Physics[FeynmanDiagrams]  compute the analytic structure of the Feynman Diagrams of a model

Calling Sequence


FeynmanDiagrams(Lagrangian, points = ..., numberoflegs = ..., numberofloops = ..., normalproducts = ..., excludepropagators = ..., fields = ...)


Parameters


Lagrangian



the interaction terms of a Lagrangian depending on quantum fields

points = ...



list of points at which the Lagrangian is evaluated; the vertices of the corresponding Feynman diagrams

numberoflegs = ...



(optional) the number of external legs of the Feynman diagrams, represented in the output as _NP(...) normal ordered products

numberofloops = ...



(optional) integer or set of integers; the number of loops of the Feynman diagrams: each line joining two vertices of the loop is a propagator, represented in the output as a list of two fields

normalproducts = ...



(optional) functions _NP(...) with fields or field names as arguments, which are the normal ordered products of unpaired fields expected in the output; the external legs of the related Feynman diagrams

excludepropagators = ...



(optional) a list of two fields representing a propagator, or a list of these lists, indicating propagators to be excluded from the output; the righthand side can also be one of the keywords free or cross

fields = ...



(optional) either a function or its name, or a set or list of them; indicates the fields of the Lagrangian





Description


•

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


Each contains integrals of timeordered products of the interaction Lagrangians evaluated at different points. For example, the second order term is of the form:


where and represent different points in spacetime.

•

Using Wick's theorems, the FeynmanDiagrams command computes the expanded form of these timeordered products of interaction Lagrangians in the integrands of each .

•

The expansion of each consists of a sum of terms, each of which is a product of certain pairings of the operator field functions entering with the normal product of the remaining unpaired operators of free fields. The pairings between fields are the propagators, the Green functions of the associated free fields.


NOTE: the current implementation of FeynmanDiagrams can expand up to (products of up to three interaction Lagrangians), and therefore can produce the analytic representation of Feynman diagrams with no more than three vertices.


The algebraic structure of the output


•

For each S[n], the algebraic structure of the output of FeynmanDiagrams is thus of the form


where are the symmetry factors, and (each one corresponds to a Feynman diagram) are functions of one or two arguments of the form

•

Given an interaction Lagrangian, by default FeynmanDiagrams will consider a field of the problem any function of spacetime variables or function name that has been set to represent a quantum operator by the Setup command. You can override that behavior with the optional argument fields = ..., where the righthand side is a function, the name of a function field, or a set or list of them.

•

To represent spinor fields, use anticommutative functions. For that purpose, you can set an anticommutative prefix, or simply load the advanced setup of Physics; both of these are done by using the Setup command. To represent the Diracconjugate of a spinor field, you can either use the Dagger command together with an anticommutative function, or just use one other anticommutative function.

•

The summation convention for spacetime, gauge, or spinor repeated indices in products entering the interaction Lagrangian is automatically used by FeynmanDiagrams, so repeated indices of the respective kinds are automatically generated when expanding the timeorder products of interaction Lagrangians.



Options


•

The optional argument numberoflegs = ..., where the righthand side is an integer between 2 and 9 or a set of them, is used to filter the output of FeynmanDiagrams to include only as many unpaired fields in normal products (the _NP(...) first argument of the returned functions) as the indicated . Recall that these unpaired fields represent external legs in the corresponding Feynman diagrams. The current implementation computes with up to nine external legs.

•

The optional argument numberofloops = ..., where the righthand side is an integer between 0 and 3 or a set of them, is used to filter the output of FeynmanDiagrams to include only as many couples of paired fields (lists in the second argument of the returned ) as the ones that can be included to form the indicated numberofloops in the corresponding Feynman diagrams. The current implementation computes with up to three loops, and the treelevel graphs correspond to numberofloops = 0.

•

The optional argument normalproducts = ..., where the righthand side is an ordered list of fields (or field names), is used to filter the output of FeynmanDiagrams to return only the terms entering having exactly those external legs; that is, having exactly the indicated normal products. Since in this implementation the number of external legs cannot be greater than 9, at most nine fields can be indicated in the arguments of _NP functions on the righthand side of this option.

•

The optional argument excludepropagators = ..., where the righthand side is either a list of two fields or a set or list of these lists, is used to filter out from the output of FeynmanDiagrams all those functions containing these propagators.


Note: FeynmanDiagrams does not compute terms involving propagators of two equal anticomutative fields taken at different points (vertices). Thus, for example, if is defined as an anticommutative field, then the output will not contain terms with a propagator .




Examples


In the Maple Standard GUI, to have the output of this help page use textbook notation, open it as a worksheet (see icon in the toolbar) and execute the input lines below after setting Physics[Setup](mathematicalnotation = true);
Load the package and set three coordinate systems to work in.
>


>


 (1) 
>


 (2) 
>


 (3) 
The expressions entering (only one vertex and so one evaluation point), representing the connected Feynman graphs for this interaction Lagrangian, are:
>


 (4) 
You can filter this result to obtain only the treelevel terms (see the Options subsection of the Description).
>


 (5) 
After defining the fields and once, you do not need to indicate them again; alternatively, you can set them as quantumoperators by using the Setup command.
>


 (6) 
The structure of (that is, the expanded form of the timeordered product of interaction Lagrangians entering the integrand of the second term in the expansion of the scattering matrix) is:
>


 (7) 
In the result above, the treelevel terms involving propagators are associated with connected, oneparticlereducible Feynman graphs, and can be obtained by specifying .
>


 (8) 
NOTE: the current implementation of FeynmanDiagrams can expand products of up to three interaction Lagrangians, so a maximum of three spacetime points are allowed, and therefore produces the analytic representation of Feynman diagrams with no more than three vertices.
Compute only the terms corresponding to Feynman diagrams with two external legs.
>


 (9) 
Compute only the terms corresponding to Feynman diagrams with two external legs corresponding to the field phi, regardless of the vertex to which they are attached (it could be X or Y).
>


 (10) 
Compute terms with two external legs corresponding only to the normal products .
>


 (11) 
Compute the terms leading to Feynman graphs with two loops.
>


 (12) 
In , among the terms resulting from expanding the time ordered product of two interaction Lagrangians (Feynman diagrams with two vertices), there are no terms involving three loops.
>


 (13) 
Compute the terms entering the integrand of , having four external legs (normal ordered products involving four fields), and excluding cross propagators; in this example, these are of the form , where each field is applied at X, Y, or Z.
>


 (14) 
Below is an example of an interaction Lagrangian with a spinorelectromagnetic vertex. First check the type of letter used to represent spinor indices (or change this setting according to your preference using Setup):
>


 (15) 
So, to represent spinor indices, use lower case latin letters, and the spinor fields will be represented by any name prefixed by , or , and the matrix indices related to the matrix product involving Dirac matrices are written explicitly in the interaction Lagrangian. Note the use of to represent the noncommutative product ( and do not commute).
>


 (16) 
Compute the expanded form of the integrand entering for this model, excluding terms (diagrams) containing propagators (connected lines) of the form .
>


 (17) 
>


Note the introduction of spacetime and spinor indices by FeynmanDiagrams; this preserves the summation convention for repeated indices by avoiding occurrences of indices more than twice in each product.


References



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


