Aspects of classical and quantum Nambu mechanics

Chatterjee, Rupak; Takhtajan, Leon A.

doi:10.1007/bf00312678

Cited by 50 publications

(62 citation statements)

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“…Other aspects of operatorial quantization of Nambu Mechanics were discussed in [1], [6], [20]. In [1], is performed an embedding of R 3 into R 6 and the original Nambu Mechanics [16] is formulated in terms of usual Hamiltonian flow with constraints.…”

Section: Difficulties With Usual Quantizationsmentioning

confidence: 99%

Deformation quantization and Nambu Mechanics

et al. 1997

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Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over R of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products.

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Section: Difficulties With Usual Quantizationsmentioning

confidence: 99%

Deformation quantization and Nambu Mechanics

et al. 1997

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“…It is surprising that this kind of generalization of Hamiltonian dynamics has not been considered so far in the theory of classical dynamical systems (Ratiu, n.d.). One of the reasons seems to be the fact that for general Lie algebras the bracket does not satisfy the so called fundamental identity (Chatterjee and Takhtajan, 1996). We shall return to this question in Sec.…”

Section: A Nambu 1973: Euler Equationsmentioning

confidence: 99%

“…The generalization of (3) which was extensively investigated in the literature under the name of a "generalized Nambu dynamics" goes in another direction (Bayen and Flato, 1975;Takhtajan, 1994;Gautheron, 1996; Chatterjee and Takhtajan, 1996). One treats the ǫ not as structure constants but as a volume form.…”

Section: A Nambu 1973: Euler Equationsmentioning

confidence: 99%

“…Obviously, the tensor has 3-indices for all Lie algebras and for this reason it is not immediately clear whether a generalization of (4) to a "generalized Nambu" n-bracket is possible. On the other hand, the structure constants occuring in (10) have a rich structure and it turns out there exists a natural generalization of (10).…”

Section: Structure Constants Revisitedmentioning

confidence: 99%

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Czachor¹

1999

International Journal of Theoretical Physics

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“…During the past two decades Nambu proposal has been a matter for many investigations [2] - [12] and the permanent interest for this issue is related with the recognition of the feasible physical richness and the mathematical beauty of ternary and higher algebraic systems. Recently such an algebraic structure has been analyzed and reformulated by Tachtajan [7] - [9] in an invariant geometrical form. He proposed the notion of Nambu-Lie "gebra", which is a generalization of Lie algebras for ternary (in general n-ary) case.…”

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confidence: 99%