Before we discuss Lorentz violation in general, it will be useful to detail a pedagogical example that will
give an intuitive feel as to what “Lorentz violation” actually means. Let us work in a field
theory framework and consider a “bimetric” action for two massless scalar fields and
,
Since in order for a physical theory to be well defined the action must be a spacetime scalar, breaking of
active Lorentz invariance is the only physically acceptable type of Lorentz violation. Sometimes active
Lorentz invariance is referred to as “particle” Lorentz invariance [172]. We will only consider
active Lorentz violation and so shall drop any future labelling of “observer”, “particle”,“active”,
or “passive” Lorentz invariance. For the rest of this review, Lorentz violation always means
active Lorentz violation. For another discussion of active Lorentz symmetry in field theory
see [240]. Since we live in a world where Lorentz invariance is at the very least an excellent
approximate symmetry, must be small in our frame. In field theoretical approaches to Lorentz
violation, a frame in which all Lorentz violating coefficients are small is called a concordant
frame [176
].
Almost all models for Lorentz violation fall into the framework above, where there is a preferred set of
concordant frames (although not necessarily a field theory description). In these theories Lorentz invariance
is broken; there is a preferred set of frames where one can experimentally determine that Lorentz violation
is small. A significant alternative that has attracted attention is simply modifying the way the Lorentz
group acts on physical fields. In the discussion above, it was assumed that everything transformed
linearly under the appropriate representation of the Lorentz group. On top of this structure,
Lorentz non-invariant tensors were introduced that manifestly broke the symmetry but the
group action remained the same. One could instead modify the group action itself in some
manner. A partial realization of this idea is provided by so-called “doubly special relativity”
(DSR) [15, 186
], which will be discussed more thoroughly in Section 3.4. In this scenario there is still
Lorentz invariance, but the Lorentz group acts non-linearly on physical quantities. The new
choice of group action leads to a new invariant energy scale as well as the invariant velocity
(hence the name doubly special). The invariant energy scale
is usually taken to be
the Planck energy. There is no preferred class of frames in these theories, but it still leads to
Lorentz “violating” effects. For example, there is a wavelength dependent speed of light in
DSR models. This type of violation is really only “apparent” Lorentz violation. The reader
should understand that it is a violation only of the usual linear Lorentz group action on physical
quantities.
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