Manifestly diffeomorphism invariant exact RG

Transcription

1 Manifestly diffeomorphism invariant exact RG II Flag Workshop The Quantum & Gravity 06/06/16 Tim Morris, Physics & Astronomy, University of Southampton, UK. TRM & Anthony W.H. Preston, arxiv: , JHEP? (2016)?

2 Manifestly diffeomorphism invariant classical exact RG II Flag Workshop The Quantum & Gravity 06/06/16 Tim Morris, Physics & Astronomy, University of Southampton, UK. TRM & Anthony W.H. Preston, arxiv: , JHEP? (2016)?

3 GR: gμν is a field QM: which must be consistent with QM Effective (quantum) field theory from 29 Gpc to GeV (x10 60 )

4 Even classical GR: back-reaction

5 Even classical GR: back-reaction Universe is not homogeneous! Earth h i ht µ i = T (0) µ hg µ i = g (0) µ g µ = g (0) µ + h µ with hh µ i =0 R (0) µ 1 2 g(0) µ R (0) = applet µ (0) +applet (0) µ apple := 8 G Could this give an effective cosmological const? & answer why now? S. Räsänen, D.J. Schwarz, C. Clarkson, G. Ellis, T. Clifton, T. Buchert, S.R. Green & R.M. Wald; TRM & A.W.H. Preston; N.J. Evans, TRM & M Scott; A.W. Preston;

6 Wilsonian RG q 0!1? S bare L S 0 = D' e S['] = D' 0 e S bare[' 0 ]

7 Wilsonian RG = 8 G Theory space... d 4 x p g g 0 + g 2 ( 2R)+g 4,1 R 2 + g 4,2 R µ R µ + g 6,1 R 3 + g 6,2 RR µ R µ + 1 apple = 1 32 G

8 Wilsonian RG = 8 G Theory space... d 4 x p g g 0 + g 2 ( 2R)+g 4,1 R 2 + g 4,2 R µ R µ + g 6,1 R 3 + g 6,2 RR µ R µ + 1 apple = 1 32 G g 2n, n 4 2n

9 Wilsonian RG Irrelevant Relevant G is irrelevant about Gaussian FP, but effective field theory over 60 orders of magnitude

10 Wilsonian RG Irrelevant Relevant G is irrelevant about Gaussian FP, but effective field theory over 60 orders of magnitude Asymptotic safety: does a non-perturbative FP exist?

11 Wilsonian RG? q 0!1? S bare L S 0 = Dg µ e S[g] = Dg 0µ e S bare[g 0 ]

12 What does q mean? Modes for what metric? Gauge fixing requires background ḡ µ Two metric theory except on shell? Diffeomorphism invariance only on shell? What does on shell mean for Wilsonian RG?

13 Manifestly diffeomorphism invariant classical exact RG = Dg µ e S[g] = Dg 0µ e S bare[g 0 ] TRM & Anthony W.H. Preston, arxiv: , JHEP (2016)

14 Scalar field theory (non-abelian) gauge theory Gravity

15 Kadanoff blocking J.I. Latorre & TRM, Exact scheme independence, JHEP 11 (2000) 004; S. Arnone, TRM & O.J. Rosten, Fields Inst. Commun. 50 (2007) 1 e S['] = D' 0 [' b [' 0 ]] e S bare[' 0 ]. = D' e S['] = D' 0 e S bare[' 0 ]

16 Kadanoff blocking e S['] = D' 0 [' b [' 0 ]] e S bare[' e S['] = x '(x) D' 0 [' b [' 0 e S bare[' 0 ] = x '(x) (x)e S['] = D' e S['] = D' 0 e S bare[' 0 ]

17 Ṡ = S (x) x '(x) (x) x '(x) General exact RG flow eqn

18 Ṡ = S (x) x '(x) (x) x '(x) Wilson/Polchinski flow eqn K.G. Wilson & J. Kogut, Phys. Rep. 12C (1974) 75; J Polchinski, Nucl. Phys. B231 (1984) 269 (x) = 1 2 y (x, y) '(y) = c(p 2 / 2 )/p 2 c(p 2 / 2 ) = S 2Ŝ Ŝ = 1 µ' c µ ' S 1 Ṡ = 1 2 ' ' 2 ' '

19 Ṡ = S (x) x '(x) (x) x '(x) Generalised scalar flow eqn S. Arnone, A. Gatti & TRM, Exact scheme dependence at one loop, JHEP 05 (2002) 059; + O.J. Rosten, at two loops, Phys. Rev. D69 (2004) (x) = 1 2 y (x, y) '(y) = c(p 2 / 2 )/p 2 c(p 2 / 2 ) = S 2Ŝ Ŝ = 1 µ' c µ ' + S 1 Ṡ = 1 2 ' ' 2 ' '

20 Requiring at classical level, S '' = Ŝ'' = p 2 /c Ṡ '' = S '' S'' agrees with =(S '' ) 1 Use this later to determine kernel

21 non-abelian gauge theory Ṡ = 1 2 S A µ { } g 1 { } A µ 2 A µ g A µ Some covariantization e.g. 2 ) 7! ( D 2 ) Gauge invariant flow equation Can solve the equation without fixing the gauge! SU(N) YM, QCD, QED, one loop & two-loop b functions TRM + S. Arnone, Yu. A. Kubyshin, J.F. Tighe, A. Gatti, O. J. Rosten: 29 papers from

22 non-abelian gauge theory D µ µ ia µ A µ =[D µ,!(x)] A R µ = 1/2 A µ =) A R µ = µ! i[a R µ,!] No wavefunction renormalisation. Only g renormalises

23 non-abelian gauge theory Ṡ = 1 2 S S[A](g) = 1 1 g 2 4 tr x A µ { } F µ c 1 D 2 g 1 { } A µ 2 A µ 2 F µ + O(F 3 )+ g A µ Ŝ

24 non-abelian gauge theory Ṡ = 1 2 S S[A](g) = 1 1 g 2 4 tr x A µ { } F µ c 1 D 2 g 1 { } A µ 2 A µ 2 F µ + O(F 3 )+ g A µ g = g 2 S 2Ŝ Ŝ

25 non-abelian gauge theory Ṡ = 1 2 S S[A](g) = 1 1 g 2 4 tr x A µ { } F µ c 1 D 2 g 1 { } A µ 2 A µ 2 F µ + O(F 3 )+ g A µ g = g 2 S 2Ŝ Ŝ S = 1 g 2 S 0 + S 1 + g 2 S 2 +

26 non-abelian gauge theory Ṡ = 1 2 S S[A](g) = 1 1 g 2 4 tr x A µ { } F µ c 1 D 2 g 1 { } A µ 2 A µ 2 F µ + O(F 3 )+ g A µ g = g 2 S 2Ŝ Ŝ S = 1 g 2 S 0 + S 1 + g 2 S 2 + := ġ = 1 g g 5 +

27 classical limit: g->0 Ṡ = 1 2 S A µ { } A µ 0 = S 0 2Ŝ S = 1 g 2 S 0

28 classical limit: g->0 Ṡ = 1 2 S A µ { } A µ Sµ AA =( µ p 2 p µ p ) c 1 p 2 / 2 Ṡµ AA = Sµ AA S AA 0 = S 0 2Ŝ S = 1 g 2 S 0

29 classical limit: g->0 Ṡ = 1 2 S A µ { } A µ Sµ AA =( µ p 2 p µ p ) c 1 p 2 / 2 Ṡµ AA = Sµ AA S AA = c/p 2 but S AA µ = µ p µ p /p 2 0 = S 0 2Ŝ S = 1 g 2 S 0

30 Regularisation Covariant higher derivatives are not enough c -1 ~ D 2(n-1) /L 2(n-1) but S 1 Tr ln D 2n = n Tr ln D 2 Embed in spontaneously broken supergroup Degrees of freedom cancel at high energies A 1 µ B µ A µ = (Parisi-Sourlas SUSY) B µ A 2 µ Massive ~ L fermionic gauge-invariant Pauli-Villars

31 Gravity Ṡ = 1 2 S g µ K µ 1 K µ g 2 g µ g K µ (x, y) = 1 p g (x y) g µ( g ) + jg µ g ( r 2 ) DeWitt parameter j!1 conformal truncation j = 1/4 unimodular

32 Gravity Ṡ = a 0 [S, ]+a 1 [ ] Ṡ = 1 2 S g µ K µ 1 K µ g 2 g µ g K µ (x, y) = 1 p g (x y) g µ( g ) + jg µ g ( r 2 ) DeWitt parameter j!1 conformal truncation j = 1/4 unimodular

33 Gravity Ṡ = a 0 [S, ]+a 1 [ ] apple( ) = 32 G =4/M 2 = apples 2Ŝ S = 1 apple S 0 + S 1 + apples 2 + apple 2 S 3 +

34 Gravity Ṡ = a 0 [S, ]+a 1 [ ] apple( ) = 32 G =4/M 2 = apples 2Ŝ S = 1 apple S 0 + S 1 + apples 2 + apple 2 S appleṡ0 Ṡ1 + appleṡ2 + apple 2 Ṡ apple 2 S 0 + S 2 +2 apples 3 + = 1 apple a 0[S 0,S 0 2Ŝ] 2a 0 [S 0 Ŝ,S 1 ]+a 1 [S 0 2Ŝ]+ apple 2a 0 [S 0 Ŝ,S 2 ] a 0 [S 1,S 1 ]+a 1 [S 1 ] + := apple = 1 2 apple apple 3 +

35 Gravity Ṡ = a 0 [S, ]+a 1 [ ] apple( ) = 32 G =4/M 2 = apples 2Ŝ S = 1 apple S 0 + S 1 + apples 2 + apple 2 S appleṡ0 Ṡ1 + appleṡ2 + apple 2 Ṡ apple 2 S 0 + S 2 +2 apples 3 + = 1 apple a 0[S 0,S 0 2Ŝ] 2a 0 [S 0 Ŝ,S 1 ]+a 1 [S 0 2Ŝ]+ apple 2a 0 [S 0 Ŝ,S 2 ] a 0 [S 1,S 1 ]+a 1 [S 1 ] + apple( ) = apple 2 =4 2 /M 2 := apple = 1 2 apple apple 3 + (apple) = apple =2apple + 1 apple apple 3 +

36 classical Gravity Ṡ = a 0 [S, ] Actions have dimn -2 = S 2Ŝ

37 classical Gravity Ṡ = a 0 [S, ] Actions have dimn -2 = S 2Ŝ = g µ = µ + h µ d p (p)s µ (p)h µ (p)+ 1 d p d q (p + q)s µ (p, q)h µ (p)h (q) d p d q d r (p + q + r)s µ (p, q, r)h µ (p)h (q)h (r)+ 3!

38 classical Gravity Ṡ = a 0 [S, ] Actions have dimn -2 = S 2Ŝ = g µ = µ + h µ d p (p)s µ (p)h µ (p)+ 1 d p d q (p + q)s µ (p, q)h µ (p)h (q) d p d q d r (p + q + r)s µ (p, q, r)h µ (p)h (q)h (r)+ 3! Diffeomorphism invariance is exact! 2p 1µ1 S µ 1 1 µ n n (p 1,,p n )= nx i=2 2i n p 1 2 Sµ 2 2 µ n n (p 1 + p 2,p 3,,p n ) +2p 1 1 ( 2 S µ 2) µ 3 3 µ n n (p 1 + p 2,p 3,,p n ) o

39 classical Gravity Two-point vertex: 1 2 h µ p 2 h µ hp 2 h +2h µ p µ p h 2h µ p µ p h + O(p 4 ) Renormalisation condition: = d 4 x p g g 0 + g 2 ( 2R)+g 4,1 R 2 + g 4,2 R µ R µ + g 6,1 R 3 + g 6,2 RR µ R µ + 1

40 classical Gravity Two-point vertex: 1 2 h µ p 2 h µ hp 2 h +2h µ p µ p h 2h µ p µ p h c 1 (p 2 / 2 ) Ṡ = 1 2 S g µ K µ 1 K µ g 2 g µ g K µ (x, y) = 1 p g (x y) g µ( g ) + jg µ g ( r 2 ) j = 1/2 = c/p 2

41 classical Gravity Two-point vertex: 1 2 h µ p 2 h µ hp 2 h +2h µ p µ p h 2h µ p µ p h c 1 (p 2 / 2 ) p 2 c 1 =1+ p2 2 d 2 ˆL = 2R R µ d( r 2 / 2 )R µ 1 2 Rd( r2 / 2 )R L = ˆL + O(R 3 )

42 classical Gravity 1X X L = a 0 [L, L 2 ˆL] L = g 2i, i O 2i, i i=0 i Ṡ = 1 2 S g µ K µ 1 K µ g 2 g µ g K µ (x, y) = 1 p g (x y) g µ( g ) + jg µ g ( r 2 ) ( r 2 )= 1X k=0 1 k! (k) (0) r 2 k Background independent calculations!

43 classical Gravity L = a 0 [L, L 2 ˆL] L = 1X i=0 X i g 2i, i O 2i, i g 0 + g 2 ( 2R)+g 4,1 R 2 + g 4,2 R µ R µ + g 6,1 R 3 + g 6,2 RR µ R µ + a 0 [O d, 1] = 1 8 (d 4) (0) O d ġ 2 = c0 (0) 2 ĝ0 =) ĝ 0 =0 ġ 0 = c0 (0) 2 g2 0

44 classical Gravity ġ 4,1 R 2 +ġ 4,2 R µ R µ = 2 c0 (0) 2 R 2 2R µ R µ

45 classical Gravity ġ 4,1 R 2 +ġ 4,2 R µ R µ = 2 c0 (0) 2 R 2 2R µ R µ g 4, / 1/ 2 at fixed point =) g 4,1 = c 0 (0)/ 2, g 4,2 = 2c 0 (0)/ 2

46 classical Gravity ġ 4,1 R 2 +ġ 4,2 R µ R µ = 2 c0 (0) 2 R 2 2R µ R µ g 4, / 1/ 2 at fixed point =) g 4,1 = c 0 (0)/ 2, g 4,2 = 2c 0 (0)/ 2 p 2 c 1 =1+ p2 2 d 2 ˆL = 2R R µ d( r 2 / 2 )R µ 1 2 Rd( r2 / 2 )R L = ˆL + O(R 3 ) consistency

47 Gravity Covariant higher derivatives are not enough Extend diffeomorphisms along fermionic directions? x A =(x µ, a ) ds 2 = dx A g AB dx B Degrees of freedom cancel at high energies (Parisi-Sourlas SUSY) 10 g µ +6g ab 16 (g µa = g aµ ) + spontaneous symmetry breaking

48 Manifestly diffeomorphism invariant exact RG Diffeomorphism invariant calculations Background independent calculations

49 Manifestly diffeomorphism invariant classical exact RG Diffeomorphism invariant calculations Background independent calculations

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