# HUP in QFT and QM:virtual particles

1. Nov 8, 2012

### haushofer

Hi,

I have a question about reconciling two pictures of virtual particles and the Heisenberg Uncertainty Principle (HUP).

In QFT "virtual particles" show up in perturbative calculations. We try to calculate an amplitude in interacting theories, this can not be done in an exact way, so we use Taylor expansions, and in this expansion intermediate states show up which we call "virtual particles".

In non-rel. QM people often say that virtual particles can exist because of the uncertainty principle between energy and time, in which one interprets the "time" in the appropriate way (see e.g. Griffiths).

My question is: how to reconcile these two pictures? If we would find the mathematical tools to calculate amplitudes in interaction theories in an exact way analytically, what would happen to these "virtual particles"? On the one hand I would say they wouldn't show up in your calculations, just as e.g. all the intermediate steps in

$$\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \ldots = 2$$

wouldn't show up; we know the answer is "2". But on the other hand, if their existence can be argued by the uncertainty principle, their existence should not depend on our ability to solve function integrals analytically in an exact way, right?

Are my analogies bad, or are the textbook statements not that accurate, or something else?

2. Nov 8, 2012

### mpv_plate

I don't think it's accurate to say that virtual particles can exist thanks to HUP. When this is stated in a text, it is misleading, I would say.

3. Nov 8, 2012

### strangerep

Yes.
IMHO, such "explanations" are rubbish. (I always become suspicious of physics authors who are subtly dismissive about mathematical rigor.) Is it only in Griffiths where you've seen such arguments, or are you thinking of other common textbooks also?

Right. Virtual particles are not real. :-)
The latter.

4. Nov 9, 2012

### tom.stoer

as strangerep says: it's rubbish
1) it's s difficult to state the time-energy HUP in non-rel. QM in a general sense; it's even more unclear how to formulate it in QFT
2) virtual particles /as defined in perturbation expansion) are not Hilbert space states but "integrals of propagators"; so you can't define Δt and ΔE (or Δx and Δp) for these expressions in the usual sense
3) somethimes people claim that virtual particles "borrow energy" or "violate energy conservation for some short time Δt"; this is nonsense as well b/c in the Feynman diagrams energy and momentum are conserved exactly at each vertex; what is violated is the mass-shell condition m² = E² - p²
4) last but not least it's b...sh.. to discuss the "existence" of virtual particles

In the Graduate College dining room at Princeton everybody used to sit with his
own group. I [Feynman] sat with the physicists, but after a bit I thought: It would be nice to see what
the rest of the world is doing, so I'll sit for a week or two in each of the other groups.
When I sat with the philosophers I listened to them discuss very seriously a book
called Process and Reality by Whitehead. They were using words in a funny way, and I
couldn't quite understand what they were saying. Now I didn't want to interrupt them in
their own conversation and keep asking them to explain something, and on the few
occasions that I did, they'd try to explain it to me, but I still didn't get it. Finally they
invited me to come to their seminar.
They had a seminar that was like a class. It had been meeting once a week to
discuss a new chapter out of Process and Reality some guy would give a report on it
and then there would be a discussion. I went to this seminar promising myself to keep my
mouth shut, reminding myself that I didn't know anything about the subject, and I was
going there just to watch.
What happened there was typical so typical that it was unbelievable, but true.
First of all, I sat there without saying anything, which is almost unbelievable, but also
true. A student gave a report on the chapter to be studied that week. In it Whitehead kept
using the words "essential object" in a particular technical way that presumably he had
defined, but that I didn't understand.
After some discussion as to what "essential object" meant, the professor leading
the seminar said something meant to clarify things and drew something that looked like
lightning bolts on the blackboard. "Mr. Feynman," he said, "would you say an electron is
an 'essential object'?"
Well, now I was in trouble. I admitted that I hadn't read the book, so I had no idea
of what Whitehead meant by the phrase; I had only come to watch. "But," I said, "I'll try
to answer the professor's question if you will first answer a question from me,
so I can have a better idea of what 'essential object' means. Is a brick an essential object?"
What I had intended to do was to find out whether they thought theoretical
constructs were essential objects. The electron is a theory that we use; it is so useful in
understanding the way nature works that we can almost call it real. I wanted to make the
idea of a theory clear by analogy. In the case of the brick, my next question was going to
be, "What about the inside of the brick?" and I would then point out that no one has
ever seen the inside of a brick. Every time you break the brick, you only see the surface.
That the brick has an inside is a simple theory which helps us understand things better.
The theory of electrons is analogous. So I began by asking, "Is a brick an essential
object?"
Then the answers came out. One man stood up and said, "A brick as an
individual, specific brick. That is what Whitehead means by an essential object."
Another man said, "No, it isn't the individual brick that is an essential object; it's
the general character that all bricks have in common their
'brickness' that is the essential object."
Another guy got up and said, "No, it's not in the bricks themselves. 'Essential
object' means the idea in the mind that you get when you think of bricks."
Another guy got up, and another, and I tell you I have never heard such ingenious
different ways of looking at a brick before. And, just like it should in all stories about
philosophers, it ended up in complete chaos. In all their previous discussions they hadn't
even asked themselves whether such a simple object as a brick, much less an electron, is
an "essential object."

We should send Griffiths et al. to those seminars 24 hours a day and 7 days a week in order to prevent them writing books!

5. Nov 9, 2012

### Sonderval

@strangerep
" Virtual particles are not real."
OTOH, Feynman said (in his lectures on gravtitation) that all particles we ever observe are "virtual", because when they are measured, they correspond to internal lines of Feynman diagrams.
But of course, in a sense, particles themselves are not real but just a convenient way of looking at fields.

6. Nov 9, 2012

### haushofer

Plenty of textbooks, like those of Beiser, but also popular literature. That's why it's hard for me to believe that all those people are sloppy, and there is no rigorous justification for it.

7. Nov 9, 2012

### haushofer

Yes, this is also how I understand it; I've even once wrote a more or less popular article about it, and saw that the editor added the "Δt and ΔE" HUP to the text which I tried to circumvent. As I said, I find it rather amazing that so many people quote this explanation for virtual particles, and it made me doubt my own understanding of QM and QFT.

In a topic about virtual particles I onced asked the question what happens with our notion of virtual particles and "our vacuum filled with virtual particles" if we were able to solve our amplitudes in an exact way, without perturbation theory. You would then agree, looking at your post, that the whole notion of virtual particles would disappear, and that they solely are artifacts of doing pertubation theory, right?

Somehow people make the notion of vp's in popular literature much more romantic. I guess a popular explanation like "artifacts of the fact that our mathematical skills are not developed enough" is not flashy enough. Thinking about it, also regarding your topic on the definition of energy and entropy in GR elsewhere, we could almost start a topic in which all these popular notions in physics literature are being critized :P

Yes, I love that story, and recently also quoted it here in some topic :D

8. Nov 9, 2012

### Sonderval

I think there is some justification for quoting the HUP in this context:
Think of the ground state of the Harmonic oscillator: I think it is permissible to say that the x-positon of the particle cannot be exactly zero because of the HUP.
If we accept this, then the fact that the probability of finding a non-zero amplitude for a field in QFT is also in some sense due to the HUP because this is (in each mode) equivalent to a H.O. The fact that this probability is smaller if k²+m² is larger fits into this picture, ath least intuitively: It is less probable to find an excitation of a higher-energy mode. (At least intuitively, this smaller probability might be thought to realte to a "shorter time", although this is nothing I would ever use to calculate something).

And with an interaction added to the theory, this probability amplitude may be described via the concept of virtual particles.

So I think there is some justificaton for this in a non-rigorous explanation, but in a good textbook you should add a bazillion of caveats.

And to finish with a question: Could one use the fall-off of a not-on-mass-shell propagator to make this more rigorous (the fall-off is faster the greater the violation of the mass-shell condition is)?

9. Nov 9, 2012

### haushofer

That means that even if we would be able to solve for an interacting theory exactly, we still would find "virtual particles"?

I'm not familiar with these things, but aren't we able to solve some 1+1 dimensional interacting QFT's exactly? Does the notion of virtual particles appear in such theories?

10. Nov 9, 2012

### haushofer

John Baez also seems to justify the relation between vp's and the HUP:

http://math.ucr.edu/home/baez/physics/Quantum/virtual_particles.html

11. Nov 9, 2012

### Naty1

That's not been the previous consensus in these forums[nor in the posts here so far]:

Here is How Zapper explains it in his blog....

[hence, no support for virtual particles.]

On QFT: explanation from prior discussions in these forums:

and a reminder from relativity:

12. Nov 9, 2012

### Sonderval

Probably I'm misunderstanding that, but that seems to contradict what I've read on QM (and it also seems to contradict all those Gedankenexperiments of Bohr and other people who are discussing the HUP). So I'd be glad if you could explain (or put a link here).
Edit: See for example this experiment:
http://en.wikipedia.org/wiki/Uncertainty_principle See einstein's slit

Last edited: Nov 9, 2012
13. Nov 9, 2012

### Naty1

Sonderval:
Likely you ARE understanding what I posted......

the best interpretation is the second quote from Zapper's blog [above]..

try these sources on for size....and make your own judgments:

I suspect the issue to which you refer is captured in post 2 and 3 here:

Is Heisenberg Uncertainty a problem with our measuring techniques

John Baez has a mathematical perspective here:

http://www.cbloom.com/physics/heisenberg.html

[Seems to me he is discussing a lower bound involving standard deviations, an ensemble of measurements, and NOT a single the measurement of conjugate observables. If so, this seems to me consistent with Zappers blog comments I quoted previously.

This is the discussion that led me to my interpretation.....

I have zero confidence that everyone accepts this interpretation.

[edit: a related discussion brought this interesting piece:

"particles *may* have well-defined positions at all times, or they may not ... the statistical interpretation does not require one condition or the other to be true."

Last edited: Nov 9, 2012
14. Nov 9, 2012

### Naty1

In rereading my notes on HUP, I came across this explanation from PAllen
that I really like [unsure which thread this is from]:

[Reading Wikipedia on HUP appears to give a different impression than these descriptions.]

Last edited: Nov 9, 2012
15. Nov 9, 2012

### Sonderval

I don't see why a momentum measurement necessarily involves two position measurements, at least not of the particle concerned. I could have a particle bounce from a ball or wall and measure the momentum change of the ball or wall at leisure. (As in the Einstein slit experiment I quoted above).

16. Nov 9, 2012

### strangerep

That's a bit misleading, imho. I suppose he's referring to how a measurement corresponds to establishing a correlation between a system's initial state and an apparatus' final state by having them interact. (Ballentine covers this reasonably well in his QM text.)
I, too, prefer to focus on thinking about fields.

17. Nov 9, 2012

### strangerep

Er,.... how do you "measure the momentum change of the ball" ?

18. Nov 9, 2012

### strangerep

I've yet to anyone seriously advocating that stuff who is also capable of explaining the notion of "unitarily inequivalent representations" and "Haag's theorem" in QFT properly. ;-)

19. Nov 9, 2012

### strangerep

The way to be rigorous is to solve the theory exactly rather than via perturbation theory. :-)

20. Nov 9, 2012

### strangerep

In the (exactly solvable) Lee model, one basically finds the correct Hilbert space for the interacting theory, rather than trying to shoehorn the interacting theory into a Hilbert space which is only appropriate for the free theory.

21. Nov 10, 2012

### tom.stoer

There are solvable models in 1+1 dim., or at least models (Schwinger model, Gross-Neveu model, 1+1 dim. QCD) where other approximations but perturbation theory are reasonable (bosonization, large-N limit, ...). In all those cases one can find non-perturbative solutions and phenomena (axial anomaly, mass generation, chiral symmetry breaking and quark condensate, mesons ins Hartree-Fock approx., baryons as topological solitons, hadrons spectrum, ...) w/o ever talking about virtual particles etc.; of course on could apply perturbation theory but nobody was interested.

But QFT was dominated by perturbation theory for decades; even standard text books like Ryder introduce QFT as PI + perturbation theory (and add some small chapters regarding topological effects, mostly in classical (!) field theory, like monopoles and instantons) Experimental physics was dominated by accelerators and scattering experiments as well. It would be interesting to scan the nobel prize list for HEP, elementary particle physics and QFT and try to find one single example which is not mostly related to perturbation theory.

If you only have a hammer, you tend to see every problem as a nail.

22. Nov 10, 2012

### Sonderval

@strangerep
I can measure the momentum change of the ball by looking at its velocity (i.e., measuring its position at two time-points), or, if I know that it was at rest initially, by looking at its change in kinetic energy. You could have the ball hit a surface and measure the dent it makes.

Of course the HUP applies to the ball as well - I just wanted to say that you need not to measure the position of an object X at two points in time to measure its momentum, you can as well transfer the momentum to another object Y and then measure the momentum of that at you leisure. Sorry if this was unclear.

Yes of course, rigorously it would be better to solve the theory exactly - I was just asking whether one could use the falling off of the propagator to give quantitative meaning to the "energy uncertainty" by using p²-m² (with p=four vector) as a measure of the "energy violation" and the fall-off length of the propagator as a measure of the uncertainty in time. Intuitively, this seems plausible to me, but I never saw something like that.

23. Nov 10, 2012

### Sonderval

I forgot, here is the exact Feynman quote on virtual particles:

24. Nov 10, 2012

### Naty1

Isn't this self contradictory?....can you observe a physical effect without an interaction?

25. Nov 10, 2012

### Sonderval

@Naty
Why should a virtual photon have anythingto do with "no interaction"? The Coulomb interaction is also exclusively due to virtual (longitudinally/timelike) polarised virtual photons.