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Noncanonical quantization of gravity. I: Foundations of affine quantum gravity. (English) Zbl 0954.83010
A model of quantum gravity ({\it affine quantum gravity}) is proposed in which canonical commutation relations are interchanged on affine commutation relations $[Q,D] = iQ$, where $D = (PQ + QP)/2$, $P$ and $Q$ are the operators connected with gravity (metric). A primary set of the normalized {\it affine coherent states} is defined by $|p,q \rangle \equiv e^{ipQ} e^{-i(\ln q) D} |\eta \rangle$, where $-\infty < p < \infty$, $0 < q < \infty$. On the basis of these definitions a path integral construction for quantum gravity is given. The choice of a metric on the classical phase space is discussed. A set of conventional local annihilation and creation operators $A(x,h)$ are introduced and local metric and scale operators defined with help of these $A(x,h)$ operators. Using $A(x,h)$ operators the local product for the gravitational field operators are deduced. In the last section the imposition of constraints is discussed. It is necessary to note that this model of quantum gravity does not engender topological changes of the underlying topological space.

83C45Quantization of the gravitational field
81V17Gravitational interaction
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