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

Calling Sequence


FeynmanDiagrams(InteractionLagrangian, points = ..., numberoflegs = ..., numberofloops = ..., externallegs = ..., excludepropagators = ..., crossedpropagators = ..., fields = ...)


Parameters


InteractionLagrangian



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; default value is 1 vertex

numberoflegs = ...



(optional) the number of external legs of the Feynman diagrams, these legs are represented in the output as _NP(...) normal ordered products; default value is all possible cases

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

externallegs = ...



(optional) the same as normalproducts = ..., the righthand side is a function _NP(...), or a set of them, where each _NP(...) contains fields or field names as arguments, which are the normal ordered products of unpaired fields expected in the output; to these unpaired fields are associated the external legs of the related Feynman diagrams

excludepropagators = ...



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

crossedpropagators = ...



(optional) it can be true or false (default), to indicate that crossed propagators (related to crossed terms in the quadratic part of the Lagrangian) are not zero and so must be taken into account

fields = ...



(optional) either a function or its name, or a set or list of them; indicates the fields of the interaction 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 conjugate or Dagger applied to the anticommutative function representing the spinor field.

•

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 the normal products (the _NP(...) appearing as first argument of the returned functions). 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 externallegs = ..., where the righthand side is an _NP function with fields (or field names) as arguments, or a list of such _NP functions, 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. Instead of externallegs you can also use as a synonym normalproducts. This option overrides the numberoflegs option.

•

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 only include terms with as many couples of paired fields (propagators, represented by lists in the second argument of the returned functions). The current implementation computes with up to three loops, and the treelevel graphs correspond to numberofloops = 0.

•

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


Note: FeynmanDiagrams does not compute terms involving propagators of two equal anticommutative 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); 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 define the vector field
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 (15) 
Check the type of letter used to represent spinor indices (or change this setting according to your preference using Setup) and set as a quantum operator:
>


 (16) 
So, to represent spinor indices, use lowercase latin letters, and the spinor fields will be represented by any name prefixed by and the matrix indices related to the matrix product involving Dirac matrices are written explicitly in the interaction Lagrangian.
>


 (17) 
Compute the expanded form of the integrand entering , only the terms with 4 external legs, so all the corresponding graphs have no loops
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 (18) 
The terms entering the integrand of and that only have 2 external legs, so all the corresponding graphs have one loop
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 (19) 
Compute the expanded form of the integrand entering for this model
>


 (20) 
Note the introduction of spacetime indices (covariant as subscripts and contravariant as superscripts) and spinor indices (all presented as subscripts) by FeynmanDiagrams; this preserves the summation convention for repeated indices by avoiding occurrences of indices more than twice in each product. To verify for correctness the indices and noncommutative products use Check
>


 (21) 
>




References



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


