Download Contemporary Problems in Mathematical Physics by Jan Govaerts, M. Norbert Hounkonnou, Alfred Z. Msezane PDF

By Jan Govaerts, M. Norbert Hounkonnou, Alfred Z. Msezane

The COPROMAPH convention sequence has now developed right into a major foreign area the place primary strategies in mathematical and theoretical physics and their physics purposes should be conceived, constructed and disseminated. simple rules for addressing a number of modern difficulties in mathematical and theoretical physics are offered in a nonintimidating surroundings. specialists give you the reader the basics to foretell new probabilities in physics and different fields.

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Examples: the sphere 5 " in Rn+1, a (one- or two-sheeted) hyperboloid in E n + 1 , a torus in ffi3, the Mobius strip in E 3 . 21. A Lie group is a group G which is at the same time a smooth manifold such that the map G x G -» G, (g,h) H-> gh~x is (infinitely often) differentiable. Examples: . Abelian Lie groups: finite dimensional vector spaces (with vector addition) , S 1 (with multiplication of complex numbers) . Matrix groups: SO(2), SO(3), SU(2), SO(l,3), GL(n,E) = GL(E"). Note that, if G is a Lie group and H a closed subgroup of G, then the quotient G/H is a smooth manifold.

It is the set of fixed points of a. The commutation relations of g are the following: [1,6] C 6, (thus 6 is a subalgebra) [e,p] = p, [p,p]Ct. Then the Lie algebra g* = t ©p*, where p* = ip, is another real form of g c , and it is noncompact (this construction is called Weyl's unitary trick). The commutation relations of g* read: [t, 6] C E, [e,p*] = p*. [p*,p*]c-e. (6 is still a subalgebra) 27 Thus classifying the real forms amounts to classify the involutive automorphisms, and again the result was obtained by Cartan.

Then one can define the unitary quasi-regular representation: [UqL(g)f}(x) = J^^f(g-lx),feSj = L2(X,dvL). (19) In these expressions, the square root factor is a Radon-Nikodym derivative, which compensates for the non-invariance of the measure. Examples: . G = SO(3), H = SO(2), X = S2. G = S0 0 (l,3), H = SO(3), X = S0 0 (l,3)/SO(3) = two-sheeted hyperboloid. 3. Two representations, T\ and Ti in the Hilbert spaces f)i,i}2, respectively, are equivalent if there exists an invertible operator S : Sji -> f)2 such that T2(g) = ST1(g)S-\ for a l l 3 € G.

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