Consistent ﬁrst order action functional for gauge theories

A novel ﬁrst order action principle has been proposed as the possible foundation for a more fundamental theory of General Relativity and the Standard Model. It is shown in this article that the proposal consistently incorporates gravity and matter ﬁelds, and guides to a new and robust path towards uniﬁcation of fundamental interactions


I. INTRODUCTION
Lorentz symmetry is a cornerstone of modern physics.The Standard Model is formulated as a quantum field theory based on the global Lorentz symmetry of Special Relativity, the fields being classified according to the representations of the (complexified) Lorentz group [1].Whilst gravity has been understood to arise from the "gauging" of the Poincaré group of the inhomogeneous Lorentz transformations in the Einstein-Cartan-Sciama-Kibble theory1 and its generalisations [3,4], this has not yet lead to a reconciliation of General Relativity and quantum mechanics.
A new take on the gauge theory of spacetime and gravity is based on precisely the homogeneous (complexified) Lorentz group2 [10].In general, gravitational models with polynomial actions can accommodate the zero ground state of the metric [14][15][16][17], which we refer to as the "pregeometric" property [18][19][20][21].The natural idea that spacetime arises via a spontaneous symmetry breaking that selects a preferred direction of time [22][23][24] is often implemented by additional fields on top of the geometry, but in-built to the Lorentz gauge theory wherein the symmetry breaking is necessary to emerge from the pregeometric state.The subtle elaboration of the mechanism entails an apparently drastically different description of gravity and spacetime, where even the Minkowski space has dynamical curvature and torsion [25].A recent Hamiltonian analysis established the consistency of the Lorentz gauge theory [13], and the possibility of a new cosmological paradigm was speculated [26].
In view of the SO (10) grand unification of the Standard Model gauge interactions [27], the new SO C (1,3) ∼ = SO C (4) gravitational gauge theory would naturally seem to fit into a yet grander SO(N) unification along the lines of the gravi-GUT proposals [28][29][30][31][32][33].However, the coupling of the Standard Model to the Lorentz gauge theory calls for a pregeometrisation of also the internal gauge field sector [34].Whereas the standard spinor actions are polynomial in the fields, 1 st order in the derivatives and possess the pregeometric property, a more fundamental action principle was required for the Yang-Mills gauge bosons.The suggested theory [34] can differ already classically from previous 1 st order formulations [35][36][37][38].
The characteristic feature of the new 1 st order gravity is the appearance of an effective dark matter source term.Interestingly, it was recently pointed out by Kaplan et al [39,40], that since unitary evolution in quantum mechanics is described by the Schrödinger equation which is 1 st order in time derivative, the classical limit of gauge theories, including gravity, could be generalised by the addition of shadow charges, whose presence reflects the fact that quantum fluctuations need not satisfy the constraints imposed by the standard, 2 nd order formulation of gauge interactions.This motivates us to consider also a modified version of the 1 st order Yang Mills theory, wherein shadow charges could arise as integration constants in the solutions to the equations of motion, analogously to the theory of gravity [34,41].
We shall focus on the conserved charges in the framework of Lorentz gauge theory from the perspective of the Noether's theorems, taking advantage of some recent developments in covariant phase space formalism [42][43][44][45][46][47][48][49].This article reports the results of our derivations.Section II presents the action, Section III covers the currents in the gravitational sector and Section IV covers the rest.All the are charges unambiguous and have a clear physical interpretation.We conclude in Section V that the consistency of the 1 st action formulation provides a valuable guiding principle in the quest for the final theory.

II. THE ACTION PRINCIPLE
We consider an action I = L with the 4-form where L G is the gravitational Lagrangian four-form polynomial in the gravitational fields which are taken to be a connection for the (complexified) Lorentz group ω ab and a scalar field φ a valued in the group's fundamental representation (which we term the khronon due to its potential to introduce a standard of time into gravitation).We choose where we have introduced the short-hand for the (proto)area-element B ab , and R ab = dω ab + ω a c ∧ ω cb is the curvature 2-form for ω ab .The ± X = (1 ∓ i⋆)X are the projectors to the self-dual (left-handed) or anti-selfdual (right-handed) sectors, ⋆ ± X = ±i ± X.It was demonstrated in [10] that (2) realizes an extension to General Relativity, when the metric tensor g is identified as g = Dφ a ⊗ Dφ a .
In (1) we take into account minimally coupled matter fields ψ which may be some p-forms.L M = L M (Dφ a , ψ, + Dψ) is the Lagrangian four-form for ψ which includes the gravitational fields, but we have excluded non-minimal couplings of − ω to ψ.We parameterise the material energy current t a and the spin current O ab , respectively, as where rep ab represents the Lorentz generator for ψ.Detailed examples are considered in IV.The variation of the total action then yields the equations of motion (EoMs) for the khronon, the gauge potential and the matter fields, respectively, and the symplectic potential = DD is the curvature 2-form operator and This shows that the action is stationary on-shell given Dirichlet boundary conditions for the variations of the gravitational and matter fields [43].There are no boundary conditions3 for the antiself-dual potential − δω ab .The EoM's E a and E ab imply that on-shell where M a is a 3-form that satisfies DM a = 0.

III. SYMMETRIES
We consider transformations δ that act on the dynamical fields.The transformation is a symmetry of L if δL = dℓ, and exact if ℓ = 0.Besides the Lorentz and diffeomorphism symmetry, the action (1) has a peculiar shift symmetry.Below we report the currents J corresponding to the 3 classes of symmetry transformations.Each current is manifestly conserved on-shell, dJ ≈ 0. For a gauge symmetry, the current is on-shell an exact form, J ≈ dj, where j is called the Noether-Wald charge [42,43].The charges are given as the integrated j of the Noether-Wald charge over a closed surface.

A. Lorentz transformation
Consider a Lorentz transformation of the fields with infinitesimal parameters λ a b , The Lorentz symmetry is exact δ λ L = 0, and we take this to be the case also independently for the matter 4-form δ λ L M = 0. Then we obtain Noether identities independently for the gravitational and matter sector.These are derived from ( 5) by considering parameters λ ab which vanish at the boundary s.t.we can neglect all the total derivatives in the variations.We obtain the 2 identities, The Noether current is an exact form on-shell J λ ≈ dj λ , where the Noether charge 2-form is now j λ = + λ ab B ab .Only the self-dual Lorentz transformations are associated with non-trivial charges.

B. Shift symmetry
The action L enjoys a shift symmetry, the invariance under constant translations of the khronon 4 , The Noether identity is trivial for this transformation.
The charge that we obtain using (7) and then (8a) describes the energy-momentum carried by the effective matter 3-form M a .This can be contrasted with Poincaré gauge theory, where the local translation is called a trivial gauge symmetry since it has zero charge.(One has to break covariance in order to extract a nonzero charge.We'll return to this point at III D.) In the Lorentz gauge theory, spacetime geometry (coframe and curvature) is generated by Lie-dragging the fundamental fields (khronon and gauge potential) covariantly5 along a vector ξ: where is the interior product on differential forms, and here and in what follows, the D is always the total covariant derivative, thus involving also internal gauge fields in the case that the fields ψ have internal gauge charge.This gauge symmetry is not exact in the sense of L being invariant under the transformation, but We obtain the Noether identity for gravity, and for the invariance of L M we get In a non-degenerate spacetime wherein e a ≡ Dφ a has an inverse @ a , these can be rewritten as where T a = De a = φ a .The Noether current vanishes identically J ξ = ξ • Θ − ξ L = 0, and thus implies that a change of coordinates is a trivial gauge transformation.
The matter sources have to be formulated consistently s.t.
which means that the Hilbert (i.e. the metrical) and the Noether (i.e. the canonical) energy-momenta are equivalent.

D. On frame-dependent charges
One can combine transformations from the above 3 classes of symmetry transformations.An example is the coordinate diffeomorphism, which is the combination of a Lorentz transformation and a proper diffeomorphism, L ξ = δ ξ + δ λ=ξ ω .The possible physical relevance of this transformation is subject to case-dependent subtleties.The way that the fields are dragged along a vector ξ has no Lorentz-covariant meaning.The corresponding charge has no Lorentz-invariant interpretation.With some manipulations, using e.g. ( 4) and assuming (17), one can verify that the Noether current from ( 18) is given, as expected, precisely by (11) with the Lorentz transformation parameter λ ab = ξ ω ab .So, the charge is frame-dependent because the parameter is non-covariant.Nevertheless, it is very well known that the currents generated by L ξ correctly describe the physical energy and momenta in many relevant special cases.This is so because energy and momentum can only be defined with respect to a reference frame, and thus it is expected that these charges are frame-dependent 6 .The basic example is the standard result in Minkowski space that the symmetry of matter actions in the fixed background under diffeomorphisms corresponding to the Killing vectors of Minkowski space can -with "improvements" -lead to the conservation of the stress energy momentum tensor and the six conservations associated with the boost and rotation Killing vectors.This can be generalised to a maximally symmetric space, available perhaps globally, locally, asymptotically, or say, as an extra-dimensional embedding.These considerations apply as such in the geometric phase of Lorentz gauge theory.

We consider fermions below in IV
we obtain the currents the EoMs and the symplectic potential In a real frame, E ψ = Ēψ .The identity ( 17) is consistent with the energy current (21a).

B. Yang-Mills fields
The 1 st order pregeometric Yang-Mills theory [34] is formulated in terms of the interface (proto)area element with the "one foot outside" and the other h a , valued in the adjoint representation of the Yang-Mills gauge group, a "vierbein" spanning an internal hyperspace 7 .We recall that D is the total covariant derivative, thus involving also the Yang-Mills gauge field A whose field strength is denoted by F .Now the field excitation * F (where * is the Hodge dual) is not postulated a priori, but the gist of this new approach to gauge interactions is that the field excitation * B = η ab * Bab ≈ * F emerges from the variational principle.An action density which achieves this is where A is the Yang-Mills gauge field, J is its material source, and • is the trace over the Lie algebra.

Standard theory
The variation yields us the EoMs, and the symplectic potential The gravitational source currents are It is not difficult to see that the internal symmetry transformation results in the expected current J g ≈ J .It has to be concluded that this prescription is the mere reformulation of the standard Yang-Mills theory.In particular, the symplectic current ( 28) assumes its expected form, and the energy current (29a) fails the consistency requirement (17).
A slightly more economic reformulation considers instead the 6 d.o.f.'s of the excitation carried in the fundamental variational d.o.f.α ab valued in the adjoints of both the Lorentz and the Yang-Mills gauge groups, s.t.h a = α a b e b .However, this would not change the conclusions.

Modified theory
A more radical alternative is to encode the variational d.o.f.'s into the isokhronon α a living in the fundamental representation of the Lorentz group and giving rise to the internal hyperspacetime h a = Dα a in an analogy to the khronon φ a in the external spacetime.Then an analogy of dark matter may also arise in form of nontrivial vacua.This describes the situation in quantum mechanics wherein the field force lines need not be strictly attached to the material source points.The case * B ≈ * F is just one of the solutions, and therefore the solution space can be constrained by phenomenological data 8 .
The variation (26) should then be reconsidered, since now the 3-form Ẽa in (27a) is closed but may not vanish on-shell.Nontrivial modifications now enter into the expression for the symplectic potential, as well as the gravitational source currents, Remarkably, the energy current (33a) identically satisfies (17).So, the results for the 3 classes of gravitational charges in III remain intact in the presence of the modified Yang-Mills interactions.
It can be verified that the internal symmetry transformation δ g α a = [g, α a ], δ g A = −Dg is associated with the current where in the last step we used the EoM's (27) (see III.C of [34]).The possible contribution to the divergence of B due to a vacuum polarisation or magnetisation (see Eq.( 50) of [34])) is cancelled by the 2 nd term in (34), and we recover the canonical gauge current.A novel property of isokhronon theory is the shift symmetry, δ χα a = χa , where D χa = 0 . ( The conserved current, is the integration form X a responsible for the possible vacuum excitation [34].It is the analogy of the integration form M a in the gravity sector 9 .An important caveat is that one is now not free to choose both integration forms independently for arbitrary solutions.Therefore this theory is probably not a viable modification of the Standard Model gauge interactions.Let us briefly speculate on a possible refinement of the unified theory, 1 st restricting to case of an Abelian gauge field A. Now, if we consider, instead of φ a , a field in (( 12 ⊗ 0) − ⊗ ( 1 2 ⊗ 1 2 ) + ) of the complex Lorentz group, and instead of the α a , a field in ((0 ⊗ 1 2 ) − ⊗ ( 1 2 ⊗ 1 2 ) + ), then both of these fields are coupled to an independent SU(2) connection.Consequently, there always exist solutions with X a = 0, apparently restoring the viable limit to standard gauge theory.However, this prescription is not without other repercussions as then the B is not a scalar but carries the SU(2)×SU(2) charges from the anti-selfdual sector of the Lorentz group.Optimistically, this hints to the structure of the gravielectroweak theory and to the geometrisation of the Higgs mechanism operated by the isokhronon in the hyperspacetime.

C. Scalar fields
Putting the above speculation aside, since the Standard Model features a Higgs scalar field, for completeness we take into account a scalar field ζ.In the 1 st order formulation, it is accompanied by a Lorentz vector z a , and a possible action is leaving open the possibility of a nontrivial potential U (ζ).We obtain the EoM the symplectic contribution and the source current whilst for scalar fields O ab = 0.

D. Cosmological constant
The perhaps simplest energy source is a cosmological constant.The contribution to the matter action is given by a Lagrangian with 2 new fields, a scalar Λ and a 3-form κ, The source contributions (4) is The EoM's for the 2 fields dictate that dκ ≈ ⋆1 and dΛ ≈ 0. Thus L M ≈ 0. In the derivation of the diffeomorphism Noether current, we have to take into account that now (17) does not hold.We obtain J ξ = Λ * ξ/2, so it would seem that the Λ does contribute.The non-trivial charge reflects the effective breaking of the longitudinal diffeomorphisms.The 3-form gauge symmetry κ → κ + k, where k is an arbitrary 2-form, has a non-trivial charge that is given as the integral of j κ = Λk/2 over a 2-surface.

V. CONCLUSION
Conserved charges lie at the heart of gauge theories.They characterise the observables of the theory and their algebra governs the structure of the theory.Charges are of paramount importance in holography and play a central role in (most approaches to) quantum gravity.In fact, the putative quantum theory might be entirely deduced from the charge algebra, according to the corner proposal and related current developments [44,45].
In this article we presented the physical charges in the new Lorentz gauge theory of spacetime and gravitation.The charges associated with the Lorentz symmetry and diffeomorphism symmetry are the direct extrapolation (e a → Dφ a ) of the results in Poincaré gauge theory.A novel feature is the "dark shadow matter" current M a associated with the shift symmetry of the action (1).
The theory was coupled to the pregeometrised Standard Model of particle physics, and it was shown that its matter fields generate consistently both the energy momenta and the angular momenta source currents.However, the most straightforward implementation of the Standard Model gauge fields inherits the issue in their usual, 2 nd order geometric formulation, which does not consistently describe the gravitational sources by the canonical Noether currents.It has often puzzled theoreticians that the canonical energy-momentum currents have the wrong expression, unless modified by some of the proposed "improvements" [1,47,[54][55][56][57][58].We considered a possible modification of the pregeometric 1 st order theory, which would provide a solution to the issue, and features the newly suggested shadow charges, associated with the shift symmetry of the 1 st order fields in the internal sector.
The modified theory is not yet a phenomenologically viable replacement of the Standard Model interactions (though it might describe hypothetical new interactions e.g. in cosmology), but calls for the elaboration towards a more final theory.We conclude that the 1 st order action principle provides a new robust framework to negotiate the unification of internal and spacetime gauge interactions and the reconciliation of gravity and quantum mechanics.