Einstein ( ) YITP
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- うまじ ねごろ
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1 Einstein ( ) YITP
2 0. massivegravity Massive spin 2 field theory Fierz-Pauli (FP ) Kinetic term L (2) EH = 1 2 [ λh µν λ h µν λ h λ h 2 µ h µλ ν h νλ + 2 µ h µλ λ h], (1) Mass term FP L mass = m2 2 (h µνh µν h 2 ), (2) ( + m 2 )h µν η µν ( + m 2 )h µ λ h λν ν λ h λµ + µ ν h + η µν λρ h λρ = 0,
3 Bargman- Wigner ( + m 2 )h µν = 0, h = 0, µ h µν = 0, (3) (Fierz-Pauli ) 4 h µν m 5
4 Fierz-Pauli 1. (BD ghost) vector propagator Proca 2. (BD) 3. graviton (vdzv ) energy-momentum tensor trace Boulware-Deser(1972) Arkani-Hamed- Georgi-Schwarz(2003) 2010 De Rham massive
5 drg, drgt de Rham-Gabadaze (2010), de Rham-Gabadaze-Tolly (2011) Galileon FP 5 graviton ADM Hamilton BFV Deser partially-massless massive
6 FP vdvz FP Propagator 1 p 2 m 2 ( 1 2 η µα η νβ η µβ η να 1 ) 3 η µν η αβ, η µν = η µν + p µp ν. m 2 ( ) Feynman-like gauge graviton propagator 1 p 2 ( 1 2 η µαη νβ η µβη να 1 ) 2 η µνη αβ, m 2 0 2
7 1. singular 1/m 2 factor 2. 1/2 1/3 (van Dam-Veltman (1970) and Zakharov (1970)) Fierz-Pauli propagator( T 2 ) Fierz-Pauli < T h(x)h αβ (y) >= η µν < T h µν (x)h αβ (y) > 0. < T µ xh µν (x)h αβ (y) > 0. transverse (T T U 0 Proca )
8 propagator de Donder (harmonic gauge) µ h µν 1 2 νh = 0, + 1 2α ( µ h µν 1 2 νh) 2, α + F.T. F µν,αβ = A(p 2 )δ µν,αβ + B(p 2 )η µν η αβ + C(p 2 )η µν p α p β +D(p 2 )η αβ p µ p ν + E(p 2 )(η µα p ν p β + η µβ p ν p α + η να p µ p β + η νβ p µ p α ) +F (p 2 )p µ p ν p α p β
9 d = 4 F (p 2 ) = p α 1 4α p2 +m α 1 2 p 2 2α m 2 α 2α 1 ( 1 2α p2 m 2 )( 2α 1 2 α p 2 3m 2 ) 1 4α p2 +m 2 1 p 2 m 2, α + f(p 2 ) m 2 p 2 m 2, vdv
10 L m = m2 2 [hµν h µν bh 2 ], (4) b = 1 Fierz-Pauli ( + M 2 )( + m 2 )h µν = 0, (5) ( + M 2 )h = 0, (6) µ h µν = a ν h, (7) M parameter a, d M 2 = db 1 (d 2)(1 b) m2, (8)
11 b (m, M) 2 Tachyon free condition 1 d b < 1, (9) 4 1/4 b < 1. a = 1 Fierz-Pauli FP
12 1. η µν 2 Fierz-Pauli ( BD, ) L (2) m = m2 2 [hµν h µν 1 2 h2 ], (10) 2. 1 Einstein 1 η µν 3. ds AdS massless (Higuchi 1987, Witten 1990?)
13 4. η µν g µν Boulware-Deser(1972)
14 ( ) 1. Fierz-Pauli mass term 6 Fierz-Pauli BD g 2. 1 gauge symmetry scalar BRS invariance 5 kugo-ojima BD Weyl Weyl
15 Dirac-Uchiyama-Freund g µν Weyl W µ, φ Weyl Dirac 1973, Uchiyama 1973, Freund 1974, Padmanabhan 1987 S 0 = a S 1 = +b d D x 1 [ g 2 d D x 1 gg µν 2 +c d D x 1 4e d D x gφ 2D D 2, g µν µ φ ν φ 1 ] (D 2) 4(D 1) Rφ2 ( µ 1 ) 2 (D 2)W µ φ gφ 2(D 4) D 2 g µν g λρ f µλ f νρ, ( ν 1 ) 2 (D 2)W ν φ
16 S 2 = ξ d D x gφ 2 [ R + 2(D 1)g λρ λ W ρ + (D 1)(D 2)g λρ W λ W ρ ] ( ) Weyl g µν = e 2Λ(x) g µν, (11) W µ = W µ µ Λ(x), (12) φ = e 1 2 (D 2) φ, (13) free a b c ξ S 0 + S 1 + S 2 = a d D x 1 2 gg µν µ φ ν φ
17 ( + ξ 1 ) (D 2) 8(D 1) a d D x gφ 2 R ( +2(D 1) ξ + 1 ) (D 2) 8 D 1 (a a) ( +(D 1)(D 2) ξ + 1 D 2 8D 1 (a a) +c d D x gφ 2D D 2 + d D x ( 1 4 d D x 1 2 gφ 2 g µν µ W ν ) d D x gφ 2 g µν W µ W ν ) gφ 2(D 4) D 2 g µν g λρ f µλ f νρ, (D 2) a ξ ξ 1 8 (D 1) a a = a + b
18 BRS Weyl BRS Kugo-Ojima G.C.T BRS δ = δ δx µ µ, δx µ = κc µ. δg µν = κ µ C λ g λν κ ν C λ g µλ, δw µ = κ µ C λ A λ, δφ = 0, δc µ = 0, δ C µ = ib µ, δb µ = 0, Weyl BRS GCT δ W g µν = 2C(x)g µν, δ W W µ = µ C(x), δ W φ = 1 (D 2)C(x)φ, 2 δ W C(x) = 0, δ W C = ib, δ W B = 0, otherwies = 0,
19 Unitary (Einstein ) Weyl Einstein gauge φ = const. = M p D 2 2, G.C.T. ( iδ ) d D x C(φ D 2 M p 2 ) = d D D 2 x(φ M p 2 )B+i d D x C( ) D 2 φc, 2 FP ghostc Nakanishi-
20 Lautrap B FP anti-ghost C = ( iδ d D x ) g C(φ D 2 M p 2 ) d D x D 2 g(φ M p 2 )B + i M p D 2 2 ) + i d D x C( ) D 2 φc, 2 d D x g CDC(φ (C C Hayashi-Kugo [] )
21 Einstein S Einstein = ξ D 2 M p d D x D gr + cm p d D x g 1 4e M D 4 p gg µν g λρ f µλ f νρ ( + d D x(d 1)(D 2) ξ + 1 ) D 2 8D 1 a D 2 M p D 2 +M p d D x ( µ ( g µν ) B ν + iκ g µν φ 2 µ Cλ ν C λ) +i d D x C( ) D 2 M (D 2)/2 p C, 2 d D x gg µν W µ W κ = M p 2, Λ cosm. = 2cM p 2
22 ξ Einstein ( ) massive 5 graviton 2 massless massive 3 Weyl Weyl Tachoyon Einsitein Plank mass Weyl W µ Einstein W µ Weyl
23 5. Boulware-Deser g g L m (a) = m2 K 2λ ( g ) 1+a, a = 0 η µν + Kh νν L (1) m (a) = 1 (1 + a)m2 2 K λ ( ) 1+a η η µν h µν = 1 (1 + a)m2 2 K λh,
24 L (2) m (a) = 1 4 (1 + a)m2 λ ( ) [ 1+a η (a + 1)η µν η ρλ η µρ η νλ η µλ η νρ] = 1 [ 2 (1 + a)m2 λ h µν h µν 1 + a 2 h2], a = +1 Fierz-Pauli a = 0 (K.S., 2005, hepth/ , ) 1 d 1 + a 2 1, 2 d d a 1,
25 4 1 2 a 1, a = 1 g 1 2 2
26 BD L m (a i ) = m2 K 2λ(a i) ( g ) 1+ai, L m = L m (a i ) = m 2 K 2λ(a i) ( ) 1+ai g, i i λ(a i ) a i λ(a i ) λ(a i ) L m = m2 K 2 da λ(a) ( ) 1+a g, (14)
27 1 η µν + Kh νν L (2) m = L (1) m = da 1 (1 + a)m2 2 K λ(a)h, da 1 ] 4 (1 + a)m2 λ(a) [(a + 1)h 2 h µν η µν, λ(a) da (1 + a)λ(a) = 0,
28 1 da (1 + a)λ(a) 0, da (1 + a) 2 λ(a) = 2 da (1 + a)λ(a), Firez-Pauli ( [ 1 2, 1) )
29 2 a = ā λ(a) = λ 0 e 1 (a ā) 2 2 σ 2, 2πσ L m = + da m2 λ 0 K 2 e 1 (a ā) 2 ( ) 2 1+a σ 2 g 2πσ = m2 ( ) 1+ā 1 K 2λ 0 g e2 (σ log g) 2 g Liouville ( )
30 3 λ(a) δ a i Riemann λ(a) λ(a) = λ i δ(a a i ) + λ(a), i a i = 0 Fierz-Pauli Kimura, Hamamoto ASG mix
31 4 y a = my y 5 λ(y) 5 ( ) 5 G MN = ( ( g) α 0 0 ( g) β g µν ),
32 Friedman Friedman L m = 1 da λ(a ) ( ) 1+a g, κ δl m = 1 κ da 1 2 (1 + a )λ(a ) ( g ) a gg µν δg µν, L = da 1 2 (1 + a )λ(a ) ( g ) a
33 FLRW ( ) ds 2 = c 2 dt 2 + a(t) 2 dσ 2, dσ 2 = dr2 1 Kr 2 + r2 (dθ 2 + sin 2 θdφ 2 ), a Hubble H = ȧ a, Friedman H 2 = 8πG 3 ρ c2 K a 2 + c2 Λ 3 c2 3 L, 3H 2 + 2Ḣ = 8πG c 2 P c2 K a 2 + c2 λ c 2 L,
34 ä a = 4πG 3 ( ρ + 3P ) c 2 + c2 3 Λ c2 3 L, w = P ρc 2, Ḣ = 4πG(1 + w)ρ + c2 K a 2, Λ L
35 FLRW g = a 3 r2 sin θ, 1 Kr 2 L = 1 2 (1 + a )λ( g) a = 1 ( 2 (1 + a )λ a 3 r2 sin θ 1 Kr 2 ) a, a = 1 Fierz-Pauli Friedman a 3 Gauss
36 enhance
37 6 BD 1 FP BD residual Deser-Waldron h µν = h µν + ( µ ν g µν ) Φ, Φ = 0 Einstein Gravity Weyl Einstein Dirac-Utiyama-Freund Stueckelberg 2
38 Weyl Stueckelberg φ a g µν = η ab µ φ a ν φ b + KH µν, FP(Feddeev-Popov) (derham ) Izawa BRS FP i C µν (2η ab µ φ a ν C b ), (derham )
39 Izawa s Procedure Stueckelberg (φ ) g µν = η ab µ φ a ν φ b + Kh µν, Stueckelberg φ a Izawa s procedure (g µν, θ λ ) (g µν = g µν (η ab µ φ a ν φ b + Kh µν ), φ a ) BRS BRS topological field theory δg µν = 0, δh µν = C µν, δφ a = C a, δ C µν = ib µν,
40 Stueckelberg Gauge fixing FP ghost terms ] iδ[ Cµν (η ab µ φ a ν φ b + Kh µν ) B µν (η ab µ φ a ν φ b + Kh µν ) = i C µν (2η ab µ φ a ν C b µ θ ν + ν θ µ ), B µν FP ( ) Weyl K Dirac-Utitama-Feund Weyl φ κ
41 λ(a) Einstein Weyl try-and-error ( 5 ) Friedman Ḣ Weyl Stuecklberg ( )
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Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x
7 7.1 7.1.1 Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 3 )=(x 0, x )=(ct, x ) (7.3) E/c ct K = E mc 2 (7.4)
Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m
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