PHYSICAL REVIEW D 102, 065015 (2020)
Lagrangian constraint analysis of first-order classical field theories with an application to gravity
† ‡ Verónica Errasti Díez ,1,* Markus Maier ,1,2, Julio A. M´endez-Zavaleta ,1, and Mojtaba Taslimi Tehrani 1,§ 1Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 Munich, Germany 2Universitäts-Sternwarte, Ludwig-Maximilians-Universität München, Scheinerstrasse 1, 81679 Munich, Germany
(Received 19 August 2020; accepted 26 August 2020; published 21 September 2020)
We present a method that is optimized to explicitly obtain all the constraints and thereby count the propagating degrees of freedom in (almost all) manifestly first-order classical field theories. Our proposal uses as its only inputs a Lagrangian density and the identification of the a priori independent field variables it depends on. This coordinate-dependent, purely Lagrangian approach is complementary to and in perfect agreement with the related vast literature. Besides, generally overlooked technical challenges and problems derived from an incomplete analysis are addressed in detail. The theoretical framework is minutely illustrated in the Maxwell, Proca and Palatini theories for all finite d ≥ 2 spacetime dimensions. Our novel analysis of Palatini gravity constitutes a noteworthy set of results on its own. In particular, its computational simplicity is visible, as compared to previous Hamiltonian studies. We argue for the potential value of both the method and the given examples in the context of generalized Proca and their coupling to gravity. The possibilities of the method are not exhausted by this concrete proposal.
DOI: 10.1103/PhysRevD.102.065015
I. INTRODUCTION In this work, we focus on singular classical field theories It is hard to overemphasize the importance of field theory that are manifestly first order and analyze them employing in high-energy physics. Suffice it to recall that each and exclusively the Lagrangian formalism. Nonsingular theo- every of the fundamental interactions we are aware of as ries are also in (trivial) reach. Throughout the paper, of yet—the gravitational, electromagnetic, strong and manifest first order shall stand for a Lagrangian that weak interactions—are described in terms of fields. depends only on the field variables and their first deriv- Correspondingly, their dynamics are studied by means of atives. This implies the equations of motion are guaranteed field theory. Most often, this is done by writing a to be second order at most. Within this framework, we Lagrangian (or a Hamiltonian) density that is a real smooth present a systematic methodology that is optimized to function of the field components (and their conjugate determine the number of field components that do propa- momenta) and that is then subjected to the principle of gate, which we denominate physical and propagating stationary action. It is customary to encounter the situation modes and degrees of freedom. To do so, we explicitly where not all of the a priori independent quantities—field obtain the constraints: specific functional relations among components and/or conjugate momenta—are conferred a the field variables and their time derivatives that avoid the dynamical evolution through the equations of motion. In propagation of the remaining field components. Our approach is complementary to the similarly aimed proce- such a case, the field theory is said to be singular or – constrained. For instance, it is well known that all gauge dures in [1 3] and is markedly distinct from, yet equivalent to, that in [4]. theories are singular. Apart from the intrinsic relevance of understanding and characterizing the constraint structure of those theories satisfying our postulates, an ulterior motivation for this *[email protected] † [email protected] investigation is to pave the way toward a consistent theory ‡ [email protected] building principle. Indeed, theoretical physics is currently §[email protected] in need of new fundamental and effective field theories that are capable of accounting for experimental data—the Published by the American Physical Society under the terms of strong CP problem, neutrino masses and the nature of the the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to dark sector, to mention but a few of the most relevant the author(s) and the published article’s title, journal citation, examples. A recurrent and challenging obstacle in the and DOI. Funded by SCOAP3. development of well-defined field theories consists in
2470-0010=2020=102(6)=065015(29) 065015-1 Published by the American Physical Society VERÓNICA ERRASTI DÍEZ et al. PHYS. REV. D 102, 065015 (2020) guaranteeing the correct number of physical modes. In this approach is complementary to the existing literature. In context, most effort is devoted to avoiding the propagation particular, it is equivalent to the recent proposal in [4],as of Ostrogradsky instabilities [5]—additional unphysical argued and exemplified in Sec. VA. degrees of freedom, which we shall denote ghosts for We proceed to employ it to analyze various well-known short. The general problem is delineated in [6] and theories: Maxwell electromagnetism, together with the numerous realizations of this idea can be found, e.g., (hard) Proca action in Sec. III and the Palatini formulation [7]. However, it is equally important to ensure the theory of gravity in Sec. IV. Their study is cornerstone to under- is not overconstrained, i.e., there are fewer than required stand the Maxwell-Proca theory [8,9] and paves the way to propagating modes. Our subsequent prescription provides a its consistent coupling to gravity. This is discussed in solid footing to this (double) end and is presented in a Sec. VB. comprehensible and ready-to-be-used manner, with the We conclude in Sec. VI, restating the instances when our goal of being useful to communities such as, but not method is most convenient and emphasizing two crucial limited to, theoretical cosmology and black hole physics. aspects that are sometimes overlooked. We describe how to convert the analytical tool here exposed into a constructive one, but the concrete realization of this B. Conventions idea is postponed to future investigations. We work on a d-dimensional spacetime manifold M of A specific materialization of the preceding general the topology M ≅ R × Σ. Namely, we assume M admits a discussion (and the one we later on employ to ground foliation along a timelike direction. This is true for all our conversion proposal) is as follows. We recall that an (pseudo-)Riemannian manifolds. For simplicity, we con- earlier version of the method here augmented and refined sider Σ has no boundary. The dimension d is taken to be already allowed for the development of the most general arbitrary but finite, with the lower bound d ≥ 2. Spacetime nonlinear multivector field theory over four-dimensional indices are denoted by the Greek letters ðμ; ν; …Þ and flat spacetime: the Maxwell-Proca theory [8,9]. There, the μν raised or lowered with the metric gμν and its inverse g .We inclusion of a dynamical gravitational field was beyond ∂ employ the standard short-hand notation ∂μ ≔ μ, where scope. The present work provides a sound footing for the ∂x μ ≔ 0 1 … d−1 ≡ 0 i 1 2 … − 1 study of singular field theories defined over curved back- x ðx ;x ; ;x Þ ðx ;xÞ, with i ¼ ; ; ;d , grounds. Thus, it paves the way for the ghost-free coupling are spacetime local coordinates, naturally adapted to the foliation R × Σ. The dot stands for derivation with respect of Maxwell-Proca to gravity. ∶ M → R Bearing in mind the above future objective and in order to time, so that for local functions f , we write _ ≔ ∂ ̈≔ ∂2 to clarify the formal presentation of the method, we f 0f and f 0f. Brackets indicating symmetrization ≔ (re)analyze the simplest spin-one and -two theories by and antisymmetrization of indices are defined as TðμνÞ 2 ≔ − 2 means of our proposed procedure: Maxwell, Proca and ðTμν þ TνμÞ= and T½μν ðTμν TνμÞ= , respectively. As Einstein’s gravity. While the former two are manifestly first is customary, summation over repeated indices should be order, the latter is not. Indeed, gravity, cast in the Einstein- understood at all times. Hilbert way, is a second-order Lagrangian for the metric, up to a noncovariant boundary term. As such, it exceeds the II. EXPOSITION OF THE METHOD domain of applicability of our approach. Favorably, this We begin by putting forward a coordinate-dependent, property can be circumvented taking advantage of the i.e., nongeometrical, Lagrangian approach to obtain all the deluge of reformulations available for the theory. Among constraints present in a manifestly first-order classical field them, we single out the Palatini formalism—see [10] for a theory. Needless to say, there exists a vast literature on the historical overview—which considers the metric and the topic: some standard references are [15]; but for its affine connection as a priori independent fields. elegance and concision, we particularly recommend [16]. Our determination of the explicit constraints present in This section serves us to fix the notation used throughout Palatini, while not yielding novel information about the the paper and provide a self-contained derivation of all our theory, conforms a remarkable piece of work. Not only is it results. We stress that, although the method is not new per carried out minutely and can be readily seen to be se, we are not aware of any reference where this material is computationally easier and shorter than the previously comprehensively presented in a ready-to-be-used manner performed Hamiltonian studies, e.g., [11–14], it also and keeping the technicalities at a bare minimum, as we provides the basis for a consistent inclusion of matter do here. fields. As such, we regard this comprehensive analysis as Our only assumptions shall be the principle of stationary an intrinsically valuable result. action and finite reducibility. The first assumption is rather obviously a very mild one, but it is worth noting that this is A. Organization of the paper not an essential requirement; for instance, see [17]. We will In the following Sec. II, we introduce the Lagrangian explain the second assumption shortly. For the time being, methodology we shall use throughout the paper. Our it suffices to note that, to our knowledge, the only known
065015-2 LAGRANGIAN CONSTRAINT ANALYSIS OF FIRST-ORDER … PHYS. REV. D 102, 065015 (2020) example of a classical field theory (of the kind here 1 n ¼ N − N1 − N2; ð2:3Þ considered) not satisfying it is bosonic string field theory, dof 2 both in its open [18] and closed [19] variants. Given a Lagrangian density L within the above postu- attributed to Dirac. Here, ðN1;N2Þ denote the number of lates, our analysis yields the constraint structure character- first- and second-class constraints, respectively. As a izing triplet reminder, first- (second-) class constraints are those which do (not) have a weakly vanishing Poisson bracket with all tðNÞ ≔ ðl; g; eÞ: ð2:1Þ of the constraints present in a given theory. Needless to say, the proven equivalence between the We stress that this is a purely Lagrangian statement, since it Lagrangian and Hamiltonian formulations of classical collects the outcome of our subsequently proposed purely theories [1,21] is a most celebrated body of work. The Lagrangian method. Here, N is the number of a priori two given prescriptions for the degree of freedom count independent field variables in terms of which L is written. in (2.2) and (2.3) are a particular materialization of this As such, N is equal to the dimension of the theory’s equivalence, which was further exploited in [2] to develop a configuration space, which we shall shortly introduce. The one-to-one mapping between the Lagrangian parameters other numbers l, g and e are defined below. ðl; g; eÞ and their Hamiltonian counterparts: On shell, we obtain l: the total number of functionally ðPÞ ðPÞ independent Lagrangian constraints. Our analysis elabo- l ¼ N1 þ N2 − N1 ;g¼ N1 ;e¼ N1; ð2:4Þ rates on the iterative algorithm presented in [1] and ðPÞ employed in Appendix A of [8]. It is the suitable gener- where N1 stands for the number of so-called primary first- alization to field theory of the coordinate-dependent class constraints, those first-class constraints that hold true method used in [2] for particle systems, which is in turn off shell. Using this information, the triplet tðNÞ defined in based on [20]. The nontrivial geometric extension to field (2.1) can be readily seen to admit the following equivalent theory of [2] was carried out in [3], where the discussion Hamiltonian parametrization: was extended to the treatment of off-shell constraints as well. Thus, our discussion is complementary to all these ðNÞ ðPÞ t ¼ðN ;N1;N2Þ: ð2:5Þ references [1–3]. 1 Off shell, we shall obtain g and e: the number of gauge An important comment is in order here. Our sub- identities and effective gauge parameters, respectively. sequently proposed Lagrangian approach to determine Gauge identities are to be understood in the usual sense, ðNÞ ∈ N ∪ 0 as (differential) relations between certain functional varia- t does not guarantee ndof f g. This means that, tions of the action that identically vanish. By effective even though all l, g and e in (2.2) are integers by definition, gauge parameters we mean the number of independent their sum need not be an even number. The reason is gauge parameters plus their successive time derivatives that simple: we put forward an analytical tool, not a mechanism to detect (or even correct) ill-posed theories. If, for some explicitly appear in the gauge transformations. We deter- L mine g and e for theories where the gauge transformations Lagrangian density , a half-integer number of physical are known a priori and provide suitable references that deal degrees of freedom is found upon correctly employing our ðNÞ with the treatment of theories where the gauge trans- prescription for t together with (2.2), then it must be formations are unknown beforehand. Notice that knowl- concluded that the theory is unphysical. The (possibly L edge of the gauge transformations for the field theory is not nontrivial) modifications required on for it to propagate an integer number of physical modes are a question beyond a necessary assumption, unlike the principle of stationary 1 action and finite reducibility. However, this information the scope of this manuscript. considerably shortens the analysis and, being a feature of For the renowned examples in Secs. III and IV,we ðNÞ all the theories we shall explicitly consider, we have opted shall minutely determine the triplet t defined in (2.1) and for only developing in detail such case. then use (2.2) to explicitly count physical modes. As such, Given the triplet tðNÞ, the physical degrees of freedom we shall perform various countings solely in Lagrangian terms. Afterward, we shall (partially) verify our results ndof in the theory under study can be counted, employing the result derived in [2]: by comparing them to a representative subset of the Hamiltonian-based literature via (2.3) and (2.4). Addi- 1 tionally, the examples of Sec. III shall be worked out in − ndof ¼ N 2 ðl þ g þ eÞ: ð2:2Þ two different (but dynamically equivalent) Lagrangian formulations, based on distinct values N and N ≠ N of We will refer to (2.2) as the master formula, the way the authors of [2] themselves do. The remarkable feature about 1This should not alarm the reader. The same is true on the the previous counting is that it is purely Lagrangian, as standard Hamiltonian formalism. In (2.3), N2 is not necessarily opposed to the usually employed Hamiltonian formula an even number, unless demands are made on the Hamiltonian.
065015-3 VERÓNICA ERRASTI DÍEZ et al. PHYS. REV. D 102, 065015 (2020)
A i A i the dimension of the configuration space. We will then see with δQ ðt1;xÞ¼0 ¼ δQ ðt2;xÞ. The above variational that, even though the constraint structure characterizing derivative is defined as triplets do not coincide, the number of propagating modes ndof does match for both descriptions: ∂L ∂L ≔ ∂ − ! 0 EA μ A A ¼ ; ð2:10Þ tðNÞ ≔ ðl; g; eÞ ≠ tðNÞ ≔ ðl; g; eÞ; ∂ð∂μQ Þ ∂Q 1 1 − N − l g e N 2 ðl þ g þ eÞ¼ndof ¼ 2 ð þ þ Þ: ð2:6Þ where the latter equality is the on-shell demand. This on- shell requirement commences the iterative algorithm we This is because ndof is a physical observable, while shall employ to determine the Lagrangian constraints pre- N; l; g; e ð Þ are not. Obviously, the same situation arises sent in the theory. Here, L ¼ L½QA is the Lagrangian in the Hamiltonian picture as well, which we briefly density. Observe that we have already restricted attention to illustrate at the end of Sec. IV. manifestly first-order field theories; i.e., we consider L In the following, we explain how to obtain the constraint A A depends only on Q and its first derivatives ∂μQ . The structure characterizing triplet tðNÞ in (2.1). study of higher-order field theories2—where L explicitly n A depends on ∂μQ , with n ≥ 2—lies beyond the scope of A. On-shell Lagrangian constraints our present investigations. We omit the possible depend- Let C be the configuration space of a classical field theory. ence of L on nondynamical field variables, such as the As usual, we take C to be a differentiable Banach manifold spacetime metric in any special relativistic theory. The said whose points are labeled by N real field variables QA: dependence can be easily incorporated to our analysis, but it does not arise in the theories we discuss in this work. C A 1 2 … ¼ spanfQ g;A¼ ; ; ;N: ð2:7Þ An important remark on notation follows. As introduced in (2.7), QA is an ordered set of a priori independent field We stress that A comprises all possible discrete indices that variables; it is neither a row nor a column vector. The same the real field variables have. For instance, if one considers is true for E in (2.10): this is the ordered set of Euler- Yang-Mills theory, A consists of both spacetime indices and A Lagrange equations for the QA field variables, not a vector. color indices. If onewishes to entertain complex Yang-Mills, We have opted for a notation where the set indices are then the real and imaginary parts of each and every Yang- always assigned the same position when ascribed to a Mills field component must be counted separately in A. So, 2 certain ordered set (for instance, upper position for the field for SUð Þ complex Yang-Mills theory in four spacetime A dimensions, we would have that N ¼ 2ð4 · 3Þ¼24. Notice variables Q and lower position for the Euler-Lagrange equations E ). The assignation is such that the Einstein that QA are real smooth functions of spacetime QA ¼ A μ summation convention employed throughout the paper is QAðx Þ, but we will suppress this dependence all along, apparent. The only quantities that will show up in this so as to alleviate notation. Thus, our notation matches that in section which have a definite character within matrix [3] and leaves out the spacetime argument compared to the calculus are the following. The various Hessians, their condensed notation introduced by DeWitt in [22] and Moore-Penrose pseudoinverses and the Jacobians are all extensively used in the literature, e.g., [23]. Then, TC is matrices. The null vectors of the Hessians are row vectors. C A _ A the tangent bundle of , which is spanned by fQ ; Q g.We Their transposed column vectors also show up. The row or A _ A ̈A refer to ðQ ; Q ; Q Þ as the generalized coordinates, veloc- column character of the ordered sets is then straightfor- ities and accelerations of the theory, respectively. wardly fixed according to dimensional analysis in all As already stated and common to most field theories, we formulas. assume that the dynamics are derivable from a principle of As a practical starting point for our iterative method, it is stationary action. In other words, the Euler-Lagrange convenient to recast the Euler-Lagrange equations (2.10) in ! 0 equations EA¼ for the field theory follow from the the form requirement that the action functional Z Z Z 2 t2 One may be tempted to evade the higher-order character of a A d 0 d−1 S ¼ S½Q ¼ d xL ¼ dx d xL ð2:8Þ theory via the Ostrogradsky prescription, i.e., introducing addi- M t1 Σ tional generalized coordinates in a manner that results in a manifestly first-order Lagrangian density. Such alteration of remains stationary under arbitrary functional variations TC must be compensated through the inclusion of Lagrange A A 0 i δQ ¼ δQ ðx ;xÞ that vanish at times t1 and t2 on the multipliers that preserve the equivalence to the original setup. To spatial slice Σ: do so consistently, one needs to either verify the so-called Z Ostrogradsky nonsingularity condition or exploit alternative δS ! methods, as detailed in [24]. In view of these nontrivial subtleties, δ δ A ≡ d δ A 0 we restrict ourselves to the study of manifestly first-order S ¼ A Q d xEA Q ¼ ; ð2:9Þ δQ M theories.
065015-4 LAGRANGIAN CONSTRAINT ANALYSIS OF FIRST-ORDER … PHYS. REV. D 102, 065015 (2020)
! ̈A manners in which the necessary stability of the functionally EB ¼ Q WAB þ αB¼0; ð2:11Þ independent Lagrangian constraints may manifest itself. where we have defined the so-called primary Hessian ≔ ∂ ∂ L WAB A_ B_ , as well as 1. Primary stage In order to determine the subset of M1 number of α ≔ ∂ ∂i L ∂i ∂ L ∂ _ A ∂i ∂j L ∂ ∂ A B ð B_ A þ B A_ Þ iQ þð B A Þ i jQ primary Lagrangian constraints out of the set of all N ∂i ∂ L ∂ A ∂ ∂ L _ A − ∂ L number of Euler-Lagrange equations in (2.11), we first þð B A Þ iQ þð B_ A ÞQ B : ð2:12Þ introduce a set of M1 number of linearly independent null γ To alleviate notation, we have introduced the following vectors I associated to the primary Hessian WAB: shorthand: A ðγIÞ WAB ¼ 0;I¼ 1; 2; …;M1: ð2:14Þ ∂ ∂ ∂ ∂ _ ≔ ∂i ≔ ∂ ≔ A _ A ; A A ; A A ; ð2:13Þ We require that these form an orthonormal basis of the ∂Q ∂ð∂iQ Þ ∂Q kernel of WAB, which amounts to imposing the normali- which we shall extensively employ henceforth. zation condition We focus on singular (or constrained) field theories γ A γJ δ J γI ≔ γ T next.3 That is, we look at field theories described by a ð IÞ ð ÞA ¼ I ; where ð IÞ ; ð2:15Þ Lagrangian density whose primary Hessian has a vanishing determinant detðW Þ¼0. This means that the rank of with T denoting the transpose operation. We stress that, AB even though in all the examples considered in Secs. III WAB (the number of linearly independent rows or columns) is not equal to its dimension N; instead, it is reduced. and IV we have chosen null vectors that are constant, this is By definition it follows that, for singular Lagrangians, not a required feature for our formalism. Rather, this is just the N number of Euler-Lagrange equations in (2.11) can be a possible choice in all the given examples that has been split into two types. First, primary equations of motion: opted for due to its computational convenience. Only the normalization (2.15) is an essential requirement for the null these are the R1 ≔ rankðWABÞ number of on-shell second-order differential equations that explicitly involve vectors. In full generality, the null vectors of all stages can A the generalized accelerations Q̈A. Second, primary have an explicit dependence on the field variables Q and A their first derivatives ∂μQ . Lagrangian constraints: these are the M1 ≔ dimðWABÞ − Then, the M1 primary Lagrangian constraints are rankðWABÞ¼N − R1 number of on-shell relations between the generalized coordinates QA and their gener- obtained by contracting the Euler-Lagrange equations EA in (2.11) with the above null vectors.4 Namely, by perform- alized velocities Q_ A. We stress an explicit dependence on ing the contraction with γ : Q_ A (QA) is not necessary for the primary Lagrangian I constraints; they can be relations between the QA’s ! φ ≡ ðγ ÞAE ¼ðγ ÞAα ¼0: ð2:16Þ (Q_ A’s) only. Consistency requires that these constraints I I A I A are preserved under time evolution. Notice that the last equality is a direct consequence of the In the following, we obtain the said constraints and on-shell demand in (2.10) or equivalently in (2.11). Hence, ensure the consistency of the field theory by means of an the primary Lagrangian constraints are on-shell constraints iterative algorithm. We refer to each iteration in the by definition. One can also see this through equivalence to algorithm as a stage. In every stage, the above specified the more familiar Hamiltonian analysis. It is common notions of equations of motion and Lagrangian constraints knowledge, e.g., [25], that primary Lagrangian constraints will arise. The algorithm closes when the preservation relate to secondary constraints in the Hamiltonian frame- under time evolution of all Lagrangian constraints is work, which are on-shell constraints by definition. guaranteed. Equivalently, when all nth stage Lagrangian The primary Lagrangian constraints in (2.16) need not be ≥ 2 constraints are stable, for some finite integer n .Annth functionally independent from each other.5 When they are, stage Lagrangian constraint is said to be stable if its time the field theory is said to be irreducible at the primary derivative does not lead to a new (i.e., functionally stage. Otherwise, the theory is reducible at the primary independent) Lagrangian constraint in the subsequent 1 (n þ )th stage. Below, we explain in detail the different 4 The complementary subset of R1 ¼ N − M1 primary equa- tions of motion can be obtained by contracting EA with the basis 3 We leave out nonsingular field theories because the sub- vectors of the image of WAB. Here, we concentrate only on the sequent analysis is redundant for them: in this case Lagrangian constraints. 5 detðWABÞ ≠ 0, which implies l ¼ 0 and one can directly move This is in contrast to the primary equations of motion, which on to Sec. II B. Within our framework, scalar field theories in flat are guaranteed by construction to be functionally independent spacetime constitute a prominent example of nonsingularity. among themselves.
065015-5 VERÓNICA ERRASTI DÍEZ et al. PHYS. REV. D 102, 065015 (2020) stage. Before we carry on, we must restrict attention to the them into secondary equations of motion and secondary functionally independent primary Lagrangian constraints Lagrangian constraints, it is convenient to write them as ! φ 0 0 1 2 … 0 ≤ I0 ¼ , where I ¼ ; ; M1 M1. Their number is given ! 0 ˜ ̈A I ˜ by M ¼ rankðJ ΛÞ, where the Jacobian matrix J Λ is E ¼ Q ðγ Þ W þ α˜ ≈0; ð2:20Þ 1 I I J A IJ J 1 defined as ∂φ where we have defined ≔ I Λ A _ A JIΛ Λ ;X¼fQ ; Q g: ð2:17Þ ∂X ˜ ≔ γ A∂ φ WIJ ð IÞ A_ J; α˜ ≔ −α AB∂ _ A∂ ∂ _ A ∂i φ This test can be easily related to the standard Hamiltonian J ð AM B_ þ Q A þð iQ Þ AÞ J: ð2:21Þ framework: it is the pullback of the phase space regularity conditions in [26]. For the theories we are concerned with We point out that, in obtaining this expressions, we have 0 in this work, we verify M1 ¼ M1. Hence, all of the primary employed the on-shell statement (2.11), so as to eliminate Lagrangian constraints in (2.16) must be considered in the from (2.20) as much dependence on the generalized 6 ̈A 8 ˜ following. accelerations Q as possible. Here, WIJ is the so-called The vanishing of all the functionally independent pri- secondary Hessian and the auxiliary matrix MAB is the mary Lagrangian constraints defines the so-called primary Moore-Penrose pseudoinverse (as detailed in [27]) of the constraint surface TC1, which is a subspace of the moduli primary Hessian. The latter is ensured to always exist and space TC0 of the field theory: be unique. Its defining relations are9 C ≔ A _ A ∈ C φ 0 ⊆ C T 1 fðQ ; Q Þ T 0j I ¼ g T 0; AB −δA γI γ A 0 AB γI 0 M WBC C þð ÞCð IÞ ¼ ;Mð ÞB ¼ : ð2:22Þ A _ A TC0 ≔ fðQ ; Q Þ ∈ TCjEA ¼ 0g ⊂ TC: ð2:18Þ To gain some more intuition into MAB, we note that it For brevity, we write constitutes a generalization of the standard matrix inverse. BC AB ! It is introduced so that WABM and M WBC are φ ∶≈ 0: ð2:19Þ AB I 1 orthogonal projections onto the image of WAB and M , respectively. For regular square matrices, the Moore- Equalities that hold true in TC1 (and not in the entire of the Penrose pseudoinverse is equivalent to the standard matrix moduli space) shall be denoted ≈ and referred to as primary inverse: M ¼ W−1 iff detðWÞ ≠ 0. 1 ˜ ˜ weak equalities. If rankðWIJÞ¼dimðWIJÞ¼M1, no secondary Lagrangian As previously noted, consistency requires us to not only constraints arise and thus the primary Lagrangian constraints enforce the primary Lagrangian constraints (2.19), but also are stable. In this case, we say that the consistency of the primary Lagrangian constraints (2.19) under time evolution is to ensure that these! are preserved under time evolution. Explicitly, E˜ ≔ φ_ ≈0. This requirement starts the second dynamically ensured,byasetofM1 (necessarily functionally J J 1 ˜ ˜ ̈A iteration in the algorithm. independent) secondary equations of motion EJ ¼ EJðQ Þ. As a result, the total number of functionally independent Lagrangian constraints present in such field theories is 2. Secondary stage 0 ! l ¼ M1.However,thisisnotwhathappensinthetheories ˜ 7 The freshly introduced demands EJ ≈ 10 are known as of our interest. the secondary Euler-Lagrange equations. In order to split Generically, the rank of the secondary Hessian is smaller ˜ than its dimension. Consequently, M2 ≔ dimðWIJÞ − 6 0 ˜ If M1 065015-6 LAGRANGIAN CONSTRAINT ANALYSIS OF FIRST-ORDER … PHYS. REV. D 102, 065015 (2020) the primary stage before. In other words, the analysis from independent secondary Lagrangian constraints is not Eq. (2.14) onward is to be repeated. (dynamically) guaranteed. Instead, it must be enforced In details, the M2 number of linearly independent null through a third iteration of the just described procedure. We vectors γ˜R of the secondary Hessian must be obtained: stress that it is essential to close the iterative algorithm in order to find the correct number l of functionally inde- I ˜ ðγ˜RÞ WIJ ¼ 0;R¼ 1; 2; …;M2; ð2:23Þ pendent Lagrangian constraints. For completeness, we provide the explicit expressions and chosen so that the normalization condition for all relevant quantities at some arbitrary stage of the algorithm in the Appendix. These have not appeared in the γ˜ I γ˜S δ S γ˜S ≔ γ˜ T ð RÞ ð ÞI ¼ R ; with ð SÞ ; ð2:24Þ literature, as far as we know. is satisfied. Then, these must be contracted with the 4. Closure of the algorithm secondary Euler-Lagrange equations in (2.20) to yield the secondary Lagrangian constraints in the theory: In full generality and as already anticipated, our algo- rithm stops when all functionally independent Lagrangian ! constraints have been stabilized. This can happen in either φ˜ ≡ ðγ˜ ÞIE˜ ¼ðγ˜ ÞIα˜ ≈0: ð2:25Þ R R I R I 1 of the following different manners: (i) Dynamical closure.—Firstly, it may happen when ðnÞ ðnÞ If the secondary Lagrangian constraints vanish when Mn ≔ dimðW Þ − rankðW Þ¼0 for some nth evaluated on the first constraint surface φ˜ ≈ 0, then the ðnÞ ≥ 2 R 1 stage Hessian W , with n . This implies that total number of functionally independent Lagrangian con- no Lagrangian constraints arise at the nth stage, 0 ðnÞ straints is l ¼ M1. Again, this is not what happens in (all of) since in this case W has full rank and hence admits the theories of our interest. no null vector. Here, the consistency under time As a consequence, we must proceed with the algorithm. evolution of the previous stage’s functionally inde- φðn−1Þ First, we need to obtain the (subset of) φ˜ R’s which are pendent Lagrangian constraints is dynami- functionally independent among themselves when evalu- cally ensured, i.e., through the (necessarily 0 ated on the first constraint surface. Their number M2 ≤ M2 functionally independent) nth stage equations of is given by motion. In other words, the functionally independent φðn−1Þ’s are stable. This closure of the algorithm is ∂ 0 ˜ ˜ exemplified in Sec. III B. M ¼ rankðJ ΛÞ; where J Λ ≔ ðφ˜ j C Þð2:26Þ 2 R R ∂XΛ R T 1 (ii) Nondynamical closure.—Secondly, it may happen 0 when Mn > 0,butMn ¼ 0, again with n ≥ 2. Λ 0 ≠ 0 and X was introduced in (2.17). When M2 , we verify This implies that the nth stage functionally inde- 0 — M2 ¼ M2 for the theories we shall consider so that they pendent Lagrangian constraints φðnÞ’s do not define are irreducible theories at the secondary stage. Thus, all a new constraint surface, so that TCn ≡ TCn−1.We secondary Lagrangian constraints in (2.25) must be con- 10 differentiate two algebraically distinct scenarios: sidered subsequently. (ii)(a) The φðnÞ’s vanish identically in the (n − 1)th The vanishing of the functionally independent secondary constraint surface: φðnÞ ≋ 0. Such φðnÞ’s are Lagrangian constraints defines the secondary constraint n−1 ! known as Lagrangian identities. Clearly, surface TC2 ⊂ TC1, which we write as φ˜ ∶≈0. Equalities R 2 Lagrangian identities are trivially stable. holding true in TC2 shall be denoted ≈ and referred to as The example of Sec. III A illustrates this 2 closure of the algorithm. secondary weak equalities. It should be obvious that the (ii)(b) The φðnÞ’s functionally depend on the secondary Lagrangian constraints are on-shell constraints (n − 1)th stage functionally independent by definition. Lagrangian constraints. Schematically, ðnÞ ðn−1Þ φ ≈ ðf1 þ f2∂iÞφ , where ðf1;f2Þ 3. Tertiary stage n−2 ˆ ≔ γ˜ I γ A∂ φ˜ are arbitrary real smooth functions of the Let WRS ð RÞ ð IÞ A_ S be the tertiary Hessian. ’ generalized coordinates and velocities When the tertiary Hessian s rank does not match its A _ A dimension, the consistency under time evolution of ðQ ; Q Þ, such that ðf1;f2Þ are naturally ˆ ˆ defined in TCn−2. Then, it readily follows M3 ≔ dimðW Þ − rankðW Þ number of the functionally RS RS that φðnÞ ≈ 0 and it is obvious that such n−1 10 0 Lagrangian constrains are stable. This clo- When 0 065015-7 VERÓNICA ERRASTI DÍEZ et al. PHYS. REV. D 102, 065015 (2020) A In all the detailed cases, the total number of functionally EA piece. However, δS ¼ 0 may also follow from the δQ independent Lagrangian constraints is given by piece. Subsequently, we briefly review the latter scenario: how the vanishing of δS may be a consequence of off-shell Xn−1 identities stemming from a strict symmetry of the action. l ¼ M0 ; ð2:27Þ a This kind of symmetry—gauge invariance—is only mani- a¼1 A fest through specific field variations δθQ , in contrast to 0 where Ma counts the number of functionally independent our previous consideration in Sec. II A of arbitrary δQA’s. ≥ 2 ath stage Lagrangian constraints and n . We are not Correspondingly, we will differentiate between δθS and δS aware of any physically relevant example of a field theory as well. where n is infinite. There are different methods to obtain the said off-shell identities, but it is not our goal to provide an overview of 5. Noteworthy considerations them here. Our subsequent discussion summarizes and We restate that it is of utmost importance to close the employs the approach put forward in [28] and later on iterative procedure in order to determine l. If the algorithm adapted to exhibit manifest covariance in [14]. This is not closed (only some or none of the constraints are adaptation makes it straightforward to apply [28] to any stabilized), one can only give a lower bound on l. While manifestly first-order classical field theory, which is our this may be enough to ensure the absence of Ostrogradsky framework. A A A instabilities [5] in the field theory, it is insufficient to Consider the field transformations Q → Q þ δθQ . A guarantee the propagation of a definite number of degrees Let the changes δθQ be of the form of freedom. In such case, one can only infer an upper bound Xn β μ μ …μ on ndof. This observation is further discussed and exem- δ A −1 s ∂ ∂ …∂ θ Ω A 1 2 s θQ ¼ ð Þ ð μ1 μ2 μs Þð β Þ ; ð2:28Þ plified in Sec. VI. s¼0 We also point out that, in general, the different stabili- zations of the functionally independent Lagrangian con- where n ∈ N ∪ f0g, β is an (possibly collective) index that β A straints that we listed are all present in a given field theory. is to be summed over and the θ ’s and Ωβ ’s are known as Namely, some functionally independent Lagrangian con- the gauge parameters and gauge generators of the trans- straints in the theory are stabilized dynamically, while formation, respectively. The θβ’s are real smooth functions μ A others are stabilized nondynamically. This is indeed what of the spacetime coordinates x , while the Ωβ ’s are defined happens in our examples of Secs. III C and IV. A in TC and as such are real smooth functions of ðQA; Q_ Þ. Besides, we warn the readers against deceiving them- The former are unspecified, while the latter are to be selves regarding the ease of the exposed iterative algorithm. determined. Introducing the above in (2.9) and operating, Even though our methodology is sound and rigorous and its one finds that logic is easy to follow, there can be no misapprehension Z as to the algebraic complexity of its implementation in d β δθS ¼ d xθ ϱβ; concrete theories, most significantly those involving grav- M ity. From this point of view, the examples in Sec. III are Xn μ μ …μ uninvolved, while that in Sec. IVA is quite challenging. ϱ ≔ ∂ …∂ Ω A 1 2 s β μ1 μs ½EAð β Þ : ð2:29Þ The example in Sec. IV B constitutes an intermediate s¼0 difficulty case. We comment further on this important A (from a practical point of view) topic in Sec. VI. If, under the field variations (2.28) for some Ωβ ’s, the At last, we remark that the algorithm just exposed does action remains invariant δθS ≡ 0, then we have that not break covariance. Namely, if a field theory within our postulates is covariant, its study under the outlined iterative ϱβ ≡ 0 ð2:30Þ methodology will preserve this feature. Nonetheless, a suitable space and time decomposition of the a priori ! holds true off shell (i.e., without making use of EA¼0). In independent field variables and an evaluation of the such a case, (2.28) and (2.30) are known as the gauge trans- Lagrangian constraints in the various constraint surfaces formations and gauge identities in the theory, respectively. will generically break manifest covariance. This should not Given (2.28), g is equal to the number of different θ be confused with the loss of covariance. parameters there present. Equivalently, g is the number of linearly independent gauge identities (2.30). On the other B. Off-shell gauge identities hand, e is equal to the total number of distinct parameters We now obtain g and e, the two remaining numbers in plus their successive time derivatives ðθ; θ_; θ̈; …Þ that the triplet tðNÞ defined in (2.1) of our interest. To begin with, appear in (2.28). Obviously, e ≥ g. we notice that in the principle of stationary action (2.9),we The recursive construction of the gauge generators A have so far only considered that δS ¼ 0 follows from the Ωβ has been a subject of vivid interest for decades. 065015-8 LAGRANGIAN CONSTRAINT ANALYSIS OF FIRST-ORDER … PHYS. REV. D 102, 065015 (2020) FIG. 1. Schematics of Sec. II. Here, eqns., Lag. consts., f.i., and num. stand for equations, Lagrangian constraints, functionally independent, and number, respectively. The computational challenge of the steps relating Lagrangian constraints to functionally independent Lagrangian constraints (represented with a double arrow), as well as the relevance of closing the iterative algorithm, are further discussed in Sec. VI. The approach in [29] is perhaps the most befitting to our own possible for the theories (within the framework of Sec. II) exposition, requiring only a suitable adaptation from particle describing the dynamics of a single vector field. Recall there systems to manifestly first-order field theories that is devoid are only two distinct types of vector fields that one can of conceptual subtleties. We shall not present the corre- entertain classically: massless and massive. For simplicity, sponding discussion here because, for the theories at hand, wewill restrict to real Abelianvector fields and focus on their the explicit form of the gauge transformations is already most elementary actions: Maxwell electromagnetism and known. This a priori knowledge allows us to effortlessly the (hard) Proca theory, respectively. We shall consider two A infer the generators Ωβ in all the subsequent examples. equivalent formulations of each of these theories, based on We stress that the determination of g and e is possible different numbers N and N ≠ N of a priori independent field and has been made systematic in theories for which the variables. Our forthcoming detailed analyses are based on gauge transformations are unknown from the onset. The the purely Lagrangian method described in the previous calculations in such theories are more involved, but there is Sec. II and thus serve to illustrate it. no theoretical obstacle that has to be overcome. To illustrate Besides and as we shall explain in Sec. VB, our forth- this point, the reader can consult [28] for the explicit coming elementary calculations turn out to be enough to derivation of the gauge generators in Yang-Mills theory and understand the complete set of manifestly first-order (self-) both the metric and Palatini formulations of general interactions among an arbitrary number of both Maxwell relativity, by means of the formalism put forward in [29]. and (generalized) Proca [30] fields in four-dimensional flat For the ease of the reader, we have schematically spacetime [8,9]. This hints to the convenience of the depicted the main line of reasoning behind this Sec. II proposed method, compared to other possible approaches, in Fig. 1. a point that shall be reinforced in the more elaborate examples of the next Sec. IV and discussed in the concluding Sec. VI. III. SIMPLE EXAMPLES: In the remaining of this section, we shall work on VECTOR FIELD THEORIES d-dimensional Minkowski spacetime, still for finite d ≥ 2. This section is devoted to the study of some of the We will choose Cartesian coordinates with the mostly N constraint structure characterizing triplets tð Þ that are positive signature, so that gμν ¼ ημν ¼ diagð−1; 1; 1; …; 1Þ. 065015-9 VERÓNICA ERRASTI DÍEZ et al. PHYS. REV. D 102, 065015 (2020) Subsequently, all spacetime indices shall be raised or The Gauss law constraint straightforwardly yields a μν ˜ lowered by ημν and its inverse η . vanishing secondary Hessian W11 ≡ 0, so that M2 ¼ M1 ¼ 1 1 and we choose ðγ˜1Þ ¼ 1. With all this information, it is a A. Maxwell electromagnetism matter of easy algebra to find the only secondary Lagrangian constraint: This renowned manifestly first-order singular field theory describes an Abelian massless vector field and its φ˜ 1≋0: ð3:5Þ linear interactions with sources in terms of N ¼ d number 1 of a priori independent field variables. As already stated, 0 0 we take the Maxwell vector field (which we denote Aμ)to Therefore, M2 ¼ and the end of the iterative algorithm is be real and consider the particularly simple case when there signalled according to the nondynamical prescription in are no sources. case (iia). We have thus found that the total number of Lagrangian constraints for Maxwell electromagnetism is l M0 M0 1 1. Lagrangian constraints just ¼ 1 þ 2 ¼ . The canonically normalized Lagrangian density of 2. Gauge identities sourceless classical electromagnetism is Maxwell’s theory enjoys an apparent Uð1Þ gauge sym- 1 μν metry. Indeed, under the transformation Aμ → Aμ þ ∂μθ, L ¼ − AμνA ; with Aμν ≔ ∂μAν − ∂νAμ ¼ −Aνμ: M 4 the Lagrangian (3.1) remains invariant. Here, θ is the only ð3:1Þ gauge parameter, while ðθ; θ_Þ are the sole two effective gauge parameters present in the fields’ transformation. The components of the Maxwell field constitute the gener- Consequently, we have that g ¼ 1 and e ¼ 2. A alized coordinates for this theory: Q ¼fAμg, so that For completeness, we point out that the said trans- A ¼ 1; 2; …;d¼ dimðCÞ ≡ N, as already announced. As formation, when compared to (2.28) immediately allows iswellknownandcanbeeasilycalculatedbymeansof(2.10), us to read off the gauge generator of the symmetry. This is the Euler-Lagrange equations following from (3.1) are ðΩAÞν ¼ −δAν. When combined with the primary Euler- Lagrange equations (3.2) as indicated in (2.29), we can μν ! EA ≡ −∂μA ¼0: ð3:2Þ right away verify the off-shell gauge identity we μν counted: ϱ ¼ ∂μ∂νA ≡ 0. If we decompose the Maxwell field into its space and ≔ ≡ 1 time components Aμ ðA0;AiÞ ðA; AiÞ with i ¼ ; 3. Physical degrees of freedom 2; …;d− 1, the Lagrangian (3.1) can be conveniently rewritten as According to our prior analysis, which shows that the constraint structure of classical electromagnetism in its 1 1 standard formulation with N ¼ d is L _ 2 ∂ 2 − 2 _ ∂ − 2 M ¼ 2 ½Ai þð iAÞ Ai iA 4 Aij; ð3:3Þ ðNÞ 1 1 2 tM ¼ðl ¼ ;g¼ ;e¼ Þ; ð3:6Þ where sum over repeated indices is to be understood and we have been careful to lower all indices with the flat metric ημν. and making use of the master formula (2.2), we count − 2 4 It is then easy to see that the primary Hessian following ndof ¼ d propagating modes. In d ¼ , these corre- 1 1 from (3.3) is WAB ¼ δAB − δA δB and therefore manifestly spond to the two polarizations of the photon. Exploiting the possesses the symmetry dictated by its very definition: equalities in (2.4), we check that our counting corresponds WAB ¼ WBA. Further, its Moore-Penrose pseudoinverse is to two first-class constraints, one primary and one secon- AB AB A B given by M ¼ δ − δ1 δ1 . Since the primary Hessian dary. Therefore, our purely Lagrangian investigation is in takes such an uncomplicated form, it readily follows that perfect agreement with the standard literature, e.g., [31].It 0 R1 ¼ 3 and thus M1 ¼ 1ð¼ M1Þ in this case. A convenient also matches the Hamiltonian definition of the Maxwell A A “ … choice for the null vector of WAB amounts to ðγ1Þ ¼ δ1 . field given in [8]: a real Abelian vector field ½ Then, the one and only primary Lagrangian constraint for the associated with two first-class constraints.” This latter theory can be effortlessly calculated to take the explicit form correspondence will play a role in Sec. VB. ! φ1 ¼ ∂ A 0∶≈0: ð3:4Þ B. The (hard) Proca theory i i 1 We turn our attention to the Proca theory next, in the This is the familiar Gauss law, telling us that, in the absence modern formulation of the original proposal in [32]. of sources, the electric field is divergenceless. Note that this Namely, we focus on the (manifestly first-order) field is an on-shell statement by construction. theory of a real Abelian vector field of mass m in the 065015-10 LAGRANGIAN CONSTRAINT ANALYSIS OF FIRST-ORDER … PHYS. REV. D 102, 065015 (2020) absence of any source described by N ¼ d a priori 2. Gauge identities independent field variables. The remark (hard) is to avoid The mass term for the Proca field explicitly breaks the ambiguity with respect to the generalized Proca theory, Uð1Þ gauge invariance of Maxwell electromagnetism. In discussed in Sec. VB. We refer to the Proca field as Bμ. our conventions, this means that there is no field trans- formation of the form (2.28) that leaves the action invariant. 1. Lagrangian constraints Therefore, there are no off-shell identities associated to (3.7) and we have g ¼ 0 ¼ e. The Lagrangian density of the said Proca theory is 1 μν 1 2 μ 3. Physical degrees of freedom L ¼ − BμνB − m BμB ; with P 4 2 Using the (hard) Proca constraint structure for N ¼ d Bμν ≔ ∂μBν − ∂νBμ ¼ −Bνμ: ð3:7Þ ðNÞ 2 0 0 tP ¼ðl ¼ ;g¼ ;e¼ Þð3:11Þ As in the Maxwell case earlier on, the components of the A Proca field are the generalized coordinates: Q ¼fBμg. obtained before in the master formula (2.2), we count − 1 We thus see that A ¼ 1; 2; …;d¼ dimðCÞ ≡ N here as ndof ¼ d degrees of freedom in the theory. By means of well. The Euler-Lagrange equations following from (3.7) (2.4), it is immediate to certify that this corresponds to two can be easily obtained as indicated in (2.10). The result is second-class constraints, as explicitly shown, for instance, in [33]. As with the Maxwell field before, we thus find μν 2 ν ! agreement with the Proca field’s definition given in [8]: E ≡ −∂μB þ m B ¼0: ð3:8Þ A “a real Abelian vector field ½… associated with two ” At this point, it is straightforward to see that the second-class constraints. We will further comment on this primary Hessian—and hence also its Moore-Penrose connection in Sec. VB later on. pseudoinverse—is the same as for the Maxwell theory 0 earlier on. This implies M1 ¼ 1ð¼ M1Þ and the associated C. The Schwinger-Plebanski A A null vector can again be chosen as ðγ1Þ ¼ δ1 . The reformulation of Maxwell and Proca primary Lagrangian constraint differs, though: In this section, we reanalyze the constraint structures of the above massless and massive vector field theories in ! 2 a formulation with N ≠ N ¼ d a priori degrees of free- φ1 ¼ ∂ B 0 − m B∶≈ 0; ð3:9Þ i i 1 dom. Specifically, we entertain the reformulation of source- less classical electromagnetism originally proposed by ≔ ≡ where we have introduced Bμ ðB0;BiÞ ðB; BiÞ. Schwinger [34] and later on popularized by Plebanski The above once more leads to a vanishing secondary [35] and employ it for the (hard) Proca theory simulta- 1 Hessian, so that M2 ¼ M1 ¼ 1 and ðγ˜1Þ ¼ 1. The secon- neously. In this setup, the real Abelian (covariant) vector dary Lagrangian constraint in this case takes the form field Cμ—be it massless or massive—and its antisymmetric (contravariant) field strength Fμν are regarded as indepen- ! 2 _ φ˜ 1 ¼ −m B ∶≈ 0: ð3:10Þ dent at the onset: 2 A ij ji i 0i i0 Q ¼fC ≡ C0;F ¼ −F ;F ≡ F ¼ −F ;Cg; Contrary to the Maxwell theory, (3.10) is obviously not a i Lagrangian identity, so the algorithm is not closing here A ¼ 1; 2; …; N ¼ dðd þ 1Þ=2: ð3:12Þ according to the prescription in case (iia). Notice as well that φ1 and φ˜ 1 are functionally independent from each The aim of this Sec. III C is to determine the constraint ðNÞ ðNÞ other, so that we are not in case (iib) of the general method structure characterizing triplets tM and tP ,soasto 0 either. Instead, we have M2 ¼ M2 ¼ 1 and we must move illustrate in a simple double example the general claim on to the tertiary stage. in (2.6). Namely, these triplets differ from the previously It is easy to check that the tertiary Hessian following determined ones tðNÞ and tðNÞ but yield the same number of ˆ 2 M P from (3.10) is W11 ¼ −m . As such, its dimension and rank propagating degrees of freedom. 0 match (M3 ¼ 0 ¼ M3) and the algorithm closes according A clarifying remark follows. Classical electromagnetism to the dynamical prescription in case (i). Namely, the as written in [34] is commonly called the manifestly first- consistency of (3.10) under time evolution is ensured via a order formulation of electrodynamics. This refers to the tertiary equation of motion and there are no tertiary order of its primary Euler-Lagrange equations, contrarily 0 0 constraints. As a result, we have obtained l ¼ M1 þ M2 þ to our convention here, where the order refers to the 0 M3 ¼ 2 functionally independent Lagrangian constraints in Lagrangian density. For us, all examples in Secs. III the (hard) Proca theory. and IV are manifestly first order and as such can be 065015-11 VERÓNICA ERRASTI DÍEZ et al. PHYS. REV. D 102, 065015 (2020) investigated by means of the methodology in Sec. II.In A ≡ A1A2, the first referring to the type of field variable view of this dissonance, we can already anticipate that there and the second to the spacetime structure of each type of will be no primary equations of motion in our subsequent field variable. In this way, A1 ¼ 1; 2; …; 4 and we have examples. The primary Euler-Lagrange equations, being ̈A 1 ≡ 2 ij ≡ ij 3 i ≡ i 4 ≡ first order, will not involve the generalized accelerations Q ½Q C; ½Q F ; ½Q F ; ½Q i Ci: and so they will all be primary Lagrangian constraints. Further, this is possible iff the primary Hessian of the ð3:16Þ theories identically vanishes, as we shall see it does. Observe that we have employed the symbol ½· to visually split the A1 index from the A2 one. 1. Lagrangian constraints Back to the algorithm and putting into practice the above Inspired by [34], we take the Lagrangian density notation, we write the primary Lagrangian constraints as μν 1 μν i 2 L L − ∂ − ∂ ½φ1 ¼∂iF − m C; ½φ2 ¼ Fij − 2∂ iCj ; V ¼ V½Cμ;F ¼ 2 ð μCν νCμÞF ij ½ _ i _ i ij 2 i ½φ3 ¼ Fi þ Ci − ∂iC; ½φ4 ¼ −F − ∂jF þ m C : 1 μν 1 2 μ i þ FμνF − m CμC with m ≥ 0 ð3:13Þ 4 2 ð3:17Þ as our starting point. When m ¼ 0, (3.13) describes Notice that φ2 − φ2 , as required by definition. ≠ 0 ½ ij ¼ ½ ji classical electromagnetism. For m , the (hard) Proca We go on to the secondary stage next. The secondary theory is portrayed. The Euler-Lagrange equations follow- ˜ ∂ φ Hessian WIJ ¼ I_ J can be portrayed in our recently ing from (3.13) are introduced notation as follows: ! 0 1 ≔−∂ μν 2 ν 0 E Cν μF þ m C ¼ ; ˜ ˜ ˜ ˜ i ð Þ ½W11 ½W12 ij ½W13 i ½W14 ! B C μν ≔ − ∂ ∂ 0 B ˜ ˜ ˜ ˜ k C EðF Þ Fμν μCν þ νCμ¼ : ð3:14Þ B ij½W21 ij½W22 kl ij½W23 k ij½W24 C W˜ ¼ B C; IJ B ˜ ˜ ˜ ˜ j C ½W31 ½W32 ½W33 ½W34 Solving the latter for Fμν and substituting the result into the @ i i jk i j i A ’ ’ i ˜ i ˜ i ˜ i ˜ j former, we recover Maxwell s (3.2) or Proca s (3.8) equa- ½W41 ½W42 jk ½W43 j ½W44 tions of motion, depending on the value of m. Then, we say ð3:18Þ both formulations, in (3.13) and in (3.1) or (3.7) as pertinent, are dynamically equivalent, as foretold. We proceed to explicitly confirm our predictions. The where, for each entry of the secondary Hessian, we have primary Hessian following from (3.13) vanishes identically placed the spacelike tensorial indices of the field variables (primary Lagrangian constraints) labeled by I (J) to the left W ≡ 0,soR1 ¼ 0 and M1 ¼ N. We can choose its AB (right). A few explicit examples that should clarify our appropriate null vectors as ðγ ÞA ¼ δ A. As a result, the I I notation are primary Lagrangian constraints coincide with the primary Euler-Lagrange equations. These can be readily ∂ φ ∂ φ seen to be functionally independent among themselves. ˜ ≔ ½ 1 ≡ ½ 1 ½W11 ∂ 1 ∂ ; Consequently, the first constraint surface TC1 coincides ½Q C with the moduli space in this case. This set of circum- ˜ ∂½φ1 ∂½φ1 ½W21 ≔ ≡ ; stances can be summarized as ij ∂½Q2 ij ∂Fij ! ∂ φ2 ∂ φ2 ! i ˜ ½ jk ½ jk 0 α γ Aα φ ∶≈0 ½W42 ≔ ≡ : ð3:19Þ ¼EA ¼ A ¼ð IÞ A ¼ I ð3:15Þ jk ∂ 4 ∂ 1 ½Q i Ci 0 ≡ 0 or simply as M1 ¼ M1 ¼ N. Notice that WAB immedi- The only nonzero components in (3.18) are ately makes its Moore-Penrose pseudoinverse vanish as well: MAB ¼ 0. We encounter this same situation of a zero j ˜ δj − ˜ j ½W43 i ¼ ¼ i½W34 ; ð3:20Þ primary Hessian in both of the theories analyzed in Sec. IV. i We briefly depart from the application of the iterative which lead to a simple secondary Moore-Penrose pseu- algorithm in order to introduce an extremely useful notation ˜ ˜ i i doinverse M with nonzero elements ½M43 ¼ −δ ¼ that will be recurrent from now on. We wish to be able to AB j j i ˜ refer to each kind of field variables in (3.12) individually. ½M34 j. This corresponds to the transpose of (3.18).Itis R ≔ ˜ 2 − 1 To this aim, we shall henceforth understand that the index A easy to see that 2 rankðWIJÞ¼ ðd Þ, which in turn 2 therein decomposes into two distinct sets of indices implies that M2 ¼ðd − 3d þ 4Þ=2. We choose the 065015-12 LAGRANGIAN CONSTRAINT ANALYSIS OF FIRST-ORDER … PHYS. REV. D 102, 065015 (2020) ˆ − 2 ˆ 2δ δ suitably normalized linearly independent null vectors for ½W11 ¼ m ; ij½W22 kl ¼ i½k l j: ð3:26Þ I I (3.18) as ðγ˜RÞ ¼ δR . 0 The above results can be employed to determine the Hence, the tertiary Hessian has full rank R3 ¼ M2 and functionally independent secondary Lagrangian constraints consequently M3 ¼ 0. Observe that this is true for both the ! ˜ ˜ I _ m ¼ 0 and the m>0 cases. The functionally independent φR∶≈0. First, we calculate φR ¼ðγ˜RÞ φI and obtain 2 secondary constraints’ consistency under time evolution is at this point dynamically ensured and the algorithm closes φ˜ ∂ _ i − 2 _ φ˜ _ − 2∂ _ ½ 1 ¼ iF m C; ½ 2 ij ¼ Fij ½iCj ; ð3:21Þ according to the prescription in case (i). The total number of functionally independent Lagrangian constraints is φ˜ − φ˜ where again the antisymmetry property ½ 2 ij ¼ ½ 2 ji d d − 1 1 if m 0; required by definition is apparent. Evaluation on the first 0 0 ð Þþ ¼ l ¼ M1 þ M2 ¼ ð3:27Þ constraint surface then gives dðd − 1Þþ2 if m>0: 2 i _ _ ½φ˜ 1 ≈m ð∂ C − CÞ; ½φ˜ 2 ≈F þ 2∂ F ; ð3:22Þ We see that the mass m gives rise to one more functionally 1 i ij 1 ij ½i j independent Lagrangian constraint, exactly as in the previous sections, where we found that l 1 for electro- which respects the noted symmetry, as it must. In more ¼ magnetism, while l 2 for the (hard) Proca theory. detail, the evaluation has been carried out as follows: by ¼ φ’ _ _ Here, the remaining constraints that l counts are asso- setting to zero all sin(3.17), solving for ðFi; CiÞ and ciated to the field strength Fμν, as a result of having plugging the resulting expressions into (3.21). Next, we promoted it to a set of a priori independent field variables. need to select only the functionally independent secondary Notice that there are dðd − 1Þ number of such supplemen- constraints. It is obvious that the mass m plays a crucial role tary constraints, 2 times the number of independent here, as could easily be anticipated in view of our results in components in Fij. This duplicity makes it manifest that the previous Secs. III A and III B. Indeed, if m 0, then ¼ these fields are superfluous when describing the dynamics one constraint identically vanishes in the first constraint of the theory. In other words, no initial data are needed for surface ½φ˜ 1 ≋0. It is thus a Lagrangian identity, meaning 1 them: Fij and F_ ij need not be specified at some initial time φ that ½ 1 is nondynamically (trivially) stabilized at the t1 when solving their associated equations of motion. Yet secondary stage in this case. We therefore see that another way to understand this is to map them to the Hamiltonian picture, where they correspond to second- M2 − 1 if m 0; 0 ¼ class constraints, as we shall shortly see. M2 ¼ ð3:23Þ M2 if m>0: 2. Gauge identities We turn to the time evolution of the functionally Consider the following transformations of the field independent secondary constraints; i.e., we commence → δ μν → μν δ μν the tertiary stage. The tertiary Hessian can be succinctly variables: Cμ Cμ þ θCμ and F F þ θF , with expressed as μν δθCμ ¼ ∂μθ; δθF ¼ 0: ð3:28Þ ˆ ˆ ½W11 ½W12 ˆ ij Here, θ is an arbitrary parameter. It can be easily checked WRS ¼ ˆ ˆ ; ð3:24Þ ij½W21 ij½W22 kl that, under the said transformations, the Lagrangian (3.13) remains invariant iff m ¼ 0. Therefore, these are the very where we have made use of the same notation as in (3.18) same gauge transformations of the massless theory that earlier on, so that we noted in Sec. III A, while the massive theory does not exhibit any kind of gauge symmetry. Straightforwardly, ∂ φ˜ ∂ φ˜ we count ˆ ½ 1 ˆ ½ 2 ij ½W11 ≡ ; ½W12 ≡ ; ∂ _ ij ∂ _ C C 1 0 2 0 ∂ φ˜ ∂ φ˜ if m ¼ ; if m ¼ ; ˆ ≡ ½ 1 ˆ ≡ ½ 2 kl g ¼ e ¼ ð3:29Þ ij½W21 ; ij½W22 kl : ð3:25Þ 0 if m>0; 0 if m>0: ∂F_ ij ∂F_ ij ˆ For completeness, we provide the gauge identity and Notice that, for m ¼ 0, the first row ½W1R should not be generators for m ¼ 0 next. Comparing (2.28) and (3.28), present, as ½φ˜ 1 weakly vanishes in this case. However, we keep it along here, so that both Maxwell and (hard) Proca we can immediately read off the nonzero generators: theories can be reanalyzed simultaneously. The nonzero μ μ μ1 δ − ∂ θ Ω 1 Ω 1 −δμ components are explicitly given by θCμ ¼ ð μ1 Þð μÞ ; ð μÞ ¼ : ð3:30Þ 065015-13 VERÓNICA ERRASTI DÍEZ et al. PHYS. REV. D 102, 065015 (2020) Notice that here we have dropped the, in this case, single- inconvenient subtleties to our aims arise in those frameworks valued β index from (2.28). Putting together (3.14) and due to their geometric construction. For instance, unlike the (3.30) as indicated in (2.29), we readily confirm the gauge metric, vielbeine are not required to be invertible. In such identity: a scenario, the strict equivalence between Palatini and Einstein’s gravity is lost due to a singular vielbein and, in ϱ ∂ Ω μ1 ≡ 0 ¼ μ1 ½EðCμÞð μÞ : ð3:31Þ general, ends up in a dynamical manifestation of torsion [40]. Similar situations might arise in other manifestly first- order formulations, like the Barbero-Holst action [41] or 3. Physical degrees of freedom background field (BF)-like models [42], which happen to We have now achieved our goal. Namely, we have enclose the most celebrated Plebanski action. For a complete shown that the constraint structure characterizing triplet review on these topics, we refer the interested reader to [43]. for (3.13) is A. Palatini in d > 2 tðNÞ ¼ðl ¼ dðd − 1Þþ1;g ¼ 1; e ¼ 2Þ if m ¼ 0; tðNÞ ¼ M The Palatini action in d>2 is a well-known (re) V ðNÞ − 1 2 0 0 0 tP ¼ðl ¼ dðd Þþ ;g ¼ ; e ¼ Þ if m> : formulation of the Einstein-Hilbert action, which is ð3:32Þ dynamically equivalent to it. This is explained shortly. Most significantly for us, Palatini is a manifestly first-order Substituting the quantities (3.32) into the master for- formulation of general relativity, which treats the spacetime Γρ Γρ mula (2.2), we count metric gμν ¼ gνμ and the affine connection μν ¼ νμ as a priori independent variables. As such, and unlike d − 2 if m ¼ 0; Einstein-Hilbert, it readily allows for the application of n ¼ ð3:33Þ dof d − 1 if m>0; the methodology introduced in Sec. II. propagating degrees of freedom. This counting coincides 1. Lagrangian constraints with the ones performed in Secs. III A and III B, where The Palatini action is of the general form given in (2.8) appropriate. We have thus verified (2.6) in two simple and its Lagrangian density can be written as [44] examples. Exploiting the equalities in (2.4), we see the following relation to the Hamiltonian side. The massless μν ρ μν ρ σ ρ σ L ¼ −ð∂ρh ÞGμν þ h ðcGρμGσν − GσμGρνÞ; theory exhibits two first-class constraints, one of which is a Pa − 1 1 primary first-class constraint, and dðd Þ second-class with c ≔ : ð4:1Þ constraints. On the other hand, the massive theory has only d − 1 second-class constraints, dðd − 1Þþ2 of them. Our purely μν ρ Here, the independent variables h and Gμν are defined Lagrangian investigation is thus in perfect agreement with exclusively in terms of the spacetime metric and affine the standard Hamiltonian literature, e.g., [36]. connection, respectively: IV. A COMPREHENSIVE CONSTRAINT qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μν μν ρ ρ ρ σ ≔ − μν ≔ Γμν − δ Γ ANALYSIS OF PALATINI THEORIES h detðgμνÞg ;G ðμ νÞσ; ð4:2Þ In the following, we apply the general framework and thus inherit their symmetry properties. presented in Sec. II to the Palatini action. We split our μν λ The primary Euler-Lagrange equations for h and Gμν calculations into the d>2 and the d ¼ 2 cases, as these are physically distinct theories. As we shall see, the former case following from (4.1) are is much more algebraically involved than the latter. ρ ρ σ ρ σ ! μν ≔ ∂ − 0 However, compared to their equivalent Hamiltonian inves- Eðh Þ ρGμν þ cGρμGσν GσμGρν¼ ; ! tigations, our Lagrangian approach shall prove much μν λ σðμ νÞ ðμ νÞσ E ρ ≔−∂ρh þ 2cG h δρ − 2Gρσh ¼ 0: ð4:3Þ simpler in both instances. ðGμνÞ λσ For concreteness, we specify our framework to be that of the metric-affine Palatini formulation of general relativity, Notice that these vanishings are on-shell statements. ordinarily ascribed to Palatini but firstly suggested by Multiplying the second set of field equations by hμν and νρ δ ρ Einstein himself [10,37]. As such, we shall study a employing the identity hμνh ¼ μ , one finds that manifestly first-order formulation of gravity based on N ¼ 2 μ μν ! dðd þ 1Þ =2 number of a priori independent degrees of 2ðc − 1ÞGμρ − hμν∂ρh ¼ 0: ð4:4Þ freedom. Even though alternative manifestly first-order ρ formulations do exist, such as the tetradic-Palatini action Solving (4.4) implies that Gμν is fixed (on shell) to be a (for example, see [38] and its recent canonical study [39]), function of hμν and its first derivatives. The substitution of 065015-14 LAGRANGIAN CONSTRAINT ANALYSIS OF FIRST-ORDER … PHYS. REV. D 102, 065015 (2020) the resulting expression into (4.1) yields the second-order (3.16) and explanations around. This notation shall prove formulation of general relativity and we say d>2 Palatini of utmost convenience. For instance, in this way, it is is dynamically equivalent to it. obvious that It is natural and convenient to decompose the variables in (4.2) as follows: h11 if d ¼ 3; G ≡ ½Q4 ≠ Q4 ¼ ð4:9Þ 00 i 0i 0 3 h ≡ h ;h≡ h ;G≡ G00; h otherwise: G ≡ G0 ;G≡ G0 ; Gi ≡ Gi ; i 0i ij ij 00 The primary Hessian following from (4.6) vanishes Gi ≡ i Gi ≡ i j G0j; jk Gjk: ð4:5Þ identically: WAB ≡ 0, as a result of having promoted the affine connection to a set of a priori independent The explicit form of the Lagrangian (4.1) in terms of the field variables. This parallels the reformulations of classical above variables is electromagnetism and the (hard) Proca theory in Sec. III C. L −_ − 2_ i − _ ij − ∂ Gi − 2 ∂ j Gi In passing, we note that the primary Hessian is symmetric Pa ¼ hG h Gi h Gij ð ihÞ ð ih Þ j WAB ¼ WBA, as it should by definition. Obviously, − ∂ jk Gi − 1 2 2 Gi − Gi 2 ð ih Þ jk þ h½ðc ÞG þ ðcG i Gi Þ rankðWABÞ¼0 and we have M1 ¼ N ¼ dðd þ 1Þ =2. This trivialization of the primary Hessian has a number þ cGiGj − Gi Gj þ2hi½ðc − 1ÞGG þ cGGj i j j i i ij of direct implications. First, it allows us to straightfor- Gj − Gj − Gj GjGk − GkGj þ cGi j Gj i Gij þ c j ik j ik wardly pick its suitably normalized null vectors to be ðγ ÞA ¼ δ A. Second, it immediately makes its Moore- þ hij½ðc − 1ÞG G þ 2ðcG Gk − G Gk Þ I I i j ði jÞk kði jÞ Penrose pseudoinverse vanish as well: MAB ¼ 0. Third, Gk Gl − Gk Gl þ c ik jl il jk : ð4:6Þ it becomes apparent that the primary Euler-Lagrange equations coincide with the primary Lagrangian con- We express the generalized coordinates of the Palatini straints. All of these constraints turn out to manifestly Lagrangian in (4.6) as be functionally independent from each other in this specific QA ¼fh; hi;hij;G;G;G ; Gi; Gi ; Gi g: ð4:7Þ theory. In other words, the moduli space is the primary i ij j jk 0 constraint surface in this case and we have M1 ¼ M1. Thus, Notice that A comprises all possible indices of our chosen field variables, so that ! ! 0 ¼ E ¼ α ¼ðγ ÞAα ¼ φ ∶≈ 0; ð4:10Þ A A I A I 1 A ¼ 1; 2; …;dðd þ 1Þ2=2 ¼ dimðCÞ ≡ N: ð4:8Þ exactly as in our examples of Sec. III C before; see (3.15). Henceforth, we shall employ the notation ½· introduced in By means of the notation employed in (3.17), the explicit Sec. III C for the collective index A above. In particular, see form of the φI’sis φ − _ ∂ Gi 2 Gi − Gi Gi Gj ½ 1 ¼ ½G þ i þ GG þ ðcG i Gi Þþ j i ; ½φ2 i − G_ ∂ Gj GG cGGj G Gj − G Gj Gj Gk ; 2 ¼ ½ i þ j i þ i þ ij þ j i ij þ k ij φ − _ ∂ Gk 2 Gk − Gk Gl Gk ½ 3 ij ¼ ½Gij þ k ij þ GiGj þ ðcGði jÞk Gkði jÞÞþ ik lj ; φ _ − 2 Gi i iGj ½ 4 ¼h ½hG þ ch i þ h Gi þ ch ij ; i ½φ5 h_ i hGi − hiG hjGi hijG chijGk ; 2 ¼ þ ½ þ j þ j þ jk φ ij _ ij 2 ðiGjÞ kðiGjÞ ½ 6 ¼ h þ ðh þ h k Þ; φ ∂ 2 j ½ 7 i ¼ ih þ ðhGi þ h GijÞ; j ½φ8 i ∂ hj − c hG hkG δj hGj hkGj hjG hjkG ; 2 ¼ i ½ ð þ kÞ i þ i þ ik þ i þ ik φ jk ∂ jk − 2 ðj δkÞ GkÞ lðj δkÞ GkÞ ½ 9 i ¼ ih ½h ðcG i þ i Þþh ðcGl i þ il Þ ; ð4:11Þ 065015-15 VERÓNICA ERRASTI DÍEZ et al. PHYS. REV. D 102, 065015 (2020) where we have defined Notice that the secondary Hessian is antisymmetric ˜ ˜ WIJ ¼ −WJI, as it should by definition. It is easy to see G ≔ c − 1 G; G ≔ c − 1 G ; ˜ ð Þ i ð Þ i that R2 ≔ rankðWIJÞ¼dðd þ 1Þ, thus yielding M2 ¼ 2 Gi ≔ Gkδi − Gi Gi ≔ Gl δi − Gi dðd − 1Þ=2. This means that R2 number of the function- j c k j j; jk c kl j jk: ð4:12Þ ally independent primary Lagrangian constraints are being dynamically stabilized at the secondary stage, while the Note that the φI’sin(4.11) are manifestly symmetric φ φ φ ij φ ji remaining M2 primary Lagrangian constraints are not where appropriate, i.e., ½ 3 ij ¼½ 3 ji, ½ 6 ¼½ 6 and jk kj stable: they lead to secondary Lagrangian constraints, ½φ9 ¼½φ9 . i i which we proceed to determine. We now turn to the secondary stage, where we inspect To this aim, we first choose the suitably normalized the consistency under time evolution of the functionally linearly independent null vectors associated to the secon- independent primary Lagrangian constraints. The secon- ˜ ∂ φ dary Hessian as dary Hessian is given by WIJ ¼ I_ J. With the conventions introduced below (3.18), we may succinctly write it as I ðγ˜RÞ ¼ð0; 0; …; 0; 1; 0; 0; …; 0Þ; ð4:16Þ 0 1 ˜ ˜ jk R B ½W11 ½W19 i C where the nonvanishing vector component is at I ¼ 2 þR. ˜ B . . . C Notice that all M2 null vectors have length M1 and their WIJ ¼ B . . . C: ð4:13Þ @ . . . A first R2 components are zero. jk ˜ jk ˜ mn All our results so far can be used to obtain the secondary i ½W91 i ½W99 l ! Lagrangian constraints φ˜ ≈0. These can readily be seen to R 1 For clarity, we provide a few examples of what is meant by be functionally independent from each other, as well as 0 our notation: with respect to the primary constraints, so that M2 ¼ M2. This means their vanishing defines the secondary constraint ∂ φ ∂½φ3 ! ˜ ≡ ½ 1 ˜ ≡ jk surface: φ˜ ∶≈0. In our ½· notation, we have ½W11 _ ; i½W23 jk _ ; R 2 ∂h ∂hi φ˜ ∂ φ i ∂ φ mn ½ 1 i j ˜ i ½ 5 jk ˜ mn ½ 9 l ≈hðτ˜1Þ þ h ðτ˜2Þ ; ½W45 ≡ ; ½W99 ≡ : ð4:14Þ 2 1 i ij ∂G_ i l ∂G_ i jk φ˜ j ½ 2 i j jk j k j ≈h ðτ˜1Þ þ h ðτ˜2Þ þ hðτ˜3Þ þ h ðτ˜4Þ ; In (4.13), the only nonzero components are 2 1 i ik i ik φ˜ jk ˜ −1 − ˜ ˜ j −2δj −j ˜ ½ 3 i ðj kÞ lðj kÞ ½W14 ¼ ¼ ½W41 ; i½W25 ¼ i ¼ ½W52 i; ≈h ðτ˜3Þ þ h ðτ˜4Þ ; ð4:17Þ 2 1 i il ˜ kl k l kl ˜ ½W36 ¼ −δ δ ¼ − ½W63 : ð4:15Þ ij ði jÞ ij where we have defined τ˜ ≔ ∂ ∂l Gk G − − 2 Gk GkG˜ l ð 1Þi iG þ k·i l þ G i GGi þðc ÞGk i þ l ik; τ˜ ≔ ∂ ∂l Gk G − Gk − Gk Gk G˜ l ð 2Þij iGj þ k·i lj þ Gj i GGij þ Gk ij Gjk i þ jl ik; τ˜ j ≔−δj·l G_ m ∂j Gl Gj − Gk·j GlGj − GmGl·j ð 3Þi i·m l þ i·l þ G i Gk i þ il l m·i; τ˜ j ≔−δj·l G_ m ∂j Gl − Gl·j Gl Gj − GmGl·j ð 4Þik i·m lk þ i·l k Gkl i þ k il kl m·i ð4:18Þ in terms of (4.12) as well as the following quantities: ∂ ≔ − 1 ∂ ∂k ≔ δk∂ − δk∂ δk·l ≔ δkδl − δkδl G ≔ Gk i ðc Þ i; i·j c i j j i; i·j c i j j i; i cðGi þ ikÞ; Gi·j ≔ Gi δj − Gj δi Gi·j ≔ Giδj − Gjδi G˜ k ≔ − 1 Gl δk − Gk k·l c k l k l; k c k k; ij cðc Þ il j ij: ð4:19Þ φ˜ jk φ˜ kj A Observe that the appropriate symmetry ½ 3 i ¼½ 3 i is expressions in terms of the generalized coordinates Q , manifest. To obtain the above, we have first computed which follow from setting to zero (4.11). φ˜ ¼ðγ˜ ÞIφ_ . Then, we have evaluated the result in the To conclude the secondary stage, we calculate the R R I ˜ first constraint surface. In practice, this means that we Moore-Penrose pseudoinverse of WIJ. It can be easily _ _ jk ˜ IJ − ˜ T have substituted ðG; Gi; …; ∂ih Þ for their suitable weak checked that this is M ¼ ðWIJÞ . 065015-16 LAGRANGIAN CONSTRAINT ANALYSIS OF FIRST-ORDER … PHYS. REV. D 102, 065015 (2020) Next, the consistency under time evolution of the above where the (d − 1) number of nonvanishing vector compo- 11 functionally independent secondary Lagrangian constraints nents are at R ¼ M3; 2M3; …; ðd − 1ÞM3. is to be inspected at the tertiary stage. The first step is to The tertiary Lagrangian constraints are given by the ˆ calculate the tertiary Hessian W ¼ðγ˜ ÞI∂_φ˜ . Employing ! RS R I S requirement φˆ ¼ðγˆ ÞRφ˜_ ≈ 0. In this case, the derivation the same conventions as in (4.13) before, we write U U R 2 0 1 with respect to time is particularly simple and coincides ˆ ˆ k ˆ kl with the naively expected one, so that B i½W11 j i½W12 j i½W13 j C ˆ B j j j C ! W B ˆ ˆ m ˆ mn C; 4:20 R _ A i _ A RS ¼ @ i½W21 l i½W22 l i½W23 l A ð Þ φˆ ¼ðγˆ Þ ½∂ Q ∂ þ Q ∂ φ˜ ≈ 0: ð4:24Þ U U i A A R 2 jk ˆ jk ˆ m jk ˆ mn i ½W31 l i ½W32 l i ½W33 l In our shorthand notation, we find it convenient to express where the nonzero components are these constraints as follows: ∂ φ˜ l φˆ j ˆ l ½ 2 k j l l j ½ 1 i _ _ j j W22 ≡ −2h cδ δ − δ δ ; ¼ hðτ˜1Þ þh ðτ˜2Þ þO½hðτ˜1Þ þh ðτ˜2Þ ; i½ k _ i ¼ ð i k i kÞ 2 i ij i ij ∂Gj mn ½φˆ 2 ∂½φ˜ 3 _ i τ˜ _ ij τ˜ O i τ˜ ij τ˜ jk ˆ mn ≡ l 2 δðm nÞðjδkÞ − δðm nÞðjδkÞ 2 ¼ h ð 1Þi þh ð 2Þij þ ½h ð 1Þi þh ð 2Þij ; ð4:25Þ i ½W33 l _ i ¼ ð i h l c l h i Þ; ∂Gjk O ∂ φ˜ lm where the operator is defined as j ˆ lm ½ 3 k ðl j mÞ mÞ j ½W23 ≡ ¼ −2h ðcδ δ − δ δ Þ i k ∂G_ i i k i k ∂ ∂ ∂ j _ l O ≔ ð∂ G_ Þ þð∂ G_ Þ þð∂ G Þ j k ∂ ∂ k l ∂ ∂ k m l ∂ φ˜ ð kGÞ ð kGlÞ ∂ð∂kGmÞ ½ 2 i ≡ lm ˆ j ¼ ½W32 i : ð4:21Þ ∂ ∂ ∂ ∂G_ k k _ l _ _ lm ∂ G G G þð k mnÞ l þ ∂ þ k ∂ ∂ð∂kGmnÞ G Gk Therefore, the tertiary Hessian obviously satisfies compo- ∂ ∂ ∂ ˆ ˆ _ G_ k G_ k nentwise the symmetry properties that ensure WRS ¼ WSR, þ Gkl þ l k þ lm k : ð4:26Þ ∂Gkl ∂G ∂G as required by definition. It is not hard to see that R3 ≔ l lm ˆ 2 rankðWRSÞ¼dðd − 3Þ=2 and hence M3 ¼ d. In more τ˜ τ˜ Recall that ð 1Þi and ð 2Þij are as introduced in (4.18). detail, the rank is equal to all nonzero rows (equivalently, Following the procedure described under (4.19), the columns) of the Hessian, minus one. There is the obvious set ˆ tertiary Lagrangian constraints in (4.25) can be evaluated of d − 1 number of zero rows given by W1 0. But the i½ R ¼ on the first constraint surface TC1. After tedious algebraic rank of the Hessian is further reduced by one because the manipulations, the weak tertiary Lagrangian constraints can i ˆ linear combination of rows i½W2R is zero. This vanishing is a be written exclusively in terms of the functionally inde- φ˜ i direct consequence of the velocity independence of ½ 2 i in pendent secondary constraints as TC1, as can be readily verified from (4.17) and (4.18).It follows that R3 number of the functionally independent φˆ ≈ − δj − 2Gj φ˜ − δj ∂ ½ 1 i ð i G i Þ½ 1 j ½ i ðGk þ kÞ secondary Lagrangian constraints are being dynamically 1 1 stabilized at the tertiary stage. The remaining M3 secondary j j k jk − δ ∂ − G ½φ˜ 2 − G ½φ˜ 3 ; Lagrangian constraints are not stable: they lead to the tertiary 2 k i ik j jk i Lagrangian constraints that we shall find next. φˆ ½ 2 i j i ij ≈2G ½φ˜ 1 þ G ½φ˜ 2 þ ∂ ½φ˜ 3 : ð4:27Þ The suitably normalized linearly independent null vec- 2 1 i i j j i tors of the tertiary Hessian can be chosen as follows. ˆ 0 Associated to i½W1R ¼ , we pick The above is a nontrivial result. Indeed, it becomes increasingly computationally challenging to evaluate γˆ R 0 0 … 0 1 0 0 … 0 ð UÞ ¼ð ; ; ; ; ; ; ; ; Þ; Lagrangian constraints on constraint surfaces as one goes with U ¼ 1; 2; …;M3 − 1; ð4:22Þ to higher stages. We elaborate on this topic and advice on how to handle the evaluations in Sec. VI. where the nonvanishing vector component is at R ¼ U. R These null vectors have length M2 and their last 3 11To determine the null vectors of Wˆ in (4.20), we consid- i ˆ RS W2 0 γˆ R i i i i i components are all zero. Corresponding to i½ R ¼ , ered the ansatz ð UÞ ¼ða ;aj;ajkÞ, with ajk ¼ akj. Then, the we select the null vector γˆ R ˆ 0 j lδj j k δj equation ð UÞ WRS ¼ results in ai ¼ cal i and ail ¼ cakl i but does not impose any condition on ai. The first equation 1 ai 0 i ≠ j j l ðγˆ ÞR ¼ pffiffiffiffiffiffiffiffiffiffið0;…;0;1;0;…;0;1;0;…;0Þ; ð4:23Þ implies j ¼ for . Setting ¼ in the second equation U¼M3 − 1 j 0 i 0 d yields aij ¼ , which in turn implies ajk ¼ for all i, j, k. 065015-17 VERÓNICA ERRASTI DÍEZ et al. PHYS. REV. D 102, 065015 (2020) δ μν θβ Ω μν − ∂ θβ Ω μν μ1 Our results in (4.27) must be further evaluated on the θh ¼ ½ð βÞ ð μ1 Þ½ð βÞ ; second constraint surface TC2, namely, in the subspace of ρ β ρ β ρ μ δθGμν ¼ θ ½ðΩβÞμν − ð∂μ θ Þ½ðΩβÞμν 1 TC1 defined by the vanishing of (4.17). We thus see that 1 ρ ∂ ∂ θβ Ω μ1μ2 þð μ1 μ2 Þ½ð βÞμν ; ð4:32Þ ½φˆ 1 ≈ 0; ½φˆ 2 ≈ 0; ð4:28Þ i 2 2 where we have introduced a bracket ð·Þ to visually split the (in general collective) indices β and A for latter conven- which implies TC3 ≡ TC2 and there are no functionally ience. In view of these transformations, the gauge gen- 0 independent tertiary constraints M3 ¼ 0. Consequently, the erators can easily be identified to be algorithm closes nondynamically, according to case (iib). ν μ μν μν μν μ1 μ1ðμ Þ μν 1 We are finally able to obtain the result of interest from ½ðΩβÞ ¼−∂βh ; ½ðΩβÞ ¼ −2h δβ þ h δβ ; the analysis here presented. The number of functionally ρ ρ ρ μ ρ μ1 μ1 ρ ½ðΩβÞμν ¼−∂βGμν; ½ðΩβÞμν 1 ¼ 2G δ − Gμνδ ; independent Lagrangian constraints for the Palatini theory βðμ νÞ β 2 ρ ρ μ μ μ μ ρ in d>2, when described in terms of N ¼ dðd þ 1Þ =2 μ1μ2 ð 1 2Þ 1 2 ½ðΩβÞμν ¼ δ μδν δβ − δ μδν δβ: ð4:33Þ number of a priori independent field variables, is equal to ð Þ ð Þ Combining (4.3) with the above as prescribed in (2.29) and 0 0 0 l ¼ M1 þ M2 þ M3 ¼ M1 þ M2 þ 0 working through, the gauge identities are obtained: d 2 d 2 2 μν μν μ ¼ ðd þ 1Þ þ ðd − 1Þþ0 ¼ d ðd þ 1Þ: ð4:29Þ ρ μν Ω ∂ μν Ω 1 2 2 β ¼ Eðh Þ½ð βÞ þ μ1 ðEðh Þ½ð βÞ Þ ρ ρ μ þ E ρ ½ðΩβÞμν þ∂μ ðE ρ ½ðΩβÞμν 1 Þ 2. Gauge identities ðGμνÞ 1 ðGμνÞ ρ μ μ ∂μ ∂μ ρ Ωβ μν 1 2 ≡ 0 It is well known (for instance, see [14]) that the Palatini þ 1 2 ðEðGμνÞ½ð Þ Þ : ð4:34Þ action corresponding to the Lagrangian density (4.1) remains invariant under the following transformations μν μν μν of its independent variables: h → h þ δθh and 3. Physical degrees of freedom ρ ρ ρ Gμν → Gμν þ δθGμν, with Putting everything together, we can finally count the number of propagating modes present in the theory. μν ρðμ νÞ μν ρ Namely, employing (4.8), (4.29) and (4.31) in the master δθh ¼ 2h ∂ρθ − ∂ρðh θ Þ; formula (2.2), we get ρ ρ ρ σ σ ρ δ μν −∂ ∂ θ δ ∂ ∂ θ − θ ∂ μν θG ¼ μ ν þ ðμ νÞ σ σG σ ρ ρ σ d þ Gμν∂σθ − 2G ∂ν θ ; ð4:30Þ n d − 3 : 4:35 σðμ Þ dof ¼ 2 ð Þ ð Þ μ 4 2 where θ are the (unspecified) gauge parameters. Notice When d ¼ , we have that ndof ¼ , corresponding to the μν νμ ρ ’ that the pertinent symmetries δθh ¼ δθh and δθGμν ¼ two massless tensor s polarizations of the graviton. For ρ d 3 δθGνμ are apparent in the precedent expressions. Of course, ¼ , the widely known triviality is recovered, with no (4.30) is just the Palatini (re)formulation of the renowned physical degrees of freedom being propagated. diffeomorphism invariance of the Einstein-Hilbert action. Our result is in perfect agreement with the counting This holds true off shell. performed in [13,14], where a purely Hamiltonian analysis It is easy to see in (4.30) that the gauge parameters θμ was done. We have thus carried out another (nontrivial) appear explicitly in all the gauge transformations ∀ μ. explicit verification of the already noted equivalence Similarly, we note that the effective gauge parameters between (2.2) and (2.3). This equivalence can be further μ μ μ verified as follows. It is explicitly shown in [13,14] that ðθ ; θ_ ; θ̈Þ are manifestly present in the gauge transforma- 3 − 1 2 ðPÞ tions ∀ μ as well. By definition, it follows that N1 ¼ d, N2 ¼ dðd Þðd þ Þ and N1 ¼ d for the d>2 Palatini theory when (4.8) holds true. Substitution 3 of these results in (2.4) readily confirms our own counting g ¼ d; e ¼ d; ð4:31Þ in (4.29) and (4.31). Besides, a direct comparison between the calculations in [13,14] and those presented in this which are the off-shell parameters we aimed to obtain in Sec. IVA unequivocally shows that our purely Lagrangian this short analysis. computation is an algebraically much simpler way to For completeness, we provide the gauge generators and derive (4.36), from which the number of physical modes confirm the gauge identities of d>2 Palatini next. A direct follows readily. comparison between (2.28) and (4.30) allows us to rewrite To sum up, we have derived the constraint structure ðNÞ 1 2 2 the latter as characterizing triplet tPa , with N ¼ dðd þ Þ = , of the 065015-18 LAGRANGIAN CONSTRAINT ANALYSIS OF FIRST-ORDER … PHYS. REV. D 102, 065015 (2020) Palatini theory in d>2 dimensions in a purely Lagrangian equivalence to Einstein’s gravity is lost. A more general approach and ratified its equivalence with a represen- yet detailed argumentation can be found in [46]. tative Hamiltonian analysis performed in the past. Correspondingly, the dynamics of the two-dimensional Mathematically, Palatini action does not constitute a smooth limit of its higher-dimensional counterpart. Namely, the Lagrangian ðNÞ 2 1 3 d 2 tPa ¼ðl ¼ d ðd þ Þ;g¼ d; e ¼ dÞð4:36Þ (4.1) in ¼ does not describe the evolution of the same family of fields as that very same Lagrangian in d>2: in the Lagrangian picture, while these are two physically different theories. The easiest way to ratify this second inequivalence is to note that the ðNÞ ðPÞ 3 −1 2 counting of degrees of freedom in (4.35), when we set tPa ¼ðN1 ¼ d;N1 ¼ d;N2 ¼ dðd Þðd þ ÞÞ ð4:37Þ d ¼ 2, yields a negative number of propagating modes, in the Hamiltonian side—recall (2.5)—both of which which is an unphysical result. Thus, a different constraint imply (4.35). structure characterizing triplet B. A special case: Palatini in d =2 ðN¼9Þ ≠ ðNÞ 1 2 2 t2 limtPa ; where N ¼ dðd þ Þ = ; ð4:39Þ Pa d→2 General relativity, in its standard second-order formu- lation, behaves drastically different in two dimensions. is then to be expected. We proceed to determine this Specifically, it can be shown that ðN¼9Þ Z t2Pa next. ffiffiffiffiffiffi 2 p− ∝ χ M SEH ¼ d x gR ð 2Þ: ð4:38Þ M2 1. Lagrangian constraints Namely, the Einstein-Hilbert action is proportional to the As a starting point, we express the generalized coor- 2 Euler characteristic χ of the spacetime manifold M2; see dinates of the Palatini theory in d ¼ as e.g., [45]. The above implies that general relativity is a A 1 11 1 1 1 topological theory in d ¼ 2 and, accordingly, propagates no Q ¼fh; h ;h ;G;G1;G11; G ; G1; G11g; degrees of freedom, a fact that we shall explicitly verify in A ¼ 1; 2; …; 9 ¼ dimðCÞ ≡ N; ð4:40Þ the following. Turning to the Palatini Lagrangian in (4.1) for d ¼ 2,we restate that this is not dynamically equivalent to two- in direct analogy to (4.7) earlier on. Next, we compute the 2 dimensional Einstein’s gravity (4.38). To see this, consider first stage quantities associated to the d ¼ version of the its corresponding Euler-Lagrange equations in (4.3). These Palatini Lagrangian (4.6). One can verify that the set of are valid for d ≥ 2. However, recall that c ≔ d − 1 −1 primary Lagrangian constraints thus obtained matches ð Þ ,so 2 that c ¼ 1 in two dimensions. In this particular case, it is the consistent d ¼ evaluation of those for a generic obvious that (4.4) cannot be solved as we said, i.e., dimension in (4.10) and (4.11). Comparatively, these two- dimensional constraints have a much simpler form, given ρ ! ρ μν μν Gμν¼Gμνðh ; ∂ρh Þ. As a result, the dynamical by the vanishing of φ φ − _ ∂ G1 2 G1 − G1 ½ 2 1 − _ ∂ G1 G1 − G1 ½ 1 ¼ ½Gþ 1 þ ðG 1 G1 Þ ; 2 ¼ ½G1 þ 1 1 þG 11 G11 ; _ 1 1 1 _ 1 1 1 ½φ3 11 ¼ −½G11 þ∂1G11 þ2ðG1G11 −G11G1Þ ; ½φ4 ¼h−2ðhG1 þh G11Þ; φ 1 ½ 5 _ 1 G1 − 11G1 φ 11 _ 11 2 1G1 11G1 2 ¼ h þh h 11; ½ 6 ¼ h þ ðh þh 1Þ; φ 1 φ ∂ 2 1 ½ 8 1 ∂ 1 − 11 φ 11 ∂ 11 −2 1 11 ½ 7 1 ¼ 1hþ ðhG1 þh G11Þ; 2 ¼ 1h hGþh G11; ½ 9 1 ¼ 1h ðh Gþh G1Þ: ð4:41Þ The demand that the above be zero constitutes a set of with the moduli space due to the primary Hessian being nine scalar primary Lagrangian constraints (M1 ¼ 9), zero, as in the d>2 case before. In other words, (4.10) whose functional independence is rather obvious holds true here as well. 0 (M1 ¼ 9)—and can be ratified through the Jacobian test The progress to the subsequent stage parallels that of the in (2.17). Therefore, such vanishing defines the primary d>2 case. The secondary Hessian is given by the skew- constraint surface TC1 of the theory, which coincides symmetric constant matrix 065015-19 VERÓNICA ERRASTI DÍEZ et al. PHYS. REV. D 102, 065015 (2020) 0 1 0 −ω 0 2. Gauge identities ˜ B C WIJ ¼ @ ω 00A; with ω ≔ diagð1; 2; 1Þ: Given the already pointed out inequivalence between the 000 d>2 and d ¼ 2 Palatini theories, it is not too surprising that the gauge transformations (4.30) preserving the action ð4:42Þ (4.6) meet a nonsmooth limit for d ¼ 2. The argument is more subtle than that of the purely on-shell inequivalence; The Hessian (4.42) has rank R2 ¼ 6 and so M2 ¼ 3. This for example, see [11]. We shall touch upon it shortly. means that six of the primary Lagrangian constraints are It has been proven, e.g., [11,47], that the two- dynamically stabilized by the (functionally independent) dimensional Palatini action is invariant under the field μν μν μν ρ ρ ρ secondary equations of motion. For the remaining three transformations h → h þδθh and Gμν → Gμν þδθGμν, primary Lagrangian constraints, the algorithm must be with pursued. μν ρðμ νÞσ We choose the suitably normalized linearly independent δθh ¼ 2ϵ h θρσ; null vectors of (4.42) as ρ ρσ σλ ρ δ μν ϵ ∂ θ 2ϵ θ θG ¼ σ μν þ Gσðμ νÞλ: ð4:47Þ I I ðγ˜ Þ ¼ δ 6 ; with R ¼ 1; 2; 3: ð4:43Þ μν R Rþ Here, ϵ stands for the two-dimensional Levi-Civita symbol (we work with the convention ϵ01 ¼ 1) and Using (4.41) and (4.43), we obtain the three secondary θμν ¼ θνμ, so there are three arbitrary gauge parameters Lagrangian constraints as the vanishing of that characterize the transformation. It is obvious that all the gauge parameters explicitly appear in the gauge trans- φ˜ ∂ _ 2 _ _ 1 _ 1 _ ½ 1 1 ¼ 1h þ ðhG1 þ h G11 þ hG1 þ h G11Þ; formations. Hence, g ¼ 3. Their first time derivatives also 1 _ 1 _ _ 11 _ 11 _ show up, adding to a total number of effective gauge ½φ˜ 2 1 ¼ 2ð∂1h − hG þ h G11 − hG þ h G11Þ; parameters e ¼ 6. We point out that g does not match the 11 _ 11 _ 1 _ 11 1 _ 11 _ ½φ˜ 3 1 ¼ ∂1h − 2ðh G þ h G1 þ h G þ h G1Þ: ð4:44Þ value predicted in (4.31) for d ¼ 2. There is a match for e, but this is purely coincidental. Notice that the above are the total time derivatives of ½φ7 1, These numbers ðg ¼ 3;e¼ 6Þ, in contrast to the naively 1 11 2 6 ½φ8 1 and ½φ9 1 in (4.41), respectively. It is easy to check expected ones ðg ¼ ;e¼ Þ from the diffeomorphism that the secondary Lagrangian constraints are functionally transformations (4.31) in d>2 Palatini, reflect the fact that dependent on the primary Lagrangian constraints. d ¼ 2 Palatini is associated to a comparatively larger Specifically, symmetry group. Its connection to the d>2 gauge group is not obvious but finds its origin in the underlying two- 1 dimensional geometry. Briefly recall the conformal flatness ½φ˜ 1 1 ¼ ∂1½φ4 − h½φ2 1 − 2h ½φ3 11 þ 2G1½φ4 1 1 1 1 of two-dimensional spacetimes, i.e., þ G11½φ5 þ 2G1½φ7 1 þ G11½φ8 1; 2 μν −2 μν 1 1 11 11 gμν Ω ημν;g Ω η ; μ;ν 0;1; : ½φ˜ 2 1 ¼ ∂1½φ5 þ 2ðh½φ1 − h ½φ3 11 − G½φ4 þG11½φ6 ¼ ¼ with ¼ ð4 48Þ − G1 φ G1 φ 11 μ ½ 7 1 þ 11½ 9 1 Þ; for some conformal factor Ω ¼ Ωðx Þ. Given this property, 11 11 1 11 μν ½φ˜ 3 1 ¼ ∂1½φ6 þ 2h ½φ1 þh ½φ2 1 the variable h introduced in (4.2) simplifies to − φ 1 2 φ 11 G1 φ 1 2G1 φ 11 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðG½ 5 þ G1½ 6 þ ½ 8 1 þ 1½ 9 1 Þ: μν μν 2 −2 μν μν h ≔ detð−gμνÞg ¼ detð−Ω ημνÞΩ η ¼ η : ð4:45Þ ð4:49Þ Therefore, the secondary constraints vanish on TC1: Consequently, in the conformal frame, hμν is flat and μν μν 1 11 detðh Þ¼−1, independent of detðg Þ. This latter equality ½φ˜ 1 1≈ 0; ½φ˜ 2 1≈ 0; ½φ˜ 3 1 ≈ 0; ð4:46Þ 1 1 1 can be expressed as an algebraic constraint: 11 1 2 and so TC2 ≡ TC1. This in turn implies that there are no hh − ðh Þ þ 1 ¼ 0; ð4:50Þ 0 functionally independent secondary constraints: M2 ¼ 0. Here, the algorithm closes nondynamically, as described in referred to as the metricity condition. We will soon get back case (iib). It follows that the total number of functionally to such condition. For a richer discussion on this topic, 0 0 independent Lagrangian constraints is l ¼ M1 þ M2 ¼ 9. though, we refer the reader to [12]. We note that this result does not correspond to setting In analogy to the higher dimensional case before, we d ¼ 2 in (4.29). provide the gauge generators and identities of d ¼ 2 065015-20 LAGRANGIAN CONSTRAINT ANALYSIS OF FIRST-ORDER … PHYS. REV. D 102, 065015 (2020) Palatini next. Comparing (2.28) to (4.47), we can conven- N ¼ 9 a priori independent field variables for d ¼ 2 iently rewrite the latter as Palatini, exactly as we did here. Following the Dirac- ðPÞ 3 6 μν αβ μν Bergmann procedure, it was shown that N1 ¼ , N1 ¼ δθh θαβ Ω ; ¼ ½ð Þ and N2 ¼ 6, which readily confirms our own independent ρ ρ ρ δ θ Ωαβ − ∂ θ Ωαβ μ1 findings. In [12], the metricity condition (4.50) was taken θGμν ¼ αβ½ð Þμν ð μ1 αβÞ½ð Þμν ; ð4:51Þ into account from the onset. As a result of incorporating with the gauge generators readily recognized as this information in the form of additional terms preceded by two Lagrange multipliers in the Hamiltonian, their setup ½ðΩαβÞμν ¼2ðϵαðμhνÞβ þ ϵβðμhνÞαÞ; had N ¼ 11 ≠ N ¼ 9 a priori independent field variables. ðPÞ ρ ρ β ρ β 5 αβ σðαj j Þ αβ μ1 ρμ1 α It was there shown that, in such formulation, N1 ¼ , ½ðΩ Þμν ¼4ϵ Gσ μδν ; ½ðΩ Þμν ¼ 2ϵ δ μδν ; ð Þ ð Þ N1 ¼ 7 and N2 ¼ 8, yielding no propagating degrees of ð4:52Þ freedom and thus corroborating again our own final result. This latter comparison provides another example for our where the bar j notation delimits the symmetrized indices. point around (2.6), this time in Hamiltonian terms. Namely, Besides the generators, the other element needed to determine the gauge identities are the primary Euler- ðNÞ ðPÞ 3 6 6 ≠ ðNÞ t2Pa ¼ðN1 ¼ ;N1 ¼ ;N2 ¼ Þ t2Pa Lagrange equations. These are given by the straightforward P 2 ð Þ 5 7 8 2 ð Þ ¼ðN1 ¼ ; N1 ¼ ; N2 ¼ Þ; ð4:56Þ evaluation of (4.3) for d ¼ . Let us refer to them as EðhμνÞ ð2Þ and E ρ , respectively. Then, their merging together with but we find that n 0 for both sets of numbers upon ðGμνÞ dof ¼ (4.52) as indicated in (2.29) yields the gauge identities for employing (2.3). As a last remark, we notice that our d ¼ 2 Palatini we were seeking, after some tedious yet calculations in this Sec. IV B are comparatively simpler elementary algebra: than those in [11,12]. Namely, our approach is certainly to be preferred if the goal is to determine the constraint αβ ð2Þ αβ μν ð2Þ αβ ρ structure of the theory and thereby manifestly count its ϱ ¼ E μν ½ðΩ Þ þE ρ ½ðΩ Þμν ðh Þ ðGμνÞ propagating modes. ð2Þ αβ ρ μ þ ∂μ ðE ρ ½ðΩ Þμν 1 Þ ≡ 0: ð4:53Þ 1 ðGμνÞ V. CONTEXTUALIZATION AND Observe that the manifest symmetry under the exchange POTENTIALITY OF OUR RESULTS α ↔ β makes the number of independent gauge identities The study of constrained systems was initiated in the coincide with our earlier counting: g ¼ 3. 1930s by Rosenfeld, in a sometimes overlooked work [48], nowadays acknowledged and revisited [49]. It was later 3. Physical degrees of freedom greatly developed during the 1950s [50] and has since been Altogether, we have now obtained the constraint struc- a very active field of theoretical research. As such, one may ture characterizing triplet of Palatini in d ¼ 2 in terms of have the impression that the investigation of manifestly Lagrangian quantities: first-order singular classical field theories must be an already closed subject. This is not true. There are ongoing ðNÞ 9 3 6 t2Pa ¼ðl ¼ ;g¼ ;e¼ Þ; ð4:54Þ advances in this fundamental topic, particularly within the Lagrangian picture. Besides the references already pro- where N ¼ 9. Employing (2.4) and (2.5), it is immediate to vided in Sec. II, the recent work [4] stands as a neat rewrite this triplet in Hamiltonian terms: example. The methodology there put forward is equivalent to our own proposal, as we shall show in the next Sec. VA. ðNÞ ðPÞ 3 6 6 To further reassure the reader of the topicality of our t2Pa ¼ðN1 ¼ ;N1 ¼ ;N2 ¼ Þ: ð4:55Þ formalism, in Sec. VBwe explain how our method lends Plugging (4.54) in the master formula (2.2), we confirm the itself to a conversion from an analytic machinery to a well-known fact that there are no physical degrees of constructive one. Indeed, the Lagrangian building principle 0 freedom propagated by the theory: ndof ¼ . We restate originally put forward in [8,9] finds in the contents of that the above cannot be obtained by simply setting d ¼ 2 Sec. II a solid footing for attempting the construction of in (4.35). novel theories. This argumentation is carried out in terms To wrap up this section, we check our results are in good of a concrete application for clarity, but the general agreement with some of the previously carried out proposal is much broader. In particular, we explain that Hamiltonian calculations. We begin our comparisons by the less elaborated upon procedure in [8,9] was cornerstone looking into the approach closest to our own, the one in for the development of the so-called Maxwell-Proca theory. μν ρ [11]. There, the quantities ðh ;GμνÞ were regarded as the This discussion justifies an interest in the calculations of 065015-21 VERÓNICA ERRASTI DÍEZ et al. PHYS. REV. D 102, 065015 (2020) Sec. III well beyond a simple exemplification of the explicit We move to Schwinger-Plebanski formulation of both usage of the proposed method. When gravity is to be electromagnetism and the (hard) Proca theory. In both involved, the examples in Sec. IV provide a useful cases, the velocity-independent primary constraints i possible basis. ½φ3 ∶≈ 0 and ½φ4 ∶≈ 0 in (3.17) are second-class i 1 1 Lagrangian constraints. This is because their stability is A. On a recent equivalent Lagrangian approach dynamically ensured, via the secondary equations of During the preparation of this manuscript, a novel motion. Thus, the algorithm closes as in case (i) for them. Lagrangian approach to obtain the functionally indepen- φ˜ ∶≈ 0 Similarly, the secondary constraints ½ 2 ij in (3.22) are dent Lagrangian constraints and count propagating modes 2 in constrained systems (of the kind here considered) stabilized dynamically at the subsequent stage. Therefore, appeared [4]. The method therein is physically equi- these and also their ascendant primary constraints ½φ2 ∶≈ 0 in (3.17) are second-class Lagrangian constraints valent to that put forward in [2,3], which—as already ij 1 — mentioned are complementary references to our own ∀ m ≥ 0. The same is true for ½φ1 ∶≈ 0 and ½φ˜ 1 ∶≈ 0 in the discussion in Sec. II. This equivalence can be easily 1 2 massive case. When m ¼ 0, we see that ½φ˜ 1 ≋0. This is a verified, as both [4] and [2,3] provide a mapping between 1 their proposed Lagrangian parameters and the usual num- Lagrangian identity that closes the algorithm nondynami- bers of different kinds of Hamiltonian constraints. We have cally, following case (iia). As a result, both the said checked this leads to a consistent mapping between their Lagrangian identity and its ascendant ½φ1 ∶≈ 0 are first- different Lagrangian parameters. 1 class Lagrangian constraints. In our understanding, the method in [4] distinguishes Turning to d ¼ 2 Palatini, we see it is rather simple to itself because it introduces the notion of first- and second- reclassify the nine functionally independent Lagrangian class (functionally independent) Lagrangian constraints. constraints we obtained into first and second class. At In our language, these are easy to identify. They are the sum the primary level, we notice that there are six velocity- of the various functionally independent Lagrangian con- dependent Lagrangian constraints among the relations that straints arising at all prior stages whose algorithm finalizes follow from requiring the vanishing of (4.41). These are nondynamically [as in cases (iia) and (iib)] and dynami- ! cally [as in case (i)], respectively. This abstract definition is ½φ ≈0, where a ¼ 1; 2; …; 6 and the tensorial indices a 1 clarified in the following, by classifying the functionally outside the square brackets ½· have been omitted. Their independent Lagrangian constraints we found in all the stability is dynamically ensured (via the secondary equa- given examples into first- and second-class Lagrangian tions of motion) and so these are second-class Lagrangian constraints. constraints. The three remaining primary Lagrangian con- In the case of Maxwell electromagnetism, the primary ! straints, ½φ ≈0 with a ¼ 7, 8, 9, are manifestly velocity Lagrangian constraint (3.4) we found is a first-class a 1 Lagrangian constraint. This is because it leads to a independent. They show a trivial stability at the secondary secondary constraint (3.5) that is identically satisfied and stage; see (4.46). Accordingly, we identify these as three so nondynamically stabilized by means of the closure (iia). first-class Lagrangian constraints. In fact, this same example is worked out in [4] as well. At last, we reclassify the functionally independent Next, consider the (hard) Proca theory. There, both the Lagrangian constraints we found for d>2 Palatini into primary (3.9) and secondary (3.10) Lagrangian constraints first- and second-class Lagrangian constraints. Recall that we determined are second-class Lagrangian constraints, 2 we obtained M1 ¼ dðd þ 1Þ =2 functionally independent since the algorithm closes dynamically at the next stage by primary Lagrangian constraints, given by the vanishing means of case (i). Such closure implies that the consistency of (4.11). Notice now that we can straightforwardly split under time evolution of the secondary constraint is deter- these primary constraints into mined through a tertiary equation of motion. ! ! ! ij M1 ¼ dðd þ 1Þnumber of velocity-dependent constraints∶ ½φ1 ∶≈ 0; ½φ ∶≈ 0; …; ½φ6 ∶≈ 0; ð5:1Þ ðvÞ 1 i 1 1 ! ! ! d 2 j jk M1 ¼ ðd − 1Þnumber of velocity-independent constraints∶ ½φ7 ∶≈ 0; ½φ8 ∶≈ 0; ½φ9 ∶≈ 0; ð5:2Þ ðnvÞ 2 i 1 i 1 i 1 where the subscripts (n)v stand for (non)velocity-dependent constraints. The consistency under time evolution of the velocity-dependent constraints is dynamically fixed at the secondary stage and so these are second-class Lagrangian constraints. The remaining velocity-independent constraints give rise to the M2 ¼ M1ðnvÞ number of functionally 065015-22 LAGRANGIAN CONSTRAINT ANALYSIS OF FIRST-ORDER … PHYS. REV. D 102, 065015 (2020) independent secondary Lagrangian constraints, which are equal to the vanishing of (4.17). Once more, it is trivial to differentiate between ! ! d 2 j jk M2 ¼ ðd − 3Þ number of velocity-dependent constraints∶ ½φ˜ 2 ∶≈0 with i ≠ j; ½φ˜ 3 ∶≈0; ð5:3Þ ðvÞ 2 i 1 i 1 ! ! i M2 ¼ d number of velocity-independent constraints∶ ½φ˜ 1 ∶≈0; ½φ˜ 2 ∶≈0: ð5:4Þ ðnvÞ i 1 i 1 The consistency under time evolution of the former is respectively. NLE encompasses a large class of theories. ensured by the tertiary equations of motion. Equivalently, The celebrated Born-Infeld theory [52] is part of it, but the algorithm closes according to the dynamical case (i) for also the more recently proposed exponential [53] and them. As a result, they are second-class Lagrangian con- logarithmic [54] electrodynamics, among others. straints. Further, the subset of M2ðvÞ number of velocity- Schematically, the Lagrangian density for NLE can be independent primary constraints they follow from are written as second-class Lagrangian constraints as well. Specifically, ! ! j jk L L ½φ8 ∶≈0 with i ≠ j and ½φ9 ∶≈0 are second class. On the NLE ¼ M þ fðAμνÞ; ð5:5Þ i 1 i 1 other hand, the above velocity-independent constraints are L trivially stable, as their time evolution yields tertiary where M is the Maxwell Lagrangian as introduced in constraints that identically vanish in TC2: recall (4.28). (3.3) and f is a smooth real function. Notice that the The algorithm closes nondynamically as in case (iib) for above depends on the Maxwell field Aμ exclusively them. Consequently, they are first-class Lagrangian con- through its field strength Aμν—up to boundary terms. straints and their ascendant primary Lagrangian constraints Indeed, it is well known [35] that a more involved ! ! i dependence is not possible, if the Uð1Þ gauge symmetry ½φ7 ∶≈ and ½φ8 ∶≈0 are first class too. i 1 i 1 is to be respected. This feature remains true even when coupling the Maxwell field to general relativity [55]. B. Relation to the Maxwell-Proca theory and beyond Only a few fine-tuned terms that contract Aμν with the As we explicitly showed in Sec. III A, in a purely Riemann tensor are possible in such a case. It is not Lagrangian formulation with as many a priori independent hard to convince oneself that the constraint structure of ðNÞ field variables as the dimension of the underlying flat NLE is characterized by the triplet tM in (3.6). In other spacetime, the constraint structure of the simplest theory words, it has the same constraint structure as classical for a single Maxwell field can be characterized by the triplet electromagnetism, in its standard formulation of tðNÞ in (3.6). The analogous investigation in Sec. III B of the Sec. III A. M The GP theory was put forward in [30] and its complete most elementary theory of one (hard) Proca field yielded the ðNÞ Lagrangian was established in [56]. Again schematically, constraint structure characterizing triplet tP in (3.11). we may express it as Employing the results of [2], we also verified the correspond- ing Hamiltonian characterization of these two triplets. We Xd ν …ν ρ …ρ thus checked that the Maxwell and (hard) Proca fields are L L T 1 n 1 n ∂ …∂ GP ¼ P þ gðBμÞþ ν1 Bρ1 ρn Bμn associated with two first- and second-class constraints, n¼1 respectively. Although usually Maxwell and Proca fields ð5:6Þ are defined in the latter Hamiltonian manner, in the following we take the former Lagrangian triplets as the vector fields’ L in d dimensions, where P is the (hard) Proca Lagrangian ν …ν ρ …ρ defining features. We stress both definitions are equivalent. in (3.7), g is a real smooth function and each T 1 n 1 n is The manifestly first-order completions of the Maxwell a certain smooth real object constructed out of the space- and (hard) Proca theories analyzed in Secs. III A and III B time metric ημν, the d-dimensional Levi-Civita tensor are nonlinear electrodynamics (NLE) and the so-called ϵμ …μ and the Proca field Bμ. Although GP has only been generalized Proca (GP) or vector-Galileon theory,12 1 d formulated for d ¼ 4, its systematic construction allows for a straightforward inferring of (5.6). Here, the underlying 12 We are aware of the recent proposal in [51].However,the key idea consists in supplementing the (hard) Proca Lagrangian there put forward is not in a manifestly first-order Lagrangian with derivative self-interaction terms of the form. The authors leave for further studies this result. In the lack of it, their theory lies beyond our framework and we cannot Proca field Bμ. This implies a nonlocal extension of the address it. notion of mass for the vector field. As such, we regard GP 065015-23 VERÓNICA ERRASTI DÍEZ et al. PHYS. REV. D 102, 065015 (2020) as an effective classical field theory.13 It can be readily At this point, it should be clear that our calculations of inferred from the calculations in [8] that the constraint ðNÞ ðNÞ tM and tP in Sec. III, elementary as they are, can be used ðNÞ structure of GP is characterized by the triplet tP in (3.11). as a basis for the construction of nontrivial theories. Having Namely, GP has the same constraint structure as the (hard) a ready-to-be-used method optimized to obtain such triplets Proca theory, when the latter is formulated as in Sec. III B. (i.e., the method explained in Sec. II and graphically Next, we consider a multifield scenario, including summarized in Fig. 1) is thus a powerful tool for the nM number of Maxwell fields, as well as nP number of development of manifestly first-order classical field theo- (generalized) Proca fields. In four-dimensional Minkowski ries where multiple fields of different spins (self-)interact. spacetime, the Maxwell-Proca (MP) theory [8,9] is the For instance, an interesting open question is that of the complete set of manifestly first-order (self-)interactions consistent coupling of the MP theory to gravity. It is in among an arbitrary number of real Abelian vector fields that principle possible to combine our calculations in all the propagates the correct number of degrees of freedom. previous sections to attempt this ambitious goal as follows. N 1 2 2 ≥ 2 These consistent interactions were derived by demanding Let 2 ¼ðnM þ nPÞd þ dðd þ Þ = , with d the that the constraint structure of each Maxwell and Proca dimension of the spacetime. A manifestly first-order L ðNÞ ðNÞ Lagrangian density MPð2ÞPa that describes the dynamics field is characterized by the triplets tM and tP , respec- N of nM number of Maxwell fields and nP number of tively. Let 1 ¼ðnM þ nPÞd. We denote the constraint N (generalized) Proca fields in the presence of Einstein’s structure characterizing the triplet of MP as tð 1Þ. Then, we MP gravity in terms of N 2 a priori independent field variables say that the building principle of the theory is based on the must be associated with a constraint structure characteriz- requirement ðN 2Þ ing triplet tMPð2ÞPa satisfying ðN 1Þ ! ðN¼dÞ ðN¼dÞ t n · t ⊕ n · t N ! MP ¼ M M P P ð 2Þ ðNÞ ⊕ ðNÞ ⊕ ðNÞ t 2 ¼nM · t nP · t t 2 ; ð5:8Þ 2 2 MPð ÞPa M P ð ÞPa ¼ðl ¼ nM þ nP;g¼ nM;e¼ nMÞ; ð5:7Þ where all the triplets on the right-hand side have already where in the last equality we have made use of (3.6) been calculated in this work; see (3.6), (3.11), (4.36) and and (3.11). (4.54). Substituting these results, we have that 2 N ! ðl ¼ n þ 2n þ d ðd þ 1Þ;g¼ n þ d; e ¼ 2n þ 3dÞ if d>2; tð 2Þ ¼ M P M M ð5:9Þ MPð2ÞPa 2 9 3 2 6 2 ðl ¼ nM þ nP þ ;g¼ nM þ ;e¼ nM þ Þ if d ¼ : The conversion of any of the above necessary conditions associated to a certain triplet tðNÞ. The reason is that such into a Lagrangian density building principle is an alge- inversion in the logic requires solving sets of coupled braically involved exercise beyond the scope of our present nonlinear partial differential equations in most cases. investigations. We thus leave it for future works. Therefore, it is overwhelmingly convenient to use all A last remark is due. As we observed at the very end of freedom of choice available in order to simplify this task Sec. II A and should be apparent from our calculations in to the utmost. For instance, one is advised to choose Sec. IVA, it is in general a conceptually clear but constant null vectors for the Hessians at all stages, if algebraically nontrivial exercise to obtain the triplet tðNÞ possible. For the concrete research project here proposed, of a given Lagrangian density L within our framework. It is it may be the case that (5.8) is not the optimal starting point. even more challenging to determine the (exhaustive) form It could happen that the equivalent demand of L from the necessary condition that it should be ! ðN 3Þ ðNÞ ⊕ ðNÞ ⊕ ðNÞ tMPð2ÞPa ¼ nM · tM nP · tP tð2ÞPa with 13 The quantization of nonlocal field theories generically leads N 1 1 2 to acausality, most often for phenomena beyond tree level. It is 3 ¼ðnM þ nP þ d þ Þdðd þ Þ= ; ð5:10Þ sometimes possible to circumvent this problem, especially in the presence of supersymmetry—for instance, see [57]. The con- with the right-hand side triplets as given in (3.32), (4.36) sistent quantization of GP has been investigated in [58]. Although the results obtained so far justify an optimistic attitude, no and (4.54), is a more befitting way to try to derive the set of complete and rigorous quantization scheme seems to have been consistent (self-)interactions of vector fields in a curved proposed so far, along the lines of the BRST and path integral background. For the reasons given at the beginning of quantization of the (hard) Proca theory in [59,60], respectively. Sec. IV, we believe that tðNÞ is indeed a beneficial basis for Therefore, we here adopt the conservative point of view that ð2ÞPa regards GP as an effective classical field theory. the gravity piece above. 065015-24 LAGRANGIAN CONSTRAINT ANALYSIS OF FIRST-ORDER … PHYS. REV. D 102, 065015 (2020) VI. CONCLUSIONS transformation from the Lagrangian to the Hamiltonian picture. Besides, as already noted in the end of Secs. IVA In the following, we summarize the results we have put and IV B, our Lagrangian approach is a computationally forward in this manuscript. Then, we proceed to discuss faster and simpler way to obtain the constraint structures of their relevance and pertinence. At last, we comment on the these theories, compared to representative Hamiltonian increasing (in n) computational difficulty of evaluating analyses. (The examples in Sec. III are so effortless Lagrangian constraints on constraint surfaces TC and n comparatively that they do not substantiate an analogous concretize the pathologies a theory may suffer from when argumentation.) the algorithm of Sec. II A is not verified to close. In more detail, implementing our algorithm in Sec. II A is considerably easier than carrying out a Hamiltonian A. Summary of results counterpart algorithm based on the Dirac-Bergman [50] In Sec. II, we have collected and complemented results procedure. As the attentive reader will have already noticed from the extensive literature on constrained systems and in our explicit examples of Sec. IV and we shall address presented a self-contained and ready to be used method to shortly, the most demanding step in our approach consists determine all the constraints in a theory. By postulation, the in evaluating the nth stage Lagrangian constraints in the theory is required to be described by a manifestly first-order (n − 1)th constraint surface, with n ≥ 1. An analogous Lagrangian. We make the mild assumptions of the principle evaluation is necessary within the Hamiltonian picture as of stationary action and finite reducibility. When the well, where two additional hurdles arise. On the one hand, theory is covariant, the iterative algorithm presented for one must classify the Dirac constraints into first and second the determination of the functionally independent class. This entails calculating the Poisson brackets of all Lagrangian constraints does not contravene this feature. Dirac constraints, a generically challenging task in field Nonetheless, manifest covariance is generically lost in our theory because nonlocal algebras usually arise,14 e.g., approach. In Secs. III and IV, we have minutely exemplified [13,14]. On the other hand, in the standard Hamiltonian the usage of our said procedure. In Sec. V, we have argued transition from one stage to the next, novel constraints for the pertinence and contemporaneity of both the general emerge and must be consistently included via Lagrange formalism and the given examples. Indeed, an equivalent multipliers. Closure of the algorithm requires the determi- but different methodology has been put forward lately [4]. nation of as many Lagrange multipliers as possible, which The examples of Sec. III constitute the foundation of the in turn implies the resolution of algebraic or even differ- also recent Maxwell-Proca theory [8,9] and those of ential equations. Even in the comparatively benign alge- Sec. IV can potentially form the basis for the consistent braic scenario, finding a solution is an increasingly (in coupling of Maxwell-Proca to gravity. stage) laborious and nontrivial task that involves inverting field-dependent matrices with complicated spatial index B. Critical discussion of results structures. The procedure explained in Sec. II presents two main For a suggestive utility of the examples in Secs. III appealing features. First, it is a coordinate-dependent and IV, the reader is referred to Sec. VB. Recall that the approach, as opposed to a geometrical one. It thus readily proposal therein is illustrative of the general theory- allows for its application, given a Lagrangian density construction idea outlined in the introduction Sec. I and satisfying the initial postulates, without having to work at the beginning of Sec. V. out any symplectic two-form. With pragmatism in mind, Sec. II has been written in a way that is (hopefully) C. Two final observations accessible to a broad audience. Even though the method In the first of our observations, we bring to light a series stands on a rigorous footing, the discussion has been made of considerations that must be taken into account when largely devoid of mathematical technicalities. applying our method. In particular, we wish to discuss the Second, it is an intrinsically Lagrangian procedure, as practical complications that field theories of the kind here opposed to a Hamiltonian or a hybrid one. The appeal of considered commonly exhibit when their Lagrangian con- this characteristic resides in the fact that, in many areas straints are to be evaluated on the suitable constraint of high-energy theoretical physics, manifestly first-order surface. classical field theories are predominantly posed and studied First, we debunk what naively may look like an in their Lagrangian formulation. This is the case for ambiguity. Recall that any constraint surface TCn for some instance in cosmology, astrophysics, black hole physics finite n ≥ 1 is defined by the weak vanishing of the and holographic condensed matter. In all these disciplines, GP, MP and allied theories, especially in the presence of 14This nonlocality is as a consequence of the distributional gravity, have been convincingly argued to be of significant nature of fields. Dirac constraints must be smeared with suitable interest, e.g., [9,30,61]. As such, our proposed procedure test functions and integrated over for a correct evaluation of the avoids non-negligible obstacles that typically arise in the Poisson brackets. 065015-25 VERÓNICA ERRASTI DÍEZ et al. PHYS. REV. D 102, 065015 (2020) functionally independent Lagrangian constraints at all prior Their raw expressions, prior to any evaluation in a con- A _ A A stages: straint surface, contain quantities f ¼ fðQ ; Q ; ∂iQ Þ that vanish in TC1. However, recognizing such f’sas ! φ ðQA; Q_ A; ∂ QAÞ∶≈ 0; primary weak zeros is a challenging task. Specifically, I i 1 ! j k A _ A A ½φˆ 1 ⊃ G f ;f¼ ∂ ½φ7 −2G ½φ7 −G ½φ8 ≈0; φ˜ ðQ ; Q ; ∂ Q Þ∶≈ 0; i ij ij ½i j ði jÞ kði jÞ 1 R i 2 ! ð6:3Þ φˆ ðQA; Q_ A; ∂ QAÞ ≈ 0; etc:; ð6:1Þ U i 3 where in the ⊃ relation we have omitted numerical factors As a direct consequence of the above, one can determine a and the φ’s are as given in (4.11). In the expression for fij, maximal set of functionally independent relations of the the first equality is nontrivial, while the subsequent primary form weak equality is obvious. An analogous situation arises with other f’s that are based on both functionally inde- ! _ B _ B A _ A A Q ≈Q ðQ ; Q ; ∂iQ Þ; pendent primary (4.11) and secondary (4.17) Lagrangian n constraints. A brute force resolution to identify all such f’s ! B B A _ A A consists in putting forward the most general ansatz com- ∂iQ ≈∂iQ ðQ ; Q ; ∂iQ Þ; n patible with the tensorial character of each of the ! QB≈QBðQAÞ: ð6:2Þ Lagrangian constraints one is trying to evaluate and n comparing it to their explicit expressions. This is indeed how we laboriously arrived at (4.27). Though it should be clear by now, we confirm the different For the second and last observation, the reader should _ A role played by the generalized velocities Q and the heed (2.2) and (2.27). We already stressed the importance spacelike derivatives of the generalized coordinates of closing the iterative algorithm for obtaining the func- ∂ A C iQ . The former are independent coordinates on T , tionally independent Lagrangian constraints toward the end while the latter are functionally related to the generalized of Sec. II A. Now, we are equipped to better grasp the coordinates QA. This clarification becomes pertinent when implications of not doing so, mentioned in the introductory evaluating the secondary Lagrangian constraints in TC1 Sec. I. Most often, failure to close the algorithm will give already. At this point (and in subsequent stages), derivatives rise to the propagation of unphysical modes. These are _ A of the form ∂iQ generically show up. In such expressions, Ostrogradsky instabilities [5], but we shall loosely refer to one must first replace the primary weak expression for the them as ghosts. Even after ensuring ghost freedom, not generalized velocity Q_ A—if pertinent—and then apply the closing the algorithm can lead to trouble: it may over- spatial derivative on it. constrain the theory, so that fewer than the desired number Having clarified this point, we notice that its consistent of degrees of freedom are propagated. implementation leads to the following nested situation. Let us consider the MP theory [8,9] discussed in _ A _ A Substitution of Q according to (6.2) in ∂iQ normally Sec. VB as a concrete framework to clarify the above B leads to the presence of terms of the form ∂iQ in (6.2). two unwanted scenarios. For our present purposes, it will These are again prone to be evaluated in TCn and can in suffice to consider the case when there are no Maxwell B 0 turn contribute terms depending on Q ’sin(6.2),etc. fields nM ¼ and there are an arbitrary but finite number of We emphasize that one must reach an expression where Proca fields nP. Recall that, in the standard formulation, we this nesting ceases to occur, before proceeding with the already saw in Sec. III B that a Proca field is associated to 0 0 algorithm. Not doing so would imply a wrong evalua- l ¼ M1 þ M2 ¼ 1 þ 1 ¼ 2 number of functionally inde- tion of the Lagrangian constraints in TCn, may lead pendent Lagrangian constraints. Bear in mind that this is to a misidentification of the functionally independent also true for a generalized Proca field. Lagrangian constraints and will almost invariably yield We denote the natural generalization of the GP theory in L L wrong results at the following (n þ 1)thstage.Infact,a (5.6) to a multifield setup as PP. PP automatically leads to 0 wrong evaluation will typically land the researcher in a M1 ¼ nP number of functionally independent primary physically inequivalent theory from the one he or she constraints. The consistency under time evolution of these 0 started with. constraints does not generically yield the M2 ¼ nP number Additionally and normally, when evaluating some of functionally independent secondary constraints one Lagrangian constraints in TCn, potentially contrived func- would naively expect. Only a fine-tuned subset of terms L tions of the previous stages’ functionally independent in PP does, precisely the terms that are part of the MP Lagrangian constraints also show up. To understand the theory. For all those terms, it was shown that no tertiary difficulty their appearance implies, consider the tertiary constraints arise (M3 ¼ 0) and the algorithm closes 2 2 Lagrangian constraints (4.25) we found for d> Palatini. dynamically giving rise to l ¼ nP. Therefore, the correct 065015-26 LAGRANGIAN CONSTRAINT ANALYSIS OF FIRST-ORDER … PHYS. REV. D 102, 065015 (2020) − ” number of physical modes ndof ¼ d nP are present in the 2018. M. T. T. would like to thank the hospitality of the theory. (To obtain this result, notice that, since there are no Max Planck Institute for Physics during his visit. gauge identities, g ¼ 0 ¼ e.) L Notice that, if one studies only the primary stage for PP, APPENDIX: FORMULAS AT AN ARBITRARY one will be deceived into thinking that the theory is valid, as L STAGE OF THE ALGORITHM it suitably extends the primary stage of GP. However, PP 0 2 has M2 065015-27 VERÓNICA ERRASTI DÍEZ et al. PHYS. REV. D 102, 065015 (2020) Finally, we introduce the auxiliary matrix MAaAb (the Using the above, the (a þ 1)th stage Euler-Lagrange Moore-Penrose pseudoinverse of WAaAb ), which always equations in (A2) can be written as exists and is uniquely determined from the relations ! ̈A γA1 γA2 … γAa α ≈ 0 EA ¼ Q ð ÞAð ÞA1 ð ÞA −1 WA A þ A ; ðA6Þ MAaAb W − δAa γAaþ1 γ Aa 0; b a a b b a AbAc Ac þð ÞAc ð Aaþ1 Þ ¼ AaAb γAaþ1 0 M ð ÞA ¼ : ðA5Þ b where the expression (A4) is to be employed for WAaAb and where we have (recursively) defined α ≔ −α Aa−1Ab−1 γ Aa−2 γ Aa−3 … γ A∂ _ − α Aa−2Ab−2 γ Aa−3 γ Aa−4 … γ A∂ _ Ab ½ Aa−1 M ð Ab−1 Þ ð Aa−2 Þ ð A1 Þ A Aa−2 M ð Ab−2 Þ ð Aa−3 Þ ð A1 Þ A _ A _ A − − α AB∂ _ ∂ ∂ ∂i φ AM B þ Q A þð iQ Þ A Ab ; ðA7Þ α α with A as given in (2.12). To obtain the presented Ab , the previous ath stage Euler-Lagrange equations are employed. These in turn depend on the (a − 1)th stage Euler-Lagrange equations and so on. This is the origin of the noted recursion. Notice we therefore explicitly employ the primary Euler-Lagrange equations and so the expression (A6) is an on-shell statement. In order to reproduce the results in Sec. II A from the above discussion, the reader only needs to do the index replacements ðA0 ≡ A; B; …Þ, ðA1 → I;J;…Þ, ðA2 → R; S; …Þ, etc., as well as take footnote 7 into account. [1] K. Kamimura, Nuovo Cimento B 68, 33 (1982). N. Mukunda, Ann. Phys. (N.Y.) 99, 408 (1976); Phys. Ser. [2] B. Díaz, D. Higuita, and M. Montesinos, J. Math. Phys. 21, 783 (1980). (N.Y.) 55, 122901 (2014). [16] J. Lee and R. M. Wald, J. Math. Phys. (N.Y.) 31, 725 (1990). [3] B. Díaz and M. Montesinos, J. Math. Phys. (N.Y.) 59, [17] J. D. Bekenstein and B. R. Majhi, Nucl. Phys. B892, 337 052901 (2018). (2015). [4] M. J. Heidari and A. Shirzad, arXiv:2003.13269. [18] I. Bengtsson, Phys. Lett. B 172, 342 (1986). [5] M. Ostrogradsky, Mem. Ac. St. Petersbourg VI, 385 (1850). [19] C. Batlle and J. Gomis, Phys. Lett. B 187, 61 (1987). [6] F. Sbisa, Eur. J. Phys. 36, 015009 (2015). [20] A. Shirzad, J. Phys. A 31, 2747 (1998); H. J. Rothe and [7] C. Deffayet, G. Esposito-Farese, and A. Vikman, Phys. Rev. K. D. Rothe, Classical and Quantum Dynamics of Con- D 79, 084003 (2009); C. Deffayet, S. Deser, and G. strained Hamiltonian Systems (World Scientific, Singapore, Esposito-Farese, Phys. Rev. D 82, 061501 (2010);S.F. 2010). Hassan, R. A. Rosen, and A. Schmidt-May, J. High Energy [21] R. Sugano, Prog. Theor. Phys. 68, 1377 (1982); J. M. Pons, Phys. 02 (2012) 026; L. Buoninfante and W. Li, Phys. Lett. J. Phys. A 21, 2705 (1988);X.Gr`acia and J. M. Pons, Ann. B 779, 485 (2018); L. Heisenberg, J. Cosmol. Astropart. Phys. (N.Y.) 187, 355 (1988). Phys. 10 (2018) 054. [22] B. S. DeWitt, Am. J. Phys. 34, 1209 (1966). [8] V. E. Díez, B. Gording, J. A. M´endez-Zavaleta, and A. [23] L. Parker and D. Toms, Cambridge Monographs on Schmidt-May, Phys. Rev. D 101, 045009 (2020). Mathematical Physics (Cambridge University Press, [9] V. E. Díez, B. Gording, J. A. M´endez-Zavaleta, and A. Cambridge, England, 2009). Schmidt-May, Phys. Rev. D 101, 045008 (2020). [24] K. Andrzejewski, J. Gonera, P. Machalski, and P. Maslanka, [10] M. Ferraris, M. Francaviglia, and C. Reina, Gen. Relativ. Phys. Rev. D 82, 045008 (2010). Gravit. 14, 243 (1982). [25] J. F. Cariñena, C. López, and N. Román-Roy, J. Math. Phys. [11] N. Kiriushcheva, S. V. Kuzmin, and D. G. C. McKeon, (N.Y.) 29, 1143 (1988). Mod. Phys. Lett. A 20, 1895 (2005). [26] O. Miskovic and J. Zanelli, arXiv:hep-th/0301256. [12] N. Kiriushcheva, S. V. Kuzmin, and D. G. C. McKeon, [27] T. L. Boullion and P. L. Odell, Generalized Inverse Matrices Int. J. Mod. Phys. A 21, 3401 (2006). (John Wiley & Sons, New York, 1971); G. H. Golub and [13] R. N. Ghalati and D. G. C. McKeon, arXiv:0712.2861. C. F. van Loan, Matrix Computations (John Hopkins [14] D. G. C. McKeon, Int. J. Mod. Phys. A 25, 3453 (2010). University Press, Baltimore, 2013). [15] E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: [28] S. Samanta, Int. J. Theor. Phys. 48, 1436 (2009). A Modern Perspective (John Wiley, New York, 1974); [29] R. Banerjee, H. J. Rothe, and K. D. Rothe, J. Phys. A 33, T. Regge and C. Teitelboim, Constrained Hamiltonian 2059 (2000). Systems (Academia Nazionale dei Lincei, Rome 1976); [30] G. Tasinato, J. High Energy Phys. 04 (2014) 067. 065015-28 LAGRANGIAN CONSTRAINT ANALYSIS OF FIRST-ORDER … PHYS. REV. D 102, 065015 (2020) [31] K. Sundermeyer, Constrained Dynamics, Lecture Notes in Helv. Phys. Acta Suppl. 4, 79 (1956); P. A. M. Dirac, Proc. R. Physics Vol. 169 (Springer-Verlag, New York, 1982); N. Soc. A 246, 326 (1958); Phys. Rev. 114, 924 (1959). Kiriushcheva, S. V. Kuzmin, and D. G. C. McKeon, Can. J. [51] C. de Rham and V. Pozsgay, arXiv:2003.13773. Phys. 90, 165 (2012). [52] M. Born, Proc. R. Soc. A 143, 410 (1934). [32] A. Proca, J. Phys. Radium 7, 347 (1936); 9, 61 (1939). [53] S. H. Hendi, Ann. Phys. (Amsterdam) 333, 282 (2013). [33] F. Darabi and F. Naderi, Int. J. Theor. Phys. 50, 3432 (2011). [54] P. Gaete and J. Helayël-Neto, Eur. Phys. J. C 74, 2816 [34] J. Schwinger, Philos. Mag. 44, 1171 (1953). (2014). [35] J. F. Plebanski, Lecture Notes, Nordita, Copenhagen, 1970. [55] G. W. Horndeski, J. Math. Phys. (N.Y.) 17, 1980 (1976). [36] K. Sundermeyer, Lect. Notes Phys. 169, 1 (1982). [56] J. B. Jimenez and L. Heisenberg, Phys. Lett. B 757, 405 [37] A. Palatini, Rendiconti del Circolo Matematico di Palermo (2016). (1884-1940), 43, 203 (1919). [57] A. Addazi and G. Esposito, Int. J. Mod. Phys. A 30, [38] L. Castellani, P. van Nieuwenhuizen, and M. Pilati, Phys. 1550103 (2015). Rev. D 26, 352 (1982). [58] A. Amado, Z. Haghani, A. Mohammadi, and S. Shahidi, [39] M. Montesinos, R. Escobedo, J. Romero, and M. Celada, Phys. Lett. B 772, 141 (2017); C. de Rham, S. Melville, Phys. Rev. D 101, 024042 (2020). A. J. Tolley, and S. Y. Zhou, J. High Energy Phys. 03 (2019) [40] R. K. Kaul and S. Sengupta, Phys. Rev. D 94, 104047 182; M. S. Ruf and C. F. Steinwachs, Phys. Rev. D 98, (2016). 025009 (2018); L. Heisenberg and C. F. Steinwachs, J. [41] S. Holst, Phys. Rev. D 53, 5966 (1996). Cosmol. Astropart. Phys. 01 (2020) 014; 02 (2020) 031; L. [42] J. Lewandowski and A. Okolow, Classical Quantum Gravity Heisenberg and J. Zosso, arXiv:2005.01639. 17, L47 (2000); R. Capovilla, M. Montesinos, V. A. Prieto, [59] Y. W. Kim, M. I. Park, Y. J. Park, and S. J. Yoon, Int. J. Mod. and E. Rojas, Classical Quantum Gravity 18, L49 (2001). Phys. A 12, 4217 (1997). [43] M. Celada, D. González, and M. Montesinos, Classical [60] J. C. Su, arXiv:hep-th/9805196. Quantum Gravity 33, 213001 (2016). [61] E. Allys, P. Peter, and Y. Rodríguez, J. Cosmol. Astropart. [44] P. Horava, Classical Quantum Gravity 8, 2069 (1991). Phys. 02 (2016) 004; E. Allys, J. P. B. Almeida, P. Peter, and [45] D. Grumiller, W. Kummer, and D. V. Vassilevich, Phys. Y. Rodríguez, J. Cosmol. Astropart. Phys. 09 (2016) 026; E. Rep. 369, 327 (2002). Allys, P. Peter, and Y. Rodríguez, Phys. Rev. D 94, 084041 [46] S. Deser, Found. Phys. 26, 617 (1996). (2016);J.B.Jim´enez and L. Heisenberg, Phys. Lett. B 770, [47] D. G. C. McKeon, Can. J. Phys. 95, 548 (2017). 16 (2017); Y. Rodríguez and A. A. Navarro, J. Phys. Conf. [48] L. Rosenfeld, Ann. Phys. (Berlin) 397, 113 (1930). Ser. 831, 012004 (2017); L. Heisenberg, J. B. Jim´enez, C. [49] D. Salisbury and K. Sundermeyer, Eur. Phys. J. H 42,23 de Rham, and L. Heisenberg, Phys. Lett. B 802, 135244 (2017). (2020); J. Sanongkhun and P. Vanichchapongjaroen, Gen. [50] P. A. M. Dirac, Can. J. Math. 2, 129 (1950); J. L. Anderson and Relativ. Gravit. 52, 26 (2020). P. G. Bergmann, Phys. Rev. 83,1018(1951); P. G. Bergmann [62] M. Crisostomi, R. Klein, and D. Roest, J. High Energy Phys. and J. Goldberg, Phys. Rev. 98, 531 (1955); P. G. Bergmann, 06 (2017) 124. 065015-29