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1 1 Chapter 1 Why supper symmetry?

2 2 Chapter 2 Representaions of the supersymmetry algebra SUSY Q a d 3 xj 0 α J x µjµ = 0 µ SUSY ( {Q A α,q βb } = 2σ µ α β P µδ A B (2.1 {Q A α,q βb } = {Q αa,q βb } = 0 [P µ,q A α ] = [P µ,q αa ] = 0 [P µ,p n ] = 0 Q SUSY P α,β α β µ,ν A,B ( SUSY N=1 SUSY A B 2.1 SUSY boson fermion N F +1-1 ( 1 NF Q α ( 1 NF Q α state = Q α ( 1 NF state (2.2 Q state ( 1 NF -1 (2.3 = ( 1 NF Q α boson = ( 1 NF fermion = 1 fermion ( 1 NF = Q α ( 1 NF boson

3 3 = Q α boson ( 1 NF = fermion ( 1 NF boson fermion Tr[( 1 NF {Q A α,q βb }] = Tr[( 1 NF (Q A αq βb +Q βb Q A α] 2 (2.3 {Q A α,q βb } = 2σ µ α β P µδ A B = Tr[ Q A α ( 1NF Q βb +Q A α ( 1NF Q βb ] = 0 (2.3 0 = Tr[( 1 NF {Q A α,q βb }] = 2σ µ α β δa B Tr[( 1NF P µ ] Tr[( 1 NF ] = 0 fermion boson susy boson fermion

4 4 Chapter 3 Component fileds spinor spinor ( ψ 4d ψ γ 0 = ψ 4d = ( ψα η β ψα T (η β ( 0 1 ( T = (η β ( 1 0 T ψα T η β ψ α 2 (SL(2,C ( 4! ψ 4d ( η β ψ α ψ 4d = ( ψα η β (ψ α ψ α (η α η α 2 spinor ( ψ 4d ψ 4d = η α ψ β ( ψ α η β = η α ψ α +ψ βη β η α ψ α η α ψ α ψ α η α ψ α η α \ / (3.1 SL(2,C SU(2 ψ 1 η 2 ψ 2 η 1 = ( ψ 1 ψ 2 ( ( η1 η 2 (3.1 ( ψ 1 ( a b ψ 2 b a ( ( a b b a ( η1 η 2 = ( ψ 1 ψ 2 ( 0 a 2 + b 2 a 2 b 2 0 ( η1 η 2

5 5 = ( ψ 1 ψ 2 ( ( η1 η 2 SU(2 ( a 2 + b 2 = 1 ( ( ε ab = εȧḃ = ε 1 0 ab = εȧḃ = ( ψ α = ε αβ ψ β ψ α = ε αβ ψ β (3.2 αβ fermion -1 (3.1 η α ψ α +ψ βη β = ε αβ η β ψ a +ε β α ψ α η β ψη ( ( ( 0 1 ψ1 0 1 = (η 1 η ψ (η 1 η ( ψ 1 ψ 2 = (η 1 ψ 2 η 2 ψ 1 ( η 1 ψ 2 +η 2 ψ 1 σ αα = ε α βε αβ σ β β = ε αβ σ β βε β α = σ α α (3.3-1 σ ( (σ α β σ β α 3.2 Algebra susy parameter space super filed component fileld component filed component filed susy multiplet A(x ψ(x ψ(x տ ւ multiplet SUSY F(x A(x,ψ(x component susy parameter odd {ξ α,ξ β } = {ξ α,q β } = 0 [P µ,ξ α ] = 0 odd even SUSY [ξ α Q α, ξ βq β ] = ξ α Q α ξ βq β ξ βq β ξ α Q α = ξ α Q α Q β ξ β ξ α Q β Qα ξ β

6 6 = ξ α Q α ε β δq δε β λ ξ λ ε αδ ξ δ Q βε αλ Q λ ξ β ε β δε β λ = ε δ βε β λ = δ δ λ ε αδ ε αλ = ε δα ε αλ = δ λ δ = ξ α Q α Q δ ξ λ +ξ α Q βq λ ξ β = ξ α β {Q α,q δ}ξ = ξ α 2σ µ α β P µξ β [ξ α Q α,ξ β Q β ] = ξ α Q α ξ β Q β ξ β Q β ξ α Q α = ξ α Q α Q β ξ β ξ α Q β Q α ξ β = ξ α {Q α,q β }ξ β = 0 [ξ α Q α,ξ βq β ] = 0 [P µ,ξ α Q α ] = [P µ,ξ α Q. α ] = 0 ψ α η α ψ α η α [a,b]=ab-ba -1 odd [ξq,ξq] = 2ξσ µ ξp µ [ξq,ξq] = [ξq,ξq] = [P µ,ξq] = [P µ,ξq] = component filed SUSY susy mass susy component filed ( componentfiled component filed multiplet (A,ψA scalar filed ψ spinor filed ( ( ( ( A e ψ (ξq+ξq A e = (ξq+ξq A (1+(ξQ+ξQA = ψ e (ξq+ξq ψ (1+(ξQ+ξQψ SUSY SUSY A [e (ξq+ξq e (ηq+ηq e (ηq+ηq e (ξq+ξq ]A = [(1+(ξQ+ξQ(1+(ηQ+ηQ (1+(ηQ+ηQ(1+(ξQ+ξQ]A = [(1+ξQ+ξQ+ηQ+ηQ+(ξQ+ξQ(ηQ+ηQ (1+ξQ+ξQ+ηQ+ηQ+(ηQ+ηQ(ξQ+ξQ]A = [ξq,ηq]a+[ξq,ηq]a+[ξq,ηq]a+[ξq,ηq]a = (2P µ ξσ µ η 2P µ ησ µ ξa (3.4

7 7 2 susy 2 susy mass 1 Q mass 1/2 Q mass 1/2 scalar filed A spinor filed ψ δ ξ A (ξq+ξqa 2ξψ δ ξ ψ = (ξq+ξqψ = 2σ µ ξp µ A+ 2ξF ξqa 2ξψ (3.5 ξqψ 2ξF ξqa 0 ξqψ 2σ µ ξp µ A F vector filed mass 1 A ψ A ψ mass 1 F F (3.4 [e (ξq+ξq e (ηq+ηq e (ηq+ηq e (ξq+ξq ]ψ = [ξq,ηq]ψ +[ξq,ηq]ψ = ξqηqψ ηqξqψ +ξqηqψ ηqξqψ = ξqη 2σ µ P µ A ηqξ 2F +ξqη 2F ηqξ 2σ µ P µ A = ξqη 2σ µ P µ A ηqξ 2F +ξqη 2F ηqξ 2σ µ P µ A δ ξ F = 2ξσ µ P µ ψ (3.6 Dirc spinor ( ( ( L = ψ 4d (i µ γ µ µψ 4d = η α ψ β i 0 (σ µ a β (σ µ βα 0 µ µ ( ψα η β = 0 L = ψ βi(σ µ βα µ ψ α +η α (i(σ µ α β µ η β µη α ψ α µψ βη β ( ψ,η globalu(1 L U(1 R symmmetry L ψ i(σ µ βα µ ψ α +µη β = 0 η i(σ µ α β µ η β µψ α = 0

8 8 odd (AB = A B AB = A B = AB spinor i(σ µ βα µ ψ α +µψ β = 0 F (3.6 δ ξ F = i 2ξ β(σ µ βα µ ψ α = 2µξ βψ β (3.5 δ ξ F = (ξ α Q α +ξ α Q α F = 2µξ βψ β F = ma (

9 9 ξ α Q α A 2ξ α ψ α (3.7 ξ α Q α A 0 (3.8 (3.8 Q α ξ α A = 0 (3.7 Q α ξ α A = ξ α Q α A = 2ξ α ψ α (ξ α Q α +ξ α Q α ψ β 2ξ β F +i 2ξ α (σ µ αβ µ A (3.9 (ξ α Q α +ξ α Q α ψ β 2ξ α ε αβ F i 2(σ µ β α ξ α µ A (3.10 (3.9 (ξ α Q α +ξ α Q α ψ β = 2ξ βf i 2((σ µ αβ ξ α µ A (3.10 i 2((σ µ βα ξ α µ A + 2ξ β F (ξ α Q α +ξ α Q α ψ β = 2ξ α ε α βf +i 2ξ α ((σ µ β α µ A = 2ξ α ε α βf +i 2ξ α (σ µ α β µ A (ξ α Q α +ξ α Q α F i 2ξ α (σ µ αβ µ ψ β (3.11 (3.11 (ξ α Q α +ξ α Q α F = i 2 µ ψ β((σ µ αβ ξ α = i 2 µ ψ β(σ µ βα ξ α mass [A] = 1 [ψ] = 3 2 [F] = [ µa ] = 2 [ µ ψ] = susy mass 2 m mass mass component filed 3.4 SUSY (component filed component filed free susy L = i µ ψσ µ ψ +A A+F F +m(af +A F 1 2 ψψ 1 2 ψψ susy susy

10 10 iσ µ µ ψ +mψ = 0 F +ma = 0 A+mF = SUSY fermion boson fermion :spinor 2 boson :scalar A,A* ψ 2 ψ,ψ L =: L : fermion boson

11 11 Chapter 4 Super fields super filed component field susy super field F(x,θ,θ F(x,θ,θ f(x+θφ(x+θχ(x+θθm(x+θθn(x+θσ µ θv µ (x+θθθλ(x+θθθψ(x+θθθθd(x θ,θ component field super F θ,θ odd 2 spinor 2 θθθ super field super space super space x,θ,θ susy G(x,θ,θ exp( ix µ p µ +iθq+iθq (4.1 susy θ,θ Hausdroff s formulae A e B = e A+B+1[A,B]+ 2 G(0,ξ,ξG(x,θ,θ = e (iξq+iξq e ( ixµ p µ+iθq+iθq = e (iξq+iξq ixµ p µ+iθq+iθq+ 1 2 [iξq+iξq, ixµ p µ+iθq+iθq] = e (iξq+iξq ixµ p µ+iθq+iθq+ 1 2 [iξq+iξq,iθq+iθq] = e (iξq+iξq ixµ p µ+iθq+iθq+ 1 2 [iξq,iθq]+1 2 [iξq,iθq] = e (iξq+iξq ixµ p µ+iθq+iθq ξσ µ θp µ+θσ µ ξp µ = e (iξq+iξq+iθq+iθq ipµ(xµ +iξσ µ θ iθσ µ ξ = G(x µ +iθσ µ ξ iξσ µ θ,θ +ξ,θ +ξ susy super space g(ξ,ξ : (x µ,θ,θ (x µ +iθσ µ ξ iξσ µ θ,θ+ξ,θ +ξ (4.2 susy θ,θ susy super space ξq+ξq = ξ α ( θ α iσµ α α θ α µ +ξ α ( θ α iθ α σ µ α β ε β α µ σ α βξ β = ξ α σ α βε β α Q α Q α Q α Q α { } Qα,Q α = 2iσ µ α α µ (4.3 { } {Q α,q β } = = 0 Q α,q β Q ε α β = θ α θ β Q diff α θ α iσµ α α θ α µ g(ξ,ξ e ξqdiff diff +ξq

12 = Q α diff θ α +iθα σ µ α α µ { } { Q diff α,q diff α f(x,θ,θ = θ α iσµ α α θ α µ, } θ α +iθα σ µ α α µ f(x,θ,θ ( { } { } { θ α, θ α +i θ α,θα σ µ α α µ +i σ µ α α θ α µ, = θ α ( { } { i θ α,θα σ µ α α µ +i σ µ α α θ α µ, ( = i(σ µ α α µ θ α σ µ α α µ θ α +θα σ µ α α µ θ α +i ( { = iσ µ α α µ +i σ µ α α θ α µ, θ α } { + σ µ α α θ α µ,θ α σ µ α α µ} f(x,θ,θ θ α } f(x,θ,θ { σ µ α α θ α µ, } f(x,θ,θ = 2iσ µ α α µf(x,θ,θ = 2σ µ α α P µf(x,θ,θ θ α } f(x,θ,θ! [2.1] SUSY SUSY G(0,ξ,ξ [4.1] g(ξ,ξ [4.2] G(0,ξ 1,ξ 1 G(0,ξ 2,ξ 2 = g(ξ 2,ξ 2 g(ξ 1,ξ 1 ( SUSY [2.1] D α θ α +iσµ α α θ α µ D α θ α iθα σ µ α α µ SUSY { { },Dα,D α f(x,θ,θ = θ α +iσµ α α θ α µ, } θ α iθα σ µ α α µ f(x,θ,θ = ( { } { i θ α,θα σ µ α α µ i σ µ α α θ α µ, θ α = 2iσ µ α α µf(x,θ,θ = 2σ µ α α P µf(x,θ,θ } f(x,θ,θ SUSY super space supaer field Super field θθ componetn field 12 F(x,θ,θ f(x+θφ(x+θχ(x+θθm(x+θθn(x+θσ µ θv µ (x+θθθλ(x+θθθψ(x+θθθθd(x super field susy component field susy δ ξ F(x,θ,θ (ξq+ξqf(x,θ,θ δ ξ f(x+θδ ξ φ(x+θδ ξ χ(x+θθδ ξ m(x+θθδ ξ n(x+θσ µ θδ ξ v µ (x+θθθδ ξ λ(x+θθθδ ξ ψ(x+θθθθδ ξ d(x super filed

微分積分サンプルページこの本の定価判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.

微分積分サンプルページこの本の定価判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです. 微分積分サンプルページこの本の定価判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)