Minimal coupling in presence of non-metricity and torsion

We deal with the question of what are and what are not minimal coupling prescriptions in presence of torsion and/or non-metricity, carefully explaining while the naive substitution $\partial\to\nabla$ is not actually a minimal coupling prescritpion. We will also investigate whether minimal coupling prescriptions at the level of the action (MCPL) or at the level of field equations (MCP) lead to different dynamics. To that end, we will first write the Euler-Lagrange equations for matter fields in terms of the covariant derivatives of a general non-Riemannian space, and derivate the form of the associated Noether currents and charges. Then we will see that, if the correct minimal coupling prescriptions is applied, for spin 0 and 1 fields the results of MCPL and MCP are equivalent, while for spin 1/2 fields there is a difference if one applies the MCP or the MCPL, since the former leads to charge violation. We will also explicitly show how the usual naive substitution $\partial\to\nabla$ is not a minimal coupling prescription for spin 0, 1/2 and 1 fields in presence of torsion and/or non-metricity.


I. INTRODUCTION
General Relativity (GR) had an important impact in the development of non-Euclidean geometries at the beginning of the twentieth century [1,2]. Einstein followed the (by then recent) developments of Riemann to shape his theory of gravity, and rapidly the idea that spacetime must have a Riemannian structure was naturally accepted despite the lack of direct experimental evidence [3]. In recent years, a renewed interest in determining whether the space-time geometry is actually Riemannian or otherwise has emerged boosted, in part, by the exploration of new gravity theories in the metric-affine formalism. The first step forward in this direction was already given by Cartan a few years after the birth of GR. He suggested that the torsion tensor could be introduced in the description of gravity, and he introduced the idea that it should be related to the intrisic angular-momentum of matter [4][5][6][7], but this idea was quickly forgotten given that the spin of the electrons was not already discovered. Weyl, Einstein, and others used Cartan's idea and introduced torsion in an unsuccesful attemnpt of unifying gravity with electromagnetism [8][9][10][11][12]. In the mid 50's of the past century, when trying to describe the continuum limit of the micro-structure of solids, the torsion tensor was found to be related to the density of dislocation deffects present in the micro-structure [13][14][15][16][17][18]. At the same time Kibble and Sciama [19][20][21] considered the description of gravity as a gauge theory of the Poincaré group, formulating the so called Einstein-Cartan-Sciama-Kibble (ECSK) theory. Soon after the formulation of ECSK theory as a gauge theory, Hehl and collaborators [22][23][24][25][26][27] worked out its geometrical formulation, showing how the spin of particles was related to torsion in ECSK, being torsion sourced by the spin-density [27][28][29]. The observable consequences of torsion have been considered in several works, concluding that they are generally suppressed by a high energy scale and therefore practically unobservable at low energies, unless a scenario with high density of spin is considered [28][29][30][31][32]. However, it * adria.delhom@uv.es has been pointed out that torsion could have important role in early universe cosmology, as it could prevent the Big Bang singularity [33]. Torsion-based theories have also received much interest recently in the context of teleparallel gravity and its functional extensions [34]. Up to date, the different attempts looking for the experimental detection of torsion have given negative results [35].
Whereas the observable effects of torsion has been fairly well studied in gravitational contexts, the other signature of non-Riemannian geometry 1 , which is non-metricity, has not been analyzed as deeply (see [36] for a short review of these results). However, there has recently been renewed interest analysing several theories that feature non-metricity and/or torsion, such as teleparallel and symmetric teleparallel theories [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53], Ricci-based gravity thoeries (which encompass f (R)) or Born-Infeld gravity) [54][55][56][57][58][59][60][61][62][63][64], or others [65][66][67][68][69]. Some of these works put forward that gravity theories with non-metricity can avoid the spacetime singularities present in GR already at a classical level [70,71]. It has also been suggested that non-Riemannian geometries could be successful in being a low-energy effective description of theories of quantum gravity, as they could be more suited in accounting for features of a quantum-spacetime that may exist at high energies [61,[72][73][74][75]. This makes clear that the existence of non-metricity should be experimentally tested at different energy scales. In order to probe non-metricity, we must first understand its experimental consequences. Recent works on metric-affine theories of gravity show how non-metricity could have important effects in scenarios with very high energy-density, giving rise for instance to effective particle interactions or shifts in the energy levels of atomic systems [61,62], which can be used to constrain the energy scale at which Ricc-based gravity theories depart form GR. Nonetheless, it remains an open question whether Riemannianity (i.e. absence of torsion and non-metricity) extends to higher energy regimes or it is a low energy limit of a more general non-Riemanian spacetime structure.
Other current lines of research also investigate the possible implications of spacetime non-metricity in classical trajectories or in the definition of geometric clocks assuming that test bodies follow affine geodesics [76][77][78] . Nonetheless, this assumption should be derived from the geometrical optics approximation of the field equations that describe matter at a fundamental level, and it is not yet clear how to do so (see [64] for a discussion). In order to derive this limit, one must first understand how the matter fields couple to generic torsion and non-metricity tensors. For minimally coupled scalar fields, it is well known that the non-Riemannian pieces of the affine connection do not couple directly in the field equations. This result is not so clear for spin 1/2 fields. Indeed it is not trivial how to generalize the Dirac equation to non-Riemannian spacetimes, as the minimal coupling prescription applied to the Minkowskian spin 1/2 field equation (MCP) gives a different result than when applied in the Minkowskian spin 1/2 Lagrangian (MCPL) [79]. Also recently, it has been claimed that the MCP and MCPL give different dynamics for matter fields in general in Riemann-Cartan space-times [80]. Since in Riemannian space-times both prescriptions lead to the same dynamics, the aim of this work is trying to shed some light into the question posed in [79,80] of whether the MCP or the MCPL is generally better suited to describe matter degrees of freedom in presence of torsion and/or non-metricity. To that end, we will discuss what is and what is not a minimal coupling prescription for matter fields of spin 0, 1/2 and 1. We will also show how both MCP and MCPL describe the same dynamics for scalar and vector fields if applied correctly, contrary to the claims of [80], and that the MCP is not consistent with charge conservation for fermionic fields, thus addressing the question raised in [79].
The structure of the paper is as follows. We will start in section II with a discussion on what does minimal-coupling mean and why the naive recipe of replacing ∂ by ∇ is not really a minimal coupling prescription when torsion and/or non-metricity do not vanish. In section III we will derive a generalised form of the Euler-Lagrange (EL) equations for arbitrary spin matter fields in a space-time with generic torsion and non-metricity tensors. As a byproduct of the derivation of the Euler-Lagrange equations, we will see how non-metricity and torsion do not affect the functional form of conserved matter currents and charges, which have the same functional dependence on the matter fields as in the Riemannian case. Then, in section IV, we will see how applying a naive MCP to the Klein-Gordon equation leads to a field equation that is not equivalent to the generalised Euler-Lagrange equations foe scalar and vector fields, while applying the proper MCP leads exactly to the generalised EL equations. We will see how for spin 1/2 fields, even if the proper MCP is applied to the Dirac equation, the resulting equation is not equivalent to the one obtained by the generalised EL equations through the correct MCPL (if one uses the hermitian Minkowskian action). Indeed, while the generalised EL equations obtained through the MCPL are perfectly consistent to describe a spin 1/2 field with interactions that conserve an internal charge, the equations resulting from applying the MCP leads to violation of this conservation. Thus, if charge conservation is assumed, the only valid minimal coupling prescription for spinor fields in presence of torsion and/or non-metricity is the MCPL. We will also explictly show how non-metricity decouples from minimally coupled matter fields, although physical effects related to non-metricty could occur doe to its non-trivial relation with the space-time metric (see [61,62]). Finally, we add some concluding remarks in section V.

II. WHAT IS MINIMAL COUPLING?
It is now convenient to discuss the meaning of minimal coupling and the subtleties behind the naive prescription commonly used to implement it. Usually, the minimal coupling prescriptions (both MCP and MCPL) are implemented by the following rule of thumb: Wherever you find a partial derivative ∂ or a Minkowski metric η in flat space-times, substitute them by the appropriate covariant derivative ∇ and space-time metric g. This sentence is a recipy that works well in absence of torsion and/or non-metricity, although it is misleading about what a minimal coupling prescription actually is, and it can lead to wrong results when torsion and/or non-metricity are non-vanishing. Indeed a more precise understanding of the minial coupling prescriptions can be achieved by arguing in the following way. As nicely argued in [81], the operator ∂ acting on tensor fields is frame dependent, i.e. different coordinate systems have associated a different ∂ operator. Therefore ∂ will yield non-coordinateinvariant objects when applyed over any tensor or spinor field. Hence, given that physical theories must be coordinate independent, we must not employ a particular ∂ operator in the construction of physical theories. Nonetheless, there are other derivative operators such as the covariant derivative ∇, the exterior differential d, the co-differential δ, or suitable combinations of these that are the same operators in all coordinate systems, and therefore both yield coordinate invariant objects when applied to tensor or spinor fields. 2 This makes ∂ unappropiate in the construction of physical theories even in flat space-times, since it will be a different operator in two non-inertially-related frames. Since usually field theories are formulated first in a Minkowskian context and in inertial coordinate frames, the symbol ∂ is commonly used, although one should recall that even in this case the derivative operators appearing in the the action and field equations are ∇, d, δ or appropriate combinations of them like the wave operator dδ + δd. Thus, strictly speaking, an MCP or MCPL prescription will be a minimal coupling prescription if it keeps track of the differential operators that are used in Minkowski space-time and uses the same operators in a general space-time. This can be summarised in the following rule of thumb: Substitute η by g and use the same differential operators in Minkowski and in general space-times (in the sense of using ∇, d, δ or the appropriate combination that is also used in the Minkowskian case).
The origin of the naive recipy for the minimal-coupling prescriptions stems from its usefulness in Riemannian spacetimes, which can be argued as follows. For spinor fields there is no ambiguity in which is the operator employed since d or δ are not well defined in this case. For scalar and vector fields (0-and 1-form fields respectively) the combinations 3 in which ∂ appears in the respective actions are such that substituting ∂ by ∇ gives the same result than using the d operator when the space-time is Riemannian. Thus for the MCPL and in Riemannian space-times the naive ∂ → ∇ gives the right answer. While this is also true for scalar fields if we aply the MCP in Riemannian space-times, as we will see later, there is already a difference if we apply the MCP as ∂ → ∇ for vector (1-form) fields: the wave operator that governs the knietic term of the field equations in Minkowski space-time can be written 4 as η = η µν ∂ µ ∂ ν , and thus it would be generalised to the operator g µν ∇ g µ ∇ g ν which gives a different result than the correct wave operator g = dδ g + δ g d when applied to (1-form) fields (see section IV C for details). Thus although the naive MCP could lead to ambiguities in the vector field description, the naive MCPL prescription of replacing ∂ by ∇ in the Minkowskian Lagrangians works perfectly well in Riemannian space-times.
Nonetheless, as we will see later in detail, both MCP and MCPL fail in being a minimal-coupling prescriptions for vector fields in non-Riemannian space-times if implemented through the naive rule ∇ → ∂. This can be understood as follows: If following the naive MCPL we forget that the field-strength of the vector field is defined by F ≡ dA and look only at its expression in some coordinate system F µν = 2∂ [µ A ν] , then the naive MCPL given by substituting ∂ → ∇ in the vector action will lead us to a wrong definition of field-strengthF µν = 2∇ [µ A ν] which introduces an extra non-minimal coupling between torsion and the vector field, as opposed to just aplying correctly the MCPL by keeping explicit track of the use of the d operator in the vector Lagrangian. When trying to use the MCPL, the naive substitution of η µν ∂ µ ∂ ν by g µν ∇ µ ∇ ν instead of the use of the correct wave operator δd + dδ (which in Minkowski space is given by η µν ∂ µ ∂ ν ) introduces non-minimal couplings to the torsion and non-metricity tensor for both scalar and verctor fields. For scalar fields, given that only first order derivatives appear in the action, there can be no naive MCPL since ∂Φ = dΦ = ∇Φ by definition. However, the naive MCP does give rise to non-minimal couplings between the scalar field and the torsion and non-metricity tensors.

III. MCPL IN NON-RIEMANNIAN SPACE-TIMES: EULER-LAGRANGE EQUATIONS AND CONSERVED CURRENTS
As it is common to work with actions and field equations written in terms of fields and their first covariant derivatives 5 , let us first derive the Euler-Lagrange (EL) equations in terms of the affine covariant derivative ∇. To that end, it will be useful to express the divergence operator associated to a volume form to the affine covariant derivative ∇ employed in the action. This relation will allow us to employ Stokes' theorem in order to derive the generalised EL equations and the conserved currents and charges associated to continuous symmetries af the action.

A. The divergence operator
Let M be an n-dimensional smooth oriented manifold with volume n-form ε = f dx µ1 ∧ ... ∧ dx µn in some chart. The divergence operator associated to ε acting on vector fields A ∈ X(M) is the function Div ε : X(M) → C ∞ (M) defined by [82] which in a coordinate chart x µ reads This definition is completely general, as it only requires the differential structure of M and a general volume form defined on it 7 ; neither a metric tensor g nor an affine structure Γ on M.
The interest of this operator relies in that it satisfies a generalised divergence theorem, i.e. it relates a vector field defined in a in a volume V with its flux through the boundary ∂V. As shown in [82], from Stokes' theorem one finds where V is the n-volume enclosed by ∂V. This is the generalised divergence theorem for n-dimensional manifolds. Now, if a metric structure g is introduced, there is a canonical choice for the volume n-form: ε ≡ dV g = g 1/2 dx µ1 ∧ ... ∧ dx µn (see footnote 4). In this case, in Lemma 16.30 of [82], it is proven that 8 where n is the unit normal to ∂V and dVg is the induced volume form on ∂V by g. Therefore, when a metric is present (and is chosen to define the volume element), the right hand side of (3) can be interpreted as the flux normal to the boundary enclosing V. Thus the generalised divergence theorem (3) can be seen as a generalised Gauss' law which relates the divergence of a vector field A inside a closed volume V with the integration over ∂V of the component of A normal to ∂V.
Note that a metric g also induces a canonical affine structure (or affine connection) Γ = C(g) on M which is said to be compatible with it: the so called Levi-Civita connection of g. However, the affine structure on M needs not be compatible with g, and it is generally independent of it. Indeed, one can have a manifold with an affine structure but no metric structure. We say that a manifold where g and Γ are compatible is a Riemannian 9 manifold (M, g, C(g)); and one where Γ and g are independent is a non-Riemannian manifold (M, g, Γ). In the following, we will generally work in a non-Riemannian manifold with the canonical volume element associated to its metric.
Any affine structure Γ defines (and is defined by) a covariant derivative ∇, which is completely specified by its connection symbols Γ µν α . The action of ∇ on (the components of) an n-form 10 f and a vector field A µ is given by 8 Here ι * S (A dVg) is the restriction of the (n-1)-form A dVg to the boundary ∂V [82]. 9 Note that matematicians call a Riemannian manifold one whith an affine structure compatible with the metric and with a metric of Riemannian signature. We will use Riemannian and non-Riemannian referring only to the (non-)compatibility of the connection and the metric and not to the signature of g, as it is often done in gravitational physics. 10 Remind that, since the space of n-forms in an n-dimensional manifols is of dimension 1, any n-form is proportional to the trivial ome dx 1 ∧ ... ∧ dx n , and therefore is specified only by one component (the proportionality factor). This component is sometimes called a tensor density of weight +1.
Therefore using (2) Div(A) can also also be written as where S µν α ≡ −2Γ [µν] α is called the torsion of the affine connection, and it identically vanishes in Riemannian manifolds. Note that (5) and (6) are true whether M has a metric structure or not.
In a non-Riemannian manifold, it is always possible to perform a decomposition of the connection symbols in the form and where Q αµν ≡ −∇ α g µν is the so-called non-metricity tensor, which identically vanishes in Riemannian manifolds, L µν α is the distortion tensor, and K µν α is called contortion tensor (see e.g. [83]). Here C µν α are the Christoffel symbols of g, which are the connection symbols of the Levi-Civita connection C(g). From (7) and the second equation in (5) (identifying the volume form f with g 1/2 ), it is possible to show that ∇ µ g 1/2 = − 1 2 Q µα α g 1/2 . Using this identity, from (6) one finds the following relation which is the sought relation between the divergence operator, the covariant derivative, and the non-metricity and torsion tensors of M. For Riemannian manifolds, where Q µνα and S µν α vanish by definition, (8) reduces to the usual expression Div(A) = ∇ g µ A µ as it must be, where ∇ g µ is the covariant derivative associated to the Levi-Civitta conection of the metric g.
Notice also that from the definition of C µν α in (7) it is possible to get the following useful relation Since from the definition of divergence operator we have (2), we have also generally that Therefore, we end up with the result that in any manifold where the volume element is given by the metric, no matter what the affine structure is, the identity (10) holds, which can be summed up in the following equation: for any (M, g, ∇). From the above discussion one can infer that, in a non-Riemannian manifold, the relation between the divergence operator (given by the metric structure) and the covariant derivative of the manifold is not the same as in a Riemannian manifold due to the fact that the affine structure has no relation with the volume form, and torsion and nonmetricity have to be taken into account. On the other hand, in a Riemanian manifold, given that ∇ = ∇ g , the affine structure and the volume element are indeed related, which translates into a direct relation between the divergence operator and covariant derivative of the manifold.

B. Euler-Lagrange equations of a Minimally coupled theory in non-Riemannian spacetimes
Clarifying the relation between the divergence operator and the affine structure will be useful to derive the generalised EL equations for any minimally coupled matter Lagrangian (although it can also be done without using the above relations). To do so, we start by applying the MCPL to the usual Minkowski Lagrangian as explained in section II, so that we start with a functional L(Ψ i , ∂Ψ i ) that defines the following action: where L[Ψ i , ∇ µ Ψ i ] is a scalar and {Ψ i } is the collection of matter fields. Let us point out that, if d instead of ∇ is employed in the construction of S m , as is the case for vector fields, we can always re-write them as covariant derivatives plus some extra terms proportional to the torsion tensor. Therefore (12) is rather general, since any minimally coupled matter action can be recast in such form.
The field equations are obtained, as usual, by applying the extremal action principle and then solving the variational problem δS m = 0 for some arbitrary variations of the matter fields δΨ i vanishing at the boundary of V. Notice though that given variation of the field δΨ i naturally introduces also a variation in its partial derivartive δ(∂Ψ i ), and the variational problems that one is used to solve are in terms of the field variables where Γ i µ are the connection coefficients in the representation of Ψ i , we can re-write the above action in terms of the independent variables {Ψ i , ∂ µ Ψ i } and proceed with standard variational methods. By explicitly substituting ∇ µ Ψ i by its expression in terms of {Ψ i , ∂ µ Ψ i } we can recast (12) as a function of the variables where the tilde here emphasizes that the functional form of S m changes when we consider it as a functional of the new variables. Now we can employ the standard methods of variational calculus to solve δS m = 0 for an arbitrary variation δΨ i that vanishes at the boundary of V, which leads to where we have used that We now want to invert the change of variables by writing ∂ µ Ψ i as a function of the old independent variables Then, we can write an arbitrary variation of ∂ µ Ψ i as a function of an arbitrary variation of the old variables: δ(∂ µΨi ) = δ(∇ µ Ψ i ) + Γ i µ δΨ i . Plugging this expression in the above equation, we end up with which by means of (8) can be recast into Since the last term is a boundary term by the generalised Gauss' law (4), it vanishes for variations δΨ i = δΨ i vanishing at the boundary of V. Thus, the above equation leads to to the non-Riemannian version of the covariant Euler-Lagrange equations, which reads Note that while in the Riemannian limit we recover the usual covariant Euler-Lagrange equations, in the general case there are, apparently, explicit couplings between the non-metricity and torsion tensors and the matter fields. However, these apparent couplings are indeed compensated by taking into account that the covariant derivative of the second term in (18) is not the one associated to the Levi-Civita connection of g. To show this, we can use (7) and split the covariant derivative in front of the second term of (18), thus re-writing the non-Riemannian Euler-Lagrange equations (18) as where (Γ i N R ) µ is the non-Riemannian part of the connection in the representation corresponding to Ψ i (i.e. the piece of Γ i µ that features torsion and non-metricity) and ∇ g is the Riemannian covariant derivative. Here we can see how the explicit couplings to non-metricity and torsion that are present in (18) actually cancel out and do not contribute to the dynamics, and the only possible source of non-Riemannian couplings comes from the (Γ i N R ) µ , and from ∇ µ Ψ i in the derivatives of the Lagrangian. This could not have been otherwise because of the following reason: Since in the action (12) the only possible non-Riemannian couplings appear through the (Γ i N R ) µ Ψ i term in the ∇ µ Ψ i variables, the only non-Riemannian terms that can show up in the field equations will also enter through (Γ i N R ) µ . From this result follows the conclusion that whether a minimally coupled (in the sense of MCPL) matter field couples or not to the non-Riemannian features of a general space-time depends only on theform of the connection in its corresponding spin representation. As we will see later, in the case of spin 0 and 1 fields, all the explicit couplings between non-metricity and torsion dissappear form the field equations if the minimal coupling prescriptions are applied correctly. On the other hand, in the case of spin 1/2 fields, a residual interaction with the totally antisymmetric part of the torsion tensor remains for the MCPL, and a more complicated coupling to the traces of torsion and non-metricity occurs for the MCP. This coupling will be seen to source a charge violating current.

C. Conserved currents and charges
As a by-product of the derivation of (18), we can also investigate whether the functional form of the Noether currents/charges associated to matter fields will be modified by non-metricity or torsion corrections. For completeness, let us briefly explain the geometrical meaning of a conserved current. By definition, a conserved vector current over a spacetime M equipped with a volume form ε is a vector field J ∈ X(M) that satisfies Div ε (J ) = 0 (20) over M. This definition is valid for Riemannian as well as for non-Riemannian spacetimes, and it has an intuitive geometrical meaning as will be clarified below. Notice that provided that the volume form is chosen to be the one given by the metric, in Riemanninan spacetimes this condition is equivalent to ∇ µ J µ = 0, while in non-Riemannian spacetimes the above condition it is not. The importance of the condition (20) relies in that it allows one to define a scalar quantity Q J on every spatial hypersurface of a Cauchy foliation of the given space-time such that Q J is invariant under time-translation (i.e. change of spatial hypersurface). This is why a vector satisfying Div ε (J ) = 0 is called a conserved current, and Q J is its associated conserved charge.
In order to clarify the geometrical meaning of the condition (20), let us first precisely formulate the existence of such conserved charge Q J in a space-time with volume form given by the metric. Consider coordinates (x 0 = t, x i ) and a foliation of spacetime given by the one-parameter family of space-like 3-surfaces Σ t normal to ∂ t . Consider also a 3-ball σ t defined in every Σ t by (x i x i ) 1/2 R, with R an arbitrarily big constant. Define the closed 4-volume B(t 1 , t 2 ) enclosed by σ t1 , σ t2 and C; where C is the union of the boundaries of each σ t for t ∈ (t 1 , t 2 ) (see fig. III C for clarification). Any vector field J defines a charge Q t J at each Σ t given by where J t = g(J, ∂ t ), ∂ t is the unit normal to σ t and dVg is the volume form induced on Σ t by dV g . Using Gauss' law (4), decomposing ∂B as 11 ∂B(t 1 , t 2 ) = (−σ t1 + C + σ t2 ), and for configurations of Ψ such that J t vanishes quickly enough at spatial infinity 12 we find This is valid for any value of t 1 and t 2 ; and in particular, for infinitesimally small values of δt = t 2 − t 1 we find Therefore a charge defined by a conserved vector current remains constant under time-evolution, i.e. we say it is conserved. The arguments within this section are independent of the choice of connection Γ, which remarks the importance of the condition Div g (J) = 0 instead of any condition involving any covariant derivative, which points out that the expression "conserved with respect to a connection", usually found in the literature, can be misleading. Indeed, these 11 The sign infront of σ t 1 is required for ∂B(t 1 , t 2 ) to have the standard induced orientation from V(t 1 , t 2 ) [82]. 12 The precise requirement is that Ψ vanish quiclky enough with increasing R so that the integal over C vanishes when R → ∞.
arguments depend only in the choice of the volume form, and not on the metric itself. Therefore, a more correct expression could be expression conserved with respect to a volume form, or a metric.
From the above discussion, it is now clear that the condition (20) implies that the change in the amount of charge Q J enclosed in the 3-volume σ t1 is given exactly by the flux of the current J through the 4-volume B, i.e. the amount of charge that exits σ t 1 in the time interval t 2 − t 1 . Therefore, total charge cannot be created or destroyed if condition (20) holds.
We can now proceed to investigate wether the functional form of Noether matter currents should be modified to account for non-Riemannian fratures. Noether currents have proven to be very useful tools for analyzing and extracting physical information of field theories. They have been used in the context of extended gravity by the use of the so called Noether symetry approach (see for instance [84][85][86][87]); and they lie at the very definition of physical charges. Noether currents are defined from a symmetry of an action, and are conserved if the corresponding symmetry is realised. Let us try to undestand whether torsion or non-metricity play any role in the physics associated to Noether currents and charges Given an action like (12) and a transformation of the matter fields δΨ i which is a continuous symmetry of that action, we can work out the functional form for the Noether current associated to this symmetry by the following argument. The first term in (17) vanishes for fields satisfying their equations of motion (18). Since δΨ i is associated to a continuous symmetry of the action (12), we have δS m = 0, hence, the second term in (17) must also vanish, yielding Therefore, as stated by the Noether theorem [88], a continuous symmetry of the matter action implies the existence of an associated conserved curent J ∈ X(M) defined by and its corresponding conserved charge given by (21). A current defined from a Lagrangian as in (23) is called Noether current, and the corresponding charge is called Noether charge. Physical charges are the Noether charges associated to some continuous global symmetry of the matter action. For instance, the electric and color charges are associated to the global U (1) EM and SU (3) C symmetries of the Standard Model action. The above equation (23) shows that Noether currents (and charges) have the same functional form in Riemannian and non-Riemannian spacetimes in terms of the covariant derivative of the matter fields. In other words, nonmetricity and torsion do not change the functional form of the currents and charges. However there could in principle be implicit corrections entering through ∇ µ Ψ i . However, we will see that in the case of scalar and spinor fields, there are no corrections due to the fact that ∇ µ Ψ i enters linearly in the action. By using the above results, we will also show later that the requirement of charge conservation in non-Riemannian space-times will serve as a discriminator between the MCP and MCPL for spin 0 and 1/2 fields.

IV. MINIMALLY COUPLED MATTER FIELDS IN NON-RIEMANNIAN GEOMETRIES
In this section we will employ the results previously derived and work out the examples of minimally coupled scalar, spinor and vector fields, focusing on the viability or distinguishability of the MCP and MCPL prescriptions, both in its naive and correct version. Also we will show whether they couple minimally to non-metricity and torsion for the different versions of each prescription. As it will become apparent later, the explicit couplings between minimally coupled matter fields and non-metricity disappear from the field equations in all the examples when the correct MCP and MCPL are applied. However it could still induce non-trivial modifications to the dynamics of matter fields through their coupling to the metric: Since non-metricity is a tensor related to both, metric and affine connection, a coupling to the metric can encode some non-metric effects, as suggested in [61,62]. Also, if the correct MCP and MCPL are applied, torsion does not couple either to minimally coupled scalar and vector fields, although its totally antisymmetric part does couple to minimally coupled spin 1/2 fields. On the other hand, we will see how the naive MCP and MCPL generally lead to non-minimal couplings between matter fields, and the torsion and non-metricity tensors.

A. The minimally coupled scalar field
The action of a complex scalar field can be written in Minkowski space-time with any of the operators ∂, d and ∇, since they are all the same when acting on scalar fields. However, the wave operator g of the Klein-Gordon equation is defined only for p-forms, so that for simplicity we will view Φ as a 0-form. However, in order to be able to use the generalised EL equations (18), we will write the action in terms of write ∇ µ Φ. Thus, by applying the MCPL, the action of a complex scalar field in a general background geometry reads as The field equations that follow after using (18) are where we have defined Γ Φ ≡ g µν ∇ µ ∇ ν Φ and the non-Riemannian current This is in apparent contradiction with the expectations that neither non-metricity, nor torsion couple to a minimally coupled scalar field, given that they do not appear in the action (24). Nonetheless, by using again the decomposition of the affine connection (7) it follows that with g = dδ g + δ g d (see footnote 3), and where the same identity holds for Φ † . Using (27), the above field equations (25) can be re-written as which, as expected, show that minimally scalar fields do not couple to torsion and non-metricity explicitly. A more clever way to find this result is by noticing that, since for scalar fields ∇ µ Φ = ∂ µ Φ, we have that the connection coefficients for the scalar representation vanish, i.e. Γ µ = 0, and in particular (Γ N R ) µ = 0 for scalar fields. Thus the scalar field equations as written in (19) are directly given by which lead directly to (28) without passing through (25). This trick does not work, however, for fields of arbitrary spin, since the quantity (Γ N R ) µ need not vanish in other representations, or seen it otherwise, ∂L/(∂∇ µ Ψ) will no longer be a vector field and then the last terms of (18) are not the divergence operator acting on ∂L/(∂∇ µ Ψ).

Now let us
analyse what happens if we apply the MCP as described in section II directly to the field equations. The scalar field equations in Minkowski space-time written in a frame-independent way are: where η = dδ η + δ η d (see footnote 3). The correct MCP tells us to replace η by g , leading to the same field equations as the correct MCPL, given by (28). Therefore, we conclude that, if applied correctly, both MCP and MCPL give the same results for scalar fields. The naive MCPL, in this case, will also be consistent with the above equations, since ∇Φ = dΦ for any affine connection, and there are only first order derivatives of the scalar field in the action (24). However, if we had applied the naive MCP in the scalar field equation, substituting η = η µν ∂ µ ∂ ν by Γ = g µν ∇ µ ∇ ν Φ, we would have arrive to the field equations which using (27) can be written as where Σ µ is the non-Riemannian current (26). It is clear that, in the case of applying the naive MCP leads to a non-minimal coupling between the scalar field and the torsion and non-metricity tensors through the current Σ µ , and therefore this cannot be regarded as a proper minimal coupling prescription.
Let us now see what happens with charge conservation if we apply the correct MCPL or the naive MCP. In Riemannian spacetimes both the correct MCPL and the naive MCP give rise to identical dynamics for scalar fields. Since the action obtained by the MCPL for a complex scalar field has a global U (1) symmetry 13 , it will have associated a conserved current given by (23). Hence, the scalar field that evolves as given by the correct as well as the naive MCP will both obey charge conservation. However, the fact that in non-Riemannian space-times the dynamics for scalar fields is different for the correct MCPL and the naive MCP spoils the above argument. In this case, the MCPL still leads to an action with a global U (1) symmetry (24), and therefore the usual conserved scalar current can be derived from the scalar action (24) by applying the Noether current formula (23), thus obtaining which is still conserved for both the correct MCPL and MCP, as can be seen by taking the divergence of J Φ and using (28). However, if one follows the naive MCP, since it differs in general from that with dynamics given by the MCPL in presence of torsion and/or non-metricity, the above current will generally not be conserved. Indeed, by taking the divergence of (33) and using the naive MCP field equations (32), we arrive to which shows that for a complex scalar field described by the naive MCP, the non-Riemannian current that couples it to nonmetricity and torsion spoils the potential U (1) symmetry of the scalar field 14 . This argument applies in a straightforward manner for any complex scalar fields which are in a given representation of some Lie group (not only U (1)). Given that the Standard Model (SM) requires for its construction that the Higgs field Lagrangian be globally invariant under SU (2) × U (1), the naive MCP prescription would enter in contradiction with the SM in presence of a (strong enough) torsion and/or non-metricity background. Since we do not have experimental data on how do matter fields behave in the presence of non-Riemannian features, there is no way to favour MCP or MCPL over other non-minimal coupling prescriptions. However, one expects the SM to work perfectly well if tiny non-metricity and/or tosion corrections are included, therefore pushing the naive MCP to an uncomfortable corner when applied to complex scalar fields. Note that the breaking of charge conservation will happen in general for any non-minimal coupling prescription that is applied directly to the field equations, since that would generically break the U (1) invariance enjoyed by the scalar action.
B. The minimally coupled spin 1/2 field Let us now examinate the case of spin 1/2 fields. Contrary to the spin 0 case, the only derivative operator that acts on spinors and is covariant under arbitrary changes of basis is ∇ (see e.g. [89,90]). Thus we conclude that the Minkowskian action is written, in a frame independent way, in terms of the covariant derivative ∇ of the purely inertial connection (which satisfies ∂ = ∇ in any inertial frame). Thus, applying the MCPL as stated in section II for the two different actions that are commonly used in the literature to describe spinor fields in Minkowski space-times are lead to where ∇ µ Ψ = ∂ µ Ψ − Γ µ Ψ and ∇ µΨ = ∂ µΨ +ΨΓ µ . There is a subtlety here: we must first specify a form for the spinor connection Γ µ . Generally, the spinor connection is defined in terms of the spin conection ω µ ab as where σ ab = 1 4 [γ a , γ b ] are the generators of the Lorentz group in the spin representation [83,91]. Thus Γ µ is completely specified once a choice for spin connection is made. In Riemannian space-times, there is a canonical lift of the Levi-Civita connection to the spin bundle leads to a unique choice for the spin connection given by where C µα ν is the Levi-Civita connection of g. The lift of the affine connection can still be done in non-Riemannian space times, leading to the canonical spin connection given by However as pointed out in [91], such lift is not sensible to some of the irreducible components of the non-metricity tensor, and one could in principle choose a non-canonical spin connection by adding terms related to non-metricity and/or torsion by hand (see for instance [91,92]). Since we are concerned with minimal coupling, we will consider only the canonical piece of the spin connection, since any extra terms would change the form of (Γ N R ) µ potentially adding non-minimal interactions.
Going back to the minimal-coupling discussion, we can see that in Riemannian space-times (where ∇ = ∇ g ), both actions are equivalent since they only differ in a boundary term which is oblivious for the field equations: and where the superindex R stands for the Riemannian. Notice that J Ψ is the Noether current (23) corresponding to a global U (1) (or other Lie group) symmetry of the spinor action (35). WhileS R Ψ is the traditional form of the action for spin 1/2 fields, the reason why S R Ψ is sometimes employed in Riemannian space-times is because it is explicitly hermitian (see e.g. [93,94]). However, in presence of torsion and/or non-metricity the two actions (35) and (36) for spin 1/2 are not dynamically equivalent anymore, since they are no longer related by a boundary term, satisfying instead the relatioñ where again we have defined a non-Riemannian current which differs from the non-Riemannian current Σ µ that appears in the scalar field case (26) in a trace of the non-metricity tensor. Since both actions are not equivalent, we must take care of using (35) in non-Riemannian space-times, since it is the hermitic while (36) is not. However, we will see that there are further consistency reasons to choose (35) over (36). To this end let us derive the field equations for Ψ andΨ that each of both actions describe. Beginning with (35) and making use (18) we arrive to the field equations where, since the gamma matrices have two spinorial indices and a Lorentz one, their covariant derivative is given by ∇ µ γ α = ∂ µ γ α + Γ µν α γ ν + [γ α , Γ µ ] (see e.g [95]). Using (39), we can compute Γ µ and then, by using the algebraic properties of the Dirac gamma matrices 15 , we can compute the covariant derivative of the gamma matrices, finding Plugging this result into the spinor field equations (43) we have have that the action (35) leads to the following equations Let us point out that these field equations were already found in [79] from the standard Riemannian equations in an alternative manner. Repeating the same procedure now with the non-hermitian action (36), we are led to the field equations A quick consistency check of both sets of field equations, (45) and (46), can be made by investigating if the on-shell relation betweenΨ and Ψ is compatible with the group theoretical definition of adjoint spinorΨ = Ψ † γ 0 . For that purpose, let us call (Ψ, Ψ) and (ψ, ψ) to a pair of arbitrary solutions of (45) and (46) respectively. Consider taking the adjoint of the equation satisfied by Ψ: Using the standard properties of the Dirac matrices, and the form of Γ µ given by (39), we can show that Γ µ † = −γ 0 Γ µ γ 0 , which leads to where Notice that the above equation is identical to the equation satisfied byΨ, thus consistently suggesting the identificationΨ = Ψ † γ 0 . Repeating the same procedure for the field equations defined by the action (36), by taking the adjoint of the field equation satisfied by ψ (i.e. the fist of (46)) we arrive at which clearly differs from the equation satisfied byψ (i.e. the second of (46)) due to the terms involving the non-Riemannian current σ µ . Therefore, the group-theoretical definition ofψ = ψγ 0 is not consistent with the dynamics given by the action (36) unles non-metricity and torsion vanish, when both (35) and (36) become dynamically equivalent. These findings imply that, in non-Riemannian space-times, the MCPL has a consistent implementation only when applied to the Minkowskian version of the action (35) and not that of (36). Thus, if generic non-minimal couplings between the spinor fields and torsion and/or non-metricity are considered, subtleties of this kind may arise if one does not apply the minimal prescriptions properly and to the correct Minkowskian action.
Since we know that within the MCPL only the action (35) is valid, let us now investigate whether spin 1/2 fields couple explicitly to non-metricity and/or torsion within this action. As already pointed out in section III for an arbitrary spin field, the only couplings between the spin 1/2 field and the non-Riemannian terms that arise in a minimal-coupling prescription come from the form of the spinor connection. Here we will again proceed by using (7) to split the ∇ µ Ψ term in (45) into its Riemannian and non-Riemannian pieces. Assuming the canonical spin connection (39) we can use decomposition of the spacetime connection (7) to find where the Riemannian covariant derivatives act on spinor fields as ∇ g µ Ψ = γ µ ∂ µ − Γ g µ Ψ and on adjoint spinor fields as Notice that Γ S is the well known interaction between the spinor fields and the totally-antisymmetric pat of the torsion tensor [27]. Using (50), the field equations for Ψ andΨ derived from the right spinor action (35), which are given by (45) become Notice that only the Levi-Civita part of ω µ ab and the totally antisymmetric part of the torsion appear in (53). The non-metricity tensor and the other Lorentz-irreducible pieces of the torsion tensor do not couple (explicitly) to spin 1/2 field if the canonical spin connection (39) is assumed. As in the case of the scalar field, a careful analysis of the action (35) (again decomposing the spinnor connection in its Levi-Civitta, torsion and non-metric parts) shows that the action only contains the Levi-Civita part of ω µ ab and the totally antisymmetrized torsion tensor. Hence, no coupling to other pieces of the torsion tensor or to the non-metricity tensor should appear in the field equations either.
Let us now analize the aplicability of the correct MCP to spin 1/2 fields. The Minkowskian field equations for a free spin 1/2 field and its adjoint written in a frame independent way are where ∇ M stands for the covariant derivative of Minkowski space-time, i.e. the one associated to a purely inertial connection (which vanishes in any inertial reference frame). Following the MCP, the non-Riemannian version of the equations is As we already computed, the adjoint of the first equation in (55) leads to the second equation of (55) if the identification Ψ =Ψ † γ 0 is made, which is perfectly consistent with the group-theoretical definition ofΨ. Therefore the MCP procedure is, in principle, also consistent to describe a field belonging to the spin 1/2 representation of the Lorentz group. Let us thus investigate the different couplings between spin 1/2 fields and the geometry if the MCP, instead of the MCPL, is applied. We can use (50) to re-write (55) as which shows how within the MCP, the non-Riemannian current σ µ defined in (42) introduces a direct coupling between spin 1/2 fields, torsion and non-metricity, therefore failing in being a minimal coupling prescription, as was the case for scalar fields. Experimentally probing this kind of couplings could help in elucidating wether the MCP or the MCPL is more suited to describe spin 1/2 fields in non-Riemannian space-times. Nevertheless, following the same reasoning as for complex scalar fields, if we assume MCP dynamics for spinor fields, which is given by (56), a direct calculation shows that conservation of the fermionic current J µ Ψ = iΨγ µ Ψ is spoiled by the non-Riemannian current σ µ , having which could lead to charge violation through the non-Riemannian current σ µ . In contrast the MCPL action for spinor fields (35) is U (1) invariant, which ensures conservation of J Ψ on-shell. Given that the global Standard Model symmetries is paramount in our current understanding of the universe, we can conclude that as for scalar fields, the MCP is again pushed to an uncomfortable corner for describing spin 1/2 field dynamics in presence of torsion and/or non-metricity. We believe that this solves the raised in [79,80] (or at least shows a physical way to distinguish between both prescriptions) regarding whether the MCP or MCPL is more appropriate for describing fermions in non-Riemannian spacetimes, favoring the MCPL if charge conservation is to be satisfied. Thus, assuming that charge conservation holds, the correct description for minimal coupling prescription for spin 1/2 fields in non-Riemannian space-times is the MCPL, which leads to the action Notice that, contrary to the findings of [80], the above action (which is their equation I.4 if we assume vanishing non-metricity) actually gives a consistent minimally coupling prescription for spin 1/2 fields, since it leads to the correct minimal coupling field equations and the covariant derivative needs no modification to achieve a consistent minimal coupling for spin 1/2 fields.

C. The minimally coupled vector field
Let us now finish by study whether the different minimal coupling prescriptions for massless vector fields make sense 16 . Since the Minkowskian action for massless vector fields is constructed only with the gauge invariant 2-form F = dA, which is a frame independent object, following the MCPL as stated in section II, the action for a minimally coupled gauge field in an arbitrary non-Riemannian space-time is given by with F µν = (dA) µν . In order to apply the EL equations (18) to the above action, we need to re-write it as a functional of {A, ∇A}. We can do this by using the relation between the exterior derivative and the covariant derivative, thus finding With the action written in this form, we can directly apply (18) and obtain the field equations for the massless vector field in non-Riemannian spaces following the MCPL, which are Again, by using (7), we can decompose the covariant derivative in its Riemannian and non-Riemannian parts, finding the identity which plugged back on the field equations (68) leads to the well known which can also be written as or in terms of the vector field Since neither d nor δ g know about the affine structure, the result is that as happens for scalar fields, vector fields that are minimally coupled according to the correct MCPL do not couple explicitly either to torsion and/or to non-metricity explicitly 17 .
Let us now see what happens if we take the naive prescription ∂ → ∇ too seriously. This leads by the naive MCPL to the non-Riemannian Lagrangian Note that whereas the correct MCPL action (60) is invariant under projective transformations Γ µν α → Γ µν α + ξ µ δ α ν , while the above action resulting of applying the naive MCPL is not. Projective symmetry has been recently proven to be relevant in order to avoid ghost degrees of freedom in the construction of metric-affine theories of gravity [63,64,97]. By using again (18) in the naive MCPL action, we arrive to the field equations which after splitting the covariant derivative can be recast into or also Again, since neither d nor δ g know about the affine structure, there can be no cancellation of the torsion terms, and therefore the naive MCPL leads to a non-minimal coupling between A and the torsion tensor, which makes the naive MCPL fail as a minimal coupling prescription.
Let us now investigate the MCP and naive MCP prescriptions. When written in a frame independent way, the source free vector field equations in Minkowski space-time are where δ η is the codiferential operator associated to the Minkowski metric η (see footnote 3), and in coordinates it reads (δ η F ) µ = ∂ α F αµ and F µν = ∂ [µ A ν] . Applying the MCP as defined in section II, in a general space-time we must use the codiferential operator associated to the space-time metric g, so that acording to the MCP the vector field equations equations in a general space-time read δ g F = 0 , F = dA (72) and in coordinates it is satisfied (δ g F ) ν = ∇ g µ F µν , thus having again that the MCP is consistent with the MCPL even in presence of non-metricity and/or torsion. In the Lorenz gauge, which in Minkowski space-time is characterised by δ η A = 0 and in general space-times is characterised by In the Lorenz gauge, characterised by δ g A = 0, the field equations are η A = 0 and g A = 0, respectively. Since we are in the Lorentz gauge, then g A = (dδ g + δ g d)A = δ g dA, thus recovering the same field equations as with the correct MCPL (66). The last equatioin can also be written in the familiar form ∇ g µ ∇ gµ A α + R g α µ A µ = 0, where R g α µ is the Ricci tensor associated to g. Let us now see what would have ended up with had we applied the naive MCP. Given that the Minkwskian field equations in any coordinates read by the naive prescription ∂ → ∇, the naive MCP would have lead us to the non-Riemannian field equations which after using (63) and the identityF µν = F µν + S µν α A α can be written as or In the Lorentz gauge, which by the naive MCP would be characterized by ∇ µ A µ this looks as We can see that this equations features several non-minimal coupling terms between torsion and non-metricity, thus showing one more time how the naive MCP is not a proper minimal-coupling prescription.

V. FINAL REMARKS
We have here dealt with the issue of minimal coupling between matter fields and the geometry in presence of torsion and/or non-metricity. Since this is a confusing issue in the literature, we first gave a definition of what we understand as a minimal coupling prescription and what should not be regarded as such. Thus, in a heuristic sense, minimal coupling prescriptions are those that change as little as possible the differentiable operators appearing in the matter field equations and/or actions. Thus to implement a minimal coupling prescription right, one should pay attention to what differential operator one is using in the Minkoskian theory, which is never ∂ since it is in general a different operator in non-inertially related frames (see section II), but rather d, δ or ∇; and be consistent when going to a general space-time in using the same operator. As we showed, If one implements the minimal coupling prescriptions in this way, the MCP and MCPL are equivalent for spin 0 and 1 fields , while only the MCPL gives a consistent minimal coupling prescription for spinor fields. This results are in contradiction with [80] due to the fact that the naive ∂ → ∇ substitution is applied there to implement MCP and MCPL in their respective naive versions. In the spin 1/2 case we also showed that charge is conserved only for the MCPL prescription, thus giving a solution to the dilemma found in [79] about whether one should use MCP or MCPL for spin 1/2 fields in non-Riemannian space-times. As a by-prduct, we also showed how the non-hermitian action (36) that is commonly used in Riemannian space-times to describe spinor fields is not equivalent to the hermitian one in presence of torsion and/or non-metricity, while in Riemannian space-times both give the same dynamics. As anticipated in section II, we also showed that the naive MCPL and naive MCP typically fail in being minimal coupling prescriptions for all matter fields, the exception being the naive MCPL for the scalar field. This exception is due to the fact that only first order derivatives of the scalar enter in the action, and for scalar field the operators d and ∇ are by definition the same operator. We hope that this discussion can thus be useful regarding the issue of coupling matter fields to torsion and/or non-metricity (see e.g. [49,68,69]).