Quantum Null Energy Condition A remarkable inequality in physics
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1 Quantum Null Energy Condition A remarkable inequality in physics Daniel Grumiller Institute for Theoretical Physics TU Wien Erwin-Schrödinger Institute, May
2 Equalities in mathematics and physics Equalities are a core part of mathematics Daniel Grumiller Quantum Null Energy Condition 2/11
3 Equalities in mathematics and physics Equalities are a core part of mathematics Example: = Daniel Grumiller Quantum Null Energy Condition 2/11
4 Equalities in mathematics and physics Equalities are a core part of mathematics Example: = Whether an equality is interesting or boring depends on context Daniel Grumiller Quantum Null Energy Condition 2/11
5 Equalities in mathematics and physics Equalities are a core part of mathematics Example: = Whether an equality is interesting or boring depends on context = smallest dimension of non-trivial rep. of monster group = first non-trivial coefficient of modular J-function equality above with this context: monstrous moonshine Daniel Grumiller Quantum Null Energy Condition 2/11
6 Equalities in mathematics and physics Equalities are a core part of mathematics Example: = Whether an equality is interesting or boring depends on context = smallest dimension of non-trivial rep. of monster group = first non-trivial coefficient of modular J-function equality above with this context: monstrous moonshine Whether an equality is useful for physics is generally unclear but often interesting equalities tend to have applications in physics Daniel Grumiller Quantum Null Energy Condition 2/11
7 Equalities in mathematics and physics Equalities are a core part of mathematics Example: = Whether an equality is interesting or boring depends on context = smallest dimension of non-trivial rep. of monster group = first non-trivial coefficient of modular J-function equality above with this context: monstrous moonshine Whether an equality is useful for physics is generally unclear but often interesting equalities tend to have applications in physics (flat space) chiral gravity is a theory with a (cosmological) horizon that has a classical entropy of 4π (in suitable units); in the full quantum theory this entropy gets shifted S = 4π + quant. corr. ( quant. corr. ) = ln ( 12.2 ) the number is interpreted as number of microstates and stems from monstrous moonshine above Daniel Grumiller Quantum Null Energy Condition 2/11
8 Equalities in mathematics and physics Equalities are a core part of mathematics Example: = Whether an equality is interesting or boring depends on context = smallest dimension of non-trivial rep. of monster group = first non-trivial coefficient of modular J-function equality above with this context: monstrous moonshine Whether an equality is useful for physics is generally unclear but often interesting equalities tend to have applications in physics (flat space) chiral gravity is a theory with a (cosmological) horizon that has a classical entropy of 4π (in suitable units); in the full quantum theory this entropy gets shifted S = 4π + quant. corr. ( quant. corr. ) = ln ( 12.2 ) the number is interpreted as number of microstates and stems from monstrous moonshine above Equalities are also core part of comparing theory with experiment Daniel Grumiller Quantum Null Energy Condition 2/11
9 Equalities in mathematics and physics Equalities are a core part of mathematics Example: = Whether an equality is interesting or boring depends on context = smallest dimension of non-trivial rep. of monster group = first non-trivial coefficient of modular J-function equality above with this context: monstrous moonshine Whether an equality is useful for physics is generally unclear but often interesting equalities tend to have applications in physics (flat space) chiral gravity is a theory with a (cosmological) horizon that has a classical entropy of 4π (in suitable units); in the full quantum theory this entropy gets shifted S = 4π + quant. corr. ( quant. corr. ) = ln ( 12.2 ) the number is interpreted as number of microstates and stems from monstrous moonshine above Equalities are also core part of comparing theory with experiment Example: g ex /2 = (073), g th /2 = (178) Daniel Grumiller Quantum Null Energy Condition 2/11
10 Inequalities in mathematics Inequalities are another core part of mathematics Daniel Grumiller Quantum Null Energy Condition 3/11
11 Inequalities in mathematics Inequalities are another core part of mathematics Many inequalities stem from simple observation that squares of real numbers cannot be negative p 2 0 p R Daniel Grumiller Quantum Null Energy Condition 3/11
12 Inequalities in mathematics Inequalities are another core part of mathematics Many inequalities stem from simple observation that squares of real numbers cannot be negative p 2 0 p R Example: given two positive real numbers a, b algebraic mean geometric mean Daniel Grumiller Quantum Null Energy Condition 3/11
13 Inequalities in mathematics Inequalities are another core part of mathematics Many inequalities stem from simple observation that squares of real numbers cannot be negative p 2 0 p R Example: given two positive real numbers a, b algebraic mean geometric mean Proof: take p = a b and get from inequality above (a b) 2 = a 2 2ab + b 2 0 Daniel Grumiller Quantum Null Energy Condition 3/11
14 Inequalities in mathematics Inequalities are another core part of mathematics Many inequalities stem from simple observation that squares of real numbers cannot be negative p 2 0 p R Example: given two positive real numbers a, b algebraic mean geometric mean Proof: take p = a b and get from inequality above add on both sides 4ab (a b) 2 = a 2 2ab + b 2 0 a 2 + 2ab + b 2 = (a + b) 2 4ab Daniel Grumiller Quantum Null Energy Condition 3/11
15 Inequalities in mathematics Inequalities are another core part of mathematics Many inequalities stem from simple observation that squares of real numbers cannot be negative p 2 0 p R Example: given two positive real numbers a, b algebraic mean geometric mean Proof: take p = a b and get from inequality above add on both sides 4ab (a b) 2 = a 2 2ab + b 2 0 a 2 + 2ab + b 2 = (a + b) 2 4ab take square root and then divide by 2 a + b ab 2 Daniel Grumiller Quantum Null Energy Condition 3/11
16 Inequalities in mathematics Inequalities are another core part of mathematics Many inequalities stem from simple observation that squares of real numbers cannot be negative p 2 0 p R Many inequalities are of Cauchy Schwarz type u v u v here u, v are some vector, is their length and the inner product Daniel Grumiller Quantum Null Energy Condition 3/11
17 Inequalities in mathematics. Inequalities are another core part of mathematics Many inequalities stem from simple observation that squares of real numbers cannot be negative p 2 0 p R Many inequalities are of Cauchy Schwarz type u v u v here u, v are some vector, is their length and the inner product e.g. triangle inequality u + v u + v graphic proof evident Daniel Grumiller Quantum Null Energy Condition 3/11
18 Inequalities in mathematics Inequalities are another core part of mathematics Many inequalities stem from simple observation that squares of real numbers cannot be negative p 2 0 p R Many inequalities are of Cauchy Schwarz type u v u v here u, v are some vector, is their length and the inner product Many inequalities from convexity (Jensen s inequality) Daniel Grumiller Quantum Null Energy Condition 3/11
19 Inequalities in mathematics. Inequalities are another core part of mathematics Many inequalities stem from simple observation that squares of real numbers cannot be negative p 2 0 p R Many inequalities are of Cauchy Schwarz type u v u v here u, v are some vector, is their length and the inner product Many inequalities from convexity (Jensen s inequality) special case of Jensen s inequality: secant always above convex curve between intersection points x 1, x 2 Daniel Grumiller Quantum Null Energy Condition 3/11
20 Inequalities in physics Interesting physical consequences from mathematical inequalities Positivity inequalities: probabilities non-negative, P 0 Daniel Grumiller Quantum Null Energy Condition 4/11
21 Inequalities in physics Interesting physical consequences from mathematical inequalities Positivity inequalities: probabilities non-negative, P 0 Example: unitarity constraints on physical parameters in quark mixing matrix if Standard Model correct then measurements must reproduce unitarity triangle Daniel Grumiller Quantum Null Energy Condition 4/11
22 Inequalities in physics Interesting physical consequences from mathematical inequalities Positivity inequalities: probabilities non-negative, P 0 Cauchy Schwarz inequalities: Heisenberg uncertainty, x p 1 2 Daniel Grumiller Quantum Null Energy Condition 4/11
23 Inequalities in physics Interesting physical consequences from mathematical inequalities Positivity inequalities: probabilities non-negative, P 0 Cauchy Schwarz inequalities: Heisenberg uncertainty, x p 1 2 Gaussian 0.8 Fourier transformed Gaussian green: localized in coordinate space (x), delocalized in momentum space (p) blue: mildly (de-)localized in coordinate and momentum space orange: delocalized in coordinate space (x), localized in momentum space (p) Daniel Grumiller Quantum Null Energy Condition 4/11
24 Inequalities in physics Interesting physical consequences from mathematical inequalities Positivity inequalities: probabilities non-negative, P 0 Cauchy Schwarz inequalities: Heisenberg uncertainty, x p 1 2 Convexity inequalities: second law of thermodynamics, δs 0 Daniel Grumiller Quantum Null Energy Condition 4/11
25 Inequalities in physics Interesting physical consequences from mathematical inequalities Positivity inequalities: probabilities non-negative, P 0 Cauchy Schwarz inequalities: Heisenberg uncertainty, x p 1 2 Convexity inequalities: second law of thermodynamics, δs 0 In gravitational context: energy inequalities Daniel Grumiller Quantum Null Energy Condition 4/11
26 Inequalities in physics Interesting physical consequences from mathematical inequalities Positivity inequalities: probabilities non-negative, P 0 Cauchy Schwarz inequalities: Heisenberg uncertainty, x p 1 2 Convexity inequalities: second law of thermodynamics, δs 0 In gravitational context: energy inequalities Definition: (local) inequalities on the stress tensor Tµν e.g. Null Energy Condition (NEC) T kk = T µν k µ k ν 0 k µ k µ = 0 Daniel Grumiller Quantum Null Energy Condition 4/11
27 Inequalities in physics Interesting physical consequences from mathematical inequalities Positivity inequalities: probabilities non-negative, P 0 Cauchy Schwarz inequalities: Heisenberg uncertainty, x p 1 2 Convexity inequalities: second law of thermodynamics, δs 0 In gravitational context: energy inequalities Definition: (local) inequalities on the stress tensor Tµν e.g. Null Energy Condition (NEC) T kk = T µν k µ k ν 0 k µ k µ = 0 Physically plausible (positivity of energy fluxes) Daniel Grumiller Quantum Null Energy Condition 4/11
28 Inequalities in physics Interesting physical consequences from mathematical inequalities Positivity inequalities: probabilities non-negative, P 0 Cauchy Schwarz inequalities: Heisenberg uncertainty, x p 1 2 Convexity inequalities: second law of thermodynamics, δs 0 In gravitational context: energy inequalities Definition: (local) inequalities on the stress tensor Tµν e.g. Null Energy Condition (NEC) T kk = T µν k µ k ν 0 k µ k µ = 0 Physically plausible (positivity of energy fluxes) Mathematically useful (singularity theorem, area theorem) Daniel Grumiller Quantum Null Energy Condition 4/11
29 Inequalities in physics Interesting physical consequences from mathematical inequalities Positivity inequalities: probabilities non-negative, P 0 Cauchy Schwarz inequalities: Heisenberg uncertainty, x p 1 2 Convexity inequalities: second law of thermodynamics, δs 0 In gravitational context: energy inequalities Definition: (local) inequalities on the stress tensor Tµν e.g. Null Energy Condition (NEC) T kk = T µν k µ k ν 0 k µ k µ = 0 Physically plausible (positivity of energy fluxes) Mathematically useful (singularity theorem, area theorem) However: all of them violated by quantum effects! Daniel Grumiller Quantum Null Energy Condition 4/11
30 Inequalities in physics Interesting physical consequences from mathematical inequalities Positivity inequalities: probabilities non-negative, P 0 Cauchy Schwarz inequalities: Heisenberg uncertainty, x p 1 2 Convexity inequalities: second law of thermodynamics, δs 0 In gravitational context: energy inequalities Definition: (local) inequalities on the stress tensor Tµν e.g. Null Energy Condition (NEC) T kk = T µν k µ k ν 0 k µ k µ = 0 Physically plausible (positivity of energy fluxes) Mathematically useful (singularity theorem, area theorem) However: all of them violated by quantum effects! Are there quantum energy conditions? Daniel Grumiller Quantum Null Energy Condition 4/11
31 Quantum energy conditions Definition: quantum energy condition = convexity condition for T µν valid for any state and any (reasonable) quantum field theory (QFT) Daniel Grumiller Quantum Null Energy Condition 5/11
32 Quantum energy conditions Definition: quantum energy condition = convexity condition for T µν valid for any state and any (reasonable) quantum field theory (QFT) Example: Averaged Null Energy Condition (ANEC) dx λ k λ T µν k µ k ν 0 valid k µ (with k µ k µ = 0) and states in any (reasonable) QFT Daniel Grumiller Quantum Null Energy Condition 5/11
33 Quantum energy conditions Definition: quantum energy condition = convexity condition for T µν valid for any state and any (reasonable) quantum field theory (QFT) Example: Averaged Null Energy Condition (ANEC) dx λ k λ T µν k µ k ν 0 valid k µ (with k µ k µ = 0) and states in any (reasonable) QFT ANEC proved under rather generic assumptions Faulkner, Leigh, Parrikar and Wang Hartman, Kundu and Tajdini Daniel Grumiller Quantum Null Energy Condition 5/11
34 Quantum energy conditions Definition: quantum energy condition = convexity condition for T µν valid for any state and any (reasonable) quantum field theory (QFT) Example: Averaged Null Energy Condition (ANEC) dx λ k λ T µν k µ k ν 0 valid k µ (with k µ k µ = 0) and states in any (reasonable) QFT ANEC proved under rather generic assumptions ANEC sufficient for focussing properties used in singularity theorems Daniel Grumiller Quantum Null Energy Condition 5/11
35 Quantum energy conditions Definition: quantum energy condition = convexity condition for T µν valid for any state and any (reasonable) quantum field theory (QFT) Example: Averaged Null Energy Condition (ANEC) dx λ k λ T µν k µ k ν 0 valid k µ (with k µ k µ = 0) and states in any (reasonable) QFT ANEC proved under rather generic assumptions ANEC sufficient for focussing properties used in singularity theorems ANEC compatible with quantum interest conjecture Daniel Grumiller Quantum Null Energy Condition 5/11
36 Quantum energy conditions Definition: quantum energy condition = convexity condition for T µν valid for any state and any (reasonable) quantum field theory (QFT) Example: Averaged Null Energy Condition (ANEC) dx λ k λ T µν k µ k ν 0 valid k µ (with k µ k µ = 0) and states in any (reasonable) QFT ANEC proved under rather generic assumptions ANEC sufficient for focussing properties used in singularity theorems ANEC compatible with quantum interest conjecture However: ANEC is non-local ( dx + ) Daniel Grumiller Quantum Null Energy Condition 5/11
37 Quantum energy conditions Definition: quantum energy condition = convexity condition for T µν valid for any state and any (reasonable) quantum field theory (QFT) Example: Averaged Null Energy Condition (ANEC) dx λ k λ T µν k µ k ν 0 valid k µ (with k µ k µ = 0) and states in any (reasonable) QFT ANEC proved under rather generic assumptions ANEC sufficient for focussing properties used in singularity theorems ANEC compatible with quantum interest conjecture However: ANEC is non-local ( dx + ) Is there a local quantum energy condition? Daniel Grumiller Quantum Null Energy Condition 5/11
38 Quantum null energy condition (QNEC) Proposed by Bousso, Fisher, Leichenauer and Wall in QNEC (in D > 2) is the following inequality T kk 2π γ S Daniel Grumiller Quantum Null Energy Condition 6/11
39 Quantum null energy condition (QNEC) Proposed by Bousso, Fisher, Leichenauer and Wall in QNEC (in D > 2) is the following inequality T kk 2π γ S Obvious observations: if r.h.s. vanishes: semi-classical version of NEC if r.h.s. negative: weaker condition than NEC (NEC can be violated while QNEC holds) if r.h.s. positive: stronger condition than NEC (if QNEC holds also NEC holds) Daniel Grumiller Quantum Null Energy Condition 6/11
40 Quantum null energy condition (QNEC) Proposed by Bousso, Fisher, Leichenauer and Wall in QNEC (in D > 2) is the following inequality T kk 2π γ S T kk = T µν k µ k ν with k µ k µ = 0 and denotes expectation value Daniel Grumiller Quantum Null Energy Condition 6/11
41 Quantum null energy condition (QNEC) Proposed by Bousso, Fisher, Leichenauer and Wall in QNEC (in D > 2) is the following inequality T kk 2π γ S T kk = T µν k µ k ν with k µ k µ = 0 and denotes expectation value S : 2 nd variation of EE for entangling surface deformations along k µ Daniel Grumiller Quantum Null Energy Condition 6/11
42 Quantum null energy condition (QNEC) Proposed by Bousso, Fisher, Leichenauer and Wall in QNEC (in D > 2) is the following inequality T kk 2π γ S T kk = T µν k µ k ν with k µ k µ = 0 and denotes expectation value S : 2 nd variation of EE for entangling surface deformations along k µ γ: induced volume form of entangling region (black boundary curve) Daniel Grumiller Quantum Null Energy Condition 6/11
43 Proofs (D > 2) For free QFTs: Bousso, Fisher, Koeller, Leichenauer and Wall, For holographic CFTs: Koeller and Leichenauer, For general CFTs: Balakrishnan, Faulkner, Khandker and Wang, Saturation of QNEC for contact terms ( Energy is Entanglement ): Leichenauer, Levine and Shahbazi-Moghaddam, Daniel Grumiller Quantum Null Energy Condition 7/11
44 Proofs and counter examples (D = 2) Ongoing work with Ecker, Stanzer and van der Schee QNEC (in CFT 2 ) is the following inequality T kk S + 6 c S 2 c > 0 is the central charge of the CFT 2 Daniel Grumiller Quantum Null Energy Condition 7/11
45 Proofs and counter examples (D = 2) Ongoing work with Ecker, Stanzer and van der Schee QNEC (in CFT 2 ) is the following inequality T kk S + 6 c S 2 c > 0 is the central charge of the CFT 2 S like anomalous operator with conformal weights (0, 0) construct vertex operator V = exp [ 6 c S] Daniel Grumiller Quantum Null Energy Condition 7/11
46 Proofs and counter examples (D = 2) Ongoing work with Ecker, Stanzer and van der Schee QNEC (in CFT 2 ) is the following inequality T kk S + 6 c S 2 c > 0 is the central charge of the CFT 2 S like anomalous operator with conformal weights (0, 0) construct vertex operator V = exp [ 6 c S] QNEC saturation equivalent to vertex operator solving Hill s equation V LV = 0 L T kk Daniel Grumiller Quantum Null Energy Condition 7/11
47 Proofs and counter examples (D = 2) Ongoing work with Ecker, Stanzer and van der Schee QNEC (in CFT 2 ) is the following inequality T kk S + 6 c S 2 c > 0 is the central charge of the CFT 2 S like anomalous operator with conformal weights (0, 0) construct vertex operator V = exp [ 6 c S] QNEC saturation equivalent to vertex operator solving Hill s equation V LV = 0 L T kk QNEC saturated for vacuum, thermal states and their descendants Daniel Grumiller Quantum Null Energy Condition 7/11
48 Proofs and counter examples (D = 2) Ongoing work with Ecker, Stanzer and van der Schee QNEC (in CFT 2 ) is the following inequality T kk S + 6 c S 2 c > 0 is the central charge of the CFT 2 S like anomalous operator with conformal weights (0, 0) construct vertex operator V = exp [ 6 c S] QNEC saturation equivalent to vertex operator solving Hill s equation V LV = 0 L T kk QNEC saturated for vacuum, thermal states and their descendants QNEC not saturated in hol. CFT 2 with positive bulk energy fluxes Daniel Grumiller Quantum Null Energy Condition 7/11
49 Proofs and counter examples (D = 2) Ongoing work with Ecker, Stanzer and van der Schee QNEC (in CFT 2 ) is the following inequality T kk S + 6 c S 2 c > 0 is the central charge of the CFT 2 S like anomalous operator with conformal weights (0, 0) construct vertex operator V = exp [ 6 c S] QNEC saturation equivalent to vertex operator solving Hill s equation V LV = 0 L T kk QNEC saturated for vacuum, thermal states and their descendants QNEC not saturated in hol. CFT 2 with positive bulk energy fluxes QNEC can be violated in hol. CFT 2 with negative bulk energy fluxes Daniel Grumiller Quantum Null Energy Condition 7/11
50 Calculating QNEC holographically calculating CFT observable holographically = some gravity calculation AdS/CFT: Maldacena hep-th/ (> citations; > 50 in May 2018) Gubser, Klebanov and Polyakov hep-th/ Witten hep-th/ holographic stress tensor: Henningson and Skenderis hep-th/ Balasubramanian and Kraus hep-th/ Emparan, Johnson and Myers hep-th/ de Haro, Solodukhin and Skenderis hep-th/ holographic entanglement entropy (HEE): Ryu and Takayanagi hep-th/ Hubeny, Rangamani and Takayanagi Swingle (possible relation between MERA and holography) Daniel Grumiller Quantum Null Energy Condition 8/11
51 Calculating QNEC holographically calculating CFT observable holographically = some gravity calculation need holographic computation of T kk well-known AdS/CFT prescription: extract boundary stress tensor from bulk metric expanded near AdS boundary Example: AdS 3 /CFT 2 ds 2 = l2 z 2 ( dz 2 +2 dx + dx ) + T ++ ( dx +) 2 + T ( dx ) 2 +O ( z 2 ) AdS 3 boundary: z 0 O(1) terms in metric: flux components of stress tensor T ±± (trace vanishes, T + = 0) l: so-called AdS-radius (cosmological constant Λ = 1/l 2 ) Daniel Grumiller Quantum Null Energy Condition 8/11
52 Calculating QNEC holographically calculating CFT observable holographically = some gravity calculation need holographic computation of T kk need holographic computation of (deformations of) EE HEE = area of extremal surface simple to calculate! also: simple proof of strong subadditivity inequalities Daniel Grumiller Quantum Null Energy Condition 8/11
53 Thermal case see work with Ecker, Stanzer and van der Schee thermal states in CFT 4 = black holes in AdS 5 paper-and-pencil calculation starts with Schwarzschild black brane ds 2 = 1 ( f(z) dt 2 z 2 + dz2 f(z) + dy2 + dx dx 2 ) 2 with f(z) = 1 π 4 T 4 z 4 Daniel Grumiller Quantum Null Energy Condition 9/11
54 Thermal case see work with Ecker, Stanzer and van der Schee thermal states in CFT 4 = black holes in AdS 5 paper-and-pencil calculation starts with Schwarzschild black brane ds 2 = 1 ( f(z) dt 2 z 2 + dz2 f(z) + dy2 + dx dx 2 ) 2 with f(z) = 1 π 4 T 4 z 4 determine area of minimal surfaces for small temperature, T l 1, and extract HEE (l = width of strip) 1 2π S l π 4 T l 4 π 8 T 8 + O ( l 8 T 12) Daniel Grumiller Quantum Null Energy Condition 9/11
55 Thermal case see work with Ecker, Stanzer and van der Schee thermal states in CFT 4 = black holes in AdS 5 paper-and-pencil calculation starts with Schwarzschild black brane ds 2 = 1 ( f(z) dt 2 z 2 + dz2 f(z) + dy2 + dx dx 2 ) 2 with f(z) = 1 π 4 T 4 z 4 determine area of minimal surfaces for small temperature, T l 1, and extract HEE (l = width of strip) 1 2π S l π 4 T l 4 π 8 T 8 + O ( l 8 T 12) do same for large temperatures, T l 1 1 2π S π 4 T 4 e 6lπT + O ( e 2 6lπT ) Daniel Grumiller Quantum Null Energy Condition 9/11
56 Thermal case see work with Ecker, Stanzer and van der Schee thermal states in CFT 4 = black holes in AdS 5 paper-and-pencil calculation starts with Schwarzschild black brane ds 2 = 1 ( f(z) dt 2 z 2 + dz2 f(z) + dy2 + dx dx 2 ) 2 with f(z) = 1 π 4 T 4 z 4 determine area of minimal surfaces for small temperature, T l 1, and extract HEE (l = width of strip) 1 2π S l π 4 T l 4 π 8 T 8 + O ( l 8 T 12) do same for large temperatures, T l 1 1 2π S π 4 T 4 e 6lπT + O ( e 2 6lπT ) use numerics for intermediate values of temperature Daniel Grumiller Quantum Null Energy Condition 9/11
57 Thermal case see work with Ecker, Stanzer and van der Schee thermal states in CFT 4 = black holes in AdS 5 1 2π ±''/π 4 T Thermal - vac Thermal Vacuum e - 6 L L π T L notational alert: L in the plot corresponds to width l Daniel Grumiller Quantum Null Energy Condition 9/11
58 Colliding gravitational shockwaves and QNEC saturation see work with Ecker, Stanzer and van der Schee plasma formation in CFT 4 = colliding gravitational shock waves in AdS 5 toy model for quark-gluon plasma formation in heavy ion collisions Daniel Grumiller Quantum Null Energy Condition 10/11
59 Colliding gravitational shockwaves and QNEC saturation see work with Ecker, Stanzer and van der Schee plasma formation in CFT 4 = colliding gravitational shock waves in AdS 5 toy model for quark-gluon plasma formation in heavy ion collisions paper-and-pencil calculations with Romatschke δ-like shocks particle production in forward lightcone of shocks shortly after collision anisotropic pressure: PL /E = 3, P T /E = +2 confirmed numerically for thin shocks by Casalderrey-Solana, Heller, Mateos and van der Schee close to shockwaves negative energy fluxes NEC violation! confirmed numerically and interpreted as absence of local rest frame by Arnold, Romatschke and van der Schee Daniel Grumiller Quantum Null Energy Condition 10/11
60 Colliding gravitational shockwaves and QNEC saturation see work with Ecker, Stanzer and van der Schee plasma formation in CFT 4 = colliding gravitational shock waves in AdS 5 toy model for quark-gluon plasma formation in heavy ion collisions paper-and-pencil calculations with Romatschke δ-like shocks particle production in forward lightcone of shocks shortly after collision anisotropic pressure: PL /E = 3, P T /E = +2 confirmed numerically for thin shocks by Casalderrey-Solana, Heller, Mateos and van der Schee close to shockwaves negative energy fluxes NEC violation! confirmed numerically and interpreted as absence of local rest frame by Arnold, Romatschke and van der Schee consider finite width gravitational shockwaves (pioneered numerically by Chesler and Yaffe ) Daniel Grumiller Quantum Null Energy Condition 10/11
61 Colliding gravitational shockwaves and QNEC saturation see work with Ecker, Stanzer and van der Schee plasma formation in CFT 4 = colliding gravitational shock waves in AdS 5 toy model for quark-gluon plasma formation in heavy ion collisions paper-and-pencil calculations with Romatschke δ-like shocks particle production in forward lightcone of shocks shortly after collision anisotropic pressure: PL /E = 3, P T /E = +2 confirmed numerically for thin shocks by Casalderrey-Solana, Heller, Mateos and van der Schee close to shockwaves negative energy fluxes NEC violation! confirmed numerically and interpreted as absence of local rest frame by Arnold, Romatschke and van der Schee consider finite width gravitational shockwaves (pioneered numerically by Chesler and Yaffe ) extract metric, holographic stress tensor and HEE numerically Daniel Grumiller Quantum Null Energy Condition 10/11
62 Colliding gravitational shockwaves and QNEC saturation see work with Ecker, Stanzer and van der Schee plasma formation in CFT 4 = colliding gravitational shock waves in AdS 5 toy model for quark-gluon plasma formation in heavy ion collisions paper-and-pencil calculations with Romatschke δ-like shocks particle production in forward lightcone of shocks shortly after collision anisotropic pressure: PL /E = 3, P T /E = +2 confirmed numerically for thin shocks by Casalderrey-Solana, Heller, Mateos and van der Schee close to shockwaves negative energy fluxes NEC violation! confirmed numerically and interpreted as absence of local rest frame by Arnold, Romatschke and van der Schee consider finite width gravitational shockwaves (pioneered numerically by Chesler and Yaffe ) extract metric, holographic stress tensor and HEE numerically check QNEC and its saturation, particularly in region of NEC violation Daniel Grumiller Quantum Null Energy Condition 10/11
63 Colliding gravitational shockwaves and QNEC saturation see work with Ecker, Stanzer and van der Schee plasma formation in CFT 4 = colliding gravitational shock waves in AdS 5 toy model for quark-gluon plasma formation in heavy ion collisions Left: energy density plot Right: black region violates NEC Daniel Grumiller Quantum Null Energy Condition 10/11
64 Colliding gravitational shockwaves and QNEC saturation see work with Ecker, Stanzer and van der Schee plasma formation in CFT 4 = colliding gravitational shock waves in AdS 5 toy model for quark-gluon plasma formation in heavy ion collisions QNEC for L (μ y = -0.5) + ''/2π - ''/2π Daniel Grumiller Quantum Null Energy Condition 10/11
65 Open issues QNEC proof for generic relativistic unitary QFT? Daniel Grumiller Quantum Null Energy Condition 11/11
66 Open issues QNEC proof for generic relativistic unitary QFT? QNEC in certain non-unitary theories (like log CFT)? Daniel Grumiller Quantum Null Energy Condition 11/11
67 Open issues QNEC proof for generic relativistic unitary QFT? QNEC in certain non-unitary theories (like log CFT)? further special features of QNEC for CFT 2? Daniel Grumiller Quantum Null Energy Condition 11/11
68 Open issues QNEC proof for generic relativistic unitary QFT? QNEC in certain non-unitary theories (like log CFT)? further special features of QNEC for CFT 2? Hawking radiation and QNEC-(non-)violation? Daniel Grumiller Quantum Null Energy Condition 11/11
69 Open issues QNEC proof for generic relativistic unitary QFT? QNEC in certain non-unitary theories (like log CFT)? further special features of QNEC for CFT 2? Hawking radiation and QNEC-(non-)violation? QNEC analogs in non-relativistic QFTs? Daniel Grumiller Quantum Null Energy Condition 11/11
70 Open issues QNEC proof for generic relativistic unitary QFT? QNEC in certain non-unitary theories (like log CFT)? further special features of QNEC for CFT 2? Hawking radiation and QNEC-(non-)violation? QNEC analogs in non-relativistic QFTs? phenomenology of QNEC-(non-)saturation? Daniel Grumiller Quantum Null Energy Condition 11/11
71 Open issues QNEC proof for generic relativistic unitary QFT? QNEC in certain non-unitary theories (like log CFT)? further special features of QNEC for CFT 2? Hawking radiation and QNEC-(non-)violation? QNEC analogs in non-relativistic QFTs? phenomenology of QNEC-(non-)saturation? experimental aspects of QNEC? Daniel Grumiller Quantum Null Energy Condition 11/11
72 Open issues QNEC proof for generic relativistic unitary QFT? QNEC in certain non-unitary theories (like log CFT)? further special features of QNEC for CFT 2? Hawking radiation and QNEC-(non-)violation? QNEC analogs in non-relativistic QFTs? phenomenology of QNEC-(non-)saturation? experimental aspects of QNEC? Thanks for your attention! Daniel Grumiller Quantum Null Energy Condition 11/11
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