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- Thread starter JohnPrior3
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haushofer

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Bear with me, I'm only an undergraduate physics student. I think my biggest area of misunderstanding how they can have negative momentum?

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They depict in a very clever way formulas that allow you to systematically calculate S-matrix elements for scattering processes in quantum field theory. The external lines depict asymptotic free states of the incoming and outgoing particles, usually plane-wave momentum eigenstates (which are distributions rather than functions by the way). These states can be identified as specific kinds of particles (say electrons) hitting a detector with a quite sharp momentum and can be counted to get measure a cross section for some process of interest (e.g., elastic electron-electron scattering), which is evaluated in QFT using the S-matrix elements which are written cleverly in terms of Feynman diagrams.

The internal lines stand for propagators. These do not symbolized particles that can somehow be detected in the above sense with real-world detectors. They are just mathematical objects used to evaluate the matrix elements.

To really understand elementary particles you have to study quantum field theory and see how the Feynman rules are derived and which meaning the physical quantities have you can define from them. The Feynman diagrams should be seen as a very clever symbolism to write down complicated formulae rather than pictures of what's going on in real-world scattering processes.

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They depict in a very clever way formulas that allow you to systematically calculate S-matrix elements for scattering processes in quantum field theory. The external lines depict asymptotic free states of the incoming and outgoing particles, usually plane-wave momentum eigenstates (which are distributions rather than functions by the way). These states can be identified as specific kinds of particles (say electrons) hitting a detector with a quite sharp momentum and can be counted to get measure a cross section for some process of interest (e.g., elastic electron-electron scattering), which is evaluated in QFT using the S-matrix elements which are written cleverly in terms of Feynman diagrams.

The internal lines stand for propagators. These do not symbolized particles that can somehow be detected in the above sense with real-world detectors. They are just mathematical objects used to evaluate the matrix elements.

To really understand elementary particles you have to study quantum field theory and see how the Feynman rules are derived and which meaning the physical quantities have you can define from them. The Feynman diagrams should be seen as a very clever symbolism to write down complicated formulae rather than pictures of what's going on in real-world scattering processes.

Thank you! I definitely need to learn more physics to fully grasp these concepts. I just love the strange phenomena in physics (basically the Standard Model in a nutshell). Hopefully one day I'll be giving explanations like that!

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I often use the following analogy. Suppose you have 1 apple. Then you can writeBear with me, I'm only an undergraduate physics student. I think my biggest area of misunderstanding how they can have negative momentum?

1 apple = 2 apples + (-1 apple)

But both 2 apples and -1 apple are virtual apples, the only real thing here is 1 apple. The virtual apples are nothing but a computational tool. Does it help?

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I often use the following analogy. Suppose you have 1 apple. Then you can write

1 apple = 2 apples + (-1 apple)

But both 2 apples and -1 apple are virtual apples, the only real thing here is 1 apple. The virtual apples are nothing but a computational tool. Does it help?

I understand what you are saying, but could you give me an example of where this would happen?

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Virtual particles are not even a disturbance in a field. They are nothing but a computational tool, not much different from the apples on the right-hand side in the post above.I understand a virtual particle is not technically a particle, but more of a disturbance in a field.

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SeeI understand what you are saying, but could you give me an example of where this would happen?

http://lanl.arxiv.org/pdf/quant-ph/0609163v2.pdf

Sec. 9.3. In particular, references [52,53] are physical examples of using "virtual particles" in CLASSICAL physics.

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tom.stoer

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In QFT a "real particle" could be associated with a state in a Hilbert space; technically a "virtual particle" is not a Hilbert space state but an integrated bunch of propagatorsBear with me, I'm only an undergraduate physics student. I think my biggest area of misunderstanding how they can have negative momentum?

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cgk

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It may also be helpful to notice that many different kinds of perturbation theories and not-quite-perturbation-theories can be written in terms of diagrams. This is by far not limited to quantum field theory, and I think it would be misguided to say that the diagrams contain any revelations about physical processes.

In some sense, the diagrams in various fields of quantum mechanics are simply depictions of terms which arise from a Wick-theorem-style contraction of creation and destruction operators. Similar diagrams can be obtained even in classical thermal perturbation theories. In all cases, they symbolize terms in expansions of expectation values or matrix elements between operators---nothing more. For example, some people in quantum chemistry use diagrams to compute matrix elements in perturbative and coupled cluster methods (see, for example, Crawford & Schaefer III - An Introduction to Coupled Cluster Theory for Computational Chemists, http://minimafisica.biodec.com/Members/k/Introduction.ps [Broken] for a introduction with little prerequisites). Others do not bother with the diagrams and evaluate the algebraic equations directly. I have yet to see someone who thinks that these kinds of diagrams describe actual physical processes... despite the fact that doing so would be at least as valid as in quantum electrodynamics.

In some sense, the diagrams in various fields of quantum mechanics are simply depictions of terms which arise from a Wick-theorem-style contraction of creation and destruction operators. Similar diagrams can be obtained even in classical thermal perturbation theories. In all cases, they symbolize terms in expansions of expectation values or matrix elements between operators---nothing more. For example, some people in quantum chemistry use diagrams to compute matrix elements in perturbative and coupled cluster methods (see, for example, Crawford & Schaefer III - An Introduction to Coupled Cluster Theory for Computational Chemists, http://minimafisica.biodec.com/Members/k/Introduction.ps [Broken] for a introduction with little prerequisites). Others do not bother with the diagrams and evaluate the algebraic equations directly. I have yet to see someone who thinks that these kinds of diagrams describe actual physical processes... despite the fact that doing so would be at least as valid as in quantum electrodynamics.

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tom.stoer

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The problem is that usually we do not care about a clear distinction between a) the theory, b) the approximation of the theory and c) the diagrams representing the approximations. So after reading some books or after the first course in QED there's the impression that the theory is identical with the approximation (e.g. perturbation theory in the coupling constant) and that the diagrams (e.g. the Feynman graphs) are identical with the approximation and with the theory itself.

But there are scenarios where these approximations break down, e.g. for non-perturbative regimes, theories which are perturbatively non-renormalizable (but well-defined non-perturbatively), ...

A very simple example is f(z) = 1/(1-z) = 1 + z + z² + z³ + ... You may introduce a graphical representation of the coefficients +1, +1, +1, ... of the series expansion, and you may discuss its physical meaning. Now what about f(z) in a domain where the above expansion becomes invalid? What is the meaning of graphical representation of a series expansion that does not exist?

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It is of course true that even the perturbation theory is not mathematically rigorous. Your example with the geometric series is even too optimistic compared with the Dyson series of perturbation theory. While the former is convergent in the disk [itex]|z|<1[/itex], the latter most probably has divergence degree 0.

It is however a good example in the sense of "resummed" perturbation theory. It's most directly related to the Dyson equation. You calculate the self-energy of, say the photon, at some order of perturbation theory and then solve for the Green's function, which schematically is a geometric series of self-energy insertions, but can as well written as the Dyson equation and be directly solved. This leads to an approximation of the full propagator of the interacting theory. There is nothing mysterious concerning the physical meaning of the diagrams, it's just a calculational tool (or just a notation of equations, if you wish) helping you to organize the calculation. You could as well go as Schwinger and never use Feynman diagrams ;-).

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Meir Achuz

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They have no physical reality, which is why they are called 'virtual'.

If perturbation theory is not used, there are usually no virtual particles.

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tom.stoer

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that's it

They have no physical reality, which is why they are called 'virtual'.

If perturbation theory is not used, there are usually no virtual particles.

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