リー環の拡大
に同型である,ただし C[λ, λ] はローラン多項式の代数であり,
∑ λ k G k ↔ ∑ λ k ⊗ G k {\displaystyle \sum \lambda ^{k}G_{k}\leftrightarrow \sum \lambda ^{k}\otimes G_{k}}
と対応する.リーブラケットは
[ P ( λ ) ⊗ G 1 , Q ( λ ) ⊗ G 2 ] = P ( λ ) Q ( λ ) ⊗ [ G 1 , G 2 ] {\displaystyle [P(\lambda )\otimes G_{1},Q(\lambda )\otimes G_{2}]=P(\lambda )Q(\lambda )\otimes [G_{1},G_{2}]}
である.この後者の視点により元は(定数!)係数が g の多項式と考えることができる.基底と構造定数のことばでは,
[ λ m ⊗ G i , λ n ⊗ G j ] = C i j k λ m + n ⊗ G k {\displaystyle [\lambda ^{m}\otimes G_{i},\lambda ^{n}\otimes G_{j}]={C_{ij}}^{k}\lambda ^{m+n}\otimes G_{k}}
である.異なる表記
λ m ⊗ G i ≅ λ m G i ↔ T i m ( λ ) ≡ T i m {\displaystyle \lambda ^{m}\otimes G_{i}\cong \lambda ^{m}G_{i}\leftrightarrow T_{i}^{m}(\lambda )\equiv T_{i}^{m}}
[ φ k ( t , x ) , π l ( t , x ) ] = i δ ( x − y ) δ k l , [ φ k ( t , x ) , φ l ( t , x ) ] = [ π k ( t , x ) , π l ( t , x ) ] = 0. {\displaystyle {\begin{aligned}{}[\phi _{k}(t,x),\pi ^{l}(t,x)]&=i\delta (x-y)\delta _{k}^{l},\\{}[\phi _{k}(t,x),\phi _{l}(t,x)]&=[\pi ^{k}(t,x),\pi ^{l}(t,x)]=0.\end{aligned}}}
The currents accordingly become operators They satisfy, using the above postulated relations, the definitions and integration over space, the commutation relations
where S10 and Heisenberg's equation of motion have been used as well as H|0⟩ = 0 and its conjugate.
Multiply this equation by f(x)f(y) and integrate with respect to x and y over all space, using integration by parts, and one finds
− i ∫∫ d x d y ⟨ 0 | C i ( x , y ) | 0 ⟩ f ( x ) ∂ f ∂ y i f ( x ) = 2 ⟨ 0 | F H F | ⟩ , F = ∫ J 0 ( x ) f ( x ) . {\displaystyle -i\iint d\mathbf {x} d\mathbf {y} \langle 0|C^{i}(\mathbf {x} ,\mathbf {y} )|0\rangle f(\mathbf {x} ){\frac {\partial f}{\partial y^{i}}}f(\mathbf {x} )=2\langle 0|FHF|\rangle ,\quad F=\int J^{0}(\mathbf {x} )f(\mathbf {x} ).
なので Πex(Gex) は Gex の本物の表現である.
ウィグナーの定理の文脈では,状況をそのようなものとして描写できる(C* を U(1) でおきかえる);SH でヒルベルト空間 H における単位球面を表し,(·, ·) をその内積とする.PH で ray space(英語版) を表し,[·, ·] で ray product(英語版) を表す.さらに波矢印で群作用を表す.すると図式
は可換である,すなわち
π 2 ∘ Π e x ( ( λ , g ) ) ( ψ ) = Π ∘ π ( g ) ( π 1 ( ψ ) ) , ψ ∈ S H {\displaystyle \pi _{2}\circ \Pi _{\mathrm {ex} }((\lambda ,g))(\psi )=\Pi \circ \pi (g)(\pi _{1}(\psi )),\quad \psi \in S{\mathcal {H}}}
である.さらに,G が [·,·] を保つ PH の対称性であるのと同様に,Gex は (·,·) を保つ SH の対称性である.π2 のファイバーはすべて円である.これらの円は U(1) の作用で不変である.これらのファイバーへの U(1) の作用は推移的で固定点がない.結論は,SH は PH 上の主ファイバー束で,構造群は U(1) である.
h k ( λ ) = ∑ n = − ∞ ∞ θ − n k λ n {\displaystyle h^{k}(\lambda )=\sum _{n=-\infty }^{\infty }\theta _{-n}^{k}\lambda ^{n}}
と書いて
e h k ( λ ) G k = 1 G + ∑ n = − ∞ ∞ θ − n k λ n G k + ⋯ {\displaystyle e^{h^{k}(\lambda )G_{k}}=1_{G}+\sum _{n=-\infty }^{\infty }\theta _{-n}^{k}\lambda ^{n}G_{k}+\dotsb }
を得る.したがって関数
h : S 1 → g ; h ( λ ) = ∑ n = − ∞ ∞ ∑ k = 1 K θ − n k λ n G k ≡ ∑ n = − ∞ ∞ λ n G n {\displaystyle h\colon S^{1}\to {\mathfrak {g}};\;h(\lambda )=\sum _{n=-\infty }^{\infty }\sum _{k=1}^{K}\theta _{-n}^{k}\lambda ^{n}G_{k}\equiv \sum _{n=-\infty }^{\infty }\lambda ^{n}G_{n}}
はリー環を構成する.
少し考えるとこれらは θ が 0 から 2π まで行くとき g 内のループであることが確かめられる.演算は g の演算によって点ごとに定義されるものである.この代数は代数
C [ λ , λ − 1 ] ⊗ g {\displaystyle C[\lambda ,\lambda ^{-1}]\otimes {\mathfrak {g}}}
⟨ 0 | ∂ C i ( x , y ) ∂ y i | 0 ⟩ = ⟨ 0 | [ J 0 ( t , x ) , ∂ J i ( t , y ) ∂ y i ] | 0 ⟩ = − ⟨ 0 | [ J 0 ( t , x ) , ∂ J 0 ( t , y ) ∂ t ] | 0 ⟩ = i ⟨ 0 | [ J 0 ( t , x ) , [ J 0 ( t , y ) , H ] ] | 0 ⟩ = − i ⟨ 0 | J 0 ( t , x ) H J 0 ( t , y ) + J 0 ( t , x ) H J 0 ( t , x ) | 0 ⟩ , {\displaystyle {\begin{aligned}\langle 0|{\frac {\partial C^{i}(\mathbf {x} ,\mathbf {y} )}{\partial _{y^{i}}}}|0\rangle &=\langle 0|[J^{0}(t,\mathbf {x} ),{\frac {\partial J^{i}(t,\mathbf {y} )}{\partial _{y^{i}}}}]|0\rangle \\&=-\langle 0|[J^{0}(t,\mathbf {x} ),{\frac {\partial J^{0}(t,\mathbf {y} )}{\partial _{t}}}]|0\rangle =i\langle 0|[J^{0}(t,\mathbf {x} ),[J^{0}(t,\mathbf {y} ),H]]|0\rangle \\&=-i\langle 0|J^{0}(t,\mathbf {x} )HJ^{0}(t,\mathbf {y} )+J^{0}(t,\mathbf {x} )HJ^{0}(t,\mathbf {x} )|0\rangle ,\end{aligned}}}
The quantities x0 and p0 are (roughly) string position from the initial condition and string momentum. If all the αμn are zero, the solution represents the motion of a classical point particle.
This is rewritten, first defining
α 0 μ = 2 α ′ a μ , α n μ = a n μ n , α − n μ = a n μ ∗ n , {\displaystyle \alpha _{0}^{\mu }={\sqrt {2\alpha '}}a_{\mu },\quad \alpha _{n}^{\mu }=a_{n}^{\mu }{\sqrt {n}},\quad \alpha _{-n}^{\mu }=a_{n}^{\mu *}{\sqrt {n}},}
and then writing
X μ ( σ , τ ) = x 0 μ + 2 α ′ α 0 μ τ + i 2 α ′ ∑ n ≠ 0 1 n α n μ e − i n τ cos  n σ . {\displaystyle X^{\mu }(\sigma ,\tau )=x_{0}^{\mu }+{\sqrt {2\alpha '}}\alpha _{0}^{\mu }\tau +i{\sqrt {2\alpha '}}\sum _{n\neq 0}{\frac {1}{n}}\alpha _{n}^{\mu }e^{-in\tau }\cos n\sigma .}
カレント代数は場の量子論において大域的ゲージ対称性の結果として生じる. Conserved currents occur in classical field theories whenever the Lagrangian respects a continuous symmetry. This is the content of Noether's theorem. Most (perhaps all) modern quantum field theories can be formulated in terns of classical Lagrangians (prior to quantization), so Noether's theorem applies in the quantum case as well. Upon quantization, the conserved currents are promoted to position dependent operators on Hilbert space. These operators are subject to commutation relations, generally forming an infinite-dimensional Lie algebra. A model illustrating this is presented below.
1 L ∑ n = − ∞ ∞ e − 2 π i n z L = 1 L ∑ n = − ∞ ∞ δ ( z + n L ) = δ ( z ) {\displaystyle {\frac {1}{L}}\sum _{n=-\infty }^{\infty }e^{\frac {-2\pi inz}{L}}={\frac {1}{L}}\sum _{n=-\infty }^{\infty }\delta (z+nL)=\delta (z)}
for z in the interval (0, L) and differentiate it to yield
− 2 π i L 2 ∑ n = − ∞ ∞ n e − 2 π i n z L = δ ′ ( z ) , {\displaystyle -{\frac {2\pi i}{L^{2}}}\sum _{n=-\infty }^{\infty }ne^{\frac {-2\pi inz}{L}}=\delta '(z),}
and finally
[ J a ( x − y ) , J b ( 0 ) ] = i ħ C a b c J c ( x − y ) δ ( x − y ) + i ħ 2 2 π δ a b C δ ′ ( x − y ) , {\displaystyle [J_{a}(x-y),J_{b}(0)]=i\hbar {C_{ab}}^{c}J_{c}(x-y)\delta (x-y)+{\frac {i\hbar ^{2}}{2\pi }}\delta _{ab}C\delta '(x-y),}
or
であることを示すことができる.ここで η は g 上の 2-コサイクルである.この 2-コサイクルはおおいに非自明な方法ではあるが ω から得ることができる
さて射影表現 Π を用いて写像 Πex を
Π e x ( ( λ , g ) ) = λ Π ( g ) {\displaystyle \Pi _{\mathrm {ex} }((\lambda ,g))=\lambda \Pi (g)}
で定義できる.それは次の性質を持つ:
Π e x ( ( λ 1 , g 1 ) ) Π e x ( ( λ 2 , g 2 ) ) = λ 1 λ 2 Π ( g 1 ) Π ( g 2 ) = λ 1 λ 2 ω ( g 1 , g 2 ) Π ( g 1 g 2 ) = Π e x ( λ 1 λ 2 ω ( g 1 , g 2 ) , g 1 g 2 ) = Π e x ( ( λ 1 , g 1 ) ( λ 2 , g 2 ) ) . {\displaystyle {\begin{aligned}\Pi _{\mathrm {ex} }((\lambda _{1},g_{1}))\Pi _{\mathrm {ex} }((\lambda _{2},g_{2}))\\&=\lambda _{1}\lambda _{2}\Pi (g_{1})\Pi (g_{2})=\lambda _{1}\lambda _{2}\omega (g_{1},g_{2})\Pi (g_{1}g_{2})\\&=\Pi _{\mathrm {ex} }(\lambda _{1}\lambda _{2}\omega (g_{1},g_{2}),g_{1}g_{2})=\Pi _{\mathrm {ex} }((\lambda _{1},g_{1})(\lambda _{2},g_{2})).\end{aligned}}}
[ J a 0 ( t , x ) , J b i ( t , y ) ] = i C a b c J c i ( t , x ) δ ( x − y ) + S a b i j ∂ j δ ( x − y ) + ⋯ . {\displaystyle [J_{a}^{0}(t,\mathbf {x} ),J_{b}^{i}(t,\mathbf {y} )]=i{C_{ab}}^{c}J_{c}^{i}(t,\mathbf {x} )\delta (\mathbf {x} -\mathbf {y} )+S_{ab}^{ij}\partial _{j}\delta (\mathbf {x} -\mathbf {y} )+\dotsb .}
The presence of the delta functions and their derivatives is explained by the requirement of microcausality that implies that the commutator vanishes when x ≠ y. Thus the commutator must be a distribution supported at x = y. The first term is fixed due to the requirement that the equation should, when integrated over X, reduce to the last equation before it.