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A third type of transformation boosting and rotating both the particle fields and the coefficients within a fixed inertial frame can also be defined [5]. It is distinct from the others and is sometimes called an (inverse) active Lorentz transformation.
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One obstacle is that the dispersion relation is a fourth-order polynomial in \°. The expressions for the eigenenergies typi-cally contain quartic roots involving X and the coefficients for Lorentz violation. A fourth-order term can therefore generate a first-order energy correction. Another difficulty is that the momentum is a variable. For example, even if the momentum component pz in the z direction occurs only at second order, when large it may dominate other first-order terms.
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84988741133
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Note that what we are calling the light cone may be unrelated to the properties of light in the present case. In the standardmodel extension, the appropriate cone is defined by the behavior of observables under observer Lorentz transformations. This is determined by the spontaneous nature of the Lorentz violation, which preserves the Minkowski metric associated with the fundamental theory at the Planck scale. Nonetheless, for simplicity we keep the terminology light cone. Note also that throughout this paper the dispersion relations are in momentum space, so all references are to the energy-momentum light cone rather than the spacetime light cone.
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53
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84988781630
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note
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The coefficients c in the Lagrangian (1) are defined to be traceless, so a model with only a nonzero COQ might appear inappropriate. However, the effect of a nonzero trace in cbv is merely to scale the normalization of the Dirac field [5]. For example, a model with a nonzero foo=c describes physics similar to a model with cw=3c/4, Cjj=c/4, j =-1,2,3. The former choice has been adopted in the text to simplify the presentation.
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If COQ is positive, the issues with stability can be eliminated by multiplication with the exponential factor exp(-cm\l) instead. Thus, consistency in the cm model can be achieved through exponential suppression of this type only for one sign of ce at a time. Note that it is common to find field theories in which consistency allows only a particular range for a parameter in the Lagrangian. For example, stability requires that the squared-mass term in a scalar field theory have only one sign.
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