*Jan SLAWIANOWSKI*

Institute for Fundamental Technological Problems,

Polish Academy of Sciences,

21 Swietokrzyska Str., 00-049 Warsaw, POLAND

E-mail: jslawian@ippt.gov.pl
**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.