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Feynman Diagrams
The scattering matrix in coordinates and momentum representation


Mathematical methods for particle physics was one of the weak spots in the Physics package. There existed a FeynmanDiagrams command, but its capabilities were too minimal. People working in the area complained about that. These diagrams are the cornerstone of calculations in particle physics (collisions involving from the proton to the Higgs boson), for example at the CERN. As an introduction for people curious, not working in the area, see "Why Feynman Diagrams are so important".


This post is thus about a new development in Physics: a full rewriting of the FeynmanDiagrams command, now including a myriad of new capabilities (mainly a. b. and c. in the Introduction below), reversing the previous status of things entirely. This is work in collaboration with Davide Polvara from Durham University, Centre for Particle Theory.


The introduction to the presentation below is as brief as it can get, emphasizing the examples instead. This material is reproducible in Maple 2019.2 after installing the Physics Updates, v.598 or higher.




At the end it is attached the worksheet corresponding to this presentation, as well as the new FeynmanDiagrams help page with all the explanatory details.



A scattering matrix S relates the initial and final states, `#mfenced(mrow(mo("⁢"),mi("i"),mo("⁢")),open = "|",close = "⟩")` and `#mfenced(mrow(mo("⁢"),mi("f"),mo("⁢")),open = "|",close = "⟩")`, of an interacting system. In an 4-dimensional spacetime with coordinates X, S can be written as:

S = T(exp(i*`#mrow(mo("∫"),mi("L"),mo("⁡"),mfenced(mi("X")),mo("ⅆ"),msup(mi("X"),mn("4")))`))


where i is the imaginary unit  and L is the interaction Lagrangian, written in terms of quantum fields  depending on the spacetime coordinates  X. The T symbol means time-ordered. For the terminology used in this page, see for instance chapter IV, "The Scattering Matrix", of ref.[1].


This exponential can be expanded as

S = 1+S[1]+S[2]+S[3]+`...`



S[n] = `#mrow(mo("⁡"),mfrac(msup(mi("i"),mi("n")),mrow(mi("n"),mo("!")),linethickness = "1"),mo("⁢"),mo("∫"),mi("…"),mo("⁢"),mo("∫"),mi("T"),mo("⁡"),mfenced(mrow(mi("L"),mo("⁡"),mfenced(mi("\`X__1\`")),mo(","),mi("…"),mo(","),mi("L"),mo("⁡"),mfenced(mi("\`X__n\`")))),mo("⁢"),mo("ⅆ"),msup(mi("\`X__1\`"),mn("4")),mo("⁢"),mi("…"),mo("⁢"),mo("ⅆ"),msup(mi("\`X__n\`"),mn("4")))`


and T(L(X[1]), `...`, L(X[n])) is the time-ordered product of n interaction Lagrangians evaluated at different points. The S matrix formulation is at the core of perturbative approaches in relativistic Quantum Field Theory, where exact solutions are known only for some 2-dimensional models.


 In brief, the new functionality includes computing:


The expansion S = 1+S[1]+S[2]+S[3]+`...` in coordinates representation up to arbitrary order (the limitation is now only the hardware)


The S-matrix element `#mfenced(mrow(mo("⁢"),mi("f"),mo("⁢"),mo("|"),mo("⁢"),mi("S"),mo("⁢"),mo("|"),mo("⁢"),mi("i"),mo("⁢")),open = "⟨",close = "⟩")` in momentum representation up to arbitrary order for given number of loops and initial and final particles (the contents of the `#mfenced(mrow(mo("⁢"),mi("i"),mo("⁢")),open = "|",close = "⟩")` and `#mfenced(mrow(mo("⁢"),mi("f"),mo("⁢")),open = "|",close = "⟩")` states); optionally, also the transition probability density, constructed using the square of the scattering matrix element abs(`#mfenced(mrow(mo("⁢"),mi("f"),mo("⁢"),mo("|"),mo("⁢"),mi("S"),mo("⁢"),mo("|"),mo("⁢"),mi("i"),mo("⁢")),open = "⟨",close = "⟩")`)^2, as shown in formula (13) of sec. 21.1 of ref.[1].


The Feynman diagrams (drawings) related to the different terms of the expansion of S or of its matrix elements `#mfenced(mrow(mo("⁢"),mi("f"),mo("⁢"),mo("|"),mo("⁢"),mi("S"),mo("⁢"),mo("|"),mo("⁢"),mi("i"),mo("⁢")),open = "⟨",close = "⟩")`.


Interaction Lagrangians involving derivatives of fields, typically appearing in non-Abelian gauge theories, are also handled, and several options are provided enabling restricting the outcome in different ways, regarding the incoming and outgoing particles, the number of loops, vertices or external legs, the propagators and normal products, or whether to compute tadpoles and 1-particle reducible terms.






[1] Bogoliubov, N.N., and Shirkov, D.V. Quantum Fields. Benjamin Cummings, 1982.

[2] Weinberg, S., The Quantum Theory Of Fields. Cambridge University Press, 2005.


Download FeynmanDiagrams_and_the_Scattering_Matrix.mw

Download FeynmanDiagrams_-_help_page.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Featured Post

Playing mini-golf recently, I realized that my protractor can only help me so far since it can't calculate the speed of the swing needed.  I decided a more sophisticated tool was needed and modeled a trick-shot in MapleSim.

To start, I laid out the obstacles, the ball and club, the ground, and some additional visualizations in the MapleSim environment.


When running the simulation, my first result wasn't even close to the hole (similar to when I play in real life!).


The model clearly needed to be optimized. I went to the Optimization app in MapleSim (this can be found under Add Apps or Templates  on the left hand side).


Inside the app I clicked "Load System" then selected the parameters I wanted to optimize.


For this case, I'm optimizing 's' (the speed of the club) and 'theta' (the angle of the club). For the Objective Function I added a Relative Translation Sensor to the model and attached a probe to the Vector Norm of the output.


Inside the app, I switched to the Objective Function section.  Selecting Probes, I added the new probe as the Objective Function by giving it a weight of 1.



Scrolling down to "Execute Parameter Optimization", I checked the "Use Global Optimization Toolbox" checkbox, and clicked Run Parameter Optimization.


Following a run time of 120 seconds, the app returns the graph of the objective function. 


Below the plot, optimal values for the parameters are given. Plugging these back into the parameter block for the simulation we see that the ball does in fact go into the hole. Success!




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