HAMILTONIAN STRUCTURE OF THE BFCG THEORY Marko Vojinovi c - PowerPoint PPT Presentation

HAMILTONIAN STRUCTURE OF THE BFCG THEORY Marko Vojinovi´ c Institute of Physics, University of Belgrade joint work with Aleksandar Mikovi´ c Lusofona University and GFMUL, Portugal

THE PROBLEM OF QUANTUM GRAVITY Why quantize gravity? • same reasons as electrodynamics (two-slit experiment, hydrogen atom, . . . ) • resolution of singularities (black holes, Big Bang, . . . ) • black hole information paradox (nonunitary evolution?) • theoretical and aesthetical reasons. . . How to quantize gravity? • perturbation theory does not work (nonrenormalizability of gravity). . . • almost zero experimental results to guide us. . . • . . . we have a problem!

LOOP QUANTUM GRAVITY The idea • Wilson loops are chosen as basic degrees of freedom, • formalized as “spin network states”, • canonically quantized. Achievements • nonperturbative quantization of GR, • kinematic sector of the theory well-defined, • lengths, areas and volumes of space quantized! Drawbacks • dynamics described only in principle, • no proof of semiclassical limit, • very limited possibility for calculations.

SPINFOAM MODELS The idea • build up on canonical LQG (use the same degrees of freedom, construct the same structure of the Hilbert space, etc.), • rewrite GR action using the Plebanski formalism, � B ab ∧ R ab + φ abcd B ab ∧ B cd , S = • discretize spacetime into 4-simplices, • perform covariant quantization of the BF sector, by providing a definition for the gravitational path integral, � � � � � � � � Z = D ω D B exp = . . . = A 2 (Λ f ) A 4 (Λ v ) , i B ∆ R ∆ v ∆ Λ f • enforce the Plebanski constraint by restricting the representations Λ and redefining the vertex amplitude A 4 .

SPINFOAM MODELS Achievements • well-defined nonperturbative quantum theory of gravity, • both kinematical and dynamical sectors under control, • can have a proper semiclassical limit. Drawbacks • geometry is “fuzzy” at the Planck scale, • has many different semiclassical limits, • matter coupling is problematic, • hard to extract any results. The reason for these drawbacks: tetrads are not explicitly present in the action!

THE BFCG ACTION One can associate the BFCG action to the Poincar´ e 2-group: � B ab ∧ R ab + C a ∧ G a , ( G a = dβ a + ω a b ∧ β b ) . S = Note that the Lagrange multiplier C a is a 1 -form and has an equation of motion ∇ C a = 0 , exactly the same as the tetrad e ! Therefore, C a ≡ e a , � • identify: KEY STEP • rename: BFCG → BFEG, and rewrite the action as � B ab ∧ R ab + e a ∧ G a . S =

THE CONSTRAINED BFCG ACTION The BFCG action can be constrained to give GR: � B ab ∧ R ab + e a ∧ G a B ab − ε abcd e a ∧ e b � � S = − φ ab . � �� � � �� � constraint topological sector Equations of motion are equivalent to: • equations that determine the multipliers and β : φ ab = R ab , B ab = ε abcd e c ∧ e d , β a = 0 • Einstein equations: ε abcd R bc ∧ e d = 0 , • no-torsion equation: ∇ e a = 0 . This is classically equivalent to general relativity!

THE MAIN BENEFITS Introduction of matter fields is straightforward: � B ab ∧ R ab + e a ∧ G a − φ ab B ab − ε abcd e a ∧ e b � � S = + � � � d + { ω, γ d } + im γ d ↔ ε abcd e a ∧ e b ∧ e c ∧ ¯ 2 e d + iκ ψ − ψ � − i 3 κ ( κ = 8 ε abcd e a ∧ e b ∧ β c ¯ 3 πl 2 ψγ 5 γ d ψ, p ) . 4 The covariant quantization is possible — spincube model: � � � � � � � � Z = D ω D B D e D β exp B ∆ R ∆ + = . . . = i e l G l ∆ l � � � � = A 1 (Λ p ) A 2 (Λ f ) A 4 (Λ v ) . Λ p v f

THE HAMILTONIAN STRUCTURE The BFCG action in components: � d 4 x ε µνρσ � � � � ∂ ρ ω ab σ + ω a cρ ω cb + e aµ ( ∂ ν β a ρσ + ω a cν β c S = ρσ ) B abµν . σ The variables: B ab e a ω ab β a µν ( x ) , µ ( x ) , µ ( x ) and µν ( x ) . Momenta and primary constraints: ≡ π ( B ) abµν ≈ 0 , ≡ π ( e ) aµ ≈ 0 , P ( B ) abµν P ( e ) aµ ≡ π ( ω ) ab 0 ≈ 0 , ≡ π ( ω ) abi − 2 ε 0 ijk B abjk ≈ 0 , P ( ω ) ab 0 P ( ω ) abi ≡ π ( β ) a 0 i ≈ 0 , ≡ π ( β ) aij + 2 ε 0 ijk e ak ≈ 0 . P ( β ) a 0 i P ( β ) aij The simultaneous Poisson brackets: d ] δ ρ { B abµν ( � x, t ) , π ( B ) cdρσ ( � x ′ , t ) } = 4 δ a [ c δ b [ µ δ σ ν ] δ (3) ( � x ′ ) , x − � { e aµ ( � x, t ) , π ( e ) bν ( � x ′ , t ) } = δ a b δ ν µ δ (3) ( � x ′ ) , x − � { ω abµ ( � x, t ) , π ( ω ) cdν ( � x ′ , t ) } = 2 δ a [ c δ b d ] δ ν µ δ (3) ( � x ′ ) , x − � b δ ρ { β aµν ( � x, t ) , π ( β ) bρσ ( � x ′ , t ) } = 2 δ a [ µ δ σ ν ] δ (3) ( � x ′ ) . x − �

THE HAMILTONIAN STRUCTURE The canonical Hamiltonian: � x ε 0 ijk � � �� d 3 � − B ab 0 i R ab jk − e a 0 G aijk − 2 β a 0 k T a ∇ i B ab jk − e a i β b H c = ij − ω ab 0 , jk The total Hamiltonian: � � µν + λ ( e ) a µ + d 3 � λ ( B ) ab H T = H c + µν P ( B ) ab µ P ( e ) a x µ + λ ( β ) a µν � + λ ( ω ) ab µ P ( ω ) ab µν P ( β ) a . Consistency of the primary constraints: P ( B ) ab 0 i = 2 ε 0 ijk S ( R ) abjk , ˙ S ( R ) abjk ≡ R abjk ≈ 0 , ˙ P ( e ) a 0 S ( G ) a ≡ ε 0 ijk G aijk ≈ 0 , = S ( G ) a , where ˙ = 2 ε 0 ijk S ( T ) ajk , P ( β ) a 0 i S ( T ) aij ≡ T aij ≈ 0 , S ( Beβ ) ab ≡ ε 0 ijk � � ˙ P ( ω ) ab 0 ∇ i B abjk − e [ ai β b ] jk = 2 S ( Beβ ) ab , ≈ 0 .

THE HAMILTONIAN STRUCTURE Determined multipliers: P ( B ) abjk ≈ 0 ˙ λ ( ω ) abi = 1 2 ∇ i ω ab 0 , P ( e ) ak ≈ 0 ˙ λ ( β ) aij = ∇ [ i β a 0 j ] − 1 2 ω ab 0 β bij , implies P ( β ) ajk ≈ 0 ˙ λ ( e ) ai = ∇ i e a 0 − ω ab 0 e bi , λ ( B ) abij = 1 � � P ( ω ) abk ≈ 0 ˙ ∇ [ i B ab 0 j ] + ω [ a c 0 B b ] c + ij 2 � � + 1 e [ a 0 β b ] ij + e [ a j β b ] 0 i − e [ a i β b ] . 0 j 4 Consistency of secondary constraints is automatic: ˙ = 2 ω [ a c 0 S ( R ) b ] c S ( R ) abij ij , ˙ = ε 0 ijk β b 0 k S ( R ) ab ij − ω a b 0 S ( G ) b , S ( G ) a = 1 ˙ S ( T ) aij 2 e b 0 S ( R ) ab ij − ω a b 0 S ( T ) b ij , S ( Beβ ) ab = 2 ε 0 ijk � � 0 S ( G ) b ] + 2 ω [ a ˙ B [ a c 0 k S ( R ) b ] c ij + β [ a 0 k S ( T ) b ] + e [ a c 0 S ( Beβ ) b ] c . ij

THE HAMILTONIAN STRUCTURE Algebra of constraints: { P ( B ) abjk , P ( ω ) cdi } 8 ε 0 ijk η a [ c η bd ] δ (3) , = { P ( e ) ak , P ( β ) bij } − 2 ε 0 ijk η ab δ (3) , = 2 ε 0 ijk � b ∂ i δ (3) + ω abi δ (3) � { S ( G ) a , P ( β ) bjk } δ a = , { S ( G ) a , P ( ω ) cdi } 2 ε 0 ijk δ a [ c β d ] jk δ (3) , = { S ( T ) aij , P ( e ) bk } j ] δ (3) + ω ab [ i δ k δ a b ∂ [ i δ k j ] δ (3) , = � � { S ( T ) aij , P ( ω ) cdk } δ (3) , δ a [ c e d ] j δ k i − δ a [ c e d ] i δ k = j { S ( Beβ ) ab , P ( e ) ci } − ε 0 ijk δ [ a c β b ] jk δ (3) , = { S ( Beβ ) ab , P ( β ) cjk } − 2 ε 0 ijk e [ ai δ b ] c δ (3) , = 2 ε 0 ijk � � { S ( Beβ ) ab , P ( ω ) cdi } δ (3) , δ a [ c B d ] bjk + δ b [ c B ad ] jk = 4 ε 0 ijk � � � δ (3) � { S ( Beβ ) ab , P ( B ) cdjk } = d ] ∂ i δ (3) + δ a [ c δ b ω a [ ci δ b d ] + δ a [ c ω bd ] i , { S ( R ) abij , P ( ω ) cdk } � j ∂ i δ (3) − δ k i ∂ j δ (3) � 2 δ a [ c δ b δ k = + d ] � � δ (3) . δ a [ c ω d ] bj δ k i − δ a [ c ω d ] bi δ k j + ω a [ ci δ b d ] δ k j − ω a [ cj δ b d ] δ k +2 i

THE HAMILTONIAN STRUCTURE First class constraints: 0 i , 0 , 0 , 0 i , P ( B ) ab P ( e ) a P ( ω ) ab P ( β ) a Second class constraints: jk , i , i , ij , P ( B ) ab P ( e ) a P ( ω ) ab P ( β ) a S ( R ) ab S ( G ) a , S ( Beβ ) ab , S ( T ) a ij , ij .

THE HAMILTONIAN STRUCTURE The gauge symmetry generator: � 1 � � 0 i − ε ab i � � 0 − ε a G a � G [ ε ab i , ε ab , ε a , ε a d 3 � ε ab ε a P ( e ) a i ] = ˙ i P ( B ) ab i G ab + ˙ + x 2 i �� + 1 0 − ε ab G ab 0 i − ε a � � � ε ab P ( ω ) ab ε a ˙ + ˙ i P ( β ) a i G a , 2 where ji + 2 ω c G abi ≡ 2 ε 0 ijk S ( R ) abjk + ∇ j P ( B ) ab 0 i , [ a 0 P ( B ) b ] c i + 2 ω c 0 − 2 e [ a 0 P ( e ) b ] 0 − 2 e [ ai P ( e ) b ] i + G ab ≡ 2 S ( Beβ ) ab + ∇ i P ( ω ) ab [ a 0 P ( ω ) b ] c cij + 2 B c [ a 0 i P ( B ) b ] c 0 i − 2 β [ a 0 i P ( β ) b ] 0 i − β [ aij P ( β ) b ] ij , + B c [ aij P ( B ) b ] 0 − 1 0 i − 1 i − ω b 2 β b 4 β b ij , G a ≡ S ( G ) a + ∇ i P ( e ) a a 0 P ( e ) b 0 i P ( B ) ab ij P ( B ) ab 0 i − 1 0 i + 1 ji − ω b ≡ 2 ε 0 ijk S ( T ) ajk + ∇ j P ( β ) a 2 e b 2 e b ij . G ai a 0 P ( β ) b 0 P ( B ) ab j P ( B ) ab