According to equation A.30, each mode of a quantum field has a ``zero-point'' energy of 21hf even when no further units of energy are present. If the field is really a vibrating string or some other material object, this isn't a problem because the total number of modes is finite: You can't have a mode whose wavelength is shorter than half the atomic spacing (see Section 7.5). But for the electromagnetic field and other fundamental fields corresponding to elementary particles, there is no obvious limit on the number of modes, and the zero-point energies can add up to something embarassing.
(a) Consider just the electromagnetic field inside a box of volume L3. Use the methods of Chapter 7 to write down a formula for the total zero-point energy of all the modes of the field inside this box, in terms of a triple integral over the mode numbers in the x, y, and z directions.
(b) There are good reasons to believe that most of our current laws of physics, including quantum field theory, break down at the very small length scale where quantum gravity becomes important. By dimensional analysis, you can guess that this length scale is of order Gℏ/c3, a quantity called the Planck length. Show that the Planck length indeed has units of length, and calculate it numerically.
(c) Going back to your expression from part (a), cut off the integrals at a mode number corresponding to a wavelength of the Planck length. Then evaluate your expression to obtain an estimate of the energy per unit volume in empty space, due to the zero-point energy of the electromagnetic field. Express your answer in J/m3, then divide by c2 to obtain the equivalent mass density of empty space (in kg/m3). Compare to the average mass density of ordinary matter in the universe, which is roughly equivalent to one proton per cubic meter. [Comment: Since most physical effects depend only on differences in energy, and since the zero-point energy never changes, a large energy density in ``empty'' space would be harmless as far as most of the laws of physics are concerned. The only exception, and it's a big one, is gravity: Energy gravitates, so a large energy density in empty space would affect the expansion rate of the universe. The energy density of empty space is therefore known as the cosmological constant. From the observed expansion rate of the universe, cosmologists estimate that the actual cosmological constant cannot be any greater than 10−7 J/m3. The discrepancy between this observational bound and your calculated value is one of the greatest paradoxes in theoretical physics.]