Finsler Structures and The Standard Model Extension

Transcription

1 Finsler Structures and The Standard Model Extension Don Colladay New College of Florida Talk presented at Miami 2011 work done in collaboration with Patrick McDonald

2 Overview of Talk Modified dispersion relations and Standard Model Extension Finsler geometry Some previously solved cases New Finsler structures from momentum-dependent couplings

3 Modified Dispersion relations and Standard Model Extension Standard Model Extension (SME): Framework within the context of conventional quantum field theory in which arbitrary background fields are coupled to standard model fields to parametrize possible Lorentz-violating processes Can assume origin of terms is some type of spontaneous symmetry breaking for a vector or tensor field, for example < B µ > ψγ 5 γ µ ψ Usually assume gauge invariance, power-counting renormalizability, and translational invariance for simplicity DC, Kostelecký PRD 1997, 1998.

4 Fermion sectors have following structure: L = i 2 ψγ µ µ ψ ψmψ Γ ν = γ ν + c µν γ µ + d µν γ 5 γ µ + e ν + if ν γ gλµν σ λµ, M = m + a µ γ µ + b µ γ 5 γ µ H µνσ µν. Dispersion relation results from vanishing determinant det(γ ν p ν M) = 0. General fourth-order polynomial roots give dispersion relation

5 General expression is of form p αp2 0 + βp 0 + γ = 0 (cubic term vanishes) where α, β and γ depend on p Quadradic expression for energy results when β = 0 Factors in absence of Lorentz violation to (p 2 0 p2 m 2 ) 2 = 0

6 Additional motivation for investigating corrections to dispersion relations is OPERA neutrino experiment (v c)/c = (2.37 ± 0.32(stat.) (sys.)) σ effect If the effect holds up, search for direction/boost dependence can help identify type of correction term required... OPERA collaboration, arxiv:

7 Finsler spaces Can be thought of as Rieman geometry without the quadradic restriction on the line interval Finsler, 1918 (thesis work) Start with distance functional F (x, u) depending on manifold location x and velocity vector u (in the tangent space). Metric on space determined by differentiation of F : g ij = u i u j ( 1 2 F 2 )

8 Specific examples: Euclidean space: F = i u 2 i g ij = δ ij Riemannian space: F = i,j u i u j r ij (x) g ij (x) = r ij (x)

9 Randers space: requires 1-form a i (x) F = u 2 i + a u Randers metric is more complicated: g ij (x, u) = (1 + a û)δ ij + a i û j + a j û i (a û)û i û j + a i a j where û is a unit vector exhibiting homogeneity property determinant can be computed as det g = (1 + a û) n+1 which is positive definite provided a < 1

10 General properties of Finsler F (x, u): Positive definite (otherwise it is called psudo-finsler...) Homogeneous function of u: F (x, λu) = λf (x, u) for λ 0 Ensures reparametrization invariance of F (γ(t), γ (t))dt Strong convexity (usually assumed...)

11 Finsler structure associated with b µ - term (M = γ 5 b µ γ µ ) Kostelecký and Russell, PLB 2010; Kostelecký PLB Dispersion relation for b µ (in Minkowski space notation) (p 2 m 2 + b 2 ) 2 4(b p) 2 + 4m 2 b 2 = 0 Velocity defined using implicit differentiation u i = p 0 u 0 p i Lagrangian given by Legendre transform L = u p

12 Can combine these five equations into an octic polynomial for L that factors (conveniently!) One of solutions is L = m u 2 (b u) 2 b 2 u 2 Gives psuedo-finsler function, can Wick rotate ( u i y) to get Finsler version F ± = y 2 ± b 2 y 2 (b y) 2 Note remarkably simple structure even though H( p) is complicated solution to fourth-order, non-factorizable polynomial

13 F ± = y 2 ± b 2 y 2 (b y) 2 This Finsler space is not a Randers space, despite being constructed using a single one-form (can be verified by computing nonvanishing Matsumoto torsion...) The calculation works with arbitrary Riemannian background metric replacing the Minkowski metric, so it can be applied to curved spacetimes as well

14 Surface F ± = constant for 3D case Nested spheroids collapse to single sphere as b 0

15 Are there other solvable cases? Some special parameter choices allow explicit solution for p 0 ( p) DC, P. McDonald, and D. Mullins JPhysA For example, the g 0ij = ɛ ijk r k term (Γ j = 1 2 goij σ oi ) yields H 2 = m 2 + p 2 (1 ± r ˆp ) 2 Note that H 2 m 2 is a homogeneous function of degree one in p. This implies that p i p i(h2 m 2 ) 1/2 = (H 2 m 2 ) 1/2.

16 Defining v i = H/ p i as usual yields the simple relation LH = m 2. Elimination of the momentum variables in terms of v yields a quadratic equation for the quantity C r = H2 H 2 m 2 = [ ] 2 v 2 ( v r) 2 ± v r v 4, that can be solved to yield the lagrangian L = m 1 1 C r.

17 Conversion to Euclidean space converts the lagrangian to a homogeneous function that can be used to define a new Finsler geometry This conversion can be accomplished using the replacement v j i uj u 0, for the three spatial velocity components yielding the parametrization invariant Finsler function F = (u 0 ) 2 + u2 C( u), where C( u) is C( u) = [ 1 (û r) 2 ± û r ] 2.

18 Surface F ± = constant Note loss of convexity as û r 0

19 Resulting metric takes form (spatial components) g ij = (û r)2 D 3 ri r j + 1 [ (D (û r) 2 ] )δ ij ± r i r j, CD where D = û r 1 (û r) 2 and ri = r i (û r)û i is the piece of r perpendicular to u. Metric is homogeneous of degree zero in u. Depends on a single one-form r(x), but space is not Randers (as with the b-case, can be formally checked by calculating nonvanishing Matsumoto torsion...) Metric is singular as û r 0

20 Conclusions Standard Model Extension parameters yield interesting examples of relatively simple Finsler spaces constructed using one-forms that are not simple Randers spaces Solvable momentum-independent couplings seem to be of general bipartite form F s = u 2 ± s ij u i u j Derivative couplings yield Finsler spaces that are not of bipartite form