# Maxwell field, ground state energy, 'The Universe in a Nutshell' help

by Leonardo Sidis
Tags: energy, field, ground, maxwell, nutshell, state, universe
 P: 546 In quantum field theory, it can be shown that a wave of a particular wavelength acts mathematically like a quantum mechanical harmonic oscillator of the associated frequency - $$\nu = \frac{c}{\lambda}$$. The energy of a quantum mechanical harmonic oscillator is given by $$E = h \nu \left (n + \frac{1}{2} \right )$$, where h is Planck's constant (~6.6261e-34 Js) and n is the number of excitations of the oscillator. In the case of the Maxwell field, n is taken to be the number of photons of that particular wavelength. If we look at the case where $$n = 0$$, we see that there is still some amount of energy, $$E_0 = \frac{h \nu}{2}$$, in this particular mode of the field, even though there are no observed waves at the particular wavelength. This analysis applies to any wavelength we can consider. But, wavelength is a continuously varying quantity. This means that, even if we though that there was a smallest possible wavelength and a largest possible wavelength, there would be an infinite number of wavelengths between those, each of which would contribute energy. Even worse, though, is that the zero point energy of a mode grows as the wavelength gets smaller. We can imagine that there might be conditions set a maximum wavelength. But, from the point of view of field theory, there shouldn't be conditions that set a minimum. This means that there must be modes which contribute arbitrarily large zero point energies. All of this together leads to the conclusion that, in the theory as it exists, there is an infinite ground state energy.