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

[Leonardo Sidis]

This is my second post concerning a question I had from Hawking book. I am completely new to this subject, and lack the mathematical background necessary to fully understand these concepts, but I hope that by posting my question I will at least not be so confused about it after receiving some responses.

Hawking says there is an infinite amount of ground state energy because there is no limit to how short the wavelengths of the Maxwell field can be.

A quote "..each wavelength would have a ground state energy. Since there is no limit to how short the wavelengths of the Maxwell field can be, there are an infinite number of different wavelengths in any region of spacetime and an infinite amount of ground state energy. Because energy density is, like matter, a source of gravity, this infinite energy density ought to mean there is enough gravitational attraction in the universe to curl spacetime into a single point, which obviously hasn't happened."

My understanding is that there are an infinite number of possible wavelengths, not infinite existing wavelengths as he seems to think there are, since there is only a finite number of waves. Wouldn't this explain the problem? Or am I missing something?

Thanks for any help.

 

[Parlyne]

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.

 

[Leonardo Sidis]

Thanks for the quick reply! So is the reason why the ground state energy is infinite is because there is a possibility for there to be an infinitely small wavelength, since, as you said, the smaller the wavelength the higher the ground state energy?