# Redundancy of Virtual Particles Borrowing Energy

1. May 22, 2012

### e2m2a

I often hear or read of virtual particles appearing because they borrow energy from the vacuum fluctuations, but they "return" the energy back to the vacuum so that there are no imbalances. Wait a minute. Doesn't the insights of relativity tell us there is no such thing as just energy or mass, but mass-energy. This mass-energy entity can manifest as either mass or energy, but it is the same beast. So, to say that particles have to exist only for a fleeting moment because they have borrowed energy from the vacuum seems redundant. If they are partices, their energy content is exactly mc sq, and so nothing has "disappeared" out of the vacuum, it's just mass-energy changed its form.

2. May 22, 2012

### Whovian

I still don't understand why this is redundant. They "borrow" the energy equal to their mass from the vacuum and "deposit" it back once they've annihilated.

This is actually more of an analogy: https://www.physicsforums.com/showthread.php?t=511176 [Broken]

Last edited by a moderator: May 6, 2017
3. May 22, 2012

### Bill_K

Despite those popular accounts, virtual particles do not "borrow" energy. Energy is conserved at all times. This is a Frequently Answered Question.

4. May 22, 2012

### Mark M

As Bill K said, no violation of the conservation of energy is present in quantum mechanics. It is a common misconception of popular science authors to say that it does.

In classical physics, the ground state of a particular field is 0. However, in quantum mechanics, the ground state is non-zero. For example, a particle in a one-dimensional well has a ground state of $$\frac {h^{2}n^{2}} {8mL^{2}}$$ Since an important feature of quantum mechanics is particle-wave duality, this energy can be interpreted as virtual particles being created and annihilated.

5. May 22, 2012

### e2m2a

Why then, do virtual particles have a "fleeting" existence? Why don't they exist magnitudes of time longer than we typically see? Is there an intrinsic instability associated with their existence?

6. May 23, 2012

### tom.stoer

Virtual particles are a mathematical construction in a certain approach (perturbation theory) to solve quantum field theories approximately. They are introduced as "internal lines in Feynman diagram" which represent a "recipe for doing calculations" and they do not - by construction - "exist" like usual particles which can be observed or detected.

Constructing these Feynman diagrams one finds - according to the "recipe" - that the energy-momentum 4-vector is strictly conserved at each vertex. So there is no "borrowing of energy".

Mathematically these virtual particles are not just single particles with definite energy and momentum, but they are representations for integrals over momentum-space! So one internal line does not correspond to one single particler but to all particles carrying arbitrary energy and momentum (up to constraints due to the energy-momentum conservation). This is the "recipe".

Due to this construction virtual particles "violate" the mass shell condition

p² - m² = 0

where p² is the 4-momentum squared and m is the rest mass (again this is only a mathematical construction)

Another major mathematical difference is that - in contrast to "usual particles" - these virtual particles do not have a Fock space representation, so they are not "quantum states".

The above mentioned perturbation expansion which introduces these virtual particles is limited to certain domains. It is e.g. useful in high-energy scattering experiments in QCD where one can study effects of "virtual gluons". But there is e.g. the low-energy regime where one is interested in quantum effects for the nucleon mass, the magnetic moment, the electromagnetic form factors etc.; in this regime the perturbation expansion breaks down, therefore in a sense these virtual particles do not even exist mathematically. It's like using 1/(1-x) = 1 + x + x² + x³ + ... for |x| < 1, then forgetting about |x| < 1 and the l.h.s. of the equation, writing down 1 + x + x² + x³ + ... and asking "in which sense x=2 does exist" in this formula.

Last edited: May 23, 2012
7. May 23, 2012

### Bill_K

Tom, I'm surprised and disappointed to hear you subscribe to this belief. It is quite untrue, backwards in fact. Just as a "real" particle is an elementary solution of the homogeneous wave equation, a virtual particle is an elementary solution of the inhomogeneous one. While a real particle must live forever without being created or destroyed, a virtual particle has a finite lifetime, created (or modified) at point A and destroyed (or modified) at point B. Every particle that has ever been observed was virtual, by definition.
No, it's just that the amplitude that Feyman diagram represents is already a sum over amplitudes, an integral over the momenta of the virtual particles (and not all of them BTW, only the ones involved in internal loops) This is a property of the diagram, not the particles themselves.
Quite true, but that does not make them an artificial construct. Photons, for example, are a useful concept in some domains and not others.
Fock space is limited to either in states or out states, but there's no reason it could not be defined in a more general context.

Is a muon a real particle or a virtual one? It lives for 10-6 sec and travels roughly half a kilometer.
Charged pion - 10-8 sec and 10 meters
Tau - 10-13 sec and 10 microns
Neutral pion - 10-17 sec and 10 Angstroms
W boson - 10-25 sec and .01 fermi
Tell me - where do you consider the cutoff to be? Which of these particles are real and which virtual? Answer: they are all virtual, in exactly the same sense, even the muon that traveled half a kilometer and the W boson that traveled .01 fermi. The difference between them is only a matter of degree.

EDIT: I should have listed the widths as well. The width of the W boson is 1 GeV. So don't expect to observe one on the mass shell! The width of the muon on the other hand is measured in nano-eVs, essentially undetectable.

Last edited: May 23, 2012
8. May 23, 2012

### tom.stoer

but it's not a Fock space state.

in a sense yes; in a detector the particle interacts with the detector, therefore it's virtual; but in the Feynman diagram the particle to be detected is an outgoing line which means it is real. That weakens the concept of real particles but does not strengthen the virtual ones ...; it shows that ontology based on Feynman diagrams is highly questionable.

but what is a virtual particle? 1/k² ?

All of them - but for tree-level diagrans you don't write down the trivial delta functions which cancel the integrals ;-)

You must not construct an universal ontology based on a mathematical approximation with limited applicability

Virtual particles i.e. internal lines in Feynman diagrams are not Fock space states, true? Are you saying that you can construct Hilbert space states which correspond to virtual particles? how?

9. May 23, 2012

### Bill_K

Let me answer my own question. The concept of a "real" particle is relative, involving the resolving power of the detector. A particle is considered to be real and on the mass shell if and only if your detector thinks it is on the mass shell. There is no other difference between real and virtual.

This indicates that Feynman diagrams are an approximation, in which certain particles are designated incoming and outgoing and imagined to be on the mass shell, when in reality they are not.

Regarding Fock space, take the simplest case of the particles I listed. Do muon states really span a Fock space? No, not in the usual sense. You can list all of the allowed momenta, and count the number of muons with each momentum value. But even for the time it lives, a muon is not (precisely) on the mass shell, and so prescribing its momentum does not determine its energy. To get a Fock space what you have to is imagine an artificially defined asymptotic region in which the interactions have been adiabatically turned off and a muon lives forever.

What about W mesons? How would you describe the space they span? They have no "in" states or "out" states to speak of. And they self-couple, which makes them hard to count. Again, to build a Fock space for W mesons you need to imagine a region where the weak coupling constant somehow goes to zero.

So my answer is: virtual particles are not mere mathematical constructs. But Feynman diagrams and Fock spaces are!

10. May 23, 2012

### e2m2a

All right. If virtual particles are "real", are they capable of interacting with other "real" particles in the classical sense? For example, can they collide with neutrons, protons, leptons, quarks and transfer momentum? Can they be represented by a world line in 4-space, such that if something interferes with their path to cause them to accelerate or de-accelerate, they experience inertial forces? Or since they are quantum particles, do classical descriptions breakdown and have no meaning in describing their dynamics?

(I think after reading the responses and the complex mathematical formalism involved in describing virtual particles, my last question is already answered.)

But, do virtual particles exist in a reality that has no connection to the common-day world we are familiar with? Or, as I alluded to, can they interact with other real particles and leave traces or fingerprints of their existence by the affect they have on less elusive particles?

11. May 24, 2012

### tom.stoer

A story told by Feynman:

Lessons learned: discussing the reality of virtual particles is like discussing the 'inside of the brick'. As soon as you are able to explain to me in which sense "real particles do exist in reality" I will try to answer your question regarding virtual particles, whether they "do ... exist in a reality that has no connection to the common-day world we are familiar with".

Don't get me wrong: I don't blame you to ask these questions; I only want to stress that perhaps you (and others) have been tempted to ask these questions by oversimplified popular 'ontological interpretations' of mathematical cookbooks.

12. May 24, 2012

### sheaf

I've read quite a few of these virtual particle debates over the years, and I wonder if it really matters. Surely all that matters is that we have a calculational mechanism to relate a theoretical idealization called an in-state to another theoretical idealization called an out-state. This is just a model, and it seems to give good answers. Isn't how we think of the intermediate elements in the model just a matter of personal taste?

13. May 24, 2012

### Haelfix

Virtual particles are a good and useful description of the world in a very specific and well defined regime of validity. As everything in physics, the ontology only works in some limited approximate sense.

Now within that ontology, the question whether you can determine whether a particle is virtual or not is wonderfully ambigous. In order to understand whether a particle is 'really there' or not, you would have to scatter another particle into the experiment to try to see if it bumps anything. But the very act of interaction conspires to do exactly the thing that you were trying to show in the first place. So it is always the case that you can reinterpret the diagram as involving real particles that decay instead of scatter.

14. May 24, 2012

### cygnet1

I'm certainly no expert in this field, but consider what happens when you fire an electron beam between two oppositely charged capacitor plates. It bends towards the positive plate. Why? No real photons can be observed passing between either plate and the beam. So what transmits the force that bends the beam? The QED explanation is that virtual photons do it. Without them, classical electrodynamics would have no quantum underpinnings. Am I correct?

Last edited: May 24, 2012
15. May 25, 2012

### e2m2a

I think I have answered my original question with a little more research. A virtual mass shell particle's existence is inseperably linked to Heisenberg's Uncertainty Principle. If it has a lot of energy, it must have a fleeting time existence, not because it has to "hurry up and return borrowed energy to the vacuum", but because of HUP its energy multiplied by its temporal existence must not exceed Planck's reduced constant divided by 2. Period.

Of course, to delve deeper one would have to ask why does nature conform to HUP in the first place?

But I am out of my depth with quantum physics. If Einstein himself never was comfortable with it, I wouldn't have chance to really understand it. But I take comfort in what Feynman said in his lectures, something to the effect, if anyone says they understand quantum physcis, then they don't understand it at all.

16. May 25, 2012

### nonequilibrium

@e2m2a: That doesn't really work. After all, the way you describe it, it should be applicable to any particle, virtual or not, so if I would give energy to a proton, it should disappear. Not the case, however.

The energy-time uncertainty relation you are referring to more exactly talks about the transition of one energy level to another (with energy difference $\Delta E$) when the system is perturbed (i.e. not exactly isolated, as indeed is the case in an interaction) and this happens in a characteristic time $\tau$ such that $\Delta E \cdot \tau$ satisfies Heisenberg's inequality.

Presumably, applying this to QFT (which I'm not well acquainted with), the clue is that this inequality links the finite lifetime of the virtual particle to it being off its mass shell (i.e. the two relevant energy levels are "existing" and "not existing"; or perhaps something similar using the Dirac sea picture).

In other words: any particle with a finite lifetime (i.e. any particle that undergoes an interaction, so basically for all practicle purposes any particle) is virtual...

EDIT: I believe my last statement is justified by what I said, but on the other hand, after reading the previous posts, I also believe tom.stoer makes very valuable points on these matters, namely that to say what I said, one has to presume that Feynman diagrams talk about things that happen, and this is indeed nothing more than an assumption, and perhaps not even the most sensible one.

Last edited: May 25, 2012
17. May 25, 2012

### e2m2a

See, Feynman was right. As soon as I thought I had a grasp on the issue, something else comes around to throw a monkey wrench in it. You are probably right. Its the ΔP and Δx
and the ΔE and the Δt that is relevant. The change, the transistion that applies to HUP.

By the way when I quoted Feynman on this subject I wasn't implying those who have posted a comment on this thread don't know what they were talking about. Far from it. They know more about the subject than I would ever know.

I think Feynman meant quantum physics is so bizzare and non-intuitive that the moment you think you have wrapped your brain around a certain problem, some other phenomenon will emerge, creating an unresolved paradox.

If you are a person who loves puzzles, paradoxes, and mysteries, then quantum physics is for you.

My take (not an academic rigorous treatment) is that a virtual particle is like an invisible boxing foe in a ring. A "real" boxing opponent named Lepton is duelling with him. Lepton can never see his opponent but he knows he is there because every so often Lepton takes a jab to his head and punch to his stomach.

What makes it more frustrating, the moment Lepton thinks he knows the exact change in his opponent's location, he can never be certain about the new strength or momentum of what his next punch is going to be, and the moment he somehow knows with exact mathematical certainty the change in the magnitude of the next punch, he can never know exactly where his opponent will be.

To complicate it even further, for an infinitesimal change in time, Lepton knows his opponent has acquired an enormous amount of fighting energy but he can't be sure how much it is, but over a large undetermined change in time his opponent's energy wanes to a small, predictable amount. (That really makes no intuitive sense.)

And then when Lepton thinks he is ready to throw his punch, his opponent could very well disappear into a burst of electro-magnetic energy, never to be seen again.

Lepton could complain to his skeptical coach that he wasn't delusional or shadow boxing, that there was a real opponent in the ring that threw real punches at him, but he could never see him.

Wherein the coach and Lepton would get into a long debate about that if you could feel the punches then he must have been real,etc.,etc.

Last edited: May 25, 2012
18. May 25, 2012

### Maui

Don't real 'particles' exist more as a catalog(shell or structure) of the effects produced by virtual particles? Aren't solidity, scattering and stability of atoms(and even mass) best understood in terms of virtual particles interactions? By a real 'particle' do you mean anything more than exchanges of virtual particles that allows a somewhat correct and visualizable picture of what is going on(and which on the whole obey certain macro scale laws - conservation of energy, etc.)?