Lattice quantum gravity with scalar fjelds Raghav G. Jha Syracuse - PowerPoint PPT Presentation

Lattice quantum gravity with scalar fjelds Raghav G. Jha Syracuse University JULY 23, 2018 Lattice 2018 (with Scott Bassler, J. Laiho and Judah Unmuth-Yockey)

Gravity on the lattice Outline Raghav G. Jha, Syracuse University 2 • Motivation for lattice gravity • Asymptotic safety conjecture • Lattice discretization • EDT coupled to scalar fjelds • Future directions

Motivation Can we understand gravity as a quantum fjeld theory in four dimensions assuming that gravity is asymptotically safe using lattice in the non-perturbative regime. Gravity on the lattice Raghav G. Jha, Syracuse University 3

Weinberg’s conjecture It is well-known that gravity is perturbatively non-renormalizable (infjnite number of parameters have to be fjxed), however Weinberg in 1979 conjectured that it might be asymptotically safe. This means, unstable directions (dimensionality of UV critical surface) Gravity on the lattice Raghav G. Jha, Syracuse University 4 • Gravity may be non-perturbatively renormalizable • Non-trivial, strongly interacting fjxed point with fjnite number of

Continuum to lattice (1) Raghav G. Jha, Syracuse University Gravity on the lattice (2) according to and Newton’s constant respectively. On a triangulation, we discretize 5 The Einstein-Hilbert action for a metric g µν has the form, 1 ∫ d 4 x √− g (2Λ − R ) S EH = 16 πG where R is the Ricci scalar and Λ and G are cosmological constant ∫ d 4 x √− g → N 4 [ T ] V 4 =

The simple form of Euclidean Einstein-Regge discrete action is given by : Raghav G. Jha, Syracuse University Gravity on the lattice (5) d-simplex is given by : (4) 6 (3) S E = − κ 2 N 2 + κ 4 N 4 where, N i is the number of simplices of dimension i . And κ 2 and κ 4 are related to the Newton’s constant G N and cosmological constant Λ respectively. κ 4 must be tuned to a critical value such that an infjnite volume can be taken. This leaves two parameters in the theory κ 2 and β . The discrete Euclidean-Regge action is, ( ) ∑ ∑ ∑ S E = − κ V 2 2 π − θ + λ V 4 8 πG , θ = cos − 1 (1/4) and λ = κ Λ . Also, the volume of a 1 where κ = √ d + 1 V d = l d √ 2 d d !

Simplifying, we get (6) Raghav G. Jha, Syracuse University Gravity on the lattice and recover the form written above in eq. (3). 7 √ √ √ ( ) 3 κ 5 3 ( 1 ) 5 cos − 1 S E = − 2 πκN 2 + N 4 + 96 λ 2 4 √ √ √ 2 πκ and κ 4 = κ 5 3 3 cos − 1 ( 1 5 We defjne new variables κ 2 = 4 ) + 96 λ 2

The dynamical triangulation approach is a modifjed version of characteristic given by Raghav G. Jha, Syracuse University Gravity on the lattice calculus due to Regge. The space in supposed to be fmat inside the 8 has 5 nodes(vertices), 5 tetrahedral faces, 10 links and 10 triangles. dimensional hinges, i.e. triangles. The angle between two tetrahedra d = 4 simplexes, the curvature being concentrated in d − 2 = 2 faces, sharing a triangle is cos − 1 (1/ d ) . Each four-dimensional simplex The manifold triangulated is usually S d , which has Euler d ( − 1) p N p = 2 − 2 h = 2 ∑ χ = 0 χ = N 0 − N 1 + N 2 − N 3 + N 4 and we have, N 3 = ( d + 1) N 4 2 N 2 = 2 N 0 + 2 N 4 − 4 N 1 = 3 N 0 + N 4 2 − 6

Problem with conformal scaling and behavior at high energies (case against asymptotic safety) The very-high energy spectrum of any d-dimensional quantum fjeld theory is that of a d-dimensional conformal fjeld theory. This is not true for gravity. Gravity on the lattice Raghav G. Jha, Syracuse University 9 In hep-th/9812237 [Banks & Aharony] and later in 0709.3555 [Shomer] claimed the following :

But why ? and d=3/2 at short distances around the UV fjxed point [VERY Raghav G. Jha, Syracuse University Gravity on the lattice (non-integer) and gravity can’t be AS [SETBACK] GOOD] So, one of the following can be possible : gravity, the high-energy spectrum must be dominated by black holes. 10 d − 1 The entropy of a renormalizable theory must scale as S ∼ E d . For d − 2 The Bekenstein-Hawking entropy area law tells us that S ∼ E d − 3 . Also note that, ( d − 1)/ d = ( d − 2)/( d − 3) for d = 3/2 . • Entropy scaling is incorrect, gravity can be AS [WAIT] • Entropy scaling and asymptotic safety (AS) scenario is correct, • Entropy scaling is correct, space-time is never fractal

Minimal coupling to scalar (no backreaction, aka quenched [Smit, deBakker 1994. Presented at Lattice Raghav G. Jha, Syracuse University Gravity on the lattice is the renormalized mass of the particle. (9) The propagator for the scalar fjeld in a fjxed background decays as, (8) 11 where, (7) Let’s start with the usual gravity action and add additional term as, 1995 Melbourne] S = S E [ g ] + S [ g, Φ] ∫ d 4 x √− g ( ) g µν ∂ µ Φ ∂ ν Φ + m 2 0 Φ 2 S [ g, Φ] = here, Φ is a test particle and the back reaction of the metric is ignored. G ( r ) = A ( r ) e − Mr where, r is the geodesic distance between two points. Here, M , which

The multiplicative renormalization follows from the shift symmetry of dimensions. For zero bare mass, there is a shift symmetry, Raghav G. Jha, Syracuse University Gravity on the lattice (10) 12 the discrete lattice action [Agishtein and Migdal, NPB 385 (1992) 395-412] and can be seen as follows, ( ) (Φ x − Φ y ) 2 + ∑ ∑ m 2 0 Φ 2 S lat = x x ⟨ xy ⟩ ( ) ∑ ( D + 1 + m 2 = 0 ) δ xy − C xy ΦΦ ⟨ xy ⟩ where, C is the simplex neighbor (or connectivity) matrix which will be discussed later, m 0 is the bare mass and D is the space-time Φ → Φ + c

Discrete laplacian D+1 Raghav G. Jha, Syracuse University Gravity on the lattice the operator corresponding to the zero eigenvalue of the Laplacian. to four same neighbors. This enables us to construct the laplacian for combinatorial case studied previously, we study degenerate (12) otherwise if x & y are nearest neighbors On each degenerate dynamical triangulation, we calculate the if x = y 13 The defjnition of discrete Laplacian is : propagator as follows, (11) G = ( − □ + m 2 0 ) − 1    □ = − 1  0  Here, D + 1 is the coordination number of a D -simplex. Unlike the triangulations in D = 4 dimension. A given four-simplex can have up confjgurations such that sum of any particular row is just m 2 0 . In the limit of vanishing bare mass ( m 0 → 0 ), we have an exact zero mode of

14 Comparing them we get, Raghav G. Jha, Syracuse University Gravity on the lattice (13) (16) , and (15) (14) L b = 1 2 ∂ µ ϕ 0 ∂ µ ϕ 0 − 1 2 m 2 0 ϕ 2 L r = 1 2 Z φ ∂ µ ϕ∂ µ ϕ − 1 2 Z m m 2 ϕ 2 φ 0 ( x ) = Z 1/2 φ ( x ) φ √ Z φ m = m 0 Z m

15 (17) Raghav G. Jha, Syracuse University Gravity on the lattice (18) have, Taking log of (16) followed by derivative w.r.t ln µ and replacing µ = 1/ a , we get: d ln ( m ) = 1 ( d ln F ) d ln a 2 d ln a where, F = Z φ / Z m The mass anomalous dimension is defjned as, γ m = d ln ( m ) d ln µ , hence we γ m = − 1 ( d ln F ) 2 d ln a

Fitting range for the scalar propagator Semi-classical regimes : inside this range. Gravity on the lattice Raghav G. Jha, Syracuse University 16 • 4k: 8 ≤ r ≤ 20 • 8k: 10 ≤ r ≤ 23 • 16k: 12 ≤ r ≤ 28 Space-time on average is S 4 in this regime. We fjt the propagators

Scalar propagator results Gravity on the lattice Raghav G. Jha, Syracuse University 17 4K, 8K, 16K at β = 0 1 0.01 0.0001 log G(r) 1x10 -6 1x10 -8 m = 1/8, 4K m = 1/16, 4K m = 1/4, 8K m = 1/8, 8K 1x10 -10 m = 1/16, 8K m = 1/4, 16K m = 1/8, 16K m = 1/16, 16K 1x10 -12 4 8 12 16 20 24 28 32 36 40 44 48 52 r

Multiplicative mass renormalization - for difgerent volumes Raghav G. Jha, Syracuse University Gravity on the lattice 18 4k, β =0 8k, β =0 1.0 1.0 0.8 0.8 0.6 0.6 m r m r 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 m 0 m 0 16k, β =0 1.0 0.8 0.6 m r 0.4 0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 m 0

Dependence on number of origins ? Gravity on the lattice Raghav G. Jha, Syracuse University 19 4k, β =0 , m 0 =0 . 05 10 1 20 origins 5 origins 10 0 G ( r ) 10 -1 10 -2 0 5 10 15 20 25 30 35 40 r

20 values Raghav G. Jha, Syracuse University Gravity on the lattice Multiplicative mass renormalization - for difgerent β 4k, β =0 4k, β =1 . 5 1.0 1.0 0.8 0.8 0.6 0.6 m r m r 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 m 0 m 0

Thank you ! Gravity on the lattice Raghav G. Jha, Syracuse University 21