Special Relativity from Soft Gravitons

Transcription

1 Special Relativity from Soft Gravitons Mark Hertzberg, Tufts University CosPA, December 14, 2017 with McCullen Sandora, PRD ( )

2 Can the laws of special relativity be violated in principle? Are they exact?

3 Special Relativity in QED QED Spin 1 Photons Coupled to Spin 1/2 Electrons

4 Special Relativity in QED QED L = 1 4 F µ µ F + (i µ D µ m)

5 Special Relativity in QED QED L = 1 4 F µ µ F + (i µ D µ m) { µ, } =2 µ µ = c c c 2 1 C A E 2 q = c 2 q 2 E 2 = c 2 p 2 + m 2 c 4

6 Violate Special Relativity in QED QED L = 1 4 F µ µ F + (i µ e D µ m) { µ e, e } =2 e µ µ = c c c 2 1 C A µ e = c 2 e c 2 e c 2 e 1 C A E 2 q = c 2 q 2 E 2 = c 2 e p 2 + m 2 c 4 e

7 Violate Special Relativity in QED c e 6= c Cherenkov radiation in vacuum (Related processes constrain faster than light neutrinos) (Cohen, Glashow)

8 Violate Special Relativity in Standard Model Colladay and Kostelecky 1998 Coleman and Glashow Lorentz violating couplings (CPT even)

9 Violate Special Relativity in Standard Model - Local - Causal - Unitary - Renormalizable (EFT) - No vacuum instability - No gauge anomalies - Same # degrees of freedom - Obeys laws of thermodynamics - (Gauge invariant) -

10 More Violations of Special Relativity L = 1 4 F µ µ F + X n n(i µ nd µ m n ) n µ! µ ( ) i nd i! f n ( i nd i ) E 2 q = K 1 (q) E 2 n = K n (p n ) K 1 (q) =c 2 q 2 + d q K n (p) =c 2 n p 2 + m 2 nc 4 n + d n p

11 In this talk Principles: Locality (and Rot. + Trans. Invariance)

12 In this talk Principles: Locality (and Rot. + Trans. Invariance) Apply to Spin 2

13 In this talk Principles: Locality (and Rot. + Trans. Invariance) Apply to Spin 2 Special Relativity E 2 = p 2 c 2 + m 2 c 4

14 Particle Spin Rotation invariance organizes particles by spin Spin s =0, 1 2, 1, 3 2, 2,...

15 Particle Spin Rotation invariance organizes particles by spin Spin s =0, 1 2, 1, 3 2, 2,... Massive s z = s, s +1,...,s 1,s Massless h = ±s

16 Particle Spin Rotation invariance organizes particles by spin Spin s =0, 1 2, 1, 3 2, 2,... Massive s z = s, s +1,...,s 1,s Massless h = ±s

17 Massless Spin 1 and Spin 2 State: hx i = (q) e iq x

18 Massless Spin 1 and Spin 2 State: hx i = (q) e iq x Spin 1 i, i q i =0 (Transverse)

19 Massless Spin 1 and Spin 2 State: hx i = (q) e iq x Spin 1 i, i q i =0 (Transverse) Spin 2 ij, ij q i =0, ii =0 (TT) Manifest rotation invariance (no need for gauge invariance for manifest symmetry)

20 Propagator for Spin 1 and Spin 2 Spin 1 i j G ij = E 2 ij q i q j q 2 K 1 (q)

21 Propagator for Spin 1 and Spin 2 Spin 1 i j G ij = E 2 ij q i q j q 2 K 1 (q) Spin 2 ij kl G ijkl = ij q i q j q 2 E 2 kl q k q k q 2 K 2 (q) (j $ k, l)

22 Action from Tree-Exchange S = Z d 4 q (2 ) 4 " J i (q) q i q j ij q 2 J j! 2 (q)+ (q) (q) K 1 (q) L 1 (q) # Dynamical Non-dynamical i spin 1 j spin 0 Magnetic+Electric (Non-Local) Electric [Coulombic] (Non-Local)

23 Locality Principle: No instantaneous action at a distance

24 Locality Principle: No instantaneous action at a distance Turn on sources at t=0 d p dt p x n(t = 0) = 0 (p 2) Acceleration Time

25 Locality S = Z d 4 q (2 ) 4 " J i (q) q i q j ij q 2 J j! 2 (q)+ (q) (q) K 1 (q) L 1 (q) # Constitutive relation q i J i (q) =M 1 (q)! (q)

26 Locality S = Z d 4 q (2 ) 4 " J i (q) q i q j ij q 2 J j! 2 (q)+ (q) (q) K 1 (q) L 1 (q) # Constitutive relation q i J i (q) =M 1 (q)! (q) d p dt p x n(t = 0) = 0 (p 2) =) Z d 4 q S = (2 ) 4 " X Poly p ( q 2 # )! p p

27 Locality S = Z d 4 q (2 ) 4 " J i (q) q i q j ij q 2 J j! 2 (q)+ (q) (q) K 1 (q) L 1 (q) # Constitutive relation q i J i (q) =M 1 (q)! (q) K 1 (q) =c 2 q 2 + q 4 P a ( q 2 ) L 1 (q) 1 = 1 q 2 + P b( q 2 ) (locality) M 1 (q) 2 =1+ q 2 P c ( q 2 )

28 Local Lagrangian for Photon Fields Field: ( i (q)â q + i (q)â q)/ p 2E q! A i (x) L = 1 2 Ȧ r A K1( ) r A + A J L 1( ) + Ȧ M 1( )r + (r A) 2 Not gauge invariant

29 Local Lagrangian for Photon Fields Field: ( i (q)â q + i (q)â q)/ p 2E q! A i (x) L = 1 2 Ȧ r A K1( ) r A + A J L 1( ) + Ȧ M 1( )r + (r A) 2 Not gauge invariant K 1 = c 2 + O( 2 ), L 1 = + O( 2 ), M 1 =1+O( )

30 Soft Gauge Invariance for Spin 1 E B L = 1 2 Ȧ + r 2 c 2 2 r A 2 + A J + O( 2 ) 4-component vector: A µ (, A) Locality =) A µ A µ µ (slowly varying ) µ (q)! µ (q)+q µ (soft )

31 Soft Gauge Invariance for Spin 2

32 Soft Gauge Invariance for Spin 2

33 Soft Gauge Invariance for Spin 2 G ijkl = ij q i q j q 2 E 2 kl q k q k q 2 K 2 (q) (j $ k, l) 2 types of non-localities Add non-dynamical h 00 (Newton) h 0i

34 Soft Gauge Invariance for Spin 2 G ijkl = ij q i q j q 2 E 2 kl q k q k q 2 K 2 (q) (j $ k, l) 2 types of non-localities Add non-dynamical h 00 (Newton) h 0i V (r) 1 r +... Find K 2 (q) =c 2 g q Locality =) h µ h µ µ µ (slowly varying µ ) µ (q)! µ (q)+q µ + q µ (soft µ )

35 Soft Gauge Invariance for Spin 1 and 2 Soft gauge invariance is required to ensure that the nondynamical fields are mixed with the propagating degrees of freedom, such that long range forces inherit the finite speed of propagation of the spin 1 or spin 2 particles

36 (Soft) Gauge Invariant Spin 2 Theories Difficult. Simple attempts fail Leading to many questions: What restrictions are placed on E? n 2 = K n (p n ) What if we include fermions and gauge/vector bosons? What constraints apply to E? q 2 = K q (q) Is the equivalence principle still required for consistency? What objects can we couple to? Need systematic analysis

37 Soft Graviton Scattering

38 Soft Graviton Scattering M N+1 = M N + X f " X i µ (q) T µ i (p i ) (E i E q ) 2 Ki (p i q) µ (q) T µ f (p f ) (E f + E q ) 2 Kf (p f + q) #

39 Soft Graviton Scattering M N+1 = M N Denominator: + X f " X i µ (q) T µ i (p i ) (E i E q ) 2 Ki (p i q) µ (q) T µ f (p f ) (E f + E q ) 2 Kf (p f + q) (E n E q ) 2 Kn (p n q) = 2 q n (p n ) #! q (E q, q) n (p n ) E n, 1 K p n

40 Soft Graviton Scattering M N+1 = M N " + X f X i µ (q) T µ i (p i ) 2 q i (p i) µ (q) T µ f (p f ) 2 q f (p f ) # Denominator: (E n E q ) 2 Kn (p n q) = 2 q n (p n )! q (E q, q) n (p n ) E n, 1 K p n

41 Unitarity and Locality Constraint µ (q)! µ (q)+q µ + q µ (soft µ )

42 Unitarity and Locality Constraint µ (q)! µ (q)+q µ + q µ (soft µ ) X i q µ T µ i (p i ) q i (p i) = X f q µ T µ f (p f ) q f (p f )

43 Unitarity and Locality Constraint µ (q)! µ (q)+q µ + q µ (soft µ ) X i q µ T µ i (p i ) q i (p i) = X f q µ T µ f (p f ) q f (p f ) Cancel graviton momenta T µ n (p n ) / µ n(p n )

44 Unitarity and Locality Constraint µ (q)! µ (q)+q µ + q µ (soft µ ) X i q µ T µ i (p i ) q i (p i) = X f q µ T µ f (p f ) q f (p f ) Cancel graviton momenta T µ n (p n )=g n (E n ) µ n(p n ) n(p n )

45 Unitarity and Locality Constraint µ (q)! µ (q)+q µ + q µ (soft µ ) X i q µ T µ i (p i ) q i (p i) = X f q µ T µ f (p f ) q f (p f ) Cancel graviton momenta X i T µ n (p n )=g n (E n ) µ n(p n ) n(p n ) g i (E i ) i (p i )= X f g f (E f ) f (p f )

46 Conservation Laws =0 X g i (E i ) E i = X f g f (E f ) E f i (Grav. Coupling) g n (E n )=apple + Q n E n

47 Conservation Laws =0 X g i (E i ) E i = X f g f (E f ) E f i (Grav. Coupling) g n (E n )=apple + Q n E n

48 Conservation Laws =0 X g i (E i ) E i = X f g f (E f ) E f i (Grav. Coupling) g n (E n )=apple + Q n E n = i X i g i (E i K p i = X f g f (E f K p f g n (E n K p n = a p n

49 Conservation Laws =0 X g i (E i ) E i = X f g f (E f ) E f i (Grav. Coupling) g n (E n )=apple + Q n E n = i X i g i (E i K p i = X f g f (E f K p f g n (E n K p n = a p n (Disp. Relation) apple E 2 n +2Q n E n = a 2 p n 2 + b n

50 Dispersion Relation apple E 2 n +2Q n E n = apple c 2 g p n 2 + b n Complete the square: E n! E n Q n apple,

51 Dispersion Relation apple E 2 n +2Q n E n = apple c 2 g p n 2 + b n Complete the square: E n! E n Q n apple, E 2 n = p n 2 c 2 g + m 2 n c 4 g Special Relativistic Dispersion Relation

52 Some Consequences - Lorentz violating Standard Model - Lorentz violating approaches to quantum gravity - Doubly/deformed special relativity - Lorentz violating alternatives to inflation Brings into doubt the viability of these models

53 Conclusions - Locality requires soft gauge invariance - For spin, special relativity is easily violated in principle apple 1 - For spin 2, special relativity is difficult to violate: - The relativistic energy-momentum relation must be exact - The leading interactions must be Lorentz invariant - It remains to be proven if all higher order interactions are too