### 2.1 Defining Lorentz violation

#### 2.1.1 Lorentz violation in field theory

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 ,

where is some arbitrary symmetric tensor, not equal to . Both and are fixed background fields. At a point, one can always choose coordinates such that . Now, consider the action of local Lorentz transformations at this point, which we define as those transformations for which is invariant, on . is a spacetime scalar, as it must be to be well-defined and physically meaningful. Scalars are by definition invariant under all passive diffeomorphisms (where one makes a coordinate transformation of every tensor in the action, background fields included). A local Lorentz transformation is a subgroup of the group of general coordinate transformations so the action is by construction invariant under “passive” local Lorentz transformations. This implies that as long as our field equations are kept in tensorial form we can freely choose what frame we wish to calculate in. Coordinate invariance is sometimes called “observer Lorentz invariance” in the literature [172] although it really has nothing to do with the operational meaning of Lorentz symmetry as a physical symmetry of nature. Lorentz invariance of a physical system is based upon the idea of “active” Lorentz transformations, where we only transform the dynamical fields and . Consider a Lorentz transformation of and ,
where is the Lorentz transformation matrix, . The derivatives transform as
from which one can easily see that since by definition . The terms are therefore Lorentz invariant. is not, however, invariant under the action of and hence the action violates Lorentz invariance. Equations of motion, particle thresholds, etc. will all be different when expressed in the coordinates of relatively boosted or rotated observers.

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].

#### 2.1.2 Modified Lorentz groups

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) [15186], 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.