Coherent states and projective representation of the linear canonical transformations

@article{Daubechies1980CoherentSA,
  title={Coherent states and projective representation of the linear canonical transformations},
  author={Ingrid Daubechies},
  journal={Journal of Mathematical Physics},
  year={1980},
  volume={21},
  pages={1377-1389}
}
  • I. Daubechies
  • Published 1 June 1980
  • Mathematics
  • Journal of Mathematical Physics
Using a family of coherent state representations we obtain in a natural and coordinate‐independent way an explicit realization of a projective unitary representation of the symplectic group. Dequantization of these operators gives us the corresponding classical functions. 
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References

SHOWING 1-10 OF 22 REFERENCES
Linear Canonical Transformations and Their Unitary Representations
We show that the group of linear canonical transformations in a 2N‐dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N
Remarks on boson commutation rules
A group theoretical derivation is given of Bargmann's representation of the boson commutation rules in an Hilbert space of analytic functions. Several interesting problems arise in the study of the
Quasi-free states of the C.C.R.—Algebra and Bogoliubov transformations
We give a complete characterization of quasi-free states (generalized free states) of the C.C.R. algebra. We prove that the pure quasi-free states areall Fock states and that any two Fock states are
An integral transform related to quantization
We study in some detail the correspondence between a function f on phase space and the matrix elements. (Qf)(a,b) of its quantized Qf between the coherent states ‖ a〉 and ‖ b〉. It is an integral
The Hamiltonians of the Schrödinger algebra
We give a complete characterization of the one‐dimensional nonconjugate subalgebras of the extended Schrodinger algebra (and of the Schrodinger algebra). This yields a classification into conjugacy
Parity operator and quantization of δ-functions
In the Weyl quantization scheme, the δ-function at the origin of phase space corresponds to the parity operator. The quantization of a functionf(υ) on phase space is the operator εf(υ/2)W(υ)dυM,
The C*-algebras of a free Boson field
AbstractWe give a systematic description of severalC*-algebras associated with a free Boson field. In this first part the structure of the one-particle space enters only through its symplectic form σ
Expression explicite de L'exponentielle gauche pour les elements finis du groupe symplectique inhomogene
We explain the special role of the inhomogenous symplectic group (linear mechanics) in the framework of Wigner isomorphism, and give an explicit representation in the set of functions on the phase
An algebraic approach to quantum mechanics
SummaryA new mathematical frame for quantum mechanics is proposed. Investigating the connection of this frame to the usual one it turns out that this approach corresponds to Haag-Kastler’s idea about
...
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