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Nonlinear realization
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<p>In mathematical physics, nonlinear realization of a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> <i>G</i> possessing a <a href="/facts/Cartan_subgroup/4mDR1t5Y">Cartan subgroup</a> <i>H</i> is a particular <a href="/facts/Induced_representation/Jh25PGDP">induced representation</a> of <i>G</i>. In fact, it is a representation of a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> g {\displaystyle {\mathfrak {g}}} of <i>G</i> in a neighborhood of its origin. A nonlinear realization, when restricted to the subgroup <i>H</i> reduces to a linear representation. </p><p>A nonlinear realization technique is part and parcel of many <a href="/facts/Field_theory_(physics)/FJdAKeRA">field theories</a> with <a href="/facts/Spontaneous_symmetry_breaking/VmXqNLlo">spontaneous symmetry breaking</a>, e.g., <a href="/facts/Chiral_model/fL5SrTxa">chiral models</a>, <a href="/facts/Chiral_symmetry_breaking/bMsTTLKw">chiral symmetry breaking</a>, <a href="/facts/Goldstone_boson/zGBpWRYU">Goldstone boson</a> theory, <a href="/facts/Higgs_field_(classical)/2mrn1nDu">classical Higgs field theory</a>, <a href="/facts/Gauge_gravitation_theory/wwRu94cD">gauge gravitation theory</a> and <a href="/facts/Supergravity/jeNk3aAB">supergravity</a>. </p><p>Let <i>G</i> be a Lie group and <i>H</i> its Cartan subgroup which admits a linear representation in a vector space <i>V</i>. A Lie algebra g {\displaystyle {\mathfrak {g}}} of <i>G</i> splits into the sum g = h ⊕ f {\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {f}}} of the <a href="/facts/Cartan_subalgebra/8gymnPU0">Cartan subalgebra</a> h {\displaystyle {\mathfrak {h}}} of <i>H</i> and its supplement f {\displaystyle {\mathfrak {f}}} , such that </p> [ f , f ] ⊂ h , [ f , h ] ⊂ f . {\displaystyle [{\mathfrak {f}},{\mathfrak {f}}]\subset {\mathfrak {h}},\qquad [{\mathfrak {f}},{\mathfrak {h}}]\subset {\mathfrak {f}}.} <p>(In physics, for instance, h {\displaystyle {\mathfrak {h}}} amount to vector generators and f {\displaystyle {\mathfrak {f}}} to axial ones.) </p><p>There exists an open neighborhood <i>U</i> of the unit of <i>G</i> such that any element g ∈ U {\displaystyle g\in U} is uniquely brought into the form </p> g = exp ( F ) exp ( I ) , F ∈ f , I ∈ h . {\displaystyle g=\exp(F)\exp(I),\qquad F\in {\mathfrak {f}},\qquad I\in {\mathfrak {h}}.} <p>Let U G {\displaystyle U_{G}} be an open neighborhood of the unit of <i>G</i> such that U G 2 ⊂ U {\displaystyle U_{G}^{2}\subset U} , and let U 0 {\displaystyle U_{0}} be an open neighborhood of the <i>H</i>-invariant center σ 0 {\displaystyle \sigma _{0}} of the quotient <i>G/H</i> which consists of elements </p> σ = g σ 0 = exp ( F ) σ 0 , g ∈ U G . {\displaystyle \sigma =g\sigma _{0}=\exp(F)\sigma _{0},\qquad g\in U_{G}.} <p>Then there is a local section s ( g σ 0 ) = exp ( F ) {\displaystyle s(g\sigma _{0})=\exp(F)} of G → G / H {\displaystyle G\to G/H} over U 0 {\displaystyle U_{0}} . </p><p>With this local section, one can define the <a href="/facts/Induced_representation/Jh25PGDP">induced representation</a>, called the nonlinear realization, of elements g ∈ U G ⊂ G {\displaystyle g\in U_{G}\subset G} on U 0 × V {\displaystyle U_{0}\times V} given by the expressions </p> g exp ( F ) = exp ( F ′ ) exp ( I ′ ) , g : ( exp ( F ) σ 0 , v ) → ( exp ( F ′ ) σ 0 , exp ( I ′ ) v ) . {\displaystyle g\exp(F)=\exp(F')\exp(I'),\qquad g:(\exp(F)\sigma _{0},v)\to (\exp(F')\sigma _{0},\exp(I')v).} <p>The corresponding nonlinear realization of a Lie algebra g {\displaystyle {\mathfrak {g}}} of <i>G</i> takes the following form. </p><p>Let { F α } {\displaystyle \{F_{\alpha }\}} , { I a } {\displaystyle \{I_{a}\}} be the bases for f {\displaystyle {\mathfrak {f}}} and h {\displaystyle {\mathfrak {h}}} , respectively, together with the commutation relations </p> [ I a , I b ] = c a b d I d , [ F α , F β ] = c α β d I d , [ F α , I b ] = c α b β F β . {\displaystyle [I_{a},I_{b}]=c_{ab}^{d}I_{d},\qquad [F_{\alpha },F_{\beta }]=c_{\alpha \beta }^{d}I_{d},\qquad [F_{\alpha },I_{b}]=c_{\alpha b}^{\beta }F_{\beta }.} <p>Then a desired nonlinear realization of g {\displaystyle {\mathfrak {g}}} in f × V {\displaystyle {\mathfrak {f}}\times V} reads </p> F α : ( σ γ F γ , v ) → ( F α ( σ γ ) F γ , F α ( v ) ) , I a : ( σ γ F γ , v ) → ( I a ( σ γ ) F γ , I a v ) , {\displaystyle F_{\alpha }:(\sigma ^{\gamma }F_{\gamma },v)\to (F_{\alpha }(\sigma ^{\gamma })F_{\gamma },F_{\alpha }(v)),\qquad I_{a}:(\sigma ^{\gamma }F_{\gamma },v)\to (I_{a}(\sigma ^{\gamma })F_{\gamma },I_{a}v),} , F α ( σ γ ) = δ α γ + 1 12 ( c α μ β c β ν γ − 3 c α μ b c ν b γ ) σ μ σ ν , I a ( σ γ ) = c a ν γ σ ν , {\displaystyle F_{\alpha }(\sigma ^{\gamma })=\delta _{\alpha }^{\gamma }+{\frac {1}{12}}(c_{\alpha \mu }^{\beta }c_{\beta \nu }^{\gamma }-3c_{\alpha \mu }^{b}c_{\nu b}^{\gamma })\sigma ^{\mu }\sigma ^{\nu },\qquad I_{a}(\sigma ^{\gamma })=c_{a\nu }^{\gamma }\sigma ^{\nu },} <p>up to the second order in σ α {\displaystyle \sigma ^{\alpha }} . </p><p>In physical models, the coefficients σ α {\displaystyle \sigma ^{\alpha }} are treated as <a href="/facts/Goldstone_boson/zGBpWRYU">Goldstone fields</a>. Similarly, nonlinear realizations of <a href="/facts/Lie_superalgebra/c1oxUoEl">Lie superalgebras</a> are considered. </p>