4 ms·

I'd be surprised if no one has done this already, but someone needs to write an article that literally enumerates out the first dozen terms of the perturbation

I'd be surprised if no one has done this already, but someone needs to write an article that literally enumerates out the first dozen terms of the perturbation expansion for a scattering matrix along with the Feynman diagram for each term. That should very clearly explain what "virtual" particles are.

geuis 6y ago
You’re gonna need something to break this down first “perturbation expansion for a scattering matrix”. I’m a semi literate layman and have no idea what you’re referring to here.
- ajkjk 6y ago
  Well -- that comment was not targeted at a layperson, I think. But the idea is: the scattering matrix (https://en.wikipedia.org/wiki/S-matrix https://en.wikipedia.org/wiki/S-matrix) is a matrix that says for each possible input and output state, what the amplitude of that transition happening. The perturbation expansion is roughly a Taylor series in the number of interactions that occur in a scattering problem. So: no interaction is order-0, exchanging one particle is order-1, etc. The first dozen terms or so then tell you the dozen or so highest-amplitude processes that occur in a given interaction. Most of these will involve virtual particles.
- frutiger 6y ago
  That's totally a fair point; the comment wasn't supposed to be the explanation itself! But let me try without actually writing such an article. The elements of the scattering matrix determine how two (or more) colliding particles will interact and emit zero or more particles as a result. These elements are defined by an integral that has no closed form. For certain fields (electromagnetic being the prominent example) the integral can be approximated by the sum of terms in an infinite series. Each term in the series includes a multiplicative factor of the coupling constant (typically denoted by alpha which is approximately equal to 1/137) raised to a certain power. Since this number is quite small, the higher the power, the less the term contributes to the overall sum. Additionally, there is a bidirectional mapping of each term in the series to a Feynman diagram (you are likely familiar with examples of these diagrams). The number of vertices in the diagram correspond to the power of the constant mentioned above. So, the terms that dominate the integral are the ones that have a small number of vertices, but you can keep going, adding more and more vertices to get a more accurate sum. These additional vertices can be added as long as you satisfy the rules of the diagram for the particular field you are considering (different fields have different constraints on invariants that must hold at each vertex). For EM, you can add additional photon/electron vertices as long as the overall electric charge is preserved. These additional photon edges in the diagram are "virtual" particles. Nothing more than a pictorial representation of a term in an infinite series that approximates an integral.
  - eperdew 6y ago
    Thank you for the explanation. This is a very comfortable level of detail in my opinion, if you were to write a full article.
- tobmlt 6y ago
  It might also help to know that the scattering matrix has classical counterparts in engineering. For example the “transfer function” in circuits and signal processing, etc, and the “response amplitude operator” (RAO) in ship design. I can speak from experience only about the RAO, but once you’ve got one built and for your system, you know quite a lot about the response of the system to input (in this case typically water waves). A higher order transfer function in ship design might be constructed to give you say, drift response (or force) as a function of wave frequency, and give you wave drift response spectra when “hit with” a wave spectrum. (First order response might regular 6DOF motions (motion spectra) in response to wave spectra) Here the analogies are pretty fun, because you are scattering and radiating waves in the water. Very physical!
tobmlt 6y ago
As a rejoinder, maybe Richard Mattuck’s book would be helpful for those interested in Feynman diagrams generally: https://www.amazon.com/Guide-Feynman-Diagrams-Many-Body-Problem/dp/0486670473 https://www.amazon.com/Guide-Feynman-Diagrams-Many-Body-Prob... Accessible to anyone who can sum a geometric series and take a Fourier transform! (Feynman diagrams for dummies basically, for any who run across this comment - yes I’m being cheeky about “dummies” but I believe it calls itself something similar in the front matter. A self interaction term, then.)
analog31 6y ago
When I was a student, one of the profs gave a talk on his work in atomic structure theory. I remember this only dimly, but the gist is that he showed slides with the first and second terms of the perturbation expansion. Then he showed a page of expressions generated by a computer algebra system. He said: "These are the third order diagrams. There are a few dozen more pages. The fourth order are unfathomable." Of course that was many years ago, and maybe they're fathomable now. There may even be code for it.

geuis 6y ago

You’re gonna need something to break this down first “perturbation expansion for a scattering matrix”. I’m a semi literate layman and have no idea what you’re referring to here.

ajkjk 6y ago
Well -- that comment was not targeted at a layperson, I think. But the idea is: the scattering matrix (https://en.wikipedia.org/wiki/S-matrix https://en.wikipedia.org/wiki/S-matrix) is a matrix that says for each possible input and output state, what the amplitude of that transition happening. The perturbation expansion is roughly a Taylor series in the number of interactions that occur in a scattering problem. So: no interaction is order-0, exchanging one particle is order-1, etc. The first dozen terms or so then tell you the dozen or so highest-amplitude processes that occur in a given interaction. Most of these will involve virtual particles.
frutiger 6y ago
That's totally a fair point; the comment wasn't supposed to be the explanation itself! But let me try without actually writing such an article. The elements of the scattering matrix determine how two (or more) colliding particles will interact and emit zero or more particles as a result. These elements are defined by an integral that has no closed form. For certain fields (electromagnetic being the prominent example) the integral can be approximated by the sum of terms in an infinite series. Each term in the series includes a multiplicative factor of the coupling constant (typically denoted by alpha which is approximately equal to 1/137) raised to a certain power. Since this number is quite small, the higher the power, the less the term contributes to the overall sum. Additionally, there is a bidirectional mapping of each term in the series to a Feynman diagram (you are likely familiar with examples of these diagrams). The number of vertices in the diagram correspond to the power of the constant mentioned above. So, the terms that dominate the integral are the ones that have a small number of vertices, but you can keep going, adding more and more vertices to get a more accurate sum. These additional vertices can be added as long as you satisfy the rules of the diagram for the particular field you are considering (different fields have different constraints on invariants that must hold at each vertex). For EM, you can add additional photon/electron vertices as long as the overall electric charge is preserved. These additional photon edges in the diagram are "virtual" particles. Nothing more than a pictorial representation of a term in an infinite series that approximates an integral.
- eperdew 6y ago
  Thank you for the explanation. This is a very comfortable level of detail in my opinion, if you were to write a full article.
tobmlt 6y ago
It might also help to know that the scattering matrix has classical counterparts in engineering. For example the “transfer function” in circuits and signal processing, etc, and the “response amplitude operator” (RAO) in ship design. I can speak from experience only about the RAO, but once you’ve got one built and for your system, you know quite a lot about the response of the system to input (in this case typically water waves). A higher order transfer function in ship design might be constructed to give you say, drift response (or force) as a function of wave frequency, and give you wave drift response spectra when “hit with” a wave spectrum. (First order response might regular 6DOF motions (motion spectra) in response to wave spectra) Here the analogies are pretty fun, because you are scattering and radiating waves in the water. Very physical!

tobmlt 6y ago

As a rejoinder, maybe Richard Mattuck’s book would be helpful for those interested in Feynman diagrams generally: https://www.amazon.com/Guide-Feynman-Diagrams-Many-Body-Problem/dp/0486670473 https://www.amazon.com/Guide-Feynman-Diagrams-Many-Body-Prob... Accessible to anyone who can sum a geometric series and take a Fourier transform! (Feynman diagrams for dummies basically, for any who run across this comment - yes I’m being cheeky about “dummies” but I believe it calls itself something similar in the front matter. A self interaction term, then.)

analog31 6y ago

When I was a student, one of the profs gave a talk on his work in atomic structure theory. I remember this only dimly, but the gist is that he showed slides with the first and second terms of the perturbation expansion. Then he showed a page of expressions generated by a computer algebra system. He said: "These are the third order diagrams. There are a few dozen more pages. The fourth order are unfathomable." Of course that was many years ago, and maybe they're fathomable now. There may even be code for it.