Clifford Algebras in Symplectic Geometry and Quantum Mechanics
نویسندگان
چکیده
The necessary appearance of Clifford algebras in the quantum description of fermions has prompted us to re-examine the fundamental role played by the quaternion Clifford algebra, C0,2. This algebra is essentially the geometric algebra describing the rotational properties of space. Hidden within this algebra are symplectic structures with Heisenberg algebras at their core. This algebra also enables us to define a Poisson algebra of all homogeneous quadratic polynomials on a two-dimensional sub-space, F of the Euclidean three-space. This enables us to construct a Poisson Clifford algebra, HF , of a finite dimensional phase space which will carry the dynamics. The quantum dynamics appears as a realisation of HF in terms of a Clifford algebra consisting of Hermitian operators.
منابع مشابه
Generalized Wavefunctions for Correlated Quantum Oscillators IV: Bosonic and Fermionic Gauge Fields
The hamiltonian quantum dynamical structures in the Gel’fand triplets of spaces used in preceding installments to describe correlated hamiltonian dynamics on phase space by quasi-invariant measures are shown to possess a covering structure, which is constructed explicitly using the properties of Clifford algebras. The unitary Clifford algebra is constructed from the intersection of the orthogon...
متن کاملProcess, Distinction, Groupoids and Clifford Algebras: an Alternative View of the Quantum Formalism
In this paper we start from a basic notion of process, which we structure into two groupoids, one orthogonal and one symplectic. By introducing additional structure, we convert these groupoids into orthogonal and symplectic Clifford algebras respectively. We show how the orthogonal Clifford algebra, which include the Schrödinger, Pauli and Dirac formalisms, describe the classical light-cone str...
متن کاملUniversality of quantum symplectic structure
Operating in the framework of ‘supmech’( a scheme of mechanics which aims at providing a concrete setting for the axiomatization of physics and of probability theory as required in Hilbert’s sixth problem; integrating noncommutative symplectic geometry and noncommutative probability in an algebraic setting, it associates, with every ‘experimentally accessible’ system, a symplectic algebra and o...
متن کاملBRST Structures and Symplectic Geometry on a Class of Supermanifolds
By investigating the symplectic geometry and geometric quantization on a class of supermanifolds, we exhibit BRST structures for a certain kind of algebras. We discuss the undeformed and q-deformed cases in the classical as well as in the quantum cases. Alexander von Humboldt fellow. On leave from Institute of Physics, Chinese Academy of Sciences, Beijing
متن کاملOn the relation of Manin’s quantum plane and quantum Clifford algebras
One particular approach to quantum groups (matrix pseudo groups) provides the Manin quantum plane. Assuming an appropriate set of non-commuting variables spanning linearly a representation space one is able to show that the endomorphisms on that space preserving the non-commutative structure constitute a quantum group. The noncommutativity of these variables provide an example of non-commutativ...
متن کامل