離散研究会2013

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2 S. Carlip, Challenges for Emergent Gravity, arxiv: [gr-qc]

3 Wheeler-DeWitt N= M abc,p abc a, b, c =1, 2,...,N M abc = M bca = M bac O(N) N.Sasakura, Quantum canonical tensor model and an exact wave function, Int.J.Mod.Phys.A28, (2013) [arxiv: [hep-th]]. M abc! J a a 0 J b b 0 J c c 0 M a 0 b 0 c 0 P abc! J a a 0 J b b 0 J c c 0 P a 0 b 0 c 0 J 2 O(N)

4 Wheeler-DeWitt

5 M ab 2-dim Simplicial Quantum Gravity (Free energy)

6 Ambjorn et al., NS, Gross, 1990 vertex propagator S = M abc M bac gm abc M ade M bef M cfd Hermiticity(generalized): M abc = M bca = M cab = M bac = M acb = M cba

7 Hermite /N Leading Group Field Theory index a! g Boulatov model, Ooguri model BF-theory Ponzano-Regge

8 Colored Tensor Model Gurau, 2009 color color 4X S = i=1 1 M i abcm i abc 2 3 gm 1 abcm 2 adem 3 bef M 4 cfd + c.c i i i i

Colored Tensor Model R.Gurau and J.P.Ryan, SIGMA 8, 020 (2012) [arxiv:1109.4812 [hep-th]].

9 Colored Tensor Model R.Gurau and J.P.Ryan, SIGMA 8, 020 (2012) [arxiv: [hep-th]]. /N Leading (leading, sub-leading) F = f 0 N D + f 1 N D 1 + Leading melonic singular Branched polymer R.Gurau and J.P.Ryan, Melons are branched polymers, arxiv: [math-ph].

10 Euclidean Dynamical Triangulation (DT) Causal Dynamical Triangulation (CDT) Ambjorn et al. time

11 CANONICAL TENSOR MODEL NS 2012 f a? f b = M abc f c {f a a =1, 2,...,N}

12 Hermiticity fa = f a (f a? f b ) = f b? f a hf a f b i = ab hf a f b? f c i = hf a? f b f c i = hf c? f a f b i Frobenius algebra Axiomatic TFT

13 f z = D (x z) z 2 R D f z1? f z2 = D (z 1 z 2 )f z1 momentum cutoff hi= Z d D x e ipx p < e ipx? e iqx = e i(p+q)x if p + q < 0 otherwise Gaussian Z f x? f y = A( ) d D z exp[ (x y) 2 (y z) 2 (z x) 2 ] f z!1 e ipx e iqx =exp[ (p 2 + q 2 +(p + q) 2 )]e i(p+q)x

14 a b c a M abc : O(N) fa 0 = J b a f b, J 2 O(N) O(N) l = M acd M bde M bef M afc M acd M bde M aef M bfc = 1 [M 2 Tr (a),m (b) ][M (a),m (b) ] M (a) bc M abc

15 proposition M abc l =0 M abc = J a a 0 J b b 0 J c c 0 M D a 0 b 0 c 0 M D abc = m a ab ac 9 m a 0 9 J 2 O(N) l l =0 [M (a),m (b) ]=0 M (a) [M (a),m (b) ]=0 9 J 2 O(N) M abc J l =0 f a? f b = m a ab f a

16 CANONICAL TENSOR MODELの局所的ハミルトニアンの決定 fa : 直感的には点各点ごとに局所的ハミルトニアン Ha 無数のパスがある大局的ハミルトニアンは直ちに相対論と矛盾するだろう時間発展初期値局所的時間発展の相互無矛盾性 [Ha, Hb ] = 0 upto kinematic symmetry fa0 = Ja b fb, J 2 so(n )

17 Canonical Tensor Model H = N a H a + N [ab] J [ab] + ND H a, J [ab], D : N a, N [ab], N : multipliers ADM H a = 1 2 P abcp bde M cde J [ab] = 1 4 (P acdm bcd P bcd M acd ) D = 1 6 M abcp abc so(n) dx dt = x2 f a! cf a Raffaelli, Sato, NS {M abc,p def } = ad be cf +(perm. of def)

18 On-shell H a, J [ab], D {H(T 1 ),H(T 2 )} = J([ T 1, T 2 ]), {J(V ),H(T)} = H(VT), {J(V 1 ),J(V 2 )} = J([V 1,V 2 ]), {D,H(T)} = H(T ), {D,J(V )} =0, H(T )=T a H a, J(V )=V [ab] J [ab], T ab P abc T c D M abb M $ P H a! H a

19 ADM!1 P xyz = c (g(x)g(y)g(z)) 1 4 exp d(x, y) 2 + d(y, z) 2 + d(z,x) 2 x, y, z D g(x) =det[g µ (x)] d(x, y) x y {H(T 1 ),H(T 2 )} = J(g ij (T j T 2 T j T 1 )) {J(V i ),H(T)} = H(V i T ) {J(V1 i ),J(V2 i )} = J(V j 1 jv i H(T )= dx T (x)h(x) 2 V j jv1 i ) Z J(V i )= dx V i (x)h i (x) Hamiltonian constraints Momentum constraints

20 GEOMETRODYNAMICS Hojman-Kuchar-Teitelboim, 1976 Canonical Tensor Model Constraint (Dirac) algebra of general relativity D Canonical Tensor Model

21 [ ˆM abc, ˆP def ]=i ad be cf + perm.of def Ĥ a = ˆP abc ˆPbde ˆMcde + i H ˆPabb ˆ J [ab] = 1 4 ˆPacd ˆMbcd ˆPbcd ˆMacd ˆD = 1 6 ˆP abc ˆMabc + i D H = (N + 2)(N + 3) 2 D = N(N + 1)(N + 2) 2

22 [Ĥ(T 1 ), Ĥ(T 2 )] = i ˆ J ([ ˆT 1, ˆT 2 ]) [ J ˆ (V ), Ĥ(T )] = iĥ(vt) [ ˆ J (V 1 ), ˆ J (V 2 )] = i ˆ J ([V 1,V 2 ]) [ ˆD, Ĥ(T )] = iĥ(t ) [ ˆD, ˆ J (V )] = 0 ˆT bc = T a ˆPabc J ˆ( ˆV )= ˆV [ab] J[ab] ˆ ( [ ˆP abc ˆPbde ˆMcde, ˆP a ) 0 b 0 b 0] (a $ a0 )=4i ˆP abc ˆPa 0 bc +2i ˆP ˆP aa 0 b bcc (a $ a 0 )=0

23 WHEELER-DEWITT Wheeler-DeWitt Ĥ a = ˆ J [ab] = ˆD =0 : P = (P abc ), ˆP abc = P abc, ˆM abc = abc, 8 >< 6 for a = b = c, (abc) = 2 for a = b 6= c, b = c 6= a, c = a 6= b, >: 1 for a 6= b, b 6= c, c 6= a. (Frobenius theorem)

24 N=2 N= H a J [ab] D so(2) J [ab] D J [ab] D

25 P 111 =1 P 112 =0 P 122 = x 1 P 222 = x @ + x 1 + x (1 2x 1 ) x 2 +3x x 1 (1 + 2x 1 +3x 1 x 1 x 1 (1 + 2x 1 ) +3x 1 x (1 + x 1 ) + 3(x x x 2 2

26 @P 112 apple 2x 1 (x 1 apple 4x 1 x 2 (x 1 +3x 2 (x (4x x 1 x x 1 1 =0 x x 2 (2x 1 1) =0 apple x 1 x 2 (x (5x 1 4 x x 1 x 2 2 x 2 2 =0 dx 2 = 2(5x 4 1 x x 1 x 2 x 2 2 ) dx 1 x 1 x 2 (x 1 1) d x 2 2 = 4(2x 1 1) dx 1 x 1 (x 1 1) x x 1 2 (5x 1 1) x 1 1 ) = f 4x1 3 + x 2 2 x 14 (x 1 1) 4 a 0 = 4x x 2 2 x 14 (x 1 1) 4 f(x) / p x

27 = b 0 p 4x13 + x 2 2 x 12 (x 1 1) 2 J [ab] D p ac bd eg fh e0 g = c 0 f 0 h 0 P aef P bgh P ce0 f 0 P dg0 h 0 0 P acd P bde P bef P afc P acd P bde P aef P bfc l l =0 favor 9 f? f =0

28 Canonical Tensor Model Wheeler-DeWitt N= favor

29 N Regular around l =0 l N> favor Group Field Theory Raffaelli, Sato, NS P abc = J a a 0 J b b 0 J c c 0 P a 0 b 0 c 0, J 2 O(N)

ファジー空間の積による局所ハミルトニアンの自然な表現 Raffaelli, Sato, NS 振動や運動がどのようにして生じるのか観測量の問題 cf. Rovelli: complete observable ヒント拘束条件 @ = const.

30 ファジー空間の積による局所ハミルトニアンの自然な表現 Raffaelli, Sato, NS 振動や運動がどのようにして生じるのか観測量の問題 cf. Rovelli: complete observable ヒント = const. つまらない =0 テンソル模型には変数が沢山ある拘束条件が一階微分で表される事を使うともっと面白い解が沢山ある = 1 (x1 ) 2 (x2 1 (x1 ) = eipx1 例えば ipx2 が解 2 (x2 ) = e 2 (x2 ) + 1 (x1 )@ 2 (x2 ) 1 (x1 ) または = x2 = t を時間と思えば = eip(x1 t) 1 2 = ip = ip 1 2 =0

PSCHG000.PS

PSCHG000.PS b c d e f bcd bcf bef cde def b c d e f bcd bcf bef cde def b c d e f bc bf cd de ef b c d e f bc bf cd de ef b c d e f bc bf cd de ef b c d e f bc bf cd de ef b c d e f bcd bcf bef cde def b c d e f bcd