Zero point energy in thermal noise spectrum?

1. Mar 26, 2013

jasonRF

This is a question about the proper form for thermal noise from a resistor. This is purely academic for me - I always work in the regime where $\hbar \omega << k T$ so the noise spectrum is simply P \approx kT. When this no longer holds, quantum effects matter of course. Then I have seen two different expressions for the spectral density. There is the Planck result
$$P = \frac{\hbar \omega}{ exp(\hbar \omega / k T) - 1}$$
that we learn about in basic modern physics. Then there is the result that uses the actual energy levels of the quantum harmonic oscillator, which adds in the zero point energy
$$P = \frac{\hbar \omega}{2} + \frac{\hbar \omega}{exp(\hbar \omega / k T) - 1}.$$
For the thermal noise problem, the Planck result shows up in Nyquist's original 1928 paper of course, while the last result shows up in Callen and Welton's 1951 paper. (my grad work was in plasma physics, so the fluctuation dissipation theorem is a tool I suposedly learned at one point - but I have only used the classical limit). I have tried google, google scholar, and stat mech books in the library at work, but have not been able to get a definitively clear understanding of which is correct, and more importantly why! Many books (Pathria 3rd edition stat mech book for example) simply sweep it under the rug with less than one sentence of discussion.

One argument I have seen for the Planck result is that the energy available for noise generation is from transitions between states, so the reference/zero-point energy is irrelevant. Seems reasonable to me. An argument the other way goes like this: the Planck result yields zero energy as T->0, violating the energy-time uncertainty principle. Given my lack of true understanding of quantum mechanics I do not know how good/bad this argument is, especially since if this expression is multiplied by the density of modes in a cavity the blackbody radiation formula once again has the ultraviolet catastrophe that Planck was trying to fix!

By the way, this is a question from an engineer who has taken only 1 quantum mechanics class ~20 years ago and has worked mostly in the classical world ever since. My stat-mech background is essentially self-taught out of Statistical and Thermal Physics by Reif - I primarily worked through the classical chapters and did solve a fair number of the problems (in other words, I learned enough to be able to survive my plasma physics courses). Perhaps I just do not understand enough physics to be able to sort this out - but I am hoping someone here can enlighten me!

Thanks!

jason

Last edited: Mar 26, 2013
2. Mar 27, 2013

Jano L.

Hi Jason,

this is a very good question.Don't worry that you should have been able to know the answer - the presence of the zero point energy term is one of the greatest difficulties in the quantum theory of EM field.

The problem is that when you sum up the zero point spectrum (which is proportional to $\omega^3$) over frequencies, you will get positive infinity, which means that something is wrong in the theory.

When the thermal spectrum is measured and compared to the Planck formula, they agree - no zero -point energy term is needed. For this reason, it is usual to think that the zero - point energy term, even if it is in the formula, is irrelevant and of no consequence.

But there are many papers which argue that the zero-point energy is real and has measurable consequences, like the Casimir and van der Waals forces, both in quantum and classical theory of electromagnetism (see the papers by Welton, Milonni and Boyer).

One possible solution of the divergence is the idea that the zero point energy term $\propto \omega^3$ is present, but only up to certain high frequency, while for frequencies higher this term drops down to zero. The total energy is then finite and the explanation of various phenomena based on the zero-point energy is preserved.

The remaining difficulty is that the cut-off would have to be at quite a high frequency and this would already require too great zero-point energy with non-negligible gravitating effects, which are not observed. For this reason the idea of zero-point field was criticized, e.g. by Jaynes, and he argued that some phenomena such as spontaneous emission and the Lamb shift, which are often thought to be tied to the fluctuations of the zero-point field, can actually be explained without them.

So the status of the zero--point term is quite problematic. If you can do without it, then it is probably better to do so.

3. Mar 27, 2013

jasonRF

Jano L.,

Thanks for the reply. I didn't realize this was an issue still up for grabs. No wonder I couldn't sort out the various literature I was able to dig up ... as someone who primarily just understands classical physics I had a tough time sorting it all out.

Best regards,

Jason

4. Mar 27, 2013

Bill_K

Can you be a bit more specific? I don't know whether to look in the Tibetan Journal of Physics or the Arkansas Academy of Sciences.

5. Mar 27, 2013

Jano L.

I did some search and come up with this list of papers. I read very few of it and there are many things I disagree with, but I think they may be an interesting reading:

Boyer, popular article on zero/point fluctuations
T. Boyer, "The Classical Vacuum," Scientific American 253, No.2, August, 70-78 (1985).

Boyer On Casimir effect in 1D
http://ajp.aapt.org/resource/1/ajpias/v71/i10/p990_s1 [Broken]

Welton on effects of fluctuations of EM Field:
http://prola.aps.org/abstract/PR/v74/i9/p1157_1

Boyer derivation of the vdW forces:
http://pra.aps.org/abstract/PRA/v7/i6/p1832_1

Boyer connection of stochastic electrodynamics with the quantum theory of Casimir effect:
http://prola.aps.org/abstract/PR/v174/i5/p1631_1

Boyer derivation of the blackbody spectrum using zero/point fluctuations
http://prola.aps.org/abstract/PR/v182/i5/p1374_1

Milonni, review on NR theory of radiation, sec. 5
http://www.sciencedirect.com/science/article/pii/0370157376900375

Milonni, Physical interpretation of the Casimir force:
http://pra.aps.org/abstract/PRA/v38/i3/p1621_1

Milonni on another way to get Casimir effect (without zero-point field)
http://pra.aps.org/abstract/PRA/v45/i7/p4241_1

Milonni and Smith on radiation reaction and vacuum fluctuations:
http://pra.aps.org/abstract/PRA/v11/i3/p814_1
http://prl.aps.org/abstract/PRL/v31/i15/p958_1

Last edited by a moderator: May 6, 2017