content_image/349146.jpg content_image/349147.jpg
when the energy and size are about 

<img src='content_image/349146.jpg'>

The quantity $_{ℏ}$/Mc is known as the Compton radius or Compton wavelength of mass M . Our inability to localize a single particle to better than its Compton radius is wellknown in particle physics. Indeed one fundamental reason that quantum field theory is used in particle physics is that it can describe the creation and annihilation of particles whereas a single particle wave function cannot. 34 

We now have two complementary minimum sizes for the volume containing a mass M : the Schwarzschild radius dictated by gravity is proportional to M , and the Compton radius dictated by quantum mechanics is inversely proportional M . The overall minimum occurs when the two are equal, which happens for 

<img src='content_image/349147.jpg'>

Thus the combination of gravity and quantum effects creates insurmountable difficulties if we attempt to shrink a volume to smaller than Planck size. 

A minor caveat should be repeated here: we have assumed that the volume is effectively 3 dimensional in that all of its dimensions are roughly comparable; if one of the dimensions is much larger or much smaller than the others the region is effectively one or two dimensional and the question of gravitational collapse is less clear, as noted in section II. 25,26 

## VI. MEASURING PROPERTIES OF A SMALL VOLUME 

For this rather generic thought experiment we use a quantum probe, such as a light pulse, to measure the size, energy content or other physical properties of a volume of characteristic size l , as shown in Fig.5. Such properties may in general fluctuate significantly in the time it takes light to cross the volume, so we would naturally want to do the measurement within that time, T ≈ l/c . For this we need a probe with frequency greater than c/l and energy greater than about E ≈ $_{ℏ}$( c/l ). 

