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Many current models which "violate Lorentz symmetry" do so via a vector or tensor field which takes on a vacuum expectation value, thereby spontaneously breaking the underlying Lorentz symmetry of the Lagrangian.
Lectures on Quantum Mechanics
Paul A. M. Dirac · 1964
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Linearization stability of nonlinear partial differential equations
Jerrold E. Marsden and Arthur E. Fischer · 1975
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The effect of gravitational interaction on classical fields: A hamilton-dirac analysis
J Isenberg and J Nester · 1977
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Gravity with a dynamical preferred frame
Ted Jacobson and David Mattingly · 2001
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Gravity, Lorentz violation, and the standard model
V. Alan Kostelecký · 2004
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Relativistic gravitation theory for the modified Newtonian dynamics paradigm
Jacob Bekenstein · 2004
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Michael D Seifert · 2009
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Michael D Seifert · 2010
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Michael D Seifert · 2010
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David Garfinkle, James Isenberg, and Jose M. Martin-Garcia · 2012
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Spontaneous breaking of Lorentz symmetry with an antisymmetric tensor
C. A. Hernaski · 2016
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Constraints and degrees of freedom in Lorentz-violating field theories
Michael D Seifert · 2019
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Dynamical Lorentz symmetry breaking in a tensor bumblebee model
J. F. Assunção, T. Mariz, J. R. Nascimento, and A. Yu Petrov · 2019
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