# Time-energy uncertainty and vacuum fluctuations

Hi,

Similar to the position-momentum uncertainty principle, there is a time-energy uncertainty of the form
$$\Delta E \Delta t \geq\frac{\hbar}{2}\enspace .$$
However, since time is not an observable, the derivation and interpretation of this inequality is somehow different compared to other uncertainty principles. Depending on the context of the derivation, the constant on the right might also be different.

In basic physics and quantum physics literature, I've seen this inequality as an "explanation" for vacuum fluctuations: Energy conservation can be violated by creating particles and antiparticles, as long as the time until they're annihilated is short enough to satisfy above inequality.

I wonder if there is any truth to that "explanation". The time-energy uncertainty can be derived from purely non-relativistic quantum mechanics which features a constant number of particles. Only with the introduction of quantum field theory the particle number becomes variable.

So is it pure coincidence (and an example of simplifying-too-much literature) that this inequality somehow also works to talk about vacuum fluctuations, or is there more to it?

## Answers and Replies

JK423
Gold Member
Hi greypilgrim,

This inequality correctly predicted the order of magnitude for the mass of the pions, the quanta that mediate the Yukawa potential. So there must be something deeper to it, but i don't know what.

I hope someone knows more cause i'm really curious myself.

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Energy conservation is not violated by vacuum bubbles. Energy and momentum is always preserved, even by virtual particles. It is the dispersion relation E2=(cp)2+(mc2)2 that may be violated by the virtual particles.

Even though time is not an observable, it is derived from observables.

All uncertainties are, in my opinion, derivatives of a/the fundamental uncertainty.

Thus the question can be framed as:

Are vacuum fluctuations responsible for the (fundamental) uncertainty?

I think the answer is yes.

Even though time is not an observable, it is derived from observables.

All uncertainties are, in my opinion, derivatives of a/the fundamental uncertainty.

Thus the question can be framed as:

Are vacuum fluctuations responsible for the (fundamental) uncertainty?

I think the answer is yes.

So you think they may the cause if the uncertainty rather than the result? That's interesting as I have started thinking along those lines too. Are there any papers out there that have crunched the equations? For example a derivation of the Schrodinger equation starting from this assumption.

Hi,

Similar to the position-momentum uncertainty principle, there is a time-energy uncertainty of the form
$$\Delta E \Delta t \geq\frac{\hbar}{2}\enspace .$$

I can see how this could apply to a particle, but to apply it to free space wouldn't the volume need to defined. Would there be any limit to the number of virtual pairs per cubic metre of space?

Energy conservation can be violated by creating particles and antiparticles, as long as the time until they're annihilated is short enough to satisfy above inequality.

That's a laymen's description of virtual particles refuted by some in these forums. It's intended to give us a somewhat classical 'picture' we can relate to the quantum world. So far as we know energy conservation is NOT violated.

here is what Lisa Randall [Physics professor, Harvard] says regarding virtual particles in her book WARPED PASSAGES... pg 225....227 my comments enclosed thus {}..

Virtual particles interact with gauge bosons and alter forces so their effect depends on distance....{sounds like they 'exist'} ...Virtual particles have the same interactions and the same charges as physical particles but they have energies that look wrong. {oops, are they ‘real’?} A virtual particle can have enormous speed but no energy...virtual particles can have any energy that is different from the energy carried by the corresponding true particle... {sounds like they are not physical} If it had the same energy it would be a true particle, not a virtual particle....the uncertainty principle allows particles to have the wrong energy...for such a short time they can never be measured...Virtual particles have measurable consequences because they influence the interactions of the real physical particles that enter and leave the interaction region...{referring to an illustration}....the photon which was exchanged to generate the classical electromagnetic force was in fact a virtual photon....it only needed to last long enough to communicate the electromagnetic force and make the real charged particles interact.

What a 'fantastic' statement...sounds like a politician describing something....yet no part was challenged when I posted it previously.

The time-energy uncertainty can be derived from purely non-relativistic quantum mechanics which features a constant number of particles. Only with the introduction of quantum field theory the particle number becomes variable.

Not really: Hawking radiation [and more generally Unruh radiation] refutes the second part of that statement. The vacuum state is observer dependent.

Discussions of direct interest:

Source of virtual particles in space:

What is a particle

missed a piece:

I've seen this inequality as an "explanation" for vacuum fluctuations.

that part seems ok:
...“quantum mechanical systems undergo fluctuations even in their ground state and have an associated zero-point energy, a consequence of their wave-like nature. The uncertainty principle requires every physical system to have a zero-point energy….
http://en.wikipedia.org/wiki/Zero-point_energy

“...Vacuum energy is the zero-point energy of all the fields in space.....the energy of the vacuum, which in quantum field theory is defined not as empty space but as the ground state of the fields......The zero-point energy is .....the {vacuum} expectation value of the Hamiltonian.... and the energy is called the vacuum energy...

JK423
Gold Member
Not really: Hawking radiation [and more generally Unruh radiation] refutes the second part of that statement. The vacuum state is observer dependent.

What do you mean by that? Hawking and Unruh radiation (two totally different things) are predictions of quantum field theory.

edit: not totally different, but closely related!

Today Hawking radiation has a number of descriptions, including Susskind's complementarity.
I thought it had originals in relativity, like this:

If we're talking about an evaporating BH...... The key thing to remember is a spacetime with an evaporating BH is a *different* spacetime from that of an "eternal" BH. The simplest way to put it is that an "eternal" BH spacetime is a purely classical spacetime, where the stress-energy tensor is literally zero everywhere--it's a "classical vacuum" spacetime. An evaporating BH spacetime is a "quantum vacuum" spacetime; the stress-energy tensor is *not* zero. Rather, the SET is that of the vacuum state of whatever quantum field is present. That changes the global properties of the spacetime.....

and this:

Hawking radiation from the cosmological horizon in a FRW universe http://arxiv.org/abs/1007.4044v3

Synopsis: It is well known that there is a Hawking radiation from the cosmological horizon of the de-sitter spacetime and the de-Sitter spacetime can be a special case of a FRW universe. Therefore, there may be a corresponding Hawking radiation in a FRW universe.

So both Hawking and Unruh radiation are related to horizons..
wiki Unruh:

An accelerating observer will perceive an apparent event horizon forming (see Rindler spacetime). The existence of Unruh radiation could be linked to this apparent event horizon, putting it in the same conceptual framework as Hawking radiation. On the other hand, the theory of the Unruh effect explains that the definition of what constitutes a "particle" depends on the state of motion of the observer.
What the above descriptions misses: acceleration of an observer OR of an horizon produces observables particles....the latter is particle production during inflationary expansion.

The complementarity viewpoints may not be shared by all:

Quoting from Birrell and Davies,
"The average wavelength of the emitted quanta is ~M, i.e comparable with the size of the hole. As it is not possible to localize a particle to within one wavelength, it is therefore meaningless to trace the origin of the particles to any particular region near the horizon." ...

here is a description that melds Qm and GR: from Steve Carlip

So.... Is there no scientific consensus on whether virtual particles are real or just useful mathematical constructs? There seems to be certain predictions like muon mass and scattering amplitudes that require them to be real even if they short-lived, off mass shell and having complex momenta. I find it hard to believe that this issue is still "up in the air".

Bill_K
So.... Is there no scientific consensus on whether virtual particles are real or just useful mathematical constructs? There seems to be certain predictions like muon mass and scattering amplitudes that require them to be real even if they short-lived, off mass shell and having complex momenta. I find it hard to believe that this issue is still "up in the air".
There's a consensus here on PF that virtual particles are merely mathematical constructs. Which I disagree with.

There's a consensus here on PF that virtual particles are merely mathematical constructs. Which I disagree with.

:thumbs: There really seems to be no consensus to this in the real/virtual world either unless anyone knows otherwise.... It may be one of those pesky interpretation type situations. I have high hopes though that one day the uncertainty relations and Schroedinger equation will be derived in quite a natural way from the virtual fluctuations, like Brownian motion. I may check in again in 10 years and see how it's going...good luck everyone :!!)

From my previous quote of Lisa Randall:

Virtual particles have measurable consequences because they influence the interactions of the real physical particles that enter and leave the interaction region.

We likely have a 'pesky' interpretational issue on what is 'real'. I kind of agree with Bill_k here but I sure do NOT know what is 'real'. But if we can attribute a 'measureable consequence' to a mathematical concept, that would seem to imply more than a 'mere mathematical construct'.

JK423
Gold Member
It's quite easy to objectively assess whether virtual particles are 'real' without any ambiguity.

The world is quantum, hence in order for something to be called 'real', the minimum requirement is for it to acquire a quantum state during its existence. Now be careful; i'm not saying that all quantum states are describing 'real' stuff (e.g. see ghosts in QFT), and also when i'm mentioning the word 'real' i'm not referring to the interpretations of quantum theory (whether the quantum state is epistemic or ontic etc). All i'm saying is that the minimum requirement for something to be considered that it's "out there", is that it's described by a quantum state. If it is described by a quantum state then let's talk about how real it is. If it's not described by a quantum state then it simply doesn't exist at all and we have nothing to talk about.

Ask yourself the question:
Are the virtual particles described by a quantum state during the time of their "existence"?

This simple question clears out any misconceptions and misunderstandings :)
(if you answer it correctly!;)

Bill_K
All i'm saying is that the minimum requirement for something to be considered that it's "out there", is that it's described by a quantum state.
Utterly naive and inadequate, JK432, and not even to the point.

How about a 'something' that cannot appear by itself, but can participate in a multiparticle state? How about a state which is a superposition of states A and state B, and the 'something' is present in A but not in B? How about a something that can be varied continuously, and can be made to approach its "real" counterpart, so closely that it is beyond experimental resolution to distinguish one from the other?

JK423
Gold Member
Utterly naive and inadequate, JK432, and not even to the point.

How about a 'something' that cannot appear by itself, but can participate in a multiparticle state? How about a state which is a superposition of states A and state B, and the 'something' is present in A but not in B? How about a something that can be varied continuously, and can be made to approach its "real" counterpart, so closely that it is beyond experimental resolution to distinguish one from the other?
No reason to be upset Bill_K

How about a 'something' that cannot appear by itself, but can participate in a multiparticle state?
Something that can participate in a multiparticle state, is described by a quantum state itself, i.e. it has a Hilbert space of its own.

How about a state which is a superposition of states A and state B, and the 'something' is present in A but not in B?

I don't understand the question. But anyway you're still talking about a quantum state, hence you're implying that this 'something' that is hiding is part of a quantum state, hence it has a Hilbert space of its own.

How about a something that can be varied continuously, and can be made to approach its "real" counterpart, so closely that it is beyond experimental resolution to distinguish one from the other?

Irrelevant. Either it's described by a quantum state (i.e. it has a Hilbert space) or not.

As you know yourself Bill_K, there is not common agreement on the definition of a virtual particle. I don't know what your view is, and it doesn't matter. Whichever your view, ask the question that i've given and you will assess whether your definition of a virtual particle gives you something 'real' or not.

Simple!

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There's a consensus here on PF that virtual particles are merely mathematical constructs. Which I disagree with.

I should probably avoid this but I can't resist.

I'd tend to agree with you. If we can observe their influence then the concept of a virtual particle seems as much of a mathematical construct as an ordinary particle. After all, we only ever observe ordinary particles through their interactions too.

In my opinion the mantra that they are purely mathematical constructs comes from a number of influences. Firstly, perturbation theory for virtual particles is less well established than the quantum mechanics of ordinary particles. Secondly, virtual particles are, an extra level removed from observations than ordinary particles. Also, some interpretations of QM hold to a version of reality that gives ordinary particles a more distinct ontological status than the virtual particles that interact with them.

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