:-) (-: :-) From Geometry to Algebra (-: :-) (-: :-) From Curved Space to Spin Model (-: :-) (-: Xiao-Gang Wen, MIT.

Transcription

1 :-) (-: :-) From Geometry to Algebra (-: :-) (-: :-) From Curved Space to Spin Model (-: :-) (-: Xiao-Gang Wen, MIT wen

2 Einstein s theory of gravity Einstein understood the true meaning of Galileo s discovery: All matter fall with the same acceleration. Bob Alice earth Gravity Acceleration Non-linear coordinate trans. x = x + g 2 t2 Geometry

3 Gravity = non-linear coordinate-trans. = curved space

4 Weyl s geometric way to understand electromagnetism Relativity of coordinates curved space gravity Relativity of units distorted unit-system electromagnetism = 1.3 HKD USD = SFR 5 = 0.2 Unit(y µ ) = Unit(xµ)[1 + (y µ xµ)aµ] Aµ is the vector potential for electromagnetic field, which is called the gauge field. Distortion in unit system = µaν ν Aµ electric and magnetic fields E = tai ia0, B = iaj j Ai = A.

5 Gauge theory is important in understanding the structure of photon. Gauge theory is also important in understanding how to make money through exchange rate (relativity of unit of money) and trading (relativity of unit of labor).

6 Since then, geometric way dominates our approach to understand nature But geometric way has a fatal flaw: The foundation of geometry, Manifold does not exist if we combine Einstein gravity with quantum mechanics. Planck length l P = cm.

7 Geometry is an emergent property This suggests that manifold and geometry may not be fundamental. The concepts of manifold/geometry may only exist at long distances. Manifold/geometry may be emergent properties from a deeper structure. What is this deeper structure??? We try to start with lattice spin model to see if we can obtain emergent geometry at long distance.

8 How to study the emergence of geometry Emergence of space: space curved space propagating distortion = gravitational waves gravitons Emergence of fiber-bundle/unit-system (gauge theory): curved fiber-bundle/unit-system propagating distortion = light waves photons Emergence of geometry = emergence of photons and gravitons The collective motions of what quantum spin model are described by Maxwell equation and Einstein equation, which lead to phonons and gravitons.

9 Emergence of photons from a spin/rotor model Rotor model on 2D-Kagome lattice (θ i,l z i ): H = J 1 (L z i ) 2 + J 2 L z i L z j J xy (e iθ i e iθ j + h.c.) θ Electromagnetic wave is a partially frozen spin wave.

10 What is spin wave? H = J 1 (L z i ) 2 + J 2 L z i L z j J xy (e iθ i e iθ j + h.c.) Small J 1 /J xy, J 2 /J xy. Rotors all want to point in one direction (which can be any direction). spin wave.

11 Quantum rotor and quantum freeze H = J 1 (L z ) 2 a rotor = a particle on a ring (θ, L z ) coordinate-momentum pair. 1 2 J 1 1 mass At temperature T, 0 < Energy = J 1 (L z ) 2 <T, L z T/J 1 Quantum rotor: θ 1/ L z J 1 /T V average /T θ Quantum freeze: small T or large J 1 (small mass 1 J 1 ), θ >2π, L z 0 Angular momentum L z is quantized as integer constraint L z =0and gauge invariance Φ 0 (θ) =Φ 0 (θ + f).

12 Two quantum rotors H =J 1 (L z 1 )2 + J 1 (L z 2 )2 +2J 2 L z 1 Lz 2 = 1 2 (J 1 + J 2 )(L z 1 + Lz 2 ) (J 1 J 2 )(L z 1 Lz 2 )2 θ 1 + θ 2 particle with small mass 1/(J 1 + J 2 ) θ 1 θ 2 particle with large mass 1/(J 1 J 2 ) Energy gap: for L z 1 + Lz 2 : J 1 + J 2,forL z 1 Lz 2 : J 1 J 2 Partial quantum freeze when J 1 J 2. Small fluctuation (θ 1 θ 2 ) 0, large fluctuation (L z 1 Lz 2 ) 1 classical picture is valid. Large fluctuation (θ 1 + θ 2 ) > 2π, small fluctuation (L z 1 + Lz 2 ) 0 constraint L z 1 + Lz 2 =0and gauge invariance Φ 0(θ 1,θ 2 )=Φ 0 (θ 1 + f, θ 2 + (θ 1,θ 2 ) (θ 1 + f, θ 2 + f) Constraint L z 1 + Lz 2 =0generate gauge transformation: e i(lz 1 +Lz 2 ) :(θ 1,θ 2 ) (θ 1 + f, θ 2 + f)

13 A lattice of quantum rotors Large J 1 /J xy, J 2 /J xy L z i more certain and θ i more uncertain quantum freeze gapped L z i =0state. Large (J 1 + J 2 )/J xy and small (J 1 J 2 )/J xy partial quantum freeze e ±iθ i conversion factor between units. θ 1 θ 2 + θ 3 θ 4 + θ 5 θ 6 0 distortion of unit system. Partially quantum-frozen spin waves satisfy Maxwell equation emergent light

14 A little Math: Emergence of photons Each link: one rotor (θ ij,l ij ). L = L ij ) 2 L ij θ ij U links vert( star J (L ij ) 2 g links all-sq 1-sq e iθij E Ȧ E 2 ( i A j j A i ) 2 + A 4 + where θ ij A, L ij E, θ ij i j

15 Three modes: helicity h =0, ±1 Two trans. h = ±1 modes: δa δθ ij 2π, δe 1 Classical picture valid. The long. h =0mode: (f, π) A = f, π = E δπ 0, δf Very quantum. Classical picture not valid. π is discrete on lattice Soft quantum modes =gap Weak fluctuations of π constraint π = E =0 Strong fluctuations of f gauge trans. A A + f E E trans. trans. long. long. k k L + Constraint and gauge trans. = Maxwell s theory of electricity and magnetism.

16 Light wave = collective motion of condensed string-net H = U ( L ij ) 2 + J (L ij ) 2 + g (e i(θ 1 θ 2 +θ 3 θ 4 ) + h.c) vert star links all-sq When J = g =0, the no string state and closed string states all have zero energy: z L =+1 z L =0 z L = 1 Open Strings Closed Strings J 1 No string state: 0 z 0 z 0 z... Closed-string state: loops of L z = ±1

17 Emergence of gravitons Each vertex: three rotors (θ aa,l aa ),aa =11, 22, 33. Each face: one rotor (θ ab,l ab ), ab =12, 23, 31. L = L ab θ ab Complicated H Total six modes (spin waves) with helicity 0, 0, ±1, ±2 L = L ab θ ab [(L ab ) 2 (L aa) 2 ] θ ab R ab 2 ( a L ab ) 2 (R aa ) 2 + E θ θ 11 θ θ 23 θ θ h=+2, 2 where R ab = ɛ ahc ɛ bdg h d θ gc. The helicity ±2 modes are classical and the classical picture is valid. h=0,0,+1, 1 Classical spin wave k

18 h =0, ±1 modes are described by (θ a,l a ): θ ab = a θ b + b θ a, L a = b L ab Quantum fluctuations: δl a =0, δθ a = L a is discrete gap. Constraint and gauge transformation: L a = b L ab =0, θ ab θ ab + a θ b + b θ a A h =0mode is described by (θ, L): L ab =(δ ab 2 a b )L, θ =(δ ab 2 a b )θ ab = R aa Quantum fluctuations: δθ =0, δl = To have gap, θ ab discretized and L ab compactified: L L + n G, θ ab =2π/n G Constraint and gauge transformation: (δ ab 2 a b )θ ab =0, L ab L ab +(δ ab 2 a b )L E h=0,0,+1, 1 h=+2, 2 graviton Partial quantum freeze

19 Low energy effective theory Symmetric tensor field theory (θ ij,l ij ) L = L ij 0 θ ij J 1 L ij L ij J 2 L ii L jj g 1 k θ ij k θ ij g 2 i θ ij k θ kj g 3 i θ ij j θ kk. Helicity modes h =0, 0, ±1, ±2. The vector constraint i L ij =0 which generates gauge transformations θ ij Wθ ij W = θ ij + i f j + j f i The gauge invariant field is a symmetric tensor field Gauge inv. Lagrangian R ij = R ji = ɛ imk ɛ jln m l θ nk L = L ij 0 θ ij α(l ij ) 2 β(l ii ) 2 γ(r ij ) 2 λ(r ii ) 2 The constraints remove h =0, ±1 modes. h =0, ±2 has ω k 2.

20 The scaler constraint R ii =(δ ij 2 i j )θ ij =0 and the corresponding gauge trans. L ij L ij (δ ij 2 i j )f Gauge inv. Lagrangian density L = L ij 0 θ ij J 2 [(L ij) (Li i )2 ] g 2 (R ij) 2 Only h = ±2 modes with ω k 4.

21 Gauge inv. Lagrangian L = d x L: L = L ij 0 θ ij J 2 [(L ij) (Li i )2 ] g 2 θij R ij Only h = ±2 modes with ω k. L + Constraint and gauge trans. = linearized Einstein gravity with θ ij g ij δ ij

22 Geometry emerges from Algebra Lattice model = Algebra, photons/gravitons = Geometry Photons/gravitons emerge from lattice model Geometry emerge from Algebra Local bosonic/spin model provides a unified origin of: (a) Gauge interaction (electromagnetism) (b) Gravity (linearized, so far) (c) Fermi statistics Gauge interaction, gravity, and Fermi statistics are properties of our vacuum Ether (our vacuum) = A local bosonic/spin model