aa r X i v : . [ h e p - ph ] J u l Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13) LORENTZ VIOLATION IN AUNIFORM GRAVITATIONAL FIELD
Y. BONDER
Physics Department, Indiana UniversityBloomington, IN 47405, USAE-mail: [email protected](Dated: IUHET 579, July 2013)
We present a method to calculate the nonrelativistic hamiltonian for the min-imal Standard-Model Extension matter sector in a uniform gravitational field.The resulting hamiltonian coincides with earlier results in the correspondinglimits but it also includes spin-dependent terms that were previously unknown.The phenomenology associated with this hamiltonian is briefly discussed.
Lorentz invariance lies at the core of our current conception of physics.Thus, it seems necessary to test empirically Lorentz invariance. TheStandard-Model Extension (SME) is a framework that includes the Stan-dard Model and General Relativity plus all possible Lorentz violatingterms, and it is widely used as a guide to search and parametrize Lorentzviolation. As was pointed out by Kosteleck´y, gravity couplings allow accessto some SME coefficients that are otherwise unobservable. Thus, to exploreall possible ways Lorentz violation could manifest, one needs to study thematter-gravity SME sector.Kosteleck´y and Tasson studied the matter-gravity SME sector andtheir analysis led to bounds on some previously unmeasured SME coef-ficients. However, in Ref. 3 they do not consider the SME coefficientsassociated with spin. Our proposal is to study the SME matter-gravity sec-tor with a different — and complementary — approach. This allows us toobtain the nonrelativistic hamiltonian for the free Dirac fermion minimalSME sector in a uniform newtonian gravitational potential, including spineffects.We now outline our work hypothesis. The fermions are assumed to betest particles, namely, the spacetime curvature generated by these parti-cles is neglected. Moreover, the background spacetime is chosen to be the roceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13) flat (and torsion-free) spacetime but as seen by a uniformly acceleratedobserver. We also assume that the observer’s acceleration can be identifiedwith a uniform newtonian gravitational potential Φ. Note that, in the situa-tion at hand, the no-go theorem precluding us from having explicit Lorentzviolations in curved spacetimes does not apply, and it is consistent to con-sider explicit Lorentz violations. This allows us to ignore the mechanismsfrom which the Lorentz violations would spontaneously emerge, simplifyingconsiderably our analysis. Furthermore, we require the SME coefficients tohave vanishing covariant derivatives, which is equivalent to the conditionthat the coefficients are constant for an inertial observer.To construct the hamiltonian we use, as a starting point, the actionof the corresponding SME sector in a curved background spacetime. Wework in Fermi-like coordinates associated with a uniformly-accelerated andnonrotating observer, as done in Ref. 5. In principle, the equations of mo-tion can be used to read off a hamiltonian, defined as the generator of timetranslations. However, one must invert a matrix contracted with the wave-function time derivative. To invert this matrix we use the field redefinitionmethod described in Ref. 3 that, in addition, ensures that the resultinghamiltonian is hermitian with respect to the standard inner product ofnonrelativistic quantum mechanics.We are interested in tabletop experiments where the particles are nonrel-ativistic. Therefore, we seek the nonrelativistic hamiltonian to first order inthe SME coefficients. To do so, we perform three Foldy-Wouthuysen trans-formations that decouple the particle and antiparticle degrees of freedom.We also remove all the unphysical terms, namely, those that can be canceledwith unitary transformations. For simplicity we only present here the non-relativistic hamiltonian for the particles (as opposed to the antiparticles)to first order in Φ; the resulting hamiltonian is H NR = A (1 + Φ) + B i σ i (1 + Φ) + 1 m C i (cid:18) p i + 12 Φ p i + 12 p i Φ (cid:19) + 1 m D ij σ j (cid:18) p i + 12 Φ p i + 12 p i Φ (cid:19) + 12 m E ij p i (1 + Φ) p j + 12 m F ijk σ k (cid:18) p i p j + 32 p i Φ p j + 32 p j Φ p i (cid:19) + 1 m G ijk σ k (cid:18) p i p j + 12 p i Φ p j + 12 p j Φ p i (cid:19) , (1)where A = m + ˆ a − m ˆ c − m ˆ e , (2) roceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13) B i = − b i + m ˆ d i + 12 ǫ ijk ( − m ˆ g jk + H jk ) , (3) C i = a i − m ˆ c i − m ˆ c i − me i , (4) D ij = − δ ij ˆ b + mδ ij ˆ d + md ij + ǫ ijk (cid:18)
12 ( ∂ k Φ) + m ˆ g k − ˆ H k (cid:19) , (5) E ij = δ ij − δ ij ˆ c − δ ik δ jl ( c kl + c lk ) , (6) F ijk = δ ij b k − δ jk b i − mδ jk ˆ d i + 12 mǫ klm δ ij ˆ g lm − ǫ ilm δ jk ( m ˆ g lm + H lm ) , (7) G ijk = mδ jk ( ˆ d i + ˆ d i ) + mǫ ikl ˆ g lj − mǫ ikl ˆ g lj . (8)In these last expressions m is the particle’s mass, p i are the componentsof the momentum operator, σ i are the Pauli matrices, ǫ ijk is the totallyantisymmetric tensor with ǫ = 1, and all the indices run from 1 to 3with the convention of summing over repeated indices. The SME coefficientsfor the sector we study are a µ , b µ , c µν , d µν , e µ , f µ g µνρ = − g νµρ and H µν = − H νµ , with Greek indices running from 0 to 3. In the case weanalyze, the SME coefficients get ‘redshifted’ by a factor (1 + Φ) − n , where n is the number of zero indices in the corresponding coefficient. Thus, toget a compact expression we introduce a caret on top of some of the SMEcoefficients with the convention that a coefficient with a caret has beenredshifted. For example, ˆ b ≡ b (1 + Φ) − and ˆ g i ≡ g i (1 + Φ) − .The nonrelativistic hamiltonian we have calculated agrees with that ofRef. 5 when the SME coefficients are set to zero and it coincides withthe result of Ref. 7 in the limit when Φ = 0. In addition, where there isoverlap, it agrees with the nonrelativistic hamiltonian presented in Ref. 3.Even though p i acts on the SME coefficients when taking the adjoint, itis possible to verify that our hamiltonian is hermitian. Moreover, a µ and e µ only appear in the combination a µ − me µ and the antisymmetric partof d µν only shows up in H µν + mε µνρσ d ρσ / ε µνρσ is the spacetimevolume form. Also, the antisymmetric part of c µν and f µ do not enter intoour hamiltonian. These facts are consistent with the analysis sketched inRef. 2 regarding the freedom to redefine the fermionic field.Note that our hamiltonian is not invariant if we add a constant to Φ.However, it is possible to see that, in the coordinates we work, the pointwhere Φ = 0 has physical meaning: it is where the observer/detector is roceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13) located. Thus, adding a constant to Φ amounts to relocating the detectorat a different height and, in order to verify that the physics is invariantunder such transformation, one needs to take into the account the redshifton the detector. In addition, the gravitational effects in the hamiltonian(1) are only relevant for experiments where the particles probe regionswith different Φ. Thus, the best candidates to search for the effects of thishamiltonian are interferometry experiments like those of Refs. 8. Also, thenonrelativistic hamiltonian for antiparticles has been calculated and it canbe obtained from equation (1) with the replacements given in Ref. 7 thatlink the particle and antiparticle hamiltonian in the nongravitational case.Furthermore, since each component of the SME coefficients gets redshiftedin a different way, our result suggests that by doing experiments at severalheights, it should be possible to disentangle the bounds that are usuallyplaced on the linear combinations of these coefficients.To summarize, we have obtained the nonrelativistic hamiltonian for theminimal matter SME sector in the presence of a uniform newtonian grav-itational potential. The spin-dependent terms of this hamiltonian are pre-sented here for the first time, and may lead to new experiments that willallow us to keep testing Lorentz invariance. Acknowledgments
I wish to thank Alan Kosteleck´y for many helpful discussions. This work wassupported by the Department of Energy under grant DE-FG02-13ER42002and by the Indiana University Center for Spacetime Symmetries.
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