Sigma Model

Notation

<<Notation`

Dag[Σ_]:=ConjugateTranspose[Σ]//Simplify

Notation[(NotationTemplateTag[(Subscript[ℼ, i_])]) ⟺ (NotationTemplateTag[pi[i_]])]
Notation[(NotationTemplateTag[(Subscript[ℼ, i_]^(,))]) ⟺ (NotationTemplateTag[pip[i_]])]
Notation[(NotationTemplateTag[(SuperDagger[Σ_])]) ⟺ (NotationTemplateTag[Dag[Σ_]])]

$Assumptions={σ[_]∈(Reals),pi[_][_]∈(Reals),pip[_][_]∈(Reals),s[_]∈(Reals),ν∈(Reals),(Subscript[f, ℼ])∈(Reals),a∈(Reals)}

Useful functions

Pickℒterm[expression_,operator_]:=(Coefficient[expression,operator]/._[x]:>0) operator

Σ - Field

Σ=σ[x] (({{1, 0}, {0, 1}}))+I ((∑_i^3)(Subscript[ℼ, i])(x) PauliMatrix[i])//Expand

(SuperDagger[Σ])

Linear Sigma Model

ℒσkin=(-(1/4) Tr[(∂_({x}))Σ.(∂_({x}))(SuperDagger[Σ])])//Expand

ℒσmass=(((μ^2)/4)Tr[Σ.(SuperDagger[Σ])])//Expand

ℒσinter=-(λ/16)(Tr[Σ.(SuperDagger[Σ])]^2)//Expand

Exponential Parametrization

Σ=(ν+s[x])U[x]

I Sum[pip[i][x]PauliMatrix[i],{i,3}]
U[x_]=MatrixExp[%]
U[x_]=Sum[MatrixPower[%%,k]/k!,{k,0,2}]

ℒσkinp=(-(1/4) Tr[(∂_({x}))Σ.(∂_({x}))(SuperDagger[Σ])])//Expand
Pickℒterm[ℒσkinp,((s'(x))^2)]
Pickℒterm[ℒσkinp,((((((Subscript[ℼ, 1]^(,))))'(x)))^2)]
Pickℒterm[ℒσkinp,((((((Subscript[ℼ, 2]^(,))))'(x)))^2)]
Pickℒterm[ℒσkinp,((((((Subscript[ℼ, 3]^(,))))'(x)))^2)]

ℒσmassp=(((μ^2)/4)Tr[Σ.(SuperDagger[Σ])])//Expand
Pickℒterm[ℒσmassp,((s[x])^2)]
Pickℒterm[ℒσmassp,(((Subscript[ℼ, 1]^(,))(x))^2)]

Nonlinear Sigma Model

((I )/(Subscript[f, ℼ])) Sum[pip[i][x]PauliMatrix[i],{i,3}];
U[x_]=Sum[MatrixPower[%,k]/k!,{k,0,2}]

ℒNLσ=((-((Subscript[f, ℼ])^2))/4)Tr[(∂_x)U[x].(∂_x)(SuperDagger[U[x]])]//Expand
Pickℒterm[ℒNLσ,((((((Subscript[ℼ, 1]^(,))))'(x)))^2)]
Pickℒterm[ℒNLσ,((((((Subscript[ℼ, 2]^(,))))'(x)))^2)]
Pickℒterm[ℒNLσ,((((((Subscript[ℼ, 3]^(,))))'(x)))^2)]

SO(N) vector model

Quit[]

Notation

<<Notation`

Notation[(NotationTemplateTag[(ϕ^(i_))]) ⟺ (NotationTemplateTag[ϕ[i_]])]
Notation[(NotationTemplateTag[(Subscript[ϕ, 0]^(i_))]) ⟺ (NotationTemplateTag[ϕ0[i_]])]
Notation[(NotationTemplateTag[(ℼ^(i_))]) ⟺ (NotationTemplateTag[ℼ[i_]])]

$Assumptions={ϕ[_][_]∈(Reals),ν∈(Reals),μ∈(Reals),λ∈(Reals),σ[_]∈(Reals),ℼ[_][_]∈(Reals)}

Useful functions

Pickℒterm[expression_,operator_]:=(Coefficient[expression,operator]/._[x]:>0) operator

Lagrangian

ℒkin=-(1/2)Sum[((∂_x)ϕ[i][x])((∂_x)ϕ[i][x]),{i,1,ℕ}]

ℒpot=((μ^2)/2)Sum[(ϕ[i][x])(ϕ[i][x]),{i,1,ℕ}]-(λ/4)(((Sum[(ϕ[i][x])(ϕ[i][x]),{i,1,ℕ}]))^2)

Vacuum

(Subscript[ϕ, 0])=={0,0,... ,0,ν}

ϕ0[i_]:=If[i==ℕ,ν,0]

ϕ[i_][x_]:=If[i==ℕ,ν+σ[x],ℼ[i][x]]

ℒkin//.ℕ->3

ℒpot//.ℕ->3
%//Expand
Pickℒterm[%,((σ[x])^2)]/.(Solve[(ν^2)==(μ^2)/λ,λ][[1]])