An algebraic formulation of quantum mechanics in fractional dimensions

Ajit Kumar

Max A. Lohe, Alagu Thilagam

We formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators P , Q satisfy the R -deformed Heisenberg relations, which depend on an operator ν. We find representations of P , Q in which the dimension d and angular momentum ℓ appear as parameters related to the eigenvalues of ν. We analyse the domain of P and find conditions which ensure that P is Hermitian. We investigate plane wave solutions and also free particle wavefunctions in fractional dimensions, and show that as a consequence of wavefunction continuity ℓ is quantized. The representations of P , Q also lead to the corresponding representations of paraboson operators which are used to solve the harmonic oscillator in dimension d , both algebraically and analytically. We demonstrate that the formalism extends also to time-dependent Hamiltonians by solving the time-dependent harmonic oscillator in any dimension d > 0 using the method of Lewis and Riesenfeld.

Ralf Otte

Quantum mechanics is formulated on complex Hilbert spaces, where the complex numbers constitute the underlying scalar algebra. In this work, the structural question is investigated whether this scalar structure is algebraically minimal or whether a consistent extension exists that preserves the analytical infrastructure of the theory. We formulate this problem as a minimality question for finite-dimensional real, commutative and associative algebras that contain the complex numbers as a subalgebra and at the same time possess a nontrivial ideal that is isomorphic to C as a real algebra and carries a complex-like exponential structure. It is shown that such structures necessarily require at least four real dimensions. The bicomplex algebra B=C ⊗R C realizes these requirements in minimal dimension and is unique up to isomorphism under the stated conditions. For the extended state space M=H ⊗C B one obtains a canonical decomposition M≅H⊕H, so that the bicomplex scalar extension structurally corresponds to two parallel complex quantum sectors. The Born rule remains unchanged. The present work provides a structural analysis of possible extensions of the scalar algebra underlying quantum mechanics.

Iwo Białynicki‐Birula, P. J. Morrison

Abstract It is shown that Nambu dynamics can be generalized to any number of dimensions by replacing the 0(3) algebra, a prominent feature of Nambu's formulation, by an arbitrary Lie algebra. For the infinite dimensional algebra of rotations in phase space one obtains quantum mechanics in the Weyl-Wigner representation from the generalized Nambu dynamics. Also, this formulation can be cast into a canonical Hamiltonian form by a natural choice of canonically conjugate variables.

Istvan Montvay