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We prove an implicit function theorem for nilpotent (not necessarily associative or Lie) algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of the classical correspondence between divisible torsion-free nilpotent groups and rational nilpotent Lie algebras. We study the related questions of the commensurators of nilpotent groups, filiform Lie algebras of maximal solvability length and partially ordered algebras. [ABSTRACT FROM AUTHOR]

Based on the concept of ontological states and their dynamical evolution by permutations, as assumed in the Cellular Automaton Interpretation (CAI) of quantum mechanics, we address the issue of whether quantum-classical hybrids can be described consistently in this framework. We consider chains of 'classical' two-state Ising spins and their discrete deterministic dynamics as an ontological model with an unitary evolution operator generated by pair-exchange interactions. A simple error mechanism is identified, which turns them into quantum mechanical objects: chains of qubits. Consequently, an interaction between a quantum mechanical and a 'classical' chain can be introduced and its consequences for this quantum-classical hybrid can be studied. We found that such hybrid character of composites, generally, does not persist under interactions and, therefore, cannot be upheld consistently, or even as a fundamental notion à la Kopenhagen interpretation, within CAI. [ABSTRACT FROM AUTHOR]

We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations e σ i in the spin representation (a.k.a. Lorentz rotors) in terms of their generators σ i : ln (e σ 1 e σ 2 ) = tanh − 1 tanh σ 1 + tanh σ 2 + 1 2 [ tanh σ 1 , tanh σ 2 ] 1 + 1 2 { tanh σ 1 , tanh σ 2 } . This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension ≤ 4 , naturally generalizing Rodrigues' formula for rotations in ℝ 3 . In particular, it applies to Lorentz rotors within the framework of Hestenes' spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex 2 × 2 matrix representation realized by the Pauli spin matrices. The formula is applied to the composition of relativistic 3 -velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle. [ABSTRACT FROM AUTHOR]

We introduce some general tools to design exact splitting methods to compute numerically semigroups generated by inhomogeneous quadratic differential operators. More precisely, we factorize these semigroups as products of semigroups that can be approximated efficiently, using, for example, pseudo-spectral methods. We highlight the efficiency of these new methods on the examples of the magnetic linear Schrödinger equations with quadratic potentials, some transport equations and some Fokker–Planck equations. [ABSTRACT FROM AUTHOR]

Based on the concept of ontological states and their dynamical evolution by permutations, as assumed in the Cellular Automaton Interpretation (CAI) of quantum mechanics, we address the issue of whether quantum-classical hybrids can be described consistently in this framework. We consider chains of ‘classical’ two-state Ising spins and their discrete deterministic dynamics as an ontological model with an unitary evolution operator generated by pair-exchange interactions. A simple error mechanism is identified, which turns them into quantum mechanical objects: chains of qubits. Consequently, an interaction between a quantum mechanical and a ‘classical’ chain can be introduced and its consequences for this quantum-classical hybrid can be studied. We found that such hybrid character of composites, generally, does not persist under interactions and, therefore, cannot be upheld consistently, or even as a fundamental notion à la Kopenhagen interpretation, within CAI.

The disentangled form of unitary operators is an indispensable tool for physical applications such as the study of squeezing properties or the time evolution of quantum systems. Here we derive a closed form disentanglement for the most general element of group ISp(2, R), whose generating Lie algebra is obtained by joining the Heisenberg-Weyl algebra to su(1,1). We attain the disentanglement formula resorting to an extension of the Truax method and check our findings through an independent factorization approach, based on the use of displacement operators. As a result we obtain a new form of factorized squeezing operators, whose action on the light vacuum state is calculated. • Analytical disentanglement of a general element of group ISp(2,R). • Group element as product of exponentials of the six algebra generators. • Extension to algebra isp(2,R) of the Truax method. • Check with a further method based on displacement operator. • Construction of generalized squeezing operator and squeezed states. [ABSTRACT FROM AUTHOR]