What is identity?: A review of `What is Reality?’ by Hayato Saigo and Sigeru Taguchi
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(2) ͔ΒͰ͋ΔɻҰํɺຊॻྔࢠΛཻࢠͷ͔؍Βଊ͑Δ͜ͱʹ൱ఆతͰ͋ Γɺʮྔࢠʹ͓͍ͯɺʮཻࢠʯͷ֓೦ͦͷͷ͕ɺैདྷͷΑ͏ͳཧ ղͷͳ͔ʹऩ·Βͳ͍ͱ͍͏͜ͱ͕໌Β͔ʹͳΓɺ ʮཻࢠʯͷ֓೦ͦͷͷ Λࠜຊతʹଊ͑͢͜ͱ͕ٻΊΒΕ͍ͯΔʯʢp. 25ʣͱड़͍ͯΔɻ5 અɺ6 અɺ7 અͰɺDHR ཧΛཻࢠతͳ͔؍ΒͰͳ͘ʮՄࣗͳٯવมΛ௨ ͯ͡ɺʮಉ͡͞ʯཱ͕͢Δʯʢp. 142ʣͱ͍͏ຊॻͰఏࣔ͞Ε͍ͯΔ͔؍ Βݕ౼͢Δɻ ຊॻධͷతೋͭ͋ΔɻҰͭʮಉ͡͞ʯʹؔΘΔຊॻͷٞΛհ͢ Δ͜ͱɺ͏Ұͭ DHR ཧʹʹ͓͍ͯຊॻͷ͔؍Βʮಉ͡͞ʯΛ ݕ౼͢Δ͜ͱʹΑͬͯຊॻͷࣹఔͷ͞Λࣔ͢͜ͱͰ͋Δɻ3 અҎ߱ DHR ཧʹؔΘΔٕज़తͳهड़͕ͳ͞ΕΔ͜ͱ͕͋ΔɻͦͷΑ͏ͳهड़Λ͢Δ తɺDHR ཧͷٕज़తͳ༰Λհ͢Δ͜ͱͰͳ͘ɺDHR ཧʹର͠ ͯຊॻͷֶతͳࢹΛద༻Ͱ͖Δ͜ͱΛࣔ͢͜ͱʹΑͬͯຊॻͰఏࣔ͞ Ε͍ͯΔֶతࢹͷద༻ൣғͷ͞Λࣔ͢͜ͱͰ͋Δɻ. 2 2 「同じさ」を中心とした本書の内容 ʮಉ͡͞ʯΛத৺ͱͨ͠ຊॻͷ༰ ຊॻɺୈ 1 ষ͔Βୈ 5 ষͷষͰߏ͞Ε͍ͯΔɻ͜ͷઅͰɺʮಉ͡͞ʯ ͱ͍͏֓೦Λத৺ʹҰষͣͭհ͢Δɻ ʮՄࣗͳٯવมΛ௨ͯ͡ɺ ʮಉ͡͞ʯ ཱ͕͢Δʯ ʢp. 142ʣͱ͍͏ओுʹ࠷ີʹؔ࿈͢Δͷɺୈ 3 ষͰ͋Δɻ ຊॻͷୈ 1 ষͰɺ౷ܭత๏ଇʹ͓͚Δʮಉ͡͞ʯΛѻ͍ͬͯΔɻ౷ܭత๏ ଇɺҰճ͖Γͷࣄͱߟ͑Δ͜ͱ͍͠ɻࢼߦΛ܁Γฦ͢͜ͱʹΑͬͯɺ ͋Δ͕ಘΒΕΔɻࢼߦΛ܁Γฦ͢߹ɺͦΕΒͷࢼߦ͕શʹಉ͡Ͱ͋ Δ͜ͱͳ͘ɺྫ͑ࢼߦΛߦ͏ॴ࣌ؒҟͳΔɻ͔͠͠ɺٻΊ͍ͨ౷ ܭత๏ଇʹͱͬͯॏཁͰͳ͍ͱΈͳ͢ҧ͍ແࢹ͠ɺॏཁͳ͕ಉ͡Ͱ͋Ε ಉ͡ࢼߦͱΈͳ͢ɻ͜ͷΑ͏ʹ͋Β͔͡Ίಉ͡ࢼߦͱԿ͔Λઃఆ্ͨ͠ Ͱɺಉ͡ࢼߦΛ܁Γฦ͢͜ͱʹͳΔɻͭ·Γɺ౷ܭత๏ଇΛಘΔͨΊʹɺಉ ͡ࢼߦͱԿ͔Λ͋Β͔͡ΊܾΊ͓͔ͯͳ͚ΕͳΒͣɺ ʮͲ͏͍͏݅Λઃ ఆ͢Δ͔ΛܾΊͳ͚Εɺ౷ܭత๏ଇͦΕࣗମ͕ఆ·Βͳ͍ʯ ʢp. 53ʣɻͦ͠ ͯɺ ʮಉ͡͞ʯʹؔ͢Δ݅Λม͑Ε౷ܭత๏ଇมΘΔɻ͜ͷΑ͏ʹʮಉ ͡͞ʯ༩͑ΒΕΔͷͰͳ͘ɺͲͷΑ͏ͳʮಉ͡͞ʯΛߟ͑Δ͔ΛܾΊ ͓ͯ͘ඞཁ͕͋Δɻ͜͏ͨ͠ࣄଶΛஶऀʮ͍͕ͳ͚Ε͑ͳ͍ʯ ʢp. 54ʣͱड़͍ͯͯɺ ʮ͍͔͚ҎલʹԿ͔͕͢Ͱʹఆ·͍ͬͯΔͱ͍͏ߟ͑ํ ɺͦ͜Ͱ͍ͯͬ͜ىΔࣄଶΛهड़͢ΔʹΉ͠ΖෆదͰ͋Δʯ ʢp. 53ʣͱ ͍͏͜ͱΛڧௐ͍ͯ͠Δɻ ୈ 2 ষͰɺֶʹ͓͚Δʮಉ͡͞ʯΛѻ͍ͬͯΔɻྫ͑ɺ ʮͨΖ͏ɺ ͳ͜ɺΏ͏͚͢ʯશһͰԿਓͰ͋Δ͔ͱ͍͏Λߟ͑Α͏ɻ͜ͷͱ͖ɺͨ 276. 2.
(3) 〈同じさ〉とは何か. Ζ͏ɺͳ͜ɺΏ͏͚͢ͱ͍͍͑ͯ͠ɺΏ͏͚͢ɺͳ͜ɺͨΖ͏ͱ͑ ͍͍ͯɻͦͯ͠ɺͲͷΑ͏ͳ͑ํͰ͋ͬͯ͑ 3 ਓͱͳΔʢp. 75ʣɻ ͜ͷ 3 ਓͱ͏ݴ͑Λग़ͨ͢Ίʹɺ۩ମతʹ͑Δ͜ͱෆՄܽͰ͋Δɻ͔͠ ͠ɺ3 ਓͱ͑ͨਓ͕ɺͨΖ͏ɺͳ͜ɺΏ͏͚͢ͱ͑ͨͷ͔ɺΏ͏͚͢ɺ ͳ͜ɺͨΖ͏ͱ͑ͨͷ͔͔Βͳ͍ɻ͜͏ͨ͑͠ํ 3 ਓͱ͍͏ ͑ͷதʹݱΕΔ͜ͱͰ͖ͣɺফ͑ͯ͠·͍ͬͯΔɻ͜ͷΑ͏ʹඞͣબ͠ ͳ͚Ε͍͚ͳ͍͕બͨ͠ޙফ͑ͯ͠·͏Α͏ͳબΛɺஶऀඇن४ తબͱΑΜͰ͍ΔɻຊষͰѻΘΕ͍ͯΔඇن४తબͷଞͷྫɺ1 + 1 ͱ ͍͏Ͱ͋Δɻ10 ਐ๏Ͱߟ͑Δͱ͜ͷ͑ 2 Ͱ͋Γɺ2 ਐ๏Ͱߟ͑Δͱ ͜ͷ͑ 10 ͱͳΓɺ͜ΕΒͷ͑ʮಉ͡ʯͰ͋Δɻ͜ͷ͑ɺʮಛఆ ͷ߸هͷબʹΑΒͳ͍ʯ͕ɺ ʮͦͷʮಉ͜͡ͱʯΛɺ͍͍ͬ͞Կͷ߸هΛ Θͣʹֶతʹѻ͏͜ͱͰ͖ͳ͍ʯ ʢp. 77ʣɻ·ͣɺྫ͑ 2 ͱ͍͏ه ߸Λબ͠ͳ͚Εɺ͜Ε͕ 10 ͱಉ͔͡Ͳ͏͔Λߟ͑Δ͜ͱͰ͖ͳ͍ͱ͍ ͏͜ͱͰ͋Δɻͭ·Γɺ͋Δ߸هΛඇن४తબΛ͢Δ͜ͱʹΑͬͯɺ ʮಉ͡ ͜ͱʯʹ͍ͭͯѻ͏͜ͱ͕Ͱ͖Δɻ ʮಉ͡͞ʯཱ͕͢Δʹඇن४తબ͕ ෆՄܽͳͷͰ͋Δɻ ୈ 3 ষͰɺৗੜ͚͓ʹ׆Δʮಉ͡͞ʯΛѻ͍ͬͯΔɻ্Ͱड़ͨΑ ͏ʹɺʮՄࣗͳٯવมΛ௨ͯ͡ɺʮಉ͡͞ʯཱ͕͢Δʯʢp. 142ʣͱ͍͏ ຊॻͷओுʹ࠷ີʹؔ࿈͢Δͷɺ͜ͷষͰ͋ΔɻৗతͳྫΛ௨ͯ͠ɺ มͷՄٯੑ͔Βʮಉ͡͞ʯΛଊ͑Δ͜ͱ͕Ͱ͖Δͱ͍͏͜ͱΛ͍ͯ͡Δɻ ྫ͑ɺ෦ʹ͋ΔصΛߟ͑Α͏ɻ෦ͷ֎ʹग़͖ͯͬͯͨͱ͖ɺصಉ ͡Ͱ͋Δͱߟ͑Δɻ͜ͷষͰѻ͏ɺ ʮصͷΑ͏ͳ͕ʮಉ͡ʯͰ͋Δ͜ ͱΛɺΘΕΘΕͲ͏ͬͯΔͷͰ͋Ζ͏͔ʯ ʢp. 94ʣͱ͍͏Ͱ͋Δɻ ʮํ͑ݟ༷ʑ͕ͩɺʪͦصͷͷʫಉ͡ͳͷͩʯͱ͍͏͑ΛஶऀͱΒ ͳ͍ɻ͜ͷΑ͏ͳ͑ʹରͯ͠ɺʮʮݱΘΕʯͱʮຊମʯΛ۠ʹີݫผ͢Δ ͳΒɺʮຊମʯݪཧ্ʮݱΘΕͳ͍ʯ͜ͱʹͳΔʯ͕ɺʮʮݱΘΕͳ͍ͷʯ ʹ͍ͭͯɺΘΕΘΕͲ͏ͬͯΔͷͩΖ͏͔ʯʢp. 94ʣͱ͍͏ٙΛ ͔͚͛Δɻͦͯ͠ɺݱΘΕʹଈͨ͠··ʮಉ͡ʯͱ͍͏͜ͱΛଊ͑Α͏ͱࢼ Έ͍ͯΔɻ͍ͯ͠Δͷɺࢹͷಈ͖ͱݱΘΕͷมԽͷରԠͱɺಈ͖Λ ͢ʹٯΔͱมԽ͕ͳʹٯΔͱ͍͏ՄٯੑͰ͋Δɻ ͋ΔํࢹΛͣΒͤɺصͷํ͑ݟҰఆͷํͰมΘΔ ͕ɺݩͷҐஔʹࢹΛͤɺͦΕʹԠͯ͡ɺصͷํ͑ݟΔɻ ͦ͠͏Ͱͳ͔ͬͨͱͨ͠ΒɺΘΕΘΕصͷ͕ܗຊʹมΘͬͯ ͠·ͬͨͱࢥ͏ͩΖ͏ɻʢp. 96ʣ صͷ͕ܗಉ͡Ͱ͋ΔͨΊʹɺݩͷҐஔʹࢹΛͤصͷࢹ͕ํ͑ݟ. 3. 277.
(4) Λಈ͔͞ͳ͔ͬͨ߹ͱมΘΒͳ͍ɺͭ·ΓɺՄͳٯม͍ͯͬͳͱΔඞཁ ͕͋Δɻୈ 3 ষͰड़ΒΕ͍ͯΔαΠίϩͷྫҾ༻͠Α͏ɻ. 1 ͷ໘͔Β 2 ͷ໘ʹݱΘΕ͕มԽ͠ɺ͙Δ͙Δճ͍ͯ͘͠ͱɺ 6 → 5 ͱมԽͯ͠࠶ͼ 1 ͷ໘͕ͯ͑͘ݟΔɻࠓʹ͖ٯճͯ͠ ΈΔɻ5 → 6 → 2 ͱมԽͯ͠ɺ࠶ͼ 1 ͕ͯ͑͘ݟΔɻԿ܁Γฦ͠ ͯɺಉ༷Ͱ͋Δɻ͜ͷΑ͏ͳͱ͖ɺզʑʮಉ͡ʯαΠίϩΛख ʹ͍ͯ͠Δͱࢥ͏ɻ࣮ࡍʹ͍ͯݟΔͷɺͦͷͭͲͷଟ༷ͳݱΘΕ ͳͷ͕ͩɺ͜ͷଟ༷ͳݱΘΕͷ࿈ͷͳ͔ʹɺʮՄٯʯͱ͍͏͜ͱ ͕͍ͯͬ͜ىΔɻʢp. 109ʣ ͜ͷྫʹ͍ͯͮجɺஶऀʮʮଟ༷ͳݱΘΕͷؒͷϓϩηεͷՄٯੑʯͦ͜ɺ ʮಉ͡ͷʯͷਖ਼ମͰ͋Δʯʢp. 109ʣͱड़͍ͯΔɻ͜ͷΑ͏ʹɺصαΠ ίϩͷࡐʹͯ͠ʮಉ͡͞ʯΛมͷՄٯੑ͔Βଊ͑Δ͜ͱ͕Ͱ͖Δͱ͍͏ ͜ͱΛઆಘతʹ͍ͯ͡Δɻ ୈ 4 ষͰɺͳͥʮࢲʯΛ࣮ମͱͯ͠ଊ͑Α͏ͱͯ͠͠·͏ͷ͔ͱ͍͏ Λߟ͢ΔɻஶऀʹΑΕɺઈ͑ͣมಈ͢Δڥզʑʹͱͬͯੜ͖ͮΒ ͍ͷͰɺզʑ͔֬͞ΛٻΊΔɻͦͷͱ͖ɺ࣌ؒతʹมԽ͢Δͷͷɺ࣌ʹ ΑΔࠩҟΛແࢹͯ͠ʮಉ͡ͷʯ͕࣌ؒΛ؏͍͍ͯΔͱߟ͑ɺ࣮ମతͳͷ ΛٻΊͯ͠·͏ɻ͔͠͠ɺ৽ͨͳݱ͕ݱΘΕͨΒɺ͜͏ͨ͠ࢼΈ࠳ં͠ ͯ͠·͏ɻୈ 3 ষͰड़ΒΕ͍ͯΔΑ͏ʹʮಉ͡͞ʯΛߟ͑ΔͨΊʹɺ࣮ ମͰͳ͘มͷ͔؍Βߟ͑Δඞཁ͕͋Δɻมͷ͔؍Βߟ͑Δͱɺ ʮಉ ͡͞ʯมͷՄٯੑͰ͋ͬͨɻ ৽ͨͳݱ͕ݱΕΔͱ͖ɺͦΕ͜Ε·ͰݟΒΕ͍ͯͨՄٯੑ Λյ͢ͷͰ͋Δͱͯ͠ɺΉ͠ΖͦΕ͕ཁૉͱͳͬͯ৽ͨͳՄٯ ੑ͕ͱͯݟΒΕ͏Δɻ͍ݹಉ͡͞ʹݻண͢Δ͜ͱɺ͜ͷΑ͏ͳߏ ͷ৽ͨͳΈʹ͑ରԠͰ͖ͳ͍ͱ͍͏͜ͱʹͳΔɻ৽ͨͳݱ ͱͱʹ৽ͨͳՄٯతߏ͕࣮ݱͷͳ͔ʹݱΘΕ͖ͯͨͱ͖ɺͦ Εʹରͯ͠ઈ͑ͣΛ։͍͍ͯ͘ͱ͍͏͜ͱͦ͜ɺ͋ΔҙຯͰΑΓ ʮ͔֬ͳʯੜ͖ํΛՄೳʹ͢Δͱ͏͍ݴΔɻʢp. 170ʣ ͜ͷҾ༻Ͱड़ΒΕ͍ͯΔ͜ͱɺ ʮಉ͡͞ʯมΘ͍ͬͯ͘ͱ͍͏͜ͱͰ ͋ΓɺҰͭͷʮಉ͡͞ʯʹݻண͠ͳ͍͜ͱɺྫ࣮͑ମతͳͷʹݻண͠ͳ ͍͜ͱͷॏཁੑͰ͋Δɻ ୈ 5 ষͰɺ ʮՊֶʹ͓͚Δ๏ଇͱ͍͏ͷʮಉ͡͞Λઃఆ͢Δ͜ͱʯʹ ґଘ͍ͯ͠Δʯ ʢp. 192ʣͱ͍͏ཱ͔ΒɺܾఆΛલఏͱ͢ΔཱΛ൷͠ ͍ͯΔɻྫ͑ɺఱจֶʮಉ͡͞ʯͷ߹͍ΛेʹߴΊΔ͜ͱʹΑܾͬͯ 278. 4.
(5) 〈同じさ〉とは何か. ఆతͳ๏ଇΛखʹೖΕͨͱ͍͑Δɻ͜ͷྫΛݟΔͱɺ ʮಉ͡͞ʯͷ߹͍Λ ߴΊ͍͚ܾͯఆతͳϞσϧΛखʹೖΕΔ͜ͱ͕Ͱ͖ΔΑ͏ʹΈ͑Δɻ͠ ͔͠ɺͨͩ୯ʹʮಉ͡͞ʯͷ߹͍ΛߴΊΔ͚ͩͰ͏·͍͘͘Θ͚Ͱͳ͍ ͱ͍͏͜ͱΛஶऀ࣍ͷΑ͏ͳྫͰઆ໌͍ͯ͠Δɻ ͋Δ͍ͯͭʹऀױʮฏۉ༨໋ʯΛߟ͑Δͱ͢Δ߹ɺͦͷऀױ ΛʮࡾेͷਓؒʯͷҰਓͱͯ͠ߟ͑Δ͔ɺʮஉੑʯͷҰਓͱͯ͠ ߟ͑Δ͔ɺ͋Δ͍ʮA ࢢࡏॅͷࡾेஉੑʯͷҰਓͱͯ͠ߟ͑Δ ͔ͳͲɺʮͲͷूஂͷҰһͱͯ͠ߟ͑Δ͔ʯʹΑͬͯͦͷਪଌ ͔ͳΒͣҟͳͬͯ͘ΔͣͰ͋ΔɻͰͱͯͬݴɺ݅ΛڧΊΕ ڧΊΔ΄Ͳྑ͍ͩΖ͍͏ͱ͍͏Θ͚ʹ͍͔ͳ͍ɻʮಉ͡͞ʯͷ ߹͍ΑΓߴ·ΔͩΖ͏ͱ͍͏ͷૉͳߟ͑Ͱ͋Δ͕ɺҰํ Ͱͦͷूஂ͕ͲΜͲΜগͳ͘ͳΔͨΊɺ๏ଇͷ৴པੑ͕֮ଋͳ͘ ͳͬͯ͘Δ͔ΒͰ͋Δɻʢp. 192ʣ զʑʮಉ͡͞ʯͷόϥϯεΛऔΔඞཁ͕͋Γɺఱจֶ͜ͷόϥϯεΛ औͬͨ݁ՌɺܾఆతʹͳͬͨΘ͚Ͱ͋Δɻ ʮࣗવʹ͓͍ܾͯఆతͳଊ͑ํ ͕ଥͰ͋Δͱ͑ݴΔͷΉ͠Ζಛघͳέʔεʯʢp. 193ʣͰ͋Γɺܾఆ͕ ઈରతͳͷͰ͋ΔΘ͚Ͱͳ͍ɻ Ҏ্ɺৗతͳྫΛத৺ʹຊॻͷʮಉ͡͞ʯͷߟ͑ํΛհͨ͠ɻ. 1. Պֶత๏ଇΛߟ͑ΔͨΊʹʮಉ͡͞ʯΛઃఆ͢Δඞཁ͕͋Γɺ͍͕ ͳ͚Ε͑ͳ͍ɻ 2. ʮಉ͡͞ʯཱ͕͢Δʹඇن४తબ͕ෆՄܽͰ͋Δɻ 3. ʮಉ͡͞ʯΛߟ͑ΔͨΊʹ࣮ମͰͳ͘มΛߟ͑Δඞཁ͕͋Γɺ ʮಉ ͡͞ʯมͷՄٯੑͰ͋Δɻ Պֶత๏ଇΛߟ͑ΔͨΊʹʮಉ͡͞ʯΛઃఆ͢Δඞཁ͕͋Δͷ͕ͩɺຊ ॻͰ͜ΕΛՄͳٯมͷ͔؍Βߟ͍͑ͯΔɻ͜ͷઅͰৗతͳྫΛ ͱʹհ͕ͨ͠ɺ͜ͷ؍ͷࣹఔͷ͞Λࣔͨ͢ΊʹɺҎԼͷઅͰ DHR ཧ ͱΑΕΔՊֶཧʹ͓͚Δʮಉ͡͞ʯΛՄࣗͳٯવมͷ͔؍Βݕ౼ ͢Δɻ. 3 3 DHR ཧʹ͓͚Δཻࢠ DHR 理論における粒子 2 અͰɺʮಉ͡͞ʯΛத৺ʹຊॻΛհͨ͠ɻ͜ͷઅҎ߱ɺҰࢠཻݟత ͳঢ়ଶΛѻ͍ͬͯΔΑ͏ʹΈ͑Δ DHR ཧ [3, 4, 5, 6] ͱΑΕΔྔࢠཧ ͷҰͭͷཧΛऔΓ্͛Δ 3 ɻͦͯ͠ɺͲͷΑ͏ͳҙຯʹཻ͓͍ͯࢠతͳঢ় ଶΛѻ͍ͬͯΔΑ͏ʹΈ͑Δͷ͔Λઆ໌͢Δɻ 5. 279.
(6) DHR ཧɺతͷྔࢠͷΈ [1] Ͱల։͞ΕΔɻతͷྔ ࢠͰɺ༗քͳۭ࣌ྖҬʹ؍ଌՄೳྔΛׂΓͯΔɻ؍ଌՄೳྔ C* ͷࣗڞݾͳݩɺঢ়ଶ C*্ͷઢܗ൚ؔͰද͞ΕΔɻঢ়ଶʹΑͬͯɺ ؍ଌՄೳྔͷظΛ͢ࢉܭΔ͜ͱ͕Ͱ͖Δɻ 2 અͷ࠷ޙͷՕॻ͖ͷ 1 Ͱड़ͨΑ͏ʹɺཧΛߟ͑Δ্Ͱʮಉ͡͞ʯΛ ઃఆ͢Δඞཁ͕͋Δɻྔࢠʹ͓͍ͯॏཁͳʮಉ͡͞ʯϢχλϦಉ 4 Ͱ ͋Δɻ2 ͭͷද͕ݱϢχλϦಉͱɺҰํͷද͚͓ʹݱΔ؍ଌՄೳྔʹର͠ ͯɺ͏Ұํͷද͍͓ͯʹݱඞͣҰͭͷ؍ଌՄೳྔ͕ରԠ͢Δͱ͍͏Α͏ ͳ݅Ͱ͋Δɻ͜ͷΑ͏ͳ݅Λຬͨ͢ͱ͖ʮಉ͡ʯͱΈͳ͞ΕΔɻ͜Εɺ తͷྔࢠʹݶΒͣྔࢠҰൠʹ͓͍ͯɺΑ͘ΘΕΔʮಉ͡͞ʯͰ ͋Δɻ ͜ͷʮಉ͡͞ʯͷΠϝʔδΛઆ໌͢ΔͨΊʹɺମͷ͞ΛఆͰنଌΔ͜ ͱΛߟ͑Α͏ɻೋछྨͷఆ͕͋نΓɺҰͭͷఆنͷ୯Ґ cmɺ͏Ұํͷఆ نͷ୯Ґੇͱ͠ɺҰੇ 3.03cm ͱ͠Α͏ɻ͕ 3.03cm ͷࢠڙͷΛ ୯Ґ͕ cm ͷఆͰنଌΔͱ 3.03cm ͱͳΓɺ୯Ґ͕ੇͷఆͰنଌΔͱҰੇͱͳ ΔɻҰରҰʹʹ͍ޓରԠ͢ΔͷͰɺͦΕͧΕͷ͞ͷද͠ํҟͳΔ͕ɺಉ ͡͞ͱ͖Ͱ͕ͱ͜͢ͳݟΔɻಉ༷ʹɺϢχλϦಉͳදݱɺද͕ݱҧͬ ͍ͯͯʹ͍ޓҰରҰʹରԠ͢ΔͷͰಉ͡ͱΈͳ͢͜ͱ͕Ͱ͖Δɻ ௨ৗͷྔࢠɺώϧϕϧτۭؒΛͱʹల։͞ΕΔɻҰํɺC*Λ ͱʹͨ͠తͷྔࢠͰώϧϕϧτۭؒΛԾఆ͠ͳ͍ɻ྆ऀΛ݁ͼͭ ͚Δͷ GNS ߏఆཧ [12, ఆཧ 3.34] Ͱ͋Δ 5 ɻC*ͷݩɺͦͷ C* ্ͷঢ়ଶ͔Β GNS ߏఆཧΛ௨ͯ͠ɺ͋Δώϧϕϧτ্ۭؒͷ࡞༻ૉͱ͠ ͯද͞ݱΕΔɻ͜ͷΑ͏ͳදݱɺϢχλϦಉੑΛͷ͍ͧͯҰҙͰ͋Δɻಛ ʹਅۭΛද͢ঢ়ଶ [1, p. 10] ͔Βߏ͞ΕΔදݱਅۭදͱݱΑΕɺDHR ཧʹ͓͍ͯॏཁͳׂΛՌͨ͢ɻ DHR ཧʹ͓͍ͯѻΘΕΔදݱɺҙͷ༗քͳۭ࣌ྖҬʹରͯ͠ɺͦͷ ྖҬͱۭؒతʹΕͨྖҬʹ͓͍ͯਅۭදͱݱϢχλϦಉͱͳΔΑ͏ͳද ͋ͰݱΔɻ͜ͷදݱɺແݶԕʹ͓͍ͯਅۭঢ়ଶͱͳΔΑ͏ͳঢ়ଶͷදͱݱ ͍͏͜ͱͰ͖Δ 6 ɻͦͯ͠ɺ͜ͷΑ͏ͳঢ়ଶͷදݱɺ؍ଌՄೳྔ͔Β؍ଌ Մೳྔͷࣗݾ४ಉͱ૾ࣸܕରԠ͍ͯͯ͠ɺ͜ͷࣸ૾ࡏہՄೳͱҠಈՄೳ ͱ͍͏ੑ࣭Λͭ [1, Thoerem 2.1.3] [7, Definitions 150 and 152]ɻ͞Βʹɺ ࡏہՄೳͰҠಈՄೳͳࣗݾ४ಉͱ૾ࣸܕਅۭදݱΛͬͯঢ়ଶΛఆٛ͢Δ͜ ͱ͕Ͱ͖Δɻͭ·ΓɺࡏہՄೳͰҠಈՄೳͳࣗݾ४ಉ૾ࣸܕɺঢ়ଶͱରԠ ͍ͯ͠Δɻ্Ͱड़ͨΑ͏ʹɺঢ়ଶͱදݱରԠ͍ͯ͠ΔͷͰɺࡏہՄೳͰ ҠಈՄೳͳࣗݾ४ಉ૾ࣸܕΛѻ͏ͱ͍͏͜ͱɺঢ়ଶΛѻ͍ͬͯΔͱ͍͏͜. 280. 6.
(7) 〈同じさ〉とは何か. ͱͰ͋Δɻ ࡏہՄೳͱ͍͏ੑ࣭ɺ༗քͳۭ࣌ྖҬ͕ଘࡏͯ͠ɺͦͷྖҬͱۭؒతʹ ΕͨྖҬͷ؍ଌՄೳྔʹରͯ͜͠ͷࣗݾ४ಉ߃͕૾ࣸܕࣸ૾ͱͳΔΑ͏ ͳੑ࣭ɺͭ·Γͦͷ༗քͳۭ࣌ྖҬҎ֎ਅۭͱΈͳͤΔΑ͏ͳੑ࣭Ͱ͋Δɻ ͦͯ͠ɺ͜ͷΑ͏ͳࣗݾ४ಉ૾ࣸܕɺͦͷ༗քͳۭ࣌ྖҬʹ͍ͯ͠ࡏہΔ ͱ͍͏ɻ͜ͷΑ͏ͳੑ࣭Λͭ߹ɺ༗քͳۭ࣌ྖҬʹ͓͍ͯͷΈਅۭͱ ҟͳΓɺͦͷ֎Ͱਅۭͱͳ͍ͬͯΔɻཻࢠͱ͍͏ͷΛ༗քͳۭ࣌ྖҬʹ͋ Δͷͱߟ͑ΔͱɺࡏہՄೳͱ͍͏ੑཻ࣭ࢠతͳੑ࣭ͷҰ෦ͱߟ͑ΒΕΔɻ ͔͠͠ɺࡏہՄೳͱ͍͏ੑ࣭͚ͩͰɺཻࢠతͳੑ࣭ͱ͍͑ͳ͍ɻͦͷ ͜ͱΛઆ໌͢ΔͨΊʹɺ2 અͰհͨ͠صͷྫͱಉ༷Ͱ͋Δ͕ɺ͜͜Ͱͷٞ ͱؔ࿈͕͔Γ͍͢ଞͷจͰड़ΒΕ͍ͯΔྫΛҾ༻͠Α͏ɻ ಈ͍͍ͯΔϘʔϧΛɺΘΕΘΕʮಉ͡ͷʯͱ͢ͳݟɻͳͥͦ Μͳ͜ͱ͕ՄೳͳͷͩΖ͏͔ɻϘʔϧࠁҰࠁҐஔΛม͍͑ͯΔɻ ͦΕΛผʑͷରͱͱ͜͢ͳݟͰ͖ΔͣͰ͋Δɻ ͔ͨ͠ʹɺϘʔϧͷҐஔ࣌ؒͱʹڞมԽ͢Δɻ͔͠͠ɺϘʔ ϧΛҐஔ A ͔ΒҐஔ B ʹ্࣋ͪ͛ͯɺ·ͨҐஔ A ʹ͢ͱ͖ɺͦ ΕԿ͠ͳ͔ͬͨͷͱಉ͡ʹͳΔɻ͜ͷΑ͏ͳͱ͖ɺΘΕΘΕ ͜ΕΛʮಉ͡Ϙʔϧʯͱ͢ͳݟɻ͜Εඞͣࣗ͠໌Ͱͳ͍ɻର Λ্࣋ͪ͛ͯɺ·ͨԼΖͨ͠Βɺผͷͷʹͳ͍ͬͯͨɺ্࣋ͪ ͛ͯԼΖͨ͠Βফ͑ͯ͠·ͬͨɺͱ͍ͬͨέʔεߟ͑ΒΕΔɻ͋ Δૢ࡞ͱผͷૢ࡞Λ߹ͨ͠ΒɺผͷͷʹͳΔํ͕ɺΉ͠Ζࣗવ Ͱ͋Δͱ͑͞ࢥ͑ͯ͘Δɻ͋Δૢ࡞Λ૬ࡴ͢Δผͷૢ࡞͕͋Δɺ͋ Δ͍ΑΓҰൠతʹɺ͋Δಈ͖Λ૬ࡴ͢Δผͷಈ͖͕͋Δͱ͍͏ك ͳࣄଶ͕ੜ͍ͯ͡Δͱ͖ɺΘΕΘΕͦ͜ʹʮಉ͡͞ʯΛݟΔͷͰ ͋Δɻ[11, p. 205] ͜ͷҾ༻จͷதͷೋஈམͷจষͷதͷϘʔϧΛཻࢠʹஔ͖ͯ͑ΈΑ͏ɻ ཻࢠΛ͋ΔྖҬ͔ΒଞͷྖҬʹҠಈ͠࠶ͼͱͷྖҬʹͨ͠ͱ͖ɺ͜Ε͕ Կ͠ͳ͔ͬͨͷͱಉ͡ʹͳΔΑ͏ͳͱ͖ɺΘΕΘΕ͜ΕΛʮಉཻ͡ࢠʯͱ Έͳ͢ɻ͜Εʹؔ࿈͢Δੑ࣭͕ҠಈՄೳͱ͍͏ੑ࣭Ͱ͋Δɻ͋Δ༗քͳۭ࣌ ྖҬʹ͍ͯ͠ࡏہΔࣗݾ४ಉ ૾ࣸܕρ Λߟ͑ɺ͜Ε͕ҠಈՄೳͱ͍͏ੑ࣭Λ ຬ͍ͯ͠Δͱ͠Α͏ɻ͜ͷͱ͖ɺҙͷ༗քͳۭ࣌ྖҬ O ʹରͯ͠ɺρ ͱ ϢχλϦಉͳࣗݾ४ಉ ૾ࣸܕρ ͕ଘࡏͯ͠ɺρ O ʹ͍ͯ͠ࡏہΔɻ͜Ε ɺ͋Δ༗քͳۭ࣌ྖҬʹ͍ͯ͠ࡏہΔ ρ Λଞͷ༗քͳۭ࣌ྖҬ O ʹҠಈ͠ ͨͱߟ͑ΒΕΔɻͦͯ͠ɺρ ͱͷ༗քͳۭ࣌ྖҬʹ͍ͯ͠ࡏہΔ ρ ͱϢ χλϦಉͳͷͰɺO ʹҠಈͯ͠ͱͷྖҬʹͨ͠߹ͱɺԿ͠ͳ͔ͬ. 7. 281.
(8) ͨ߹ಉ͡ʹͳΔɻ͜ͷΑ͏ʹɺࡏہՄೳͰҠಈՄೳͳࣗݾ४ಉ૾ࣸܕɺ ͍Ζ͍Ζͳۭ࣌ྖҬʹ͍ͯͯ͠ࡏہɺ͜ͷҾ༻จͷҙຯͰʮಉ͡ͷʯͱ Έͳ͢͜ͱ͕Ͱ͖ΔɻDHR ཧͰѻ͏දݱɺࡏہՄೳͰҠಈՄೳͳࣗݾ४ ಉͱ૾ࣸܕਅۭදͰݱද͢͜ͱ͕Ͱ͖Δͷ͔ͩΒɺཻࢠతͳঢ়ଶΛѻ͏ཧ ͱߟ͑Δ͜ͱ͕Ͱ͖Δɻ ͋Δঢ়ଶʹରԠ͢Δࣗݾ४ಉ૾ࣸܕͷͱͰ༗քͳۭ࣌ྖҬʹ͓͍ͯૢ࡞ Λߦ͏ͱɺҰൠʹҧ͏ঢ়ଶʹରԠ͢Δࣗݾ४ಉͳʹ૾ࣸܕΔɻDHR ཧͷ ΈͰɺૢ࡞Λߦ͏ͱɺࡏہՄೳͰҠಈՄೳͳࣗݾ४ಉ ૾ࣸܕρ ɺҧ͏ ࡏہՄೳͰҠಈՄೳͳࣗݾ४ಉ ૾ࣸܕρ ʹม͞ΕΔɻρ ͱ ρ Λ͚ؔͮΔ བྷ࡞༻ૉͱΑΕΔ࡞༻ૉͷू߹ (ρ, ρ ) ͱॻ͔ΕΔ [1, p. 73] [7, Definition 154]ɻ ͜ͷઅͰड़ͨ DHR ཧͷղऍΛ·ͱΊΔͱɺ࣍ͷΑ͏ʹͳΔɻҰൠʹ ɺཻࢠతͳঢ়ଶʹରԠͨࣗ͠ݾ४ಉ ૾ࣸܕρ ʹ (ρ, ρ ) ͷཁૉ͕࡞༻͠ɺଞ ͷཻࢠతͳঢ়ଶʹରԠͨࣗ͠ݾ४ಉ ૾ࣸܕρ ʹม͞Ε͍ͯ͘ɻρ ͱ ρ ͕Ϣ χλϦಉͱ͍͏ಛघͷ߹ɺҧ͏ۭ࣌ྖҬʹ͍ͯͯ͠ࡏہʮಉ͡ʯཻ ࢠతͳঢ়ଶʹରԠͨࣗ͠ݾ४ಉͱ૾ࣸܕղऍ͞ΕΔɻ. DHR 理論における対象と射 4 4 DHR ཧʹ͓͚Δରͱࣹ ຊॻʹ͓͍ͯॏཁͳׂΛՌ͍ͨͯ͠ΔͷͷҰͭݍͰ͋Δɻ͜ͷઅ Ͱɺຊॻͷ͔؍Β DHR ཧͱݍͷؔΛߟ͑Δɻ ·ͣɺؔ͢ʹݍΔຊॻͷ؍తͳઆ໌ΛҾ༻͢Δɻ ͋ΔݱΘΕ͔ΒผͷݱΘΕͷʮมԽʯͱݺΕ͍ͯͨͷɺ ͍͓ͯʹݍɺʮࣹʯʢmorphismʣͱݺΕΔҰछͷʮҹʯͱ͠ ͯѻΘΕΔɻͱݍɺ͋ΔछͷࣹͷωοτϫʔΫͰ͋Δɻʢp. 99ʣ. 3 અͰड़ͨ͜ͱݍͷݴ༿Ͱड़Δ͜ͱ͕Ͱ͖Δɻ͋ΔࡏہՄೳͰҠ ಈՄೳͳࣗݾ४ಉ ૾ࣸܕρ ρ ରͰ͋Γɺρ ͔Β ρ ͷมΛද͢བྷ࡞ ༻ૉͷू߹Λ (ρ, ρ ) ͱද͠هɺ͜Ε͕ʮࣹʯͱͳΔ [7, Definition 154]ɻ͜͏ ͨ͠ݍΛ E ͱॻ͘ 7 ɻ3 અͰड़ͨΑ͏ʹɺࡏہՄೳͰҠಈՄೳͳࣗݾ४ಉܕ ࣸ૾ɺཻࢠతͳঢ়ଶʹରԠ͢ΔͷͰ͋ͬͨɻ͕ͨͬͯ͠ɺE ཻࢠతͳঢ় ଶΛରɺཻࢠతͳঢ়ଶ͔Βཻࢠతͳঢ়ଶͷมΛࣹͱ͢ΔΑ͏ͳͱݍΈ Δ͜ͱ͕Ͱ͖Δɻ ࣮ࡍʹɺཻࢠతͳঢ়ଶͷΈ͕୯ಠͰ͋ΔΘ͚Ͱͳ͘ɺͦΕʹର͢Δ༷ʑ ͳ࡞༻͕͋Γɺଞͷཻࢠతͳঢ়ଶʹมԽ͍ͯ͘͠ɻ͜ͷΑ͏ͳมԽΛଊ͑Δͨ Ίʹɺཻࢠతͳঢ়ଶʹରԠͨ͠ର ρ Λݻఆ্ͨ͠Ͱɺؔख (ρ, −) : E → Set Λఆٛ͠Α͏ [2, Example 1.2.8.d] [14, pp. 73-76]8 ɻ͜ͷؔखɺཻࢠతͳ 282. 8.
(9) 〈同じさ〉とは何か. ঢ়ଶʹରԠͨ͠ର σ ʹରͯ͠ (ρ, σ) ΛׂΓͯΔɻDHR ཧͷΈͰड़ Δͱɺ͋Δ༗քͳۭ࣌ྖҬʹࢠཻͨ͠ࡏہతͳঢ়ଶʹରԠͨ͠ର ρ Λݻ ఆ্ͨ͠Ͱɺρ ͔Βଞͷཻࢠతͳঢ়ଶʹରԠͨ͠ର σ ͷมԽͷू߹Λߟ ͍͑ͯΔɻ͜Εɺρ Λத৺ͱࣹͨ͠ͷωοτϫʔΫɺͭ·Γɺρ ͔Βଞͷཻ ࢠతͳঢ়ଶʹରԠͨ͠ରʹͳΔΑ͏ͳ࡞༻ͷωοτϫʔΫΛߟ͍͑ͯΔ͜ ͱʹͳΔɻ. 5 5 DHR ཧʹ͓͚Δࣗવม DHR 理論における自然変換 3 અͱ 4 અͰɺDHR ཧΛཻࢠʹؔ͢ΔཧͰ͋Δ͔ͷΑ͏ʹհ͖ͯ͠ ͨɻ4 અͰಋೖͨؔ͠ख (ρ, −) ཻࢠతͳঢ়ଶ ρ ͔Βଞͷཻࢠతͳঢ়ଶʹͳ ΔΑ͏ͳ࡞༻ͷωοτϫʔΫͰ͋ͬͨɻ͔͠͠ɺຊॻͷཱ͔ΒݟΕɺ͜ Εཻࢠͱ͍͏ʮ੩తɾݻணతͳʮઈରੑʯΛମʹܥʮಡΈࠐΜͰ͠·ͬͯ ͍Δʯʯʢp. 179ʣͱ͍͑ΔͩΖ͏ɻ2 અͷ࠷ޙͷՕॻ͖ͷ 3 Ͱड़ͨΑ͏ ʹɺʮಉ͡͞ʯΛߟ͑ΔͨΊʹ࣮ମͰͳ͘มΛߟ͑Δඞཁ͕͋Δɻ DHR ཧͱҰݟແؔͳຊॻͷ AɺBɺC ͷձͷྫҾ༻͠Α͏ʢpp. 144-145ʣɻ A ʮࢲ͜ͷຊ͕͖ͳΜͰ͢ɻʯ B ʮࢲͰ͢ʂʯ C ʮࢲͦ͏Ͱͳ͍͔ͳɾ ɾ ɾʯ ͜ͷձࣗମʹҧײΛ͡ײΔਓ͍ͳ͍ͩΖ͏ɻ͔͠͠ɺ࣍ͷΑ͏ʹࢲ Λ W ʹม͑Δͱɺʮࢲʯͱ͍͏ݴ༿ͷಛҟੑ͕Θ͔Δɻ. A ʮW ͜ͷຊ͕͖ͳΜͰ͢ɻʯ B ʮW Ͱ͢ʂʯ C ʮW ͦ͏Ͱͳ͍͔ͳɾ ɾ ɾʯ ͜ͷձΛͨݟਓɺW ͱ͍͏Ͱ߸هಉҰਓΛ૾ͯ͠͠·͏ɻͦͷͨ Ίɺ͜ͷձͷҙຯΑ͔͘Βͳ͍ɻ͔͠͠ɺલͷձΛͨݟਓɺʮࢲʯ ͱ͍͏Ͱ߸هಉҰਓΛ૾͢ΔͷͰͳ͘ɺͦͷͭͲҧ͏ਓΛࢦ͍ͯ͠ Δͱߟ͑ΔɻA B C ʮࢲʯͱ͍͏͜ͱ͕Ͱ͖Δ͕ɺʮࢲʯͱ͍͏ݴ༿ ͷதͦΕͧΕҟͳ͍ͬͯͯஔ͖͑ՄೳͰ͋Δɻ W ͱʮࢲʯͷҧ͍ɺW ͋ΔݸମΛݻఆతʹࢦ͍ࣔͯ͠Δͷʹରͯ͠ɺ ʮࢲʯͱͱ͜͏ݴʮແʹݶ։͔Εͨஔ͖͑Մೳੑͷͳ͔ʹࣗࣗΛஔ͘ʯ ʢp. 147ʣͱ͍͏͜ͱͰ͋Δɻ͜ͷΑ͏ʹɺஶऀʹΑΕʮࢲʯมஔ͖ ͑Մೳੑͱͯ͠ଊ͑ΒΕΔͷͰ͋ΓɺԿΒ͔ͷ࣮ମͱͯ͠ʮࢲʯͱ͍͏ ࣮ݱΛଊ͑Δ͜ͱͰ͖ͳ͍ 9 ɻஶऀɺ࣍ͷΑ͏ʹड़͍ͯΔɻ. 9. 283.
(10) ʮࢲʯͱ͍͏ݴ༿Ͱ͍ݴද͞Ε͍ͯΔͷɺ ʮݸମత࣮ମʯͱ ͯ͋͠ΔΘ͚Ͱͳ͘ɺ ʮීวత࣮ମʯͱͯ͋͠ΔΘ͚Ͱͳ͍ɻΉ ͠ΖɺͦͷΑ͏ʹʮ࣮ମʯతͳߟ͑ํʹͱΒΘΕ͍ͯΔͱɺ ʮࢲʯͱ ͍͏ͷͷຊ࣭Λ͔ͭΉ͜ͱ͕Ͱ͖ͳ͘ͳͬͯ͠·͏ͷͰ͋Δɻ͜ Ε·Ͱల։͖ͯͨ͠ߟ͑ํʹै͏ͳΒɺʮࢲʯͱ͍͏ͷʮஔ͖ ͑ʯʮมʯͷͳ͔ʹ͋ΔɻݸମੑීวੑΛઈରԽͤͣʹɺ ʮஔ͖͑ʯͱ͍͏ग़དྷࣄͷ͏ͪͰɺͦͷ྆ऀ͕ෆՄʹབྷΈ߹ͬ ͍ͯΔ͞·Λ͖ͱͨͬͱͯݟɺ ʮࢲʯͱ͍͏ͷͷຊ࣭͕͔ͭΊΔɻ ʢp. 159ʣ ஶऀʮࢲʯʹؔ͢Δ͕ٞɺཻࢠΛ࣮ମͱͯ͠ଊ͑Δ͜ͱͷʹؔ ͢Δٞͱࣅ͍ͯΔͱࢦఠ͍ͯ͠Δʢp. 158ʣɻ࣮ࡍɺ͜ͷҾ༻จͷதͷʮࢲʯ Λʮཻࢠʯʹஔ͖͑Δͱɺ͜ͷҾ༻จʮཻࢠʯʹ͍ͭͯड़͍ͯΔ͜ͱ ʹ͕ͩͭ͘ؾΖ͏ɻຊॻͷཱ͔Βߟ͑Δͱɺมͷ͔؍ΒཻࢠΛߟ͑Δ ඞཁ͕͋Δɻ 4 અͰɺରɺࣹɺؔख͔͠ߟ͍͑ͯͳ͔͕ͬͨɺʮʮࣗવมʯΛఆٛ ͍͔ͨ͠Βͦ͜ݍ͕ੜ·Εͨʯʢp. 127ʣͱ͍͏͔؍Βߟ͑ͯɺରɺ ࣹɺؔखΛߟ͑Δ͚ͩͰෆेͰ͋Δͱ͍͑Δɻୈ 3 ষʹ͓͍ͯɺࣗવม ͍ͯͭʹ࣍ͷΑ͏ʹड़ΒΕ͍ͯΔɻ ؔखͱʮ͔ݍΒݍͷ͚ؔͮʯͰ͋Γɺࣗવมͱʮؔ ख͔Βؔखͷ͚ؔͮʯɺͭ·Γʮ͚͔ؔͮΒ͚ؔͮͷؔ ͚ͮʯͰ͋Δɻʢp. 118ʣ ʮݸʑͷݱΘΕʯ=ؔख͕ੜ·ΕΔ͜ͱ͕ඇن४తબͰ͋Γɺ ͔ͦ͠Ε͕ʮͦͷؔखͰͳͯ͘Α͔ͬͨʯ ʮผͳؔखʹมՄೳʯ ͱ͍͏͜ͱΛ௨ͯ͡ඇن४ੑ͕ʮফ͞ΕΔʯͱ͍͏͜ͱɺͦͷߏ =ग़དྷࣄ͕ʢ...ʣࣗવมʹରԠ͍ͯ͠ΔͷͰ͋Δɻ ʢpp. 129–130ʣ. 4 અͰड़ͨؔख (ρ, −) Λߟ͑Α͏ɻ͜ͷؔखɺཻࢠతͳঢ়ଶʹରԠ͢ Δࣗݾ४ಉ ૾ࣸܕρ Λɺρ ͔Βҧ͏ཻࢠతͳঢ়ଶʹରԠ͢Δࣗݾ४ಉ૾ࣸܕ ͷมͷू߹ͷ͚ؔͮͰ͋ͬͨɻ͜ͷؔखΛߟ͑ΔͨΊʹɺ·ͣର ρ ΛબͿඞཁ͕͋ͬͨɻ͔͠͠ɺρ ͨ·ͨ·બ͞Εͨʹ͗͢ͳ͍ͷͰɺ ؔख (ρ, −) Ͱͳ͘ɺଞͷର σ ʹͮ͘جଞͷؔख (σ, −) ͰΑ͔ͬͨɻ͜ ͷඇن४ੑΛফͨ͢Ίʹɺؔख (σ, −) ͔Βؔख (ρ, −) ͷ͚ؔͮɺͭ·Γ ؔख (σ, −) ͔Βؔख (ρ, −) ͷࣗવม 10 Λߟ͑ͳ͚ΕͳΒͳ͍ [14, pp. 127-130]ɻ. 284. 10.
(11) 〈同じさ〉とは何か. ຊॻͰɺ ʮʮ࠲ඪมʯΛҰൠԽͨ͠ͷ͕ɺؔख͔Βؔखͷ͚ؔͮʹ ʮࣗવมʯͱݺΕΔͷʯʢp. 125ʣͱड़͍ͯΔɻ ؔखΛߏங͢Δ͜ͱΛʮ࠲ඪԽʯͱᄻ͑ͨɻ͋Δʮ࠲ඪԽʯ͔ Βผͳʮ࠲ඪԽʯͷ͚ؔͮɺ͢ͳΘͪ࠲ඪมΛߟ͑Δ͜ͱ͕ ॏཁͳҙࣝͰ͋ΔΑ͏ʹɺ͋Δʮؔखʯ͔Βผͳʮؔखʯͷ ʮࣗવมʯΛߟ͑Δ͜ͱ·ͨॏཁͰ͋Δɻʢp. 125ʣ ࠲ඪมͰɺ͋Δ࠲ඪʹ͓͚ΔΛଞͷ࠲ඪʹ͓͚Δʹ͚ؔͮΔͩ ͚Ͱͳ͘ɺ͋Δ࠲ඪʹ͓͚Δ͚ؔͮଞͷ࠲ඪʹ͓͚Δ͚ؔͮʹؔͮ ͚͍ͯΔɻಉ༷ʹࣗવม͋Δؔखʹ͓͚Δ͚ؔͮΛଞͷؔखʹ͓͚Δ ͚ؔͮʹࣗવʹ͚ؔͮΔ 11 ɻ ͜͜Ͱɺର ρ Λબͼؔख (ρ, −) Λߟ͑Δ͜ͱ͕࠲ඪԽʹ͋ͨΔɻର ρ ΛબͿͱ͍͏બඇن४తબͰ͋Δɻଞͷର σ Λબͼଞͷ࠲ඪԽͰ͋ Δؔख (σ, −) Λߟ͑Δ͜ͱ͕Ͱ͖ΔɻຊॻͷཱʹཱͭͳΒɺؔख (σ, −) ͔Βؔख (ρ, −) ͷࣗવมΛߟ͑Δ͜ͱ͕ॏཁʹͳΔɻ6 અͰɺ͜ͷΑ͏ ͳࣗવมΛݕ౼͢Δɻ. 6 6 DHR ཧʹ͓͚ΔถాຒΊࠐΈ DHR 理論における米田埋め込み. 5 અͰड़ͨΑ͏ʹɺຊॻͷཱʹཱͭͳΒɺؔख (σ, −) ͔Βؔख (ρ, −) ͷ͚ؔͮͰ͋ΔࣗવมΛߟ͑Δඞཁ͕͋Δɻ͔͠͠ɺDHR ཧର ρ ͔Β σ ͷࣹ (ρ, σ) Λओʹߟ͑Δ͕ɺ໌ࣔతʹ͜ͷΑ͏ͳࣗવมߟ͑ͳ ͍ɻͦͷͨΊɺDHR ཧҰݟຊॻͷཱͱંΓ߹͍͕ѱ͍Α͏ʹΈ͑Δɻ ͔͠͠ɺ ถాຒΊࠐΈ [2, Proposition 1.5.2] [14, pp. 131-135] ʹͱͮ͘جɺ ͦ͏Ͱͳ͍ɻถాຒΊࠐΈʹؔͯ͠ɺஶऀ࣍ͷΑ͏ʹड़͍ͯΔɻ ؔखͱɺ·͞ʹݱʹݍ͕ݍΕΔɺͦͷʮݱΘΕʯͰ͋Δɻͦ ͯ͠ɺͦͷʮݱΘΕͷมԽʢಈ͖ɾϓϩηεʣʯ͕ࣗવม͏͍ͱ ͜ͱʹͳΔɻ͜ͷΑ͏ʹͯ͠ɺؔखΛରͱ͠ɺࣗવมΛࣹͱ͢ ΔݍΛߟ͑Δ͜ͱ͕Ͱ͖Δɻ͜ΕΛؔखͱݍΑΜͰ͍Δɻ࣮ɺຊ ࣭తʹ͋ΒΏΔݍ͋ΔؔखݍͷҰ෦ͱߟ͑ΒΕΔ͜ͱ͕ΒΕ ͍ͯΔʢถాຒΊࠐΈʣɻ͜͏ͯ͠ɺݍͷΈ͕៉ྷʹҰपཐ ટ֊ஈΛొΔʢ͋Δ͍Լͱ߱Γ͍ͯ͘ʣɻʢp. 127ʣ ͜ͷҾ༻ͷ༰Λ DHR ཧͷΈͰड़Δͱɺର ρ ͔Βର σ ͷ ࣹɺؔख (σ, −) ͔Βؔख (ρ, −) ͷࣗવมͱಉ͡ͱΈͳͤΔɻ͏গ͠ ஸೡʹड़ΔͱɺNat((σ, −), (ρ, −)) Λ (σ, −) ͔Β (ρ, −) ͷࣗવม͠ͱ ͯɺN : E → Set ͱ͍͏ؔखΛ N (σ) = Nat((σ, −), (ρ, −)) ͱఆٛ͢Δͱɺ. 11. 285.
(12) ؔख N ͔Βؔख (ρ, −) ͷՄࣗͳٯવม͕ଘࡏ͢Δ 12 ɻʮՄࣗͳٯવม Λ௨ͯ͡ɺʮಉ͡͞ʯཱ͕͢Δʯʢp. 142ʣͱ͍͏ຊॻͷཱ͔ΒΈΔͱɺ Nat((σ, −), (ρ, −)) ͱ (ρ, σ) ʮಉ͡ʯʹͳΔɻ ͜ͷΑ͏ʹɺࣗવมʮʮؔख͔Βؔखͷ͚ؔͮʯɺͭ·Γʮؔͮ ͚͔Β͚ؔͮͷ͚ؔͮʯʯ ʢp. 118ʣͱ͍͏நͷߴ͍ͷͰ͋Δ͕ɺ ͜ͷΑ͏ʹநͷߴ͍͚ؔͮɺର͔Βରͷ͚ؔͮͰ͋Δࣹͱ ಉ͡Ͱ͋Δͱ͍͏͜ͱͰ͋ΔɻҾ༻จͷൺᄻʹैͬͯड़Δͱɺࣹͱ͍͏ ໘͔Βग़ൃ͚ͯؔͮ͠ͱ͍͏໊ͷ֊ஈΛొ͍ͬͯͨΒɺ͍ͭͷؒʹ͔Լʹ ߱Γ͍ࣹͯͯͱ͍͏໘ʹ͍ͬͯͨͱ͍͏ΤογϟʔͷֆͷΑ͏ͳͰ͡ײ ͋Δɻࠓͷ߹ͩ·͠ֆͰͳ͘ɺԼͱ্͕Մࣗͳٯવมͷ͔؍Βಉ ͡ʹͳ͍ͬͯΔɻ DHR ཧओʹɺର ρ ͔Βର σ ͷࣹ (ρ, σ) ΛௐΔ͕ɺࠓड़ͨ Α͏ʹɺถాຒΊࠐΈʹΑΕ͜Ε (σ, −) ͔Β (ρ, −) ͷࣗવมͱʮಉ ͡ʯͰ͋Δɻ͕ͨͬͯ͠ɺ͜ͷΑ͏ͳࣗવมΛ໌ࣔతʹѻ͍ͬͯͳͯ͘ɺ ถాຒΊࠐΈͷ͔؍Βߟ͑ΔͱɺDHR ཧຊॻͷཱͱંΓ߹͍͕ѱ͍ Θ͚Ͱͳ͍ 13 ɻ ஶऀɺม࣍ͯؔ͠ʹͷΑ͏ʹड़͍ͯΔɻ ʮมʯΛ·ͨʮઈରతʯͳͷͱଊ͑ͨͳΒɺͨͩ Λҧ͏ϨϕϧʹઌૹΓ͚ͨͩ͠ͷ͜ͱͱͳΔͷ໌Β͔Ͱ͋Δɻ ཁ͢Δʹ͜͜Ͱ͍͍ͨͷ࣍ͷΑ͏ͳ͜ͱͰ͋Δɻ ʮมʯʹ ࣠ΛҠͨ͠ͱ͖ʹɺΘΕΘΕ͕ಘͨͷ୯ͳΔΑΓߴ࣍ͷʮ֬ ͔͞ʯͰ͋ΓʮίϯτϩʔϧՄೳੑʯͳͷͩΖ͏͔ʁͦ͜ͰΘΕΘ Ε͕ಘͨͷɺΉ͠Ζʮݻఆͨ͠ͷʹݻண͠ͳ͍ʯͱ͍͏͜ͱɺ ͔ͦ͜Βʮղ์͞ΕΔʯͱ͍͏͜ͱɺʮࣗ༝ʹͳΔʯͱ͍͏͜ͱͳ ͷͰͳ͍ͩΖ͏͔ʁʢpp. 172-173ʣ ถాຒΊࠐΈΛ௨ͯ͠ɺཻࢠతͳඳ૾ʹݻண͔ͤͣͦ͜Β։์͞Εࣗ༝ʹ ͳΔ͜ͱ͕Ͱ͖Δͱ͍͑ΔͩΖ͏ɻ. 7 7 DHR ཧʹ͓͚Δ༷ʑͳಉ͡͞ DHR 理論における様々な同じさ ຊॻͷୈ 4 ষͰɺʮՄࣗͳٯવมΛ௨ͯ͡ɺʮಉ͡͞ʯཱ͕͢Δͱ͍ ͏͖Ͱ͋Δʯʢp. 142ʣͱड़ΒΕ͍ͯΔɻDHR ཧʹ͓͍ͯɺՄͳٯ ࣗવม͕ॏཁͳׂΛՌ͍ͨͯ͠Δɻ 4 અͰߟ͑ͨରɺ͋Δ༗քͳۭ࣌ྖҬͰͷΈਅۭͱҟͳΔΑ͏ͳঢ়ଶ Ͱ͋ͬͨɻ͞Βʹɺ2 ͭͷ༗քͳۭ࣌ྖҬ͕͋Γɺͦ͜ʹ͓͍ͯͷΈਅۭͱҟ. 286. 12.
(13) 〈同じさ〉とは何か. ͳΔΑ͏ͳঢ়ଶΛߟ͑Δɺͭ·ΓɺೋͭͷରΛߟ͑Δ͜ͱࣗવͰ͋Ζ͏ɻ ͜Εɺ4 અͰड़ͨ ݍE ͷੵ E × E Λߟ͑Δ͜ͱʹରԠ͢Δɻ ೋͭͷ༗քͳۭ࣌ྖҬ O1 ͱ O2 Λߟ͑Α͏ɻO1 ͱ O2 ͰͷΈਅۭͱҟͳ ΔΑ͏ͳঢ়ଶͰ͋ͬͨͱͨ͠Βɺ͜ͷঢ়ଶɺO1 ͱ O2 ΛؚΉΑ͏ͳۭ࣌ྖ Ҭ O ʹ͓͍ͯͷΈਅۭͱҟͳΔঢ়ଶͱΈͳ͢͜ͱ͕Ͱ͖Δɻ͕ͨͬͯ͠ɺE ͱ E × E ແؔͰͳ͘ɺԿΒ͔ͷ͕ؔ͋Δ͜ͱ͕༧͞ΕΔɻ ͜ͷͱ͖ɺE × E ͔Β E ͷ͚ؔͮɺͭ·Γɺؔख ⊗ Λߟ͑Δ͜ͱʹͳ Δ [7, Definitions 170, 176, and 178]ɻ͜ͷؔखɺρ1 ͱ ρ2 ͱ͍͏ E ͷର ʹରͯ͠ ρ1 ⊗ ρ2 ͱ͍͏ E ͷରΛରԠͤ͞ΔΑ͏ͳؔख [7, Proposition 175 and Definition 176] Ͱ͋Δɻ͜ͷؔखͷૢ࡞ɺೋͭͷରΛͻͱ·ͱΊʹ ͢ΔΑ͏ͳૢ࡞Ͱ͋Δɻͦͯ͠ɺࣹͷରԠఆٛ͞Ε͍ͯΔ [7, Proposition 177 and Definition 178]ɻ ؔख ⊗ Λఆٛ͢ΔͱɺෳͷؔखΛߟ͑Δ͜ͱ͕Ͱ͖Δɻྫ͑ɺ3 ͭͷ ର ρ1 ɺρ2 ɺρ3 Λߟ͑Α͏ɻ·ͣɺρ2 ͱ ρ3 ΛҰͭͷର ρ2 ⊗ ρ3 ͱΈͳ͠ ͨ ޙρ2 ⊗ ρ3 ͱ ρ1 ΛҰͭͷର ρ1 ⊗ (ρ2 ⊗ ρ3 ) ͱ͢ͳݟ߹ͱɺρ1 ͱ ρ2 ΛҰ ͭͷର ρ1 ⊗ ρ2 ͱΈͳͨ͠ ޙρ1 ⊗ ρ2 ͱ ρ3 ΛҰͭͷର (ρ1 ⊗ ρ2 ) ⊗ ρ3 ͱ ͢ͳݟ߹ಉ͡Ͱ͋Δͱߟ͑ΒΕΔɻͳͥͳΒɺ͜ΕΒ 3 ͭͷରͷଊ ͑ํ͕ҧ͏͚ͩͰ͋Δ͔ΒͰ͋ΔɻDHR ཧʹ͓͍ͯ͜ͷ 2 ͭͷؔखՄٯ ͳࣗવม͚ͭͼ݁ͰΒΕΔ [7, Definitions 171, Proposition 181]ɻ ਅۭΛද͢ର ι ͱ ρ1 ͕༗քͳۭ࣌ྖҬ O1 ͰͷΈਅۭঢ়ଶͱҟͳΔର Λߟ͑Α͏ɻ͜ΕΒΛͻͱ·ͱΊʹͨ͠ରɺρ1 ͱ͍͏ঢ়ଶͱಉ͡Ͱ͋Δ ͱ͍͏͜ͱͰ͋Δͱߟ͑ΒΕΔɻ࣮ࡍɺDHR ཧʹ͓͍ͯ͜ΕΒՄࣗͳٯ વม͚ͭͼ݁ͰΒΕΔ [7, p. 792]ɻρ1 ⊗ ι ͱ ρ1 ಉ༷Ͱ͋Δɻ ͜ΕΒͷʮಉ͡͞ʯɺಉ͡Ͱ͋Δ͜ͱ͕༧͞ΕΔʮಉ͡͞ʯͰ͋ͬͨɻ Ұݟɺಉ͡Ͱ͋ΔΑ͏ʹࢥΘΕͳ͍ʮಉ͡͞ʯ͋Δɻ͜Εɺཻࢠʹؔ ΘΔɻDHR ཧͷΈͰɺਅۭঢ়ଶʹରԠͨ͠ର ι ͔ΒཻࢠʹରԠ͠ ͨର ρ¯ ͱཻࢠʹରԠͨ͠ର ρ ͕ੜ͢ΔมԽΛදࣹ͢ R ∈ (ι, ρ¯ ⊗ ρ) ɺ ཻࢠʹରԠͨ͠ର ρ¯ ͱཻࢠʹରԠͨ͠ର ρ ͕ফ໓ͯ͠ਅۭঢ়ଶʹରԠ ͨ͠ରʹͳΔมԽΛදࣹ͢ R∗ ∈ (¯ ρ ⊗ ρ, ι) ͳͲ͕ଘࡏ͢Δ͜ͱ͕ࣔ͞ΕΔ 14 [1, Proposition 4.7.4] ɻρ¯ ͱ ρ ͕ੜͨ͠Γফ໓ͨ͠Γ͢ΔࣹͷৼΔ͍͔ Β ρ¯ ρ ͷཻࢠͱղऍ͞ΕΔɻDHR ཧͷΈͰɺཻࢠੜফ ໓ͱ͍ͬͨมͷ͔؍ΒఆࣜԽ͞Ε͍ͯΔɻ ͜ͷΑ͏ͳ ρ ͷཻࢠʹରԠ͢Δର ρ¯ ͱͦΕΒʹؔΘΔࣹ͕ଘࡏ͢Δͱ ͖ɺҙͷର ρ1 Λݻఆ্ͨ͠Ͱɺؔख (ρ1 ⊗ ρ, −) ͱؔख (ρ1 , − ⊗ ρ¯) Λߟ͑ Δɻ͜ͷͱ͖ɺؔख (ρ1 ⊗ ρ, −) ͔Βؔख (ρ1 , − ⊗ ρ¯) ͷՄࣗͳٯવม 15 ͕. 13. 287.
(14) ଘࡏ͢Δ [8, Lemma 2.1] ͷͰɺʮՄࣗͳٯવมΛ௨ͯ͡ɺʮಉ͡͞ʯཱ͕ ͢Δʯ ʢp. 142ʣͱ͍͏ຊॻͷཱ͔Βͯݟɺ͜ΕΒͷؔखʮಉ͡ʯͱͳΔɻ ಛʹɺρ1 ΛਅۭʹରԠͨ͠ର ι ͱ͠ɺ− ʹ ρ ΛೖΕΔͱɺ(ρ, ρ) ͱ (ι, ρ ⊗ ρ¯) ʮಉ͡ʯͰ͋Δɻ6 અͰड़ͨඳ૾ʹै͏ͱɺρ ͔Βଞͷରͷม ͱρ ͔Βଞͷରͷมͷ͚ؔͮɺρ ⊗ ρ¯ ͔Βଞͷରͷม ͱι ͔Βଞ ͷରͷมͷ͚ؔͮͱʮಉ͡ʯͱ͍͏͜ͱʹͳΔɻ ͜ͷΑ͏ʹɺDHR ཧʹ͓͍༷ͯʑͳՄࣗͳٯવมɺͭ·Γɺ༷ʑͳ ʮಉ͡͞ʯ͕߹తʹಉ͍ͯ͠ډΔɻ͜ͷ͔؍ΒͯݟɺՄࣗͳٯવมͷ ͔؍Βಉ͡͞Λଊ͑ΔຊॻͷཱॏཁͰ͋Δͱ͍͑ΔͩΖ͏ɻ. 8 8 おわりに ͓ΘΓʹ ຊॻͰఏࣔ͞Ε͍ͯΔ࣮ݱͷଊ͑ํΛɺʮಉ͡͞ʯΛத৺ʹհ͖ͯͨ͠ɻ ຊॻʹΑΕɺ ʮಉ͡͞ʯઃఆ͢Δඞཁ͕͋ΔɻஶऀͦͷதͰมͷՄ ٯੑͷ͔؍Βʮಉ͡͞ʯΛଊ͑Δ͜ͱΛॏཁͰ͋Δͱߟ͓͑ͯΓɺ2 અͰ صαΠίϩΛࡐʹͱͬͨٞΛհͨ͠ɻ 3 અͱ 4 અͰɺDHR ཧཻ͕ࢠʹؔ͢ΔཧͷΑ͏ʹΈ͑Δ͜ͱΛड़ ͨɻҰํɺຊॻͰೋॏεϦοτͷ࣮࣍ͯؔ͠ʹݧͷΑ͏ʹड़͍ͯΔɻ ࣮ମԽ͞Εཻͨࢠͷඳ૾ͰཧղͰ͖ͳ͍ग़དྷࣄʹ͍ͭͯͷ࣮ ͳݧͷ͔ͩΒɺ݁Ռ͕ʮύϥυΩγΧϧʯʹ͑ݟΔͷવͰ͋Δɻ લఏ͞Ε͍ͯΔ࣮ݱͷඳ૾ࣗମΛͱͯۛ͠ຯͤͣʹɺ͔͋ͨ ࣗવࣗମ͕ύϥυΩγΧϧͰ͋Δ͔ͷΑ͏ʹड़ཱͯΔͷɺࣗવ ͦͷͷΛ͍ͨͣΒʹਆൿԽ͢Δ͜ͱʹ΄͔ͳΒͳ͍ɻʢp. 40ʣ. 5 અɺ6 અɺ7 અͰɺҰࢠཻݟతͳཧʹΈ͑ຊॻͷཱͱંΓ߹͍͕ѱ ͍Α͏ʹΈ͑Δ DHR ཧʮՄࣗͳٯવมΛ௨ͯ͡ɺʮಉ͡͞ʯཱ͕͢ Δʯ ʢp. 142ʣͱ͍͏͔؍Βଊ͑Δ͜ͱ͕Ͱ͖Δ͜ͱΛ֬ೝͨ͠ɻ͜ͷΑ͏ ʹɺຊॻͰఏࣔ͞Ε͍ͯΔߟ͑ํద༻ൣғ͕͘ॏཁͰ͋Δͱ͍͑Δɻ 3 અҎ߱ͰɺDHR ཧΛࡐʹͱΓຊॻͷ༰Λհͨ͠ͷͰɺຊॻ͕ ཧֶʹͷΈؔΘΔ༰Λѻ͍ͬͯΔΑ͏ʹͨ͡ײਓ͍Δ͔͠Εͳ͍ɻͦ ͷΑ͏ͳ͜ͱͳ͘ɺຊॻͷओுͷద༻ൣғͷ͞Λࣔͨ͢ΊʹɺҰݟຊॻ ͷཱͱંΓ߹͍͕ѱ͍Α͏ʹΈ͑Δ DHR ཧΛඇن४తʹબ͚ͨͩ͠ Ͱ͋Δ 16 ɻஶऀͪΖΜଞͷՊֶཧ೦಄ʹஔ͍͍ͯΔɻଞͷՊֶཧ ʹଓ͢ΔώϯτΛগͳ͘ͱࡾͭߟ͑ΒΕΔɻ ҰͭΊʮ͍͕ͳ͚Ε͑ͳ͍ʯ ʢp. 54ʣʹ͓͚Δʮ͍ʯͷॱংͰ ͋Δ 17 ɻຊॻͰɺ͜ͷ͜ͱΛࣾձՊֶͰΘΕΔख๏Ͱ͋ΔΞϯέʔτΛ ͱʹߟ͍ͯ͠Δɻྫ͑ɺઌʹʮڕͷྫΛ͍͋͛ͯͩ͘͞ʯɺ࣍ʹʮϖο 288. 14.
(15) 〈同じさ〉とは何か. τͷྫΛ͍͋͛ͯͩ͘͞ʯͱฉ͘ͱɺڕͷྫͱͯ͠ϚάϩͳͲΛ͑Δ֬ ͕ߴ͍ͷʹରͯ͠ɺॱংΛม͑ͯઌʹʮϖοτͷྫΛ͍͋͛ͯͩ͘͞ʯɺ࣍ ʹʮڕͷྫΛ͍͋͛ͯͩ͘͞ʯͱฉ͘ͱʮάοϐʔʯͱ͑Δ͕֬ߴ͘ͳ Δʢp. 212ʣɻ͍ʹઌཱ͕ͬͯ͑͋ΔͷͰ͋Εɺ͜ͷ͜ͱઆ໌͕͠ ͘ɺ ʮ͍͕ͳ͚Ε͑ͳ͍ʯͱ͍͏ຊॻͷओுΞϯέʔτʹؔΘΔ ͜ͱΛ͍ࣔࠦͯ͠Δͱ͍͑ΔͩΖ͏ɻຊॻͷෟͰɺ ʮ͍ʯͷॱং͕ॏཁ ͱͳΔඇՄ֬ͷΈͰनਓͷδϨϯϚʹ͓͚Δਓؒͷ࣮ݱͷҙࢥܾ ఆͷ͋ΓํΛߟ͍ͯ͠Δ͕ɺ͜ΕຊॻͷओுΛࣾձՊֶʹଓ͢ΔࢼΈ ͱ͑ݴΔɻ ೋͭΊ࣮ମʹݻண͠ͳ͍ͱ͍͏࢟Ͱ͋Δɻ࣮ମʹݻண͠ͳ͍ͱ͍͏࢟ ຊॻͰపఈ͞Ε͍ͯͯɺྔࢠʹؔΘΔٞʹ͓͍ͯΑ͘ग़ͯ͘Δ ಈؔͱ͍͏༻ޠɺຊॻͰΘΕ͍ͯͳ͍ɻ͋͘·ͰධऀͷਪଌͰ͋ Δ͕ɺྔࢠྗֶ͕ಈͱ͍͏࣮ମΛѻ͏ཧͰ͋ΔͱಡऀʹࢥΘͤΔ͜ͱΛ ආ͚ΔͨΊΘͳ͔ͬͨͷͰͳ͍͔ͱࢥ͏ɻྔࢠྗֶʹ͓͍ͯݱΕΔඇՄ ֬Λਓؒͷҙࢥܾఆʹద༻͢Δͱ͍͏ൃΛ͢ΔͨΊʹɺྔࢠྗֶ ΛཻࢠಈͳͲͷ࣮ମͷཧͰ͋Δͱߟ͍͑ͯͨΒ͍͠Ͱ͋Ζ͏ɻਓؒ ͷҙࢥܾఆʹؔΘΔͳͲͷ࣮ମ͕ྔࢠྗֶʹै͏ͷ͔ͱ͍͏͕ग़ͯ͘ Δ͔ΒͰ͋ΔɻඇՄ֬Λਓؒͷҙࢥܾఆʹద༻͢Δͱ͍͏ൃɺૢ ࡞ͷඇՄੑʹ࣮͠ମʹݻண͠ͳ͍͔Βͦ͜ग़͖ͯͨͱ͍͑Δɻ·ͨɺ2 અͰհͨ͠صαΠίϩͷྫͰɺصαΠίϩͱ͍࣮ͬͨମͰͳͦ͘ ΕΒʹؔΘΔมͷ͔؍Βʮಉ͡͞ʯΛߟ͍ͯͨ͠ɻ͔ͩΒͦ͜ 6 અ 7 અͰѻͬͨΑ͏ʹɺصαΠίϩͱ͕ؔͳ͍Α͏ʹΈ͑Δ DHR ཧʹՄ ͳٯมࢹ͏͍ͱΛଓͰ͖ͨͱ͍͑Δɻ ࡾͭΊɺ͜ͷॻධͰѻͬͨʮಉ͡͞ʯͷ͔؍ΒͷྨͰ͋ΔɻDHR ཧ Ͱɺ7 અͰड़ཻͨࢠʹؔΘΔࣹ [7, Definition 214] [8, Definition 3.4] Λͱʹ࣍ͱݩΑΕΔෆมྔΛఆٛ͠ɺͦΕʹ͍ͯͮجରΛྨ͢Δɻͦ ͯ͠ɺͦͷதʹϑΣϧϛΦϯϘιϯͱಉ༷ͷৼΔ͍Λ͢Δର·ؚ ΕΔɻྨͱ͍͏ߦҝɺଞͷՊֶཧʹ͓͍ͯΈΒΕΔɻྨΛѻ͏Պ ֶཧͱຊॻͰߟ͑ΒΕ͍ͯΔ࣮ݱͷҰൠߏΛଓ͢Δํ๏ͷҰͭͱͯ͠ ɺมͷ͔؍Βʮಉ͡͞ʯΛઃఆ͠ɺ͜ͷʮಉ͡͞ʯʹ͍ͯͮجྨ͢ Δͱ͍͏͜ͱͰ͋Ζ͏ɻ ຊॻɺ༷ʑͳΛԣஅ͢ΔΑ͏ͳ࣮ݱͷ౷Ұతͳଊ͑ํΛಡऀʹఏࣔ ͍ͯ͠Δɻ ʮʪ࣮ݱʫͱԿ͔ʯͱ͍͏ʹڵຯ͋Δ༷ʑͳͷʹऀڀݚɺ という問題に興味がある様々な分野の研究者 Ұಡɺͦͯ͠ख़ಡΛનΊ͍ͨɻ に、一読、そして熟読を薦めたい。. 15. 289.
(16) ँࣙ 謝辞 ຊߘɺจ෦ՊֶলՊֶڀݚඅิॿۚج൫( ڀݚC) ʢ՝൪߸ɿ20K00279ʣ ͷॿΛड͚͍ͯΔɻ 注 1. ͜ͷॻධͰɺຊॻͱ͍͏දهৗʹจ[ ݙ13]ɺஶऀͱ͍͏දهจݙ [13] ͷஶऀΛࢦ͢ɻ 2. ҎԼɺຊॻ [13] ͷҾ༻ͷࡍɺϖʔδͷΈΛද͢هΔɻ. 3. DHR ཧ͕ల։͞Ε͍ͯΔจ[ ݙ5, 6] ͷλΠτϧʹʮཻࢠͷ౷ܭʯͱ ͍͏୯·ؚ͕ޠΕ͍ͯΔɻ 4. ϢχλϦಉͷදݱɺࣗવಉͱΈΔ͜ͱ͕Ͱ͖Δ [14, pp. 141-145][10, ୈ 7 ষ]ɻ 5. GNS ߏఆཧʹΑΔͱɺC* A ্ͷঢ়ଶ ψ ʹରͯ͠ɺώϧϕϧτۭؒ Hψ ͱ A ͷ८ճද ݱπψ ɺώϧϕϧτ্ۭؒͷ८ճϕΫτϧ xψ ͕ଘࡏͯ͠ɺ ҙͷ A ∈ A ʹରͯ͠ ψ(A) = xψ , πψ (A)xψ ͱͳΔɻ͞Βʹɺ{Hψ , πψ , xψ } ϢχλϦಉੑΛͷ͍ͧͯҰҙͰ͋Δɻ 6. ʮແݶԕʹ͓͍ͯਅۭঢ়ଶͱͳΔΑ͏ͳঢ়ଶʯͱ͍͏ํ͍ݴɺ[5] ͷ Proposition A.1 ʹ͍͍ͯͮجΔɻ 7. ରΛΪϦγΞจࣈͰද͢هΔ͜ͱʹҧײΛ͡ײΔਓ͍Δ͔Ε ͳ͍͕ɺจ[ ݙ1] ͰࡏہՄೳͰҠಈՄೳͳࣗݾ४ಉ૾ࣸܕΛΪϦγΞจࣈͰ ද͍ͯͯ͠هɺจ[ ݙ8] ͰରΛΪϦγΞจࣈͰද͍ͯ͠هΔͷͰɺ͜ͷॻ ධͰରΛΪϦγΞจࣈͰද͢هΔɻ 8. ؔख (ρ, −) : E → Set ɺର σ ʹରͯ͠. (ρ, −)(σ) := (ρ, σ) ͱఆٛ͞ΕΔɻSet ू߹ͷ͋ͰݍΔɻࣹ A ∈ (σ , σ ) ʹରԠ͢Δࣹ (ρ, A) : (ρ, σ ) → (ρ, σ ) ҙͷ B ∈ (ρ, σ ) ʹରͯ͠ (ρ, A)(B) := A ◦ B ͱఆٛ͞ ΕΔ [2, Example 1.2.8.d]ɻ͜ͷؔखɺ− ʹೖΕΔର σ ΛબͿͱɺࣹͷू ߹ (ρ, σ) ͕ఆ·ΔΑ͏ͳؔखͰ͋Δ [2, Example 1.2.8.d] [14, pp. 73-76]ɻ 9. ຊॻͰɺ ʮࢲʯͷஔ͖͑Մೳੑͷٞʹଓ͍ͯɺΧϯτʹΑΔʮͦΕ ͕ීวత๏ଇͱͳΔ͜ͱΛಉ࣌ʹཉ͠͏ΔΑ͏ͳ֨ʹैͬͯͷΈߦಈͤΑʯ ͱ͍͏ಓಙ๏ଇஔ͖͑Մೳੑͷ͔؍Βߟ͍ͯ͠Δʢpp. 149-157ʣɻྫ 290. 16.
(17) 〈同じさ〉とは何か. ͑ʮଞਓΛࡴͯ͠ͳΒͳ͍ʯͱ͍͏͜ͱΛʮ͕ࣗࡴ͞Εͨ͘ͳ͍ʯͱ ͍͏͜ͱͷஔ͖͑Մೳੑͱͯ͠ଊ͍͑ͯΔɻͦͯ͠ɺ ʮΈ͔ͣΒ͕ஔ͖͑ ෆՄೳͰ͋Δͱ͍͏͜ͱͦΕࣗମͷஔ͖͑ՄೳੑΛཧղͨ͠ͱ͖ɺΘΕΘ Εࣗݾͷଘࡏͷࠜຊతͳྙཧੑʹͮ͘ؾʯ ʢp. 157ʣͱड़͍ͯΔɻ͜ͷΑ ͏ʹʮࢲʯͷஔ͖͑ՄೳੑͷٞɺཻࢠͷٞͷΈͳΒͣྙཧֶͷٞ ʹͭͳ͕͍ͬͯΔɻ 10. ࣹ A ∈ (ρ, σ) ʹରͯ͠ɺࣗવม( A, −) : (σ, −) ⇒ (ρ, −) ɺҙͷ ର σ ∈ E ͱ B ∈ (σ, σ ) ʹରͯ͠ɺ(A, −)σ (B) = B ◦ A ͱఆٛ͞ΕΔ [2, Example 1.3.6.c]ɻؔख (ρ, −) Λରͱ͠ɺࣗવม( A, −) Λࣹͱ͢ΔݍΛ ߟ͑Δ͜ͱ͕Ͱ͖ɺ͜ͷΑ͏ͳݍؔखͱݍΑΕΔʢcf. p. 127ʣɻ 11. ࣹ A ∈ (ρ, σ) ʹରͯ͠ɺࣗવม( A, −) : (σ, −) ⇒ (ρ, −) Λߟ͑Δɻ ҙͷ B ∈ (σ, σ )ɺC ∈ (σ , σ ) ʹରͯ͠ɺ. (ρ, C) ◦ (A, −)σ (B) = (A, −)σ ◦ (σ, C)(B) ͱͳ͍ͬͯΔɻ͜ͷࣜͷҙຯͰɺࣗવม( A, −) (σ, −) ʹ͓͚Δ͚ؔͮ Λ (ρ, −) ʹ͓͚Δ͚ؔͮʹࣗવʹ͚͍ؔͮͯΔɻ 12. ؔख N ͔Βؔख (ρ, −) ͷՄࣗͳٯવม͕ଘࡏ͢Δͱ͍͏ࣄ࣮ɺถ ాͷิͷ͋ͰܥΔɻ[2, Theorem 1.3.3 and Proposition 1.5.2] Λͱʹɺ֓ ུΛड़͓ͯ͜͏ɻρ Λରͱ͢Δͱɺؔख N : E → Set ҙͷର σ ʹ ରͯ͠ N (σ) = Nat((σ, −), (ρ, −)) ͱఆٛ͞ΕΔɻ͜͜ͰɺNat((σ, −), (ρ,−)) ɺ(σ, −) ͔Β (ρ, −) ͷࣗવม͋ͰΔɻࣹ A ∈ (σ, σ ) ʹରͯ͠ɺ ରԠ͢Δࣸ૾ N (A) : Nat((σ, −), (ρ, −)) → Nat((σ , −), (ρ, −)) ɺҙͷ α ∈ Nat((σ, −), (ρ, −)) ʹରͯ͠ɺN (A)(α) = α ◦ (A, −) ͱఆٛ͞ΕΔɻ ࣸ૾ θρ,σ : Nat((σ, −), (ρ, −)) → (ρ, σ) Λɺα ∈ Nat((σ, −), (ρ, −)) ʹର ͯ͠ θρ,σ (α) := ασ (1σ ) ͱఆٛ͢Δɻҙͷ A ∈ (σ, σ ) ͱ α ∈ Nat((σ, −), (ρ, −)) ʹରͯ͠ɺ. θρ,σ ◦ N (A)(α) = (ρ, A) ◦ θρ,σ (α) ͱͳΔͷͰɺθρ,σ σ ʹؔͯؔ͠ख N ͔Βؔख (ρ, −) ͷࣗવม͋ͰΔɻ ର σ ͱ B ∈ (ρ, σ) ʹରͯ͠ɺࣸ૾ τ (B)σ : (σ, σ ) → (ρ, σ ) Λҙͷ A ∈ (σ, σ ) ʹରͯ͠. τ (B)σ (A) := (ρ, A)(B) = A ◦ B 17. 291.
(18) ͱఆٛ͢Δɻҙͷ C ∈ (σ , σ )ɺA ∈ (σ, σ ) ʹରͯ͠ɺ. (ρ, C)τ (B)σ (A) = τ (B)σ (σ, C)(A) ͱͳΔͷͰɺτ (B) ࣗવม͋ͰΔɻ ҙͷ B ∈ (ρ, σ) ʹରͯ͠. θρ,σ (τ (B)) = τ (B)σ (1σ ) = 1σ ◦ B = B Ͱ͋Γɺҙͷ α ∈ Nat((σ, −), (ρ, −)) ͱ A ∈ (σ, σ ) ʹରͯ͠. τ (θρ,σ (α))σ (A) = A ◦ ασ (1σ ) = (ρ, A) ◦ ασ (1σ ) = ασ ◦ (σ, A)(1σ ) = ασ (A) ͱͳΔͷͰɺθρ,σ Մ͋ͰٯΔɻ ͕ͨͬͯ͠ɺ ʮՄࣗͳٯવมΛ௨ͯ͡ɺ ʮಉ͡͞ʯཱ͕͢Δʯ ʢp. 142ʣ ͱ͍͏ຊॻͷཱ͔ΒͯݟɺN (σ) = Nat((σ, −), (ρ, −)) ͱ (ρ, σ) ʮಉ͡ʯ Ͱ͋Δͱ͍͑Δɻ 13. DHR ཧʹ͓͍ͯɺࣹབྷ࡞༻ૉͰ͋ͬͨɻབྷ࡞༻ૉࣗମ͕ࣗવมͱ Έͳ͢͜ͱͰ͖Δ [14, pp. 141-145][10, ୈ 7 ষ]ɻ 14. ¯ ∈ (ι, ρ ⊗ ρ¯) ͕ଘࡏͯ͠ɺ ର ρɺର ρ¯ɺࣹ R ∈ (ι, ρ¯ ⊗ ρ)ɺࣹ R ¯ ∗ ⊗ 1ρ ◦ 1ρ ⊗ R = 1ρ , R. ¯ = 1ρ¯ R∗ ⊗ 1ρ¯ ◦ 1ρ¯ ⊗ R. ͕Γཱͭ [8, p. 106]ɻલऀͷࣜɺཻࢠ ρ ͕͋Δͱ͜Ζʹɺਅۭ͔Βཻ ࢠ ρ¯ ͱཻࢠ ρ ͕৽ͨʹੜͨ͠ޙɺͱ͔Β͋ͬͨ ρ ͱ৽ͨʹੜ͞Εͨ ཻࢠ ρ¯ ͕ফ໓͠ਅۭͱͳΓɺ৽ͨʹੜ͞Εཻͨࢠ ρ ͚͕ͩΔͱ͍͏มԽ ɺͱ͔Β͋ͬͨ ρ ͕ԿมΘΒͣ ρ ͷ··Ͱ͋Δͱ͍͏มԽͱಉ͡Ͱ͋ Δ͜ͱΛ͍ࣔͯ͠Δɻ 15. ¯ ∈ (ι, ρ ⊗ ρ¯) ɺ ର ρɺର ρ¯ɺࣹ R ∈ (ι, ρ¯ ⊗ ρ)ɺࣹ R ¯ ∗ ⊗ 1ρ ◦ 1ρ ⊗ R = 1ρ , R. ¯ = 1ρ¯ R∗ ⊗ 1ρ¯ ◦ 1ρ¯ ⊗ R. ΛΈͨ͢ͱ͢Δɻ্ͷ٭Ͱड़ͨΑ͏ʹɺͦͷΑ͏ͳରͱࣹଘࡏ͢Δɻ ର ρ1 ͱ ρ2 ʹରͯ͠ɺࣸ૾ iρ1 ,ρ2 : (ρ1 ⊗ ρ, ρ2 ) → (ρ1 , ρ2 ⊗ ρ¯) Λҙͷ ࣹ T ∈ (ρ1 ⊗ ρ, ρ2 ) ʹରͯ͠. ¯ iρ1 ,ρ2 (T ) := T ⊗ 1ρ¯ ◦ 1ρ1 ⊗ R 292. 18.
(19) 〈同じさ〉とは何か. ͱఆٛ͠ɺࣸ૾ jρ1 ,ρ2 : (ρ1 , ρ2 ⊗ ρ¯) → (ρ1 ⊗ ρ, ρ2 ) Λࣹ T ∈ (ρ1 , ρ2 ⊗ ρ¯) ʹର ͯ͠ jρ1 ,ρ2 (T ) := 1ρ2 ⊗ R∗ ◦ T ⊗ 1ρ ͱఆٛ͢Δɻ͜ͷͱ͖ɺA ∈ (ρ2 , ρ2 ) ʹରͯ͠. (ρ1 , A⊗1ρ¯)◦iρ1 ,ρ2 = iρ1 ,ρ2 ◦(ρ1 ⊗ρ, A), (ρ1 ⊗ρ, A)◦jρ1 ,ρ2 = jρ1 ,ρ2 ◦(ρ1 , A⊗1ρ¯) ͱͳΔͷͰɺiρ1 ,ρ2 ͱ jρ1 ,ρ2 ࣗવม͍ͯͬͳͱΔɻ·ͨɺ. iρ1 ,ρ2 ◦ jρ1 ,ρ2 (T ) = T ,. jρ1 ,ρ2 ◦ iρ1 ,ρ2 (T ) = T. ͱͳΔͷͰɺiρ1 ,ρ2 Մ͋ͰٯΔ [8, Lemma 2.1]ɻ͕ͨͬͯ͠ɺʮՄࣗͳٯવ มΛ௨ͯ͡ɺ ʮಉ͡͞ʯཱ͕͢Δʯ ʢp. 142ʣͱ͍͏ຊॻͷཱ͔Βͯݟɺ ؔख (ρ1 ⊗ ρ, −) ͱؔख (ρ1 , − ⊗ ρ¯) ʮಉ͡ʯͰ͋Δͱ͍͑Δɻ 16. ຊॻධͰɺՊֶཧͱͯ͠ DHR ཧΛऔΓ্͕͛ͨɺ͜ͷཧΛઈର ࢹ͢Δ͜ͱຊॻͷཱʹ͢Δɻஶऀʮֶऀ͕ɺࡏݱଘࡏ͢ΔՊֶత ݟΛઈରࢹͯͦ͠Εʹґ͠ڌΑ͏ͱ͢ΔͳΒɺΉ͠ΖͷՊֶऀ͕ͦΕΛ ෆదͩͱͩ͢ͳݟΖ͏ʯʢp. 185ʣͱड़͍ͯΔɻ࣮ࡍɺDHR ཧ͕ѻ͑ ͳ͔ͬͨࣗൃతରশੑͷഁΕѻ͑ΔΑ͏ͳཧ֦ு͢ΔࢼΈ͕͋Δ [9]ɻ 17. ຊॻͰɺ ʮ͍ʯΑΓҰൠతʹૢ࡞ͷ͔؍Βɺૢ࡞ͷॱংͷඇՄੑ ʹؔͯ͠ड़͍ͯΔʢpp. 215-217ʣɻͦͯ͠ɺՄ͋Ͱ࡞ૢͳΔ؍ Λఏ͍ࣔͯ͠Δɻྫ͑ɺ2 Λʮ2 Λ͔͚Δʯɺ3 Λʮ3 Λ͔͚Δʯͱ͍͏ૢ ࡞ͱΈΕɺ2 × 3 = 3 × 2 ͱ͍͏͜ͱɺ͜ΕΒͷૢ࡞͕Մ͋ͰΔͱ͍͏ ͜ͱΛද͍ͯ͠Δɻͭ·ΓɺʮͷγεςϜૢ࡞ͷγεςϜͷಛघྫʯʢp. 216ʣͰ͋Δɻ ຊॻͰ͞ΒʹՄ࡞ૢͳͷू·ΓΛɺC*ʹ͓͚ΔήϧϑΝϯτ-φ ΠϚϧΫͷఆཧ [12, ఆཧ 3.7] ͷ͔؍Βߟ͍ͯ͠ΔɻήϧϑΝϯτ-φΠϚ ϧΫͷఆཧʹΑΕ C*ͷͯ͢ͷ͕ͯ͢ݩՄ͋ͰΕɺ͜ΕΒͷݩ ෳૉΛͱͯ͠ͱΔ࿈ଓؔͱΈͳ͢͜ͱ͕Ͱ͖Δɻಛʹɺ؍ଌՄೳྔ ʹରԠ͢ΔΑ͏ͳࣗڞݾͳݩ࣮Λͱͯ͠ͱΔ࿈ଓؔͱͱ͜͢ͳݟ ͕Ͱ͖Δɻ͜ͷ͜ͱ͔Βɺ ʮՄੈͳքͱɺͯ͢ΛΛʹ͖ؐͰݩ ΔੈքͰ͋ΓɺඇՄੈͳքͱɺʹ͍ؐͳ͖ͰݩΑΓҰൠͳૢ࡞Λؚ Ήੈքʯʢp. 216ʣͱ͍͑Δɻ จݙ. [1] H. Baumg¨artel. Operatoralgebraic Methods in Quantum Field Theory. Akademie Verlag, Berlin, 1995. 19. 293.
(20) [2] F. Borceux. Handbook of Categorical Algebra: Volume 1, Basic Category Theory. Cambridge University Press, 1994. [3] S. Doplicher, R. Haag, and J. E. Roberts. Fields, observables and gauge transformations I. Communications in Mathematical Physics, Vol. 13, No. 1, pp. 1–23, 1969. [4] S. Doplicher, R. Haag, and J. E. Roberts. Fields, observables and gauge transformations II. Communications in Mathematical Physics, Vol. 15, No. 3, pp. 173–200, 1969. [5] S. Doplicher, R. Haag, and J. E. Roberts. Local observables and particle statistics I. Communications in Mathematical Physics, Vol. 23, No. 3, pp. 199–230, 1971. [6] S. Doplicher, R. Haag, and J. E. Roberts. Local observables and particle statistics II. Communications in Mathematical Physics, Vol. 35, No. 1, pp. 49–85, 1974. [7] H. Halvorson and M. M¨ uger. Algebraic quantum field theory. In J. Butterfield and J. Earman, editors, Philosophy of Physics, pp. 731–922. Elsevier, Amsterdam, 2007. [8] R. Longo and J. E. Roberts. A theory of dimension. K-theory, Vol. 11, No. 2, pp. 103–159, 1997. [9] খౢઘ. ͩΕ͕ྔࢠΛ͔ͨݟ. தଜҰ, தଜప, ลܟೋʢฤʣ, ͩΕ ͕ྔࢠΛΈ͔ͨ, pp. 65–107. ຊධࣾ, 2004. [10] ݍͷา͖ํҕһձ. ݍͷา͖ํ. ຊධࣾ, 2015. [11] ాޱໜ, ߕڷਓ. ݍʹΑΔݱֶͷਂԽ — ࣹͷҰݩɾϞφυϩ δʔɾࣗݾ. ݱࢥ, Vol. 48, No. 9, pp. 202–214, 2020. [12] ക֞ᆹय़, ߹จ༤, େխଇ. ෮༻࡞ ץૉೖʕHilbert ۭؒΑΓ von Neumann . ཱڞग़൛, 2003. [13] ߕڷਓ, ాޱໜ. ʪ࣮ݱʫͱԿ͔ — ֶɾֶ͔Β࢝·Δੈք૾ ͷస. ஜຎॻ, 2019. [14] ߕڷਓ, ೳඒेࡾ. ݍͷಓҊ. ٕज़ධࣾ, 2019. ʢຊେֶʣ. 294. 20.
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