From Heisenberg’s uncertainty principle we learn that the uncertainty in time (the time interval of a measurement) multiplied by the uncertainty in the energy that we measure is always larger than Planck’s constant. Now the argument goes that since one cannot make measurements that are more accurate than what the uncertainty principle allows, nature is free to violate energy conservation on shorter time scales. Particles with non-zero mass (hence, non-zero energy based on E=m c^2) can pop in and out of existence, as long as they exist for times shorter than Planck’s constant divided by their implied energy fluctuation.
This reasoning is severely flawed. It assigns to nature the ability to violate more than just energy conservation. In fact, it assumes nature can violate the mathematical foundation of the Heisenberg uncertainty principle itself. The notion of a vacuum fluctuation says that a (virtual) particle associated with a particular energy can exist for a time that is shorter than allowed by the uncertainty relation. That is to say that one can produce a function with a frequency that exist for a shorter time than allowed by the time-bandwidth product. This is mathematically impossible and therefore contradicts the mathematical foundation of the Heisenberg uncertainty principle.
Fourier theory teaches us that a function and its Fourier transform are both complete representations of the information. The Fourier transform of a function does not provide additional information over and above the information that is already contained in the function itself. Since quantum theory is based on such a Fourier relationship, the same is true in nature. The temporal behaviour of a quantum system contains all the information in the system. There is no additional information that one can obtain by considering the energies of particles or anything else in that system. Smaller time intervals would obviously contain less information than larger time intervals. One cannot increase this amount of information by looking at the energy or frequency spectrum in that time interval. The same goes for the spectrum: by looking at smaller frequency (or energy) ranges one ends up with less information than what is available in larger frequency ranges and again, looking at the temporal variation within that frequency range cannot increase the amount of available information. This fact is encapsulated in the time-bandwidth product and thus also in the Heisenberg uncertainty principle.
What the time-bandwidth product (and by implication the Heisenberg uncertainty principle) actually tells us is that there is a fundamental limit in the amount of information contained in the description of any mathematical quantity over small intervals. Since nature is described by mathematics, the immediate implication is that if this limitation exists in mathematics then it also exists in nature. Nature cannot fluctuate over energy ranges that violate the Heisenberg uncertainty principle because it does not have enough information to enable such a fluctuation.
So, there is no such thing as vacuum fluctuations that can hide within the Heisenberg uncertainty principle. Now perhaps this is merely a poor choice of terminology, because, although it does not actually `fluctuate,’ nature does allow something to exist over those small intervals or small ranges, something that has a measurable effect on how nature behaves. This plays the role of `virtual’ particles, but I’ll leave this topic for later.