N ov 1 99 6 Classical Gravity on Fuzzy Space - Time
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چکیده
A review is made of recent efforts to find relations between the commutation relations which define a noncommutative geometry and the gravitational field which remains as a shadow in the commutative limit. The position x and the momentum p of a classical particle can be simultaneously measured and (x, p) defines a point in classical phase-space. The set of polynomials in the variables (x, p) can be added and multiplied; they form a commutative algebra. When the particle is quantized the points disappear; because of the Heisenberg uncertainty relations the two operatorsˆx andˆp can no longer be simultaneously measured. However, it still makes sense to consider the algebra of polynomials in them. It is a noncommutative algebra which Dirac [1] referred to as the 'quantum algebra'. A vector in classical phase space can be naturally identified with a derivation of the algebra of polynomials and as such can be generalized to the quantized case. Dirac called it a 'quantum differentiation'. From the correspondence principle we see that lim ¯ h→0
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تاریخ انتشار 1996