represents a Feynman integral appearing in the expansion of the Scattering matrix (in coordinates representation) or in its matrix elements (in momentum representation) - Maple Programming Help

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Physics[FeynmanIntegral] - represents a Feynman integral appearing in the expansion of the Scattering matrix (in coordinates representation) or in its matrix elements (in momentum representation)

Calling Sequence

FeynmanIntegral(integrand, [[X1], [X2], ... [Xn]])

FeynmanIntegral(integrand, [[p1], [p2], ... [pn]])

Parameters

integrand

-

the integrand that appears in the expansion of the Scattering matrix S=1+S1+S2+... in coordinates representation, or its matrix elements f|S|i in momentum representation.

[[X1], ... [Xn]]

-

a list of lists, here each Xj is a label representing a coordinates system; it can also be one set using Setup or Coordinates. In this case the integral is in coordinates representation and displayed with a preceding factor inn!, where n is the order of Sn in the expansion of S.

[[p1], ... [pn]]

-

a list of lists, here each pj is the name of a tensor, automatically defined as such when the integral appears in the output of FeynmanDiagrams; it can also be defined using Define. In this case, the integral is in momentum representation and has no preceding factor.

Description

• 

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

S=TⅇiLXⅆX4

  

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 chapters IV and V, "The Scattering Matrix" and "The Feynman Rules and Diagrams", in ref.[1]. This exponential can be expanded as

S=1+S1+S2+S3+...

  

where

Sn=inn!TLX__1,,LX__nⅆX__14ⅆX__n4

  

and TLX1,...,LXn is the time-ordered product of n interaction Lagrangians evaluated at different points. Note the factor inn! in this definition of Sn used here.

• 

The FeynmanIntegral(integrand, [[X1], [X2], ... [Xn]]) command is thus a computational representation for Sn in coordinates representation. Note the factor inn! included in the definition used here for Sn. That factor is automatically displayed when the second argument, [[X1], [X2], ... [Xn]], is a list of lists of coordinate systems. For details on the algebraic structure of integrand see FeynmanDiagrams.

• 

The FeynmanIntegral(integrand, [[p1], [p2], ... [pn]]) command, where [[p1], [p2], ... [pn]] is a list of lists of spacetime tensors, is a computational representation for the integrals in momentum representation that enter the S-matrix elements f|S|i with initial and final states i and f, respectively with s initial particles with defined momentum pi and r final particles with defined momentum pf.

• 

To obtain the S-matrix elements f|S|i in momentum representation for each Sn entering the expansion of S in coordinates representation see sec. 20.1 of ref.[1].

• 

In both coordinates and momentum representation, to each element in the list of lists of the second argument in FeynmanIntegral(integrand, [[..], [..], ... [..]]) corresponds a 4-dimensional integral, and the number of elements in that list of lists indicates the number of vertices of the Feynman diagrams corresponding to the Feynman integral.

• 

The inert form of this command, %FeynmanIntegral, is used by the FeynmanDiagrams command to represent each Sj entering S=1+S1+...+Sn in coordinates representation, as well as the integrals in momentum representation entering f|S|i. For details on the algebraic structure of integrand in both representations see FeynmanDiagrams.

Examples

Load the package, set three coordinate systems and set φ to represent a quantum operator

withPhysics:

Setupmathematicalnotation=true,coordinates=X,Y,Z,quantumoperators=φ

Systems of spacetime coordinates are:X=x1,x2,x3,x4,Y=y1,y2,y3,y4,Z=z1,z2,z3,z4

_______________________________________________________

coordinatesystems=X,Y,Z,mathematicalnotation=true,quantumoperators=φ

(1)

Let L be the interaction Lagrangian

LλφX4

LλφX4

(2)

The 2nd term of the expansion S=1+S1+S2+... in coordinates representation is represented using the inert form of the FeynmanIntegral command. All the corresponding Feynman diagrams contain two vertices

S2=FeynmanDiagramsL,numberofvertices=2,diagrams

S2=%FeynmanIntegral16λ2_GF_NPφX,φX,φX,φY,φY,φY,φX,φY+96λ2_GF_NPφX,φY,φX,φY,φX,φY,φX,φY+72λ2_GF_NPφX,φX,φY,φY,φX,φY,φX,φY,X,Y

(3)

From the diagrams above, the possible configurations involve 2, 4 or 6 external legs, with respectively 2, 1 and 0 loops. To get the Feynman integral with the configuration that has only n external legs use the numberofexternallegs option of FeynmanDiagrams

%evalS2,legs=2=FeynmanDiagramsL,numberofvertices=2,numberofexternallegs=2

%evalS2,legs=2=%FeynmanIntegral96λ2_GF_NPφX,φY,φX,φY,φX,φY,φX,φY,X,Y

(4)

Corresponding to this result, one possible specific process involving 2 external legs is that where there is an initial state i with 1 incoming particle, and a final state f with 1 outgoing particle. The S-matrix element f|S|i for that process is expressed in terms of a FeynmanIntegral in momentum representation via

%eval%Bracketf,S2,i,loops=2=FeynmanDiagramsL,incomingparticles=φ,outgoingparticles=φ,numberofloops=2

%eval%Bracketf,S2,i,loops=2=%FeynmanIntegral%FeynmanIntegral38Iλ2DiracP__2+P__1π7E__1E__2p__22m__φ2+IPhysics:-FeynmanDiagrams:-εp__32m__φ2+IPhysics:-FeynmanDiagrams:-εP__1p__2p__32m__φ2+IPhysics:-FeynmanDiagrams:-ε,p__2,p__3

(5)

The Feynman integrals, whose corresponding diagrams have 3 loops, for the same process φφ

%eval%Bracketf,S2,i,loops=3=FeynmanDiagramsL,incomingparticles=φ,outgoingparticles=φ,numberofloops=3

%eval%Bracketf,S2,i,loops=3=%FeynmanIntegral%FeynmanIntegral%FeynmanIntegral932λ3DiracP__2+P__1π11E__1E__2p__32m__φ2+IPhysics:-FeynmanDiagrams:-εp__42m__φ2+IPhysics:-FeynmanDiagrams:-εp__52m__φ2+IPhysics:-FeynmanDiagrams:-εp__3p__4p__52m__φ2+IPhysics:-FeynmanDiagrams:-εP__1+P__2+p__3+p__4+p__52m__φ2+IPhysics:-FeynmanDiagrams:-ε,p__3,p__4,p__5+2%FeynmanIntegral%FeynmanIntegral%FeynmanIntegral932λ3DiracP__2+P__1π11E__1E__2p__32m__φ2+IPhysics:-FeynmanDiagrams:-εp__42m__φ2+IPhysics:-FeynmanDiagrams:-εp__52m__φ2+IPhysics:-FeynmanDiagrams:-εp__3p__4p__52m__φ2+IPhysics:-FeynmanDiagrams:-εP__1+p__4+p__52m__φ2+IPhysics:-FeynmanDiagrams:-ε,p__3,p__4,p__5+%FeynmanIntegral%FeynmanIntegral12048λDiracP__2+P__1%FeynmanIntegral576λ2p__22m__φ2+IPhysics:-FeynmanDiagrams:-εp__2p__4p__52m__φ2+IPhysics:-FeynmanDiagrams:-ε,p__2π11E__1E__2p__42m__φ2+IPhysics:-FeynmanDiagrams:-εp__52m__φ2+IPhysics:-FeynmanDiagrams:-εP__1+p__4+p__52m__φ2+IPhysics:-FeynmanDiagrams:-ε,p__4,p__5

(6)

For details about the integrands entering these Feynman integrals see FeynmanDiagrams.

See Also

Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Physics/Setup, Physics[Dgamma], Physics[FeynmanDiagrams], Physics[Psigma]

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.

  

[3] Berestetskii, V.B., Lifshitz, E.M., Pitaevskii, L.P., Quantum Electrodynamics. Vol.4, Course of Theoretical Physics, 2nd edition, Pergamon Press, 1982.

  

[4] Xiao, B., Wang, H., Zhu, S., A simple algorithm for automatic Feynman diagrams generation. Computer Physics Communications, Volume 184, Issue 8, (2013).

Compatibility

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The Physics[FeynmanIntegral] command was introduced in Maple 2020.

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For more information on Maple 2020 changes, see Updates in Maple 2020.