Abstract
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 vartheta _W 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.
Highlights
The idea of formulating the basic interactions among elementary particles in terms of vector potentials, generalizing electromagnetism, is traced back to the pioneers papers by Yang and Mills [1], Utiyama [2] and Kibble [3]
Theorem (Noether): If Y 1 is a symmetry of PC, the quantity JY 1 ≡ iY 1 PC − αY 1 is conserved along the solutions
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
Summary
The idea of formulating the basic interactions among elementary particles in terms of vector potentials, generalizing electromagnetism, is traced back to the pioneers papers by Yang and Mills [1], Utiyama [2] and Kibble [3]. L = i ψγ μ∂μψ − mψψ, which is invariant under the rigid (or global) group U (1), that is, under the transformation ψ(x) → ψ (x) = e−iαψ(x), x ∈ M the space-time, Page 3 of 85 304 we require L to be minimally modified, to Lso as to be invariant under the corresponding gauge (or local) transformation ψ → ψ = e−iα(x)ψ. Note that the term in the original Lagrangian ψγ μ∂μψ, due to the derivative acting on the local parameter α(x), transforms as ψγ μ∂μψ → ψγ μ∂μψ−i ψγ μ∂μα(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. For non-Abelian symmetries, associated with a (let us say) compact group G, we generalize the discussion above:. ∂μφa where by ∂μφa we mean something like θb(a)(φ)∂μφb, associated with the canonical (left or right) 1-form on the Lie group G
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