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Volumn 63, Issue 6, 2001, Pages 19-

Stability, causality, and Lorentz and [Formula Presented] violation

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EID: 85037874319     PISSN: 15507998     EISSN: 15502368     Source Type: Journal    
DOI: 10.1103/PhysRevD.63.065008     Document Type: Article
Times cited : (52)

References (68)
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    • L. R. Hunter et al., in (Formula presented) and Lorentz Symmetry, edited by V. A. Kostelecký (World Scientific, Singapore, 1999).
    • L. R. Hunter et al., in (Formula presented) and Lorentz Symmetry, edited by V. A. Kostelecký (World Scientific, Singapore, 1999).
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    • R. E. Stoner, in (Formula presented) and Lorentz Symmetry, edited by V. A. Kostelecký (World Scientific, Singapore, 1999).
    • R. E. Stoner, in (Formula presented) and Lorentz Symmetry, edited by V. A. Kostelecký (World Scientific, Singapore, 1999).
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    • Similar results are likely to hold for any SO(2,1) violation in (2+1)-dimensional theories. See, e.g.
    • Similar results are likely to hold for any SO(2,1) violation in (2+1)-dimensional theories. See, e.g.
  • 47
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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
    • 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.
  • 49
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    • One obstacle is that the dispersion relation is a fourth-order polynomial in (Formula presented). The expressions for the eigenenergies typically contain quartic roots involving (Formula presented) 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 (Formula presented) in the z direction occurs only at second order, when large it may dominate other first-order terms.
    • One obstacle is that the dispersion relation is a fourth-order polynomial in (Formula presented). The expressions for the eigenenergies typically contain quartic roots involving (Formula presented) 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 (Formula presented) in the z direction occurs only at second order, when large it may dominate other first-order terms.
  • 51
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    • See, e.g., G. Sansone and J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable, Vol. 1 (Noordhoff, Groningen, 1960).
    • See, e.g., G. Sansone and J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable, Vol. 1 (Noordhoff, Groningen, 1960).
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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 standard-model 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.
    • Note that what we are calling the light cone may be unrelated to the properties of light in the present case. In the standard-model 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.
  • 53
    • 85037910748 scopus 로고    scopus 로고
    • The coefficients (Formula presented) in the Lagrangian (1) are defined to be traceless, so a model with only a nonzero (Formula presented) might appear inappropriate. However, the effect of a nonzero trace in (Formula presented) is merely to scale the normalization of the Dirac field
    • The coefficients (Formula presented) in the Lagrangian (1) are defined to be traceless, so a model with only a nonzero (Formula presented) might appear inappropriate. However, the effect of a nonzero trace in (Formula presented) is merely to scale the normalization of the Dirac field 5. For example, a model with a nonzero (Formula presented) describes physics similar to a model with (Formula presented). The former choice has been adopted in the text to simplify the presentation.
  • 56
    • 85037906805 scopus 로고    scopus 로고
    • If (Formula presented) is positive, the issues with stability can be eliminated by multiplication with the exponential factor (Formula presented) instead. Thus, consistency in the (Formula presented) model can be achieved through exponential suppression of this type only for one sign of (Formula presented) 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.
    • If (Formula presented) is positive, the issues with stability can be eliminated by multiplication with the exponential factor (Formula presented) instead. Thus, consistency in the (Formula presented) model can be achieved through exponential suppression of this type only for one sign of (Formula presented) 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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