Geometry of the Fundamental Interactions: On Riemann's Legacy to High Energy Physics and Cosmology
M. D. Maia
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The Yang-Mills theory of gauge interactions is a prime example of interdisciplinary mathematics and advanced physics. Its historical development is a fascinating window into the ongoing struggle of mankind to understand nature. The discovery of gauge fields and their properties is the most formidable landmark of modern physics. The expression of the gauge field strength as the curvature associated to a given connection, places quantum field theory in the same geometrical footing as the gravitational field of general relativity which is naturally written in geometrical terms. The understanding of such geometrical property may help one day to write a unified field theory starting from symmetry principles.
Of course, there are remarkable differences between the standard gauge fields and the gravitational field, which must be understood by mathematicians and physicists before attempting such unification. In particular, it is important to understand why gravitation is not a standard gauge field.
This book presents an account of the geometrical properties of gauge field theory, while trying to keep the equilibrium between mathematics and physics. At the end we will introduce a similar approach to the gravitational field.
Rαβγ ε + Rαεβγ + Rαγ εβ = 0 (2.8) (2.9) (2.10) (2.11) Finally the covariant derivative of Riemann’s tensor gives the Bianchi’s identities Rαβγ ε;μ + Rαβεμ;γ + Rαβμγ ;ε = 0 (2.12) Ricci’s curvature tensor is derived from Riemann’s tensor by a contraction Rαε = g βγ Rαβγ ε (2.13) On the other hand, the contraction of Ricci’s tensor gives the scalar curvature (or the Ricci scalar curvature). R = g αβ Rαβ (2.14) We shall return to the Riemann tensor in the latter sections, showing that it has
(θ )) = f (θ, θ ) (3.1) Since these charts cover the whole group, they form an atlas similar to the case of differentiable manifolds. Then the above condition (3.1) must be satisfied for all elements of the group (such condition does not exist in differentiable manifolds). When the parameters vary continuously within a given interval on IR N and (3.1) is a homeomorphism we have a continuous group. This is less demanding than the differentiable manifold structure where the relation between the
coordinate transformation, required in conformity with the diffeomorphism invariance and with the transformation of coordinates in a volume integral. The meaning of the Einstein– Hilbert principle seems clear, although it is seldom mentioned: It means that the space–time based on Riemann’s geometry of the curvature is the smoothest possible. The concept of time in general relativity is that of a mere coordinate. Together with the principle of general covariance, time has lost its special
charged particles was intermediated by photons. The second important contribution to gauge theory from the period 1918 to 1919 was the theorem by Emmy Noether showing how to construct the observables of a physical theory, starting from the knowledge of its Lagrangian and its symmetries . Although the motivations and results of Weyl and Noether were independent, they met at the point where Noether introduced a matrix-vector quantity (a vector whose components are matrices) to obtain the
type of field. We will denote this variation generically by 8.1 Noether’s Theorem for Coordinate Symmetry 109 δA Ψ = Ψ (x) − ψ(x) Since this algebraic variation depends on each type of field, we may start by examples. The algebraic variation of a scalar field is by its own definition equal to zero. On the other hand, the algebraic variation of a vector field A given by its contravariant components is given by the contravariant tensor transformation. For an infinitesimal transformation like