A Non-Technical (Honest!) Explanation of the Problem of Quantum Gravity. Richard Woodard University of Florida

Transcription

1 A Non-Technical (Honest!) Explanation of the Problem of Quantum Gravity Richard Woodard University of Florida

2 Sketch of General Relativity 1. Gravitational Field: g µν (t,x) ds 2 = g µν dx µ dx ν 2. g µν affects fields thru Minimal Coupling -( / ct) (-g) ½ µ [(-g) ½ g µν ν ] 3. Other fields affect g µν thru Einstein Eqn G µν = 8πG T µν Key Principle: Energy gravitates 4. Eqn also predicts gravitational radiation Purely grav. DOF s not fixed by matter

3 Some Quantization Unavoidable (Key Idea #1) Dual role of force fields: Mediate interactions Harbor new quanta Cf. EM E= ρ ǫ E B µ 0 ǫ 0 t = J B E+ B t =0 B(t, x)= + d 3 k t k x (2π) 3ei d 3 k (2π) 3ei k x )] 0 dt sin[ck(t t i k J(t, k) ckǫ 0 { B 0 ( k)cos(ckt) i E k 0 ( k)sin(ckt) } ck 1 st term quantized because of matter, whether or not there are photons

4 Matter quantized whether or not photons & gravitons are! Cf Hydrogen: H = p 2 /2m e Φ(q) Φ(q) = e/(4πǫ 0 q) an operator from q Same for General Relativity g µν = (functional of T µν ) + (gravitons) Fields in T µν are certainly quantum! Allowing quantum matter to interact gravitationally causes problems with or without gravitons

5 How bad is it? Approximate with Perturbation theory QED: (0 th ) [1 + a 1 α + a 2 α ] QCD: (0 th ) [1 + b 1 α S + b 2 α S ] QGR: (0 th ) [1 + c 1 G + c 2 G ] Could we distinguish 2 different {c i } s? G ~ l 3 /mt 2 c 1 = # E 2 / c 5 c 1 G = # E 2 (G/ c 5 ) ~ # (E/10 19 GeV) 2 NB even E = 1 TeV c 1 G ~ 10-32

6 Two Problems of QG (Key Idea #2) 1. Get Finite Results 2. Subject them to meaningful experimental scrutiny

7 Physics behind the s: Recall the QM Harmonic Osc. H = p 2 /2m + ½mω 2 q 2 q(t) = q 0 cos(ωt) + p 0 /mω sin(ωt) cos(ωt) = ½ (e iωt + e -iωt ) q(t) = ½(q 0 + ip 0 /mω)e -iωt + ½(q 0 ip 0 /mω) e iωt = [a e -iωt + a e iωt ]/(2ω} ½ Mode coordinate: a = (ω/2) ½ (q 0 + ip 0 /mω) Mode function: ε(t) = e -iωt /(2ω) ½ q(t) = a ε(t) + a ε * (t)

8 Field Theories have Modes E.g., EM for 0 x i L E 1 (t,x) = Σ k [a k ε k (t,x) + a k ε k *(t,x)] where k = π/l (n 1,n 2,n 3 ) & ω = πnc/l ε k =[π 3 /2ωL 3 ] ½ sin(k 1 x 1 )cos(k 2 x 2 )cos(k 3 x 3 )e -iωt a k = L dx 1 L dx 2 L dx 3 ε k *(0,x) {ω E 1 (0,x) + ic [ 2 B 3 (0,x)- 3 B 2 (0,x)]} Two modes for every (n1,n2,n3)

9 Conspiracy of Four Principles 1. Continuum Field Theory Modes 2. Q. Mechanics Can t have q 0 =p 0 =0 Each mode has ½ ω + interactions Changes shift energies (Casimir & Lamb) 3. General Relativity Energy gravitates 4. Perturbation theory shifts add Too many modes interacting too strongly (Key Idea #3)

10 Renormalization 1: Classical EM in a medium Dumb way: ǫ 0 div(e) = ρ free + ρ bnd ρ bnd = Σ atm q [δ 3 (x-x atm ) δ 3 (x-x atm - x)] - ǫ div(e) + O( x) Smart way: [ǫ 0 + ǫ]div(e) = ρ free + O( x)

11 Renormalization 2: QED in vacuum 2 Field = e {Source + 0-pt} modes k with ω = [c 2 k 2 + m 2 c 4 / 2 ] ½ EM couples to charge same modes Field distorts 0-point = d 3 k 1 (2π) 3 2ω e Λ 2 term vanishes & e R =e/[1 - #e 2 ln(λ 2 )] 2 Field = e R {Source + Finite part of 0-pt} { #+# 2 c 2 ω = e { #Λ 2 +#ln(λ 2 ) 2 +finite } (Field) } (Field)

12 Renormalization 3: Quantum General Relativity [ 2 + CC] Field = 8πG {Source + 0-point} modes k with ω = [c 2 k 2 + m 2 c 4 / 2 ] ½ GR couples to stress-energy ~ ω 2 Field distorts 0-point = d 3 k 1 (2π) 3 2ω ω2 { #+# 2 4 c 2 ω2+# c 4 ω = { #Λ 4 +#Λ 2 2 +#ln(λ 2 ) 4 +finite } (Field) Λ 4 renormalizes CC & Λ 2 renormalizes G but ln(λ 2 ) introduces a NEW 4 term } (Field)

13 What s wrong with higher s? Newton: mẍ = F(x,ẋ) L = L(x,ẋ) Q = x and P = L/ ẋ H(Q,P) = Pẋ(Q,P) L(Q,ẋ(Q,P)) Ostrogradsky: d 4 x/dt 4 = f(x,ẋ,ẍ,d 3 x/dt 3 ) L = L(x,ẋ,ẍ) Q 1 = x, Q 2 = ẋ, P 2 = L/ ẍ, P 1 = L/ ẋ - d/dt L/ ẍ H(Q 1,Q 2,P 1,P 2 ) = P 1 Q 2 + P 2 ẍ(q 1,Q 2,P 2 ) L(Q 1,Q 2,ẍ(Q 1,Q 2,P 2 )) Why physics is based on 2 nd order eqns!

14 Key Idea #4 Pert. Quantum Gravity differs because: Other forces couple the same modes GR couples more strongly to large k modes This requires 4 th order counterterms which would make the universe blow up instantly

15 Divergent Opinions Key Idea #5 1. Relativists (love General Relativity) Perturbation theory is wrong! Nonlinear grav. ints cancel the s 2. Particle Theorists (love Pert. Theory) General Relativity is wrong! Superstrings have ± 0-point energies and interact more weakly at large k

16 Repartee with Relativists Fact: GR is weak & QGR unobs. small Question: How can PT be wrong? 1. Correct series nonanalytic in G Eg (0 th ) {1 + GE 2 / c 5 ln(ge 2 / c 5 ) +...} 2. 0 th order may diverge for G 0

17 Charged shell of radius R 0 (ADM 1960) Without GR: mc 2 = m 0 c 2 + q 2 /8πǫ 0 R renormalize with m 0 c 2 = m obs c 2 q 2 /8πǫ 0 R With GR: mc 2 =m 0 c 2 +q 2 /8πǫ 0 R Gm 2 /2R m= Rc2 G Gm 0 Rc 2 + Gq2 Perturbative Result: 4πǫ 0 R 2 c 2 Oscillating series of ever-higher s q2 4πǫ 0 G

18 All Proposed Fixes Involve E < 0 Key Idea #6 1. Relativist s dream: gravity regulates Negative grav. int. energy cancels s nonperturbatively 2. Particle Theorist s dream: superstrings SUSY adds E > 0 fermions which contribute -½ ω 3. Pert. Gravity s wish: induce higher s E < 0 particles

19 The Other Problem Experiments! Theoretical theory (Wizards dueling) Study weak ints using unique features µ - e - ν e ν µ doesn t occur in QED Unique features of gravity Negative interaction energy M=0 gravitons without conformal invariance Photons can t distinguish g µν (x) from Ω 2 (x) g µν (x) Cosmology is a natural venue (Key Idea #7) ds 2 = -dt 2 + a 2 (t) dx 2 = a 2 [-dη 2 + dx 2 ] Gravity is long range and knows about a(t)

20 Conclusions 1. Some quantization of GR unavoidable This causes problems w/o gravitons 2. Goals of Quantum Gravity are: Get finite results Test them 3. Perturbative s from too many modes interacting too strongly 4. Perturbative Quantum GR differs because: Other forces couple same to all modes GR couples more strongly to high k modes

21 More Conclusions 5. Divergent Opinions: Relativists: GR is right & PT is wrong! Particle Theorists: PT is right & GR is wrong! 6. All fixes involve some negative energy Gravitational interaction energy Super-partner 0-point energy E < 0 for 4 counterterms 7. Phenomenolgy uses unique properties Negative interaction energy M=0 gravitons w/o conformal invariance 8. Cosmology a natural venue