# Computations for Quantum Vacuum Fluctuations

For quite some time the notion of quantum vacuum fluctuations is bothering me.

What exactly is the theoretical origin of this notion? This notion has become quite common in physics and is used to predict quite sophisticated effects. For example the Hawking radiation, where a particle pair pops into existence from the vacuum at the horizon of a black hole and one falls into the black hole and the partner escapes. So far I've failed to find a computation for this process of particle creation and annihilation in the vacuum from nothing.

In many books these notions are motivated somewhat vague using the Heisenberg time uncertainty relation

$$\Delta t \Delta E > h$$

For a short time interval $\Delta t$ the conservation of energy can be violated by $\Delta E$ and therefore we can have a particle pair with energy $\Delta E$ pop into existence from nothing, as long as they vanish after $\Delta t$. This is quite a nice idea, but their is no "time operator" in quantum mechanics and therefore the interpretation of the equation $\Delta t \Delta E > h$ isn't as straight-forward as $\Delta p \Delta x > h$ etc. It is possible to find arguments that point into the direction energy violation for a short period of time, but I never found them fully satisfactory when it comes to particle creation and annihilation from nothing.

I think the appropriate framework for this kind of thing should be Quantum Field Theory, because particle creation and annihilation is what QFT is all about.

What process or computation is the theoretical basis for Vacuum Fluctuations? Maybe the disconnected Feynman graphs that accompany regular Feynman graphs? Nevertheless, these do not happen in a vacuum but only as attachment to regular processes. Are there computations of the form

$$<0| \mathrm{ \ Vacuum \ Fluctuations \ } |0>$$

Any idea or reference would be awesome!

• I would recommend reading up on either the Casimir effect, where vacuum fluctuations can be experimentally measured, or the Schwinger effect, where an external electric field can rip the virtual pairs into existence. BTW, in terms of Feynman diagrams the virtual pairs are represented by loops. – Surgical Commander Mar 24 '15 at 9:58
• What process is the theoretical basis for Vacuum Fluctuations? My understanding is that we can only measure frequency to a certain degree of accuracy, This implies that we can measure time only to a certain degree of precision also. So that always leaves us, using the uncertainty principle quoted in your post, with some uncertainity regarding the amount of energy associated with the fluctation. To my knowledge, the Uncertainty Principle is simply a law of nature, similiar to the speed of light value being ~300 000 km/s. It's a given, with no proven theoretical basis as of today. – user74893 Mar 24 '15 at 11:01
• @irishphysics Correct, and the underlying principle is that the universe appears to be built on (matter) waves. Hence it is the Fourier transform that mediates between observables, both in the proper Heisenberg uncertainty relations and in the improper one between time and energy (which is angular frequency times $hbar$). – pyramids Mar 24 '15 at 11:18
• I might write an answer later, but for now I just want to point out that if you look up pedagogical treatments of "virtual particles" you will probably find information relevant to this question. What you're calling "vacuum fluctuations" and what people call "virtual particles" are very closely related concepts. – DanielSank Mar 24 '15 at 22:51
• @DanielSank. If you do write an answer could you please include a reference to the words somewhat vague in the OP. I had thought that the uncertainty principle was not at all vague, but acuriousmind's comments mean that I am now not at all certain that the uncertainty principle is as straightforward as I originally thought, if you get my drift. What's incorrect or trivial in calling them quantum fluctuations? I can ask this as a separate question if required. Thank you – user74893 Mar 25 '15 at 0:34