What is Really Meant by "Virtual Particles"?

Discussion in 'Physics & Math' started by exchemist, Oct 6, 2014.

  1. exchemist Valued Senior Member

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    In the process of answering queries on another forum, I came across the following, entirely non-mathematical, explanation of virtual particles. I thought it was so clear and interesting that it might be worth sharing more widely: http://profmattstrassler.com/articl...ysics-basics/virtual-particles-what-are-they/

    The essential point I take from this is that the term "virtual particles" is in some ways an unfortunate misnomer, since what they really are is a disturbance in the relevant field that is for convenience modelled mathematically as if they were particles. So for example a "virtual photon" isn't a photon at all, it is a disturbance in the EM field that QED models as if it were a photon for calculation purposes.

    However, I'd quite like to check whether my understanding of this is right, as QFT (understandably) did not form part of the QM supplementary course I took as part of my Chemistry degree, all those years ago.

    And I'd also be interested in other comments from real physicists of course.
     
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  3. Fednis48 Registered Senior Member

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    At last, a question in my specialty!

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    That's a really good link you shared, and the analysis is solid. It's more thorough than anything I could come up with, so instead of rehashing it I'll touch on the math that it glosses over.

    As is described in the link, there is an intimate connection between particles and fields in QFT. A stable, propagating excitation of a field is a particle, and every particle has a corresponding field or fields. The simplest example is a photon, which is a stable excitation of the electromagnetic field. Because of this, we can describe the electromagnetic field in one of two ways: as a collection of photons/particles ("first quantization") or with a field operator ("second quantization"). The relation between the two looks something like \(\hat{E}\propto\hat{a}+\hat{a}^{\dagger}\), where \(\hat{E}\) is the field operator and \(\hat{a}\) and \(\hat{a}^{\dagger}\) are the so-called annihilation and creation operators, respectively, named such because they act to remove or add a photon to a mode. The proper formula has several other coefficients and has to be summed over all possible frequencies of light, and I may have a sign error, but the important part for this analysis is that the field looks like a sum of creation and annihilation operators.

    This whole structure is perhaps easier to think of in a cavity, which in its simplest form is a pair of mirrors, facing each other, that physicists sometimes use to "store" light. Because of the boundary conditions associated with reflecting surfaces, the field amplitude has to go to zero at both mirrors, so the only photons that can "fit" in the cavity are ones whose wavelength is an integer fraction of the cavity length. (For example, if the cavity's length is \(L\), it could contain some combination of photons of wavelength \(L\), \(\frac{L}{2}\), \(\frac{L}{3}\), etc. but it could not contain any photons of wavelength \(2L\), \(\frac{3L}{5}\), etc.) If we wanted to describe an electromagnetic field inside a cavity, we could do it one of two ways. We could just write out the field amplitude as a function of position, or we could repeatedly add photons with various phases and wavelengths to build up the overall waveform we want. This process is exactly analogous to a Fourier transform, if you have any experience with those. The free-space case gets a little messier because of subtleties in normalizing a wavefunction with no hard boundary, but the idea is the same.
    So what does this have to do with virtual particles? The dynamics of any system are governed by its Hamiltonian, which is the operator describing its energy. In the simple case of one electron moving in a field, the interaction energy is proportional to \(\hat{E}\cdot\hat{x}\), where \(\hat{x}\) is the electron's position operator. Written in the first quantization picture, this looks like creation and annihilation operators multiplied by the position operator. If we expand the picture to two electrons, the Hamiltonian shows both electrons coupled to the same creation and annihilation operators, but not directly to each other. We can then calculate the time-evolution of the system by repeatedly applying the Hamiltonian, and we get terms that look like \(\hat{x_1}\hat{a}\hat{a}^{\dagger}\hat{x_2}\).
    Reading the operators from right to left, this looks like electron 2 created a photon by perturbing its position, and electron 1 annihilated that photon to perturb its own position. It looks like the electrons have exchanged a photon!
    But all such terms have just as many \(\hat{a}\)s as \(\hat{a}^{\dagger}\)s, so at no point is a photon actually created. Instead, the result is an effective direct coupling between the positions of the two electrons, mediated by the field.

    In the context of field-mediated interactions like this, treating things in terms of virtual particles can be misleading. But in other contexts, the idea of "virtual excitations" can be extremely useful. For instance, in my research, we need to get the valence electrons in alkali atoms up above the n=80 atomic orbital, when their ground states are usually around n=5 or 6. That takes a lot of energy, so one common solution is to use two lower-frequency lasers whose frequencies add up to the required amount. To get the atoms to absorb one photon from each laser simultaneously, we can tune the lasers close to an intermediate state, such as n=7. When we repeatedly apply the Hamiltonian, we get terms that look like the atoms are absorbing one photon to go from n=6 to n=7, then another to go from n=7 to n=80+. But throughout the experiment, the intermediate state is only slightly populated; what we've really done is create a compound two-color laser that couples n=6 to n=80+ more or less directly. When we've got the setup down, we can even use a technique called "adiabatic elimination" to remove n=7 from our formula entirely and treat it all as a single coupling laser in future calculations. But thinking of it as a two-step process, with a virtual excitation in the middle, makes it a lot simpler and more intuitive to formulate.

    Whew - long rant. Hope some of it is interesting!
    edit: Damned TeX! I put serious effort into this, too.

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    Last edited: Oct 6, 2014
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  5. exchemist Valued Senior Member

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    Fednis48 thank you very much for this. It makes interesting reading. The relation between these creation and annihilation operators and the field operator is obviously the key to the equivalence. I'm rusty on all this - my degree was almost 40 years ago and I never had to use QM in my career, but I do recall something of the operator formulation of QM, so it is not totally alien. And yes Fourier series are familiar.

    Your example of the double excitation of alkali metals is very interesting, though I shall have to read what you've written several times. I'm not sure I can picture what may be happening if the n=7 state is not a real intermediate state. Is it that the n=7 eigenstate somehow couples the two photons together so they act if they are one when it comes to exciting the n=80 state, without ever physically populating n=7? Or something? And is this is known phenomenon, with a name? (I recall something about 2 photon processes and "pooling" of energy from photochemistry, I think, but I'd need to look it up.)
     
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  7. Dr_Toad It's green! Valued Senior Member

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    Thanks from me, too. Very nice response. Even in my near-mathlessness I grok. (Where's the "Om" smilie?)
     
  8. Fednis48 Registered Senior Member

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    Glad you liked it! If you want more technical details, http://web.stanford.edu/~rsasaki/AP387/chap3 seems like a pretty good overview without cutting corners like I did.

    As far as the two-photon transition stuff, I don't have a good source on it, because the technique was already very well established when I started my research, so I just worked with my adviser's notes. I have always heard to it referred to as "two-photon transitions" though. As best I understand it, the technique works because atomic orbitals are a complete basis one can use to describe an electron's position wavefunction. If we go back to the second quantization (field picture), an electric field deforms the electron cloud, and an "excitation" occurs when a precisely calibrated field drags the electron from one orbital into another. In order to do this, the field needs to be in some sense "in sync" with the electron's natural oscillations, otherwise the two sources of dynamics will clash and nothing will be accomplished. This is what it means to be on resonance. The most straightforward way to be in sync with the electron's natural oscillations is to just set your laser to the frequency difference between the original and target orbitals, but that's not the only way. When searching for more complicated ways, like two-photon transitions, thinking about virtual excitations in the first quantization picture is a useful way to grasp the electron cloud's response to time-dependent deforming pressures. We need the intermediate state to adequately describe the cloud as it's transforming - and indeed, the probability of finding the electron in n=7 is nonzero throughout the process - but the overlap \(|\langle n=7|\psi\rangle|^2\) remains small enough that it would be a mistake to say that the the atom excites into n=7 and then excites into n=80+ as a separate process.
     
  9. exchemist Valued Senior Member

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    OK, I'll definitely have a look at this 2 photon transition stuff. But meanwhile you have nicely answered my question in the OP about virtual particles and have given an inkling of how this arises mathematically, which is excellent.

    (And we've managed it all before the nutters turn up to derail the thread, which is even better!

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  10. exchemist Valued Senior Member

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    I've had a look at this now. None of my old books mentioned 2 photon processes. There is an article about it on Wiki but I don't find it very helpful in explaining the transition process. Though I was interested to learn that a woman called Maria Goeppert-Mayer (who later got a Nobel prize for work on nuclear physics) predicted it in 1931. One thing I did see that made a lot of sense is that the selection rules would involve a change of not 1 but either 2 or zero units of spin.

    Another article I found makes the interesting point that a 2 photon process can proceed EITHER via an intermediate eigenstate of the atom or molecule OR by means of a virtual state. The point is that the virtual state process specifically does not (so they say) involve an eigenstate of the atom! If you have a virtual state very close to an eigenstate then it starts to acquire some eigenstate character and this shows up in the lifetime and hence the degree of population the state exhibits. The plot thickens……..
     
    Last edited: Oct 7, 2014
  11. paddoboy Valued Senior Member

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    :Speculative mode on:
    I wonder if these disturbances could in anyway be linked to strings, as in "string theory" or one of its many derivitives.
     
  12. exchemist Valued Senior Member

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    That's well outside my competence, which is basically that of an undergraduate quantum chemist. As I understand it the disturbance is in one of the fields of QFT. But I do not know how these fields fit into string theory at all.

    Bring on the real physicists….
     
  13. Fednis48 Registered Senior Member

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    In my official capacity as a real physicist, let me say that I have no friggin' idea how string theory works. All I know is that QFT can be formulated perfectly well without string theory, so my best guess is that you could link these disturbances to strings, but you wouldn't predict anything new or different as a result. Which, come to think of it, is a frequent complaint about string theory in general.

    Also, thanks for reminding me that you don't need an intermediate state for a two-photon transition. I'll have to think for a while about a good physical interpretation of such a situation.
     
  14. krash661 [MK6] transitioning scifi to reality Valued Senior Member

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    amusing
    think about,
    band gap
    two photon emission
    emission and absorption

    electrons emit and obsorb particles ,also split(coupling constant)( abundance of strings)
    "integral part of string theory", " mode" and " polarization "
    excitation upconversion process.
     
    Last edited: Oct 8, 2014
  15. exchemist Valued Senior Member

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    Er, live long and prosper?
     
  16. danshawen Valued Senior Member

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    For every photon emission event that is observable within the light cone of a particular point in space-time of a particular reference frame, literally an infinitude of similar events will not be observable either because they are emitted in a direction that never approaches the observer, are blocked, absorbed, or bent by something along the path that connects them to the observer, or having been observed once for a short period in the past, will never be observable again. If these are not what is meant by "virtual" photons, perhaps they should be. Virtual energy is just as substantial as energy you can observe, even if you can never determine its cause, proper frame, or suitable accelerated or inertial frame from which to observe them.

    I agree, you don't really need string theory to realize this, but it might work also, and I'm told the calculations are a little easier. These virtual particle ideas kind of makes sense of Feynman's line integrals "around the moons of Jupiter", a staple of QED calculations, which I now realize is a holdover from earlier ideas about SED (stochastic electrodynamics).
     
  17. Layman Totally Internally Reflected Valued Senior Member

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    Virtual particles just have a different mass than the other normal versions of that particle when they are not isolated. Then they can't say it is just a certain particle, because it has a different mass. It is unknown why virtual particles have different masses.
     
  18. exchemist Valued Senior Member

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    Suggest you read the link I originally posted. They are not particles at all.
     
  19. Layman Totally Internally Reflected Valued Senior Member

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    Blasphemy! No one actually knows if they are real particles are not or even which of the "real" particles are actually particles.
     
  20. exchemist Valued Senior Member

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    Have you read it?

    I thought not.
     
  21. Layman Totally Internally Reflected Valued Senior Member

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    I had a hard time loading it at the time, but I just looked over it. I still raise the blasphemy flag on this one. It assumes that the Standard Model is the final theory. I will give you an example. The Higgs Boson has more mass and is bigger than the particles even used to make it. It would be impossible for atoms to have Higgs Bosons sitting around inside of them. Then the Higgs Boson cannot exist by itself, but that does not make it a virtual particle. Then there is no such thing as virtual Higgs Bosons. I don't think a virtual Higgs Boson could ever be discovered.

    I have read books on quantum physics written by people that work or have worked with particle accelerators. Then none of them claimed that particles can actually be physical particles or knowing if virtual particles are any more/less real than other particles. For all we know, particles may be forced to have different values when joined together, and they cannot remain stable isolated with these values. In that case, they could be just as real as other particles.
     
  22. Layman Totally Internally Reflected Valued Senior Member

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    In my own personal opinion, the only particles that could actually be real particles, the real stuff of substance that things are made of, would be the particles that have a virtual particle counterpart.
     
  23. Dr_Toad It's green! Valued Senior Member

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    The particle's mass is not what you think it is, and you're right about particles having different values when joined. That can equate to differences in mass, charge, or spin between the composite and the constituents.

    "Particles" are sometimes stable excitations of their respective fields, and sometimes ephemeral. It isn't intuitive, and can sometimes be confounding.

    I'm out of my depth, but I hope that helps.
     

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