成蹊大学理工学研究報告
J. Fac. Sci.Tech., Seikei Univ.
Vol.45 No.1 (2008)pp.43-48

ύϥ౷‫͏͕ͨ͠ʹܭ‬௒ରশͳྔࢠྗֶ໛‫ܕ‬ҏ౻Ү෉ ∗A Supersymmetric Model of Quantum Mechanics with Parastatistical Variables
Ikuo Ito*
AbstractA supersymmetric model of quantum mechanics with a bose-like and a fermi-like variables isconstructed. Generalizing the procedure for constructing a supersymmetric oscillator model, thesupercharge Q is deﬁned by including a superpotential which is an odd function of the coordinate
variable q. The Hamiltonian H is obtained by calculating an anticommutator of the supercharge
Q and Q† .
Keywords: Supersymmetry, Parastatistics, Quantum Model ɹ(Received March 24, 2008)1ʹ͕ͨ͠͏͕ɼࡾ߲ؒͷ୅਺

Introduction[f, [f † , f ] ] = 2f௒ରশੑ͸Ϙʔζ౷‫ࣗ͏͕ͨ͠ʹܭ‬༝౓ͱϑΣϧϛ౷‫ʹܭ‬͕ͨ͠͏ࣗ༝౓ͱͷؒͷରশੑͰ͋Δɻ΋ͱ΋ͱ͸ɼૉཻࢠ͚ͩΛԾఆͯ͠΍͸Γྔࢠ࿦Λߏ੒͢Δ͜ͱ͕Ͱ͖Δɻ͜ͷͷଟ༷ͳϘκϯͱϑΣϧϛΦϯͷؒͷରশੑͱͯ͠ߟ͑ΒΕΑ͏ͳࣗ༝౓ΛϑΣϧϛతม਺ͱ͍͏ɻϑΣϧϛతม਺͋Δ

ͨ [1],[2],[3] ͕ɼۙ೥Ͱ͸௒ରশͳ৔ͷཧ࿦͕ૉཻࢠͷඪ४͍͸Ϙʔζతม਺ʹ͕ͨ͠͏ෳ਺ͷࣗ༝౓Λߟ͑ͨͱ͖ɼͦ໛‫ܕ‬Λ௒͑Δ໛‫ܕ‬ͷ༗ྗͳީิͱͯ͠‫ݕ‬౼͞Ε͍ͯΔ [4]ɻΕΒͷ͕ͨ͠͏౷‫ܭ‬Λ௨ৗͷϘʔζ౷‫ܭ‬ɼϑΣϧϛ౷‫۠ͱܭ‬ྔࢠྗֶͷ໛‫Ͱܕ‬௒ରশੑΛௐ΂Δ͜ͱ͸ɼͱ͘ʹͦͷผͯ͠ύϥ౷‫[ ͏͍ͱܭ‬6],[7]ɻҎԼͰ͸Ϙʔζతม਺ɼϑΣഁΕͷ‫ߏػ‬Λཧղ͢Δͷʹ໾ʹཱͭɻ࠷ॳʹͦͷΑ͏ͳ‫ڀݚ‬ϧϛతม਺ͷ͜ͱΛ·ͱΊͯύϥม਺ͱΑͿɻΛߦͬͨͷ͸ E.Witten Ͱ͋Δ [5]ɻ௨ৗͷ࠲ඪͱӡಈྔΛύϥม਺Λ‫ؚ‬Έ௒ରশෆมͳྔࢠྗֶͷ໛‫ܕ‬͸ɼ࠷ॳʢϘʔζม਺ʣͱάϥεϚϯ୅਺ʹ͕ͨ͠͏ϑΣϧϛม਺Λ

Rubakov ͱ Spiridonov ʹΑͬͯߏ੒͞Εͨ [8]ɻͦΕ͸Ϙʔ‫ؚ‬Ή௒ରশͳ໛‫͍͓ͯʹܕ‬ɼ௒ϙςϯγϟϧ͕ͲͷΑ͏ͳੑζม਺ͱύϥม਺Ͱ͋ΔϑΣϧϛతม਺Λ‫ؚ‬Μͩ΋ͷͰɼͦͷ࣭Λ΋ͭ΋ͷͰ͋Ε͹ɼ‫ܥ‬ͷ௒ରশੑ͕อͨΕΔ͔ɼ͋Δ͍εϖΫτϧ΍ഁΕͷύλʔϯ͕ৄ͘͠ௐ΂ΒΕͨ [9],[10],[11]ɻ

͸ (ࣗൃతʹ) ഁΕΔ͔Λٞ࿦ͨ͠ɻ͔͠͠ɼ͜ΕΒͷ໛‫Ͱܕ‬͸Ϙʔζͷࣗ༝౓͕ύϥม਺Ͱ͸௒ରশͳྔࢠྗֶ໛‫ܕ‬͸Ϙʔζతม਺ͱϑΣϧϛతม਺ͷͳ͍ͨΊʹɼ‫ͳີݫ‬ҙຯͰͷ௒ରশੑʹͳ͍ͬͯͳ͍ɻ͜Ε৔߹ʹ΋ߏ੒Ͱ͖Δɻ௨ৗͷϘʔζม਺͸[b, b† ] = 1ʹରͯ͠ɼϘʔζɼϑΣϧϛ͕͍ͣΕ΋ύϥม਺Ͱ͋Δ௒ରশͳ໛‫ܕ‬͸ RIKKYO άϧʔϓʹΑͬͯௐ΂ΒΕͨ [12]ɻͨ

(1)ͩ͠ɼ΋ͬͱ΋؆୯ͳৼಈࢠ໛‫ݶʹܕ‬ΒΕ͓ͯΓɼεϖΫτ(ҎԼͰ͸͢΂ͯ ~ = m = 1 ͷ୯Ґ‫ܥ‬ΛͱΔ) ͱ͍͏ަ‫ؔ׵‬ϧͷରশੑ͸͋ΔҙຯͰࣗ໌Ͱ͋Δɻ·ͨ௒ରশੑͷഁΕͷ܎ʹ͕ͨ͠͏͕ɼ͜Εʹର͠ɼࡾ߲ؒ୅਺
†[b, {b , b}] = 2b‫͍ͯͭʹߏػ‬͸Θ͔Βͳ͍ɻ1͜ͷϊʔτͰ͸ɼϘʔζతม਺ͱϑΣϧϛతม਺Λ‫ؚ‬Έɼ(2)ҰൠͷϙςϯγϟϧΛ࣋ͭ௒ରশྔࢠྗֶ໛‫ܕ‬Λߏ੒͢Δɻ͚ͩΛԾఆͯ͠ྔࢠ࿦Λߏ੒͢Δ͜ͱ͕Ͱ͖Δ [6],[7]ɻ͜ͷ͜ͷͨΊͷํ਑͸࣍ͷ௨ΓͰ͋Δɻ·ͣɼୈ 2 ষͰϘʔζΑ͏ͳࣗ༝౓ΛϘʔζతม਺ͱ͍͏ɻతม਺ͱϑΣϧϛతม਺ؒͷࡾ߲ؒ୅਺ؔ܎ΛɼOhunuki-·ͨɼϑΣϧϛΦϯม਺͸௨ৗͷ൓ަ‫܎ؔ׵‬{f † , f } = 1(4)Kamefuchi[7] ʹ͕ͨͬͯ͠ಋೖ͢Δɻ͜ΕΒͷؔ܎Λ΋ͱ(3)∗ ੒᪟େֶཧ޻ֶ෦‫ڞ‬௨‫ڭૅج‬त (ito@st.seikei.ac.jp)

1 ͜ΕΒͷ໛‫ܕ‬ͷ review ͸ [13] ʹ·ͱΊΒΕ͍ͯΔɻ

1

─43─

ʹͯ͠ɼୈ 3 ষҎ߱Ͱ௒ిՙ Q ͓Αͼ‫ܥ‬ͷϋϛϧτχΞϯ

[b, [f † , f ] ] = 0,H Λɼ࣍ͷ৚݅Λຬͨ͢Α͏ʹߏ੒͢Δɿ[b† , [f † , f ] ] = 0(17)͜͜Ͱ η = + ·ͨ͸ − ͰɼͦΕͧΕަ‫ [( ੵࢠ׵‬, ])ɼ൓ަ

1) ΂͖ྵੑ

‫ {( ੵࢠ׵‬, }) Λද͢΋ͷͱ͢Δɻ·ͨɼ֤ࣜͰ‫ʹ͍ޓ‬Τϧ

2Q2 = Q† = 0(5)ϛʔτ‫ڞ‬໾ͷؔ܎ʹ͋Δ΋ͷ͸ϖΞʹͯ͋͠Δɻ͜ΕΒͷࡾ߲ؒ୅਺͸ɼࣗ༝౓ b, f Λ Green ੒෼ʹల։ͨ͠ͱ͖ɼͦ2) ϋϛϧτχΞϯ†{Q, Q } = H(6)[Q, H] = 0(7)ͷ੒෼ಉ͕࢜ަ‫ͨ·܎ؔ׵‬͸൓ަ‫܎ؔ׵‬ΛΈͨ͢͜ͱɼ͓ΑͼͦΕ͕ b, f ͷࡾ߲ؒ୅਺Ͱ༩͑ΒΕΔ͜ͱɼͱ͍͏ೋͭͷ3) ௒ରশෆมੑཁ੥͔Βಋ͔ΕΔ΋ͷͰ͋Δɻ͜Ε͚ͩͷ৚͔݅Β͸ɼη ͷූ߸͸Ұҙతʹܾ·Βͳ͍ɻ(8)-(12) ͓Αͼ (13)-(17) ͸ɼ͢΂͕ͯಠཱͳؔ܎ͱ͍͏(7) ͸ Q ͱ H ͕ (5) ͱ (6) Λຬͨ͢Α͏ʹߏ੒͞Ε͍ͯΕ͹ɼࣗಈతʹຬͨ͞ΕΔɻ1)-3) Λ Q, Q ͓Αͼ H ͕ͭ͘Θ͚Ͱ͸ͳ͍ɻͨͱ͑͹ɼ(10) ͱ (11) ͷؔ܎͔ࣜΒ (12) ͕Δ௒ରশ୅਺ͱ͍͏ɻୈ 3 ষͰ͸௒ରশৼಈࢠ໛‫[ ܕ‬12] ΛಘΒΕΔɻ࣮ࡍɼ(10) ͱ (11) ͷ͸͡Ίͷࣜ͸ͦΕͧΕ†ߏ੒͢Δɻ௒ిՙͱ b, f ͱͷ୅਺͸ɼࡾ߲ؒ୅਺͔Βࣗવʹb(b† f + ηf b† ) − η(b† f + ηf b† )b = 2fಋ͔ΕΔ͜ͱ͕Θ͔Δɻͦͷ͋ͱɼҰൠͷϙςϯγϟϧͷ৔b† (bf + ηf b) − η(bf + ηf b)b† = −2f߹΁ͷ֦ுΛࢼΈΔɻୈ 4 ষͰ͸ɼͦͷͨΊͷ४උͱͯ͠ɼ͜͜Ͱ η 2 = + Ͱ͋Δ͜ͱΛ࢖͍ɼลʑ଍͋͠ΘͤΔͱϘʔζతม਺ b, b† Λ௨ৗͷϘʔζม਺ͷ৔߹ͱಉ༷ɼ࠲ඪ qͱӡಈྔ p ͷҰ࣍݁߹ͱߟ͑ɼৼಈࢠ໛‫ܕ‬Λ q, p ͓ΑͼϑΣbb† f − f b† b + b† bf − f bb† = 0ϧϛతม਺ f Ͱॻ͖௚͢ɻଓ͘ୈ 5 ষͰɼ௒ϙςϯγϟϧW (q) Λಋೖ͠ɼ௒ిՙͷ΂͖ྵੑ͔Β W (q) ʹର͢Δ੍‫ݶ‬͜ Ε ͸ (12) ͷ ͸ ͡ Ί ͷ ࣜ ʹ Ұ க ͢ Δ ɻಉ ༷ ʹ ͠ ͯ ɼΛௐ΂Δɻ௒ରশ୅਺͕ຬͨ͞ΕΔͨΊʹ௒ϙςϯγϟϧ͸

(15),(16),(17) ͸‫ʹ͍ޓ‬ಠཱͰ͸ͳ͍͜ͱΛ͔֬ΊΔ͜ͱ͕q ͷ‫͚ͳͰ਺ؔح‬Ε͹ͳΒͳ͍͜ͱ͕Θ͔Δɻୈ 6 ষͰ͸ɼ

Ͱ͖Δɻ௒ిՙΛ΋ͱʹϋϛϧτχΞϯ H Λߏ੒͢Δɻ௨ৗͷϘʔζม਺ɼϑΣϧϛม਺͸ɼҎ্ͷؔ܎Ͱ η = +ͱͨ͠΋ͷΛຬͨ͢ɻͨͱ͑͹ɼ(10) ͷࠨลͰ [b, [b† , f ]η ]−η͸ɼη = + Ͱ͋Ε͹ɼb, b† ͱ f ͕Մ‫͋Ͱ׵‬Δ͜ͱΛ࢖ͬͯ2[b, {b† , f }] = [b, 2b† f ] = 2f [b, b† ] ͱͳΔɻ௨ৗͷϘʔζม਺ࡾ߲ؒ୅਺Ͱ͸ަ‫( ܎ؔ׵‬1) ͕੒Γཱ͔ͭΒɼ͜Ε͸ 2f ʹͳΓɼ͔֬2Ohunuki-Kamefuchi[7] ʹ͕ͨͬͯ͠ɼಉ͡Φʔμʔ ͷʹ (10) ͷӈลʹͳ͍ͬͯΔɻb ͱ f ͷؒͷࡾ߲ؒ୅਺͸࣍ͷ‫Ͱܗ‬༩͑ΒΕΔ΋ͷͱ͢Δɿ
[b† , [b† , f † ]η ]−η = 0[b, [b, f ]η ]−η = 0,††[b, [b, f ]η ]−η = 0,[b, [b† , f ]η ]−η = 2f,(8)†[b , [b , f ]η ]−η = 0(9)[b† , [b, f † ]η ]−η = −2f †(10)†[b , [b, f ]η ]−η = −2f,††[b, [b , f ]η ]−η = 2f†[f, {b, b }] = 0,

†

†

†

[f , {b, b }] = 0͜Εʹରͯ͠ɼ΋͠ η = − ͱ͢Δͱɼ(10) ͷࠨล͸{b, [b† , f ]} Ͱ͋Δɻb ͱ f ͕Մ‫͋Ͱ׵‬Ε͹ɼ͜Ε͸ 0 ͷ͸͕ͣͩɼͦ͏͢Δͱ (10) ͷӈลͱ͸Ұக͠ͳ͍ɻ͕ͨͬͯ͠ɼ(10)ͷؔ܎͸௨ৗͷϘʔζม਺ɼϑΣϧϛม਺ʹରͯ͠͸੒Γཱͨͳ͍͜ͱʹͳΔɻಉ͡Α͏ʹͯ͠ɼ(15) ͷࠨลͰ [f, [f † , b]η ]η
͸ɼη = + Ͱ͋Ε͹ {f, {f † , b}} = {f, 2f † b} = 2b{f, f † }(11)ͱͳΓɼ௨ৗͷϑΣϧϛม਺ͷަ‫( ܎ؔ׵‬3) Λ࢖͏ͱɼ2b ʹ(12)ͳͬͯ (15) ͷӈลʹ౳͍͠ɻ͜Εʹରͯ͠ɼη = − ͷ৔߹ʹ͸ࠨล͕ [f, [f † , b] ] Ͱɼ௨ৗͷϘʔζม਺ɼϑΣϧϛม਺Ͱ͸ f † ͱ b ͕Մ‫͔ͩ׵‬Βɼ͜ͷؔ܎͸੒Γཱͨͳ͍ɻ

[f, [f, b]η ]η = 0,[f † , [f † , b† ]η ]η = 0(13)[f, [f, b† ]η ]η = 0,[f † , [f † , b]η ]η = 0(14)[f, [f † , b]η ]η = 2b,[f † , [f, b† ]η ]η = 2b†(15)ަ‫܎ؔ׵‬͸੒Γཱͨͳ͍ͷͰɼ͜ͷΑ͏ͳʮҟৗͳʯࡾ߲ؒ

[f, [f † , b† ]η ]η = 2b† ,[f † , [f, b]η ]η = 2b(16)୅਺Ͱ͋ͬͯ΋ໃ६͸ͳ͍ɻη = − ͷͱ͖ʹɼͲͷΑ͏ͳҎ্ͷ͜ͱ͔Β η = − ͷ৔߹͸͍Θ͹ʮҟৗͳʯ୅਺ؔ܎ͱ‫͑ݴ‬ΔɻΦʔμʔ͕ 2 Ҏ্ͷ৔߹ʹ͸ɼೋ߲ؒͷ (൓)໛‫͕ܕ‬Մೳͳͷ͔͸‫ڵ‬ຯ͋Δͱ͜ΖͰ͋Δ͕ɼ͜ͷϊʔτͰ͸ɼη = + ͷූ߸ʹ‫ݶ‬Δ͜ͱʹͯٞ͠࿦ΛਐΊΔɻ32 ύϥม਺͸Φʔμʔͱ‫ݺ‬͹ΕΔඇෛ੔਺ n = 0, 1, 2, · · · ʹΑͬͯಛ௃෇͚ΒΕΔɻϘʔζతม਺ɼϑΣϧϛతม਺ͷ‫਺ݸ‬ԋࢉࢠͷ

3 η = − ͷͱ͖ʹ΋ୈ 3 ষͷํ਑ʹ͕ͨͬͯ͠௒ిՙΛఆٛ͠ɼ‫ݻ‬༗஋εϖΫτϧ͸ͦΕͧΕ௒ରশͳৼಈࢠ໛‫ܕ‬Λߏ੒͢Δ͜ͱ͕Ͱ͖Δɻ͔ͦ͠͠ͷͱ͖ͷϋ

n/2, n/2 + 1, · · · , n/2 + k, · · · ɹ (Ϙʔζతม਺)ϛϧτχΞϯ͸ (1/2)[b, b† ] + (1/2){f, f † } ͱͳΔɻ΋͠ɼb ͱ f−n/2, −n/2 + 1, · · · , n/2 − 1, n/2 ɹ (ϑΣϧϛతม਺)͕௨ৗͷϘʔζม਺ͱϑΣϧϛม਺ͳΒɼ͜ͷϋϛϧτχΞϯ͸ 0ͱ͘ʹ n = 1 ͸௨ৗͷϘʔζม਺ɼϑΣϧϛม਺ʹରԠ͢Δɻ

ͱͳΔɻ

2─44─͜ͷΑ͏ʹࡾ߲ؒ୅਺͸ɼ௒ରশৼಈࢠ໛‫ܕ‬ͷ b, b† , f, f † ͱ௒ରশৼಈࢠ໛‫ܕ‬

3Q, Q† , HB , HF ͱͷަ‫ੵࢠ׵‬ΛܾΊ͍ͯΔɻલষͰઃఆͨ͠ (η = + ͷͱ͖ͷ) ࡾ߲ؒ୅਺Λ΋ͱʹɼҎ্ͷؔ܎Λ࢖ͬͯɼ௒ରশ୅਺ 1)-3) ͕੒Γཱͭ͜ͱΛ௒ରশੑΛ΋ͭৼಈࢠ໛‫ܕ‬Λߏ੒͢ΔɻϘʔζతม਺ɼϑΣࣔͦ͏ɻϧϛతม਺ͦΕͧΕͷৼಈࢠͷϋϛϧτχΞϯΛϘʔζม਺ɼ1) ϕΩྵੑɿϑΣϧϛม਺ͷͱ͖ʹ฿ͬͯHB ≡1 †{b , b},
2HF ≡

1 †
[f , f ]2௒ిՙͷೋ৐͸

(18)

Q2 =

ͱ͢Δɻb, b† ͷ୅਺ (2) ͔Β[b, HB ] = b,[b† , HB ] = −b†

Qb† f = [Q, b† ]f + b† {Q, f } − b† f Qԋࢉࢠʹͳ͍ͬͯΔɻಉ༷ʹɼf, f † ͷ୅਺ (4) ͔Β[f, HF ] = f,

[f , HF ] = −f(29)

ӈล (ɹ) ಺ͷୈҰ߲͸

(19)ΑͬͯɼHB ͷ‫ݻ‬༗ϕΫτϧʹରͯ͠ b† , b ͸ͦΕͧΕঢ߱†”i“ †Qb f + Qf b†2

͜͜Ͱɼ(23) ͱ (25) Λ࢖͑͹ӈลͷୈҰ߲ͱୈೋ߲͸ 0 ͱ†

ͳΔɻΑͬͯ

(20)

Qb† f = −b† f QHF ͷ‫ݻ‬༗ϕΫτϧʹରͯ͠ f † , f ͸ͦΕͧΕঢ߱ԋࢉࢠͰ(29) ͷӈล (ɹ) ಺ୈೋ߲΋ಉ༷ʹͯ͋͠Δɻ͜ΕΒͷ࿨ΛͱͬͨQf b†

H = HB + H F= {Q, f }b† − f [Q, b† ] − f b† Q= 0 · b† − f · 0 − f b† Q(21)͕͕ͨͬͯ͠Ϙʔζతม਺ͱϑΣϧϛతม਺ͱ͔ΒͳΔৼಈࢠ໛‫ܕ‬ͷϋ

i
Q2 = − (b† f + f b† )Q = −Q2 = 02ϛϧτχΞϯͰ͋Δ͕ɼ͜Ε͕࣮ࡍʹ௒ରশੑΛ΋ͭ͜ͱΛࣔͦ͏ɻͱͳΓɼ΂͖ྵੑ͕ࣔ͞Εͨɻ͜ͷ‫͕ܥ‬௒ରশੑΛ࣋ͭͷͰ͋Ε͹ɼ௒ిՙ Q ͕͋ͬͯɼࣜ(5)-(7) ͷ௒ରশ୅਺Λຬͨ͢ɻ௨ৗͷϘʔζม਺ͱϑΣϧϛม਺͔ΒͳΔ௒ରশৼಈࢠ໛‫Ͱܕ‬͸ɼ௒ిՙ͸ Q = ib† f, Q† =2) ϋϛϧτχΞϯɿ2−if † b Ͱ͋Δɻೋ߲ؒͷަ‫܎ؔ׵‬Λ࢖ͬͯ Q2 = Q† = 0Λࣔ͢͜ͱ͕Ͱ͖Δɻ·ͨɼ(6) ʹΑͬͯϋϛϧτχΞϯ͸

Q ͱ Q† ͷੵΛͭ͘Δͱ(21) Ͱ༩͑ΒΕΔɻ͔͠͠ɼҰൠͷΦʔμʔͰ͸ɼೋ߲ؒͷ

QQ† = −ؔ܎͕ͳ͍ͷͰɼ͜ͷ··Ͱ͸௒ిՙʹͳΒͳ͍ɻͦ͜Ͱɼ௒ిՙͱͯ͠

Q=i †{b , f },2”i“ †Qf b + Qbf †2(30)ӈลͷ (ɹ) ಺ୈҰ߲͸iQ† = − {f † , b}
2Qf † b = {Q, f † }b − f † [Q, b] − f † bQ

(22)ͱ͠Α͏ɻ͢Δͱɼࡾ߲ؒͷ୅਺ؔ܎ (9),(10),(14),(15) ͸ɼ͜͜Ͱ (24) ͱ (26) Λ࢖͏ͱ௒ిՙͱ b, b† ͋Δ͍͸ f, f † ͱͷަ‫ੵࢠ׵‬Λද͍ͯ͠ΔɻQf † b = ib† b − f † (−if ) − f † bQ(9) ͸[b, Q† ] = 0ɼ [b† , Q] = 0(23)[b, Q] = if ɼ [b† , Q† ] = if †(24){f, Q} = 0ɼ {f † , Q† } = 0(25)ಉ༷ʹͯ͠ɼ(30) ͷӈลʢɹʣ಺ୈೋ߲͸(10) ͸Qbf †= −if f † + b(ib† ) − bf † Q(14)͕ͨͬͯ͠QQ†(15) ͸{f, Q† } = −ibɼ {f † , Q} = ib†(26)·ͨɼ(12) ͱ (17) ͸ɼৼಈࢠϋϛϧτχΞϯͱͷަ‫܎ؔ׵‬†[b, HF ] = [b , HF ] = 0”i“ †i{b , b} + i[f † , f ] − {f † , b}Q
2= HB + HF − Q† Q=−ͱͳΓɼϋϛϧτχΞϯ͸ H = HB + HF ͱͳΔ͜ͱ͕ࣔΛ͋ͨ͑Δɻ

[f, HB ] = [f † , HB ] = 0= [Q, b]f † + b{Q, f † } − bf † Q

͞Εͨɻ(27)3) ௒ରশෆมੑɿ(28)

3

─45─

͕ಋ͚Δɻ·ͨɼ(37),(38) ͸࣍ͷؔ܎ࣜͱಉ౳Ͱ͋Δɻ͢Ͱʹ 1),2) ͕͋ΔͷͰ (7) ͸ࣗಈతʹ੒Γཱ͕ͭɼ(23)-(26)ͷؔ܎Λ࢖͏͜ͱͰɼ͜ΕΛ௚઀ࣔ͢͜ͱ͕Ͱ͖Δɻ·ͣ

[f, {q, p}] = [f † , {q, p}] = 01
1
[Q, HB ] = [Q, bb† ] + [Q, b† b]22(40){f, [q, p]} = 4if + 2(pf q − qf p)ͰɼӈลͷୈҰ߲͸††††{f , [q, p]} = 4if + 2(pf q − qf p)

1
1
1[Q, bb† ] = [Q, b]b† + b[Q, b† ]222(42)

͞Βʹɼ(34) Λ (13)-(17) ʹ୅ೖͯ͠੔ཧ͢Δͱ(23)ɼ(24) Λ࢖͑͹ɼ͜Ε͸ −(i/2)f b† ʹ౳͍͠ɻಉ༷ʹ
ͯ͠

1
1
i
1
[Q, b† b] = [Q, b† ]b + b† [Q, b] = − b† f2
2
2
2{f, {f, q}} = 0,{f † , {f † , q}} = 0(43){f, {f, p}} = 0,{f † , {f † , p}} = 0(44){f, {f † , q}} = 2q,[q, [f † , f ] ] = 0

(45)

††

{f, {f , p}} = 2p

͕ͨͬͯ͠

i
[Q, HB ] = − {f, b† } = −Q2(41)[p, [f , f ] ] = 0(46)࠷‫ʹޙ‬ɼb ͷࡾ߲ؒ୅਺ (2) Λ p, q Ͱॻ͖௚ͤ͹(31)[q, p2 ] = 2ip,·ͨɼ1
1[Q, f † f ] − [Q, f f † ]22Ͱɼӈลͷަ‫ੵࢠ׵‬͸ (25),(26) Λ࢖ͬͯ[Q, HF ] =[p, q 2 ] = −2iq(47)ೋ൪໨ͷؔ܎͔Βɼq 2 ͷؔ਺ F (q 2 ) ͱ p ͱͷަ‫ੵࢠ׵‬͸ɼ௨ৗͷྔࢠ࿦ͱಉ༷ɼF ͷಋؔ਺Λ༩͑Δ͜ͱ͕Θ͔Δɻ4

11i
1[Q, f † f ] = {Q, f † }f − f † {Q, f } = b† f2222[p, F (q 2 )] = −2iqF  (q 2 )11i
1[Q, f f † ] = {Q, f }f † − f {Q, f † } = − f b†
2222
͕ͨͬͯ͠ɼi[Q, HF ] = {f, b† } = Q(32)2(31),(32) Λ͋ΘͤΔͱ [Q, H] = 0 ͱͳΓɼ(22) Λ௒ిՙͱ5(48)ϙςϯγϟϧ໛‫΁ܕ‬ͷ֦ு͜ͷষͰ͸ɼલষͰಋ͍ͨࡾ߲ؒ୅਺Λ࢖͍ɼ௒ରশͳৼಈࢠ໛‫ܕ‬Λϙςϯγϟϧ໛‫֦ʹܕ‬ு͢Δ͜ͱΛߟ͑Δɻͦ͢Δৼಈࢠ໛‫ܕ‬͸௒ରশෆมͰ͋Δ͜ͱ͕ࣔͤͨɻͷͨΊʹɼ·ͣৼಈࢠ໛‫ܕ‬ͷϋϛϧτχΞϯ (21) Λ (34) Λ࢖ͬͯ q, p Ͱॻ͖௚͢ɻ͢Δͱ

4H=࠲ඪදࣔͷࡾ߲ؒؔ܎

iQ = √ {f, q − ip}
2 2ͱӡಈྔʹ͋ͨΔม਺ q, p Ͱஔ͖‫ͨ͑׵‬΋ͷͰॻ͖௚͢ɻ

1ip = √ (b − b† )2(33)ϙςϯγϟϧͱΈͳͤ͹ɼ͜ͷ෦෼Λద౰ͳϙςϯγϟϧͰ

1b = √ (q − ip)2†ஔ͖‫͑׵‬Δ͜ͱͰҰൠͷϙςϯγϟϧ໛‫ܕ‬Λߏ੒Ͱ͖Δɻ͠

(34)

͔͠ɼϙςϯγϟϧΛ͏·͘બ͹ͳ͍‫ݶ‬ΓɼͦͷΑ͏ͯ͠ಘΒΕͨϋϛϧτχΞϯͷ௒ରশੑ͸ඞͣ͠΋อূ͞Εͳ͍ɻΛɼ(8)-(11) ʹ୅ೖ͠ɼ੔ཧ͢ΔͱҎԼͷؔ܎͕ಘΒΕΔɻ
†[q, {q, f }] = 0,[q, {q, f }] = 0(35)[p, {p, f }] = 0,[p, {p, f † }] = 0(36)[q, {p, f }] = 2if,[q, {p, f † }] = 2if †(37)[p, {q, f }] = −2if,[p, {q, f † }] = −2if †(38)͢ͳΘͪɼ௒ిՙ͕ଘࡏͯ͠ɼ(6) ͷؔ܎Λຬ͔ͨ͢Ͳ͏͔Θ͔Βͳ͍ɻ

4 Φʔμʔ 2 Ҏ্ͷ৔߹ɼq Λର֯Խ͢ΔදࣔͰӡಈྔԋࢉࢠ͸p = −i͜ΕΒͷ͏ͪɼ(35) ͓Αͼ (36) ͔Β
22[f, q ] = [f, p ] = 0,(50)͜͜ͰɼϋϛϧτχΞϯ (49) ͷ͏ͪ (1/2)q 2 ͸ௐ࿨ৼಈࢠͷ͋Δ͍͸ɼ‫ʹٯ‬ղ͍ͨ

1b = √ (q + ip),2(49)ಉ༷ʹͯ͠௒ిՙ (22) ΋ q, p Ͱද͢͜ͱ͕Ͱ͖ͯલͷষͰ༩͑ͨࡾ߲ؒ୅਺Λɼม਺ b Λ‫ڞʹ͍ޓ‬໾ͳ࠲ඪ

1q = √ (b + b† ),21 21(p + q 2 ) + [f † , f ]22†2cd+i Rqdqͱॻ͚Δ͜ͱ͕Θ͔͍ͬͯΔɻ͜͜Ͱɼc ͸࣮ఆ਺ɼR ͸࠲ඪͷූ
†

2[f , q ] = [f , p ] = 0߸Λม͑ΔԋࢉࢠͰ೚ҙͷؔ਺ G(q) ʹରͯ͠ RG(q) = G(−q) Ͱ(39)͋Δɻ͜ͷද͕ࣜ (48) ͷؔ܎Λຬ͍ͨͯ͠Δ͜ͱ͸༰қʹ͔֬Ί
Δ͜ͱ͕Ͱ͖Δɻ

4

─46─

ͦ͜Ͱɼॱ൪Λม͑ɼ௒ిՙ͔Βग़ൃͯ͠ϋϛϧτχΞϯӈลͷୈҰ߲ͱୈೋ߲͸ͦΕͧΕ q ʹ͍ͭͯͷ‫حۮ‬ੑ͕ҟͳΛߏ੒͠Α͏ɻͦΕʹ͸ࣜ (50) Ͱɼq − ip ͷ q Λ௒ϙςϯ

Δ͔ΒɼQ2 = 0 ͱͳΔʹ͸ u(q 2 ) ≡ 0 Ͱͳ͚Ε͹ͳΒͳ͍ɻγϟϧ W (q) Ͱஔ͖‫ͨ͑׵‬͢ͳΘͪɼ௒ϙςϯγϟϧ͸ W (q) = qv(q 2 ) Ͱͳ͚Ε͹ͳͷ‫ʹܗ‬௒ిՙΛԾఆ͢Δɻࣜ (51) ͷ Q ͕௒ిՙͰ͋ΔͨΊΒͳ͍͜ͱ͕Θ͔ͬͨɻ͕ͨͬͯ͠௒ిՙ͸”
1 “Q= √π + iv(q 2 )χ2ʹ͸ɼ΂͖ྵੑΛຬ͍ͨͯ͠ͳ͚Ε͹ͳΒͳ͍ɻͦͷͨΊʹɼ·ͨɼͦͷ‫ڞ‬໾͸

iQ = √ {f, W (q) − ip}
2 2(51)”
1 “Q† = √ π − iv(q 2 )χ†2

௒ϙςϯγϟϧ W (q) ͕ຬͨ͢΂͖৚݅Λ‫ٻ‬ΊΑ͏ɻͦ͏͠
ͨ͋ͱͰɼ(6) ʹΑͬͯϋϛϧτχΞϯΛߏ੒͢Δɻ·ͣ௒ϙςϯγϟϧΛ

(65)ͷΑ͏ʹॻ͚Δɻ2

2W (q) ≡ u(q ) + qv(q )௒ిՙ Q ͱྗֶతม਺ f, f † , q, p ͱͷ୅਺͸ (35)-(47) Λ(52)࢖ͬͯ࣍ͷΑ͏ʹͳΔɻ

ͱॻ͍͓ͯ͘ɻ͜͜Ͱɼu(q 2 ), v(q 2 ) ͸͍ͣΕ΋ q 2 ͷؔ਺Ͱ{Q, f } = {Q† , f † } = 0f, f † ͱՄ‫͋Ͱ׵‬Δɻ‫ࢉܭ‬Λ‫ݟ‬௨͠Α͘͢ΔͨΊʹπ≡(64)

1{f, p},2χ≡

1{f, q}2i{Q, f † } = √ (qv − ip),2(53)i{Q† , f } = − √ (qv + ip) (67)2ii[Q, q] = − √ f, [Q† , q] = − √ f † ,(68)221
1[Q, p] = − √ (2qv  χ + vf ), [Q† , p] = √ (2χ† qv  + vf † )22(69)

ͱ͓͚͹i
Q = √ (2u(q 2 )f + v(q 2 )χ − iπ)
2(66)

(54)

ͱॻ͚Δɻπ ΍ χ ͷ୅਺͸લষͰಋ͍ͨࡾ߲ؒ୅਺ (35)-(47)͔Βπ 2 = χ2 = {π, χ} = 0π†2†2=χ= {π † , χ† } = 0†2†{π, π } = p ,(55)6(56)
2{χ, χ } = qi
{χ, π } = {p, q} − [f † , f ]2i{π, χ† } = {p, q} + [f † , f ]2·ͨ u(q 2 ), v(q 2 ) ʹରͯ͠͸ (48) Λ࢖ͬͯ†(57)(64),(65) Ͱ༩͑ΒΕͨ௒ిՙ Q, Q† ͷ൓ަ‫ੵࢠ׵‬Λ‫ࢉܭ‬(58)ͯ͠ϋϛϧτχΞϯΛ‫ٻ‬ΊΑ͏ɻπ ͱ χ Λ࢖ͬͨදࣔͰ

(59)

H = {Q, Q† }”1“{π, π † } + v 2 {χ, χ† } + i{vχ, π † } − i{π, vχ† }ɹ=2ӈลͷ (ɹɹ) ಺ୈҰ߲ͱୈೋ߲͸ࣜ (57) Λ࢖ͬͯ[π, u] = −2iu χ,[π, v] = −2iv  χ(60)[π † , u] = −2iu χ† ,[π † , v] = −2iv  χ†(61){π, π † } = p2 ,͕੒Γཱͭ͜ͱ͕Θ͔Δɻ

{vχ, π † } = v{χ, π † } + [π † , v]χ

{f, χ} = {f, π} = 0,{f † , χ† } = {f † , π † } = 0(62){f † , π} = {f, π † } = p,{f † , χ} = {f, χ† } = q,(63)ͱ௚͓͖ͯ͠ɼ(58) ͱ (61) Λ࢖͏ͱ„«i{vχ, π † } = v − [f † , f ] + {p, q} − 2iv  χ† χ2(54) ͔ΒɼQ2 ͸
1“ 2 2Q2 = −4u f + v 2 χ2 − π 22”ɹɹɹɹɹ+ 2{uf, vχ} − 2i{uf, π} − i{vχ, π}ಉ༷ʹͯ͠ୈ࢛߲͸{π, vχ† } = v´
1` 2 24u f − 2i[π, u]f − i[π, v]χ
2«i †[f , f ] + {p, q} − 2iv  χχ†2(70)΋ͬͱ΋؆୯ͳͷ͸ v(q 2 ) ≡ 1 ͷ৔߹Ͱɼ͜ͷͱ͖ (70) ͸͜͜Ͱ (60) ʹΑͬͯ [π, u], [π, v] Λ u, v ͷಋؔ਺Ͱॻ͖௚௒ରশৼಈࢠ໛‫ܕ‬ͷϋϛϧτχΞϯ (21) ʹ‫ؼ‬ண͢Δɻ͠ɼ͞Βʹ χ2 = 0 Ͱ͋Δ͜ͱΛ࢖͏ͱ2 2„Ҏ্Λ·ͱΊΔͱ”1“ 2H=p + v 2 q 2 + 2v  [χ† , χ] + v[f † , f ]2ӈลͰ (55) ͓Αͼ (62) Λ࢖͏ͱɼ2v 2 {χ, χ† } = v 2 q 2ୈࡾ߲͸͞Βʹɼπ, χ ͱ f ͱ͸࣍ͷ͕ࣜ੒Γཱͭɻ

Q2 = −ϋϛϧτχΞϯ

ϋϛϧτχΞϯ (70) ͸ɼࡾ߲ؒ୅਺ʹΑͬͯಋ͔Ε͕ͨɼ

Q = −2u f + 2u χf௨ৗͷϘʔζม਺ɼϑΣϧϛม਺ʹରͯ͠͸͞Βʹೋ߲ؒ

5

─47─

ͷަ‫܎ؔ׵‬ΛԾఆ͢Δ͜ͱͰ؆୯ʹͳΔɻ(70) Ͱ [χ† , χ] ͸
2[10] I.Ito and T.Kon, in Proc. of 2nd Workshop on†

q [f , f ] ʹ‫ؼ‬ண͢Δ͔Βɼ͜ͷͱ͖ʹ͸
”1“ 2H1 =p + v 2 q 2 + (2q 2 v  + v)[f † , f ]2Thermal Field Theories and Their Applications, eds.
H.Ezawa et.al. (North-Holland, 1991) p.521(71)[11] I.Ito and T.Kon, Int.J.Mod.Phy. A7 (1992) 3997

ͱͳΓɼ͜Ε͸ Witten ͷղੳͨ͠௒ରশϙςϯγϟϧ໛‫ܕ‬[12] M.Hama, A.Koike, M.Sawamura and H.Suzuki,

Ͱ W (q) = qv(q 2 ) ͱͨ͠΋ͷʹ΄͔ͳΒͳ͍ɻ͔͠͠ɼҰ

Mod.Phys.Lett. A6 (1991) 2019ൠͷΦʔμʔͷͱ͖ʹ͸ [χ† , χ]/q 2 ͱ [f † , f ] ͱ͸‫ʹ͍ޓ‬ಠཱ[13] I.Ito, Bull. of Seikei University 28 (1996) #2

ͳԋࢉࢠͰ͋Δɻ͜ͷ͜ͱ͸࣍ͷΑ͏ͳߟ࡯͔Β΋Θ͔ΔɻΦʔμʔ n ͷϑΣϧϛతม਺ʹ͍ͭͯ (1/2)[f † , f ] ͷ‫ݻ‬༗[14] S.N.Biswas and A.Das, Mod.Phys.Lett. A3 (1988)549

஋͸ −n/2, −n/2 + 1, · · · , n/2 − 1, n/2 ͷΑ͏ʹ n + 1 ‫͋ݸ‬Δɻͱ͜ΖͰ„[χ† , χ]q2[15] J.Beckers and N.Debergh, Mod.Phys.Lett. A4 (1989)«2†2=1209; Mod.Phys.Lett. A4 (1989) 2019({χ† , χ})2 − 2χ† χ2 χ† − 2χχ χ(q 2 )2

͸ (55),(57) Λ࢖͏ͱ͜ͷӈล͸ 1 ʹ౳͍͠ɻ͕ͨͬͯ͠[χ† , χ]/q 2 ͱ͍͏ԋࢉࢠͷ‫ݻ‬༗஋͸ ±1 ʹͳΔɻn = 1 ͷͱ͖ʹ͸ɼ[f † , f ] ͷ‫ݻ‬༗஋͕ ±1 ʹͳͬͯҰக͢Δɻ͜Ε͸্Ͱ‫ͨݟ‬Α͏ʹೋͭͷԋࢉࢠ͕ಉ͡΋ͷʹͳΔ͔ΒͰ͋Δɻ͔͠͠ɼn(≥ 2) ͷͱ͖ʹ͸‫ݻ‬༗஋ͷεϖΫτϧ͕ҟͳͬͨ΋ͷʹͳΓɼ͕ͨͬͯ͠ೋͭͷԋࢉࢠ͸‫ʹ͍ޓ‬ಠཱͰ͋Δɻ͜ͷϊʔτͰ͸ɼ௒ిՙ͕Ұ‫ݸ‬ͷεʔύʔϙςϯγϟϧͰදͨ͠໛‫͍ͯͭʹܕ‬ϋϛϧτχΞϯΛ‫ٻ‬Ίͨɻ͔͠͠ɼ͜Ε͕΋ͬͱ΋Ұൠతͳ௒ରশϙςϯγϟϧ໛‫͏͍ͱܕ‬Θ͚Ͱ͸ͳ͘ɼෳ਺‫ݸ‬ͷεʔύʔϙςϯγϟϧΛ‫ؚ‬Ή௒ରশ໛‫ߏ͕ܕ‬
੒Ͱ͖ΔՄೳੑ͕͋Δɻ࣮ࡍɼϘʔζม਺ͱΦʔμʔ 2 ͷϑΣϧϛతม਺͔ΒͳΔ Rubakov ͱ Spiridonov ͷ௒ରশ໛‫[ ܕ‬8] Ͱ͸ɼ௒ిՙ͸ೋͭͷεʔύʔϙςϯγϟϧΛ‫ؚ‬ΜͰ͍Δɻύϥม਺ͷߏ੒͸ҧ͍ͬͯΔ΋ͷɼ௒ిՙ (64) Λ͞ΒʹҰൠԽͨ͠໛‫ܕ‬ͷՄೳੑΛ‫͍ͯࠦࣔ͘͠ڧ‬Δɻࢀߟจ‫ݙ‬[1] H.Miyazawa, Phys.Rev.170 (1968) 1568
[2] D.V.Volkov and V.P.Akulov, JETP Lett. 16 (1972)
438[3] J.Wess and B.Zumino, Nucl.Phys. B70 (1974) 39[4] e.g. H.P.Nilles, Phys.Rept. 110 (1984) 1[5] E.Witten, Nucl.Phys. B188 (1981) 513[6] H.S.Green, Phys.Rev. 90 (1953) 270[7] Y.Ohunuki and S.Kamefuchi, Quantum Field Theoryand Parastatistics, University of Tokyo Press, (1982)[8] V.A.Rubakov and V.P.Spridonov, Mod.Phys.Lett.
A3 (1988) 1337
[9] F.Ardlan and F.Mansouri, Phys.Rev.Lett. 56 (1986)
2456; Phys.Lett. 176B (1986) 996─48─