U AdS d+1 /CF T d. AdS d+1

Transcription

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2 U U U AdS d+1 /CF T d U AdS d+1

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5 "i #i ± #i "i p 2 ( #i + "i)( #i + "i) 2

6 A := tr HA c A A c S A := tr HA A log A H = H A H A c A A c "ih" + #ih# 2 ( #i + "i)( #i + "i) 2 S left = log 2 S left =0 S (0,L) = c 3 log L S (0,L) = Constant

7 Z CFT [ (x)] = Z gravity [ (x) =lim z!0 z O d (x, z)] CFT d CFTi $ gravityi AdS d+1

8 S A = Area( ) 4G N A AdS d+1

9 E 0 i E 0 i AdS AdS CFT 1 CFT 2 i = X i e E i 2 Ei i E i i S CFT1 = CFT 1 = 2 G N CFT 1 CFT 2

10 I(A, B) (ho A O B i ho A iho B i) 2 2 O A 2 O B 2 CFT 1 CFT 2 I(A, B) =S A + S B S A[B ho A O B i e md(a,b) I(A, B) (ho AO B i ho A iho B i) 2 2 O A 2 O B 2 / e md(a,b)

11 S A = hh A i A = e H A S A = hh A 2 zh i i + d +1 z H i i j H i j H ij =0

12 z ds 2 AdS = dz2 + dx 2 z 2 z z 0 z O(x) O(y) O(x) O(y) z 0 µ 1 z AdS d+1

13 Z Z = D (l) IR( (l)) UV ( (l)) Z IR( (l)) = DM l e R / Z DM l e S(l) (l)o e S(l) = Z (l) IR( (l)) UV ( (l)) AdS d+1 D (l)e R (l)o UV ( (l))

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16 0 0 X 1... X 1 2 U 0 1(2) U 2 0(2) i 0i U 00(1) 1 2 1i 2i X 1 2 w w w U 1 2 (1) i 2 i 3 i 4 i w w (2) = X a a aiha P (2) = X a: a6=0 aiha

17 S half apple log = S = c A 3 Log L L c 3 = Constant 0

18 X 1 2 X 1 2 U 0 1 i w 1i 2i 2 1i 2i u U i 2 i = U U

19 u u w w (2) = X a a aiha P (2) = ˆ0 ˆ X a: a6=0 aiha

20 MERA i = X X X U 00 1, 2 U 0, 1 1, 2 U 2,0 3, 4 1, 2 1,..., 4 U 0 1 1, 2 U 2, 3 3, 4 U 4,0 1,..., 6 5, 6 1,..., 6 i U U U U

21 = Log 2 L 2Log apple S[ (2Log 2L) A ]+Log 2 L 2Log 0ih0 = = (2log 2L) A (1) A S A apple S[ (1) A ]+2Log A S A[B apple S A + S B S A log

22 1 2 X 1 2 U 0 1 i w 2 1i 2i 0i 1 2 X 1 2 0i i w U i 2i 0i 0i 0i (1) MERA i = X 1, 2 U 00 1, 2 (2) MERA i = X 1, 2 U 00 1, 2 0, 0, 1, 2, 0, 0i X 1,..., 4 U 0, 1 1, 2 U 2,0 3, 4 0, 1,..., 4, 0i 0i 1 2 0i 0i 0i

23 vacuumi 0i 0i 0i vacuumi i Z 0 vacuumi = P exp( i 1 du ˆK(u)) i

24 e H Bi hb O(x)O(y)... Bi 'hb O(x) BihB O(y) Bi... O(x) O(y) 2 S( )! S( )+M D 2 Z d D x ( (x) (x)) 2 M (x) (x) =0 S A = 1 3 log 4 a a

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26 S A = Area( ) 4G N S A apple log A A

27 Area( ) 4G N = S A log

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29 ( )i = Bi (0)i = vacuumi (u)i

30 0 (0)i = Bi 0 (u)i 0 (0)i = vacuumi

31 S A log A A

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33 A i A = ta i S A = X S Ai

34 Area( ) 4G N = X i S i = X i i Area( i ) 4G N i i i Area(minimal) log 4G N Area( ) log 4G N = log dimh

35 ds 2 = dt 2 + dx 2 + dy 2 t =0 L i S A = L 4G N ds 2 = R 2 ( dt 2 + cosh 2 td 2 S ) 2 i L t =0 S A = L 4G N

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38 U : V 0! V U U =1 U 0 au = a a h0 a(l) a (0) 0i = 1 2 a h0 0 0 a( L 2 ) 0 a(0) 0 0 i = 1 L 2Log 2 a h0(log 2L) Log 2 L (Log 2 L) a (0) 2 0 (Log2L) i 00 a a = 1 2 Log 2 a V 0 0 a V a

39 MPSi = X Tr(A 1 1 A A N N 1, 2..., N ) 1, 2..., N i ( X A A =1) b A b,c c 1 2 i

40 1 2 S A (L) const

41 i tr right ih = X 1applea a a left iha left D D left i, 2 left i,..., D left i 0 i = X 1appleaappleD,1applebappleD a 0 lefti = X s X 1appleaappleD M s 1s 2 ab a left i s 1 i s 2 i b right i A aa 0 (s) a left i si

42 tr right ih = X 1applea a a left iha left! X D D appleaappleD a a left iha left A = X 1appleaappleN 1 N aiha S = c A = Log N 3 Log L N L c 3 = N