Quantum Renormaliza/on Group: From QFT to Quantum Gravity. Sung- Sik Lee McMaster University Perimeter Ins/tute

Transcription

1 Quantum Renormaliza/on Group: From QFT to Quantum Gravity Sung- Sik Lee McMaster University Perimeter Ins/tute

2 AdS/CFT correspondence [Maldacena] Non- trivial evidences for supersymmetric gauge theories Holography is believed to be a general framework for a large class of QFT s There is no proof of the conjecture, and no systema/c way to derive the dual theory for a given QFT / quantum many- body system

3 AdS/CFT Dic/onary [Gubser, Klebanov, Polyakov; Wi]en] Dφ(x)e is D[φ(x)]+i J n (x)o n = Dj(x, z)e is D+1[j(x,z)] e is D+1[ j(x,z)] j n (x,z=0)=j n (x) jn (x,z=0)=j n (x) This talk : Can one explicitly construc/on holographic duals from QFT?

4 Other Related works : E. T. Akhmedov, Phys. Le]. B 442 (1998) 152 S. R. Das and A. Jevicki, Phys. Rev. D 68 (2003) R. Gopakumar, Phys. Rev. D 70 (2004) ; ibid. 70 (2004) I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, J. High Energy Phys. 10 (2009) 079. R. Koch, A. Jevicki, K. Jin and J. P. Rodrigues, arxiv: I. Heemskerk and J. Polchinski, arxiv: T. Faulkner, H. Liu and M. Rangamani, arxiv: M. Douglas, L. Mazzucato, and S. Razamat, Phys. Rev. D 83 (2011)

5 Matrix model Ac/on is constructed from single- trace operators Z[J(x)] = Dφ e i dxl L = J n (x)o n + J mn (x)o m O n +... O n : complete set of single- trace operators e.g. tr[φ n ], tr[φ µ ν φ], tr[φ( µ1 µ2... µi φ)...( ν1 ν2... νi φ)],.. Any local operator allowed by symmetry can be wri]en as a polynomial of single- trace operators and their deriva/ves

6 Conven/onal RG : Classical RG multi trace operators dj nm... (x, z) dz = β[j n,j nm,...] z : length scale (RG /me) tr[φφφφφφφφ] subspace of single trace operators tr[φφφ]tr[φφφ] For a give ini/al condi/on (and RG prescrip/on), there is a unique RG trajectory Even though one starts with the single- trace operators at a given scale, mul/- trace operators are generated

7 Conven/onal RG : Classical RG multi trace operators subspace of single trace operators Z = e 0 dzdd xf[j n (x,z), J n,...] RG flow The par//on func/on is given by a (D+1)- dimensional integra/on of a (local) func/onal of scale dependent couplings

8 AN ALTERNATIVE APPROACH : QUANTUM RG (HOLOGRAPHY)

9 Matrix model Ac/on is constructed from single- trace operators Z[J(x)] = Dφ e i dxl L = J n (x)o n + J mn (x)o m O n +...

10 Step 1 : remove mul/- trace operators by introducing auxiliary fields Z[J(x)] = Dj n (1) Dp (1) n Dφ e i dxl L = j (1) n (p (1) n O n ) J n p (1) n + J nm p (1) n p (1) m +... J n (1) : Lagrangian mul/plier that plays the role of dynamical source that enforces the constraint p n (1) = O n P n (1) : dynamical operator

11 Step 2 : Integrate out high energy mode φ < : k < Λe dz, Z[J(x)] = φ > : Λe dz < k < Λ Dj n (1) Dp (1) n Dφ < e i dxl = J nm p (1) n p (1) m + p (1) n (j (1) n J n )+dzl c [j (1) ] (j (1) n + dza n [j (1) ])O n + dzb nm [j (1) ]O n O m Casimir energy : poten/al energy for dynamical source Double- trace operators

12 Step 3 : remove double- trace operators by introducing a second set of auxiliary Z[J(x)] = fields Dj (1) n Dp (1) n Dj (2) n Dp (2) n Dφ < e i dxl = J nm p (1) n p (1) m + p (1) n (j (1) n J n )+dzl c [j (1) n ] j n (2) (p (2) n O n ) (j n (1) + dza n [j (1) ])p (2) n + dzb nm [j (1) ]p (2) n p (2) m Low energy fields has only single- trace operators Quadra/c term in p n The double trace operators generated out of quantum correc/on provide the kine/c energy for the dynamical source

13 Step 4 : repeat 2-3 again and again Z[J(x)] = j n (0) (x) =J n (x) l=1 L = J nm p (1) n p (1) m + Dj n (l) (x)dp (l) n (x) e i d D xl i=1 dza n [j (i) ]p (i+1) n p (i) n (j (i) n j n (i 1) )+dzl c [j n (i) ] + dzb nm [j (i) ]p (i+1) n A set of dynamical sources and dynamical operators are introduced at each step of RG at the expense of decima/ng high energy mode bit by bit p (i+1) m

14 Extra dimension as a length scale z IR UV z=0 0 dz Boundary condi/on j n (0) (x) =J n (x) j (l) n (x), p (l) n (x) z l = ldz j(x, z), p(x, z)

15 Quantum fluctua/ons in RG path multi trace operators Only single- trace operators appear Quantum fluctua/ons in the RG trajectory : Quantum RG subspace of single trace operators

16 Quantum Renormaliza/on Group Wavefunc/on in the space of theory with single- trace operators Ψ[J n (x), 0] amplitude Ψ[J n (x),z]=e ihz Ψ[J n (x), 0] Space of sources for Single- trace operators Quantum RG trajectory = Evolu/on of wavefunc/on of couplings under a change of scale

17 Quantum beta func/on Z = lim T < Ψ f e it Ĥ Ψ i > Wavefunc/on for D - dimensional space/me dependent sources It is useful to view the scale parameter z as `/me Dynamical sources and dynamical operators are conjugate to each other [ĵ l (x), ˆp m (x )] = 1 δ l,mδ(x x ) = 1 N 2 Par//on func/on is wri]en as a transi/on amplitude of D- dimensional quantum wavefunc/on of coupling constants The Hamiltonian generates scale transforma/on for dynamical couplings The Heisenberg equa/on : quantum beta func/on

18 Quantum RG In general, there is no interpreta/on of quantum RG in terms of classical RG e.g. RG for Grassmanian sources (e.g. electron star) In some large N limit, one can use the saddle point approxima/on, which then can be interpreted as the classical RG The form of the bulk theory is sensi/ve to the regulariza/on scheme in QFT In this prescrip/on, one can construct simple holographic ac/on (e.g. scalar field in AdS) under some assump/ons on CFT

19 D- dimensional O(N) matrix field theory single- trace operators O [q+1;{µ i j }] = 1 ( )( ) [Φ N tr µ 1 1 µ µ 1 p1 Φ µ 2 1 µ µ 2 p2 Φ... ( µ q 1 µ q 2.. µ q pq Φ )] Space/me dependent sources Z[J ]= DΦ exp [in 2 d D x { } ϕ : N x N tracelss symmetric real matrix field ( ) ] J m O m + V [O m ; J {m i},{νj i} ] V [O m ; J {m i},{ν i j } ]= q=1 mul/- trace deforma/on ( )( ) ( ) J {m i},{νj i} O m1 ν ν 1 p1 O m2 ν ν 2 p2 O m3... ν q.. 1 νpq q O m q+1

20 Z[J ]= Bulk ac/on : (D+1)- dimensional gravity DJ(x, z)dπ(x, z) e i ( [ d D xdz ] S = N 2 ( ) S UV [π(x,0)]+s[j(x,z),π(x,z)]+s IR [J(x, )] ) J(x,0)=J (x) [( z J n )π n α(x, z)h N µ (x, z)h µ ] Hamiltonian constraint : replaced by the usual derivative in the canonical variables. M H = õν [J(x)]π [ [2,µν] B µνλσ [J(x)] π [2,µν] π [2,λσ] ] G { } G C 0 [J(x)] + C 1 [J(x)]R +..., H µ Hthe higher dimensional terms that involve covariant de [ ( ) = 2 ν π [2,µν] ν J [q,{µ1 1 µ1 2...µa b 1 νµa b+1...}] π [q,{µ 1 1 µ µa b 1 µµa b+1...}] Momentum constraint : [q,{µ i j }] [2,µν] a,b +( µ J [q,{µi j }] )π [q,{µ i j }] ]. SL (2012)

21 Local RG Space/me dependent coarse graining φ Λ +... φ φ Λ(x) φ Λ(x) = Λe α(x)dz Speed of coarse graining By construc/on, Z is independent of α(x,z) Choosing different RG scheme α(x,z) : choosing different gauge

22 Shiu One does not have to choose the coordinate of the low energy field as the coordinate of the high energy mode length scale Shiu of the coordinate of the low energy field rela/ve to the coordinate of the high energy field µ N dz low energy high energy

23 Diffeomorphism = Freedom to choose different local RG schemes r Length scale dz Ndz D- dim manifold with same z (a) x

24 < H M (x, z) >= 1 Z First- class constraints Independence of par//on func/on on RG schemes (speed of RG and shius) è (D+1)- constraints δz δn M (x, z) =0. shift play the role of Lagrangian mu M=0, 1, 2,, (D- 1), D z < H M(x, z) >= r this to be true for any choices of H =0, H µ =0 N D (x, z) α(x, z) and H D H. The (D+1)- constraints are in the (classically) boundary field theory. first- class The H d D yn M (y, z) {H M (x, z), H M (y, z)} =0., we have {H M (x, z), H M (y, z)} =0

25 Summary D- dimensional QFT can be mapped into a (D+1)- dimensional quantum theory of gravity based on a local RG Quantum beta func/on Example of emergent gravity Diffeomorphism = freedom to choose different RG schemes Construc/ve proof of the Maldacena s conjecture for N=4 SU(N) gauge theory? Applica/ons to concrete condensed ma]er systems?