Singular Homogeneous Plane-Waves, Matrix Big-Bangs and their non-abelian Theories

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1 Singular Homogeneous Plane-Waves, Matrix Big-Bangs and their non-abelian Theories Lorenzo Seri Time & Matter 2010, Budva, Montenegro October 5, 2010 Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 1 / 22

2 The basic question Hawking - Penrose (late 60): Under natural assumptions we get always singular cosmologies; and so? POSSIBLE ANSWERS This is simply a failure of classical GR, and it indicates we must switch to some kind of quantum theory Hic sunt leones: it is impossible to get through the singularity, asking this is not physical sensible Temporal evolution and time itself are not anymore fundamental (not even suitable) concepts as we re going towards the singularity: sticking to those one gets troubles??? Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 2 / 22

3 What is a space-time singularity? Subtle point: In General Relativity we are not allowed to include points where the observables show some kind of illness Definition A space-time is said to be geodesically complete if every geodesic can be extended to arbitrary values of its affine parameter. This needs some refinement... Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 3 / 22

4 How did string come into play (2002) Time-dependent orbifolds as string backgrounds: hep-th/ , hep-th/ , hep-th/ (2005) Matrix Big Bang: hep-th/ (2008) Generalisation to backgrounds with some curvature: arxiv: Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 4 / 22

5 Discrete Light-Cone Quantization (I) The Infinite Momentum Frame is defined by compactifying x so that P = a (p a) = N/R s taking an infinite longitudinal boost, sending N so that m a (p a ) and E = P + (p a ) 2 + m2 a 2 (p a a ) Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 5 / 22

6 Discrete Light-Cone Quantization (I) The Infinite Momentum Frame is defined by compactifying x so that P = a (p a) = N/R s taking an infinite longitudinal boost, sending N so that m a (p a ) and E = P + (p a ) 2 + m2 a 2 (p a a ) For the Light-Cone Frame, analogously, x + is chosen as time, x is compactified so that P = a (p a) = N/R s the energy for every N is H lc = P + = a (p a ) 2 + m2 a 2(p a ) an infinite null boost is operated, sending N Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 5 / 22

7 Discrete Light-Cone Quantization (II) Advantages of DLCQ: p is conserved, so the Hilbert space decomposes into superselection sectors labeled by N For every fixed N we have a system with Galilean symmetry In order of having a non-negative Hamiltonian (p a ) > 0, so modes with negative p are decoupled Seiberg and Sen explained how the DLCQ of String Theory (flat background) is given by N D0 branes. Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 6 / 22

8 Not just strings in String Theory Dp-brane: (p + 1)-dimensional object which open strings are attached to. Taking N coincident D-branes, one gets a non-abelian gauge theory on the world-volume, while the transverse directions become themselves N N matrices Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 7 / 22

9 The Background In arxiv: the DLCQ procedure is generalised to the background { ds 2 = dx + dx + A ab (x + )x a x b (dx + ) 2 + δ ab dx a dx b e 2φ = (x + ) 3b b+1 b 1 where A ab (x + ) = ma(ma 1) x + 2 δ ab and a m a(m a 1) = 3b b+1. Homogeneous : invariant under (x +,x,x a ) (γx +,γ 1 x,x a ). Why these metrics? Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 8 / 22

10 The Penrose limit Penrose limit procedure: Given a metric g µν, choose a null geodesic γ(u) and embed it in a twist-free congruence of null geodesics ds 2 = dudv+a(u, V, Y k )dv 2 +2b i (U, V, Y k )dy i dv+g ij (U, V, Y k )dy i dy j Redefine again the coordinates as Compute the limit metric (U, V, Y k ) = (u, λ 2 v, λy k ) d s 2 = lim λ 0 λ 2 ds 2 γ = 2dudv + ḡ ij (u)dy i dy j Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 9 / 22

11 The Penrose limit of Szekeres and Iyer metrics (I) Metric with power-law singularities: ds 2 = [x(u,v)] r dudv + [x(u,v)] s dω 2 d where x(u,v) = 0 identifies the singularity surface. This class includes Friedmann-Robertson-Walker cosmological model, Schwarzschild black hole... Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 10 / 22

12 The Penrose limit of Szekeres and Iyer metrics (I) Metric with power-law singularities: ds 2 = [x(u,v)] r dudv + [x(u,v)] s dω 2 d where x(u,v) = 0 identifies the singularity surface. This class includes Friedmann-Robertson-Walker cosmological model, Schwarzschild black hole... If the singularity is space-like, the Penrose limit is a HPW when the strict Dominant Energy Condition is fulfilled Definition DEC: For every future directed time-like vector v µ the vector T µ ν v ν is a future directed time-like vector. This means that the speed of the energy flow of matter is always less than the speed of light Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 10 / 22

13 The Penrose limit of Szekeres and Iyer metrics (II) Figure: Taken from hep-th/ Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 11 / 22

14 The Background In arxiv: the DLCQ procedure is generalised to the background { ds 2 = dx + dx + A ab (x + )x a x b (dx + ) 2 + δ ab dx a dx b e 2φ = (x + ) 3b b+1 b 1 where A ab (x + ) = ma(ma 1) x + 2 δ ab and a m a(m a 1) = 3b b+1. Homogeneous : invariant under (x +,x,x a ) (γx +,γ 1 x,x a ). This isometry enables us to implement the DLCQ procedure......but in principle any PW possesses the homothety (x +,x,x a ) (x +,λ 2 x,λx a ) Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 12 / 22

15 The non-abelian theory One is brought to study the non-abelian time-dependent theory ( S BC = d 2 σtr 1 4 g 2 YM ηαγ η βδ F αβ F γδ 1 2 ηαβ δ ab D α X a D β X b g2 YM δ acδ bd [X a,x b ][X c,x d ] + 1 ) 2 A ab(τ)x a X b where g YM e φ and the Xs are matrix coordinates. We are interested in the case g YM 0 as t 0, so we have to admit tachyonic masses. Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 13 / 22

16 A toy model We will focus on the handy model (arxiv: ) S = 1 dttr ( Ẋ Ẏ 2 ω X (t) 2 X 2 ω Y (t) 2 Y 2 + λ t 2q [X,Y ] 2 ) where ω X,Y (t) 2 = ω 2 n X,n Y p(p+2) 4t 2. This is built: compactifying on a world-sheet circle; restricting ourselves to two transverse directions (represented by two su(2) matrices); picking a Fourier mode for each direction. Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 14 / 22

17 A toy model We will focus on the handy model (arxiv: ) S = 1 dttr ( Ẋ Ẏ 2 ω X (t) 2 X 2 ω Y (t) 2 Y 2 + λ t 2q [X,Y ] 2 ) where ω X,Y (t) 2 = ω 2 n X,n Y p(p+2) 4t 2. This is built: compactifying on a world-sheet circle; restricting ourselves to two transverse directions (represented by two su(2) matrices); picking a Fourier mode for each direction. Out of generality we worked with p and q unrelated. Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 14 / 22

18 Classical analysis: early times Y 1 20 Y 1 20 Y X X X Figure: Numerical integration of toy model projected in the X 1, Y 1 plane for t i = 100 and t f = 10, 1 and.001 respectively. As t 0 the fields decouple and each one is effectively governed by the equation d 2 dt X p(p+2) 2 4t X = 0 2 Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 15 / 22

19 Classical analysis: late times We redefine our time and coordinates as X = t q 3 W, Y = t q 3 Z, T t 2 3 q+1 so that ( 1 dw 2 2 dt Tr dt + dz 2 dt m 2 ( W 2 + Z 2 )T 4q/(2q+3) ( ) 4 ( W 2 + Z 2 ) ) q(2q + 3) p(p + 2) 9 4T 2 λ [W,Z] 2 Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 16 / 22

20 Classical analysis: late times Y 1 10 Y 1 10 Y X X X Figure: Numerical integration of toy model projected in the X 1, Y 1 plane for t i = 100 and t f = 500, 1000 and 5000 respectively. As t the system behaves chaotically, driven by the potential λ [W, Z] 2 Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 17 / 22

21 Quantum analysis: early times From perturbation theory In our model the quartic interaction can be treated as a perturbation as t 0 (Condition 2q 2p > 1 always satisfied) From computational approach We studied the spreading of wave packets initially located near the origin as t 0: their evolution is slow enough that they don t feel the minimum due to the quartic interaction Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 18 / 22

22 Conclusions and future perspectives Near the singularity the off-diagonal terms are activated, but on the other hand the quartic interaction is negligible! For late times the evolution is chaotic, even if the system stays in commuting configurations for long periods. Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 19 / 22

23 Conclusions and future perspectives Near the singularity the off-diagonal terms are activated, but on the other hand the quartic interaction is negligible! True both at a classical and at a quantum mechanical level For late times the evolution is chaotic, even if the system stays in commuting configurations for long periods. Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 19 / 22

24 Conclusions and future perspectives Near the singularity the off-diagonal terms are activated, but on the other hand the quartic interaction is negligible! True both at a classical and at a quantum mechanical level For late times the evolution is chaotic, even if the system stays in commuting configurations for long periods. True classically, and it seems reasonable also in quantum mechanics... (Compare with analogous models) Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 19 / 22

25 Conclusions and future perspectives Near the singularity the off-diagonal terms are activated, but on the other hand the quartic interaction is negligible! True both at a classical and at a quantum mechanical level For late times the evolution is chaotic, even if the system stays in commuting configurations for long periods. True classically, and it seems reasonable also in quantum mechanics... (Compare with analogous models) Open question: knowing that near t = 0 only the quadratic term matters what can we learn about the singularity? Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 19 / 22

26 Conclusions and future perspectives Near the singularity the off-diagonal terms are activated, but on the other hand the quartic interaction is negligible! True both at a classical and at a quantum mechanical level For late times the evolution is chaotic, even if the system stays in commuting configurations for long periods. True classically, and it seems reasonable also in quantum mechanics... (Compare with analogous models) Open question: knowing that near t = 0 only the quadratic term matters what can we learn about the singularity? Open question: how exactly is the flat space perturbative String Theory recovered as t? Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 19 / 22

27 Five string theories, one M theory In a string theory there are two fundamental parameters: l s and g s. We know several string theories but actually they are all related!!! T-duality: R l2 s R S-duality: g s 1 g s Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 20 / 22

28 Generalisation of the Seiberg-Sen approach General procedure to get a Matrix String Theory: 1 Coordinate transformation aligning an almost light-like circle with a small spatial one 2 Dynamical transformation that rescales the energies 3 Rescaling of coordinates (and masses) These transformations are supposed to leave invariant the form of the metric Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 21 / 22

29 Quantum analysis: perturbation theory Schrödinger solutions for the inverted harmonic oscillator: ( ) ψ(x,t) = x,t s = N F 1/2 e ix2ḟ /2F x e iϕ(t) H s F with F(t) = ωth ν (ωt). The wave-functions are singular as t 0: the s states, actually, are not! Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 22 / 22

30 Quantum analysis: perturbation theory Schrödinger solutions for the inverted harmonic oscillator: ( ) ψ(x,t) = x,t s = N F 1/2 e ix2ḟ /2F x e iϕ(t) H s F with F(t) = ωth ν (ωt). The wave-functions are singular as t 0: the s states, actually, are not! T.d. perturbation: Ŵ(t) = t 2q Tr[X,Y ] 2 Full Schrödinger solutions: t = s M b sm (t) s M db s (1) M (t) = λ dt i s M Ŵ s M b s M (t) s M Lorenzo Seri (SISSA - UNG) Non-Abelian Theories of Matrix Big-Bangs T&M Budva, MNE 22 / 22