Abstract
The operadic Lax representations of the harmonic oscillator are used to construct the quantum counterparts of 3d real Lie algebras in the Bianchi classification. The Jacobi operators of these quantum algebras are studied. It is shown how the energy conservation is related to the Jacobi identity and how the quantization leads to an anomaly – the quantum violation of the Jacobi relations. By using the nonvanishing quantum Jacobi operators, the derivative quantum algebra for a triple of 3d real Lie algebras is defined. It is proposed that the derivative algebra is the 3d real Heisenberg algebra. From this it follows that in this model only the discrete values of the spatial coordinates are physically allowed.
Highlights
Introduction and outline of the paperIn Hamiltonian formalism, a mechanical system is described by the canonical variables qi, pi and their time evolution is prescribed by the Hamiltonian equations dqi = ∂H, dt ∂pi dpi dt = − ∂H ∂qi (1.1)By a Lax representation [3] of a mechanical system one means such a pair (L, M ) of matrices L, M that the above Hamiltonian system may be represented as the Lax equation dL dt ML LM (1.2)from the algebraic point of view, mechanical systems may be represented by linear operators, i.e by linear maps V → V of a vector space V
From the algebraic point of view, mechanical systems may be represented by linear operators, i.e by linear maps V → V of a vector space V
If an operadic system depends on time one can speak about operadic dynamics [4]
Summary
In [5, 6, 7], the low-dimensional binary operadic Lax representations for the harmonic oscillator were constructed. The operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of some real three dimensional Lie algebras. It turns out that the Jacobi identity is violated in these quantum algebras and to inquire situation we use in Sec. 9 and Sec. 10 the semiclassical approximation. In this approximation one can explicitly see how the quantum mechanical fundamental canonical commutation relations spoil the Jacobi identity on the quantum algebras
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