2013 9 27-30,
S. Carlip, Challenges for Emergent Gravity, arxiv:1207.2504 [gr-qc]
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.A28,1350111 (2013) [arxiv:1305.6389 [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)
Wheeler-DeWitt
M ab 2-dim Simplicial Quantum Gravity (Free energy)
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
Hermite /N Leading Group Field Theory index a! g Boulatov model, Ooguri model BF-theory Ponzano-Regge
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. 2 3 1 4 4 i i i i
Colored Tensor Model R.Gurau and J.P.Ryan, SIGMA 8, 020 (2012) [arxiv:1109.4812 [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:1302.4386 [math-ph].
Euclidean Dynamical Triangulation (DT) Causal Dynamical Triangulation (CDT) Ambjorn et al. time
CANONICAL TENSOR MODEL NS 2012 f a? f b = M abc f c {f a a =1, 2,...,N}
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
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
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
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
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 )
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)
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
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 1 @ j T 2 T 2 @ j T 1 )) {J(V i ),H(T)} = H(V i @ i T ) {J(V1 i ),J(V2 i )} = J(V j 1 Z @ jv i H(T )= dx T (x)h(x) 2 V j 2 @ jv1 i ) Z J(V i )= dx V i (x)h i (x) Hamiltonian constraints Momentum constraints
GEOMETRODYNAMICS Hojman-Kuchar-Teitelboim, 1976 Canonical Tensor Model Constraint (Dirac) algebra of general relativity D Canonical Tensor Model
[ 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
[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
WHEELER-DEWITT Wheeler-DeWitt Ĥ a = ˆ J [ab] = ˆD =0 : P = (P abc ), ˆP abc = P abc, ˆM abc = i (abc) @ @P 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)
N=2 N= H a J [ab] D so(2) J [ab] D J [ab] D
P 111 =1 P 112 =0 P 122 = x 1 P 222 = x 2 apple @ @ @ + x 1 + x 2 +2 =0 @P 111 @x 1 @x 2 apple @ @ (1 2x 1 ) x 2 +3x 1 @P 112 @x 1 apple @ 3 + x 1 (1 + 2x 1 ) @ +3x 1 x 2 @P 111 @x 1 apple @ x 1 (1 + 2x 1 ) +3x 1 x 2 @P 112 @ @x 2 =0 @ @x 2 + 5(1 + x 1 ) =0 @ + 3(x 2 1 + x 2 2 ) @ +5x 2 =0 @x 1 @x 2
@, @P 111 @ @P 112 apple 2x 1 (x 1 apple 4x 1 x 2 (x 1 1) @ @x 1 +3x 2 (x 1 1) @ @x 1 + 3(4x 1 3 +2x 1 x 2 2 1) @ @x 2 +5x 1 1 =0 x 2 2 ) @ @x 2 +5x 2 (2x 1 1) =0 apple x 1 x 2 (x 1 1) @ @x 1 + 2(5x 1 4 x 1 3 +2x 1 x 2 2 x 2 2 ) @ @x 2 =0 dx 2 = 2(5x 4 1 x 3 2 1 +2x 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 2 2 + 4x 1 2 (5x 1 1) x 1 1 ) = f 4x1 3 + x 2 2 x 14 (x 1 1) 4 a 0 = 4x 1 3 + x 2 2 x 14 (x 1 1) 4 f(x) / p x
= 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
Canonical Tensor Model Wheeler-DeWitt N= favor
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. 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 @ @ 1 2 = ip = ip 1 2 =0