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Matthew Lake

Title: The black hole - uncertainty principle correspondence from extended de Broglie relations

Abstract: The Compton wavelength, which gives the minimum radius within which the mass of a particle may be localized due to quantum mechanical effects, and the Schwarzschild radius, which gives the maximum region within which the mass of a black hole may be localized due to classical gravity, become coincident for objects with rest mass equal to the Planck mass, $m_P$. On a $(\log(r),\log(m))$ plot, the two lines intersect near the Planck point $(l_P,m_P)$, at which quantum gravity effects become significant.

Since canonical (non-gravitational) quantum mechanics is based on the concept of wave-particle duality, encapsulated mathematically in the de Broglie relations, these relations should break down near $(l_P,m_P)$, and it is unclear what physical interpretation can be given quantum particles with $E >> m_Pc^2$, since these correspond to wave modes with wavelengths $\lambda << l_P$ or time periods $T << t_P$ in the standard theory.

We therefore propose a modification of the standard de Broglie relations, which gives rise to a modified evolution equation and a modified expression for the Compton wavelength, which may be extended into the region $E >> m_Pc^2$. For the proposed modifications, we recover the expression for the Schwarzschild radius for $E >> m_Pc^2$ and the usual Compton formula for $E << m_Pc^2$. Importantly, the sign of the inequality flips at $m \sim m_P$, as required, and we interpret the additional terms in the modified de Broglie relations as representing the self-gravitation of the wave packet.