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Physics

Maple provides a state-of-the-art environment for algebraic computations in Physics, with emphasis on ensuring that the computational experience is as natural as possible. The theme of the Physics project for Maple 2020 has been the consolidation of the functionality introduced in previous releases, together with significant enhancements to further strengthen the functionality especially in

• 

Particle Physics: the Scattering matrix in coordinates and momentum representation and related Feynman Diagrams,

• 

General Relativity: the slicing and spatial gauge conditions of the 3+1 decomposition of spacetime, and numerical relativity

• 

The connection between different tensors and related differentiation operations

• 

Simplification of tensorial expressions, now involving spinor, su2 and su3 tensor indices

 

Overall, the enhancements throughout the entire package further extend the range of Physics-related algebraic computations that can be done naturally in a worksheet. The presentation below illustrates both the novelties and the kind of mathematical formulations that can now be performed.

As part of its commitment to providing the best possible environment for algebraic computations in Physics, Maplesoft launched a Maple Physics: Research and Development website in 2014, which enabled users to download research versions, ask questions, and provide feedback. The results from this accelerated exchange with people around the world have been incorporated into the Physics package in Maple 2020.

 

Feynman Diagrams - the scattering matrix in coordinates and momentum representation

The slicing and spatial gauge conditions of the 3+1 decomposition of spacetime

The new explore option of TensorArray

Differentiating the spacetime metric, the tetrad, and their determinants, with respect to each other

Simplifying spinor, su2, su3 and gauge indices

Psigma definition and algebra and KroneckerDelta as a metric for spinor, su2 and su3 indices

Miscellaneous

See Also

Feynman Diagrams - the scattering matrix in coordinates and momentum representation

 

  

Mathematical methods for particle physics were only partially developed in the Physics package in previous releases. There existed a FeynmanDiagrams command, but its capabilities were too minimal. 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 not working in the area, see "Why Feynman Diagrams are so important".

  

In connection, for Maple 2020 a full rewriting of the FeynmanDiagrams command, now including a myriad of new capabilities (mainly a. b. and c. and related options - see below), reversing the previous status of things entirely.

  

 

  

A scattering matrix S relates the initial and final states, i and f, of a quantum interacting system. In a 4-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 chapter IV, "The Scattering Matrix", of 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. 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:

a. 

The expansion S=1+S1+S2+S3+... in coordinates representation up to arbitrary order (the limitation is now only the hardware)

b. 

The S-matrix element f|S|i in momentum representation up to arbitrary order for given number of loops and initial and final particles (the contents of the i and f states); optionally, also the transition probability density, constructed using the square of the scattering matrix element f|S|i2, as shown in formula (13) of sec. 21.1 of ref.[1].

c. 

The Feynman diagrams (drawings) related to the different terms of the expansion of S or of its matrix elements f|S|i.

  

Interaction Lagrangians involving derivatives of fields, typically appearing in non-Abelian gauge theories, are handled, and several options are provided enabling restricting the outcome 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 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 expansion of S 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

%evalS,`=`order,3=FeynmanDiagramsL,diagrams

%evalS,order=3=1+%FeynmanIntegralλ_GF_NPφX,φX,φX,φX,X+%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+%FeynmanIntegral10368λ3_GF_NPφX,φY,φZ,φZ,φX,φY,φX,φY,φX,φZ,φY,φZ+1728λ3_GF_NPφX,φX,φY,φY,φZ,φZ,φX,φZ,φX,φY,φZ,φY+2592λ3_GF_NPφX,φX,φY,φY,φX,φZ,φX,φZ,φZ,φY,φZ,φY+576λ3_GF_NPφX,φX,φX,φY,φY,φZ,φZ,φZ,φX,φY,φY,φZ+10368λ3_GF_NPφX,φY,φX,φY,φX,φZ,φX,φZ,φY,φZ,φY,φZ+3456λ3_GF_NPφX,φX,φX,φY,φX,φZ,φY,φZ,φY,φZ,φY,φZ,X,Y,Z

(3)

The expansion of S  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 and normal products shown.

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 above that correspond to diagrams with 1 loop use numberofloops = 1 (for tree-level, use numberofloops = 0)

%evalS,`=`order,3,`=`loops,1=FeynmanDiagramsL,numberofloops=1,diagrams

%evalS,order=3,loops=1=%FeynmanIntegral72λ2_GF_NPφX,φX,φY,φY,φX,φY,φX,φY,X,Y+%FeynmanIntegral1728λ3_GF_NPφX,φX,φY,φY,φZ,φZ,φX,φZ,φX,φY,φZ,φY,X,Y,Z

(4)

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

A scattering process with matrix element f|S|i 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 particles is equal to 4. For example, one with 2 incoming and 2 outgoing particles. The transition probability for that process is given by

%eval`#mfenced(mrow(mo("⁢"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo(",",mathcolor = "olive"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo("⁢"),mo("|"),mo("⁢"),mi("S"),mo("⁢"),mo("|"),mo("⁢"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo(",",mathcolor = "olive"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo("⁢",mathcolor = "olive")),open = "⟨",close = "⟩")`,`=`loops,1=FeynmanDiagramsL,incomingparticles=φ,φ,outgoingparticles=φ,φ,numberofloops=1,diagrams

%eval#mfenced(mrow(mo("⁢"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo(",",mathcolor = "olive"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo("⁢"),mo("|"),mo("⁢"),mi("S"),mo("⁢"),mo("|"),mo("⁢"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo(",",mathcolor = "olive"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo("⁢",mathcolor = "olive")),open = "⟨",close = "⟩"),loops=1=%FeynmanIntegral98λ2DiracP__3P__4+P__1+P__2π6E__1E__2E__3E__4p__22m__φ2+IPhysics:-FeynmanDiagrams:-εP__1P__2p__22m__φ2+IPhysics:-FeynmanDiagrams:-ε,p__2+%FeynmanIntegral98λ2DiracP__3P__4+P__1+P__2π6E__1E__2E__3E__4p__22m__φ2+IPhysics:-FeynmanDiagrams:-εP__1+P__3p__22m__φ2+IPhysics:-FeynmanDiagrams:-ε,p__2+%FeynmanIntegral98λ2DiracP__3P__4+P__1+P__2π6E__1E__2E__3E__4p__22m__φ2+IPhysics:-FeynmanDiagrams:-εP__1+P__4p__22m__φ2+IPhysics:-FeynmanDiagrams:-ε,p__2

(5)

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 f|S|i is related to the transition probability density dw (formula (13) of sec. 21.1 of ref.[1]) by

dw=2π3s4n__1...n__sFpi,pf2δi=1spif=1rpf d 3p1 ...d 3pr

where n__1...n__s represent the particle densities of each of the s particles in the initial state i, the δ (Dirac) is the expected singular factor due to the conservation of the energy-momentum and the amplitude Fpi,pfis related to f|S|i via

f|S|i=Fpi,pfδi=1spif=1rpf

To directly get the probability density dw instead off|S|iuse the option output = probabilitydensity

FeynmanDiagramsL,incomingparticles=φ,φ,outgoingparticles=φ,φ,numberofloops=1,output=probabilitydensity

Physics:-FeynmanDiagrams:-ProbabilityDensity4π2%mulni,i=1..2absF2DiracP__3P__4+P__1+P__2%muldP_f3,f=1..2,F=%FeynmanIntegral98λ2π6E__1E__2E__3E__4p__22m__φ2+IPhysics:-FeynmanDiagrams:-εP__1P__2p__22m__φ2+IPhysics:-FeynmanDiagrams:-ε,p__2+%FeynmanIntegral98λ2π6E__1E__2E__3E__4p__22m__φ2+IPhysics:-FeynmanDiagrams:-εP__1+P__3p__22m__φ2+IPhysics:-FeynmanDiagrams:-ε,p__2+%FeynmanIntegral98λ2π6E__1E__2E__3E__4p__22m__φ2+IPhysics:-FeynmanDiagrams:-εP__1+P__4p__22m__φ2+IPhysics:-FeynmanDiagrams:-ε,p__2

(6)

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 S up to order = 3, restricting the outcome to the terms corresponding to diagrams with only 2 external legs

%evalS,`=`order,3,`=`legs,2=FeynmanDiagramsL,numberofexternallegs=2,diagrams

%evalS,order=3,legs=2=%FeynmanIntegral96λ2_GF_NPφX,φY,φX,φY,φX,φY,φX,φY,X,Y+%FeynmanIntegral3456λ3_GF_NPφX,φX,φX,φY,φX,φZ,φY,φZ,φY,φZ,φY,φZ+10368λ3_GF_NPφX,φY,φX,φY,φX,φZ,φX,φZ,φY,φZ,φY,φZ,X,Y,Z

(7)

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 the process %Bracketφ,S,φcorresponding to the first integral (1 term with symmetry factor = 96) is then

FeynmanDiagramsL,incomingparticles=φ,outgoingparticles=φ,numberofloops=2,diagrams,output=probabilitydensity

Physics:-FeynmanDiagrams:-ProbabilityDensity12%mulni,i=1..1absF2DiracP__2+P__1%muldP_f3,f=1..1π,F=%FeynmanIntegral%FeynmanIntegral38Iλ2π7E__1E__2p__22m__φ2+IPhysics:-FeynmanDiagrams:-εp__32m__φ2+IPhysics:-FeynmanDiagrams:-εP__1p__2p__32m__φ2+IPhysics:-FeynmanDiagrams:-ε,p__2,p__3

(8)

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

%eval%Bracketφ,S,φ,`=`loops,2=FeynmanDiagramsL,incoming=φ,outgoing=φ,numberofloops=2,indices

* Partial match of 'indices' against keyword 'putindicesinsquareofmomentum'

%eval%Bracketφ,S,φ,loops=2=%FeynmanIntegral%FeynmanIntegral38Iλ2DiracP__2~kappa+P__1~kappaπ7E__1E__2p__2μp__2~mum__φ2+IPhysics:-FeynmanDiagrams:-εp__3νp__3~num__φ2+IPhysics:-FeynmanDiagrams:-εP__1βp__2βp__3βP__1~betap__2~betap__3~betam__φ2+IPhysics:-FeynmanDiagrams:-ε,p__2,p__3

(9)

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 (7)

%evalS3,`=`legs,2,`=`loops,3=FeynmanDiagramsL,legs=2,loops=3

* Partial match of 'legs' against keyword 'numberoflegs'

* Partial match of 'loops' against keyword 'numberofloops'

%evalS3,legs=2,loops=3=%FeynmanIntegral3456λ3_GF_NPφX,φX,φX,φY,φX,φZ,φY,φZ,φY,φZ,φY,φZ+10368λ3_GF_NPφX,φY,φX,φY,φX,φZ,φX,φZ,φY,φZ,φY,φZ,X,Y,Z

(10)

%eval%Bracketφ,S,φ,`=`loops,3=FeynmanDiagramsL,incomingparticles=φ,outgoingparticles=φ,numberofloops=3

%eval%Bracketφ,S,φ,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

(11)

Consider the interaction Lagrangian of Quantum Electrodynamics (QED). To formulate the problem start defining the vector field Aμ.

DefineAμ

Defined objects with tensor properties

Aμ,γμ,P__1μ,P__2μ,P__3μ,P__4μ,σμ,μ,gμ,ν,p__1μ,p__2μ,p__3μ,p__4μ,p__5μ,εα,β,μ,ν,Xμ,Yμ,Zμ

(12)

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 below by ψ, so set ψ as an anticommutativeprefix, and set A and ψ as quantum operators

Setupspinorindices=lowercaselatin_is,anticommutativeprefix=ψ,op=A,ψ

* Partial match of 'op' against keyword 'quantumoperators'

_______________________________________________________

anticommutativeprefix=ψ,quantumoperators=A,φ,ψ,spinorindices=lowercaselatin_is

(13)

The matrix indices of the Dirac matrices are written explicitly and use conjugate to represent the Dirac conjugate ψj&conjugate0;

L__QEDαconjugateψjXDgammaμj,kψkXAμX

L__QEDαψjX&conjugate0;ψkXAμXγμμj,k

(14)

Compute S2, only the terms with 4 external legs, and display the diagrams: they have no loops

%evalS2,`=`legs,4=FeynmanDiagramsL__QED,numberofvertices=2,numberoflegs=4,diagrams

%evalS2,legs=4=%FeynmanIntegral2α2Dgamma~muj,kDgamma~alphai,l_GF_NPψkX,AμX,conjugateψiY,AαY,ψlY,conjugateψjX+α2Dgamma~muj,kDgamma~alphai,l_GF_NPconjugateψjX,ψkX,conjugateψiY,ψlY,AμX,AαY,X,Y

(15)

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])

%evalS2,`=`legs,2=FeynmanDiagramsL__QED,numberofvertices=2,numberoflegs=2,diagrams

%evalS2,legs=2=%FeynmanIntegral2α2Dgamma~muj,kDgamma~alphai,l_GF_NPψkX,conjugateψiY,AμX,AαY,ψlY,conjugateψjXα2Dgamma~muj,kDgamma~alphai,l_GF_NPAμX,AαY,ψlY,conjugateψjX,ψkX,conjugateψiY,X,Y

(16)

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:

%evalS2,`=`legs,2,`=`products,_NPA,A=FeynmanDiagramsL__QED,numberofvertices=2,numberoflegs=2,normalproduct=_NPA,A

* Partial match of 'normalproduct' against keyword 'normalproducts'

%evalS2,legs=2,products=_NPA,A=%FeynmanIntegralα2Dgamma~muj,kDgamma~alphai,l_GF_NPAμX,AαY,conjugateψjX,ψlY,ψkX,conjugateψiY,X,Y

(17)

The corresponding S-matrix elements in momentum representation

%eval%BracketA,S,A,`=`loops,1=FeynmanDiagramsL__QED,incomingparticles=A,outgoing=A,numberofloops=1,diagrams

%eval%BracketA,S,A,loops=1=%FeynmanIntegral116Physics:-FeynmanDiagrams:-PolarizationVectorAνP__1_conjugatePhysics:-FeynmanDiagrams:-PolarizationVectorAαP__2_m__ψKroneckerDeltal,n+p__2βDgamma~betal,nα2Dgamma~alphan,iDgamma~num,lP__1τ+p__2τDgamma~taui,m+m__ψKroneckerDeltai,mDiracP__2+P__1π3E__1E__2p__22m__ψ2+IPhysics:-FeynmanDiagrams:-εP__1+p__22m__ψ2+IPhysics:-FeynmanDiagrams:-ε,p__2

(18)

where ϵA is the corresponding polarization vector.

The self-energy of the electron:

%eval%Bracketψ,S,ψ,`=`loops,1=FeynmanDiagramsL__QED,incoming=ψ,outgoing=ψ,numberofloops=1

%eval%Bracketψ,S,ψ,loops=1=%FeynmanIntegral18Physics:-FeynmanDiagrams:-UspinorψiP__1_conjugatePhysics:-FeynmanDiagrams:-UspinorψlP__2_g_α,ν+p__2νp__2αm__A2α2Dgamma~alphal,mDgamma~nun,iP__1β+p__2βDgamma~betam,n+m__ψKroneckerDeltam,nDiracP__2+P__1π3p__22m__A2+IPhysics:-FeynmanDiagrams:-εP__1+p__22m__ψ2+IPhysics:-FeynmanDiagrams:-ε,p__2

(19)

 

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

Setupmassless=A

* Partial match of 'massless' against keyword 'masslessfields'

_______________________________________________________

masslessfields=A

(20)

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

%eval%Bracketψ,S,ψ,`=`loops,1=FeynmanDiagramsL__QED,incoming=ψ,outgoing=ψ,numberofloops=1

%eval%Bracketψ,S,ψ,loops=1=%FeynmanIntegral18Physics:-FeynmanDiagrams:-UspinorψiP__1_conjugatePhysics:-FeynmanDiagrams:-UspinorψlP__2_g_α,να2Dgamma~alphal,mDgamma~nun,iP__1β+p__2βDgamma~betam,n+m__ψKroneckerDeltam,nDiracP__2+P__1π3p__22+IPhysics:-FeynmanDiagrams:-εP__1+p__22m__ψ2+IPhysics:-FeynmanDiagrams:-ε,p__2

(21)

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 Bμ,a where a is a SU2 index (see eq.(12) of sec. 19.4 of ref.[1]). The interaction Lagrangian can be entered as follows

Setupsu2indices=lowercaselatin_ah,massless=B,op=B

* Partial match of 'massless' against keyword 'masslessfields'

* Partial match of 'op' against keyword 'quantumoperators'

_______________________________________________________

masslessfields=A,B,quantumoperators=A,B,φ,ψ,ψ1,su2indices=lowercaselatin_ah

(22)

DefineBμ,a,quiet:

F__Bμ,ν,ad_μBν,aXd_νBμ,aX

F__Bμ,ν,aμBν,aXνBμ,aX

(23)

Lg2LeviCivitaa,b,cF__Bμ,ν,aBμ,bXBν,cX+g24LeviCivitaa,b,cLeviCivitaa,e,fBμ,bXBν,cXBμ,eXBν,fX

Lgεa,b,cμBν,aXνBμ,aXBμbμbXBνcνcX2+g2εa,b,cεa,e,fBμ,bXBν,cXBμeμeXBνfνfX4

(24)

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

FeynmanDiagramsL,incomingparticles=B,B,outgoingparticles=B,B,numberofloops=0,output=probabilitydensity,factor, diagrams

* Partial match of 'factor' against keyword 'factortreelevel'

Physics:-FeynmanDiagrams:-ProbabilityDensity4π2%mulni,i=1..2absF2DiracP__3~sigmaP__4~sigma+P__1~sigma+P__2~sigma%muldP_f3,f=1..2,F=18IP__1~kappaP__2~kappaP__4~kappag_~lambda,~tau+P__1~lambda+P__2~lambda+P__3~lambdag_~kappa,~taug_~kappa,~lambdaP__3~tauP__4~tauLeviCivitaa1,a3,hP__1~beta+12P__2~betag_~alpha,~sigma+12P__1~sigma+12P__2~sigmag_~alpha,~beta12g_~beta,~sigmaP__1~alpha+2P__2~alphaLeviCivitaa2,d,gg_σ,τKroneckerDeltaa2,a3P__1χP__2χP__1~chiP__2~chi+IPhysics:-FeynmanDiagrams:-ε116ILeviCivitaa1,a3,gP__1~beta+P__3~betaP__4~betag_~lambda,~tau+P__1~lambdaP__2~lambdaP__3~lambdag_~beta,~tau+g_~beta,~lambdaP__2~tau+P__4~tauLeviCivitaa2,d,hP__1~sigma+P__3~sigmag_~alpha,~kappa+2P__1~kappa+P__3~kappag_~alpha,~sigma+g_~kappa,~sigmaP__1~alpha2P__3~alphag_σ,τKroneckerDeltaa2,a3P__1χ+P__3χP__1~chi+P__3~chi+IPhysics:-FeynmanDiagrams:-ε116ILeviCivitaa3,g,hP__1~betaP__3~beta+P__4~betag_~kappa,~tau+P__1~kappaP__2~kappaP__4~kappag_~beta,~tau+g_~beta,~kappaP__2~tau+P__3~tauLeviCivitaa1,a2,dP__1~sigma+P__4~sigmag_~alpha,~lambda+P__1~alpha2P__4~alphag_~lambda,~sigma2P__1~lambda12P__4~lambdag_~alpha,~sigmag_σ,τKroneckerDeltaa2,a3P__1χ+P__4χP__1~chi+P__4~chi+IPhysics:-FeynmanDiagrams:-ε116IKroneckerDeltad,hg_~alpha,~betag_~kappa,~lambda2g_~alpha,~kappag_~beta,~lambda+g_~alpha,~lambdag_~beta,~kappaKroneckerDeltaa1,g2KroneckerDeltad,gg_~alpha,~betag_~kappa,~lambda12g_~beta,~kappag_~alpha,~lambda12g_~alpha,~kappag_~beta,~lambdaKroneckerDeltaa1,h+KroneckerDeltag,hKroneckerDeltaa1,dg_~alpha,~betag_~kappa,~lambda+g_~alpha,~kappag_~beta,~lambda2g_~alpha,~lambdag_~beta,~kappag2conjugatePhysics:-FeynmanDiagrams:-PolarizationVectorBκ,hP__3_conjugatePhysics:-FeynmanDiagrams:-PolarizationVectorBλ,a1P__4_Physics:-FeynmanDiagrams:-PolarizationVectorBα,dP__1_Physics:-FeynmanDiagrams:-PolarizationVectorBβ,gP__2_π2E__1E__2E__3E__4

(25)

To simplify the repeated indices, use 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:

Check,all

The repeated indices per term are: ...,...,..., the free indices are: ...

a1,a2,a3,α,β,χ,d,g,h,κ,λ,σ,τ,

(26)


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

Coordinatesclear,Z

Unaliasing Z previously defined as a system of spacetime coordinates

(27)

Setupquantumoperators=W,Z

quantumoperators=A,B,W,Z,φ,ψ,ψ1

(28)

DefineWμ,Zμ

Defined objects with tensor properties

Aμ,Bμ,a,γμ,P__1μ,P__2μ,P__3μ,P__4μ,σμ,Wμ,Zμ,μ,gμ,ν,p__1μ,p__2μ,p__3μ,p__4μ,p__5μ,ψj,εα,β,μ,ν,Xμ,Yμ

(29)

CompactDisplayW,ZX

WXwill now be displayed asW

ZXwill now be displayed asZ

(30)

F__Wμ,νd_μWνXd_νWμX

d_μWνX,Xd_νWμX,X

(31)

F__Zμ,νd_μZνXd_νZμX

d_μZνX,Xd_νZμX,X

(32)

L__WZIgcosθ__wDaggerF__Wμ,νWμXDaggerWμXF__Wμ,νZνX+WνXDaggerWμXF__Zμ,ν

Igcosθ__w`*``*`d_μDaggerWνX,Xd_νDaggerWμX,X,W~muX`*`DaggerWμX,d_~muWνX,Xd_νW~muX,X,Z~nuX+`*`WνX,DaggerWμX,d_~muZ~nuX,Xd_~nuZ~muX,X

(33)

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. There, due to the use of size simplification and the option factortreelevel, the result is a product of 10 factors each of  which is nested several times, totaling a computational length of 23,750.  Expanding that product the computational length is 325,761 and the number of terms is 480, giving an idea of how fast the size of the S-matrix elements grow with the number of terms in the interaction Lagrangian.

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.

The slicing and spatial gauge conditions of the 3+1 decomposition of spacetime

 

A key feature of the 3+1 decomposition of Einstein's equations is the freedom in the choice of coordinates and foliation (3D hypersurface), represented by the freedom of choice for the Lapse and Shift functions, frequently referred as the slicing and spatial gauge conditions respectively, entering the 3+1 equations. Only a few of the possible choices are actually useful for the purpose of numerical simulations of the solutions of the 3+1 equations. Depending on the problem, finding suitable conditions is still a research topic.

Some Lapse and Shift conditions, however, are well understood and useful in different contexts. Those conditions, entailing non-trivial algebraic relationships, are difficult to remember or to input for further manipulations, and crucial in that they determine the actual form of the 3+1 equations.

In connection, for Maple 2020 the Physics:-ThreePlusOne package includes a new command LapseAndShiftConditions, that return the equations of the most common slicing and spatial gauge conditions, optionally attempting to solve them for the Lapse and Shift.

restart;withPhysics: withThreePlusOne;

_______________________________________________________

Setting lowercaselatin_is letters to represent space indices

Defined as 4D spacetime tensors see ?Physics,ThreePlusOne,γμ,ν,μ,Γμ,ν,α,Rμ,ν,Rμ,ν,α,β,βμ,nμ,tμ,Κμ,ν

Changing the signature of spacetime from - - - + to + + + - in order to match the signature customarily used in the ADM formalism

Default differentiation variables for d_, D_ and dAlembertian are:X=x,y,z,t

Systems of spacetime coordinates are:X=x,y,z,t

_______________________________________________________

ADMEquations,Christoffel3,D3_,ExtrinsicCurvature,Lapse,LapseAndShiftConditions,Ricci3,Riemann3,Shift,TimeVector,UnitNormalVector,gamma3_

(34)

To see the conditions implemented in this first version of LapseAndShiftConditions, you can enter the command with no arguments

LapseAndShiftConditions

advectiveonepluslog,generalizedharmonicslicing,geodesicslicing,harmoniccoordinates,harmonicslicing,kdriver,maximalslicing,onepluslog

(35)

To  see the condition equations behind these keywords in this first version of LapseAndShiftConditions, use

for condition in LapseAndShiftConditions do       _______________________________________;       condition = LapseAndShiftConditionsconditionod;

_______________________________________

advectiveonepluslog=%LieDerivativeUnitNormalVectorLapse=2%ExtrinsicCurvaturetrace%d_0LapseShift~j%d_jLapse=2Lapse%ExtrinsicCurvaturetrace

_______________________________________

generalizedharmonicslicing=%d_0Lapse=Lapse2%fLapse%ExtrinsicCurvaturetraceShift~i=0

_______________________________________

geodesicslicing=Lapse=1Shift~i=0

_______________________________________

harmoniccoordinates=%d_0LapseShift~j%d_jLapse=Lapse2%ExtrinsicCurvaturetrace%d_0Shift~iShift~j%d_jShift~i=Lapse2%gamma3_~i,~j%d_j%lnLapse+%gamma3_~j,~k%Christoffel3~i,j,k

_______________________________________

harmonicslicing=%d_0Lapse=Lapse2%ExtrinsicCurvaturetraceLapse=%Cx,y,z,t%gamma3_determinantShift~i=0

_______________________________________

kdriver&equals;%d_0%ExtrinsicCurvaturetrace&equals;%c%ExtrinsicCurvaturetrace%d_0Lapse&equals;%epsilon%d_0%ExtrinsicCurvaturetrace&plus;%c%ExtrinsicCurvaturetrace0<%c&comma;0<%epsilon

_______________________________________

maximalslicing&equals;%ExtrinsicCurvaturetrace&equals;0&comma;%d_0%ExtrinsicCurvaturetrace&equals;0D_~muD_&mu;Lapse&equals;Lapse%ExtrinsicCurvature&mu;&comma;&nu;%ExtrinsicCurvature~mu&comma;~nu&plus;4&pi;%EnergyMomentum&nu;&comma;&alpha;%gamma3_&mu;&comma;~nu%gamma3_~mu&comma;~alpha&plus;%EnergyMomentum~mu&comma;~nu%UnitNormalVector&mu;%UnitNormalVector&nu;

_______________________________________

onepluslog&equals;%d_0Lapse&equals;2Lapse%ExtrinsicCurvaturetraceLapse&equals;1&plus;ln%gamma3_determinantShift~i&equals;0

(36)

 

In the compact equations above, these condition equations for the Lapse and Shift include other tensors of the ThreePlusOne package, expressed in terms of their inert forms (displayed in grey) to allow for further manipulations. To retrieve these equations using these keywords, you can pass any them complete or, if you don't remember the exact spelling, just a portion of them suffices. For example,

LapseAndShiftConditionsgeneralized

* Partial match of 'generalized' against keyword 'generalizedharmonicslicing'

%d_0Lapse&equals;Lapse2%fLapse%ExtrinsicCurvaturetraceShift~i&equals;0

(37)

These conditions form a system of equations for the Lapse and Shift that, depending on the 4D metric set, may be solvable in exact form. For example, the metric set at this point is the default metric when Physics loads

g_

g_&mu;&comma;&nu;&equals;1000010000100001

(38)

and for this metric two exact solutions for the Lapse and Shift are

LapseAndShiftConditionsgeneralized&comma;output&equals;solution

* Partial match of 'generalized' against keyword 'generalizedharmonicslicing'

α=_F1x&comma;y&comma;z&comma;β11=0&comma;β22=0&comma;β33=0,α=RootOff_Z+1&comma;β11=0&comma;β22=0&comma;β33=0

(39)

where the RootOf in the second set implies on

DEtoolsremove_RootOfα=RootOff_Z+1

fα+1=0

(40)

These results can be obtained step by step manipulating the equations (37) themselves, in which case it is more convenient to have them evaluated first. For that purpose use the option evaluate, and set the Lapse and Shift to arbitrary so that the ExtrinsicCurvature (or any other tensor of ThreePlusOne) is not prematurely evaluated in terms of the Minkowski metric (38) (comments about this further below)

Setuplapse&equals;arbitrary

* Partial match of 'lapse' against keyword 'lapseandshift'

_______________________________________________________

lapseandshift=arbitrary

(41)

Now, instead of Κtrace= 0 we have the trace expressed in terms of arbitrary Lapse and Shift

ExtrinsicCurvaturetrace

β11xαα2+β22yαα2+β33zαα2+β11xα+β22yα+β33zαtαα2

(42)

and so, recomputing the equations (37) using the evaluate option, we have

EQ  LapseAndShiftConditionsgeneralizedharmonicslicing&comma; eval

* Partial match of 'eval' against keyword 'evaluate'

diffLapse&comma;t&equals;Lapse2Shift~1Shift~1diffLapse&comma;x&plus;Lapse2Shift~2Shift~2diffLapse&comma;y&plus;Lapse2Shift~3Shift~3diffLapse&comma;z&plus;diffLapse&comma;tfLapseShift~1&equals;0Shift~2&equals;0Shift~3&equals;0

(43)

 

Substitute the equations for the Shift, βj&equals;0, into the equations for the Lapse α

convertEQ2&comma;setofequations

β11=0&comma;β22=0&comma;β33=0

(44)

subs&comma;EQ1

tα=tαfα

(45)

It is now possible to see that this system has two solutions: either fα&equals;1 (the second solution in (39), resulting in (40)) or the Lapse α does not depend on t (the first solution in (39)).


To understand the need, in (41), to set the lapseandshift to arbitrary, check what happens with the default value Setuplapse&equals;standard

Setuplapse&equals;standard

* Partial match of 'lapse' against keyword 'lapseandshift'

_______________________________________________________

lapseandshift=standard

(46)

Trying to reproduce the computation above, the first step works

LapseAndShiftConditionsgeneralizedharmonicslicing&comma; eval

* Partial match of 'eval' against keyword 'evaluate'

diffLapse&comma;t&equals;Lapse2Shift~1Shift~1diffLapse&comma;x&plus;Lapse2Shift~2Shift~2diffLapse&comma;y&plus;Lapse2Shift~3Shift~3diffLapse&comma;z&plus;diffLapse&comma;tfLapseShift~1&equals;0Shift~2&equals;0Shift~3&equals;0

(47)

but because of Setuplapse&equals;standard, in the next step the derivative of the Lapse is computed rewriting the Lapse in terms of the Minkowski metric (38), for which the Lapse is actually equal to 1

Lapsedefinition

α=1μtμμt

(48)

SumOverRepeatedIndices

α=1

(49)

and so, the step that results in (45), here results in nothing

subs&comma;EQ1

0=0

(50)

The key observation is understanding what is defined (expressed) in terms of what. Given a 4D metric gμ,ν, when Setuplapseandshift&equals;standard the Lapse and Shift are defined in terms of the components g0,μ. That is why in (49) we have α=1. On the other hand, when Setuplapseandshift&equals;arbitrary the Lapse and Shift are not expressed in terms of the metric, they remain arbitrary, allowing for solving the condition equations in order to then express the components g0,μ in terms of the Lapse and Shift.

An example where the conditions can be solved exactly and the spacetime is curved: set the Schwarzschild metric in spherical coordinates

g_sc

_______________________________________________________

Systems of spacetime coordinates are:X=r&comma;θ&comma;φ&comma;t

Default differentiation variables for d_, D_ and dAlembertian are:X=r&comma;θ&comma;φ&comma;t

The Schwarzschild metric in coordinates r&comma;θ&comma;φ&comma;t

Parameters: m

_______________________________________________________

g_&mu;&comma;&nu;&equals;rr&plus;2m0000r20000r2sin&theta;20000r&plus;2mr

(51)

Consider the harmonicslicing conditions

LapseAndShiftConditionsharmonicslicing

Christoffel~4&comma;&nu;&comma;~nu&equals;0%d_0Lapse&equals;Lapse2%ExtrinsicCurvaturetraceLapse&equals;%Cr&comma;&theta;&comma;&phi;&comma;t%gamma3_determinantShift~i&equals;0

(52)

Note that in spite of having Setuplapseandshift&equals;standard, in the internal intermediate computations of LapseAndShiftConditions Setuplapseandshift&equals;arbitrary is used, so the problem can be formulated and tackled in the right way, resulting in the right conditions equations when using the option evaluate, allowing to compute a solution

LapseAndShiftConditionsharmonicslicing&comma;evaluate

0&equals;0diffLapse&comma;t&equals;2sin&theta;12Lapse2r&plus;m12rrShift~2diffLapse&comma;&theta;&plus;2sin&theta;Shift~312Lapse2r&plus;m12rrdiffLapse&comma;&phi;&plus;2sin&theta;Shift~112Lapse2r&plus;m12rrdiffLapse&comma;