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 asked for more functionality. These diagrams are the cornerstone of calculations in particle physics (collisions involving from the electron 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), reversing the previous status of things entirely. This is work in collaboration with Davide Polvara from Durham University, Centre for Particle Theory.

The complexity of this material is high, so 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 they are attached the worksheet corresponding to this presentation and a PDF version of it, as well as the new FeynmanDiagrams help page with all the explanatory details.

A scattering matrix  relates the initial and final states,  and , of an interacting system. In an 4-dimensional spacetime with coordinates ,  can be written as:

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

This exponential can be expanded as

where

and  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.

In connection, the FeynmanDiagrams  command has been rewritten entirely for Maple 2020. In brief, the new functionality includes computing:

The expansion  in coordinates representation up to arbitrary order (the limitation is now only your hardware)

The S-matrix element  in momentum representation up to arbitrary order for given number of loops and initial and final particles (the contents of the  and  states); optionally, also the transition probability density, constructed using the square of the scattering matrix element , 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 .

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.

Examples

For illustration purposes set three coordinate systems , and set  to represent a quantum operator

Let  be the interaction Lagrangian

The expansion of  in coordinates representation, computed by default up to order = 3 (you can change that using the option order = n), by definition containing all possible configurations of external legs, displaying the related Feynman Diagrams, is given by

The expansion of   in coordinates representation to a specific order shows in a compact way the topology of the underlying Feynman diagrams. Each integral is represented with a new command, FeynmanIntegral , that works both in coordinates and momentum representation. To each term of the integrands corresponds a diagram, and the correspondence is always clear from the symmetry factors.

In a typical situation, one wants to compute a specific term, or scattering process, instead of the S matrix up to some order with all possible configurations of external legs. For example, to compute only the terms of this result that correspond to diagrams with 1 loop use numberofloops = 1 (for tree-level, use numerofloops = 0)

In the result above there are two terms, with 4 and 6 external legs respectively.

A scattering process with matrix element  in momentum representation, corresponding to the term with 4 external legs (symmetry factor = 72), could be any process where the total number of incoming + outgoing parties is equal to 4. For example, one with 2 incoming and 2 outgoing particles. The transition probability for that process is given by

When computing in momentum representation, only the topology of the corresponding Feynman diagrams is shown (i.e. the diagrams associated to the corresponding Feynman integral in coordinates representation).

The transition matrix element  is related to the transition probability density  (formula (13) of sec. 21.1 of ref.[1]) by

where  represent the particle densities of each of the s particles in the initial state , the  (Dirac) is the expected singular factor due to the conservation of the energy-momentum and the amplitude is related to  via

To directly get the probability density  instead ofuse the option output = probabilitydensity

In practice, the most common computations involve processes with 2 or 4 external legs. To restrict the expansion of the scattering matrix in coordinates representation to that kind of processes use the numberofexternallegs option. For example, the following computes the expansion of  up to order = 3, restricting the outcome to the terms corresponding to diagrams with only 2 external legs

This result shows two Feynman integrals, with respectively 2 and 3 loops, the second integral with two terms. The transition probability density in momentum representation for a process related to the first integral (1 term with symmetry factor = 96) is then

In the above, for readability, the contracted spacetime indices in the square of momenta entering the amplitude F (as denominators of propagators) are implicit. To make those indices explicit, use the option putindicesinsquareofmomentum

This computation can also be performed to higher orders. For example, with 3 loops, in coordinates and momentum representations, corresponding to the other two terms and diagrams in (1.7)

A corresponding S-matrix element in momentum representation:

Consider the interaction Lagrangian of Quantum Electrodynamics (QED). To formulate this problem on the worksheet, start defining the vector field .

Set lowercase Latin letters from i to s to represent spinor indices (you can change this setting according to your preference, see Setup ), also the (anticommutative) spinor field will be represented by , so set  as an anticommutativeprefix, and set  and  as quantum operators

The matrix indices of the Dirac matrices  are written explicitly and use conjugate  to represent the Dirac conjugate

Compute , only the terms with 4 external legs, and display the diagrams: all the corresponding graphs have no loops

The same computation but with only 2 external legs results in the diagrams with 1 loop that correspond to the self-energy of the electron and the photon (page 218 of ref.[1])

where the diagram with two spinor legs is the electron self-energy. To restrict the output furthermore, for example getting only the self-energy of the photon, you can specify the normal products you want:

The corresponding S-matrix elements in momentum representation

In this result we see  spinor (see ref.[2]), and the propagator of the field  with a mass . To indicate that this field is massless use

Now the propagator for  is the one of a massless vector field:

The self-energy of the photon:

where  is the corresponding polarization vector.

When working with non-Abelian gauge fields, the interaction Lagrangian involves derivatives. FeynmanDiagrams  can handle that kind of interaction in momentum representation. Consider for instance a Yang-Mills theory with a massless field  where a is a SU2 index (see eq.(12) of sec. 19.4 of ref.[1]). The interaction Lagrangian can be entered as follows

The transition probability density at tree-level for a process with two incoming and two outgoing B particles is given by

To simplify the repeated indices, us the option simplifytensorindices. To check the indices entering a result like this one use Check ; there are no free indices, and regarding the repeated indices:

This process can be computed with 1 or more loops, in which case the number of terms increases significantly. As another interesting non-Abelian model, consider the interaction Lagrangian of the electro-weak part of the Standard Model

This interaction Lagrangian contains six different terms. The S-matrix element for the tree-level process with two incoming and two outgoing W particles is shown in the help page for FeynmanDiagrams .

References

[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.

Compute Derivatives: Yes or No

Numerical Evaluation

Two Hankel Transform Definitions

More integral transform results

Mellin and Hankel transform solutions for Partial Differential Equations with boundary conditions

Compute derivatives: Yes or No.

For historical reasons, previous implementations of these integral transform commands did not follow a standard paradigm of computer algebra: "Given a function , the input  should return the derivative of ". The implementation instead worked in the opposite direction: if you were to input the result of the derivative, you would receive the derivative representation. For example, to the input  you would receive . This is particularly useful for the purpose of using integral transforms to solve differential equations but it is counter-intuitive and misleading; Maple knows the differentiation rule of these functions, but that rule was not evident anywhere. It was not clear how to compute the derivative (unless you knew the result in advance).

To solve this issue, a new command, setup, has been added to the package, so that you can set "whether or not" to compute derivatives, and the default has been changed to computederivatives = true while the old behavior is obtained only if you input . For example, after having installed the Physics Updates,

the current settings can be queried via

and so differentiating returns the derivative computed

while changing this setting to work as in previous releases you have this computation reversed: you input the output (1.3) and you get the corresponding input

Reset the value of computederivatives

In summary: by default, derivatives of integral transforms are now computed; if you need to work with these derivatives as in  previous releases, you can input . This setting can be changed any time you want within one and the same Maple session, and changing it does not have any impact on the performance of intsolve, dsolve and pdsolve to solve differential equations using integral transforms.

Numerical Evaluation

In previous releases, integral transforms had no numerical evaluation implemented. This is in the process of changing. So, for example, to numerically evaluate the inverse laplace transform ( invlaplace  command), three different algorithms have been implemented: Gaver-Stehfest, Talbot and Euler, following the presentation by Abate and Whitt, "Unified Framework for Numerically Inverting Laplace Transforms", INFORMS Journal on Computing 18(4), pp. 408–421, 2006.

For example, consider the exact solution to this partial differential equation subject to initial and boundary conditions

Note that these two conditions are not entirely compatible: the solution returned cannot be valid for  and  simultaneously. However, a solution discarding that point does exist and is given by

Verifying the solution, one condition remains to be tested

Since we now have numerical evaluation rules, we can test that what looks different from 0 in the above is actually 0.

Add a small number to the initial value of t to skip the point

The default method used is the method of Euler sums and the numerical evaluation is performed as usual using the evalf command. For example, consider

The Laplace transform of F is given by

and the inverse Laplace transform of LT in inert form is

At  we have

This result is consistent with the one we get if we first compute the exact form of the inverse Laplace transform at t = 1:

In addition to the standard use of evalf to numerically evaluate inverse Laplace transforms, one can invoke each of the three different methods implemented using the MathematicalFunctions:-Evalf  command

Regarding the method we use by default: from a numerical experiment with varied problems we have concluded that our implementation of the Euler (sums) method is faster and more accurate than the other two.

Two Hankel transform definitions

In previous Maple releases, the definition of the Hankel transform was given by

where  is the  function. This definition, sometimes called alternative definition of the Hankel transform, has the inconvenience of the square root  in the integrand, complicating the form of the hankel transform for the Laplacian in cylindrical coordinates. On the other hand, the definition more frequently used in the literature is

With it, the Hankel transform of  is given by the simple ODE form . Not just this transform but several other ones acquire a simpler form with the definition that does not have a square root in the integrand.

So the idea is to align Maple with this simpler definition, while keeping the previous definition as an alternative. Hence, by default, when you load the inttrans package, the new definition in use for the Hankel transform is

You can change this default so that Maple works with the alternative definition as in previous releases.  For that purpose, use the new inttrans:-setup command (which you can also use to query about the definition in use at any moment):

This change in definition is automatically taken into account by other parts of the Maple library using the Hankel transform. For example, the differentiation rule with the new definition is

This differentiation rule resembles (is connected to) the differentiation rule for BesselJ, and this is another advantage of the new definition.

Furthermore, several transforms have acquired a simpler form, as for example:

Let's compare: make the definition be as in previous releases.

The differentiation rule with the previous (alternative) definition was not as simple:

And the transform (3.5) was also not so simple:

Reset to the new default value of the definition.

More integral transform results

The revision of the integral transforms includes also filling gaps: previous transforms that were not being computed are now computed. Still with the Hankel transform, consider the operators

Being able to transform these operators into algebraic expressions or differential equations of lower order is key for solving PDE problems with Boundary Conditions.

Consider, for instance, this ODE

Its Hankel transform is a simple algebraic expression

An example with formula_plus

In the case of hankel , not just differential operators but also several new transforms are now computable

Mellin and Hankel transform solutions for Partial Differential Equations with Boundary Conditions

In previous Maple releases, the Fourier and Laplace transforms were used to compute exact solutions to PDE problems with boundary conditions. Now, Mellin and Hankel transforms are also used for that same purpose.

Example:

As usual, you can let pdsolve choose the solving method, or indicate the method yourself:

It is sometimes preferable to see these solutions in terms of more familiar integrals. For that purpose, use

An example where the hankel transform is computable to the end: