Posted by Steve Simon on January 17, 2013, 4:33 pm, in reply to "Infinite zero point energy"
Very good question.
For the case of sound waves, it turns out that there is no real divergence. As Debye guessed, there are in fact only a finite number of degrees of freedom for motion of atoms in the solid (3N degrees of freedom). Debye imposed this limit on the number of degrees of freedom ad-hoc, but we will see later on in the course that he was right to do this. So the zero point energy actually gives a finite result not an infinite result.
On the other hand, for electromagnetism (Planck's calculation) the situation is really disturbing. You can think about electromagnetic waves up to infinite frequency... so there really are an infinite number of electromagnetic wave modes in a box. This means that the blackbody energy of the light in the box follows a T^4 law up to infinite temperature. Presumably this also means that there is genuinely infinite zero point energy inside the box!!
It turns out however, that it is very hard to construct an experiment that will directly measure (or observe) this infinite energy! This is what seems to save the day --- you can't tell the infinite energy is there! [For example, even before you put the box there, there was also an infinite zero point energy, and you don't change the total energy much by adding the box!].
However, such divergences do lead to some serious issues with general relativity and gravity. Many of these puzzles remain unresolved!!
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