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- W2890172889 abstract "Our study concerns a novel scheme for the quantization of positive-definite matrix degrees of freedom of the type associated with the spacial part of the metric tensor of general relativity. Standard canonical schemes for the quantization of such objects, wherein each matrix element is quantized independently, run into difficulties because the spectrum of the resulting 'quantum matrix' is not, in general, positive definite. We demonstrate that this situation can be overcome by abandoning canonical quantization in favor of 'affine quantization', that is, utilizing a quantum kinematical structure based the affine group, rather than the Heisenberg group. Our approach involves a generalization of the one-dimensional affine group, the group of translations and dilations of the real line. A multidimensional generalization of the one-dimensional affine Lie algebra possesses a representation in which, crucially, the spectrum of the matrix operator of interest is positive definite. The corresponding Hilbert space admits a group-defined coherent state representation, and we develop the formalism of this representation, including the formulation of coherent state overlap functions as suitably regularized path integrals. The coherent state label space, endowed with a suitable ray metric, may be viewed as a Riemannian manifold whose scalar curvature is shown to be a negative constant. This symmetry leads naturally, through the procedures of metrical quantization, to acceptable Hilbert spaces of high dimension. The incorporation of constraints into affine quantum mechanics represents an important problem, especially from the point of view of quantum gravity. We outline a relatively new technique, involving projection operators, that overcomes many of the difficulties associated with more standard approaches, and provide a simple example. ( en )" @default.
- W2890172889 created "2018-09-27" @default.
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- W2890172889 date "2008-01-01" @default.
- W2890172889 modified "2023-09-24" @default.
- W2890172889 title "Affine Quantization of Metric Variables" @default.
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