Problems of affine symmetry in physics

Abstract:
Usually symmetry of physical models is based on Lie groups preserving some scalar products, i.e., bilinear-symmetric or sesquilinear-hermitian forms. Typical examples are O(k,l), U(k,l), etc., used either as external symmetries acting on independent variables or internal symmetries affecting dependent field variables (e.g., in gauge theories). We are suggesting some models based on “amorphous” groups like GL(n,R), Gaf(n,R), GL(n,C), Pr(n,R) = SL(n+1,R), Pr(n,C) = SL(n+1,C). They are not associated with any scalar product concept, thus, they are “amorphous”, and in this sense more fundamental than “rigid” models with fixed (pseudo)metrical geometries. Discussed are both theoretical formal schemes and possibilities of physical applications in field theory, mechanics of continua (including ones with defects), nuclear dynamics and astrophysics. On the level of Hamiltonian mechanics the resulting models are somehow related to the theory of integrable lattices. Momentum mappings and relative equilibria are discussed. There are also alternative approaches to gravitation and quantum mechanics based on linear groups.