HUP in QFT and QM:virtual particles

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
[tex]
\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \ldots = 2
[/tex]
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,

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).

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?

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: ...

[I]In the Graduate College dining room at Princeton everybody used to sit with his
own group...

We are really using the quantum-mechanical approximation method known as perturbation theory. In perturbation theory, systems can go through intermediate "virtual states" that normally have energies different from that of the initial and final states. This is because of another uncertainty principle, which relates time and energy.

In the pictured example, we consider an intermediate state with a virtual photon in it. It isn't classically possible for a charged particle to just emit a photon and remain unchanged (except for recoil) itself. The state with the photon in it has too much energy, assuming conservation of momentum. However, since the intermediate state lasts only a short time, the state's energy becomes uncertain, and it can actually have the same energy as the initial and final states. This allows the system to pass through this state with some probability without violating energy conservation.

Some descriptions of this phenomenon instead say that the energy of the system becomes uncertain for a short period of time, that energy is somehow "borrowed" for a brief interval. This is just another way of talking about the same mathematics. However, it obscures the fact that all this talk of virtual states is just an approximation to quantum mechanics, in which energy is conserved at all times. The way I've described it also corresponds to the usual way of talking about Feynman diagrams, in which energy is conserved, but virtual particles can carry amounts of energy not normally allowed by the laws of motion.

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

....the HUP isn't about the knowledge of the conjugate observables of a single particle in a single measurement. .... there's nothing to prevent anyone from knowing both the position and momentum of a particle in a single measurement with arbitrary accuracy that is limited only by our technology. However, physics involves the ability to make a dynamical model that allows us to predict when and where things are going to occur in the future. While classical mechanics does not prohibit us from making as accurate of a prediction as we want, QM does!

The HUP isn't about the knowledge of the conjugate observables of a single particle in a single measurement. The uncertainty theorem is about the statistical distribution of the results of future measurements. The theorem doesn't say anything about whether you can measure both at the same time. That is a separate issue.

There is not a definite line differentiating virtual particles from real particles — the equations of physics just describe particles (which includes both equally). The amplitude that a virtual particle exists interferes with the amplitude for its non-existence; whereas for a real particle the cases of existence and non-existence cease to be coherent with each other and do not interfere any more. In the quantum field theory view, "real particles" are viewed as being detectable excitations of underlying quantum fields. As such, virtual particles are also excitations of the underlying fields, but are detectable only as forces but not particles. They are "temporary" in the sense that they appear in calculations, but are not detected as single particles. Thus, in mathematical terms, they never appear as indices to the scattering matrix, which is to say, they never appear as the observable inputs and outputs of the physical process being modeled. In this sense, virtual particles are an artifact of perturbation theory, and do not appear in a non-perturbative treatment.

We're familiar with other cases where geometric circumstances create real (not virtual) particles e.g. Hawking radiation at BH horizon and Unruh radiation caused by acceleration or felt by an accelerated observer. So it seems that expansion of geometry itself, especially inflation, can produce matter. ....Quantum fluctuations in the inflationary vacuum become quanta [particles]
at super horizon scales.....it seems that expansion of geometry itself, especially inflation, can produce matter....The evolution of quantum fluctuations is from their birth [at Planck Scale] in the inflationary vacuum and their subsequent journey out to superhorizon scales where they become real life perturbations....[particles at detection]
....., is perhaps my favorite calculation in physics.

"there's nothing to prevent anyone from knowing both the position and momentum of a particle in a single measurement with arbitrary accuracy that is limited only by our technology"

Probably I'm misunderstanding that, but that seems to contradict what I've read on QM....

PAllen: If you are measuring position and momentum of the 'same thing' at two different times, the measurements are necessarily timelike. The measurements occur at two times on the world line of the thing measured. This order will never change, no matter what the motion of the observer is. If, instead, they occur for the same time on the "thing's" world line, they are simultaneous for the purposes of the uncertainty principle.


to measure a particle's momentum, we need to interact with it via a detector, which localizes the particle. So we actually do a position measurement (to arbitrary precision). Then we calculate the momentum, which requires that we know something else about the position of the particle at an earlier time (perhaps we passed it through a narrow slit). Both of those position measurements, and the measurement of the time interval, can be done to arbitrary precision, so we can calculate the momentum to arbitrary precision. From this you can see that in principle, there is no limitation on how precisely we can measure the momentum and position of a single particle.

Where the HUP comes into play is that if you then repeat the same sequence of arbitrarily precise measurements on a large numbers of identically prepared particles (i.e. particles with the same wave function, or equivalently particles sampled from the same probability distribution), you will find that your momentum measurements are not all identical, but rather form a probability distribution of possible values for the momentum. The width of this measured momentum distribution for many particles is what is limited by the HUP. In other words, the HUP says that the product of the widths of your measured momentum probability distribution, and the position probability distribution associated with your initial wave function, can be no smaller than Planck's constant divided by 4 times pi.

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.

[...] particles themselves are not real but just a convenient way of looking at fields.

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. [...]