Spacetime foam and modified dispersion relations

Transcription

1 Institute for Theoretical Physics Karlsruhe Institute of Technology Workshop Bad Liebenzell, 2012

2 Objective Study how a Lorentz-invariant model of spacetime foam modify the propagation of particles

3 Spacetime foam Quantum mechanics = quantum fluctuations General relativity: t E ħ 2 Energy Geometry Quantum gravity (?) = spacetime foam: quantum fluctuations of the geometry and topology of spacetime

4 Spacetime foam Scale of the fluctuations is expected to be the order of the Planck lenght δl m We take into account only topological fluctuations (defects, wormholes,...) We consider defects to be point-like λ photon δl

5 CPT Anomaly 1 Chiral gauge theory e.g. SO(10) (SO(10) SU(3) SU(2) U(1)) A single static topological defect (linear defect, wormhole) CPT anomaly In the U(1) subgroup of electromagnetism the anomalous term turns out to be S CP T = 1 d 4 x f M (x, A µ )ɛ µνρσ F µν F ρσ (1) 32π 1 F. R. Klinkhamer and C. Rupp, Space-time foam, CPT anomaly, and photon propagation, Phys. Rev. D 70 (2004) [hep-th/ ].

6 CPT Anomaly For a large number of defect no exact calculation is practicable We assume the abelian anomalies to add up incoherently into a background field: g(x) = λ n ɛ n h(x x n ) ɛ n = ±1 = g(x) 0, h(x x n ): anomalous contribution from the defect at x n The action for the photon field is S Aµ = 1 } d 4 x {F µν F µν + g(x)ɛ αβγδ F αβ F γδ 4 (2)

7 Sprinkling Sprinkling is described as a Poisson process: Divide spacetime into small boxes of volume V Place a sprinkled point into each box with probability P = ρ V The limit V 0 corresponds to the Poisson process The result is a Poisson distribution of points into spacetime: P n (V ) = (ρv )n e ρv n! P n (V ) is the probability to find n points into the 4-volume V The mean value is n(v ) = n n P n (V ) = ρ V = ρ represents the density of sprinkled points

8 Sprinkling The Poisson process depends only on the spacetime volume V = it is Lorentz invariant It has been proved that even the individual realizations of the process are Lorentz invariant 2 Lorentz invariance here has the following meaning: The discrete set of sprinkled points must not, in and of itself, serve to pick out a preferred reference frame 2 L. Bombelli, J. Henson and R. D. Sorkin, Discreteness without symmetry breaking: A Theorem, Mod. Phys. Lett. A 24 (2009) 2579 [gr-qc/ ].

9 Sprinkling Example of sprinkling in 2d Minkowski spacetime: (a) β = 0 (b) β = 0.7 Figure: A sprinkling of points as it looks in two different inertial frames The mean density ρ is the same in both frames

10 Sprinkling A not-invariant distribution: (a) β = 0 (b) β = 0.7 Figure: A regular distribution of points as it looks in two different inertial frames ρ is uniform in the first case but not in the second

11 Model Our model is based on the effective action: { S = d 4 x 1 4 F µνf µν + m2 ( 0 µ φ µ φ m 2 2 1φ 2) + } (3) +φ ɛ n δ 4 (x x n ) λ 4 φɛαβγδ F αβ F γδ n=1 the topological point-like defects are represented by delta functions the interaction between defects and photons is mediated by a scalar field

12 Model Solving the scalar field equation (for λ 1) we obtain the photon action S = 1 } d 4 x {F µν F µν + g(x)ɛ αβγδ F αβ F γδ 4 in the anomalous term the factor g(x) is ( d 4 k g(x) = (2π) 4 h(k) ) ɛ n e ikxn n e ikx where h(k) = λ φ (k) = λ m 2 0 (k2 m iɛ)

13 Model The distribution of defects enters in the product g(q)g(p): g(q)g(p) = λ 2 h(q)h(p) e i(q+p)xn + ɛ n ɛ m e iqxn e ipxm n n m the sum over n m averages to zero, while e i(q+p)xn ρ d 4 x e i(q+p)x = (2π) 4 ρ δ 4 (q + p) n sprinkling ensures that dn = ρd 4 x independently of the frame = g(q)g(p) = (2π) 4 λ 2 ρ h(q) 2 δ 4 (q + p) (4)

14 Calculations Photon propagator, perturbative expansion: Ω A µ (a)a ν (b) Ω = 0 A µ (a)a ν (b) 0 i d 4 x 0 A µ (a)a ν (b)h int (x) 0 d 4 x d 4 y 0 A µ (a)a ν (b)h int (x)h int (y) (5) H int (x) = L int (x) = 1 2 g(x)εαβρσ α A β (x) ρ A σ (x) Ω A µ (a)a ν (b) Ω = O(0) + O(λ) + O(λ 2 ) +... = = D µν (a b) + B µν (a b) + C µν (a b) +... (6)

15 Calculations O(λ 0 ) : D µν (a b) = g µν F (k) F (k) = i k 2 + iɛ O(λ 1 ) : B µν (a b) = 0 ( g(x) = 0) O(λ 2 ) : C µν (a b) = λ 2 Π µν (k) = 3!ρ δ α [γ δρ ηδ β ν] g µβ d 4 k (2π) 4 F (k)π µν (k) e ik(a b) d 4 q (2π) 4 k αk η (k ρ q ρ )(k γ q γ ) 2 φ (q) F (k q)

16 Calculations (Π µν (k)) We apply the following techniques to manipulate Π µν (k) : Passarino-Veltman reduction: ( Π µν (k) = g µν k ) µk ν k 2 Π(k) Π reg (k) = 3ρ 16π 2 m 4 0 Π(k) = 4ρ 1 m 4 0 k 2 + iɛ Dimensional regularization: d 4 q q 2 k 2 (q k) 2 1 (2π) 4 (q 2 m iɛ)2 (k q) 2 + iɛ { 1 ˆε + 4 ( ) 2 ) 3 m2 1 k 2 log m2 1 µ 2 1 m2 1 k 2 log (1 } k2 m 2 iɛ 1 Renormalization (Πren (k) = Π reg (k) Π reg (m 1 )): 3ρ Π ren (k) = 16π 2 m 4 0 { m 2 ( ) 2 ) } 1 k m2 1 k 2 log (1 k2 m 2 iɛ 1 1

17 Modified dispersion relations To the 2nd order the resummed photon propagator is D µν (k) = g µν F (k) ( 1 λ ren Π ren (k) + λ 2 renπ ren (k) ) D µν (k) = ig µν k 2 (1 + λ 2 renπ ren (k)) The dispersion relations of the possible wave modes correspond to the poles of the propagator The dispersion equation is k 2 (1 + λ 2 renπ ren (k)) = 0 (7)

18 Modified dispersion relations, λ real Π ren (k) is regular in k 2 = 0 lim k 2 0 k2 Π ren (k) = 0 one solution is the standard dispersion relation k 2 (1 + λ 2 renπ ren (k)) = 0 has no other physical solutions The only possible dispersion relation is the conventional one, k 2 = 0 The foam background does not introduce any modification

19 Modified dispersion relations, λ imaginary λ iλ H int (x) H int (x) = ih int(x) H int(x) = i 2 g(x)εαβρσ α A β (x) ρ A σ (x) The new Hamiltonian is not more Hermitian, but it is still PT symmetric (PT H = H) It has been shown 3 that non-hermitian but PT -symmetric Hamiltonians can coherently describe physical systems The dispersion equation is now k 2 (1 λ 2 renπ ren (k)) = 0 3 C. M. Bender, S. Boettcher and P. Meisinger, PT symmetric quantum mechanics, J. Math. Phys. 40 (1999) 2201 [quant-ph/ ].

20 Modified dispersion relations, λ imaginary In this case a second physically acceptable solution appears: k 2 = α(γ)m 2 1 (8) where γ λ 2 ρ, α [0, 1] Α Γ Γ Figure: Several data points for the behaviour of α(γ)

21 Modified dispersion relations, percolative foam Does α(γ) describe a phase transition? { k 2 = 0 γ < γ c k 2 = α(γ)m 2 1 γ > γ c (9) γ ρ suggests a relation with percolation phase transition Percolation: Percolation theory studies the formation and properties of clusters of objects randomly distributed in space. It exhibits a phase transition: For densities smaller than a critical value ρc there are only clusters of finite size For densities larger than ρ c clusters of infinite size also appear

22 Modified dispersion relations, percolative foam If there is a phase transition then α(γ) is the order parameter = α(γ) (1 γ c /γ) β (10) The critical exponent in 4D percolation is β = Α Γ Γ Figure: α(γ) interpolated by Eq. (10) with β = 0.64

23 Summary We studied the propagation of photons through a Lorentz invariant foam of topological point-like defects The topological anomaly is encoded in the anomalous term L CP T = 1 4 g(x)ɛαβγδ F αβ F γδ Lorentz invariance is garanteed by the sprinkling process We found that: For a real coupling constant the foam of defects does not introduce any modification For an imaginary coupling a photon mass seems to emerge m photon = α(γ) 1/2 m 1 The behaviour of the photon mass respect to the density of defects suggests a relation with the percolation phase transition, but this issue deserves further studies