Comentario sobre el artículo de Mangano

undated (1996-2003 by context) · Notas de física · 155 words · recovered from win/Escritorio/alejo/mathrew/mangano.tex

 The author suggests to build non commutative symplectic spaces
by introducing a dimensional parameter $\lambda_P^2$ which replaces
$\hbar$ in the Feymann recipe. With this goal,
the paper extends to explain how the {\it generating functional} 
technique can be used to study such spaces: quantization defines
a one parametric family of algebras $A(\tau)$, got from $2^n$
generators $X^\mu(\tau)$, whose commutation rules can be 
indirectly studied through linear functionals
$\rho_{x_0}([X^\mu(\tau_1)...X^\nu(\tau_k)])\equiv
(-i\lambda_P^2)Z(x_0,0)^{-1} {\partial \over \partial J_\mu (\tau_1)
}...{\partial\over\partial J_\nu(tau_k)} Z(x_0,J) |_{J=0}, $
where $Z(x_0,J)$ is the generating functional defined as usual. 

The parameter $\Lambda_P$ has obviously dimensions of length. In the
commutative limit, $\lambda_P \to 0$, the state $\rho_{x_0}$ becomes
the equivalent of Dirac' delta measure, or evaluation map, over the
algebra of commutative functions.
The production of such functionals for the case $R^2$ is developed in detail,
and then generalization
to $R^{2n}$ and extensions to generic $M^{2n}$ manifolds are sketched in
the rest of the paper.
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