@MISC{Frampton08innocuousimplications, author = {Paul H. Frampton}, title = {Innocuous Implications of a Minimum Length in Quantum Gravity}, year = {2008} }

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Abstract

A modification to the time-energy uncertainty relation in quantum gravity has been interpreted as increasing the duration of fluctuations producing virtual black holes with masses greater than the Planck mass. I point out that such virtual black holes have an exponential factor arising from the action such that their contribution to proton decay is suppressed, rather than enhanced, relative to Planck-mass black holes. frampton@physics.unc.edu In a remarkable paper Sakharov [1] not only provided his well-known requirements for baryogensis in the early universe but also made the first theoretical estimate of the proton lifetime. Ignoring all factors of order O(1), as I shall do throughout, his formula had the form τp ∼ M4 Planck M 5 p (1) and gives a value τp ∼ 10 45 y. This contains only the contribution of quantum gravity and the lifetime in Eq. (1) is far too long for practical measurement. The present emprical lower bound [2] on τp is only τp ∼ 5 × 10 33 y. Although Sakharov did not use the language of spacetime foam, the Euclidean path integral approach to quantum gravity provides the means to make a similar estimate of the proton lifetime [3]. The amplitude for producing a black hole through a fluctuation of the space-time metric leads to a density of Planck-size black holes of order one, in dimensionless units. The estimate for the proton decay rate arises from looking at virtual black holes (hereafter VBHs) with masses in the vicinity of the Planck mass. VBHs underly, from this viewpoint, the proton lifetime formula found by Sakharov as in Eq.(1). To arrive at Eq.(1) employing parallel methods to insights in [3], there is, in general, an exponential tunneling factor [4]. When the VBH has a mass close to the Planck mass, the exponential factor is of order one, O(1), and so does not survive in Eq.(1). For the case of VBHs which are significantly heavier than the Planck mass, MV BH = ηMPlanck with η ≫ 1 there was recently an interesting paper [5] which merits further study. It arrives at the following formula for the proton lifetime