A Research of Star Products and Also Quantum Time Maps

The connection between a star product on a symplectic complex and a symplectic association on that complex shows up in numerous connections. Specifically, when one studies lands of invariance of star products, outcomes are much simpler when there is an invariant association. We demonstrate here that there is a characteristic class of star products which characterize a novel symplectic association. We consider the invariance of such products and the conditions for them to have a minute guide.

ASSOCIATIONS

The connection between the thought of star product on a symplectic complex and symplectic associations recently shows up in the fundamental paper of Bayen, Flato, Fronsdal, Lichnerowicz furthermore Sternheimer, and was further improved by Lichnerowicz who demonstrated that any purported Vey star product (i.e. a star product described by bidifferential drivers whose vital images at every request harmonize with those of the Moyal star product) confirms a novel symplectic association. Fedosov gave a development of Vey star products starting from a symplectic association and an arrangement of shut two structures on the complex. It was demonstrated that any star product is proportionate to a Fedosov star product. By the by, numerous star products which show up in regular settings (cotangent bunches, Kaehler manifolds. . . ) are not Vey star products (however are regular in the sense described above). The point of this area is to generalise the consequence of Lichnerowicz and to show that on any symplectic complex, a regular star product confirms a novel symplectic association. Given any torsion free linear associationon (M, P), the term of order 1 of a natural star product can be written where and the term of order 2 can be written in a chart whereand whereis skewsymmetric. Remark that E is not uniquely defined; two choices differ by an element. Observe that the first lines in the definition of C2 for two such different choices only differ by an element in. Indeed

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and (adEi) m, (adX) { , } are in, so also is ((adX) o (ad Ei) m). Changing the torsion free linear connection gives a modification of the terms of the second line of C2; writing, this modification involves terms of order 2 in one argument and 1 in the other given by as well as terms of order 1 in each argument, wheredenotes a cyclic sum over the indicated variables. Notice that the terms above coincide with the terms of the same order in the cobound- ary of the operator If the Poisson tensor is invertible (i.e. in the symplectic situation), the symbol of any differential operator of order 3 can be written in this form E', hence we have: Recommendation - A star product on a symplectic complex , so that Ci is a bidifferential driver of request 1 in every contention and C2 of request at generally 2 in every contention,

Comments - This shows, in particular, that any natural star producton a symplectic manifolddetermines a unique symplectic connection.

A derivationis said to be essentially inner or Hamiltonian if D =ad* u for some. We denote by Inn(M, *) the essentially inner deriv ations of *. It is a linear subspace of Der(M, *) and is the quantum analogue of the Hamiltonian vector fields. a Lie group almost *- Hamiltonian if eachis essentially inner, and call a linear choice of functions satisfyinga (quantum) Hamiltonian. We say the action is *-Hamiltonian if can be chosen to makea homomorphism of Lie algebras. Whenthis map is called a quantum time map. Considering the mapgiven by and defining a bracket onby then, by associativity of the star product, a is a homomorphism of Lie algebras whose image is Inn(M, *). Since, Inn(M, *) is an ideal in Der(M, *) and so there is an induced Lie bracket on the quotient Der(M, *)/Inn(M, *).

of derivations modulo inner derivations, Der(M,

proof - The first part is well known; let us recall that locally any derivation Der(M, *) is inner, and that the ambiguity in the choice of a corresponding function u is locally constant so that the exact 1-forms du agree on overlaps and yield a globally defined (formal) closed 1-form aD. The mapdefined by ifis a linear isomorphism with the space of formal series of closed 1-forms and maps essentially inner derivations to exact 1-forms inducing a bijection . Let Di and D2 be two derivations of. But [ui,u2]* does not change if we add a local constant to either function, so is the restriction to U of a globally defined function which depends only on Di and D2.

This shows thatand hence that the induced bracket onis zero. □ The kernel of a consists of the locally constant formal functionsand hence: Remark - If * is a differential star product on a symplectic manifoldthen there is an exact sequence of Lie algebras where. Result -Let G be a Lie assembly of symmetries of a star product * on and the prompted minuscule movement. Assuming that or at that point the activity is essentially *-Hamiltonian. Surely, by definition, the movement is practically *- Hamiltonian if . This is the situation under either of the two conditions.

PRODUCT WITH AN INVARIANT ASSOCIATION

Letbe endowed with a differential star product *, Recognize a polynomial math 0 of vector fields on M comprising of inferences of * and accept that there is a symplectic companionship which is invariant under . is identical, through an equivariant proportionality to a Fedosov star product constructed from and an arrangement of invariant shut 2-structures which give an agent of the trademark class of *. Watch that for anyif and only if Henceforth the Lie variable based math comprises of inward inductions for * if and just if this is correct for the Fedosov star product and this correct if and just if there exists an arrangement of capacities such that Thus Specifically, this yields Hypothesis - Let G be a reduced Lie gathering of symplectomorphisms of and the comparing Lie variable based math of symplectic vector fields on M. Recognize a star product * on M which is invariant under G. The Lie algebra comprises of inward inferences for * if and just if there exists an arrangement of functions and a representative of the trademark class of * such that

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