Commenter miller asks:
It’s often said that virtual particles can “borrow” energy, as long as it’s for a short enough time to be compatible with the uncertainty principle. This never made sense to me, because the uncertainty principle says that product of uncertainty in energy and uncertainty in time is greater than h-bar over 2, not less than. Please explain.
The relevant equation is in the graphic at the top of this blog, just to the right of the title– the one with ΔEΔt. It’s easy to get turned around with this, due to the slightly unfamiliar business of working with inequalities.
The right way to think about this is to imagine moving the ΔE to the right side of the equation, giving it the form:
Δt ≥ h/2ΔE
What this is telling you is that if you want to detect the presence or absence of a particle whose rest mass energy (E = mc2) is equal to ΔE, you need to look for at least a time Δt. You can look for longer if you like, but the minimum observation time needed to ensure that the uncertainty in your measurement is less than the energy of the particle. If you look for less time, your energy uncertainty will be bigger than the mass energy, and you can’t be sure whether the particle was really there or not. (Or, more precisely, you can’t be sure it wasn’t a zero mass, zero energy particle, which is close enough to not existing for blog purposes.)
I talk about this in the Bunnies Made of Cheese chapter of How to Teach Physics to Your Dog, and because I should really take a break from typing on the computer to let my neck loosen up a bit, I will shamelessly quote myself:
How does this get us bunnies made of cheese? Well, let’s think about applying this uncertainty principle to empty space. If we look at some small region over a long period of time, we can be quite confident that it is empty. Over a short interval, though, we can’t say for certain that it isn’t empty. The space might contain some particles, and in quantum mechanics, that means it will.
Uncertainty about the emptiness of space isn’t as strange as it may seem at first. If a physicist or a stage magician gives a dog a box to inspect at leisure, she can conclusively state that the box is empty. She can sniff in all the corners, check for false bottoms, and make absolutely sure that there’s nothing hiding in some little recess. If she’s allowed only a brief peek or a quick sniff inside the box, though, she can’t be as confident that the box is empty. There might be something tucked into a corner that she wasn’t able to detect in that short time.
The amount of time needed to determine whether the box is empty also depends on the size of the thing you might expect to find. You don’t need to look for very long to determine whether the box contains Professor Schrödinger’s famous cat, but if you’re attempting to rule out the presence of a much smaller object–a crumb of a dog treat, say–a more thorough inspection is required, and that takes time .
The same idea applies to empty space in quantum physics, via the energy-time uncertainty relationship. When we look at an empty box over a long period of time, we can measure its energy content with a small uncertainty, and know that there is only zero-point energy–no particles are in the box. If we only look over a short interval, however, the uncertainty in the energy can be quite large. Since energy is equivalent to mass through Einstein’s famous E = mc2, this means that we can’t be certain that the box doesn’t contain any particles. And as with Schrödinger’s cat, if we don’t know the exact state of what’s in the box, it’s in a superposition of all the allowed states. The cat is both alive and dead, and the box is both empty, and full of all manner of particles, at the same time.
That’s the basic idea. So, a virtual particle can “borrow” some energy provided it doesn’t stick around long enough for any canine physicists to measure its energy with enough precision to say that it’s there. The energy-time uncertainty principle gives you the minimum time needed to make that measurement, which is the maximum time that a particle can get away with “borrowing” energy from the vacuum before it has to disappear again.
I hope that makes sense. Or at least as much sense as possible under the circumstances.
Last year I asked my intro particle physics professor that question, and I never really got a satisfactory explanation. I’ve been wondering ever since. Thanks, Chad!
Sorry Chad, but this is bogus. Energy cannot be “borrowed,” even by virtual particles, and putting quotes around it does not help. The explanation from your book is literally correct, it’s the terminology I’m objecting to. Because unfortunately it’s something that’s easy for non-physicists to latch on to and misinterpret.
Energy is locally conserved. You can borrow sugar, but you cannot “borrow” energy any more than you can borrow angular momentum or electric charge. Energy (actually stress-energy) is the source of the gravitational field, and consistency in the form of gauge invariance requires that it must be coupled to a locally conserved source.
This idea of “borrowing” often arises in discussion of a short-lived state or resonance which will exhibit a certain width or spread about a central energy Eo. That is, it was formed with a reaction energy E less than Eo, lived for a short time and then decayed, again with energy E. However that does not mean that in between it temporarily “borrowed” energy Eo – E. Throughout the process it has the same energy E that it started with.
I also don’t see how exactly uncertainty principle explains “borrowing” of energy.
AFAIK the uncertainty principle is about fundamental limitation on measurements which is due to wavelike nature of matter and fields/quantum nature of interactions. But the fact that we don’t have enough time to measure energy with low uncertainty doesn’t explain how a system can temporarily violate energy conservation.
A good example of such borrowing of energy would be a collision of two particles whose total energy is E1 resulting in them combining into a third particle whose rest energy alone E2=mc^2 is already larger then E1, this particle then decays after a short time to two other particles whose total energy E3 is equal to E1 and less then E2.
Such a process indicates that the intermediate particle borrowed energy from the vacuum since the two particles didn’t have enough energy to produce it otherwise and such a process can only happen if the amount of borrowed energy E2-E1 multiplied by intermediate particles lifetime is lower then h/2.
One common explanation I’ve seen is based on vacuum fluctuations – a particle pair pops into existence transfers some energy to the process and then takes the energy back and vanishes again. (This explanation would make more sense, to me at least, if the popping-vanishing was interpreted as being due to non-cancellation of background fields as I’ve explained in the comment on the “what do you want to read about” post, but that seems to be a non-mainstream way of looking at it.)
OTOH Bill K above, if I understand him right, seems to be saying that this intermediate particle simply has lower rest energy and therefore lower mass and so no borrowing of energy really takes place.
So how exactly does uncertainty principle enter the picture and explain the borrowing and the time limit? We don’t really measure the energy of the intermediate particle for example, the fact that it was created can be inferred from the type of decay.
I used the “borrowing” terminology because that’s what was used in the question. It’s not the way I would usually phrase it (as you can see from the book excerpt). If you’re looking for someplace to say that the energy was “borrowed” from, the traditional answer is vacuum fluctuations– it’s part of the energy that’s always present in the field.
You can also turn things around and say that as long as the particles in question appear and disappear sufficiently quickly, the uncertainty in their mass energy is great enough that it includes zero, or at least something less than the vacuum energy. In which case you have particles that might’ve had no mass at all, and energy conservation isn’t necessarily violated. I find that more confusing to explain than the other way around, which is why I didn’t go that route.
As for how the virtual pairs get created, I think the full picture involves considerably more than just energy-time uncertainty (this is not an area I have much background in, being a low-energy experimentalist). The uncertainty argument is just a convenient heuristic to understand the end result. Sort of like the way you need to use the Schrodinger equation to find the ground state wavefunction for a square well or a harmonic oscillator, but you can get a reasonable estimate of the energy from uncertainty arguments.
“Because unfortunately it’s something that’s easy for non-physicists to latch on to and misinterpret.” it’s done on purpose. “physicists” block other people out of their realm by creating “misunderstandings” and using cryptical lingo – not even math’s free from this. therefore “physics” is more mysticism than explanation. they need this misunderstandings so that they need time to “explain” these misunderstandings and crude language (including the symbols) -> time for which they get paid. when physics leaves the realm of self-explanation it’s simply because a bunch of people wants to keep others out in order to profit from this blockage. they simply do the same thing bureaucrats do to ensure being paid for their artificially constructed problems and the paperwork around it.