Structure of gauge theories

Elementary interactions are formulated according to the principle of minimal interaction although paying special attention to symmetries. In fact, we aim at rewriting any field theory on the framework of Lie groups, so that, any basic and fundamental physical theory can be quantized on the grounds of a group approach to quantization. In this way, connection theory, although here presented in detail, can be replaced by “jet-gauge groups” and “jet-diffeomorphism groups.” In other words, objects like vector potentials or vierbeins can be given the character of group parameters in extended gauge groups or diffeomorphism groups. As a natural consequence of vector potentials in electroweak interactions being group variables, a typically experimental parameter like the Weinberg angle ϑW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta _W$$\end{document} is algebraically fixed. But more general remarkable examples of success of the present framework could be the possibility of properly quantizing massive Yang–Mills theories, on the basis of a generalized Non-Abelian Stueckelberg formalism where gauge symmetry is preserved, in contrast to the canonical quantization approach, which only provides either renormalizability or unitarity, but not both. It proves also remarkable the actual fixing of the Einstein Lagrangian in the vacuum by generalized symmetry requirements, in contrast to the standard gauge (diffeomorphism) symmetry, which only fixes the arguments of the possible Lagrangians.

we require L to be minimally modified, toL so as to be invariant under the corresponding gauge (or local) transformation ψ → ψ = e −iα(x) ψ.
Note that the term in the original Lagrangianψγ μ ∂ μ ψ, due to the derivative acting on the local parameter α(x), transforms as ψγ μ ∂ μ ψ →ψγ μ ∂ μ ψ−iψγ μ ∂ μ α(x)ψ so that we should require an extra field that includes a derivative of the local coefficient in its transformation law under U (1), that is and replacing In the same way, for non-Abelian symmetries, associated with a (let us say) compact group G, we generalize the discussion above: with Lie algebra generators satisfying [T a , T b ] = C c ab T c and modifying the usual derivative with the covariant derivative where by ∂ μ φ a we mean something like θ (a) b (φ)∂ μ φ b , associated with the canonical (left or right) 1-form on the Lie group G.

Basics on differential geometry
This first section is devoted to a presentation from scratch of those mathematical ingredients that are required to a sound understanding of a general setting of basic physical interactions in Nature. Here, we follow rather standard textbooks on Differential Geometry [4][5][6][7][8][9][10] and Lie Groups [11][12][13].

Differentiable manifolds
Let S be a set. A local chart on S is a pair (U, ϕ)/ U ⊂ S ϕ is a bijection U ↔ V, an open subset of some vector space F .
An atlas is a family A of local charts (U i , ϕ i i ∈ I ) Fig. 1). The two chosen charts correspond to the stereographic projection from both north and south poles. Coordinates from the south pole corresponding to the point ζ ∈ S 1 will be noted x, whereas those obtained from the same point through the north projection will be y. Looking at Fig. 4, the point ζ is characterized by the angle φ, ζ = e iφ , or by the projection angles ψ and θ corresponding to the two coordinate systems. The relationship among the three angles is: Writing x, for instance, in terms of φ, that is, sinφ = 4−x 2 4+x 2 , we can express the relation between both local coordinates as: Clearly, polynomial functions are not, in general, allowed, since positive powers in y lead to negative ones in x. However, we can find rational functions which are analytic as seen from both local charts, for instance 4x 4 + x 2 ←→ 4y 4 + y 2 (3) and, more generally, the Chebyshev Polynomials T n ( 4x 4+x 2 ), and 2 nd -class Chebyshev Polynomials −4+x 2 4+x 2 U n ( 4x 4+x 2 ), which constitute a basis for the analytical functions that are well defined on the manifold S 1 (Fig. 4).

Tangent Space
Tangent curves: A (differentiable) curve c at m ∈ M is a (differentiable) application from I ⊂ R to M such that c(0) = m (Fig. 5). We say that two curves c 1 , c 2 at m ∈ U ⊂ M are equivalent, c 1 ∼ c 2 , if ϕ • c 1 and ϕ • c 2 are tangent at ϕ(m) in the sense of R n , i.e., This equivalence condition is independent of the local chart (U, ϕ). We define the Tangent Space at m ∈ M as the space of equivalence classes of tangent curves at m, that is, Also note that in R n there is a natural representative for each [c] n ; that is to say, [ϕ •c] ϕ(m) has a preferred member: Locally the following expressions are correct: Coordinates at T (U ) ⊂ T (M): so that, a tangent vector is written as: Vector fields on M, X (M): They are mappings associating a tangent vector on the tangent space to any point on the manifold, that is: is internal (although none of the summands, separately, are a derivation), bilinear (R-linear) and anti-symmetric, and it is named Lie bracket, mainly when acting on X (M). In local coordinates, the Lie bracket is written as: It satisfies the following four properties characterizing a Lie Algebra: Given a basis {X (i) }, we have: where C k i j are constants called structure constants. The tangent map is a Lie algebra homomorphism: That is, Here, we have assumed that f has an inverse, but even if it is not invertible, Y can be defined so that Y • f = f T • X and still named the transformed vector field of X . In this case, it also holds that Example: Lie algebra of rotations in R 3 H, P (i) = 0, H, J (i) = 0, H, K (i) = P (i) P (i) , J ( j) = η k i j. J (k) , P (i) , K ( j) = 0 Example: Lie algebra of the Poincaré group in R 3 × R: (x i = −x i , x 0 = x 0 ≡ ct) (we just write the differences with the Galilei group) Example: The diffeomorphism algebra of R Example: The diffeomorphism algebra of S 1 The difference is just the way we wright the generators, ζ ∈ C / |ζ | = 1, L n = ζ n+1 ∂ ∂ζ , ∀n ∈ Z .
Tensor fields on M: Associated with the vector space T m (M) on any m ∈ M, it is possible to construct the entire tensor space T m (M) r s ≡ T r s (T m (M)), that is, the space of the tensor of { r s }-type (r-times contravariant, s-times covariant): Tensor fields are then defined in an analogous manner to the vector fields:

Differential calculus
Interior product i X : Given a vector field on M, X ∈ X (M), the interior product by X is defined as the following endomorphism of (M): Properties: in particular, h may be the restriction to an open set, h = | U , ⇒ the interior product commutes with the restriction to U , that is, i X is a local operator; note that h * α(X ) = α(h T X ).
This is a consequence of i X being local. In fact, locally, any differential form can be written as a product of functions, α i 1 i 2 ,...i p , and differentials of functions, du i k .
Exterior Differential D : Let α ∈ (M) p with p ≥ 1. We define the exterior differential Dα ∈ (M) p+1 as: For p = 1, the expression (22) reduces to In . From now on, D will be named d since it extends the ordinary differential.
is called the p th -cohomology group of M.
Lie Derivative L X : Combining the interior product and the exterior differential, we define the Lie derivative by the vector field X as the following endomorphism in (M) preserving the order of the differential forms: This operator is local so that In particular, if p = 1: (A more general definition of L X , nextly).

Integration of vector fields
Let X ∈ X (M) be a vector field. Then, there exists an open set V ⊂ R × M {0} × M and a differentiable mapping / is the one-parameter group generated by X . We usually call e t X ≡ ϕ t and say that X is the infinitesimal generator of . Formally, d dt e t X | t=0 = X .

Exercise:
Proposition (Frobenius Lemma): Given X ∈ X (M), written in a coordinate system Before going to a general proof, let us give an instructive simple example.
Integral curves We perform the change of variables in R × R 3 × R 3 : We shall proceed to a constructive proof in Physical terms (Mechanics à la Cartan) leaving a more formal proof to the seasoned reader. In Mechanical terms, Frobenius Lemma would say that a vector field (associated with a dynamical system) under the change of variables, constituting the Hamilton-Jacobi Transformation for the Principal Hamilton function (in the language of canonical transformations in Analytical Mechanics [14,15]). After this transformation, the new variables {K i , P j } behave as constant Canonical Coordinates and Momenta. In fact: The vector field (we have omitted the index i) provides the uniparametric group ϕ in terms of which we construct, explicitly, the change of variables where we have assumed that the possible component of X in ∂ ∂t , X 0 , does not vanish in those local coordinates and the entire vector field has been divided by X 0 . Applying the tangent coordinate transformation we arrive at:

Lie groups
A group is a composition law on a set G / S) where G is a group (with the composition law * ) and S is a differentiable structure on G, respect to which the mappings * : G × G → G and −1 : Equivalently, the maps L g ≡ * | {e}×G , R g ≡ * | G×{g} are required to be differentiable. The transformations L g a → g * a ; R g a → a * g are called left-translation; right-translation and do commute: The tangent space at the identity, T e (G), is called the Lie Algebra. G is an ordinary manifold, so that we may define any object as in M. In particular, vector fields X : In the same way, X ∈ X (G) is right-invariant if R * g X = X , that is, The set of left-invariant vector fields will be named X L (M) " " " right-invariant " " " " " X R (M) .
Proof Given an element in T e (G), X e , we construct X on G in the form X (g) = L T g X e . This vector field, so built, is in X L (G). In fact, Lie algebra structure: X L ,R where subalgebras of X (G) and isomorphic to G ≡ T e (G). Let us denote χ this isomorphism. We can translate the Lie bracket from X L ,R (G) to G: where Z L ,R G is the translated of Z by L g , R g , respectively. This way, we have Note that in terms of vector fields, from L to R there is a global minus sign on C i jk : In practice, where L g is the mapping here a plays the role of x in a function f (x) and g that of f . Similar comment holds for L ↔ R.
(after writing X L (g ) we can rename g by g).

Some examples
The group SU (2) is a double covering of the group SO(3) of rotations in the space R 3 . We shall parameterize the group by the components of a vector in the direction of the rotation axis and a module related to the rotation angle; that is, g ≡ { i }, | | = 2sin φ 2 . A rotation in R 3 with this parameterization is written as: From the product of two rotations R( )R( ) = R( ), we deduce the composition law: Now we proceed to compute the left and right generators: Example II G ≡ Galilei Group Galilei transformations [16,17] relate inertial reference systems, that is, reference systems where the Newton Laws hold. We shall write the transformations in R × R 3 × R 3 , parameterized by the ten parameters {B, A, V, R( )} corresponding to a translation in time, translation in space, change in velocity and rotation of the axis. By composing two of them, a composition law is obtained from which we compute left and right generators:

The adjoint representation: Killing form
An action of G on a manifold M is a Lie group homomorphism from G to the group of diffeomorphisms of M: such that the mapping provides a Lie algebra homomorphism: As a particular case, M can be G, G or G * With a given g ∈ G, we associate the (nonlinear, in general) mapping on G, Now, we take the tangent of ade at e: It defines an action of G on G named Adjoint Representation of G: The tangent of Ad at the identity g = e is called Adjoint Representation of G and noted ad. ad is a Lie algebra homomorphism that turns out to be The Killing form is defined as: It is bilinear, symmetric and satisfies: which means, somehow, that Ad is unitary with respect to the scalar product k (ad is Hermitian). In coordinates, k i j = C k im C m jk .
To be precise, k is a scalar product only when |k| ≡ detk = 0, which happens iff G is semisimple, that is, it contains no Abelian invariant subgroup. If |k| = −1, G is also compact. Invariant Forms: They are dual to left-and right-invariant vector fields. If {X L ,R (i) } is a basis of X L ,R , {θ (i)L ,R } will be the dual basis, that is, θ (i)L (X L ( j) ) = δ i j . They are explicitly calculated as: Invariance properties: The set of invariant forms are codified by a single 1-form, that is, the Canonical1-form: Note that θ L ,R is a G-valued 1-form, the θ (i)L ,R being ordinary R-valued 1-forms.
Note also that θ L ,R (Z L ,R G ) = Z ≈ Z L ,R G , that is to say, θ L ,R is the G-valued 1-form that is the identity on X L ,R (G).
Exercise: Compute k i j and θ L ,R for G = SU (2) and realize that θ (i)L ,R (X L ,R ( j) ) = δ i j and that

Central extensions of Lie groups
We say thatG is an extension of the Lie group G by H if H is a normal subgroup (that is, invariant under conjugation: ghg −1 ) and Note that G is not necessarily subgroup ofG. G is a central extension of G by H , if H is Abelian and is in the center ofG (that is, the elements in H commute with all the elements inG). Very special situation appears when H = U (1) [18].
Central Extensions of G by U (1): In that case, the group law forG can be written as follows: where the local exponent ξ(g , g) is named 2-cocycle of G valued on U (1). The properties which establish the 2-cocycle definition can be derived from the condition of the expression above being a group law forG: Coboundaries: A cocycle ξ cob satisfying is called coboundary. Coboundaries define trivial extensions. In fact, a change of variableŝ g = gĝ =ĝ ĝ ⇒ ζ = e −iη ζζ =ζ ζ destroys the central extension turningG into G × U (1). The function η is the generating function of the coboundary. The name cocycle comes from the fact that the set of central extensions of G by U (1) are parameterized by the 2 nd -cohomology group of G with values on U (1) (according to Bargmann): that is to say, cocycles that are not a coboundary.
"Pseudo-cohomology": However, there are coboundaries which are generated by a linear function on G and they do modify the structure constants of the Lie algebra, as if they were "true" cocycles [19,20]. This subset of coboundaries (in fact a subgroup of B 2 ) defines a (true) cohomology group H 2 (G C , U (1)) of a contracted group G C of G.
The typical situation could be that of a family of generating functions η on G that go badly under a certain lie group contraction, that is, η → ∞ in a contraction limit, but ξ cob ≡ δη has a well-defined limit.
Paradigmatic Example: The Poincaré group with η = mcx 0 . G C is then the Galilei group and δη a non-coboundary cocycle for c → ∞.

Principal bundles
In this subsection, we shall follow the presentation of principal bundles given by Koszul [21] (see also [9]).
A principal bundle is a differentiable manifold P on which a Lie group G acts from the right, along with a differentiable mapping p from P onto a differentiable manifold M such that: ∀m ∈ M, there exists U m and a diffeomorphism γ satisfying: The application p is called projection, M base, G structure group, p −1 fiber over m (Figs. Homomorphism between principal bundles: A homomorphism H between two principal bundles P, P with the same structure group G is a (differentiable) mapping It is clear that H takes fibers into fibers defining h : M → M : do not depend on the particular element ξ taken on p −1 ( p(ξ )), that is, they define properties that characterize the so-called 1-cocycle on {U α } valued on G.
The name transition functions comes from the fact that they address the change between local charts: in as well as Denoting h β α (m) ≡ ρ β (H (σ α (m))), we have: to be compared with (Fig. 9). The functions h β α : U α ∩ h −1 (U β ) → G satisfy: The pair (h, {h β α }) defines H globally on P.
Remark If H is an isomorphism of P such that h is the identity on M, the expressions above reduce to: which express the relationship between the transition functions corresponding to two isomorphic principal bundles, ({U α }, g αβ ) and ({U γ }, g γ δ ). As a Corollary, the transition functions g αβ and g γ δ corresponding to two isomorphic bundles, subordinated to the same covering {U α }, are related by a family of functions {h α : U α → G} such that: In fact: it suffices to define h α ≡ h αα . We come from motivating the following cohomological characterization of Principal Bundles on M, that is, theȞ 1 (M, {U α }; G}: Non-equivalent principal bundles on M, with structure group G, are characterized by 1-cocycles g αβ , that is, satisfying: which are not coboundaries, that is, In the limit of refinement of {U α }, with a minimum of elements and minimal intersection, it defines theČech Cohomology SpaceȞ 1 (M; G). Proof (just sketched): We construct ≡ ∪ α U α × G and take quotient by the following equivalence relation ∼ : The quotient / ∼ is P, the projection being p(α, m, a) = m.

Associated vector bundles
Let P p → M be a principal bundle characterized by ({U α }, g αβ ), and λ : G −→ G L(F) a linear representation of the structure group G on a vector space F. The set ({U α }, λ• g αβ ≡ḡ αβ ) constitutes a 1-cocycle relative to {U α } and valued on G L(F).
defines a vector bundle E π → M, with fiber F associated with P trough the representation λ.
→ M be an associated bundle with fiber L by means of the representation λ, and let (E) be the linear space of sections of E, that is, mapping from M to E such that π E • σ = I M . The following commutative diagram corresponds to a homomorphism between vector bundles: The mapping β : Definition A differentiable function on P with values on L satisfying the condition is called G-function and we say that ψ ∈ L G (P).

Proposition The application β : (E) → L(P) is injective and verifies
From now on, we shall identify sections of a vector bundle E with G-functions on the principal bundle P from which E is an associated vector bundle.
Vector fields on a Principal Bundle. The different structures of the base manifold M and the fiber G of a principal bundle P manifest themselves in the behavior of the components of a vector field, in a (principal-bundle) local chart, under a change of coordinates. As we shall see, the components along the fiber keep some identity as vertical components, whereas those along the base cannot be considered as horizontal since this character changes in going from one chart to other. The reason for that lies in the expression of the change of chart: Denoting the local coordinates as (x μ , s a ), we have: Then, a vector field X on P, will be written alternatively as and the tangent application to the change of coordinates above reads This way, even though X a = 0, in the new basis X acquires a non-null vertical component In other words, the property of X being "horizontal" is not preserved under a change of coordinates.
Only vertical vector fields preserve their structure in changing coordinates. Therefore, it makes sense to define the vertical subspace T v ξ (P) at a point ξ ∈ P. The submodule X v (P) admits a basis made of generators of the action of G on P:

Connections on principal bundles
In the last section, we motivated the need for some extra structure in order to define properly the notion of horizontality as regards the components of the vector fields on a Principal Bundle. This extra structure corresponds to a connection.
A connection on a principal bundle is a 1-form on P, , X v (P)-valued, such that: That is to say, is a projection of X (P) onto X v (P) invariant under G. This allows us to define a horizontal submodule X h (P) such that: It is a G-valued 1-form on P with the properties: Transformation properties ofγ : γ is a 1-form on P, G-valued. Locally we may characterize γ by means of a set {γ α } of 1-forms on M. In fact, given {U α } M, we define: Proposition On the intersection U α ∩ U β , we have: Example 1 Case of G = G L(n) as the structure group of the Reference Bundle. We shall use the matrix elements as coordinates, so that we have: We compute the explicit expression of the left translation and its tangent: where we have relaxed the notation so as to identify s i j and s(g) i j , as usual. Then, we obtain: and finally, and in global and symbolic form, though rather standard, Let us compute g * αβ θ on the intersection U α ∩ U β . For simplicity, we denote {x i } the coordinates on U α and {x j } those on U β , then Denoting k i j the components of γ : where we have computed Ad(g −1 )Z as g −1 Zg, as corresponding to the action of a linear group. The symbols k i j are non-tensorial (due to the affine term in the transformation law) and are called Christoffel Symbols.
Example 2 Case of structure group G = U (1). This is a very special, though simple, case relevant in both gauge theory and quantization. The elements of the group are parameterized globally by ζ ∈ C, / |ζ | = 1, and locally by ζ = e iφ . The canonical 1-form and the transition one are: and the transformation rule,

Derivation law on an A-module M:
We provide the more general (algebraic) definition and then specify the more relevant cases. Let K be a commutative ring A be an algebra over the ring K M be a module over A (A − module) .
A derivation law on M is a mapping Two derivation laws differ in an element of It must be remarked that a derivation law is not tensorial since the elements in Hom K (M, M) are only linear with respect to the scalars in K . Conversely, the elements in Hom A (M, M) are linear with respect to "scalars" in A, so that the difference of two derivations laws is a tensor. This extent will be nitid in the following A derivation law then turns out to be a derivation law for vector fields (usually referred to as "connection"): Taking a local basis in X (U ), X (i) , we have: If now P is a principal bundle over M with structure group G, ρ is a linear representation of G on G L(F), and E is a vector bundle associated with P, through the representation ρ, with any connection γ on P we may associate the following derivation law on (E), ∇ γ : A (a) μ ≡ vector potentials or Yang-Mills fields.

Variational calculus
After a more traditional exposition of variational calculus as in standard textbooks [22], we recommend intermediate texts as [23,24] and, finally, more formal papers as [25] and references therein.

Jet bundles
sections about x. In x (E), we define the following equivalence relation 1 ∼: and consider the quotient space J 1 ∼, and the natural projection Given a section ψ : M → E, we can define its 1-jet extension which is an immersion of (E) → (J 1 (E)).
The structure 1-forms {θ α }, characterize the jet extension of sections and vector fields: In the same way, given X ∈ X (M), j 1 (X ) ∈ X (J 1 (E)), is the only field that projects on X and preserves the 1-form system {θ α }: The jet extension is a Lie algebra homomorphism: Lagrangian (density): A Lagrangian density is a real function L on J 1 (E). Then, we define the Action functional where ω is a volume n-form on M and π 1 * is its pull-back to J 1 (E). (Usually, M is the Minkowski space-time and ω = dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 ).

Hamilton principle
The Ordinary Hamilton Principle establishes that the critical sections of the variational problem are the points of (E) where δS, the "differential" of S , is zero, that is: As is well known, critical sections satisfy the Euler-Lagrange equations: where d dx μ stands for "derivative with respect to x μ along the section ψ." Exercise: Derive the Euler-Lagrange equations!! Hint: Realize that the variations δx μ , δψ α and δ∂ μ ψ α correspond to the components X μ , X α , X α μ ofX . The fact that ψ α μ behaves as ∂ μ ψ α under variation is related to the fact thatX is j 1 (X ).

Modified Hamilton principle: the Poincaré-Cartan form
The Modified Hamilton Principle assumes the independent variation of ψ α and ψ β ν . That means that we look for critical sections in the module (J 1 (E)), rather than (E), where the variations are caused by arbitrary X 1 ∈ X (J 1 (E)) that are no longer jet extensions.
The Modified Hamilton Action L 1 : where the Poincaré-Cartan(-Hilbert) form is a (n =dim M)-form defined by where θ μ ≡ i ∂ ∂ x μ ω. The Poincaré-Cartan n-form can also be written as When L is regular, that is, det where π μ α ≡ ∂L ∂ψ α μ are the covariant momenta and the form PC can be written as Remark The Poincaré-Cartan form might be redefined as for future relationships.
The Modified Hamilton Principle defines critical sections as those sections ψ 1 ∈ (J 1 (E)) on which the functional derivative of S 1 , δS 1 , is zero: The equations of motion above generalize the Euler-Lagrange ones in the sense that if L is regular Let us remark that PC reduces to Lω on jet extensions since PC = π μ α θ α ∧ θ μ + Lω and θ α | j 1 (ψ) = 0.
In the regular case, i X 1 d PC | ψ 1 = 0 can be taken into the Hamiltonian form: We shall remark that in the Ordinary Variational Calculus people define only π α ≡ π 0 α , the time component, and the non-covariant Hamiltonian H = π α ψ α 0 − L (π αψ α − L). The extra Hamiltonian equations we have, simply provide the definition of covariant momenta. Note: The non-covariant Hamiltonian H will be obtained in our scheme as the time component of the conserved current associated with the invariance under time translations (see later).

Symmetries and the Noether Theorem: Hamilton-Jacobi transformation and Solution Manifold
A symmetry of the variational problem is a vector field Y 1 ∈ X (J 1 (E)) such that We actually say that Restricting this expression to solutions, we have: Note that J Y 1 is an (n − 1)-form and we can define the dual current (we shall omit the In terms of J , the constancy of i Y 1 PC − α Y 1 along solutions becomes where is a Cauchy surface, is a constant. It is named conserved charge associated with the symmetry.

Examples
The free Galilean particle: The expression above concerning the trajectories of the free particle can be read as an invertible transformation in R × R 3 × R 3 to be referred to as the This Hamilton-Jacobi transformation permits the pass to the Solution Manifold parameterized by the basic constants of motion.
After this (H-J) transformation, PC comes down to the Solution Manifold, except for a total differential: Its differential is that is, the symplectic form on the Solution Manifold. ≡ P i d K i is the Potential 1-form or Liouville 1-form.
Reminder: At this moment, we must remind the reader some few words on Symplectic Manifold (to be completed with traditional references like Ref. [5] and/or Ref. [4]): Since det(ω) = 0, given f : S → R, the equation Symmetries of the free particle: Note that all Noether invariants are written in terms of the basic ones K, P.

The free scalar field (Klein-Gordon)
Klein-Gordon fields are sections φ of the line (R for real fields, C for charged ones) vector bundle over Minkowski space-time M [24] (see for instance [26][27][28] for a more physically minded presentation) The Euler-Lagrange equations lead to: where is a Cauchy surface, usually R 3 (x 0 = 0). The "constants" a(k), a * (k) parameterize the Solution Manifold.
Space-time symmetry: The K-G Lagrangian is invariant under the Poincaré group generated by with jet extension: In particular In the same way The space-time symmetries play the analogous role of time translations in Mechanics and the corresponding Noether invariants do not contribute to the Solution Manifold, that is to say: SM cannot be parameterized by Noether invariants associated with space-time symmetries. "Internal" symmetries: (Such symmetries are rarely reported in Literature and considered as "hidden symmetries" [27]) The following vector fields on the bundle E are non-trivial symmetries: with Noether invariants Alternatively, the "configuration space" counterparts are with Noether invariants Q π(y) = ϕ(y) Q ϕ(y) = π(y) .
Note that π(x) =φ(x 0 = 0, x) , ϕ(x) = φ(x 0 = 0, x) and that k and y in the subscript are indices, whereas x μ is the variable in the base manifold M of E. The Hamilton-Jacobi transformation: Passing to the Solution Manifold By writing the Klein-Gordon solutions in a proper way, and adding the trivial transformation x 0 = χ 0 , the following transformation (H − J ) has an inverse ((H − J ) −1 ) : The tangent H-J transformation becomes: Acting with H-J on the objects on (E) we arrive at the Solution Manifold endowed with a symplectic structure and Hamiltonian symmetries. In fact, the "integral on the Cauchy surface" of PC comes down to the SM except for a total differential after applying the H-J transformation: and the differential dϑ PC actually comes down defining the Symplectic form: ≡ δ Hamiltonian vector fields: constitute the basic local symmetries .
as vector fields on the bundle J 1 (E) . (132) By the way, even in the massive case, Y f = f ∂ ∂φ is a symmetry of the Lagrangian if f is a solution of the Klein-Gordon equation: When f is not a solution, symmetry under such a vector field Y f requires the introduction of compensating Yang-Mills fields.

Current algebra (on the example of the massless Klein-Gordon field)
We write the complete symmetry of the Klein-Gordon field in the form of a semi-direct product group: The space-time rigid symmetry provides charges: The internal symmetries lead to rigid charges: and as well as local ones associated with the Hamiltonian vector fields (131): In other words, given a rigid symmetry, the integrand of the corresponding Noether invariants, that is, the zero th component of the currents, j 0 , are in turn Noether invariants of a current algebra!!.

Symmetry and quantum theory
Canonical quantization proved to be inadequate very soon for dealing with nonlinear systems in general, except for certain perturbative conditions. See, for instance the historical paper on "No-Go theorems" [29] as well as, more recently [30,31]. Here, we shall adopt a symmetrybased algorithm more appropriate to formulate basic physical systems irrespective of their (non-)linear character, provided that we are able to parameterize their Solution Manifold by means of Noether charges associated with symmetries [32][33][34][35][36].

Group Approach to Quantization
The basic idea of GAQ consists in having two mutually commuting copies of the Lie algebrã G ofG a central extension by U (1): Then, a copy, let us say X R (G), constitutes the (pre-)Quantum Operators acting by usual derivations on complex U (1)-functions onG. The other copy, now X L (G), is used to reduce the (pre-)quantum representation in a compatible way ⇒ true Quantization In fact, given a group law, g = g g, we have two actions: and they do commute: This property also implies: The left-invariant form θ L(U (1)) plays the role of generalized Poincaré-Cartan form or quantization form .
The classical Noether invariants are iX R a , as they are invariant along the equations of motion, that is,X L a in the characteristic subalgebra G : Wave functions ψ are U (1)-functions (ψ(g) = ζ (g), ζ ∈ U (1)) invariant under the right action of a polarization subgroup P: P is a maximal subgroup of G containing the characteristic subgroup G and excluding the U (1) central subgroup , G acts on ψ from the left,ĝ ψ(g) = ψ(L g g), providing an irreducible representation ofG. At the infinitesimal level, the U (1)-function condition ψ(g) = ζ (g) is written as ψ = iψ, where stands forX which is the central generator of the groupG , ζ ∈ U (1).
Starting from a complex function ψ(g, ζ ) onG, we must impose the Polarization conditions in the form:X L a ψ = 0.
X L ∈ P generate a left-invariant subalgebra P containing G and excluding the vertical generator .
If such a polarization subalgebra does not exist, then we may search for a higher-order subalgebra in the left-enveloping algebra substituting a first-order one. On the space of polarized wave functions, the right-invariant generatorsX R a operate defining the true quantum operators associated with the group variable a,â. They generate a unitary and irreducible representation of the groupG, that is, a quantization of the physical system with basic symmetryG

Non-relativistic harmonic oscillator (1+1 dimension)
Group law: (central extension by U (1) of the Newton group) There is no first-order real polarization!
Polarization Equations: = (ζ, t, a, a * ) Configuration space: Higher-order Polarization where H n are the Hermite polynomials.

Relativistic harmonic oscillator (1+1 dimension)
What is a relativistic harmonic oscillator? A dynamical system characterized by a symmetry that contract to that of the non-relativistic harmonic oscillator in the non-relativistic limit and that contract to the symmetry of the free relativistic particle in the limit of zero frequency [37]. Here is the proposed Lie algebra: where Left generators: Configuration space: Higher-order Polarization Restoring the rest energy, that is, where N ≡ mc 2 hω and ≡ 1 is the D'Alembert operator in antide Sitter space-time. The evolution equation is solved by power series expansion: where H N n is a polynomial in the variable ξ ≡ mω h x satisfying The polynomials H N n are the Relativistic Hermite Polynomials!! The energy operator provides the value E N n = ( 1 2 +N + n)hω. For N ≡ mc 2 hω → ∞, H N n → H n (Hermite Polynomials). The value N = 1 2 corresponds to the extreme relativistic regime.

Particle moving on SU (2): PNLσ M
The standard classical approach to a particle moving on a Riemann manifold with metric g i j (x) is established by the Lagrangian (see [38,39] and references therein): where e (a) i are the vierbeins defining the metric The form θ (i) = θ (i) j d j ≡ θ R(i) is the right-invariant canonical 1-form (we could have used the left forms since L is chiral.
The inverse "vierbeins" are the right-invariant vector fields The momentum, Hamiltonian and Poincaré-Cartan form are and the solutions to the equations of motion (ω ≡ 2 m H = θ i θ i ): where ε i ≡ i (0),ε i ≡˙ i (0) are constants of motion parameterizing the Solution Manifold. Note that θ i ≡ ϑ i is also constant of motion. The symplectic form on the SM turns out to be In local (Darboux) coordinates, we have Note that the Hamiltonian, in coordinates (ε i , ϑ j ) will be free from normal-order ambiguities as regards quantization. The basic symmetries are the Hamiltonian vector fields associated with ε i , ϑ j and ρ ≡ They lead to the Poisson algebra (beyond Heisenberg-Weyl): Remark: The (Hamiltonian) function ϑ i generate (Killing) symmetries of the Lagrangian, whereas ε k only of the Poincaré-Cartan form, that is, ε k generate pure contact symmetries. Group Approach to Quantization now proceeds by exponentiating the Poisson algebra above arriving at the SU (2)-sigma group centrally extended by U (1): The characteristic subalgebra and polarization are: On the Quantum Solution Manifold,G/G , the quantization form is up to a total differential) .

The Klein-Gordon field
Typical infinite-dimensional systems in Physics appear as mappings from a space-time manifold M into a non-(necessarily)Abelian group target G [36]: If g is an element in Diff(M) the following semi-direct group law holds: Here M is the Minkowski space-time (x 0 ≡ ct, x), and Diff(M) is restricted to the Poincaré subgroup or just Translations parameterized by (a 0 ≡ cb, a). G is simply the complex (or real) vector space parameterized by φ.
A natural parameterization of the Klein-Gordon group is associated with a factorization of M as × R (Cauchy surface times Time): we have parameters b, a; ϕ(x),φ(x) (the Lorentz subgroup of the Poincaré group can be easily added).
It should be stressed that the action of a on ϕ(x) just consists in moving the argument by a: ϕ(x) → ϕ(x − a), whereas the action of b requires the knowledge of the equation of motion (although not necessarily their actual solutions).
Therefore, we write for ϕ (b(x)): so that, the Complete Group Law becomes: Notice that we can read from the group law the expression of the evolved fields φ(x 0 , x),φ(x 0 , x) in terms of the initial conditions ϕ(x),φ(x): Left-invariant algebra: Quantization form: Commutators: The Characteristic subalgebra is G = X L b ,X L a , so that where the subscript 0 refers to the initial value in the integration of (generalized) equations of motion corresponding to G .
Covariant Formulation. The construction above can be repeated in a form more convenient for the interaction. Now the fields will be defined on the entire Minkowski space-time but supposed to be solutions of the equations of motion where the invariant function Pauli-Jordan (x) verifies where ε(k 0 ) is the sign function, ω(k) ≡ |k| and V is the (infinite) volume of "time." Left-generators: (formally distinguishing between ∂ μ φ and φ μ ) where we have disregarded the infinite volume V . Commutators: Textually, X L φ(x) ,X L φ(y) = 0, unless we interpret that φ μ = ∂ μ φ (something that happens along the physical trajectories) and in this case we would have the arbitrary-time commutator: where we have "redefined" the fields so as to make explicit the mass m 2 .
This computation renders clear the necessity that (x − y) satisfies the equation of motion.

The Dirac field
(just sketched, of the Lorentz subgroup discarded) It is customary to use the invariant function S(x) ≡ (iγ μ ∂ μ + m) (x), which satisfies the Dirac equation: Remark: If we consider ψ andψ as Fermionic variables, then the relative sign in the cocycle would be + Left-generators: Right ones: Arbitrary-time commutators:

Gauge theory of internal symmetries
Internal symmetries refer to transformations moving only the internal (fiber) components of a matter field [1,28,41,42]. In our language, they are generated by vector fields of the form: Here, {ϕ α } are the coordinates of the fiber of E π → M on the space-time M, usually the Minkowski space with coordinates {x μ }, μ = 0, 1, 2, 3. The generators above are supposed to close a (finite-dimensional) algebra: to be referred to as the rigid or global symmetry algebra. The Minimal Interaction Principle establishes that a matter Lagrangian L matt invariant under a rigid group G can be converted into a new one,L matt , invariant under the corresponding local (usually called gauge) group G(M), that is, a group generated by F (M) ⊗ G, F (M) being the algebra of functions on M, G the Lie algebra of G. The Lie algebra of G(M) satisfies: We must introduce extra compensating fields A (a) μ , the gauge vector bosons, transforming under G(M) as: This way, the complete generators of G(M), acting on ϕ α and A (a) μ , are: The transformation properties of A

Utiyama's Theorem
We establish this theorem in two parts, the first of which refers to the matter field Lagrangian,L matt , whereas the second tell us about the Lagrangian, L 0 , governing the (free) gauge fields themselves. Utiyama's Theorem I: The new LagrangianL matt describing the dynamics of the matter fields along with their interaction with the vector potentials A is invariant under the local group G(M), that is, Proof Consider the following change of variables χ: Eur. Phys. J. Plus (2021) 136:304 and the Jacobian: After this change of variables, We must now compute f (a) X (a)Lmatt : Thinking of A (a) μ as connections, we may say that under the Minimal Coupling, the covariant "derivative" of ϕ α substitutes the ordinary one in L matt : On jet extensions, Notice that under G(M), ϕ α μ transforms as a tensor: We have introduced new fields A (a) μ which must be controlled by a given Lagrangian ν,σ ) so that the total Lagrangian will beL matt + L 0 (211) named curvature tensor or intensity tensor.
Proof We have to solve the equation for arbitrary f (a) , which implies that (a) in turns implies that L 0 is invariant under the rigid group G.
with general solution f = f (y + 1 2 kx 2 ). Therefore, The additional condition for L 0 of being invariant under the rigid Poincaré group (or any other kinematical space-time rigid symmetry) means a further restriction: L 0 must be a scalar. For internal symmetries the Yang-Mills Lagrangian where k ab is the Killing metric, is usually adopted. The Euler-Lagrange equations of the total Lagrangian L tot = L matt (ϕ α , ϕ μν ) corresponding to the independent variables ϕ α and A (a) μ are: In particular, for the Yang-Mills Lagrangian L 0 we have: or, using the covariant derivative notation, D μ , where the currentĴ is defined aŝ It is worth noticing that the Euler-Lagrange equations of the Lagrangian L tot , after the change of variables used for proving Utiyama's theorem, would be those of the free fields φ and B. In fact, L tot = L matt (φ, φ μ ) + L 0 (B, F), without interacting term!!. However, this is a consequence of the fact that the mentioned change of variables does not preserve the structure 1-forms of the jet-bundle; variational calculus is not invariant under an arbitrary change of variables.

Some remarks on Local vs Gauge symmetries.
Let us test explicitly the Gauge symmetry of L tot under the group G(M) and compute the corresponding Noether invariants, as an exercise. Generator of G(M) (no sum on (a)): confirming that the symmetry above is gauge indeed.
However, let us also demonstrate that there are local symmetries which are not gauge, that is, their associated Noether charges are non-trivial. To this end, consider the massless Klein-Gordon field: The generator X ≡ ∂ ∂φ is a symmetry. In fact,X = ∂ ∂φ , so thatX L = 0. But is the local generator X f = f (x) ∂ ∂φ a symmetry? We compute the corresponding jet extension and the Lie derivative of the Lagrangian: and we realize that only if f is a solution of the equations of motion, f μ φ μ is a gradient, that is, h μ = f μ φ. But in this case, the Noether charge is a non-trivial quantity (see the symmetries parameterizing the solution manifold of the Klein-Gordon field).

Example of the Dirac field
Free Dirac field L D ≡ L matt = iψγ μ ψ μ − mψψ The Euler-Lagrange equations of motion become: It is assumed (as corresponding to the Ordinary Hamilton Principle) that ψ μ = ∂ μ ψ but not derived from the equations of motion. The Poincaré-Cartan form is derived in the standard manner: Remark H is not the ordinary Hamiltonian driving the time evolution. Evolution is driven by the Noether invariant associated with the invariance under time translation P (0) ≡ j 0 (0) = iψγ · ∇ψ + mψψ. Even more, if we rewrite the Poincaré-Cartan form in the way where a "conserved current," the "conserved charge," dσ μ T μ PC plays the role of a Quantum-Mechanics Poincaré-Cartan form: with H = iψ † α · ∇ψ + mψ † βψ.
Coupled Dirac field: The Poincaré-Cartan form associated with the coupled Lagrangian becomes: Integrating againˆ PC over the Cauchy surface, we have:

Brief report on the Group Quantization of Electrodynamics
One way of proceeding in facing the quantization of a system whose full symmetry (to be precise, the basic symmetry evolved in time) is unknown consists in quantizing the basic symmetry that characterizes the Solution Manifold and then realizes the right-enveloping algebra, which preserves the representation space (Hilbert space) of the basic algebra of quantum operators. In other words, the exponential of the complete Hamiltonian will act perturbatively on the wave functions defined on the Solution Manifold (The complete Hamiltonian is a constant of motion in any isolated system and, thus, it is well defined on the SM). This procedure, proposed here, is related to the approach followed in "Landau's series" text books when dealing with formal perturbation theory in that which concerns with exact propagators and exact vertices in the Heisenberg picture (see [44]).
Another way would be that of closing perturbatively the classical Poisson algebra, exponentiating the approximate algebra at each order and applying GAQ at the corresponding order. This more precise method will not be considered here.
Quantum Basic Symmetry: General case (Space-time symmetry excluded; internal indices of the matter fields are not explicit) Since we aim at representing just the basic symmetry on SM and then realize the quantum evolution perturbatively, we ignore the semi-direct action of the Poincaré group and think of the arguments of the fields, x, on the Cauchy surface, only as (infinitely many) indices. In the same way, spatial derivatives do act as infinitesimal translations on those indices, whereas time derivative of the fields correspond to different field coordinates with initial values on SM. Roughly spiking, ∂ i φ is not independent of φ, although ∂ 0 φ indeed is. Nevertheless, we intend to take the Lorentz covariance as far as possible in the proposed group law: It must be stressed that the co-cycleξ matt can be written as if it where the sum of the co-cycle for the free matter ξ matt plus an interaction term proportional to the coupling constant, that is:ξ but the "interaction" term, itself, is not a co-cycle. The reason for this fact is that the unextended group, for whichξ matt is a co-cycle, is a deformation of the direct product of the unextended groups corresponding to the free matter and free gauge fields. We leave as an exercise the verification of the co-cycle condition (55) forξ matt and ξ 0 , that is,

Scalar Electrodynamics
For Scalar Electrodynamics, we have: Left generators and form: Right generators: with structure constants which are the opposite to the left ones. Noether invariants: Note that the commutators [X R A μ ,X R φ ] = −eX R φ μ will only imply the quantum commutators: where the association of right generators with quantum operators is:

Time Evolution from the Solution Manifold
The methodology to be here sketched is quite general and can be applied to any physical system whose basic operators do not close algebra in "finite" dimension with the Hamiltonian.
In that which follows we shall consider the time evolution of either a classical function f (q, p) on the classical Solution Manifold or a function f (q,p) of quantum operatorŝ q,p represented on (polarized) wave functions of classical variables (q or p, or some combination). In the same way, a bracket [ , ] will mean Poisson bracket as regarding classical evolution, or quantum commutators in the case of the quantum evolution. Schematically: (Time evolution by Magnus Series [45]) With a given function on the solution manifold we associate the following "evolutive" version: where B k are Bernoulli numbers and the "powers" of ad f means Magnus series (versus Dyson-like series) offers "unitarity" even at finite orders (at the classical level we would say "symplecticity").
For t-independent Hamiltonians, as corresponds to objects on SM, we arrive at a rather simpler formula: which constitutes the Inverse Hamilton-Jacobi transformation by H.
In particular, we can compute the "arbitrary-time" commutator of two (field) operators or the exact propagator of the fieldÂ as where T stands for "time-order" in the traditional way, to be further developed in terms of the free propagator D(x, x ).

Massive Gauge Theory
Weak Interactions were originally described by a "current-current" term in the Lagrangian to account for the property of being very local. To turn them into a gauge theory would require a very massive intermediate particle, a fact which makes quite difficult the corresponding renormalizability beyond the Abelian case [46]. To avoid this difficulty, a mechanism [47,48], imported from solid-state physics, was introduced in Particle theory [49]. For a review, we recommend Ref. [50]. Group Law: where g μ (x) above is not, necessarily, ∂ μ g(x). Equivalently, we may define new coordinates: Notice that, now, A μ is not, necessarily, Explicitly, the left-invariant canonical 1-form on the group G 1 (M) is written as μ (x)}, the group law reads: Note also that the group G(M) is naturally contained in G 1 (M) by means of the jet extension: In fact, if the element A μ (x) in the group law corresponds to a jet extension Then, if we think of A μ in the group law as an ordinary Yang-Mills physical field, of g as an ordinary gauge transformation, to be call g, and of A μ as the transformed of A μ , A μ , we can read: just as corresponds to the transformation law of a physical Yang-Mills field. Ordinary connections can be derived from G 1 (M) by simply taking the quotient by G(M) (that is to say, by j 1 (G(M)) ∈ G 1 (M)).
However, we should not take the mentioned quotient but, rather, A μ and θ μ will live together and they will combine in the proper way in due time.

Massive Gauge Theory
We may repeat Utiyama's theory on the grounds of some exotic matter g a (x). The action of G on the scalar fields g a (x) proceeds as the own right action with generators X L (b) . This way, the generators of G(M) on (g a , A and the minimal coupling is realized as to be compared with the standard expression for ordinary fields It should also be compared the expressions of the group generators acting on g and ϕ: the main difference being that now X Lb (a) (g) is an invertible function (though nonlinear, in general) of g. In fact, the inverse matrix is [X Lb which is an affine coupling (it is not linear in g as ϕ μ + A μ ϕ was in ϕ). So then, giving dynamics to the "exotic matter" g a through a kinetic term in the Lagrangian, L "matt" , of the form the Minimal Coupling Principle provides mass to the fields A μ without damaging gauge invariance !!. (In the expression above, η μν stand for the metric in the space-time manifold M and k ab for the Killing metric in G).
In fact, L "matt" becomesL "matt" : which contains the mass term 1 It is a Minimal coupling with affine character. This Lagrangian L "matt" addresses part of the Non-Abelian Stueckelberg Lagrangian in massive gauge theory: After the change of variablesÃ μ = U † (A μ − θ μ )U , that is, the "unitary gauge", this Lagrangian is written as corresponding to a Non-Abelian Proca Field.

Standard attempt to the quantization of massive gauge theory: Nonlinear Sigma Model (N-LSM)
The , then chiral) is usually referred to as σ −Lagrangian, the origin of the name being traced back to the low-energy models for strong interactions, where a set of field (σ, π ), SU (2)-valued obeyed a Lagrangian of this kind.
The Euler-Lagrange equations for A similar scheme, but with external scalar fields φ a behaving as our g a , had been considered in the Literature in an attempt to make the massive gauge theory renormalizable. This scheme is called non-Abelian Stueckelberg formalism as it generalizes the Abelian case, Page 59 of 85 304  the Massive Electrodynamics, introduced by this physicist. The main difference is that the Abelian case is renormalizable under Canonical Quantization whereas the non-Abelian one is not [51,52].
Canonical Quantization renders divergent the amplitude for processes of the form (L stands for the longitudinal components of A μ ) where (A + μ , A − μ , A 0 μ ) are the gauge fields associated with a "root" of the semi-simple group G (Fig. 10).
In the specific case of the Standard Model, it would read: and the infinite contribution has to be substracted by means of processes involving the Higgs field (Figs. 11, 12):

Brief note on the Higgs-Kibble Mechanism
A conventional field has a self-interacting potential V (φ) = m 2 |φ| 2 (like the spring potential V (x) = kx 2 ) (Fig. 13 but that of the Higgs field is a bit different, V H (φ) = μ 2 |φ| 2 + λ|φ| 4 , corresponding to an imaginary mass (Fig. 14). The minimum of the potential is degenerated, at a distance v 2 ≡ − μ 2 λ of the origin, which implies that we have to decide which one should be the best !!.
"We break down" the symmetry by moving the origin to one of the local minima: When φ couples to a field W μ according to the Minimal Interaction Principle, the interaction term |φ| 2 turns to v 2 W 2 + ... giving mass to W μ .
In the same way, coupling φ to a fermion ψ à la Yukawa, that is, κφψψ, the displacement of φ leads to the mass term κvψψ, where κ is a constant, providing the mass κv to the fermion.
General case: G semi-simple group of dimension r H ∈ G of dimension s, preserving the vacuum φ representing G in dimension n η (Higgses) n − (r − s) massive real fields ξ (Goldstone bosons) r − s massless real fields, which will be gauged away A μ (massless vector bosons s A μ (massive vector bosons) r − s When n = r we shall have as many η's as A μ 's (n-(n-s) = s), provided that n ≥ r , of course.

Group Quantization of Non-Abelian Stueckelberg Field Model for Massive Gauge
Theory: Thinking of SU (2) The original non-Abelian Stueckelberg model was addressed by the Lagrangian given above (260) μ were made of external scalar fields φ (a) (x) behaving under the group G just like the own group parameters ϕ a (x) do. Here we just turn φ a into group parameters ϕ a ≡ g a and find the complete group law bearing the corresponding Solution Manifold as a co-adjoint orbit. Then, we apply GAQ instead of CQ [53].
Inspired on the symmetry of the particle S 3 -sigma model we directly guess the proper symmetry for the Massive Yang-Mills field theory associated with SU (2) gauge group (generalizations for other semi-simple groups are also possible).
The σ -sector is the more relevant one. The˜ SU (2) local group law (a central extension by U (1) of a group SU (2) local ) for elements of the formǓ ≡ (U, U μ U −1 , z ν ) ∼ (ϕ a , θ (b) μ , z ν ), U ∈ SU (2) local can be written in the form: x ∈ ≡ Cauchy Surface .
Note that (ϕ, θ, z) is a non-central extension of (ϕ, θ ) by z. *** Remark: The unextended local group SU (2) local can be formally rewritten as: where R(ϕ) is the adjoint rotation in SU (2). For λ = 1, we obtain the (generalized) gauge symmetry of Massive Yang-Mills fields, whereas for λ = 0 we recover the ordinary gauge symmetry of the Massless ones.*** Complete Group Law: (Including the Yang-Mills fields) ByǓ , we shall understandǓ ≡ (U, Lie algebra commutators: Sigma Sector It should be remarked that X θ (a) 1,2,3 are non-basic generators; they are derived, as operators, from X ϕ a . Note also that the parameters z μ (x) do not contribute to the SM. Adding Vector Bosons: (only nonzero commutators)

Massive Yang-Mills fields interacting with Fermionic Matter
As commented above, the relevant modification concerning the quantization of massive Yang-Mills interaction lies on the vector boson sector. Let us justify this fact by looking at the group action of this sector on the (Fermionic) matter fields.
The group law of the massive gauge symmetry must be completed with the Fermionic sector in the way: (U acting on ψ is assumed to be the standard linear action. The arguments of the fields are omitted whereas no confusion could arise) Then, the full expression of the generators of the symmetry group (omitting the local indices for the sake of simplicity) is as follows: In fact, a proper geometric analysis of the possible mixture of the involved U (1) subgroups, the Cartan subgroups, concludes that ϑ W should be quantized with a non-trivial ground value of 30 o (Fig. 15).
Graphically, this can be easily depicted by looking at the possible closed geodesic curves on the Cartan Torus taking into account that the "velocity" in one direction is twice than in the other ** Another remarkable fact related to the group approach to quantization of the electroweak interactions is that the mass generation in the Stueckelberg-like treatment involves the vector potentials but not, a priori, the Fermionic matter. Then, we have to be able to provide some group-theoretical algorithm to give mass to fermions.
In fact, as will be widely developed in the last chapter, devoted to possible generalizations of the gauge formulation of Gravitation, we resort to another mixing of the rigid symmetry, that time involving the Electromagnetic U (1) group and the Translation subgroup of the Poincaré group: This mixing leads to a momentum operator P 0 = P 0 + κ Q, combining the old energy and electric charge, so that the new mass operator for a charged fermion ψ is: Then, for "originally" massless particles (m 0 = 0) we get This mass-generation mechanism might be further developed involving more "sophisticate" mixings.
where we have introduced the notation A μσ with X (a) loses consistence. In fact, it is possible to sum up all the h (a) 's in a simpler quantity, precisely k ν μ = δ ν μ + h (a)ν μσ X σ (a) . The objects k ν μ will recover an algebraic role as associated with the symmetry group under a slightly different viewpoint (see below) and, for the time being, they simplify in general the transformation properties. In fact, the variation of k ν μ , δk ν μ , restrict to: Let us repeat Utiyama's Theorem I, very briefly, in terms of k ν μ : Given L matt (ϕ α , ϕ β μ ) invariant under G, the minimally coupled Lagrangian L matt (ϕ α , ϕ α μ , A (a) μ , k ν μ ) ≡ L matt (ϕ α , k ν μ (ϕ α ν + A (a) ν X α (a)β ϕ β )) , ≡ det(q), leads to an invariant actionŜ matt = ω L matt . In fact, the change of variables accomplishes the same task as before, that is, f (a) X (a) L matt (ϕ, ϕ ν , A, k) = f (a)X (a) L matt (φ, φ ν ) .
We must find now the structure of the Lagrangian driving the dynamics of the fields (A and F is the already known object be determined. Among all possible Lagrangians there is one that reproduces the Hilbert-Einstein Lagrangian (except for a total derivative). This Lagrangian is called Teleparallelism Lagrangian, and is given by: η λν η σ θ η μρ + 1 2 δ θ μ η νλ δ σ ρ − 1δ σ μ δ θ ρ η νλ , (337) where the numerical coefficients have been determined by hand in order to achieve our purpose, that is: It must be noticed that the equations of motion of a particle, derived from the Gauge Theory are: and it turns out to be equivalent to those of geodesic motion in the pseudo-Riemannian geometry addressed by (L−C) : although the formers do not correspond to a geodesic motion.
According to the general theory, L 0 = L 0 (T σ μν , F The equations of motion become: from which we conclude that the source for the torsion is the spin of the matter. Comparison with the standard theory: We shall limit ourselves to the case of absence of matter. In the vacuum case, the equation of motion for the field A (σρ) μ , becomes: and can be solved explicitly in terms of the Cartan torsion T σ μν : with A vacuum (σρ)μ ≡ A (λθ )vacuum μ η λσ η θρ , T σρλ ≡ T μ ρλ η μσ , that is to say, A vacuum (σρ)μ are the so-called Ricci rotation coefficients in the standard theory.
Then, in the vacuum, we arrive at The second equation implies that σ μν is symmetric and, therefore, it coincides with the Levi-Civita connection associated with g μν . Likewise, R μν coincides with the Ricci tensor providing the ordinary Einstein equations. Remark on the "gauge theory" of the Lorentz group: The Lorentz group is not an invariant subgroup of the Poincaré group and if we desire to keep the rigid invariance under the whole Poincaré group, making local the Lorentz subgroup entails necessarily the local character of the Translation subgroup and, then, of the total Poincaré group.

Beyond the Poincaré group as rigid symmetry
Naively, the more natural generalization of the Poincaré group as the starting rigid symmetry is the group G L(4, R), which had been considered in Literature long ago. It leads to Edington Geometry. The simplest and best motivated generalization is that addressed by the Weyl group, made of Poincaré and Dilatations transformations.
Even more interesting proves to be the generalization of GR combining the Weyl group with the mass-generating scheme, discussed above, giving dynamics to only the field associated with the dilatation parameter [56,57]. This constitutes some sort of "Stueckelberg" model for the Weyl group (Brief comments): We consider the Weyl group as G and start from a very special "matter" Lagrangian constituted by the partial-trace σ -Lagrangian associated with the dilatation subgroup of W . That is to say: The minimal coupling principle entails the minimal substitution: where θ (dil) μ is just ∂ μ ϕ dil . As far as the Lagrangian L 0 is concerned, we resort to the simplest, yet new possibility: σρ η μσ η νρ = k σ μ k ρ ν F (μν) + F (dil) μν F (dil) σρ g μσ g νρ ,

No-go theorems on symmetry mixing
The possibility of unifying internal gauge interactions with Gravity, as a gauge theory associated with a space-time symmetry group, was tied to the existence of a finite-dimensional global symmetry group containing the Poincaré group and an internal unitary (compact) group in a non-trivial way, that is, not a tensor product. This possibility was soon discarded by the publication of a series of papers establishing the now known as "No-Go theorems" on symmetries (see, in particular, [59,60]). The situation is quite different in dealing directly with infinite-dimensional groups where those theorems do not apply.

Electrogravity mixing
Thinking of Quantum Theory as a more exact theory than Classical Theory, and starting from the rigid symmetry of "quantum matter" we arrive at a non-trivial consequence consisting in a non-trivial mixing of space-time and internal gauge interactions. A first attempt was given at the Quantum Mechanical level [61], and then this idea was extended to field theory in the form of a generalized gauge theory [58]. Let us substitute the U (1)-extended Poincaré group,P, by the standard Poincaré group P. The Lie algebra ofP is: [ M μν , P ρ ] = η νρ P μ − η μρ P ν − (λ μ η νρ − λ ν η μρ ) ≡ C σ μν, ρ P σ + C μν, ρ , (388) where C μν, ρ ≡ λ ν η μρ − λ μ η νρ , is the (central) generator of U (1), and λ μ is a vector in the Poincaré co-algebra belonging to a certain co-adjoint orbit.
We shall take λ μ in the simplest, though non-covariant, way: the constant κ being the mixing parameter. Then, the new structure constants are C μ, σρ ≡ −κ(η ρμ δ 0 σ − η σ μ δ 0 ρ ), and give rise to the following curvature components: Note that F ( ) μν involves, apart from the free term A μ,ν − A ν,μ , the potentials A j μ associated with translations, which are omitted in the standard theory. Besides, the electromagnetic strength of gravitational origin find its source in the Coriolis-like gravitational potentials; that is to say, those of rotating massive bodies.
The geodesic motion, for instance, can be derived by considering matter Lagrangian corresponding to a single particle: L matt = 1 2m p μ p ν η μν . We easily arrive at:  , corresponding to the ordinary electromagnetic field added with the new mixing term. We refer the reader to Ref. [58] for specific details.