Gravity and the cosmological constant as superconducting phenomena - PowerPoint PPT Presentation

Gravity and the cosmological constant as superconducting phenomena Gianluca Calcagni August 28th, 2008

Based on S. Alexander, G.C., Superconducting loop quantum gravity 1 and the cosmological constant [ 0806.4382 ]. S. Alexander, G.C., Quantum gravity as a Fermi liquid 2 [ 0807.0225 ].

Aims of the talk

Aims of the talk To throw a rock.

Aims of the talk To throw a rock. Not to hide the hand.

The rock

The rock Setup: Loop quantum gravity with Λ , no matter, and the Chern–Simons state as ground state.

The rock Setup: Loop quantum gravity with Λ , no matter, and the Chern–Simons state as ground state. Assumption: Deform the topological sector (then Λ becomes dynamical).

The rock Setup: Loop quantum gravity with Λ , no matter, and the Chern–Simons state as ground state. Assumption: Deform the topological sector (then Λ becomes dynamical). Result 1: spacetime degenerate ( 1 + 1 dimensions), Hamiltonian modified by a quantum counterterm.

The rock Setup: Loop quantum gravity with Λ , no matter, and the Chern–Simons state as ground state. Assumption: Deform the topological sector (then Λ becomes dynamical). Result 1: spacetime degenerate ( 1 + 1 dimensions), Hamiltonian modified by a quantum counterterm. Result 2: Gravity behaves as a Fermi liquid, in particular BCS.

BCS theory

BCS theory Bardeen, Cooper, and Schrieffer, Phys. Rev. 108 , 1175 (1957).

BCS theory Bardeen, Cooper, and Schrieffer, Phys. Rev. 108 , 1175 (1957). Nobel Prize in 1972 (Bardeen’s second!).

BCS theory Bardeen, Cooper, and Schrieffer, Phys. Rev. 108 , 1175 (1957). Nobel Prize in 1972 (Bardeen’s second!). FERMI SURFACE Perturbative vacuum Fermi−Dirac statistics Fermionic Occupation Number n(k) CONDENSATION Nonperturbative vacuum 1 Bose−Einstein statistics Momentum k

BCS theory Bardeen, Cooper, and Schrieffer, Phys. Rev. 108 , 1175 (1957). Nobel Prize in 1972 (Bardeen’s second!). FERMI SURFACE Perturbative vacuum Fermi−Dirac statistics Fermionic Occupation Number n(k) CONDENSATION Nonperturbative vacuum 1 Bose−Einstein statistics Momentum k 1958-1971: 1 Nobel Prize for studies on condensed matter (Landau).

BCS theory Bardeen, Cooper, and Schrieffer, Phys. Rev. 108 , 1175 (1957). Nobel Prize in 1972 (Bardeen’s second!). FERMI SURFACE Perturbative vacuum Fermi−Dirac statistics Fermionic Occupation Number n(k) CONDENSATION Nonperturbative vacuum 1 Bose−Einstein statistics Momentum k 1958-1971: 1 Nobel Prize for studies on condensed matter (Landau). 1973-2007: 10 Prizes awarded in this field.

The ripples (to be explored)

The ripples (to be explored) Λ is exponentially suppressed, nonperturbative 1 phenomenon; at small scales gravity is perturbative , like in QCD confinement.

The ripples (to be explored) Λ is exponentially suppressed, nonperturbative 1 phenomenon; at small scales gravity is perturbative , like in QCD confinement. Geometrical measurements amount to counting Cooper 2 pairs.

The ripples (to be explored) Λ is exponentially suppressed, nonperturbative 1 phenomenon; at small scales gravity is perturbative , like in QCD confinement. Geometrical measurements amount to counting Cooper 2 pairs. Classically, Cooper pairs are microscopic nonlocal d.o.f. 3 living on the dS boundary (wormholes).

The ripples (to be explored) Λ is exponentially suppressed, nonperturbative 1 phenomenon; at small scales gravity is perturbative , like in QCD confinement. Geometrical measurements amount to counting Cooper 2 pairs. Classically, Cooper pairs are microscopic nonlocal d.o.f. 3 living on the dS boundary (wormholes). Four dimensions recovered by the spin network defined by 4 the superfluid theory.

The ripples (to be explored) Λ is exponentially suppressed, nonperturbative 1 phenomenon; at small scales gravity is perturbative , like in QCD confinement. Geometrical measurements amount to counting Cooper 2 pairs. Classically, Cooper pairs are microscopic nonlocal d.o.f. 3 living on the dS boundary (wormholes). Four dimensions recovered by the spin network defined by 4 the superfluid theory. Matter is ‘hidden’ in gravity? 5

Nonlocal degrees of freedom on dS horizons

The hand: 1. Loop quantum gravity

The hand: 1. Loop quantum gravity α τ i dx α and Ashtekar variables: connection C -field A ≡ A i real triad E i α .

The hand: 1. Loop quantum gravity α τ i dx α and Ashtekar variables: connection C -field A ≡ A i real triad E i α . Scalar , vector , and Gauss constraints: � � B k + Λ ǫ ijk E i · E j × 3 E k H = = 0 , ( E i × B i ) α = 0 , G i = D α E α V α = i = 0 .

The hand: 1. Loop quantum gravity α τ i dx α and Ashtekar variables: connection C -field A ≡ A i real triad E i α . Scalar , vector , and Gauss constraints: � � B k + Λ ǫ ijk E i · E j × 3 E k H = = 0 , ( E i × B i ) α = 0 , G i = D α E α V α = i = 0 . Quantum theory: E → ˆ α , ˆ i = − δ/δ A i A i α multiplicative. E α

The hand: 1. Loop quantum gravity α τ i dx α and Ashtekar variables: connection C -field A ≡ A i real triad E i α . Scalar , vector , and Gauss constraints: � � B k + Λ ǫ ijk E i · E j × 3 E k H = = 0 , ( E i × B i ) α = 0 , G i = D α E α V α = i = 0 . Quantum theory: E → ˆ α , ˆ i = − δ/δ A i A i α multiplicative. E α Constraints annihilated by the Chern–Simons state � i θ θ ≡ 6 π 2 � � S 3 tr ( A ∧ dA + 2 Ψ CS = exp 3 A ∧ A ∧ A ) , i Λ , 8 π 2

The hand: 1. Loop quantum gravity α τ i dx α and Ashtekar variables: connection C -field A ≡ A i real triad E i α . Scalar , vector , and Gauss constraints: � � B k + Λ ǫ ijk E i · E j × 3 E k H = = 0 , ( E i × B i ) α = 0 , G i = D α E α V α = i = 0 . Quantum theory: E → ˆ α , ˆ i = − δ/δ A i A i α multiplicative. E α Constraints annihilated by the Chern–Simons state � i θ θ ≡ 6 π 2 � � S 3 tr ( A ∧ dA + 2 Ψ CS = exp 3 A ∧ A ∧ A ) , i Λ , 8 π 2 Different sectors of Euclidean gravity ( θ → i θ ) connected by large gauge transformations .

The hand: 2. Deformation of θ

The hand: 2. Deformation of θ We deform the topological sector as θ → θ ( A ) ,

The hand: 2. Deformation of θ We deform the topological sector as θ → θ ( A ) , thus breaking large-gauge U ( 1 ) invariance (analogy with Peccei–Quinn invariance in QCD).

The hand: 2. Deformation of θ We deform the topological sector as θ → θ ( A ) , thus breaking large-gauge U ( 1 ) invariance (analogy with Peccei–Quinn invariance in QCD). Λ is promoted to an evolving functional Λ( A ) .

The hand: 2. Deformation of θ We deform the topological sector as θ → θ ( A ) , thus breaking large-gauge U ( 1 ) invariance (analogy with Peccei–Quinn invariance in QCD). Λ is promoted to an evolving functional Λ( A ) . No matter introduced by hand!

The hand: 2. Deformation of θ We deform the topological sector as θ → θ ( A ) , thus breaking large-gauge U ( 1 ) invariance (analogy with Peccei–Quinn invariance in QCD). Λ is promoted to an evolving functional Λ( A ) . No matter introduced by hand! The only sectors compatible with this step and the Gauss constraint are degenerate : det E = 0 , no metric!

The hand: 3. Jacobson sector (rk E = 1 )

The hand: 3. Jacobson sector (rk E = 1 ) E.o.m. for A can be written as the ( 1 + 1 ) -dimensional Dirac equation γ 0 ˙ ψ + γ z ∂ z ψ = 0

The hand: 3. Jacobson sector (rk E = 1 ) E.o.m. for A can be written as the ( 1 + 1 ) -dimensional Dirac equation γ 0 ˙ ψ + γ z ∂ z ψ = 0 , where  iA 1  1 A 1  2  ψ ≡  .   A 2  1 iA 2 2

The hand: 3. Jacobson sector (rk E = 1 ) E.o.m. for A can be written as the ( 1 + 1 ) -dimensional Dirac equation γ 0 ˙ ψ + γ z ∂ z ψ = 0 , where  iA 1  1 A 1  2  ψ ≡  .   A 2  1 iA 2 2 E E V E

The hand: 3. Jacobson sector (rk E = 1 ) E.o.m. for A can be written as the ( 1 + 1 ) -dimensional Dirac equation γ 0 ˙ ψ + γ z ∂ z ψ = 0 , where  iA 1  1 A 1  2  ψ ≡  .   A 2  1 iA 2 2 E E V E A model for V interactions and physical interpretation naturally emerge at quantum level.

The hand: 4. Suppression of Λ

The hand: 4. Suppression of Λ A quantum counterterm in H modifies the e.o.m. for A as γ 0 ˙ ψ + γ z ∂ z ψ + im ψ = 0 .

The hand: 4. Suppression of Λ A quantum counterterm in H modifies the e.o.m. for A as γ 0 ˙ ψ + γ z ∂ z ψ + im ψ = 0 . Mass term m = − 2 i ¯ ψγ 5 ∂ z ψ .

The hand: 4. Suppression of Λ A quantum counterterm in H modifies the e.o.m. for A as γ 0 ˙ ψ + γ z ∂ z ψ + im ψ = 0 . Mass term m = − 2 i ¯ ψγ 5 ∂ z ψ . The simplest nonperturbative solution requires Λ = Λ 0 exp ( − ¯ ψγ 5 γ z ψ )

The hand: 4. Suppression of Λ A quantum counterterm in H modifies the e.o.m. for A as γ 0 ˙ ψ + γ z ∂ z ψ + im ψ = 0 . Mass term m = − 2 i ¯ ψγ 5 ∂ z ψ . The simplest nonperturbative solution requires Λ = Λ 0 exp ( − ¯ ψγ 5 γ z ψ ) j 5 α associated with a chiral transformation of the fermion ψ and not conserved in the presence of m .