots
Journal: Advances in Mathematical Physics
 (2011)

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## 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.


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## Body

## 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 Section2 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 group𝒫1,3 (known also as inhomogeneous Lorentz group ℐℒ [2–11]). This group is obtained by combining Lorentz transformations Λ and space-time translations aT, (2.1)(a,Λ)≡aTΛ:E1,3∋xμ⟶Λνμxν+aμ∈E1,3.
The group’s composition law (a1,Λ1)(a2,Λ2)=(a1+Λ1a2,Λ1Λ2) generates the semidirect structure of 𝒫1,3, (2.2)P1,3=T1,3⊙L,
where 𝒯1,3 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 𝔼1,3 with metric (2.3)ημν=diag⁡(1,-1,-1,-1).
The condition of the metric to be invariant under Lorentz transformations Λ takes the form (2.4)ΛρμημνΛσν=ηρσ.
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,(2.6)T(Λ)ψ(x)=(τ(Λ)ψ)(Λx)≡ψΛ(Λx).
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)ωμνeμν) where the Lorentz generators are given by (2.7)(eμν)σρ=-ημρηνσ+ημσηνρ
and ωμν=-ωνμ are six independent parameters, the parametrization of T reads (2.8)T(Λ(ω))=exp⁡(-i2ωμνsμν).
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 (2.9)T†(Λ)H=HT-1(Λ)⟺sμν†H=Hsμν.
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 operator𝒪 [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,(2.11)(τ(Λ)Oτ-1(Λ))(Λx)(τ(Λ)ψ)(Λx)=T(Λ)O(x)ψ(x).
Using (2.6) we obtain (2.12)(τ(Λ)Oτ-1(Λ))(Λx)T(Λ)ψ(x)=T(Λ)O(x)ψ(x).
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 (2.14)O(Λx)T(Λ)=T(Λ)O(x)
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 transformationT(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):xμ→xμ+δωμνxν, the substantial variation is given by [13] (2.15)δ0ψ(x)≡ψ′(x)-ψ(x)=-i2δωρσMρσψ(x),
where Mρσ=ℓρσ+sρσ, ℓρσ=i(xρ∂σ-xσ∂ρ). The corresponding finite proper Lorentz transformation can be written as (2.16)τ(Λ(ω))=exp⁡(-i2ωμνMμν),
and the multiplicative structure of the group generates the adjoint action (2.17)Adτ(Λ):Mμν⟶τ-1(Λ)Mμντ(Λ)=ΛμρΛνσMρσ.
Due to (2.9) the generators sρσ fulfill sρσ†H=Hsρσ. They depend on the spin of the field but not on the coordinates xμ. Therefore, we have [ℓμν,sρσ]=0. If a generic element of the translation group is written as (2.18)exp⁡(+iaμPμ),
the commutator relations of the Lie algebra are given by (2.19)[Mμν,Mρσ]=i(ημσMνρ+ηνρMμσ-ημρMνσ-ηνσMμρ),[Mμν,Pρ]=i(ηνρPμ-ημρPν),[Pμ,Pν]=0.
The Casimir operators of the algebra are P2=PμPμ and W2=WμWμ, where (2.20)Wμ=+12ϵμνρσMνρPσ
is the Pauli-Lubanski pseudovector, [Pμ,Wν]=0. In coordinate representation we have Pμ=i∂μ, and the finite Poincaré transformation has the form (2.21)τ(a,Λ):ψ(x)⟶(τ(a,Λ)ψ)(x)=T(Λ)ψ(Λ-1(x-a)).
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 operator𝒪 in the above sense we consider the operator of the wave equation. The Dirac-type wave equation we will consider has the form (3.1)D(∂)ψ(x)≡(iβμ∂μ-ρ)ψ(x)=0,
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 (3.2)DΛ(Λ∂)ψΛ(Λx)=0
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), (3.3)(τ(Λ)Dτ-1(Λ))(Λ∂)(τ(Λ)ψ)(Λx)=T(Λ)D(∂)ψ(x)=0,
and leads to the explicit Lorentz transformations (3.4)βΛμ=ΛρμT(Λ)βρT-1(Λ),ρΛ=T(Λ)ρT-1(Λ).
The Lorentz invariance is given by the substitution (3.5)D(∂)ψ(x)=0⟶(2.6)D(∂)ψΛ(x)=0.
or (3.6)T-1(Λ)βμT(Λ)=Λρμβρ,T-1(Λ)ρT(Λ)=ρ.
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 (3.7)[βμ,sρσ]=i(ημρβσ-ημσβρ),[ρ,sρσ]=0.
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 (3.8)βμ†H=Hβμ,ρH=Hρ.
Writing ψ̅=ψ†H, one obtains the adjoint equation (3.9)ψ̅D(-∂←)=ψ̅(-iβμ∂←μ-ρ)=(HD(∂)ψ)†=0.


## 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 (4.1)AdV(A):p1,3⟶p1,3d(A)=V(A)p1,3V-1(A).
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 (4.2)V(A):D(∂)ψ(x)=0⟶Dd(∂,A)Ψ(x,A)=0,
where 𝒟d(∂,A)=𝒱(A)𝒟(∂)𝒱-1(A) and (4.3)Ψ(x,A)=V(A)ψ(x)
(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 (4.4)
is commutative, that is,(4.5)Ψλ(A+∂λ)=G(λ)Ψ(A).
According to (2.13), the “dynamical” interaction 𝒟d is gauge invariant under the gauge transformation A→Aλ≡A+∂λ if the diagram (4.6)is commutative, that is,(4.7)Dd(A+∂λ)Ψλ(A+∂λ)=G(λ)Dd(A)Ψ(A).
Together with (4.5) we obtain 𝒟d(A+∂λ)G(λ)Ψ(A)=G(λ)𝒟d(A)Ψ(A) or (4.8)Dd(A+∂λ)G(λ)=G(λ)Dd(A)
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. Specifying𝒱(A) by Gauge Invariance

At this point we specify𝒱(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 p1,3d(A), (5.1)p1,3d(A+∂λ)G(λ)=G(λ)p1,3d(A).
By using (4.1) and multiplying by G(λ)-1 from the right we obtain (5.2)V(A+∂λ)p1,3V-1(A+∂λ)=G(λ)V(A)p1,3(G(λ)V(A))-1.
This means that the first claim is fulfilled if (5.3)V(A+∂λ)=G(λ)V(A).
On the other hand, with (4.3) and (4.5) we obtain (5.4)Vλ(A+∂λ)ψ(x)=G(λ)V(A)ψ(x)
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 operator𝒱(A) should be of Lorentz type, that is, for the generators sμν of the Poincaré algebra p1,3 one has (5.5)V(A)sμνV-1(A)=Vρμ(A)Vσν(A)sρσ
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 (5.6)Vμρ(A)Vσμ(A)=Vρμ(A)Vσμ(A)=ηρσ.
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 (5.7)Vμν(A)=ημν-qkPGμν-q22kP2A2kμkν,
where q is the electric charge of the particle and Gμν=kμAν-kνAμ. 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 kP≡kμPμ commutes with any other and has a special role in the theory. For particles with nonzero mass one has kμPμ≠0. 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 (5.8)V0(A)=exp⁡{-i∫dξ2kP(2q(AP)-q2A2)},Vs(A)=exp⁡{-iq2kPGμνsμν}.
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,(5.9)V†(A)H=HV-1(A).
Considering the nonsingular transformation of Dirac-type wave equation (5.10)V(A):(βμPμ-m)ψ=0⟶(Γμ(A)Πμ(A)-m)Ψ(A)=0,
with the help of (5.8) the “dynamical” counterparts to the operator Pμ=i∂μ can be calculated to be Πμ(A)=𝒱(A)Pμ𝒱-1(A), (5.11)Pμ⟶Πμ(A)=Pμ+kμq2kP(qA2-2AP-F),(5.12)P2⟶Π2(A)=(P-qA)2-qF
(F≡sμνFμν) while the “dynamical” counterpart to βμ is given by Γμ(A)=𝒱(A)βμ𝒱-1(A), (5.13)Γμ(A)=Vνμ(A)βν=βμ-qkP(q2kPA2kμkν+Gμν)βν.
In terms of Πμ(A) and Γμ(A) we have (5.14)Dd(A)Ψ(A)=(Γμ(A)Πμ(A)-m)Ψ(A)=0.
However, expressed in terms of Dμ=Pμ-qAμ and βμ, we obtain (5.15)Dd(A)Ψ(A)≡(βμDμ-q2kPkF-m)Ψ(A)=0,
where k≡βμkμ. 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 theγ-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, (5.16)Aμ(ξ)⟶Aμ(ξ)+∂μλ(ξ),
becomes (5.17)V(A)⟶V(A+∂λ)=e-iqλV(A).
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 of𝒱(A) we obtain  (1)
the invariance of the wave function under gauge transformations,(6.1)Ψλ(A+∂λ)=Vλ(A+∂λ)ψ=V(A+∂λ)ψ=Ψ(A+∂λ),
that is, Ψλ=Ψ,(2)
the explicit shape ofG(λ) in (4.5),
(6.2)Ψλ(A+∂λ)=V(A+∂λ)ψ=e-iqλV(A)ψ=e-iqλΨ(A),
that is, G(λ)=e-iqλ,(3)
the invariance of𝒟d under gauge transformations from (4.7) and
(6.3)Dd(A+∂λ)Ψλ(A+∂λ)=Dd(A+∂λ)e-iqλΨ(A),
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(6.4)Dd(A)Ψ(A)=(βμDμ-q2kPkF-m)Ψ(A)=0
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.



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