Über den Satz von Weil-Cartier

A generalisation of the process of normal quantisation is studied which deals with a physical system whose phase space is given by a coadjoint orbit of a locally compact Lie group G. Under certain conditions, an algebra structure-implemented by a Fronsdal *-product-is assigned to the family of physical observables in such a way that the normal quantisation provides a representation of the algebra. This method, then, provides a useful means of determining a physically sensible Fronsdal *-product for a given system.It is well-known that the traditional formalism of classical mechanics, where the phase space for the physical system is described by a symplectic manifold (M, co) and the observables jtf form a Lie subalgebra of C°°(M) under the Poisson bracket {,}, is not one which can be conveniently reconciled to quantum mechanics,, Indeed Moyal [10] has shown in the case M = R 2n that, in order for classical mechanics and quantum mechanics to be correctly related, the Lie algebra structure {,} of stf should be deformed to a new structure {*} (the Moyal bracket), which is related to the Poisson bracket {,} in the sense that {fag} = {f,g} + 0(fi 2 ) forf 9 g<=3/ and \f*g] = \f, g]for/e stf and ge j/ 03 where j/ 0 is the Lie subalgebra of C°°(M) spanned by the constant functions and the 2/2 coordinate functions f 1? . ". , £ 2n of R 2n , Indeed it has been shown that the algebra structure of pointwise multiplication on C°°(M) can be deformed to a new associative structure * (the Moyal product) on C°°(M) such that f*g=

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“…is given by (12) " dt ^ ' s (Hannabuss [3]). Let K be the stabiliser of/ under the (7-coadjoint action,, …”

Section: §3 Notation and Elementary Resultsmentioning

confidence: 99%

Some Aspects of Normal Quantisation

Hannabuss¹,

Hennings²

1985

Publ. Res. Inst. Math. Sci.

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“…Construction of the Map U Let us now suppose that a is a strongly nondegenerate multiplier on the locally compact separable abelian group G, and let f : G->(5 be the associated isomorphism. As in [3], [4], [12], for any subset E of G we define…”

Section: Jjr 2nmentioning

confidence: 99%

The Weyl Transformation and Quantisation for Locally Compact Abelian Groups

Hennings¹

1985

Publ. Res. Inst. Math. Sci.

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“…Now let /c be any function in o~ (Q), where Then O' is in ~r (G,/'), and clearly O and/~ may be so chosen that O' takes the value 1, say, at (0, 0) ~ G x G*. Now define, for 1 <~ p < 0% :P (G, /3 as the space of all complexvalued functions 0 on G x G* satisfying (1) Proof of (4). and by (6) above the inversion formula (4) is proved.…”

Section: Theta Functionsmentioning

confidence: 99%

Theta functions and symplectic groups

Reiter

1984

Monatshefte f�r Mathematik

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Abstract.A. WEIL introduced in 1964 theta functions on locally compact commutative groups (cf.[5] III). It is shown here that the use of certain function spaces on such a group G, and the consideration of theta functions on the product G x G, gives some insight into the structures involved, also in connection with Poisson's formula.Of the themes that play a prominent part in A. WEIL'S paper 'Sur certains groupes d'op&ateurs unitaires' (1964 b in [5] III), three will be discussed here (cf. the words of Democritus '~ ~T~Z~ z~p+~ ~6 vo3 (I) The role of the space 5 r (G) of Schwartz--Bruhat functions on a locally compact commutative group G in connection with A. Weil's methods, and other function spaces.(II) The Hilbert space of Weil's general theta functions (relative to a closed subgroup F of G), and its isomorphism with L 2 (G).(III) The connection between the proofs ofWeil's Th6or6me 1 and Th6or~me 4 in his paper.

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Über den Satz von Weil-Cartier

Cited by 16 publications

References 5 publications

Some Aspects of Normal Quantisation

Some Aspects of Normal Quantisation

The Weyl Transformation and Quantisation for Locally Compact Abelian Groups

Theta functions and symplectic groups

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