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According to the representation theory of the Lorentz group, Lorentz covariant quantities are built out of scalars, four-vectors, four-tensor s, and spinors.
The space-time interval is a Lorentz-invariant quantity, as is the Minkowski norm of any four-vector.
Equations which are true in any inertial reference frame are also said to be Lorentz covariant (some use the term invariant here). Lorentz covariant equations can always be written in terms Lorentz covariant quantities. According to the principle of relativity all fundamental equations of physics must be Lorentz covariant.
Note: this usage of the term covariant should not be confused with the related concept of a covariant vector. On manifoldIn mathematics, a manifold ''M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. Therefors, the words covariantIn category theory, see covariant functor. In tensor analysis, a covariant coordinate system is reciprocal to a corresponding contravariant coordinate system. Roughly speaking, a covariant tensor is a vector field that defines the topology of a space; it and contravariantContravariant is a mathematical term with a precise definition in tensor analysis. It specifies precisely the method (direction of projection) used to derive the components by projecting the magnitude of the tensor quantity onto the coordinate system bein refer to how objects transform under general coordinate transformations. Confusingly, both covariantIn category theory, see covariant functor. In tensor analysis, a covariant coordinate system is reciprocal to a corresponding contravariant coordinate system. Roughly speaking, a covariant tensor is a vector field that defines the topology of a space; it and contravariantContravariant is a mathematical term with a precise definition in tensor analysis. It specifies precisely the method (direction of projection) used to derive the components by projecting the magnitude of the tensor quantity onto the coordinate system bein four-vectors can be Lorentz covariant quantities.
These questions will all remain open as long as the classical limits of various LQG models (see below for the sources of variation) cannot be calculated.
Mathematically LQG is local gauge theory of the self-dual subgroup of the complexified Lorentz group, which is related to the action of the Lorentz group on Weyl spinors commonly used in elementary particle physics. This is partly a matter of mathematical convenience, as it results in a compact SO(3) or SU(2) gauge group as opposed to the non-compact SO(3,1) or SL(2.C). The compactness of the Lie group avoids some thus-far unsolved difficulties in the quantization of gauge theories of noncompact lie groups, and is responsible for the discreteness of the area and volume spectra. The infamous Immirzi parameter is necessary to resolve an ambiguity in the process of complexification. These are some of the many ways in which different quantizations of the same classical theory can result in inequivalent quantum theories, or even in the impossibility to carry quantization through.
It should be pointed out that the reasons why one can't distinguish between SO(3) and SU(2) or between SO(3,1) and SL(2,C) at this level is that the respective Lie algebras are the same. In fact, all four groups have the same complexified Lie algebra, which makes matters even more confusing (these subtleties are usually ignored in elementary particle physics). The physical interpretation of the Lie algebra is that of infinitesimally small group transformations, and gauge bosons (such as the graviton) are Lie algebra representations, not Lie group representations. What this means for the Lorentz group is that, for sufficiently small velocity parameters, all four complexified Lie groups are indistinguishable in the absence of matter fields.
To make matters more complicated, it can be shown that a positive cosmological constant can be realized in LQG by replacing the Lorentz group with the corresponding quantum group. At the level of the Lie algebra, this corresponds to what is called q-deforming the Lie algebra, and the parameter q is related to the value of the cosmological constant. The effect of replacing a Lie algebra by a q-deformed version is that the series of its representations is truncated (in the case of the rotation group, instead of having representations labelled by all half-integral spins, one is left with all representations with total spin j less than some constant). It is entirely possible to formulate LQG in terms of q-deformed Lie algebras instead of ordinary Lie algebras, and in the case of the Lorentz group the result would, again, be indistinguishable for sufficiently small velocity paramenters.
In the spin-foam formalism the Barrett-Crane model , which was for a while the most promising state-sum model of 4D Lorentzian quantum gravity, was based on representations of the noncompact groups SO(3,1) or SL(2,C), so the spin foam faces (and hence the spin network edges) were labelled by positive real numbers as opposed to the half-integer labels of SU(2) spin networks.
These and other considerations, including difficulties interpreting what it would mean to apply a Lorentz transformation to a spin network state, led Lee Smolin and others to suggest that spin network states must break Lorentz invariance. Lee Smolin and Joao Magueijo then went on to study doubly-special relativity, in which not only there is a constant velocity c but also a constant distance l. They showed that there are nonlinear representations of the Lorentz lie algebra with these properties (the usual Lorentz group being obtained from a linear representation). Doubly-special relativity predicts deviations from the special relativity dispersion relation at large energies (corresponding to small wavelengths of the order of the constant length l in the doubly-special theory). Giovanni Amelino-Camelia then proposed that the mystery of ultra-high-energy cosmic rays might be solved by assuming such violations of the special-relativity dispertion relation for photons.
Phenomenological (hence, not specific to LQG) constraints on anomalous dispersion relations can be obtained by considering a variety of astrophysical experimental data, of which high-energy cosmic rays are but one part. Current observations are already able to place exceedingly stringent constraints on these phenomenological parameters.