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Noncommuting position variables due to quantum gravity 
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#1
Jan2913, 10:50 AM

P: 20

I've read that if one wants to quantize gravity , there must be a smallest length scale " The planck length " . If I want to measure the position of a point particle then in conventional Quantum mechanics I'll find it at the point $(x,y,z)$ at sometime $t$ with an arbitrary momentum but this can't be though in quantum gravity since the smallest length scale that can exist is the planck scale so the particle must be at a neighbourhood of (x,y,z) of area that's equal to planck length . So if we measured the X operator to find the eigenvalue x the particle must be spread in the eigenspace of the Y and Z operators that's X,Y,Z are not commuting operators So we must have [X,Y] not equal to zero .
Is this line of reasoning correct ? Have something of this sort been worked out ? It seems that many things in Quantum mechanics should be modified if we want to incorporate gravitational effects 


#2
Jan2913, 02:20 PM

Sci Advisor
P: 8,665

http://arxiv.org/abs/hepth/0106048
Noncommutative Field Theory Michael R. Douglas (Rutgers, IHES), Nikita A. Nekrasov (IHES, ITEP) We review the generalization of field theory to spacetime with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, both on the classical and quantum level. 


#3
Jan2913, 02:49 PM

P: 20

Sounds interesting . What mathematical knowledge required to be able to understand that paper ?



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