著者中川弘一

(1)

Dp‑ブレイン上のボソン的開弦の量子エンタングルメントについて

著者中川弘一

雑誌名星薬科大学一般教育論集

号 37

ページ 1‑16

発行年 2019‑12‑10

URL http://id.nii.ac.jp/1240/00000824/

(2)

- ブレイン上のボソン的開弦の量子エンタングルメントについて

中川　弘一

（星薬科大学　物理学研究室）

Dp - ϒϨΠϯ্ͷϘιϯత։ݭͷ

ྔࢠΤϯλϯάϧϝϯτʹ͍ͭͯ

த઒ɹ߂Ұ

੕ༀՊେֶɹ෺ཧֶݚڀࣨ

֓ཁ

ڞมతͳ։ݭͷ৔ͷཧ࿦Λ༻͍ͯɼଟॏDp-ϒϨΠϯ্ͷϘιϯత։ݭʹର͢Δ

ྔࢠΤϯλϯάϧϝϯτͷݚڀ͕ɼࢀߟจݙ[1]ͰߦΘΕͨɽ͜ͷݚڀͰ͸Dp-^ϒ ϨΠϯ͕෇͍͍ͯΔ௒ฏ໘ʹਨ௚ͳۭؒ࠲ඪͷҰͭΛબͼɼͦͷ௒ฏ໘Λ൒෼ʹ෼

ׂ͠ɼFockۭؒදࣔʹ͓͚Δݭͷ೾ಈؔ਺Λ༻͍Δ͜ͱʹΑΓɼΤϯλϯάϧϝϯ τɾΤϯτϩϐʔ͕ܭࢉ͞Εͨɽͦͷ݁ՌɼΤϯλϯάϧϝϯτɾΤϯτϩϐʔ͸෼

ׂ͞Εͨ௒ฏ໘ͷ(p−1)-࣍ݩతͳڥքͷ໘ੵʹൺྫ͠ɼࢵ֎ʢUV^{ʣྖҬͳΒͼʹ}

੺֎ʢIRʣྖҬʹ͓͍ͯൃࢄ͢Δ͜ͱ͕Θ͔ͬͨɽ͔͠͠ɼओཁͳൃࢄ͸Τϯλϯά ϧϝϯτɾΤϯτϩϐʔʹର͢ΔλΩΦϯͷد༩ʹΑΔ΋ͷͰɼ௒ରশͳݭཧ࿦Ͱ͸

ղফ͢ΔͰ͋Ζ͏ͱߟ͑ΒΕɼλΩΦϯʹΑΔൃࢄΛผʹ͢ΔͱɼDp-^{ϒϨΠϯ্ͷ} Ϙιϯత։ݭʹର͢ΔΤϯλϯάϧϝϯτɾΤϯτϩϐʔ͸ɼ2≤p≤dcl−2^ͷͱ

͖ʹ༗ݶͰɼp= 1, dcl−1ͷͱ͖ʹର਺తʹൃࢄ͢Δ͜ͱ͕ٞ࿦͞ΕͨɽຊߘͰ͸

ࢀߟจݙ[1]ʹج͖ͮɼҎ্ͷܭࢉͱٞ࿦ʹ͍ͭͯͷৄࡉͳղઆΛߦ͏ɽ

1 ^ং࿦

ΤϯλϯάϧϝϯτɾΤϯτϩϐʔ[2]͸ɼزԿֶతͳΤϯτϩϐʔ[3–9]ͱ΋Α͹Εɼ

෺ੑ෺ཧֶ[10–12]ɼྔࢠ৔ཧ࿦[13–19]ɼྔࢠ৘ใཧ࿦[20,21]ɼྔࢠॏྗͱϒϥοΫϗʔ ϧ෺ཧֶ[22–31]͓Αͼݭཧ࿦ʹ͓͚ΔAdS/CFTରԠ[32–36]Λแׅ͢Δɼཧ࿦෺ཧֶ

ͷ޿͍ൣғΛ෴͍ͭ͘͢ɼ࠷ۙͷൃలͷதͷয఺ͱͳͬͨɽϒϥοΫϗʔϧͷཧ࿦ʹؔ͢

ΔBekenstein [37, 38]^ͱHawking [39]ͷಠ૑తͳ࿦จ͕ൃද͞ΕͯҎདྷɼϒϥοΫϗʔ ϧͷ೤ྗֶ͸ɼ40೥Ҏ্΋ͷؒɼॏྗɼྔࢠ࿦ɼ೤ྗֶ͓Αͼ৘ใཧ࿦ͷؒʹ͋Δجૅత ͳؔ܎ੑΛཧղ͢Δ͜ͱʹ޲͚ͯͷ޿ൣғͳݚڀͷഎޙͰओͳݪಈྗͱͳ͖ͬͯͨɽ ͜

(3)

ͷݚڀͷૅੴͱͳΔ΋ͷ͕ɼϒϥοΫϗʔϧͷྔࢠঢ়ଶ਺Λද͠ɼBekenstein-Hawking Τϯτϩϐʔͱͯ͠஌ΒΕΔɼϒϥοΫϗʔϧɾΤϯτϩϐʔͰ͋Δɽ

ϒϥοΫϗʔϧɾΤϯτϩϐʔʹىҼ͢Δجຊతࣗ༝౓Λݟग़ͨ͢Ίʹɼଟ͘ͷ౒ྗ

͕ͳ͞Ε͖ͯͨ[40]ɽ͜ΕΒͷ౒ྗ͸ɼ࠷ऴతʹɼզʑΛҰ؏ͨ͠ॏྗͷྔࢠ࿦ʹಋ͍

ͯ͘ΕΔՄೳੑΛൿΊ͍ͯΔɽ2ͭʹ෼ׂ͞Ε্ۭͨؒͷྔࢠ৔ཧ࿦Λఆٛ͢Δ͜ͱ ʹΑͬͯ[41]ɼϒϥοΫϗʔϧɾΤϯτϩϐʔͷىݯΛઆ໌͢ΔͨΊʹɼΤϯλϯάϧ ϝϯτɾΤϯτϩϐʔ͸ͦͷݚڀ෼໺ʹ͓͍ͯ஫໨͞ΕΔΑ͏ʹͳͬͨ[3, 4]^ɽ͜Ε͸ɼ

Bekenstein-HawkingΤϯτϩϐʔ͕ϒϥοΫϗʔϧɾϗϥΠζϯͷ໘ੵʹൺྫ͢Δ͜ͱ

ͱಉ༷ʹɼΤϯλϯάϧϝϯτɾΤϯτϩϐʔ͕ɼۭؒΛ2ͭͷ෦෼ۭؒʹ෼ׂ͍ͯ͠

Δɼڥք໘ͷ໘ੵʹൺྫ͢ΔͨΊͰ͋Δɽ͔͠͠ͳ͕Βɼہॴతͳ৔ͷྔࢠ࿦Λ༻͍ͯ

ܭࢉ͞ΕͨΤϯλϯάϧϝϯτɾΤϯτϩϐʔ͸ɼ༗ݶͳBekenstein-Hawking^Τϯτϩ ϐʔͱ͸ରরతʹɼ௨ৗ௨Γͷࢵ֎ྖҬʹ͓͚Δೋ఺૬ؔؔ਺ͷൃࢄతͳৼ෣͍ʹΑΓɼ ຊ࣭తʹൃࢄ͢Δɽจݙ[6, 23]ʹ͓͍ͯɼBekenstein-HawkingΤϯτϩϐʔ͸ݹయ࿦త ͳϨϕϧͰͷϒϥοΫϗʔϧɾΤϯτϩϐʔͰ͋Δͷʹର͠ɼΤϯλϯάϧϝϯτɾΤϯ τϩϐʔ͸ϒϥοΫϗʔϧɾΤϯτϩϐʔʹର͢ΔୈҰ࣍ͷྔࢠิਖ਼ʹ͋ͨΔ͜ͱ͕ࢦఠ

͞Εͨɽ͜ͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔͷࢵ֎ൃࢄ͸ྔࢠ৔ཧ࿦ͷ͘ΓࠐΈͷ ॲํʹΑΓରॲͰ͖Δͱߟ͑ΒΕΔɽͭ·ΓɼΤϯλϯάϧϝϯτΤϯτϩϐʔͷൃࢄ

͢Δྔࢠิਖ਼͸͘Γࠐ·ΕͨNewtonఆ਺ʹٵऩ͢Δ͜ͱ͕Ͱ͖ΔͰ͋Ζ͏ͱߟ͑ΒΕ

Δ[23, 42]ɽͨͩ͠ɼॏྗ࡞༻ʹର͢Δ1ϧʔϓͷྔࢠิਖ਼ͱൃࢄΤϯλϯάϧϝϯτɾ

ΤϯτϩϐʔΛൺֱ͢Δͱɼඇۃখ݁߹͕͋Δ৔߹ɼΤϯλϯάϧϝϯτɾΤϯτϩϐʔ ͷ͘ΓࠐΈͱχϡʔτϯఆ਺ͷ͘ΓࠐΈͱͷؒʹෆ੔߹͕ݟ͔ͭΔՄೳੑ͕͋Δ[43, 44]ɽ

͜ΕΛඇۃখ݁߹ͷύζϧͱΑͿɽ

ϒϥοΫϗʔϧͷΤϯτϩϐʔʹର͢ΔΑΓ༗๬ͳΞϓϩʔν͸ɼݭཧ࿦ʹΑͬͯఏڙ

͞ΕΔՄೳੑ͕͋Δ[23]ɽݭཧ࿦͸ࢵ֎ྖҬͰ༗ݶͰ͋Δͱߟ͑ΒΕ͍ͯΔͷͰɼݭཧ࿦

ʹ͓͚ΔΤϯτϩϐʔ͸༗ݶͰ͋Δͱߟ͑ΒΕΔɽπϦʔϨϕϧʢϧʔϓμΠϠάϥϜΛ

ؚ·ͳ͍ʣͰͷݭཧ࿦͸ɼϒϥοΫϗʔϧʹର͢Δ͋Δ༗ݶͳΤϯλϯάϧϝϯτɾΤϯ τϩϐʔͱNewtonఆ਺Λಉ࣌ʹ༩͑ΔͰ͋Ζ͏ͱߟ͑ΒΕΔɽBekenstein-HawkingΤ ϯτϩϐʔ͸ɼ͋ΔΫϥεͷۃݶϒϥοΫϗʔϧͱۙࣅతͳۃݶϒϥοΫϗʔϧΛهड़͢

ΔBPSιϦτϯͷඍࢹతঢ়ଶΛ਺্͑͛Δ͜ͱ[45, 46]ʹΑͬͯಘΒΕͨ͜ͱʹ΋஫໨

(4)

͢΂͖Ͱ͋Ζ͏[47–49]ɽ͕ͨͬͯ͠ɼݭཧ࿦͸ϒϥοΫϗʔϧɾΤϯτϩϐʔΛجૅత ͳϨϕϧͰཧղ͢ΔͨΊͷຊ࣭తͳख͕͔ΓΛఏڙ͢ΔՄೳੑ͕͋Δɽ

ϒϥοΫϗʔϧɾΤϯτϩϐʔʹؔ͢Δ࠷ۙͷਐల͸AdS/CFT^{ରԠʹ༝དྷ͢Δͱ}

͜Ζ͕େ͖͍[50]^ɽAdS/CFT^{ରԠʹΑΔͱɼ}AdSۭؒͷڥք໘্Ͱఆٛ͞Εͨڞܗ

৔ཧ࿦ʢCFT^ʣ͸ɼAdSۭؒͷόϧΫʹ͓͚Δॏྗཧ࿦ͱ౳ՁͰ͋Ζ͏ͱߟ͑ΒΕΔɽ AdS/CFTରԠͷख๏ʹ͕͍ͨ͠ɼRyu^ͱTakayanagi^͸ɼAdS^{ۭؒͷۭؒతͳڥք໘}

্ͷดͨ͡෦෼ۭؒΣʹؔ͢ΔΤϯλϯάϧϝϯτɾΤϯτϩϐʔSΣΛఆٛͰ͖ΔΑ

͏ʹΣΛબͿͱɼͦͷΤϯτϩϐʔ͸ɼΣͱಉ͡ڥքΛڞ༗͢ΔɼAdS^ۭؒͷόϧΫ ʹ͓͚Δۃখ໘Γ^ͷ໘ੵA(Γ)^ͱɼCFTͱ૒ରͳॏྗཧ࿦ʹ͓͚ΔNewton^ఆ਺GN

ʹΑܾͬͯ·ΔͰ͋Ζ͏ͱ͍͏͜ͱΛఏএͨ͠ɽͭ·ΓɼSΣ =A(Γ)/4GN,∂Σ =∂Γ ͱ͍͏͜ͱͰ͋Δɽ͜ͷఏএ͞Εͨؔ܎ࣜSΣ =A(Γ)/4GN͸ϒϥοΫϗʔϧʹؔ͢Δ Bekenstein-HawkingͷΤϯτϩϐʔެࣜSBH=ABH/4GNʢABH͸ϒϥοΫϗʔϧɾ ϗϥΠζϯͷ໘ੵʣͱࠅࣅ͍ͯ͠Δ͜ͱ͸஫໨͢΂͖Ͱ͋Ζ͏ɽ࣮ࡍɼڥք໘ͷΤϯλϯ άϧϝϯτɾΤϯτϩϐʔ͸ɼRyu-Takayanagiͷؔ܎ࣜΛ༻͍ͯɼBekenstein-Hawking ͷΤϯτϩϐʔͱͯ͠ද͞ΕΔ͜ͱ͕ɼจݙ[51]Ͱ໌Β͔ʹࣔ͞Ε͍ͯΔɽ

Bekenstein-HawkingΤϯτϩϐʔͱΤϯλϯάϧϝϯτɾΤϯτϩϐʔʹؔ͢Δݚڀ

͸ଟ਺ଘࡏ͢Δ͕ɼϒϥοΫϗʔϧͷBekenstein-HawkingΤϯτϩϐʔͷݪҼͱͳΔɼ جຊతࣗ༝౓͕ԿͰ͋Δ͔ͱ͍͏ٙ໰͸ະղܾͷ··ʹͳ͍ͬͯΔɽ͜ͷٙ໰͸ɼݭͷ׬

શͳࣗ༝౓ΛߟྀʹೖΕͨɼݭͷ৔ͷཧ࿦ʹ͓͍ͯΑΓྑ͘ཧղ͞ΕΔͱߟ͑ΒΕΔɽͦ

ͷͨΊຊߘͰ͸ɼࢀߟจݙ[1]ͰߦΘΕͨɼڞมతݭͷ৔ͷཧ࿦[52–55]Λ༻͍ͨDp-ϒ ϨΠϯ্ͷϘιϯత։ݭʹର͢ΔΤϯλϯάϧϝϯτɾΤϯτϩϐʔͷݚڀΛϨϏϡʔ͢

Δ͜ͱʹ͢Δɽ

Dp-ϒϨΠϯ͸ɼۭؒతp࣍ݩͷ௒ฏ໘ʹΑͬͯهड़͞ΕΔɼ֦͕ΓΛ΋ͬͨ෺ମͰɼ

ͦͷ্ʹ։ݭͷ୺఺Λ۩͍͑ͯΔɽ͜͜Ͱ͸ɼ1ຕͷ௒ฏ໘্ʹہࡏ͢ΔଟॏDp-ϒϨΠ ϯΛߟ͑ɼͦͷ௒ฏ໘্ͷ͋Δํ޲ʹԊͬͯͦͷ௒ฏ໘Λ2ͭʹ෼ׂ͢Δɽͦ͜ͰɼFock

ۭؒදࣔʹ͓͚Δݭͷ৔Λ΋͍ͪͯɼݭͷີ౓ߦྻΛఆٛ͢Δɽͦͯ͠ɼ։ݭʹର͢Δ ΤϯλϯάϧϝϯτɾΤϯτϩϐʔ͸ϨϓϦΧɾτϦοΫΛద༻͢Δ͜ͱʹΑͬͯಘΒΕ Δɽͦͷ݁Ռͱͯ͠ಘΒΕͨΤϯλϯάϧϝϯτɾΤϯτϩϐʔ͸෼ׂ͞Εͨ௒ฏ໘ͷ (p−1)࣍ݩڥք໘ͷ໘ੵʹൺྫ͠ɼ༧૝௨Γൃࢄ͍ͯ͠Δɽ͔͠͠ɼͦͷUVྖҬ͓Α

(5)

ͼIRྖҬʹ͓͚Δओཁͳൃࢄ͸ओʹ։ݭʹؚ·Ε͍ͯΔλΩΦϯϞʔυʹΑΔ΋ͷͰ͋

Γɼ௒ରশੑΛ۩͑ͨݭཧ࿦ʢ௒ݭཧ࿦ʣʹ͓͍ͯ͸ফ͑Δ͜ͱ͕༧૝Ͱ͖ΔɽλΩΦϯ ͷد༩ʹΑΔ͜ΕΒͷൃࢄΛআ͘ͱɼDp-ϒϨΠϯ্ͷϘιϯత։ݭʹର͢ΔΤϯλϯά ϧϝϯτɾΤϯτϩϐʔ͸ɼ2≤p≤dcl−2 = 24^{ʹ͓͍ͯ༗ݶͰ͋Γɼ}p= 1,25^ʹ͓

͍ͯର਺ൃࢄ͢Δɽͦͷ݁Ռɼ͋Δݭͷॏ͍ྭىঢ়ଶͷແݶౝ͸ཧ࿦ͷUV^͓ΑͼIR^ྖ Ҭͷৼ෣Λେ͖͘มԽͤ͞Δ͜ͱ͕༧૝Ͱ͖Δɽ

2 Dp - ϒϨΠϯ্ͷϘιϯత։ݭͱͦͷີ౓ߦྻ

Dp-ϒϨΠϯ͸ɼۭؒతp࣍ݩͷ௒ฏ໘ʹΑͬͯهड़͞Εɼͦͷ্ʹ։ݭͷ୺఺Λ۩

͍͑ͯΔͱ͍͏͜ͱ͸࣍ͷΑ͏ͳ͜ͱͰ͋Δɽ։ݭ࠲ඪX^µ,µ= 0,1,· · ·, pɼͷ୺఺͸

Neumannڥք৚݅

∂X^µ

∂σ

��

�σ=0, π= 0, for µ= 0,1,· · ·, p (1) Λຬͨ͠ɼ։ݭ࠲ඪXⁱ,i=p+ 1,· · ·, dɼͷ୺఺͸Dirichletڥք৚݅

Xⁱ��

�σ=0, π= 0, for i=p+ 1,· · ·, d (2)

Λຬ͍ͨͯ͠Δ΋ͷͱ͢Δɽ͜ͷͱ͖ɼ։ݭ࠲ඪX^µ,µ= 0,1,· · ·, p^͸Dp-^ϒϨΠϯͷ

ੈքମੵʹ઀͢Δํ޲Λ޲͖ɼҰํɼ։ݭ࠲ඪXⁱ,i=p+ 1,· · ·, d=dcl−1^͸ͦͷ઀

ฏ໘ʹର͠ਨ௚ͳํ޲Λ޲͍͍ͯΔ͜ͱʹͳΔɽ͜͜Ͱ͸1≤p≤dcl−1^ͷDp-^ϒϨΠ ϯΛߟ͑Δ͜ͱʹ͢Δɽڥք৚݅(1)^ͱ(2)^{ΑΓɼ։ݭ࠲ඪ}X^I, I= 0,1,· · ·, d^͸ɼৼ ಈͷج४ϞʔυΛ༻͍ͯɼ

X^µ(σ) =x^µ+√ 2∑

k=1

x^µ_kcos (kσ), µ= 0,1,· · ·, p, (3) Xⁱ(σ) =√

2∑

k=1

xⁱ_ksin (kσ), i=p+ 1,· · ·, d (4)

ͱల։ͨ͠ܗʹද͢͜ͱ͕Ͱ͖Δɽ͜ͷͱ͖ʹɼ։ݭ࠲ඪXⁱ, i=p+ 1,· · ·, dʹ͸θϩ Ϟʔυؚ͕·Ε͍ͯͳ͍͜ͱʹ஫ҙ͢Δͱɼ։ݭͷ୺఺͕nຕͷଟॏDp-ϒϨΠϯʹͭ

͍͍ͯΔ৔߹ʹ͸ɼ։ݭͷ৔Ψ͸U(n)܈ͷΠϯσοΫε0,1,· · ·, n²−1Λ΋ͪɼ Ψ[X] = 1

√2Ψ⁰[X] + Ψ^a[X]T^a, a= 1,· · ·, n²−1 (5)

(6)

ͱද͞ΕΔɽ͜͜ͰɼT^a, a= 1,· · ·, n²−1͸SU(n)܈ͷੜ੒ࢠͰ͋Δɽ

BRSTෆมͳݭͷ৔ͷཧ࿦ͷ࡞༻͸ɼWittenͷcubic։ݭͷ৔ͷཧ࿦[56]Λ֦ு͢Δ

͜ͱʹΑΓɼ

SBRST=

∫ tr

(

Ψ∗QΨ +2g

3Ψ∗Ψ∗Ψ )

(6)

Ͱ༩͑ΒΕΔɽͦΕ͸ɼݭ࠲ඪX^Iͷ௨ৗͷج४Ϟʔυల։Λ(3)^ࣜͱ(4)^{ࣜͷΑ͏ͳ΋}

ͷʹஔ͖׵͑Δ͚ͩͰ͋Δɽ͜͜Ͱ͸ɼࣗ༝ݭཧ࿦ʹର͢ΔΤϯλϯάϧϝϯτɾΤϯτ ϩϐʔͷܭࢉʹݶఆ͢ΔͨΊɼg= 0ͱऔΔ͜ͱʹ͢ΔɽϑΣϧϛΦϯతΰʔετθϩ ϞʔυΛੵ෼͢Δͱɼݻ༗࣌ήʔδͰͷݭͷ৔ͷ࡞༻[52]

S0=

∫ tr Ψ(

L0+L^gh₀ )

Ψ, (7)

L0+L^gh₀ =p^µp^νηµν+∑

k=1

k a^†I_ka^J_kηIJ+

∑2

i=1

∑

k=1

ka^gh†_ik a^gh_ik−1 (8)

͕ಘΒΕΔɽ͜͜Ͱɼa^gh_ik, i = 1,2͸BRSTΰʔετ࠲ඪͷFourier੒෼Ͱ͋Γɼ {a^gh_ik, a^†gh_jℓ }=δijδkℓΛຬͨ͢.௨ৗͷBRSTΰʔετ࠲ඪ͸a^gh_ik,i= 1,2Λ༻͍ͯɼ

bzz(σ) =b0

2 +1 2

∑

k=1

(

a^gh_1ke^−ikσ−ia^†gh_2k e^ikσ)

, (9)

b_z¯_¯_z(σ) =b0

2 +1 2

∑

k=1

(a^gh_1ke^ikσ−ia^†gh_2k e^−ikσ)

, (10)

c^z(σ) =c0

2 +1 2

∑

k=1

(a^†gh_1k e^ikσ+ia^gh_2ke^−ikσ)

, (11)

c^z^¯(σ) =c0

2 +1 2

∑

k=1

(a^†gh_1k e^−ikσ+ia^gh_2ke^ikσ)

(12)

ͱల։͢Δ͜ͱ͕Ͱ͖Δɽ

ΤϯλϯάϧϝϯτɾΤϯτϩϐʔΛܭࢉ͢ΔͨΊʹ͸ɼ։ݭΛදݱ͢Δɼໃ६ͳ͘ఆ

·ͬͨہॴ৔ͷ࡞༻ૉΛݟ͚ͭΔඞཁ͕͋ΔɽͦͷͨΊʹ͸ݭͷ৔ͷFockۭؒදݱ

|Ψ⟩= ∑

{N_k^B,N_k^gh,k=1,2,3,··· }

∑

a

Ψ^a_{NB

k,N_k^gh}(x^µ)T^a|{N_k^B, N_k^gh, k= 1,2,3,· · · }⟩

=∑

a

(

ϕâ(x) +Aâ_µ(x)a^µ†₁ +φâ_i(x)aî†₁ +· · ·)

T^a|0⟩ (13)

(7)

͕ద͍ͯ͠ΔͰ͋Ζ͏ɽ͜͜Ͱɼϕâ,Aâ_µ, φâa= 0,1,· · ·, n²−1͸ɼͦΕͧΕɼλΩΦ

ϯ৔ɼYang-Millsήʔδ৔ͱ࣭ྔθϩͷϕΫτϧ৔ʹରԠ͢Δɽ͜ͷͱ͖ɼॏ͍ߴεϐ

ϯ৔ͷແݶͷౝ͸লུ͞Ε͍ͯΔɽ(13)ࣜΛ࢖ͬͯɼݭͷਅۭ೾ಈ൚ؔ਺͕

Φ[Ψ] =⟨0|Ψ⟩= 1 Z

∫ Φ

{Nk,Ngh

k }(0,x¹,···,x^p)=Ψ

{Nk,Ngh

k}(x¹,···,x^p) Φ{Nk,Ngh

k }(−∞,x¹,···,x^p)=0

D[Φ]e^−S⁰^(Φ) (14)

ͱॻ͚Δ͜ͱ͕Θ͔Δɽ͜͜ͰɼZ͸ن֨Խఆ਺Ͱ͋Δɽݭͷ৔ʹର͢Δਅۭີ౓ߦྻ͸

͜ͷجఈʹ͓͍ͯ

ρ[Ψ,Ψ^′] =⟨Ψ|0⟩⟨0|Ψ^′⟩= Φ[Ψ]^∗Φ[Ψ^′] (15) ͱఆٛͰ͖Δɽ͜Ε͸ɼ௨ৗͷྔࢠ৔ʹର͢Δਅۭີ౓ߦྻͷఆٛͷ࢓ํͱಉ༷Ͱ͋Δɽ

・・・・・・

_

ⁿ^ᯛࡢ^Dp^ࣈࣞ࢖ࣥ

O

㛤ᘻ 㛤ᘻ

ਤ1 ೋ෼ׂ͞ΕͨDp-ϒϨΠϯ্ͷ։ݭ

ͭ͗ʹɼਤ1^{ʹ͋ΔΑ͏ʹɼ}p࣍ݩͷ௒ฏ໘Λ൒෼ʹ෼͚ɼx¹ >0^{ͷ൒௒ฏ໘}A^ͱ x¹ <0^{ͷ൒௒ฏ໘}Bʹ͢Δɽ͕ͨͬͯ͠ɼݭͷ೾ಈ൚ؔ਺͸௚࿨Ψ = ΨA⊕ΨBͱ ද͢͜ͱ͕Ͱ͖Δɽͦ͜ͰɼB^{্Ͱ͸Ұக͢Δ}2^{ͭͷ೾ಈ൚ؔ਺}Ψ = ΨA⊕ΨBͱ Ψ^′= Ψ^′_A⊕ΨBΛߟ͑Δɽͦͷͱ͖ɼ൒௒ฏ໘Aʹ͓͚Δॖ໿ີ౓ߦྻ΋ݭͷਅۭ೾ಈ

൚ؔ਺ͷ৔߹ͱಉ༷ʹఆٛͰ͖ΔɽBʹ͍ͭͯͷऔΓಘΔ͢΂ͯͷ൚ؔ਺ΨBʹ͍ͭͯ

(8)

ੵ෼͠ɼ

ρA(Ψ,Ψ^′) =

∫

D[ΦB]Φ[ΨA⊕ΨB]^∗Φ[Ψ^′_A⊕ΨB]

= 1 Z

∫Φ

{Nk,Ngh

k}(0⁺,x¹,···,x^p)=Ψ

{Nk,Ngh

k}(x¹,···,x^p) Φ{Nk,Ngh

k }(0⁻,x¹,···,x^p)=Ψ^′

{Nk,Ngh

k}(x¹,···,x^p)

D[Φ]e^−S⁰^(Φ) (16)

ΛಘΔɽ݁Ռͱͯ͠ɼॖ໿ີ౓ߦྻͷn৐ͷτϨʔεtrρⁿ_A͸ɼnຕͷॏͳͬͨRiemann ໘্ͷܦ࿏ੵ෼ͱͯ͠ɼ

trρⁿ_A= ZA(n)

ZA(1)ⁿ (17)

ͱද͞ΕΔɽnʹ͍ͭͯͷղੳ઀ଓΛ͠ɼn= 1Ͱͷඍ෼ΛͱΔͱɼ։ݭͷ৔ʹର͢ΔΤ ϯλϯάϧϝϯτɾΤϯτϩϐʔ

S_ent= lim

n→1

[

−∂

∂ntrρⁿ_A ]

= lim

n→1

[

−∂

∂n

{lnZ_A(n)−nlnZ_A(1)}]

(18)

͕ಘΒΕΔɽ͜ͷํ๏͸ϨϓϦΧ๏ͱΑ͹ΕΔɽ

3 ։ݭͷ৔ʹର͢ΔΤϯλϯάϧϝϯτɾΤϯτϩϐʔͷ ܭࢉ

Fockۭؒදݱʹ͓͚Δ։ݭͷ৔͸λΩΦϯɼ࣭ྔθϩͷ৔͓Αͼແݶݸͷॏ͍৔͔Β

੒͍ͬͯΔɽFockۭؒදݱʹ͓͍ͯɼ։ݭͷ੒෼৔ʹର͢Δࣗ༝৔ͷ࡞༻͸ݸ਺࡞༻ૉ

NB+Ngh−1ͷݻ༗஋ʹΑͬͯ༩͑ΒΕΔ࣭ྔΛ΋ͬͨεΧϥʔ৔ͷ࡞༻

S0=

∫ ∑

{Nk^B,N_k^gh,k=1,2,3,··· }

Ψ^a†

{N_k^B,N_k^gh}

(p²+NB+Ngh−1) Ψ^a_{NB

k,N_k^gh}, (19) NB=∑

k=1

ka^I†_ka^I_k, I= 0,1,· · ·, d, (20)

N_gh=∑

k=1

k(

a^†gh_1k a^gh_1k+a^†gh_2k a^gh_2k)

(21)

ͱಉ͡Ͱ͋Δ͜ͱʹ஫ҙ͢Δͱɼ੒෼৔Ψ^a

{N_k^B,N_k^gh}ͷ౷ܭੑ͸ɼ

Fgh=∑

k=1

(

(22)

(9)

Λ୲͏શΰʔετ਺ʹΑܾͬͯ·Δɽ͜ͷΑ͏ʹɼ։ݭʹର͢ΔΤϯλϯάϧϝϯτɾΤ ϯτϩϐʔ͸ॏ͍εΧϥʔ৔ͷ݁ՌΛ௚઀Ԡ༻͢Δ͜ͱͰಘΒΕΔɽ

ॏ͍εΧϥʔ৔ʹର͢ΔΤϯλϯάϧϝϯτɾΤϯτϩϐʔͷܭࢉ͸จݙ[13]^ʹݟΒΕ Δɽ͜ͷܭࢉʹଟগͷमਖ਼Λ͢Δ͜ͱͰɼ։ݭͷ৔ͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔͷ ܭࢉʹ༻͍Δ͜ͱ͕Ͱ͖Δɽ(p+1)^{࣍ݩʹ͓͚Δ࣭ྔ}mͷࣗ༝εΧϥʔ৔ʹର͠ɼn^ຕͷ

ॏͳͬͨRiemann໘্Ͱఆٛ͞Εͨ෼഑ؔ਺Z(n)^͸lnZ(n) =−¹₂ln Det[

−∆ +m²] ͱॻ͘͜ͱ͕Ͱ͖Δɽn= 1^ۙ๣Ͱn^{Λղੳ઀ଓ͢Δͱɼ}n^{ຕͷॏͳͬͨ}Riemann^໘͸

ܽଛ֯δ= 2π(1−n)^ͷԁਲ਼ۭؒRⁿ^{ʹͳΔɽ෼഑ؔ਺}Z(n)͸ɼԁਲ਼ۭؒʹ͓͚Δॏ͍

৔ʹର͢ΔGreen^ؔ਺Gn(x,x^′)^{ͱ࣍ͷΑ͏ͳؔ܎}

∂

∂m²lnZ(n) =−1 2

∫

Rⁿ

d^p+1x lim

x^′→xGn(x,x^′) (23)

͕͋Γɼ͜͜Ͱɼx= (x⁰, x¹,· · ·, x^p) = (x⁰, x¹,x_⊥)Ͱ͋Δɽ Greenؔ਺Gn(x,x^′)ͷදࣜ͸

Gn(x,x^′) = 1 2πn

∫ d^p−1p_⊥ (2π)^p−1

×

∑∞ ℓ=0

dℓ

∫_∞

0

dq qJ_ℓ/n(qr)J_ℓ/n(qr^′) q²+m²+p²_⊥ cos

(ℓ n(θ−θ^′)

)

e^ip^⊥^·(x^⊥^−x^′^⊥⁾ (24)

ͱද͞Εɼ͜ͷࣜʹ͓͍ͯɼJ^͸ୈ1^छBessel^{ؔ਺Ͱ͋Γɼ}ℓ≥1^ʹର͠d0= 1,dℓ= 2^Ͱ

͋Δɽ·ͨɼ(r, θ)^͸2^࣍ݩ(x⁰, x¹)ฏ໘্ʹ͓͚Δۃ࠲ඪͰ͋Γ[13]^ɼp_⊥= (p²,· · ·, p^p) Ͱ͋ΔɽಉҰ఺ۃݶx^′→x^ʹ͓͚ΔGreen^{ؔ਺ͷදࣜΛ࢖͏ͱɼ}

∂

∂m²ln trρⁿ= ∂

∂m²ln Z(n) Z(1)ⁿ =−1

2 {∫

Cⁿ

d^p+1xGn(x,x)−n

∫

d^p+1xG1(0) }

=−1−n² 24n A_⊥

∫ d^p−1p_⊥ (2π)^p−1

1

m²+p²_⊥ (25)

͕ಘΒΕɼ͜͜ͰɼA_⊥^͸(x⁰, x¹)ฏ໘ʹਨ௚ͳڥք௒ۂ໘ͷ໘ੵͰɼA_⊥=∫ d^p−1x_⊥ Ͱ͋Δɽn^{Ͱඍ෼ͯ͠ɼ}m²Ͱੵ෼Λ࣮ߦ͢Δͱɼॏ͍৔ͷΤϯλϯάϧϝϯτɾΤϯτ ϩϐʔ

S_A=−1 12A_⊥

∫ d^p−1p_⊥ (2π)^p−1ln(

m²+p²_⊥)

=A_⊥ 12

∫ ds s

∫ d^p−1p_⊥ (2π)^p−1exp{

−s(

p²_⊥+m²)}

=A_⊥ 12

1 (8π²)^p−1²

∫_∞

0

dt 1

t^p+1² e^−2πm²^t (26)

(10)

͕ಋ͔ΕΔɽ͜͜Ͱɼt=s/(2π)^Ͱ͋Δɽ

͜ͷ݁Ռ͸FockۭؒදݱͰͷݭͷ৔ʹద༻ՄೳͰ͋Δɽ͜ͷ݁ՌΛɼϘιϯɾηΫ λʔ(N_k^gh= 0, k= 1,2,· · ·)ʹ͓͚Δ։ݭͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔʹ௚઀

ద༻͢Δͱɼ

S^։ݭ_A =A_⊥ 12

∫ ds s

∫ d^p−1p_⊥ (2π)^p−1Tr exp{

−s(

p²_⊥+NB−1)}

(27)

ͱͳΓɼ͜ͷͱ͖ɼ‘Tr’^͸Fock^ۭؒͳΒͼʹU(n)܈ۭؒͰͷτϨʔεΛද͢ɽϘιϯ తௐ࿨ৼಈࢠͷ୅਺[57]^͔Β

Tre^−sN^B=n² ∑

{N_k^B}

exp {

−s∑

k=1

kN_k^B }

=n²∏

k=1

( 1 1−e^−sk

)d+1

=n²e⁻^d+1²⁴^s 1 η(_is

2π

)d+1 (28)

ͱͳΓɼ͜ͷͱ͖ɼη(τ)^͸η(τ) :=e^{iπτ /12}∏

k=1

(1−e^2πikτ)

Ͱఆٛ͞ΕΔDedekind^ͷη

ؔ਺Ͱ͋Δɽ͜ͷΑ͏ʹɼϘιϯɾηΫλʔʹ͓͚Δ։ݭͷΤϯλϯάϧϝϯτɾΤϯτ ϩϐʔ

S_A^։ݭ= A_⊥ 12

n² (8π²)^p−1²

∫ ∞ 0

dt t

1 t^p−1² exp

{(

1−d+ 1 24

) 2πt

} 1

η(it)^d+1 (29)

͕ܭࢉ͞Εɼ͜͜Ͱɼt=s/(2π)^Ͱ͋Δɽ

ΰʔετɾηΫλʔͷد༩Λߟྀ͢Δͱɼ։ݭͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔ͸

S_A^։ݭ=A_⊥ 12

∫ ds s

∫ d^p−1p_⊥ (2π)^p−1Tr exp{

−s(

p²_⊥+NB+Ngh−1)}

(−1)^F^gh (30) ͱͳΓɼ͜ͷͱ͖ɼ

Ngh=∑

k=1

k(

, Fgh=∑

k=1

(

a^†gh_1k a^gh_1k+a^†gh_2k a^gh_2k) (31)

Ͱ͋Δɽ(30)ࣜͰ͸ɼΰʔετ࠲ඪͷϑΣϧϛΦϯతͳ౷ܭੑʹ஫ҙͯ͠ɼ(−1)^F^gh^Ҽࢠ Λಋೖͨ͠ɽެࣜ

∑

{Ngh}

e^−sN^gh(−1)^F^gh=∏

k=1

(1−e^−sk)2

=e¹²^sη (is

2π )2

(32)

(11)

Λ༻͍ͯɼΤϯλϯάϧϝϯτɾΤϯτϩϐʔS_A^։ݭΛ

S^։ݭ_A =A⊥

12 n² (8π²)^p−1²

∫ ∞ 0

dt t

1 t^p−1² exp

{(25−d 24

) 2πt

} 1

η(it)^d−1 (33) ͱॻ͘͜ͱ͕Ͱ͖Δɽd=d_crtical−1 = 25ͷϘιϯతݭཧ࿦ʹରͯ͠͸

S_A^։ݭ=A_⊥ 12

n² (8π²)^p−1²

∫ _∞

0

dt t

1 t^p−1²

1

η(it)²⁴ (34)

ͱͳΔɽ

(34)^{ࣜͷඃੵ෼ؔ਺ͷ੺֎}(IR)^ྖҬͱࢵ֎(UV)^{ྖҬͰͷৼ෣͸}p^{ʹඇৗʹڧ͘ґଘ}

͍ͯ͠Δ͜ͱ͕Θ͔Δɽඃੵ෼ؔ਺ͷIR^Ͱͷৼ෣͸Dedekind^ͷη^{ؔ਺ͷ઴ۙల։}[57]

η(it) =e⁻¹²^π^t(

1−e^−2πt−e^−4πt+· · ·)

, t→ ∞ (35)

͔ΒಡΈऔΔ͜ͱ͕Ͱ͖Δɽ઴ۙྖҬʹ͓͍ͯɼඃੵ෼ؔ਺͸

1 t^p+1²

1

η(it)²⁴ = 1 t^p+1²

{

e^2πt+ 24 +O(e^−2πt)}

(36)

ͱͳΔɽ(34)ࣜɼ(36)ࣜͱ(26)ࣜͷॏ͍εΧϥʔ৔ͷΤϯλϯάϧϝϯτɾΤϯτϩ ϐʔΛൺֱ͢Δͱɼओཁൃࢄ͸λΩΦϯͷد༩ʹΑΔ΋ͷͰ͋Δ͜ͱ͕Θ͔Δɽ͜ͷओཁ

ൃࢄͷଞʹɼ։ݭʹ͍ͭͯͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔ͸ߴʑର਺తʹൃࢄ͢

ΔɽͦͷIRྖҬͰͷৼ෣͸ɼp≥2ʹରͯ͠͸༗ݶ஋Ͱ͋Γɼp= 1ʹରͯ͠͸ର਺ൃࢄ

Ͱ͋Δɽ͜ͷ݁Ռ͸ɼແݶछྨ͋Δॏ͍ঢ়ଶ͕ΤϯλϯάϧϝϯτɾΤϯτϩϐʔͷIR

ྖҬͰͷৼ෣Λվળ͢ΔͰ͋Ζ͏ͱ͍͏༧૝ͱໃ६͠ͳ͍͜ͱ͕Θ͔Δɽ

UV^ྖҬ͸t = 0ۙ๣ͷྖҬʹରԠ͢ΔɽDedekind^ͷη ^{ؔ਺ͷϞδϡϥʔม׵}

η(−1/τ) = (−iτ)^1/2η(τ)^Λ࢖༻͠ɼs= 1/t^ʹΑΓS_A^։ݭʹର͢Δੵ෼Λॻ͖׵͑Δͱɼ S_A^։ݭ=A_⊥

12 n² (8π²)^p−1²

∫ _∞

0

ds s¹²^(p−27) 1

η(is)²⁴ (37)

ΛಘΔɽs→ ∞^ʹରԠ͢ΔUVྖҬͰͷඃੵ෼ؔ਺͸઴ۙతʹ s¹²^(p−27) 1

η(is)²⁴ =s¹²^(p−27){

e^2πs+ 24 +O(e^−2πs)}

(38)

ͷΑ͏ʹల։Ͱ͖Δɽओཁൃࢄ͸࠶ͼɼ“^{ดݭνϟϯωϧ}”ʹ͓͚ΔλΩΦϯͷد༩ʹؼ

͠ɼ௒ରশͳݭཧ࿦ʹ͓͍ͯ͸ແ͘ͳΔ͜ͱͰ͋Ζ͏ɽλΩΦϯ͔Βͷد༩Λআ͘ͱɼΤ ϯλϯάϧϝϯτɾΤϯτϩϐʔ͸p≤24^ʹରͯ͠UV^{ྖҬͰ༗ݶ஋ΛͱΓɼ}p= 25^ʹ ରͯ͠͸ର਺తʹൃࢄ͢Δɽ

(12)

4 ^݁࿦ͱٞ࿦

[1]^ͰߦΘΕͨɼ1≤p≤25^{ʹର͢Δଟॏ}Dp-ϒϨΠϯ্ͷϘιϯత։ݭʹର͢ΔΤϯ λϯάϧϝϯτɾΤϯτϩϐʔͷܭࢉʹ͍ͭͯਫ਼ࠪͨ͠ɽہॴతͳ৔ͷ࡞༻ૉΛఆٛ͢Δ

ͨΊʹɼڞมతݭͷ৔ͷཧ࿦ͱ։ݭͷ೾ಈ൚ؔ਺ͷFock^{ۭؒදࣔΛ࠾༻ͨ͠ɽ}Dp-^ϒϨ Πϯͷۭؒ࣍ݩΛߏ੒͢Δp࣍ݩ௒ۂ໘͸൒෼ʹ෼ׂ͞Εͨɽաڈʹͳ͞Εͨ[13]^ɼॏ͍

৔ͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔͷܭࢉ๏Λ௚઀։ݭͷ৔ͷΤϯλϯάϧϝϯτɾ Τϯτϩϐʔͷܭࢉʹద༻͢Δ͜ͱʹΑΓɼ։ݭͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔͷ ܭࢉΛ਱ߦ͢Δ͜ͱ͕Ͱ͖ͨɽͦΕΛ࣮ߦ͢ΔͨΊʹ͸ଟগͷमਖ਼͕ඞཁͰ͋ͬͨɽΤϯ τϩϐʔ͸ɼ༧૝௨Γɼ2ͭʹ෼ׂͨ͠p࣍ݩ௒ۂ໘ͷڥքͷ໘ੵʹൺྫ͍ͯ͠Δ͜ͱ͕

͔֬ΊΒΕͨɽ͔͠͠ɼͦͷUVྖҬ͓ΑͼIRྖҬͷৼ෣͸ہॴత৔ͷཧ࿦ͷ৔߹ͱ͸

શ͘ҟͳ͍ͬͯͨɽͦͷৼ෣͸UVྖҬͱIRྖҬͷ྆ํʹ͍ͭͯൃࢄ͍͕ͯͨ͠ɼ͜Ε Βͷൃࢄ͸λΩΦϯͷد༩ʹؼ͢͜ͱ͕Ͱ͖ɼ௒ରশͳཧ࿦ʹ͓͍ͯ͸ແ͘ͳΔͰ͋Ζ͏

ͱ༧૝Ͱ͖ΔɽλΩΦϯʹΑΔओཁൃࢄΛআڈ͢Ε͹ɼ։ݭͷΤϯλϯάϧϝϯτɾΤϯ τϩϐʔ͸ہॴత৔ͷཧ࿦ͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔΑΓ΋ྑ͍ৼ෣Λ͢Δɽ

։ݭͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔ͸ɼ2≤p≤dcl−2 = 24ͷ৔߹ʹ͸UVྖҬ ͱIRྖҬͷ྆ํʹ͍ͭͯ༗ݶ஋ʹͳΓɼp= 1,25ͷ৔߹ʹ͸ߴʑର਺తʹൃࢄ͢Δɽ[1]

ͷݚڀͰɼ։ݭͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔͷUVྖҬฒͼʹIRྖҬͷৼ෣ʹ

͍ͨ͠ɼ։ݭʹؚ·ΕΔແݶछྨͷॏ͍৔͕ڧ͘ӨڹΛٴ΅͍ͯ͠Δ͜ͱ͕໌Β͔ʹࣔ͞

Εͨɽ

ࠓޙͷల๬ʹؔ͢Δ͍͔ͭ͘ͷҙݟΛ੔ཧ͓ͯ͘͠ɽଟॏDp-ϒϨΠϯ্ͷ௒ରশͳ

։ݭʹର͢ΔΤϯλϯάϧϝϯτɾΤϯτϩϐʔͷܭࢉ͸ɼ͍ۙকདྷʹ਱ߦ͞Εͳ͚͹ͳ Βͳ͍ॏཁͳٸ຿Ͱ͋Δɽ௒ରশͳݭཧ࿦Ͱ͸ɼΤϯλϯάϧϝϯτɾΤϯτϩϐʔ͸༗

ݶ͔·ͨ͸ߴʑର਺ൃࢄΛ͢Δ͜ͱ͕ظ଴Ͱ͖ΔɽDp-ϒϨΠϯ্ͷ։ݭͷΤϯλϯάϧ ϝϯτɾΤϯτϩϐʔ͸։ݭͷ̍ϧʔϓৼ෯[58]ͱྨࣅ͍ͯ͠Δ͕ɼ͜͜Ͱܭࢉͨ͠։

ݭͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔ͸̍ϧʔϓิਖ਼Ͱ͸ͳ͘ɼπϦʔϨϕϧͷྔͰ͋

Δɽ͜ͷΑ͏ʹɼ͜͜Ͱͷܭࢉ݁Ռ͸ɼSusskindͱUglum [23]ʹΑΔɼϒϥοΫϗʔ ϧɾΤϯτϩϐʔ͸ݭཧ࿦ͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔͱͯ͠ཧղͰ͖Δͱ͍͏

11

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ఏҊΛࢧ࣋͢Δ΋ͷͰ͋Δɽ

࣭ྔθϩͷήʔδ৔͸੒෼ͷҰͭͱͯ͠ݭͷ৔ͷதʹؚ·Ε͍ͯΔɽ͔͠͠ɼήʔδର শੑ͸BRSTܗࣜʹ͓͍ͯ׬શʹݻఆ͞Ε͍ͯΔɽ΋͠ɼΤϯλϯάϧϝϯτɾΤϯτϩ ϐʔΛܭࢉ͢Δͱ͖ʹBRSTΰʔετɾηΫλʔΛߟྀ͢ΔͳΒ͹ɼ੒෼৔Ψ^a

{Nk^B,N_k^gh}

͸࣭ྔNB+Ngh−1Λ΋ͬͨεΧϥʔ৔ͱͯ͠ѻΘΕΔ͸ͣͰ͋Δɽ͜͜Ͱ͸ɼࣗ༝ͳ ݭͷ৔ͷཧ࿦ͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔ͚͕ͩٞ࿦͞Εͨɽ͔͠͠ɼ͜ͷܭࢉ

๏Λ֦ு͠ɼ૬ޓ࡞༻ͷೖͬͨݭͷ৔ͷཧ࿦ͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔΛݚڀ

͢Δ͜ͱ͸ͦ͏೉͘͠͸ͳ͍Ͱ͋Ζ͏ɽ։ݭͷ৔ͷཧ࿦ͷcubic૬ޓ࡞༻͸Τϯλϯάϧ ϝϯτɾΤϯτϩϐʔʹର͠ݹయ࿦త͔ͭྔࢠ࿦తͳิਖ਼Λੜ੒͢Δ͜ͱͰ͋Ζ͏ɽ

·ͨɼ௿ΤωϧΪʔۃݶͰݱ৅࿦తʹ΋ڵຯਂ͍৔ͷཧ࿦తͳϞσϧΛੜ੒͢ΔΑ͏

ͳɼΑΓෳࡶͳܗঢ়ͷDp-ϒϨΠϯʹ෇͍ͨ։ݭͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔ Λݚڀ͢Δ͜ͱ΋Ͱ͖ΔͰ͋Ζ͏ɽ͔͠͠ɼࠓޙͷൃలʹ͓͍ͯඇৗʹॏཁͳ͜ͱ͸ڞม తͳดݭͷ৔ͷཧ࿦ͷ࿮૊ΈͷதͰΤϯλϯάϧϝϯτɾΤϯτϩϐʔΛݚڀ͢Δ͜ͱͰ

͋Ζ͏ɽ͜ͷ఺Ͱɼݻ༗࣌ؒήʔδͰͷڞมతͳดݭͷ৔ͷཧ࿦[59]͕։ݭͷΤϯλϯ άϧϝϯτɾΤϯτϩϐʔΛݚڀ͢ΔͨΊʹ͸ܽ͘͜ͱ͕Ͱ͖ͳ͍ಓ۩ͱͯ͠໾ʹཱͭ͜

ͱͰ͋Ζ͏ɽ࠷ۙͷݚڀͷதͰɼݻ༗࣌ؒήʔδͰͷڞมతͳดݭͷ৔ͷཧ࿦͕௿Τωϧ ΪʔۃݶͰͷॏྗࢠͷࢄཚৼ෯Λੜ੒͢Δ͜ͱ͕੒ޭཪʹࣔ͞Εͨɽ։ݭͷ৔ͷཧ࿦ͷΤ ϯλϯάϧϝϯτɾΤϯτϩϐʔͱดݭͷ৔ͷཧ࿦ͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔ Λൺֱ͢Δ͜ͱ͸ɼؔ࿈ͨ͠໰୊ʹώϯτΛ༩͑ΔͰ͋Ζ͏ͱߟ͑ΒΕΔɽ

͘͝࠷ۙͰ͸ɼBalasubramanianͱParrikar [60]ʹΑΓɼޫԁਲ਼ήʔδͰͷ৔ͷཧ࿦

Λ࢖ͬͨɼ։ݭͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔ͕ܭࢉ͞Εͨɽ൴Βͷ݁Ռ͸ɼD25 - ϒϨΠϯΛຬ্ۭͨؒ͢Ͱͷ։ݭͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔʹରԠ͍ͯ͠Δɽ

͞Βʹɼจݙ[61–63]Ͱ͸ɼҟͳΔηοςΟϯάͰͷݭཧ࿦ͷΤϯλϯάϧϝϯτɾΤϯ τϩϐʔ͕ݚڀ͞Εͨɽ

Ҏ্ͷΑ͏ͳཧ࿦త࿮૊ΈΛɼ༗ݶԹ౓ܥʹ֦ு͢Δͱ͍͏ํ޲ੑ΋ߟ͑ΒΕΔɽैདྷ ͷ༗ݶԹ౓ʹ͓͚Δݭཧ࿦ͷ1ϧʔϓৼ෯͸೤৔μΠφϛοΫε(TFD)Λ༻͍ͯܭࢉ͞

ΕɼԹ౓ͷม਺ʹ͍ͭͯղੳ઀ଓ͢Δ͜ͱʹΑΓɼͦͷղੳੑ͕ٞ࿦͞Εͨ[64–67]ɽͦ

ͷ݁Ռɼݭཧ࿦ͷHagedornԹ౓ΑΓ௿͍Թ౓ྖҬͰ͸ɼ1ϧʔϓৼ෯ͷղੳੑ͕ྑ͘ͳ Δ͜ͱ͕ࣔ͞ΕͨɽҰํɼTFDʹΑΔɼ༗ݶԹ౓ʹ͓͚ΔεϐϯܥͷΤϯλϯάϧϝϯ

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τɾΤϯτϩϐʔͷݚڀ΋ͳ͞Ε͍ͯΔ[68, 69]^{ɽ͜Ε౳ʹ฿ͬͯɼ}TFD^{ͷ࿮૊Έͷத} Ͱɼ༗ݶԹ౓ʹ͓͚Δݭཧ࿦ͷΤϯλϯάϧϝϯτɾΤϯτϩϐʔΛܭࢉ͠ɼԹ౓ྖҬʹ

͓͚Δͦͷৼ෣͍Λٞ࿦ͯ͠ΈΔ͜ͱ͸େมڵຯਂ͍͜ͱͰ͋Γɼࠓޙͷ༗๬ͳ՝୊ͷҰ

ͭͱͯ͠ڍ͛Δ͜ͱ͕Ͱ͖Δɽ

ࢀߟจݙ

[1] T. Lee, Phys. Lett. B782, 589 (2018).

[2] J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod.82, 277 (2010).

[3] L. Bombelli, R. Koul, J. Lee, and R. Sorkin, Phys. Rev. D34, 373 (1986).

[4] M. Srednicki, Phys. Rev. Lett.71, 666 (1993).

[5] C. Holzey, F. Larsen, and F. Wilczek, Nucl. Phys. B424, 443 (1994).

[6] C. Callan and F. Wilczek, Phys. Lett. B333, 55 (1994).

[7] J. S. Dowker, Class .Quant. Grav.11, L55 (1994).

[8] F. Larsen and F. Wilczek, Annals Phys.243, 280 (1995).

[9] H. Casini, Class. Quant. Grav.21, 2351 (2004).

[10] T. J. Osborne and M. A. Nielsen, Phys. Rev. A66, 032110 (2002).

[11] A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature416, 608 (2002).

[12] G. Vidal, J. I. Latorre, E. Rico and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003).

[13] P. Calabrese and J. L. Cardy, J. Sat. Mech.0406, P06002 (2004).

[14] H. Casini and M. Huerta, J. Stat. Mech.0801, P01012 (2008).

[15] H. Casini and M. Huerta, J. Phys. A42, 504007 (2009).

[16] M. P. Hertzberg and F. Wilczek, Phys. Rev. Lett.106, 050404 (2011).

[17] M. P. Hertzberg, J. Phys. A46, 015402 (2013).

[18] S. Seki and S.-J. Sin, Phys. Lett. B735, 272 (2014).

[19] G. Grignani and G. W. Semenoﬀ, Phys. Lett. B772, 699 (2017).

[20] C. H. Bennett and D. P. DiVincenzo, Nature (London)404, 247 (2000).

(15)

[21] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Commu- nication (Cambridge University Press, Cambridge, 2000).

[22] G. ’tHooft, Nucl. Phys. B256, 727 (1985).

[23] L. Susskind and J. Uglum, Phys. Rev. D50, 2700 (1994).

[24] D. Kabat, Nucl. Phys. B453281, (1995).

[25] S. N. Solodukhin, Phys. Rev. Lett.97, 201601 (2006).

[26] D. V. Fursaev, JHEP0609, 018 (2006).

[27] M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939.

[28] M. Van Raamsdonk, Gen. Rel. Grav.42, 2323 (2010).

[29] B. Czech, J. L. Karczmarek, F. Nogueira, and M. Van Raamsdonk, Class. Quant.

Grav.29, 235025 (2012).

[30] B. Swingle and M. Van Raamsdonk, (2014)Universality of gravity from entan- glement, arXiv:1405.2933.

[31] V. Balasubramanian, B. D. Chowdhury, B. Czech, and J. de Boer, JHEP01, 048 (2015).

[32] S. Ryu and T. Takayanagi, Phys. Rev. Lett.96, 181602 (2006).

[33] S. Ryu and T. Takayanagi, JHEP0608, 045 (2006).

[34] T. Faulkner,The entanglement Renyi entropies of disjoint intervals in AdS/CFT, [hep-th] arXiv:1303.7221.

[35] T. Faulkner, A. Lewkowycz and J. Maldacena, JHEP11, 074 (2013).

[36] T. Faulkner, M. Guica, T. Hartman, R. C. Myers, and M. Van Raamsdonk, JHEP03, 051 (2014).

[37] J. D. Bekenstein, Phys. Rev. D7, 2333 (1973).

[38] J. D. Bekenstein, Phys. Rev. D9, 3292 (1974).

[39] S. W. Hawking, Comm. Math. Phys.43, 199 (1975).

[40] R. M. Wald, Living Rev. Rel.4, 6 (2001).

[41] S. Solodukhin, Living Rev. Rel.14, 8 (2011).

[42] D. V. Fursaev and S. N. Solodukhin, Phys. Lett. B365, 51 (1996).

(16)

[43] S. Solodukhin, Phys. Rev. D52, 7046 (1995).

[44] A. O. Barvinsky and S. N. Solodukhin, Nucl. Phys. B479, 305 (1996).

[45] M. Cvetic and D. Youm, Nucl. Phys. B472, 249 (1996).

[46] D. Youm, Phys. Rept.316, 1 (1999).

[47] A. Strominger and C. Vafa, Phys. Lett. B379, 99 (1996).

[48] C. G. Callan and J. M. Maldacena, Nucl. Phys. B472, 591 (1996).

[49] G. Horowitz and A. Strominger, Phys. Rev. Lett.77, 2368 (1996).

[50] J. M. Maldacena, Adv. Theor. Math. Phys.2, 231 (1998).

[51] T. Azeyanagi, T. Nishioka and T. Takayanagi, Phys. Rev. D77, 064005 (2008).

[52] T. Lee, Ann. Phys. 183 (1988) 191.

[53] T. Lee, Phys. Lett. B768, 248 (2017).

[54] T. Lee, Jour. Kor. Phys. Soc.71, 886 (2017).

[55] T. Lee, Chin. Phys. C42, 113105 (2018).

[56] E. Witten, Nucl. Phys. B268, 253 (1986).

[57] M. B. Green, J. H. Schwarz and E. Witten,Superstring Theory Volume 1 and 2, (Cambridge University Press, Cambridge, 1987).

[58] T. Lee, K. S. Viswanathan, and Y. Yang, Jour. Kor. Phys. Soc.42, 34 (2003).

[59] T. Lee, EPJ Web of Conferences168, 07004 (2018).

[60] V. Balasubramanian and O. Parrikar, Phys. Rev. D 97, 066025 (2018).

[61] S. He, T. Numasawa, T. Takayanagi and K. Watanabe, JHEP1505, 106 (2015).

[62] L. A. Pando Zayas and N. Quiroz, JHEP1501, 110 (2015).

[63] L. A. Pando Zayas and N. Quiroz, JHEP1611, 023 (2016).

[64] Fujisaki H. and Nakagawa K.,Prog. Theor. Phys.82, 236; 1017 (1989) ;83, 18 (1990) ;Europhys. Lett.14, 639 (1991) ;20, 677 (1992) ;28, 1; 471 (1994) ;35, 493 (1996) ;Soryushiron Kenkyu(Kyoto)94, D34 (1997) ;Cosmological Constant and the Evolution of the Universe, ed. by Sato K.et al. (Universal Academy Press, Tokyo, 1996) p. 249.

[65] Fujisaki H., Nakagawa K. and Sano S., Nuovo Cim. A 110, 161 (1997) ; So- ryushiron Kenkyu(Kyoto)98, A12 (1998) ;Hoshi J. Gen. Educ. 24, 29 (2006).

(17)

.

[66] Fujisaki H., Sano S. and Nakagawa K.,Particle Cosmology, ed. by Sato K.et al.

(Universal Academy Press, Tokyo, 1998) p. 247.

[67] Nakagawa K.,Prog. Theor. Phys.85, 1317 (1991) . [68] Y. Hashizume and M. Suzuki, Physica A392, 3518 (2013) . [69] K. Nakagawa, Prog. Theor. Exp. Phys.2015, 021A01 (2015).

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著者 中川 弘一

Dp‑ブレイン上のボソン的開弦の量子エンタングル メントについて