Restricted Gravity A New Approach to Quantum Gravity Y. M. Cho - PowerPoint PPT Presentation

Restricted Gravity —A New Approach to Quantum Gravity— Y. M. Cho School of Electrical and Computer Engineering Ulsan National Institute of Science and Technology and School of Physics and Astronomy College of Natural Science, Seoul National University Korea February 27, 2012 Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 1 / 47

Motivation and Plan Problems of spin-two graviton The metric is a classical concept which allows precise mesurement, but 1 quantum gravity requires a quantum field which requires intrinsic fuzziness — Geroch. The metric can not describe the gravitational coupling to fermions 2 ( ¯ ψγ a ∂ µ ψ ) × e µ a . This tells that the tetrad (4 spin-one fields e µ a ) is more fundamental than the metric. So we need a new paradigm for quantum gravity. Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 2 / 47

Motivation Is Einstein’s theory the simplest possible generally invariant theory? 1 Yes?.....No! What is the simpler theory? 2 Restricted gravity which describes the core dynamics of Einstein’s theory. How can we obtain such gravity? 3 Making Abelian projection to Einstein’s theory. How can we describe the graviton in this theory? 4 By a spin-one Abelian gauge field. Quantum gravity Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 3 / 47

Plan Treat Einstein’s theory as a gauge theory of Lorentz group. Make the 1 Abelian projection to decompose the connection to the restricted part and the valence part. Remove the valence part to separate the core dynamics of Einstein’s 2 theory. Obtain the restricted gravity. Express the restricted gravity by an Abelian gauge theory, and show 3 that the graviton can be described by a massless spin-one gauge field. Recover Einstein’s theory adding the valence part. Establish the 4 Abelian dominance in Einstein’s theory. Example: Restricted QCD Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 4 / 47

Abelian Decomposition: SU(2) QCD A) Abelian decomposition Let (ˆ n 1 , ˆ n 2 , ˆ n 3 = ˆ n ) be an orthonormal basis, and select ˆ n to be the Abelian (i.e., the color) direction. Make the Abelian projection n 2 = 1) n + g � D µ ˆ n = ∂ µ ˆ A µ × ˆ n = 0 . (ˆ n − 1 A µ → ˆ � n · � A µ = A µ ˆ g ˆ n × ∂ µ ˆ n. ( A µ = ˆ A µ ) With this we have the Abelian (Cho-Faddeev-Niemi or Cho-Duan-Ge) decomposition n − 1 � n + � n · � A µ = A µ ˆ g ˆ n × ∂ µ ˆ X µ , (ˆ X µ = 0) . Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 5 / 47

Under the infinitesimal gauge transformation A µ = 1 δ � gD µ � α, δ ˆ n = − � α × ˆ n, we have A µ = 1 δ ˆ ˆ δ � α × � D µ � α, X µ = − � X µ . g ˆ A µ has the full SU(2) gauge degrees of freedom, and forms an SU(2) 1 connection space by itself. � X µ transforms covariantly. 2 Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 6 / 47

B) Restricted QCD (RCD) ˆ A µ is essentially Abelian, but has a dual structure F µν = ∂ µ ˆ ˆ A ν − ∂ ν ˆ A µ + g ˆ A µ × ˆ A ν = ( F µν + H µν )ˆ n, F µν = ∂ µ A ν − ∂ ν A µ , H µν = − 1 g ˆ n · ( ∂ µ ˆ n × ∂ ν ˆ n ) = ∂ µ C ν − ∂ ν C µ , C µ = 1 g ˆ n 1 · ∂ µ ˆ n 2 . So ˆ F µν is described by two Abelian potentials, the “electric” A µ and the “magnetic” C µ . Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 7 / 47

C µ = − 1 � Let g ˆ n × ∂ µ ˆ n and find H µν = ∂ µ � � C ν − ∂ ν � C µ + g � C µ × � C ν = H µν ˆ n. Moreover, � C µ with ˆ n = ˆ r describes precisely the Wu-Yang monopole, n represents the non-Abelian monopole topology Π 2 ( S 2 ) . where ˆ Define the restricted QCD by L RCD = − 1 ˆ 2 F µν . 4 It has the full non-Abelian gauge invariance and thus inherits all topological properties of QCD, but is much simpler than QCD. Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 8 / 47

C) Abelian dominance Find F µν = ˆ � F µν + ( ˆ D µ � X ν − ˆ D ν � X µ ) + g � X µ × � X ν , L QCD = − 1 µν = − 1 µν − g � ˆ F µν · ( � ˆ X µ × � F 2 F 2 X ν ) 4 4 2 X µ ) 2 − g 2 − 1 4( ˆ D µ � X ν − ˆ D ν � 4 ( � X µ × � X ν ) 2 . So QCD can be viewed as RCD made of ˆ A µ which has the valence gluons as colored source. The valence gluons play no role in confinement, because they are the colored source which have to be confined. Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 9 / 47

D) Monopole dominance The Abelian projection separates the monopole potential gauge independently. The one-loop effective action of QCD shows that the monopole condensation plays the central role in color confinement. The monopole dominance in the color confinement has been confirmed by recent KEK lattice calculations based on Abelian projection. Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 10 / 47

Figure: The monopole dominance based on Abelian projection in lattice QCD. Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 11 / 47

Vacuum Decomposition A) Vacuum potential Impose the vacuum isometry ∀ i D µ ˆ n i = ( ∂ µ + g � A µ × ) ˆ n i = 0 , ∀ i [ D µ , D ν ] ˆ n i = � � F µν × ˆ n i = 0 ⇒ F µν = 0 . Construct the most general vacuum potential n k = − 1 A µ → ˆ � Ω µ = C k k µ ˆ 2 gǫ (ˆ n i · ∂ µ ˆ n j ) ˆ n k . ij Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 12 / 47

With S 3 compactification of R 3 , we have the multiple vacua | n � classified by the Hopf invariant Π 3 ( S 3 ) ≃ Π 3 ( S 2 ) which represents the knot topology of ˆ n = ˆ n 3 , � n = − g 3 ǫ αβγ ǫ ijk C i α C j β C k γ d 3 x. ( α, β, γ = 1 , 2 , 3) 96 π 2 With ˆ Ω µ , the restricted potential ˆ A µ admits further decomposition A µ = ˆ ˆ Ω µ + � B µ = ( A µ + � � B µ , C µ ) ˆ n, Ω µ = 1 gD (0) ( D (0) δ ˆ δ � α × � = ∂ µ + g ˆ � α, B µ = − � B µ , Ω µ × ) . µ µ So ˆ Ω µ (just like ˆ A µ ) forms its own SU(2) connection space. Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 13 / 47

Figure: The structure of non-Abelian connection space: It has two proper subspaces made of the restricted potentials ˆ A µ and the vacuum potentials ˆ Ω µ which form their own non-Abelian connection spaces. Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 14 / 47

B) Vacuum tunneling The multiple vacua | n � are physically (as well as topologically) inequivalent, but are unstable under the quantum fluctuation. They are connected by the vacuum tunneling through the instantons. The vacuum tunneling assures the existence of the θ -vacuum in QCD � e inθ | n � . | Ω � = n The SU(2) results directly applies to Einstein’s theory because SU(2) is the rotation subgroup of Lorentz group. Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 15 / 47

Einstein’s Theory: Gauge Theory of Lorentz Group Einstein’s theory can be viewed as a gauge theory of Lorentz group, and the local Lorentz invariance assures the general invariance. In the presence of spinor field one must have the local Lorentz invariance. This necessitates a gauge theory of Lorentz group, where the tetrad (not the metric) plays the fundamental role. Constructing a gauge theory of Lorentz group is a natural way to rediscover Einstein’s theory. Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 16 / 47

Introduce a coordinate basis ∂ µ and an orthonormal basis e a c [ ∂ µ , ∂ ν ] = 0 , [ e a , e b ] = f ab e c , e a = e µ ∂ µ = e a a ∂ µ , µ e a . ( µ, ν ; a, b, c = 0 , 1 , 2 , 3) Let J ab = − J ba be the generators of Lorentz group, [ J ab , J cd ] = η ac J bd − η bc J ad + η bd J ac − η ad J bc mn = f J mn , ab,cd where η ab = diag ( − 1 , 1 , 1 , 1) is the Minkowski metric. Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 17 / 47

With the 3-dimensional rotation and boost generators L i and K i we have [ L i , L j ] = ǫ ijk L k , [ L i , K j ] = ǫ ijk K k , [ K i , K j ] = − ǫ ijk L k . Notice that 1. The Lorentz group is non-compact, so that the invariant metric is indefinite. 2. The Lorentz group has the well-known invariant tensor ǫ abcd which allows the dual transformation. 3. The Lorentz group has rank two, so that it has two commuting Abelian subgroups. Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 18 / 47

Remember that 1. In the gauge formalism of Einstein’s theory the spin connection ω ab corresponds to the gauge potential Γ ab µ , and the curvature µ ab ab tensor R corresponds to the field strength F µν . µν 2. In Einstein’s theory the metric g µν propagates, but in gauge theory the potential Γ ab propagates. µ ( R = e µ ab a e ν ab 3. The Einstein-Hilbert action is linear in R b R µν ) , µν ab but in gauge theory the Yang-Mills action is quadratic in F µν ( F 2 = F ab ab µν F µν ) . Y. M. Cho (Seoul National University) Restricted Gravity February 27, 2012 19 / 47