Quantization of Affine Bodies: Theory and Applications in Mechanics of Structured Media

Abstract

Discussed is kinematics and dynamics of bodies with affine degrees of freedom, i.e., homogeneously deformable "gyroscopes". The special stress is laid on the status and physical justification of affine dynamical invariance. On the basis of classical Hamiltonian formalism the Schroedinger quantization procedure is performed. Some methods of the partial separation of variables, analytical treatment and search of rigorous solutions are developed. The possiblity of applications in theory of structured media, nanophysics, and molecular physics is discussed.

Mechanics of systems of affine bodies. Geometric foundations and applications in dynamics of structured media

“…When dealing with objects like molecules, fullerens, graphens, and clusters, one must use the quantized version of the model. Certain primary ideas in this direction were presented in [22,23].…”

Quantized mechanics of affinely rigid bodies

“…But, as said above, quite independently of those fundamental problems, almost every mechanical and field‐theoretical scheme admits those 2 formulations: classical and quantum. For example, large molecules or fullerenes may be described in a satisfactory way in both frameworks, every one to be used in its specific domain of applications (see, for instance, our most recent paper about classical dynamics of fullerenes and some earlier papers about quantized mechanics of collective and internal affine modes). Moreover, such micro‐ and nano‐objects are physically placed somewhere in the convolution region of 2 theories.…”

“…The configuration space of an affinely rigid body may be identified with LI( U , V )× M , the Cartesian product of the internal configuration space and the manifold of translational degrees of freedom . When we choose some orthonormal Cartesian coordinates in M , N , namely, x i , a K , then the induced coordinates in the configuration space are x i , and the configuration manifold itself is identified with the semi‐direct product .…”

“…The formulas (16) tell us that L(k), R(k) are unitary with respect to the scalar product based on the Haar measure µ on G. Therefore we have that L (exp (q µ E µ )) = exp (q µ L µ ), R (exp (q µ E µ )) = exp (q µ R µ ), and that L µ , R µ are formally anti-self-adjoint:…”

Ground states in complex bodies

“…• Bodies with affine structure: In the special case in which the manifold of substructural shape coincides with the linear space of second-rank tensors, the substructure is called affine [44,53]. The scheme is suitable to cover various cases such as the one of bodies with dense polymeric linear chains discussed above or fullerene-reinforced composites.…”