aa r X i v : . [ h e p - ph ] S e p CLASSICAL LAGRANGE FUNCTIONS FOR THE SME ∗ N. RUSSELL
Physics Department, Northern Michigan UniversityMarquette, MI 49855, USAE-mail: [email protected]
A technique is presented for finding the classical Lagrange function correspond-ing to a given dispersion relation. This allows us to study the classical analogueof the Standard-Model Extension. Developments are discussed.
1. Introduction
The Standard-Model Extension, or SME, is a framework for Lorentz vi-olation that is set up in the context of effective field theory in flat andcurved spacetime. In the fermion sector, the SME is established by system-atically adding Lorentz-breaking operators of increasing mass dimension tothe Lagrange density. For the minimal SME, the dimension-three operatorshave coefficients a ν , b ν , H µν , and the dimension-four operators have coeffi-cients c µν , d ν , e ν , f ν , and g λµν . Numerous experiments have placed boundson these coefficients and ones from the other sectors of the SME. The dispersion relation F ( m, a, b, c, d, e, f, g, H ; p ) = 0 (1)for a mass m fermion propagating in these background fields is known. To find it, one seeks plane-wave solutions to the modified Dirac equation.In the resulting equation, a 4 × b ν = 0, and only c µν = 0, ittakes the forms 0 = ( − p + b + m ) − b · p ) + 4 b p , (2)0 = p ( δ + 2 c + c T c ) p − m . (3) ∗ Presented at the Fifth Meeting on CPT and Lorentz Symmetry, Bloomington, Indiana,June 28-July 2, 2010
In our conventions, light has unit speed and the metric has diagonal entries(+1 , − , − , − p ν . In particular, note thatEq. (2) is quartic, and that Eq. (3) is quadratic. The full function F involvesnumerous contractions among the SME coefficients, and factorization is notstraightforward. Dispersion relations also exist for classical point particles. For a conven-tional particle of mass m , the classical Lagrange function is L = − m √ u ν u ν . (4)The canonical momentum is p ν = − ∂L∂u ν = mu ν √ u · u , (5)and the corresponding dispersion relation is found by eliminating the four-velocity: p = m . (6)This proceedings contribution addresses the question of finding the clas-sical Lagrange function corresponding to the quantum-mechanics-derivedSME dispersion relation (1). Effectively, this means we are seeking a methodof constructing the classical Lagrange function from a given dispersion re-lation.
2. Finding the Lagrange function
We seek a Lagrange function L that yields the spacetime coordinates x ν fora particle of mass m as a function of a curve parameter λ . To meet the basicSME requirement of conserved energy and conserved linear momentum, L cannot depend on time x or position ( x , x , x ). So, it can take the form L = L ( u ν , λ ), where u ν ≡ dx ν /dλ . The trajectory is found by extremizationof the action, which is the integral of L ( u ν , λ ) dλ along a path. To ensurethat the result is independent of the choice of curve parameter λ , L mustalso have no explicit λ -dependence and must be homogeneous of degree onein u ν . Using Euler’s theorem for homogeneous functions, this implies that L ( u ) = ∂L∂u ν u ν ≡ − p ν u ν . (7)This equation shows that the Lagrange function can be found if the canon-ical momenta p ν ( u ) are known. To establish a match between a classical point particle and the quantum-mechanical analogue of a particle, we require that the classical velocity dx j /dx is the same as the group velocity of a plane-wave packet. Notingthat the classical velocity can be related to the four velocity in the chosencurve parametrization, dx j dx = dx j dλ / dx dλ = u j u , (8)we find the group-velocity condition to be u j u = − ∂p ∂p j , (9)where j is an index for the three cartesian spatial directions. The partialderivative on the right-hand side of Eq. (9) can be evaluated by implicitdifferentiation of the given dispersion relation.We can now state a method for finding the Lagrange function for agiven dispersion relation. The dispersion relation (1) and the group-velocityconditions (9) form a set of four equations in the 8 variables u µ , p ν ; we mustsolve this system for the 4 variables p ν , in terms of the four variables u µ and then substitute into Eq. (7) to get the Lagrange function.With sufficiently small Lorentz breaking, solutions for L must exist be-cause they can only be a perturbation of the conventional ones.An alternative is to consider the set of 5 equations (1), (7), and (9) inthe 9 variables u µ , p ν , and L ; use 4 of them to eliminate p ν , and considerthe resulting polynomial equation in L with u µ -dependent coefficients. Ofthe multiple solutions, only real ones can be relevant, and they correspondvery roughly to particles, antiparticles, and differing spin-like states.
3. The b ν background As an example, we consider the case of the SME b ν background. The pro-cedure discussed above leads to an octic polynomial in L , expressed here infactorized form:0 = (cid:0) − b ( b · u ) + b L − m ( b · u ) (cid:1) × (cid:16) b u − ( b · u ) + ( L + m √ u ) (cid:17) × (cid:16) b u − ( b · u ) + ( L − m √ u ) (cid:17) . (10)For vanishing b ν , the second and third factors give the expected limits of L , which are Eq. (4) and its negative-mass counterpart. We disregard thefirst term as unphysical. The solutions for the positive-mass case are: L = − m √ u ∓ p ( b · u ) − b u . (11) The corresponding canonical momenta are p ν = mu ν √ u ± ( b · u ) b ν − b u ν p ( b · u ) − b u . (12)This reveals that the conserved 3-momentum p j is not collinear with the3-velocity u j , and that the one can be nonzero when the other vanishes.Another property is the dependence of the conserved energy E ≡ p on thedirection of the 3-velocity. This differs from the conventional case, wherethe dependence is only on the magnitude of the 3-velocity.Equation (12) is given in parametrization-independent form. While theproper-time interval along the curve dτ = ( η µν dx µ dx ν ) / can always bechosen as the curve parameter, other choices may be more convenient fordifferent SME terms. In this case, the proper-time choice dλ = dτ leads to u ν u ν = 1, and provides a convenient simplification.
4. Spacetime torsion
The spacetime torsion tensor, denoted T µαβ , has a close relationship tovarious terms in the SME. To appreciate the rich structure underlying its24 independent components, we may draw upon the standard 4 + 4 + 16irreducible decomposition with components A ν , T ν , M λµν . The axial part A ν can be defined by A µ ≡ ǫ αβγµ T αβγ . (13)If torsion is present, this axial component enters the Dirac equation as acoupling to spin. It allows limits to be placed based on experiments seekingspacetime anisotropies and Lorentz violation. Recent work has investigated nonminimally-coupled torsion and its rela-tion to the SME background fields. Using results from experiments testingLorentz symmetry, the first limits were placed on 19 of the 24 independentcomponents of the torsion tensor. These are measured in the standardinertial reference frame used for SME experiments. The experiments in-volved were a dual-maser system at the Harvard-Smithsonian Centerfor Astrophysics, and a spin-polarized torsion pendulum at the Universityof Washington in Seattle.Using the conventions of Ref. 10, the correspondence between axial tor-sion and the b ν background is b ν = − A ν . (14) Exploiting this relationship derived in the quantum-mechanical context, wemay deduce from Eq. (11) the Lagrange function for a classical particle ina constant minimal-torsion background. For a point particle of mass m inMinkowski spacetime, the result is L = − m √ u ∓ p ( A · u ) − A u . (15)We remark that this Lagrange function is valid at all orders in the axialtorsion tensor. We may readily deduce the dispersion relation and canonicalmomentum, and the results are identical to those given in Eqs. (2) and (12),up to factors of − /
5. Discussion
Our main result is a method that, in principle, can generate the classicalLagrange function corresponding to the full minimal-SME dispersion rela-tion. Finding the full Lagrangian is technically difficult because it involvessolving a polynomial of high order. A useful illustrative example is the caseof the b ν background. Using this, we have deduced the Lagrange functionfor a point particle in the presence of minimal torsion. There are manyother avenues for investigation, some of which are discussed elsewhere. References
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