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Comments on ``A note on first-order formalism and odd-derivative actions'' by S. Deser

We argue that the obstacles to having a first-order formalism for odd-derivative actions presented in a pedagogical note by Deser are based on examples which are not first-order forms of the original actions. The general derivation of an equivalent first-order form of the original second-order action is illustrated using the example of topologically massive electrodynamics (TME). The correct first-order formulations of the TME model keep intact the gauge invariance presented in its second-order form demonstrating that the gauge invariance is not lost in the Ostrogradsky process.

In a recent pedagogical note [1], S. Deser discusses obstacles to the first-order formulations of theories with an odd number of derivatives occurring in the actions, arguing that firstorder formulations might destroy gauge invariance. The title of the note is very general and creates impression that there is a problem in applying first-order formulations to all oddderivative actions, which we argue below to be incorrect. The simplest counter-example is a pure Yang-Mills action which contains odd-derivative terms, but its first-and second-order formulations are equivalent and both of them are gauge invariant.
The simplest illustration of an odd-derivative action in [1] is topologically massive electrodynamics (TME) in 3D, where the Lagrangian in the standard second-order form is 1 where A µ is the abelian vector field and is just a short notation, not an independent variable. This sum of the Maxwell Lagrangian and a Chern-Simons term has been proposed in three dimensions and exhaustively analyzed in its second-order form in [2]. (We follow the notation of [1] which is slightly different from that of [2].) The first-order formulations of (1) are discussed in [1]. However, neither of the proposed first-order formulations of (1) reproduces the equations of motion that follow from (1) and thus should not be used as grounds for any comparison between the first and secondorder forms and, in particular, to reach any conclusion about the loss of gauge invariance in this transition.
In his introduction, the author of [1] refers to a first-order form as a particular case of the general Ostrogradsky procedure for lowering the number of derivatives appearing by adding new variables. 2 This general procedure in which new variables are added is a well defined operation that leads to an equivalent variational problem of the lower order Lagrangian through using extra fields as Lagrange multipliers and, as we will demonstrate, does not destroy gauge invariance. In the case of (1) this procedure gives the first-order Lagrangian In (4) there are two new independent antisymmetric tensor fields: F µν , W µν . The coefficient in front of the last term is chosen so as to yield the conventional first-order form in the limit of m = 0 (i.e. the first-order formulation of the Maxwell Lagrangian). Let us demonstrate the equivalence of (1) and (4). Performing a variation of (4) with respect to all independent variables we obtain the system of equations Solving (5,6) for F µν and W µν and substituting them into (7) This is also a general result: if an algebraic equation of motion (such as (5)) can be solved for a field, this field can be eliminated from the Lagrangian. 4 As before, one can repeat the calculations of eqs. (4-7) and show explicitly that elimination of W µν from the equations of which can be used to recover both eqs. (3) and (1).
The actions of eqs. (4) and (8) are both true first-order formulations of (1) and these actions, not the actions considered in [1], lead to the equations of motion (3) which arise in the second-order form of TME [2].
Having true first-order formulations, we can check whether there is indeed a loss of gauge invariance in first-order formulations and see if this obstacle is an artifact of working with a wrong formulation of first-order TME. It would seem unreasonable to expect a loss of gauge invariance for formulations that lead to the same equations of motion (3), which are invariant under the abelian gauge transformation Of course, there is a difference between the TME model and ordinary gauge theories such as Maxwell or Yang-Mills (YM). The last two are invariant exactly under the corresponding gauge transformations in both first-and second-order formulations, whereas the second-order TME is invariant only up to a surface term. Variation of (1) leads to the expression that, by addition of zero in the form 0 ≡ −mǫ µνα ∂ µ ∂ α A ν φ, can be presented as a surface 4 Not all actions, especially with higher-order derivatives, satisfy such a condition and it is better to start from an unambiguous form such as (4). Also, in the case of non-linear theories, this longer way, may even be preferable, because a term with derivatives will be present only in terms bilinear in the fields and all non-linearity becomes purely algebraic (i.e. without derivatives). 5 Alternatively, a different zero, 0 ≡ −mǫ µνα A ν ∂ µ ∂ α φ, can be added; this gives a different surface term An invariance up to a surface term requires us to impose a condition on the asymptotic behavior of gauge parameters, whereas in the case of an exact gauge invariance (for such Lagrangians as Maxwell and Yang-Mills) gauge parameters are free from such restrictions.
The consequences of this difference is an interesting question to discuss but it is beyond the scope of our short comment. This "up to surface term invariance" is also present in the first-order formulations of TME.

It is not difficult to check that the Lagrangian (4) is invariant under transformations
and (8) under The Lagrangians in both first-order formulations are invariant up to a surface term which is the same surface term as the one appearing in the second-order formulation. Moreover, all equations of motion (5-7) and (9,10) are exactly invariant under (14) and (15), in full agreement with the results of the second-order formulation, thus demonstrating that the presence of the Chern-Simons term does not create any "fundamental distinction" between the first-and second-order formulations of the model. A detailed canonical analysis of the first-order formulation of TME using the Dirac constraint formalism including a derivation of the transformations of Eqs. (14,15) is the subject of a forthcoming paper [6].
The second example considered in [1] is the "first-order form" of 3D topologically massive gravity (TMG) [2]. In this case the combination (Eqs. (5b+7a) of [1]) used by the author is again not a first-order formulation of the original model. The equations of motion for these formulations of [1] are different because affinity, according to [1], is no longer a metric one in the so-called "fully Palatini model". 6 The true first-order formulation of this model requires more involved calculations compared to the TME model, in particular, because the derivatives of the metric up to third order are present in the original action and this means that more additional fields are required to convert it into an equivalent first-order form. Such calculations would be another illustration of the general procedure for lowering the order of derivatives appearing in the action by means of Lagrange multipliers.