Conserved and Non-Conserved Noether Currents from the Quantum

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1 Introduction 1

2 Symmetries and extended symmetries 3 2.1 Functional integral, partition function and eﬀective action 4 2.2 Symmetry transformations of microscopic action 5 2.3 Symmetry transformation of integral measure 5 2.4 Continuous symmetries of eﬀective actions 6 2.5 Extended symmetries 7

3 Conserved and non-conserved Noether currents 7 3.1 Noether current from microscopic action versus Noether current from quantum eﬀective action 7 3.2 Noether current from local transformations 8 3.3 Noether currents from external gauge ﬁelds 9

4 Space-time symmetries and extended symmetries 10 4.1 General coordinate transformations 11 4.2 General connection 12 4.3 Variation of the quantum eﬀective action 16 4.4 Weyl gauge transformations 18 4.5 Local Lorentz transformations 19 4.6 Conservation laws in the tetrad formalism 22 4.7 General linear frame change transformations 24

5 Example: scalar ﬁeld theory with non-minimal coupling 30

6 Conclusions 32

1 Introduction

The relation between the microscopic formulation of a quantum field theory, and the macroscopic formulation which includes the effect of quantum and statistical fluctuations, can be nicely discussed in terms of actions. The microscopic action S[χ] defines a theory at a microscopic scale or at very high momenta where quantum fluctuations are suppressed. This is the object that enters the functional integral. In classical situations where quantum fluctuations are negligible, the microscopic action yields directly the classical Euler-Lagrange equations and for this reason it is sometimes called classical action. The microscopic action depends of course on the field values but is otherwise independent of the state. Initial conditions enter only as boundary conditions for solutions to the equations of motion. In contrast to this, the one-particle irreducible or quantum effective action (see e. g. [1, 2]) depends on field expectation values and yields different expressions for which all quantum corrections have already

– 1 – been taken into account. For example, the propagators and vertices obtained from functional derivatives of the one-particle irreducible effective action around a vacuum solution yield the full correlation functions and S-matrix elements when used in tree diagrams [1, 2]. Similarly, from the variation of the quantum effective action with respect to the fields one obtains renormalized equations of motion. It becomes already clear from these statements that the determination of the quantum effective action itself is typically a formidable task. In particular, it differs from the microscopic action through both perturbative and non-perturbative quantum and statistical corrections. There are several methods to take these corrections into account; one of them is the functional renormalization group [3–6]. One should remark here that the quantum effective action can, and will, contain terms of higher derivatives in the fields and possibly non-local terms. All terms allowed by symmetries can appear and are typically present. Scaling arguments based on the classification of operators into relevant, irrelevant and marginal can sometimes be used in the vicinity of renormalization group fixed points, but that is not the generic situation. One should also remark that the quantum effective action depends in general on the quantum state through the boundary conditions of the functional integral. In the present paper we are interested in the relation between Noether currents [7] and the quantum effective action. Because it differs in a rather non-trivial way from the microscopic action, also the corresponding Noether currents can differ substantially. In the presence of quantum anomalies it is possible that a current is conserved at the classical level, but not conserved when quantum fluctuations have been taken into account. This motivates a detailed study of how Noether currents follow in full glory from the quantum effective action. Both expectation values of currents, as well as their correlation functions, are actually of interest. One reason is fluid dynamics. Experience shows that in the macroscopic regime, i. e. for large time intervals and long distances, one can approximate quantum field dynamics often rather well by a variant of fluid dynamics [8–10]. This follows the rational that those degrees of freedom are important over long time intervals that are preserved from relaxation by conservation laws [11–13]. In practise, for a relativistic fluid, it is by reasons of causality not possible to take only the strict hydrodynamical degrees of freedom (which are directly governed by conservation laws, e. g. energy- and momentum density) into account, but some non-hydrodynamical fields must be propagated, as well [14–18]. For example, in Israel-Stewart theory [19, 20] for a fluid with conserved energy-momentum tensor but no additional conserved quantum numbers these are the shear stress and bulk viscous pressure. Equations of motion for the latter are typically postulated in a phenomenological way. We will actually argue that these identities should follow from the additional non-conservation laws for the dilatation current and shear current. In some regards, the equations we find are close to those proposed in ref. [21]. An extension of fluid dynamic equations to include the spin tensor has been proposed and discussed in several recent publications [22–28], partly with the motivation to explain the measurement of the polarization of the hyperon Λ produced in high energy nuclear collisions [29] (see ref. [30] for a review). Correlation functions of various conserved and non-conserved currents are also of high interest. For example, they can be used to describe critical behaviour in the vicinity of phase transitions. In the context of linear or weakly non-linear response theory around equilibrium states they can be used to obtain transport properties, e. g. through the Kubo relations [31–39]. In the present paper we have two main goals. The first is to establish a general formalism for obtaining currents associated to different symmetry transformations for the quantum effective action. Our focus will

– 2 – be on expectation values of such currents, but the formalism can in a similar way also be used for correlation functions. Our discussion will include traditional symmetry transformations that leave the action invariant, but also what we call extended symmetry transformations [40–42]. The latter do not leave the action invariant but change it in a specific way by a term that is actually known in the macroscopic theory. We will argue that such transformations are still very powerful. Associated to such extended symmetry transformations are non-conserved Noether currents. We will discuss different examples for normal and extended symmetries related to the geometry of space-time for a relativistic quantum field theory. These transformations encompass local changes of coordinates (diffeomorphisms), but also transformations in the frame and spin bundles such as local Lorentz transformations, dilatations, and shear transformations. The associated currents we discuss encompass two versions of the energy-momentum tensor (a generalization of the canonical energy-momentum tensor to the quantum effective action and the symmetric energy-momentum tensor), the spin current, the dilatation or Weyl current, and the shear current. The latter three form together a tensor of rank three known as the hypermomentum tensor [43–47]. Hypermomentum was initially investigated mainly in the context of modified theories of gravity, because it constitutes natural sources for non-Riemannian geometrical structures such as torsion and non- metricity [43–57]. We will also essentially obtain the currents of hypermomentum from varying these geometrical structures and observing the response of the quantum effective action. However, we want to argue that hypermomentum is not only of interest in the context of somewhat speculative extensions of Einsteins theory of general relativity. In fact we believe that the spin current, dilatation current and shear current can all be non-zero for generic interacting quantum field theories in non-equilibrium situations. Even if these currents are apparently absent at the classical level, they can arise from quantum fluctuations, similar to quantum anomalies. We will also discuss an (approximate) effective action for a scalar field with non-minimal coupling to gravity to gravity to underline this point.

2 Symmetries and extended symmetries

Symmetries play an important role in quantum field theory, for at least three reasons. First, they constrain substantially the form of a microscopic action S[χ] and define in this sense to a large extend a microscopic theory. Together with renormalizability, symmetries provide the main guiding principle for the construction of microscopic physics theories. The second reason is that symmetries constrain also to a large extend the quantum effective action Γ[φ] where φ = χ . One may think that this is less important because Γ[φ] is anyway derived from S[χ] h i (we recall the construction below), but in practice the effective action Γ[φ] can usually not be obtained exactly, so that insights gained through symmetry considerations are particularly important and powerful. In the absence of anomalies or explicit symmetry breaking, only terms that are allowed by the symmetries are allowed to appear in the effective action. It is this aspect of symmetries (and extended symmetries) that will be discussed in the present section. The third important aspect of symmetries are the conservation laws they lead to. This will be discussed in the subsequent section. While traditionally a symmetry is a transformation that leaves the action invariant, we will in the present paper also discuss more general transformations (or formally Lie group / Lie algebra actions) for which this is not the case, but that are nevertheless rather useful because they change the action in a

– 3 – speciﬁc way that allows to make powerful statements, as well. We refer to such transformations as extended symmetries (see below).

2.1 Functional integral, partition function and effective action We consider a theory for quantum fields χ(x) which we do not specify in further detail here. In practise χ(x) can stand for a collection of different fields and may encompass components transforming as scalars, vectors, tensors, or spinors. The theory is described by a microscopic action S[χ]. The latter enters the partition function iS[χ]−i {J(x)χ(x)} Z[J]= Dχ e Rx . (2.1) Z In eq. (2.1) we have introduced sources J(x) for the fundamental fields χ(x). More generally one may also introduce sources for composite operators as will be discussed in more detail below. For example, the µν metric gµν (x) acts like an external source for the energy-momentum tensor T (x). We are using in (2.1) for relativistic theories the abbreviation

d d = d x√g = d x det(gµν (x)). (2.2) x − Z Z Z q The space-time metric gµν (x) has signature ( , +, +, +). − The functional integral Dχ is defined as usual. Even though we do not write this explicitly, a microscopic theory S[χ] is first defined in the presence of ultraviolet (and possibly infrared) regularization R and for renormalizable theories there is a renormalization procedure that allows to make the corresponding ultraviolett scale arbitrarily large, in particular much larger than any other relevant mass scale. It is through this procedure that a more rigorous definition of the partition function must be done. It is worth to note here that the partition function in (2.1) depends also on the quantum state or density matrix ρ through the boundary conditions in the temporal domain. This is in particular important in non-trivial situations such as at finite temperature, density, or out-of-equilibrium. For initial value problems one should work with the Schwinger-Keldysh double time path formalism. For the present paper we keep the boundary condition, and therefore the state dependence, implicit. This could be easily changed, however. From the partition function one defines the Schwinger functional W [J]= i ln Z[J] and from there the quantum effective action or one-particle irreducible effective action Γ[φ] as a Legendre transform,

Γ[φ] = sup J(x)φ(x) W [J] . (2.3) − J Zx The effective action Γ[φ] depends on φ, which is the expectation value of the field χ. To see this one evaluates the supremum by varying the source field J(x) leading to

1 δ 1 i δ Dχ χ(x) eiS[χ]−i R Jχ φ(x)= W [J]= Z[J]= . δJ(x) Z[J] δJ(x) iS[χ]−i R Jχ g(x) g(x) R Dχ e One also writes this asp p R φ(x)= χ(x) , h i with the obvious deﬁnition of the expectation value in the presence of sources J. In addition, W [J] and h·i Γ[φ] depend on additional sources that have beed introduced for composite operators, such as the metric gµν (x).

– 4 – An interesting property of Γ[φ] is its equation of motion. It follows from the variation of (2.3) as

δ Γ[φ]= g(x)J(x). (2.4) δφ(x) p In particular, for vanishing source J = 0, one obtains an equation that resembles very much the classical equation of motion, δS/δχ = 0. However, in contrast to the latter, (2.4) contains all corrections from quantum ﬂuctuations! Another interesting property is that tree-level Feynman diagrams become formally exact when propagators and vertices are taken from the eﬀective action Γ[φ] instead of the microscopic action S[χ].

2.2 Symmetry transformations of microscopic action Both the microscopic action S[χ] and the quantum effective action Γ[φ] are functionals of fields. When one speaks of a symmetry transformation of the action one means in practise a symmetry transformation of the fields on which the action depends. A symmetry of the microscopic action means an identity of the form S[gχ]= S[χ]. (2.5) Here the group element g G is acting on the fields (not necessarily linearly) and the symmetry implies ∈ that the action is unmodified by this transformation. (More generally, the right hand side could differ from the left hand side by a constant or boundary term that does not affect the equations of motion.) So far, G could be either a finite group, an infinite discrete group or a continuous Lie group. In the latter case one can compose finite group transformations out of infinitesimal transformations. One can write for the latter j χ(x) gχ(x)= χ(x)+ dχ(x)= χ(x)+ idξ (Tjχ)(x), (2.6) → where Tj is an appropriate representation of the Lie algebra acting on the fields χ (at this point not necessarily linearly). The microscopic action transforms as

j d δS[χ] j

S[gχ]= S ½ + idξ T χ = S[χ]+ d x idξ (T χ)(x) , (2.7) j δχ(x) j Z One also abbreviate the right hand side of (2.7) as S[χ]+ dS[χ] and a continuous symmetry corresponds then to the statement dS[χ] = 0.

2.3 Symmetry transformation of integral measure The functional integral measure can also be transformed. One says that it is invariant if

Dχ = D(gχ). (2.8)

An example for this would be a generalization of a change of integration variable xj f j(~x), where one → would in general expect a Jacobian determinant, dN x = det(∂xj/∂f k) dN f(x). The above equation tells | | that this determinant is unity. (More generally one may also allow a field independent constant that can be dropped for many purposes.) Again this goes for elements g G of finite, discrete or continuous groups ∈ G. It can happen that one finds a symmetry of a microscopic action S[χ] but that the properly defined functional integral measure is not invariant. In that case one speaks of a quantum anomaly.

– 5 – In the following we will use a somewhat generalized notion of anomaly which also applies to transformations that are not necessarily symmetries of the microscopic action but may correspond to extended symmetries (see below). It is a priori not clear how the functional integral measure behaves with respect to these transformations. This will be an important question to be addressed in the future.

2.4 Continuous symmetries of eﬀective actions We now specialize to continuous transformations which we can study in the inﬁnitesimal form (2.6). After a change of integration variable χ gχ we write the Schwinger functional (2.1) as →

j j j iS[χ+idξ Tj χ]−i R J(x)(χ(x)+idξ Tj χ(x)) Z[J]= D χ + idξ Tjχ e x . Z j We now assume the invariance of the measure D χ + idξ Tjχ = Dχ, or, in other words, the absence of an anomaly. This leads for small dξj to

iS[χ+idξj T χ]−i J(x) χ(x)+idξj T χ(x) Z[J]= Dχ e j Rx ( j ) Z 1 δ j iS[χ]−i R J(x)χ(x) = Dχ 1+ S[χ]+ J(x) dξ Tjχ(x) e x . x − g(x) δχ(x) Z " Z ( ! )# The leading term on the right hand sidep is just Z[J] itself. Subtracting it we ﬁnd using (2.4) the Slavnov- Taylor identity

d δ j d δ j dS[χ] = d x S[χ] idξ Tjχ(x) = d x Γ[φ] idξ Tjχ(x) . (2.9) h i δχ(x) δφ(x) h i Z Z An important class of transformations is such that the Lie algebra generators Tj act on the ﬁelds χ in a linear way. In that case one can write

Tjχ(x) = Tj χ(x) = Tjφ. (2.10) h i h i In that case the right hand side of (2.9) can be written as dΓ[φ] and one has

dS[χ] = dΓ[φ]. (2.11) h i In particular, the most important case is here that the microscopic action is invariant, dS[χ] = 0, from which it follows that also the eﬀective action is invariant, dΓ[φ] = 0, or

Γ[gφ]=Γ[φ]. (2.12)

In summary, in the absence of a quantum anomaly, and for a linear representation of the Lie algebra on the fields, we conclude that the effective action Γ[φ] shares the symmetries of the microscopic action S[χ]. This is very useful in practice because it constrains very much the form the effective action can have. This is important for example for proofs of renormalizability or also for solving renormalization group equations in practice [1].

– 6 – 2.5 Extended symmetries Interestingly, eq. (2.11) can also useful when the microscopic action S[χ] is not invariant, i. e. dS[χ] = 0. 6 For example, if dS[χ] is linear in the field χ, one can infer that the effective action Γ[φ] must change under a corresponding transformation of the expectation value field φ in an analogous way such that eq. (2.11) remains fulfilled. This can also constrain the form of Γ[φ] substantially [40–42]. When dS[χ] is non-linear in the fields χ, eq. (2.11) is less useful to constrain the form of Γ[φ], because the left hand side involves then expectation values of composite operators that are not easily available on the level of the effective action Γ[φ]. For example, the connected two-point correlation function χχ χ χ h i − h ih i involves the inverse of the second functional derivative of Γ[φ]. In such a case one may however use eq. (2.11) to calculate the expectation value on the left hand side through a simple transformation of the effective action Γ[φ]. Another useful case is when the left hand side of eq. (2.11) is linear in composite operators that are available on the level of the effective action Γ[φ] because external sources have been coupled to them. An example would be when dS[χ] involves the energy-momentum tensor which can be obtained from a functional derivative of Γ with respect to the metric gµν (x). All this assumes invariance of the functional integral measure, though. One can also often make use out of anomalous symmetries, i. e. symmetry transformations where the functional measure is not invariant, if the corresponding Jacobian determinant can be absorbed into a change of the action but we will not discuss this further here.

3 Conserved and non-conserved Noether currents

We now come to the second implication of symmetries besides constraints to the form of the microscopic and quantum effective action, namely conservation laws. We discuss two methods that can be used to extract conserved currents from the quantum effective action and will argue that the second method is superior to the first. For the first method one makes the symmetry transformations space-time dependent, while for the second method one also introduces an appropriate external gauge field. Before we go into this, let us first discuss why it is useful to obtain a Noether current from the effective action instead of the microscopic action.

3.1 Noether current from microscopic action versus Noether current from quantum effective action On the classical level of a field theory one can obtain the Noether currents from the microscopic action S[χ] through the standard textbook procedure. Sometimes such an expression is useful also for a quantum description as a starting point to calculate its expectation value when quantum and statistical fluctuations are taken into account. Alternatively one can obtain the expectation value of a Noether current directly from the quantum effective action as will be discussed below. However, one should be aware of the fact that the difference between the classical or microscopic action and the quantum effective action is a rather non-trivial result of quantum and possibly statistical fluctuations. In particular, quantum fluctuations are present on all scales and need proper regularization and renormalization procedures. It is well known that quantum fluctuations can modify a theory substantially, for example the propagating degrees of freedom can change from fundamental fields to composite

– 7 – fields for bound states [58–60]. In this sense it is possible that Noether currents differ substantially in form when derived from microscopic actions or quantum effective actions, respectively. There are powerful non-perturbative theoretical techniques to calculate the quantum effective action, such as the functional renormalization group. It allows to take quantum fluctuations step-by-step into account and to find suitable approximations at each scale. The above arguments show that it can be rather advantageous to calculate Noether currents directly from the quantum effective action, instead of aiming at an expression in terms of the microscopic action and expectation values in terms of complicated composite operators. One should note here, however, that Noether currents that follow from the quantum effective action are usually not monomials or polynomials in the field expectation values and their derivatives. There can be additional terms that do not vanish, even when the field expectation values do. For example, at finite temperature, the energy-momentum tensor is non-zero even when the field expectation values vanish. Also, one should keep in mind that the effective action is state-dependent and accordingly the Noether currents derived from it also are. Again, the finite temperature state may serve as an example.

3.2 Noether current from local transformations Consider the transformation of the ﬁelds

j φ(x) φ(x)+ idξ (x) Tjφ(x), (3.1) → j where dξ (x) are space-time position-dependent, infinitesimal parameters. The generators Tj act linearly on the field expectation value fields φ(x) such that eq. (2.10) holds. The transformation law (3.1) is inherited from the corresponding transformation of the microscopic fields in eq. (2.6). The quantum effective action Γ[φ] changes to linear order in dξj like [61]

j d j µ j 1 µν j Γ[φ+idξ Tjφ]=Γ[φ]+ d x√g j(x) dξ (x)+ j(x) µ dξ (x)+ (x) µ ν dξ (x)+ . . . . (3.2) I J ∇ 2Kj ∇ ∇ Z If the transformation (3.1) is a global symmetry of the eﬀective action, this implies that j(x) = 0 so that I the expansion on the right hand side of (3.2) starts with the second term which has one derivative. However, one can also consider transformations that are not global symmetries such that j(x) is non-vanishing. One I can now write eq. (3.2) after partial integration as

d j 1 δΓ µ 1 µν d x√g dξ (x) iTjφ(x) j(x)+ µ (x) µ ν (x)+ . . . = 0. (3.3) √g δφ(x) −I ∇ Jj − 2∇ ∇ Kj Z Surface terms have been dropped here. Because dξj(x) is arbitrary, this implies

1 δΓ µ 1 µν iTjφ(x) j(x)+ µ (x) µ ν (x)+ . . . = 0. (3.4) √g δφ(x) −I ∇ Jj − 2∇ ∇ Kj Using the ﬁeld equation (2.4) one obtains a set of conservation-type relations

µ 1 µν µ (x)+ ν (x) . . . = iJ(x)Tj φ(x) j(x). (3.5) ∇ −Jj 2∇ Kj − −I For a transformation which deﬁnes a global symmetry such that j(x) = 0 and for vanishing source J = 0 I the right hand side vanishes and the expression in brackets on the left hand side of (3.5) deﬁnes then a set of conserved Noether currents.

– 8 – Let us note here that a relation of the type (3.5) as derived from a quantum effective action may also be useful when j(x) is non-vanishing, as long as this field is known, for example because an external I source is coupled to it. In that case one may call the expression in brackets on the left hand side of (3.5) a set of non-conserved Noether currents corresponding to an extended symmetry. A problem with the above derivation is that the expansion on the left hand side of (3.5) does in general not terminate. This makes the entire construction somewhat implicit. A notable exception is when Γ[φ] = S[φ] is the microscopic or classical action which contains at most second derivatives of the fields such that the equations of motion are partial differential equations of at most second order. In this case the above construction leads to the standard Noether currents of the classical theory. In contrast, when the effective action Γ[φ] differs from S[φ] by the effect of quantum fluctuations, one can not assume that only low orders of a derivatives of the fields are present. In fact, the quantum effective action contains in general all orders of a derivative expansion, as well as non-perturbative terms. These remarks show that an alternative approach is needed to obtain Noether currents from the quantum effective action when the latter cannot be assumed to have a derivative expansion terminating at finite order. To such a construction we turn next.

3.3 Noether currents from external gauge fields Let us now introduce an external gauge field for the local transformation (3.1). All derivatives of the field χ(x) in the microscopic action and of the expectation value field φ(x) in the quantum effective action Γ[φ] are now replaced by covariant derivatives,

j Dµφ(x)= µ iA (x)Tj φ(x). (3.6) ∇ − µ When the generators Tj are not commuting, the external gauge ﬁeld is non-abelian. In that case we may introduce structure constants through the relation

j [Tk, Tl]= ifkl Tj. (3.7)

The transformations of the expectation value ﬁelds φ(x) continue to be of the form (3.1), while the (non-abelian) gauge ﬁelds transforms as usual according to

j j j k l j j j A (x) A (x)+ f A (x)dξ (x)+ µdξ (x)= A (x) + (Dµdξ) . (3.8) µ → µ kl µ ∇ µ In the last equation we defined the covariant derivative of dξj in a variant of the adjoint representation. j j We will call an infinitesimal transformation dξ (x) “global” when (Dµdξ) (x) = 0. We assume now for simplicity that this gauged transformation is free from any anomalies, so that the analysis in section 2 goes through for this transformation, as well. In particular, when the microscopic action S[χ, A] has a symmetry or extended symmetry under the transformation (3.8), this will also be the case for the quantum effective action Γ[φ, A]. An important consequence is that derivatives of fields φ can appear in Γ[φ, A] only as covariant derivatives of the form (3.6). One should note here, however, that Γ[φ, A] can contain covariant derivatives of arbitrary order, and also non-local gauge invariant terms. The gauge field has been introduced in such a way that a local transformation (with space-time- j j dependent dξ (x)) is in fact equivalent to a global transformation when Aµ(x) is transformed, as well. In

– 9 – j j other words, the only eﬀect of a locally varying dξ (x) is taken up by the transformation behavior of Aµ(x). One has now for the transformation of the eﬀective action

j j j k l j d j Γ[φ + idξ Tjφ, A + f A dξ + µdξ ]=Γ[φ]+ d x√g j(x) dξ (x) . (3.9) µ kl µ ∇ I Z In contrast to (3.2) no higher order derivatives are present on the right hand side of (3.9). This is a consequence of the external gauge ﬁeld, and the fact that all derivatives have been replaced by covariant derivatives (3.6). Note that this includes possible regulator terms that have been added to make the functional integral well deﬁned. We can write now

d 1 δΓ j 1 δΓ j k l j d x√g iTjφ(x) j(x) dξ (x)+ f A (x)dξ (x)+ µdξ (x) = 0. √g δφ(x) −I √g j kl µ ∇ Z ( δAµ(x) ) (3.10) Using partial integration, the ﬁeld equation (2.4), and the fact that dξj(x) is arbitrary, this implies now with the deﬁnition µ 1 δΓ Jj (x)= j , (3.11) √g δAµ(x) the covariant conservation-type relation

µ µ l k µ DµJ (x)= µJ (x)+ f A (x)J (x)= iJ(x)Tj φ(x) j(x). (3.12) j ∇ j jk µ l −I µ The first equation defines a covariant derivative Dµ for the current Jj (x) in a variant of the adjoint representation of the gauge symmetry. j If there is a global symmetry, i. e. if the effective action is invariant for (Dµdξ) = 0, one has j(x) = 0 j I and for vanishing external sources J(x) = Aµ(x) = 0, this is indeed a covariant conservation law of the µ standard form µJ = 0. More general, the “source term” on the right hand side of (3.12) might be ∇ j non-vanishing. The equation is then still potentially very useful, as long as the source term is known explicitly. In summary, the above discussion gives a recipe to derive conservation-type equations by introducing gauge fields associated with fields transformations in the partition function and taking functional derivatives with respect to it. When the transformation corresponds to a real symmetry one obtains in this way a conserved Noether current for which all quantum corrections have been taken into account. When the transformation is instead an extended symmetry one obtains a conservation-type equation with a source term on the right hand side. It should also be seen as a macroscopic equation of motion for which quantum corrections have been taken into account.

4 Space-time symmetries and extended symmetries

In the following we discuss a number of transformations related to space-time geometry. Following the general principles introduced in section 3.3 we introduce appropriate (external) gauge ﬁelds and discuss what kind of conservation laws follow from their variation. We start our discussion with general coordinate transformations which can be seen as a localized version of translations. Subsequently we discuss local changes of frame in the tangent space of the space-time manifold. This will be done ﬁrst for the restricted

– 10 – set of orthonormal frames where the transformations can be seen as local versions of Lorentz transformations. Subsequently we turn to general linear local frame changes which include besides local Lorentz transformations also local dilatations as well as shear transformations. We argue that the latter two should be understood as extended symmetries in general. In all cases we discuss what are the associated conserved and non-conserved Noether currents.

4.1 General coordinate transformations We start with a one-particle irreducible or quantum effective action Γ[φ, g] that depends besides the field expectation values φ(x) on the space-time metric gµν . For the present subsection we assume that φ(x) contains only fields of integer spin or, in other words, that fermionic fields have been fully integrated out from the partition function at vanishing source. This simplifies somewhat the discussion in the sense that we do not yet have to introduce a tetrad. The matter fields φ(x) could be scalars, vectors or tensors with respect to general coordinate transformations. Under a general coordinate transformation or diffeomorphism xµ x′µ(x), the metric transforms like → ρ σ ′ ′ ∂x ∂x gµν (x) g (x )= gρσ(x). (4.1) → µν ∂x′µ ∂x′ν Changing afterwards the coordinate label from x′µ back to xµ gives the transformation rule

ρ σ ′ ∂x ∂x ′ ′ ′ gµν (x) g (x)= gρσ(x) g (x ) g (x) . (4.2) → µν ∂x′µ ∂x′ν − µν − µν For an infinitesimal transformation x′µ = xµ εµ(x) this reads − ρ ρ ρ gµν (x) gµ(x)+ ε (x)∂ρgµν (x) + (∂µε (x)) gρν (x) + (∂νε (x)) gµρ(x) → (4.3) = gµν (x)+ εgµν (x). L µ We are using here the Lie derivative ε in the direction ε (x). More general, any coordinate tensor field L transforms under infinitesimal general coordinate transformations with the corresponding Lie derivative ε. This fixes in particular how the components of the matter fields φ(x) transform. L In the following we will also need the covariant derivative. In the present section it is based on the Levi-Civita connection given by the Christoffel symbols of second kind,

ρ ρ 1 ρλ Γ = = g (∂µgνλ + ∂ν gµλ ∂λgµν ) . (4.4) µ ν µν 2 − For future reference we note also the variation of the Levi-Civita connection, which can be written as

ρ 1 ρλ δΓ = g ( µδgνλ + νδgµλ λδgµν ) . (4.5) µ ν 2 ∇ ∇ −∇ ρ ρ Note that in contrast to the Christoﬀel symbol Γµ ν itself, its variation δΓµ ν is actually a coordinate tensor. The change of the metric in eq. (4.3) can also be written as

gµν (x) gµν (x)+ µεν(x)+ νεµ(x). (4.6) → ∇ ∇

– 11 – This illustrates that the metric can be seen here as the gauge field of general coordinate transformations. Variation of the effective action Γ[φ, g] with respect to the metric at stationary matter fields, δΓ/δφ = 0, yields the energy-momentum tensor, 1 δΓ[φ, g]= ddx√g T µν (x)δg (x). (4.7) 2 µν Z In fact, T µν (x) as defined by this expression should be seen as the expectation value of the symmetric energy-momentum tensor (see also below) in the state that defines the effective action Γ[φ, g]. The variation includes the connection, with (4.5) obeyed. Inserting (4.3) in (4.7) shows that invariance under general coordinate transformations yields the covariant conservation law for the energy momentum tensor

µν µT (x) = 0. (4.8) ∇ In this sense, one may see the covariant conservation of energy and momentum as a special case of the general principles discussed in section 3.3.

4.2 General connection The Levi-Civita connection is uniquely determined by being both metric compatible and torsion free. It seems that the space-time we inhabit fulfills these two conditions to an excellent approximation. Never- theless, it is interesting to relax these constraints and to study more general connections. Usually this is done in order to understand and constrain alternative theories of gravitation in more detail. For us the purpose is different: We are interested in constraining the form of the effective action for matter fields and to derive conservation-type relations. A very interesting possibility to this end is to study the quantum field theory in a geometry characterized by a general affine connection and to take functional derivatives of the quantum effective action with respect to the connection field.

Parallel transport. As a starting point for the deﬁnition of a covariant derivative one may take the notion of a parallel transport. The rule is here that a vector ﬁeld U µ(x) counts as parallelly displaced from a position xµ to xµ + dxµ when it changes by

ρ ρ ρ σ µ dU (x)= Γ (x) ∆U Bµ(x)δ U (x)dx . (4.9) − µ σ − σ The square bracket on the right hand side contains two terms. The first is a geometric part proportional to ρ the affine connection Γµ σ(x) which generalizes the Levi-Civita connection. For the second term we take ρ the field U to have the (momentum or mass) scaling dimension or conformal weight ∆U . The Weyl gauge ρ field Bµ(x) performs an additional local scaling of the field U (x). In contrast to the first term in (4.9), the second term or dilatation term is also present for scalar fields ϕ(x) when they have a non-vanishing scaling dimension ∆ϕ and when the Weyl gauge field Bµ(x) is non-vanishing.

Co-covariant derivative. The so-called co-covariant derivative [62, 63] associated to the parallel transport (4.9) is given by

ρ ρ ρ ρ σ µU (x)= ∂µU (x)+ Γ (x) ∆U Bµ(x)δ U (x). (4.10) ∇ µ σ − σ

– 12 – In particular this vanishes when U ρ(x) is parallelly transported according to (4.9). Eq. (4.10) is easily generalized to other tensor fields in a coordinate basis. For example, the co-covariant derivative of a tensor ρ field χ λ(x) with scaling dimension ∆χ would be ρ ρ ρ σ τ ρ ρ µχ (x)= ∂µχ (x)+Γ (x)χ (x) Γ (x)χ (x) ∆χBµ(x) χ (x). (4.11) ∇ λ λ µ σ λ − µ λ τ − λ For a scalar field ϕ(x) the co-covariant derivative is given by

µϕ(x)= ∂µϕ(x) ∆ϕBµ(x)ϕ(x). (4.12) ∇ − The co-covariant derivative has its name because it is covariant with respect to both general coordinate transformations x x′(x) and local scaling or Weyl gauge transformations → −∆φζ(x) 2ζ(x) φ(x) e φ(x), gµν (x) e gµν (x). (4.13) → → Aﬃne connection. Generalizing beyond the Levi-Civita connection (4.4) one may write the aﬃne connection as ρ ρ ρ 1 ρλ ρ Γ = + N = g (∂µgσλ + ∂σgµλ ∂λgµσ)+ N , (4.14) µ σ µσ µ σ 2 − µ σ ρ where Nµ σ is known as the deviation or distortion tensor. (It transforms indeed as a tensor under general ρ coordinate transformations, in contrast to Γµ σ.)

Co-covariant and Levi-Civita covariant derivatives. We will use in the following a notation where µ denotes the co-covariant derivative as introduced in (4.10), while µ is the ordinary covariant derivative ∇ ∇ based on the Levi-Civita connection (4.4). Equation (4.10) can also be written as

ρ ρ ρ ρ σ µU (x)= µU (x)+ N (x) ∆U Bµ(x)δ U (x). (4.15) ∇ ∇ µ σ − σ Non-metricity. The co-covariant derivative of the metric itself is given by

µgρσ(x)= [Nµρσ(x)+ Nµσρ(x)] ∆gBµ(x)gρσ(x). ∇ − − (4.16) = [Nµρσ(x)+ Nµσρ(x)] + 2Bµ(x)gρσ(x). − In the second line we have used that the metric gµν (x) has conformal weight ∆g = 2, as follows from eq. − (4.13). The first term on the right hand side of (4.16), namely the combination 1 1 Bµρσ(x)= [Nµρσ(x)+ Nµσρ(x)] = µgρσ(x) (4.17) 2 −2∇ is known as the non-metricity tensor. It is obviously symmetric in the last two indices. (Our convention differs by the factor 1/2 on the right hand side of (4.17) from other places in the literature.) It will be convenient below to further split the non-metricity tensor according to ρ ˆ ρ ρ Bµ σ(x)= Bµ σ(x)+ Bµ(x) δ σ, (4.18) ρ ρ where Bˆµ σ(x) is trace-less and sometimes called proper non-metricity tensor, Bˆµ ρ(x) = 0, and Bµ(x)= ρ (1/d)Bµ ρ(x) corresponds to the trace of the non-metricity tensor and is the Weyl vector or Weyl gauge field introduced already in eq. (4.9). Note that the full co-covariant derivative of the metric in (4.16) is in fact given by the proper non- metricity tensor 2Bˆµρσ(x). −

– 13 – Torsion. Consider the commutator of two co-covariant derivatives acting on a scalar ﬁeld ϕ(x),

ρ µ ν ϕ(x) ν µϕ(x)= T (x) ρϕ(x) ∆ϕ [∂µBν(x) ∂ν Bµ(x)] ϕ(x). (4.19) ∇ ∇ − ∇ ∇ − µν ∇ − − This contains two kinds of field strengths. One is the torsion tensor which is formally defined through the following combination of vector fields with vanishing scaling dimension, T (U, V )= U V V U [U, V ]. ∇ − ∇ − In components it is given by the anti-symmetric part of the affine connection,

T ρ (x)=Γ ρ (x) Γ ρ (x)= N ρ (x) N ρ (x). (4.20) µσ µ σ − σ µ µ σ − σ µ

The second term in (4.19) is the combination Bµν (x) = ∂µBν(x) ∂νBµ(x) known as the segmental − curvature tensor (see also below).

Decomposition of distortion tensor. Using eqs. (4.17) and (4.20) we may write the distortion tensor as

ρ 1 ρ ρ ρ ρ ρ ρ Nµ σ = Tµ σ Tσµ + T µσ + Bµ σ + Bσµ B µσ 2 − − (4.21) ρ ρ =Cµ σ + Dµ σ.

The combination 1 C ρ = T ρ T ρ + T ρ (4.22) µ σ 2 µ σ − σµ µσ is known as the contorsion tensor. It is anti-symmetric in the last two indices, Cµρσ = Cµσρ, so it does − not contribute to the non-metricity in (4.17). In contrast, the combination

ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ D = B + B B = Bˆ + Bˆ Bˆ + Bµδ + Bσδ B gµσ (4.23) µ σ µ σ σµ − µσ µ σ σµ − µσ σ µ − which we may call con-metricity tensor, is symmetric in µ and σ and does not contribute to torsion in eq. (4.20). We may therefore write the torsion tensor in terms of contorsion as

T ρ (x)= C ρ (x) C ρ (x), (4.24) µσ µ σ − σ µ and the non-metricity tensor in terms of the con-metricity tensor as 1 B (x)= [D (x)+ D (x)] . (4.25) µρσ 2 µρσ µσρ

The proper non-metricity Bˆµρσ(x) corresponds to the symmetric and trace-less part of con-metricity with respect to the last two indices. However, con-metricity has also an anti-symmetric part. The Weyl gauge ﬁeld can also be obtained directly from the trace of con-metricity as 1 B (x)= D ρ (x). (4.26) µ d µ ρ With this, the trace of the complete aﬃne connection (4.14) can be written as

ρ 1 Γµ ρ(x)= ∂µ g(x)+ dBµ(x). (4.27) g(x) p p

– 14 – In this sense the Weyl gauge ﬁeld is actually determined by the aﬃne connection and the metric,

1 ρ 1 Bµ(x)= Γµ ρ(x) ∂µ g(x) . (4.28) d " − g(x) # p p ρ For our purposes it is particularly useful to work with contorsion Cµ σ, the Weyl gauge ﬁeld Bµ and ρ proper non-metricity Bˆµ σ as the ﬁelds that parametrize the distortion tensor, so that the full connection becomes

ρ 1 ρλ ρ Γµ σ = g (∂µgσλ + ∂σgµλ ∂λgµσ)+ Cµ σ 2 − (4.29) ρ ρ ρ ρ ρ ρ + Bˆ + Bˆ Bˆ + Bµδ + Bσδ B gµσ. µ σ σµ − µσ σ µ − Variation of aﬃne connection. The full variation of the aﬃne connection is now given by

ρ 1 ρλ ρ ρ δΓ = g ( µδgσλ + σδgµλ λδgµσ)+ δC + δD . (4.30) µ σ 2 ∇ ∇ −∇ µ σ µ σ The covariant derivative on the right hand side uses the Levi-Civita connection. In particular it follows ρ from (4.30) that all components of the connection ﬁeld Γµ σ can be varied free of constraints when this variation is understood as a superposition of the variation of the Christoﬀel symbols due to a variation of the metric and variations of the torsion tensor and non-metricity tensor. In some situations one may further restrict this and demand for example that the non-metricity vanishes.

Curvature tensor. One may deﬁne the curvature tensor by the commutator of covariant derivatives of vector ﬁelds with vanishing scaling dimension,

R(U, V )W = U V W V U W W. (4.31) ∇ ∇ − ∇ ∇ − ∇[U,V ] In components,

ρ ρ ρ ρ λ ρ λ R σµν =∂µΓν σ ∂νΓµ σ +Γµ λΓν σ Γν λΓµ σ − − (4.32) ρ ρ ρ ρ λ ρ λ =R + µN νN + N N N N . σµν ∇ ν σ −∇ µ σ µ λ ν σ − ν λ µ σ ρ This is obviously anti-symmetric in the last two indices. In the second line of (4.32), R σµν is the standard Riemann tensor based on the Levi-Civita connection and the covariant derivatives are also based on the Levi-Civita connection. It is also useful to have the variation of (4.32) at hand. It can be written as

ρ ρ ρ λ ρ δR = µδΓ νδΓ + T δΓ , (4.33) σµν ∇ ν σ − ∇ µ σ µν λ σ with torsion as in (4.24). We are using here the co-covariant derivative with vanishing scaling dimension ρ for the variation of the connection δΓν σ.

Ricci scalar. There is a unique complete contraction of (4.32) which forms the analog of the Ricci scalar ρσ R = R ρσ,

ρσ ρσ ρσ ρ λσ ρ λσ R = R ρσ =R + ρNσ σNρ + Nρ λNσ Nσ λNρ ∇ −∇ − (4.34) ρσ σρ ρ ρ λσ ρ λσ =R + 2 ρC + 2 ρBˆ 2(d 1) ρB + N N N N . ∇ σ ∇ σ − − ∇ ρ λ σ − σ λ ρ

– 15 – Let us note that in the presence of torsion one can alternatively define a version of the curvature tensor ρ ρ ρ ρ on the transposed affine connection Γ˘µ σ =Γµ σ T µσ =Γσ µ, but we will not discuss this further. Also, − in contrast to Riemann geometry, Rρσµν is in general not anti-symmetric in the first two indices. Based on (4.32) one may define different contractions. One is the segmental curvature tensor

1 ρ 1 ρ ρ Bµν = R = ∂µΓ ∂νΓ = ∂µBν ∂ν Bµ. (4.35) d ρµν d ν ρ − µ ρ − ρ ρ The other two possibilities are Rµν = R µρν and Rµ νρ which both equal the standard Ricci tensor in the absence of non-metricity and torsion. The variation of the Ricci scalar is given by

µν νσ ρ νσ ρ λ σ ρ δR = R δgµν + g ρδΓ g νδΓ + T δΓ . (4.36) − ∇ ν σ − ∇ ρ σ ρ λ σ On the right hand side one may use (4.30) for further simplifcation. In the absence of contorsion and non-metricity this reduced to the standard identity

µν µ ν µν ρ δR = R δgµν + [ δgµν g ρδgµν ] . (4.37) − ∇ ∇ − ∇ ∇ 4.3 Variation of the quantum effective action In the following we will investigate how a quantum effective action for matter fields reacts the contorsion and non-metrcity as external sources and specifically what kind of equations can be derived from transformations for which the connection acts as a gauge field. ρ Let us write the variation of the action with respect to the metric gµν and the connection Γµ σ as

d 1 µν 1 µ σ ρ δΓ= d x√g U (x)δgµν (x) S (x)δΓ (x) . (4.38) 2 − 2 ρ µ σ Z µν The variation with respect to gµν (x) at fixed connection defines a symmetric tensor U (x), while the µ σ variation with respect to the connection at fixed metric defines a tensor field S ρ (x). The latter is known as hypermomentum current [43, 44, 46, 47]. We have assumed in (4.38) that the Weyl gauge field Bµ(x) has been expressed through eq. (4.28) in terms of the affine connection and the metric. µ σ It is conventional and convenient to further decompose the hypermomentum current S ρ (x) according to µ σ µ σ µ σ µ σ σµ µσ S ρ (x)= Q ρ (x)+ W (x) δρ + S ρ (x)+ S ρ(x)+ Sρ (x). (4.39) µ σ µρσ µσρ Here S ρ (x) is anti-symmetric, S (x)= S (x), in the last two indices and known as the spin current. − It can be written in terms of the hypermomentum as 1 Sµρσ = (S µρσ S µσρ). (4.40) 2 − µ σ µρσ µσρ µ ρ In contrast, Q ρ (x) is symmetric in the last two indices, Q (x)= Q (x), and traceless, Q ρ (x) = 0, and known as the (intrinsic) shear current. Finally, W µ(x) is the (intrinsic) dilatation current or Weyl current. With these definitions we follow ref. [43–45]. The combination of shear current and dilatation current can be written in terms of the hypermomentum current as 1 Qµρσ + W µgρσ = (S µρσ + S ρσµ S ρµσ + S µσρ + S σρµ S σµρ). (4.41) 2 − −

– 16 – Using eqs. (4.30), (4.18) and (4.39) one can write (4.38) as

d 1 µν 1 µρσ δΓ= d x√g U δgµν S ( µδgσρ + σδgµρ ρδgµσ) (2 − 4 ∇ ∇ −∇ Z (4.42) 1 µ σ ρ 1 µ σ ˆ ρ d µ S ρ δCµ σ Q ρ δBµ σ W δBµ . − 2 − 2 − 2 ) The first line gives the full variation of the effective action with respect to the metric at fixed (typically vanishing) distortion tensor. Partial integration and the use of δgµν = δgνµ allows to write this part as

1 d µν 1 ρµν ρνµ µνρ νµρ µρν νρµ δΓ= d x√g U + ρ (S + S + S + S S S ) δgµν . (4.43) 2 4∇ − − Z Because this must equal 1 δΓ= ddx√g T µν δg , (4.44) 2 µν Z we ﬁnd for the energy-momentum tensor the decomposition

µν µν 1 ρµν ρνµ µνρ νµρ µρν νρµ T =U + ρ (S + S + S + S S S ) 4∇ − − (4.45) µν 1 ρµν ρ µν =U + ρ (Q + W g ) . 2∇ Let us note here that U µν(x) is in general not conserved by itself. The contributions from the shear current and dilatation current in (4.45) are needed to obtain a conservation law. However, these two terms come unavoidably with derivatives. Taking the trace of (4.45) we obtain the divergence-type relation

ρ 2 µ µ ρW = (T U ). (4.46) ∇ d µ − µ Similarly, by subtracting the trace we ﬁnd

ρµν µν µν 2 σ σ µν ρQ = 2(T U ) (T U )g . (4.47) ∇ − − d σ − σ We will argue below that these two relations should be understood as conservation-type relations for non- conserved Noether currents associated to extended symmetries. While variants of eq. (4.46) have been discussed in the context of dilatation and conformal symmetry (see also section 4.4 below), eq. (4.47) is new to the best or our knowledge. The decomposition in (4.45) is particularly interesting from the point of view of relativistic fluid dynamics and its derivation from quantum field theory. The first part contains the equilibrium part of the energy-momentum tensor, while the second term is by construction at least one order higher in derivatives and can give a non-equilibrium part of the energy-momentum tensor. In the following we will investigate different transformations in the frame bundle for which the affine connection acts as a gauge field, in more detail. This will lead to further insights into the physics significance of the spin current, dilatation current and shear current. We start with Weyl transformations and local Lorentz transformations and turn then to shear transformations before we combine everything into general linear transformations.

– 17 – 4.4 Weyl gauge transformations It is interesting at this point to discuss dilatations or Weyl gauge transformations in more detail. The transformations of the metric and matter ﬁelds in (4.13) get supplemented by a transformation of the Weyl gauge ﬁeld so that the complete transformation is

−∆φζ(x) 2ζ(x) φ(x) e φ(x), gµν (x) e gµν (x), Bµ(x) Bµ(x) ∂µζ(x). (4.48) → → → − Interestingly, the general connection in (4.29) is left unchanged by this transformation because contributions from the Levi-Civita part and the Weyl non-metricity part cancel. Let us also note here that √g edζ √g and T µν e−(2+d)ζ T µν so that the energy-momentum tensor → → with two upper indices has the scaling dimension ∆T µν =2+ d. For the effective action we find from (4.38) the following change under an infinitesimal Weyl transformation 1 δΓ= ddx√g U µν (x)δg (x)= ddx√g U µ (x)δζ(x). (4.49) 2 µν µ Z Z µ For a generic quantum field theory U µ(x) is non-vanishing and the effective action is not invariant under Weyl transformations. An exception is a scale-invariant theory at a renormalization group fixed point µ where U µ(x) = 0. (Even then there are corrections to the right hand side in curved space due to the conformal anomaly.) µ It is interesting to note that a symmetry under dilatations implies U µ(x) = 0 and not directly a µ vanishing trace of the energy-momentum tensor T µ(x) = 0. The latter condition would be implied by a symmetry of the theory under the larger group of conformal transformations (again up to anomalous corrections arising in curved space). µ Assume now that we consider a theory that is invariant under scaling transformations so that U µ(x)= 0. Equation (4.46) implies then µ d µ T µW = 0, (4.50) µ − 2∇ µ d µ In this context, V = 2 W is known as the virial current [64]. It was shown in ref. [65] that under the − µ ρµ condition that the virial current is itself a divergence, V = ρσ , one can actually define an “improved” ∇ energy-momentum tensor which is then trace-less. In practise this improvement can be done by changing the way the theory couples to space-time curvature, more specifically the Ricci scalar and Ricci tensor. In fact, it has been shown [66] that if the theory has a conformal symmetry in flat space one can couple it the Ricci tensor in such as way that the energy-momentum tensor following through eq. (4.7) is in fact the “improved” energy-momentum tensor. Assuming now that our theory is conformal and that this kind of improvement has been done implies that the energy-momentum tensor is trace-less (in flat space), and from (4.50) it follows that in this case µ µW = 0. (4.51) ∇ To summarize, the (intrinsic) dilatation or Weyl current W µ is in general not conserved and fulfills the divergence-type relation (4.46). Because the right hand side is known (or calculable) one should understand W µ as a non-conserved Noether current. For a scale-invariant system the divergence-type relation simplifies to (4.50). Finally, for conformal systems one has an actual conservation law for the Weyl current (4.51). One should mention here, however, that oftentimes for a conformal field theory the Weyl current actually simply vanishes, W µ = 0.

– 18 – Finally, let us mention that the conservation law associated to full dilatation symmetry in Minkowski space (for a review see ref. [67]) has an additional part due to the scaling of coordinates. It can be written as µ µν d µ J = xνT W , (4.52) D − 2 and is indeed conserved when eq. (4.50) is fulﬁlled.

4.5 Local Lorentz transformations As a next step we want to investigate local changes of frame that leave the space-time metric invariant. One can also understand them as a local version of Lorentz transformations. Mathematically, these transformations correspond to changes of basis in the frame bundle restricted to orthonormal frames. Orthonormal frames are anyway needed to describe fermionic fields because the standard version of the Clifford algebra uses them. (For an alternative approach see ref. [68] and references therein.) A choice of frame is usually parametrized in terms of the tetrad field, through a formalism we recall below. The A µ tetrad can be defined formally as a Lorentz vector valued one-form Vµ (x)dx . The latin index A is here a Lorentz index (in a sense to be made more precise below), while the Greek index µ is a standard coordinate index. The tetrad parametrizes the change of basis in the frame bundle, and its associate bundle, from A A µ the holonomic or coordinate frame to an orthonormal frame. More precisely, θ (x)= Vµ (x)dx could be A seen as a new basis for one-forms, out of which any one-form can be composed, ω(x)= ωA(x)θ (x). µ We also introduce the inverse tetrad V A(x) such that

A ν ν A µ A Vµ (x)V A(x)= δµ ,Vµ (x)V B(x)= δ B. (4.53)

µ The inverse tetrad can be seen as constituting a new basis for vectors, vA(x) = V A(x)∂µ such that any A 1 A vector ﬁeld can be written locally as U(x)= U (x)vA(x). The dual basis for one-forms is precisely θ (x). With Minkowski metric ηAB = diag( 1, +1, +1, +1) one can write the coordinate metric gµν (x) as − A B gµν (x)= ηABVµ (x)Vν (x). (4.54)

Under a coordinate transformation or diffeomorphism xµ x′µ(x) on the coordinate side, the tetrad → transforms like a one-form ∂xν V A(x) V ′A(x′)= V A(x). (4.55) µ → µ ∂x′µ ν Changing afterwards the label or integration variable from x′µ back to xµ gives the transformation rule ∂xν V A(x) V ′A(x)= V A(x) V ′A(x′) V ′A(x) . (4.56) µ → µ ∂x′µ ν − µ − µ For an infinitesimal transformation x′µ = x′µ εµ(x) this reads − A A ν A ρ A A A V (x) V (x)+ ε (x)∂ν V (x) + (∂µε (x)) V (x)= V (x)+ εV (x). (4.57) µ → µ µ ρ µ L µ µ We are using here again the Lie derivative ε in the direction ε (x). L From (4.54) and (4.13) one finds that under a Weyl transformation one has

V A(x) eζ(x)V A(x), (4.58) µ → µ 1 V µ x We are following here the conventions of ref. [69]. Other authors refer to A( ) as the tetrad ﬁeld.

– 19 – so that the tetrad has the scaling dimension ∆V = 1. (Obviously the inverse tetrad has the opposite − scaling dimension.) In addition to coordinate and Weyl transformations one may also consider local Lorentz transformations or changes of the orthonormal frame acting on the tetrad according to

V A(x) V ′A(x) = ΛA (x) V B(x), (4.59) µ → µ B µ A where Λ B(x) is at every point x a Lorentz transformation matrix such that

A C Λ B(x)Λ D(x)ηAC = ηBD. (4.60)

In other words, at every space-time point x the matrices ΛA (x) are elements of the group SO(1, d 1). B − Note that these local Lorentz transformations are intrinsic or internal, i. e. they do not act on the space- time argument x of a ﬁeld as a conventional Lorentz transformation would do. In inﬁnitesimal form, the local Lorentz transformation (4.59) reads

V A(x) V ′A(x)= V A(x)+ dωA (x)V B(x), (4.61) µ → µ µ B µ where dωAB(x)= dωBA(x) is anti-symmetric and inﬁnitesimal. − Coordinate vector and tensor ﬁelds can be transformed using the tetrad and its inverse to become scalars under general coordinate transformations, e. g.

B B µ AB A B µν ϕ (x)= Vµ (x)ϕ (x), χ (x)= Vµ (x)Vν (x)χ (x). (4.62)

The results are then Lorentz vectors and tensors, respectively. In other words these objects have now been fully transformed to the orthonormal frame. At this point it is worth to note that an action that is stationary with respect to coordinate tensor fields like χµν (x) is also stationary with respect to the resulting Lorentz tensor field χAB(x). More generally, a field Ψ might transform in some representation with respect to the local, internal R Lorentz transformations or changes of orthonormal frame,

Ψ(x) Ψ′(x)= L (Λ(x))Ψ(x), (4.63) → R or infinitesimally ′ i AB Ψ(x) Ψ (x)=Ψ(x)+ dωAB(x)M Ψ(x). (4.64) → 2 R One would also like to have a covariant derivative with respect to the local Lorentz transformations. This leads to the spin connection. The spin covariant derivative Dµ is defined such that for the spinor field Ψ(x) transforming under local Lorentz transformations according to (4.63) one has

µ B µ V (x)DµΨ(x) Λ (x)V (x)L (Λ(x))DµΨ(x). (4.65) A → A B R In other words, the covariant derivative of some ﬁeld transforms as before, with an additional transformation matrix for the new index, but without any extra non-homogeneous term. The full covariant derivative is now Dµ = µ + Ωµ(x). (4.66) ∇

– 20 – Here, µ is the co-covariant derivative as introduced in section 4.2 including the Weyl gauge field, the ∇ affine connection for coordinate indices (see below for some restrictions) and Ωµ depends on the Lorentz µ representation of the field the derivative acts on. We also use the abbreviation DA = V A(x)Dµ. The spin connection Ωµ(x) must transform like a non-abelian gauge field for local Lorentz transformations,

′ −1 −1 Ωµ(x) Ω (x)= L (Λ(x))Ωµ(x)L (Λ(x)) [∂µL (Λ(x))] L (Λ(x)). (4.67) → µ R R − R R A A A We also write this for an inﬁnitesimal Lorentz transformation Λ B(x)= δ B + dω B(x) as

′ i AB i AB Ωµ(x) Ω (x)= Ωµ(x)+ dωAB(x) M , Ωµ(x) M ∂µdωAB(x). (4.68) → µ 2 R − 2 R This is the transformation rule for a non-abelian gauge ﬁeld associated to SO(1, d 1). Quite generally, − one may write the spin connection as i Ω (x)=Ω (x) M AB, (4.69) µ µAB 2 R where ΩµAB(x) is anti-symmetric in the Lorentz indices A and B and now independent of the representation . Sometimes it is also called spin connection. As an examples we note here the covariant derivative of a R Lorentz vector with upper index and scaling dimension ∆A

B B B C B DµA (x)= ∂µA (x)+Ω (x)A (x) ∆ABµ(x)A (x). (4.70) µ C − A At present, the spin connection Ωµ B(x) could be of quite general form, as long as it is anti-symmetric A in the last two indices. However, in practise, it is most useful to deﬁne the spin connection Ωµ B such that the fully covariant derivative of the tetrad vanishes,

A A A B ρ A A DµV = ∂µV +Ω V Γ V + BµV = 0. (4.71) ν ν µ B ν − µν ρ ν This leads to a consistent formalism where derivatives of coordinate and Lorentz tensors are compatible. One may solve this relation for the spin connection, leading to

A A ν A ν ρ ρ A ν Ω = µV V = ∂µV V + (Γ Bµδ )V V . (4.72) µ B − ∇ ν B − ν B µν − ν ρ B µ ν From ∂µηAB = µηAB = µ(V A(x)V B(x)gµν (x)) = 0 one can show that ΩµAB in eq. (4.72) is indeed ∇ ∇ ρ anti-symmetric as long as the connection Γµν is weakly metric-compatible, in the sense

µgρσ = 2Bˆµρσ = 0. (4.73) ∇ − Proper non-metricity is therefore not permitted in the present formalism. However, both the contorsion ρ tensor Cµ σ(x) and the Weyl gauge field Bµ(x) are allowed to be non-zero. Finally we note a useful identity for the variation of the spin connection that can be easily derived from (4.72), A A ν ρ ρ A ν δΩ (x)= DµδV V + δ(Γ Bµδ )V V . (4.74) µ B − ν B µν − ν ρ B We use here the fully covariant derivative Dµ and the variation of the affine connection as specified in A (4.30) (for vanishing proper non-metricity). Note that in contrast to the spin connection Ωµ B(x) itself, A which is a gauge field, its variation δΩµ B(x) transforms simply as a tensor with one upper and one lower A A index under local Lorentz transformations. Under coordinate transformations both Ωµ B(x) and δΩµ B(x) transform as one-forms.

– 21 – 4.6 Conservation laws in the tetrad formalism Let us now investigate what kinds of conservation-type relations we can obtain from the effective action Γ[φ, V, Ω,B]. We take the latter to depend on matter fields φ(x) which can be taken to be local Lorentz A vectors, tensors and spinors. In addition the action depends on the tetrad field Vµ (x) which also replaces the metric everywhere. All derivatives of fields are assumed to be fully covariant derivatives DA which AB depend on the spin connection Ωµ and the Weyl gauge field Bµ. When we take the spin connection to be independent of the tetrad and the Weyl gauge field, this is due to the possibility of non-vanishing contorsion. Because of relation (4.72) or (4.74) the spin connection and can be varied independent of the tetrad and the Weyl gauge field only through a variation of contorsion. Even at vanishing physical torsion and contorsion, it is useful to consider a variation with respect to it. This is similar to varying the metric as done in section (4.1) even though the latter is subsequently fixed, for example to describe Minkowski space. For stationary matter fields δΓ/δφ = 0, the variation of the effective action is

d µ A 1 µ AB d µ δΓ= d x√g T (x)δV (x) S (x)δΩ (x) W (x)δBµ(x) . (4.75) A µ − 2 AB µ − 2 Z µ The field T A(x) is defined through a variation with respect to the tetrad at fixed spin connection and Weyl gauge field. We will argue below that it is actually the canonical energy-momentum tensor. The µ variation with respect to the spin connection with fixed tetrad defines the spin current S AB(x). Finally the variation with respect to the Weyl gauge field at fixed tetrad and spin connection defines a field W µ(x). All µ µ µ three fields T A(x), S AB(x) and W (x) transform under coordinate transformations and local Lorentz A AB transformations as indicated by their indices. The reason is that the variation δVµ (x), δΩµ (x) and δBµ(x) are all transforming as tensors in this sense and the variation of the action itself must be a scalar. We should also state here that a full variation of the effective action with respect to the tetrad (with the spin connection taken to obey relation (4.72) at fixed contorsion) leads to the energy momentum tensor as a mixed coordinate and Lorentz tensor, while a related variation of the Weyl gauge field gives the dilatation current as in (4.42), d µ A d µ δΓ= d x√g T (x)δV (x) W (x)δBµ(x) . (4.76) A µ − 2 Z Using (4.74) we can relate the quantities in (4.75) and (4.76) and find

µν µν 1 ρµν µνρ νµρ T (x)= T (x)+ ρ [S (x)+ S (x)+ S (x)] . (4.77) 2∇ One can recognize this as the Belinfante-Rosenfeld form of the energy-momentum tensor with the first term T µν (x) being the canonical energy-momentum tensor and T µν(x) its symmetric relative. Note that the µν µν expression in square brackets in (4.77) is anti-symmetric in ρ and µ. This implies µT (x)= µT (x), ∇ ∇ so that both tensors are conserved. (This conservation law can be directly obtained from general coordinate transformations as usual.) In other words, canonical energy-momentum tensor follows from a variation of the action with respect to the tetrad at fixed spin connection, while the symmetric energy-momentum tensor follows from a related variation but at contorsion kept fixed. One also finds for the two versions of the dilatation currents defined by (4.75) and (4.76) the relation 2 W µ(x)= W µ(x)+ Sρµ (x). (4.78) d ρ

– 22 – By construction the action is invariant under local Lorentz transformations. We consider now such a transformation in infinitesimal form. The matter fields are still assumed to be stationary, δΓ/δφ = 0, so that it suffices to consider the variations of the tetrad and spin connection. We first consider a variation where only the tetrad is being varied, and the spin connection is taken as dependent according to eq. (4.72) at vanishing contorsion. One finds

d µA B δΓ= d x√g T (x)Vµ (x)δωAB (x). (4.79) Z Because this must vanish for arbitrary δωAB(x) one ﬁnds that the energy-momentum tensor is symmetric,

T AB(x)= T BA(x). (4.80)

However, one may also do the calculation in an alternative way where the spin connection is ﬁrst varied independent of the tetrad and we use then (4.61) and (4.68),

µ 1 µ δΓ= ddx√g T (x)δV A(x) S (x)δΩ AB(x) A µ − 2 AB µ Z d µ A B 1 µ A C A C A = d x√g T (x)δω (x)V (x) S (x) δω (x)Ω (x) Ω (x)δω (x) ∂µδω (x) . A B µ − 2 AB C µ B − µ C B − B Z h (4.81)i

Using partial integration one can rewrite this as

d BA 1 µAB δΓ= d x√g T (x) DµS (x) δωAB(x). (4.82) − 2 Z

For this to vanish for arbitrary δωAB(x) the expression in square brackets must be symmetric. Because SµAB = SµBA is anti-symmetric, we ﬁnd for the divergence of the spin current − µρσ σρ ρσ µS (x)= T (x) T (x). (4.83) ∇ − This is the conservation-type relation we were looking for. We argue that the spin current SµAB(x) should be seen as a non-conserved Noether current associated to an extended symmetry. The transformation in eq. A (4.81) is not a full symmetry in the sense of section 2 because a global transformation with Dµδω B(x) = 0 does not make the action stationary as long as T AB(x) = T BA(x). Nevertheless, eq. (4.83) is still a very 6 useful identity as long as the right hand side is known. This is indeed the case, because it follows from a variation of the quantum eﬀective action according to eq. (4.75). We emphasize again that the spin current is in general not conserved. What needs to be conserved as a consequence of full Lorentz symmetry in Minkowski space (also including a coordinate transformation) is the sum of spin current and orbital angular momentum current,

M µAB(x)= xA(x)T µB(x) xB(x)T µA(x)+ SµAB(x). (4.84) − A A A We assume here Dµx (x) = Vµ (x) (which essentially deﬁnes what is meant by x (x) in non-cartesian µAB coordinates) one has indeed DµM (x) = 0 as a consequence of (4.83) and the conservation law µA DµT (x) = 0.

– 23 – 4.7 General linear frame change transformations Mathematically, the frame bundle allows for changes of basis transformation that are more general than the restriction to orthonormal frames we discussed above. The full group of local transformations is the general linear group GL(d), which contains SO(1, d 1) as a subgroup, but encompasses also dilatations − and shear transformations. In the following we discuss first briefly the Lie algebra of GL(d) and decompose it into the different generators. Subsequently we discuss general frame change transformations and the conservation-type relations that follow from them. We will consider the general linear group as an extension of the Lorentz group. Accordingly we introduce in addition to the generators M AB for infinitesimal Lorentz transformations also generators SAB for shear transformations and D for dilatations. Note that we are using indices A and B as for an orthonormal frame to label the generators. AB BA AB The generators for shear transformations are symmetric, S = S and trace-less, S ηAB = 0. For d space-time dimensions one has d(d 1)/2 generators M AB, d(d + 1)/2 1 generators SAB and 1 − − generator D. Indeed, these make up the d2 generators of the general linear group GL(d). Without the generator for dilatations D, the generators M AB together with SAB generate the Lie algebra of the special linear group SL(d). In terms of matrices in the fundamental representation one may take (M AB)C = i(ηAC δB D − D − ηBC δA ) as usual for the generators of Lorentz transformations, DA = iδA for the generator of dilata- D B − B tions, and (SAB)C = i(ηAC δB + ηBC δA (2/d)ηAB δC ) for the generators of shear transformations. D − D D − D The Lie brackets are

M AB, M CD = i ηBC M AD ηAC M BD + ηBDM CA ηADM CB , − − − AB CD BC AD AC BD BD CA AD CB M ,S = i η S η S + η S η S , − − − (4.85) SAB,SCD = i ηBC M AD + ηAC M BD ηBDM CA ηADM CB , − − − AB AB M , D = S , D = [D, D] = 0.

Obviously, M AB andD each generate sub-groups, while the SAB alone do not. The center of the Lie algebra (4.85) is generated by D. It will sometimes be convenient to split GL(d) into to abelian subgroup of dilatations and the remaining group SL(d). Previously we have already discussed a group of transformations consisting of SO(1, d 1) and the − dilatations in terms of orthonormal fields. We will now extend first the indefinite orthogonal group SO(1, d − 1) to the larger group SL(d) and subsequently also add the dilatation part. Similar to the discussion of orthonormal frames in section 4.5 we introduce now a frame field or soldering form that parametrizes the change from a coordinate basis to a more general frame that we may a µ call an unimodular frame. It can be introduced as a vector valued one form eµ (x)dx . The smaller case latin index a is now belonging to a frame that is in general neither holonomic (induced by a coordinate system), nor orthonormal. We may also introduce the inverse frame field such that

a ν ν a µ a eµ (x)e a(x)= δµ , eµ (x)e b(x)= δ b. (4.86)

a A The frame ﬁeld eµ (x) behaves with respect to coordinate transformations very similar as the tetrad Vµ (x) and we do not discuss this further.

– 24 – The metric gµν (x) in the coordinate frame is expressed through the frame ﬁeld as

a b µ ν gµν (x) =g ˆab(x)eµ (x)eν (x), gˆab(x)= gµν (x)e a(x)e b(x). (4.87)

Here we introduce the metric in the unimodular frameg ˆab(x). It has the property

gˆ = det gab(x) = 1, (4.88) − but can otherwise we a quite general symmetric matrix. In this sense eq. (4.87) generalizes eq. (4.54). In order to discuss how general linear transformations act on the frame field, let us first exclude dilatations, which need a separate discussion because their generator is in the center of the algebra (4.85). Excluding them means here to restrict to transformations with unit determinant, i.e. to restrict from GL(d) to SL(d). Such a special linear transformation acts on the frame field according to

e a(x) e′ a(x)= M a (x) e b(x), (4.89) µ → µ b µ where M a (x) is at every point x a matrix with unit determinant, M(x) SL(d). Similarly one can b ∈ transform other vector ﬁelds and tensors with upper indices. Covectors and tensor ﬁelds with lower indices transform with the transpose of the inverse of M(x). For example the unimodular metric transforms as

′ −1 c −1 d gˆab(x) gˆ (x) = (M ) (x)(M ) (x)ˆgcd(x). (4.90) → ab a b This makes sure that contractions of upper and lower indices can be done consistently. Two remarks are in order here:

ab (i) Becauseg ˆab(x) and its inverseg ˆ (x) are not invariant symbols with respect to SL(d), some care is needed when using them to pull indices down or up.

(ii) In a theory with spinor fields one would now have to work with three different frames and corresponding indices. Besides the coordinate frame and the general frame one also needs there an orthonormal frame where the Clifford algebra is rooted. An extension of spinor representations from SO(1, d 1) − to SL(d) is not easily possible. (In principle it is possible to define the operation of general linear transformations on the Clifford algebra by employing a basis for the latter in terms of p-forms [70], but that has substantial implications we do not discuss further here.) The transition from the or- a a µ thogonal frame to the general frame is then mediated by eB (x) = eµ (x)V B(x). We will largely avoid this technical complication here and assume similar as in section 4.1 that all fermionic fields have been integrated out, already. We are then left with fields of integer spin that can be organized into scalar, vector and tensor representations under Lorentz transformations. These representations can be extended to special linear transformations in a rather direct way.

Weyl gauge transformations of frame ﬁeld. Under a Weyl gauge transformation the frame ﬁeld must transform analogously to the tetrad (see eq. (4.58))

e a(x) eζ(x)e a(x). (4.91) µ → µ Combining this with the SL(d) transformation in (4.89) leads to the general linear group GL(d).

– 25 – Representations. We consider now ﬁelds φ in some representation of these generators so that an R inﬁnitesimal transformation reads

′ i AB i AB φ(x) φ (x)= φ(x)+ dωAB(x)M φ(x)+ dζAB(x)S φ(x)+ idζ(x)D φ(x). (4.92) → 2 R 2 R R As an example, a vector ﬁeld ϕa(x) is in the fundamental representation with respect to Lorentz and shear transformations and would transform for dζ = 0 according to

ψa(x) ψ′a(x)= ψa(x)+ dωa (x)ψb(x)+ dζa (x)ψb(x). (4.93) → b b a A a B a We use here dω b(x) = dω B(x)eA (x)e b(x) etc. Note that Lorentz boosts parametrized by dω b(x) and a shear transformations parametrized by dζ b(x) are represented in a closely related way. More formally, the fundamental representation has the generators

AB c Ac B Bc A (M (x)) d = i e (x)e d(x) e (x)e d(x) , F − − (4.94) (SAB(x))c = i eAc(x)eB (x)+ eBc(x)eA (x) (2/d)ηAB δc . F d − d d − d Note that the generators depend here on the space-time position x. In a similar way one can ﬁnd other tensor representations. For example, a covector ﬁeld would transform as

′ a a χb(x) χ (x)= χb(x) dω (x)χa(x) dζ (x)χa(x). (4.95) → b − b − b A general (n,m)-tensor representation of SL(d) changes under a ﬁnite group transformation as

φa1···an (x) M a1 (x) M an (x)(M −1)d1 (x) (M −1)dm (x) φc1···cn (x). (4.96) b1···bm → c1 · · · cn b1 · · · bm d1···dm Dilatations. Let us now discuss dilatations. Because they are in the center of the algebra (4.85) one can assign in principle an arbitrary charge to some field ϕ(x). Usually this is done such that a scalar field with the (momentum) scaling dimension ∆ϕ would transform under an infinitesimal dilatation like

ϕ(x) ϕ(x) dζ(x)∆ϕϕ(x). (4.97) → − In a similar way, any (n,m)-tensor ﬁeld in an orthonormal frame would transform under dilatations according to its scaling dimension ∆φ (written here for a ﬁnite transformation),

A1···An A1···An φ (x) exp( ζ(x)∆φ) φ (x). (4.98) B1···Bm → − B1···Bm Dilatations in the unimodular frame are as in an orthonormal frame, because the transition matrix B µ B e (x) = e (x)V (x) has the scaling dimension ∆ B = 0. This implies in particular that the metric in a a µ e a the unimodular frameg ˆab(x) also has vanishing scaling dimension and for a general tensor one has

a1···an a1···an φ (x) exp( ζ(x)∆φ) φ (x). (4.99) b1···bm → − b1···bm Covariant derivative. In order to make derivatives transform in the appropriate representation of GL(d), we need to deﬁne an appropriately generalized covariant derivative. We will write the latter as

Dµ = ∂µ + Ωµ(x). (4.100)

– 26 – Equation (4.69) is now generalized to

i i i Ω (x)=Ω (x) M AB(x)+ SAB(x)+ ηABD . (4.101) µ µAB 2 R 2 R d R

The general linear connection ΩµAB(x) is now not anti-symmetric as the spin connection in the last two indices any more, but has also a symmetric and trace-less contribution which determines the shear transformation sector, as well as a trace which governs dilatations. The trace is directly related to the Weyl gauge ﬁeld by 1 B (x)= Ω A (x). (4.102) µ d µ A For tensor representations of GL(d) one can directly work with

a Aa B Ωµ b(x)=ΩµAB(x)e (x)e b(x), (4.103)

a a a c c a a so that for example Dµχ (x)= ∂µχ (x)+Ωµ c(x)χ (x) Ω (x)χ (x) Bµ(x)∆χχ (x). b b b − µ b c − b Gauge transformations. The general linear connection is now a gauge ﬁeld for the group GL(d). As such, it transforms as

a ′ a a c −1 d a −1 c Ω (x) Ω (x)= M (x)Ω (x)(M ) (x) [∂µM (x)] (M ) (x). (4.104) µ b → µ b c µ d b − c b For an inﬁnitesimal transformation this reads

a ′ a a a c a c a Ω (x) Ω (x)=Ω (x)+dω (x)Ω (x) Ω (x)dω (x) ∂µdω (x) µ b → µ b µ b c µ b − µ c b − b a c a c a +dζ c(x)Ω (x) Ωµ c(x)dζ b(x) ∂µdζ b(x) µ b − − (4.105) a ∂µdζ(x)δ − b a a a a =Ω (x) Dµ(dω + dζ + dζδ ). µ b − b b b a Similar to the spin connection, the general linear connection Ωµ b(x) should be deﬁned such that the fully covariant derivative of the frame ﬁeld vanishes,

a a a b ρ a Dµe = ∂µe +Ω e Γ e = 0. (4.106) ν ν µ b ν − µν ρ Note that the right hand side also naturally contains the Weyl gauge ﬁeld with the right prefactor. This has the advantage that derivatives of tensors can be consistently evaluated in terms of the coordinate or the general linear frame. One may solve eq. (4.106) for the general linear connection, leading to a a ν ρ a ν Ω = (∂µe ) e +Γ e e . (4.107) µ b − ν b µν ρ b One may check that (4.107) transforms also correctly, i. e. according to (4.105) and that in contrast to the spin connection (4.72), Ωµab(x) is now not anti-symmetric in a and b any more. Also, with eq. (4.28) one can see that eq. (4.102) in indeed fulﬁlled. a An equation analogous to (4.74) also holds for the variation δΩµ b(x),

a a ν ρ a ν δΩ (x)= (Dµδe ) e + δΓ e e . (4.108) µ b − ν b µν ρ b

– 27 – ρ One may use here (4.30) for the variation δΓµν . It is important to note here that the general linear ρ connection (4.104) is consistently defined also with non-vanishing contorsion Cµ σ(x) and non-metricity ρ a Bµ σ(x). In this sense the variation δΩµ b(x) in (4.108) is free of algebraic constraints, even if the frame a field eν (x) is kept fixed. From eqs. (4.87), (4.106) and (4.17) one finds

Dµgˆab(x)= 2Bµab(x). (4.109) −

The metricg ˆab(x) is only covariantly constant for vanishing non-metricity. A non-vanishing Weyl gauge ﬁeld Bµab = Bµgab is already a deviation from this.

Cartan’s structure equations. For completeness we also note Cartan’s ﬁrst structure equation for torsion, a a a a b a b T = ∂µe ∂νe +Ω e Ω e . (4.110) µν ν − µ µ b ν − ν b µ Using eq. (4.106) one can see that this is indeed in agreement with eq. (4.20). In particular the right hand side vanishes in situations without space-time torsion. Similarly, Cartan’s second structure equations yields the curvature tensor,

a a a a c a c R = ∂µΩ ∂νΩ +Ω Ω Ω Ω . (4.111) bµν ν b − µ b µ c ν b − ν c µ b σν µ b a The Ricci scalar is given by R = g e aeσ R bµν. a Some possible choices for the frame field. The unimodular frame field eµ (x) can be left open, but it can also be fixed in different ways. Two possibilities are particularly interesting.

1. An orthonormal frame is a special case of an unimodular frame. This is obtained by fixingg âb(x)= ηab a a to the Minkowski metric. The frame field is then a tetrad field, eµ (x)= Vµ (x) with all the properties discussed in sections 4.5 and 4.6.

a a 2. One can also set eµ (x) = δµ /χ(x). Here we have introduced a kind of external dilaton field χ(x) with scaling dimension ∆χ = 1 such that it transforms under Weyl transformations according to χ(x) e−ζ(x)χ(x). Accordingly the frame field has the correct scaling dimension. For this choice → 2 the coordinate metric is of the form gµν (x) =g ˆµν (x)/χ(x) . Because ofg ˆ = detg âb(x)=1 one has −2d − g(x)= det gµν (x)= χ(x) . − Response to local general frame transformations. Let us now discuss the response of a quantum field theory to general linear changes of frame. We will again employ the quantum effective action which a depends on matter field expectation values φ(x), the frame field eµ (x) and the general linear connection a Ωµ b(x). In addition it also depends on the metric in the unimodular frame gab(x). Because the metric ab has fixed determinant,g ˆ = detg âb(x) = 1, its variation is trace-free,g ˆ (x)δgâb(x) = 0. − a Because of (4.108) and (4.30) the general linear connection and the frame field eµ (x) are only independent when contorsion and non-metricity are allowed to vary. In this sense one should understand a a a variation where Ωµ b and eµ are taken to be independent at an intermediate step. We also need to carry the metric gab(x) because it is needed to construct the coordinate metric gµν (x) according to eq. (4.87). Also, gab and its inverse may of course appear in the effective action.

– 28 – We write that eﬀective action as Γ[φ, e, Ω, gˆ], (4.112) and for stationary matter ﬁelds it has the variation

d µ a 1 µ b a 1 ab δΓ= d x√g T (x)δe (x) S (x)δΩ (x)+ Uˆ (x)δgâb(x) . (4.113) a µ − 2 a µ b 2 Z µ This defines a field T a which must be the canonical energy-momentum tensor in the unimodular frame. This becomes clear when one compares with (4.75) and realizes thatg âb could be kept fixed at the Minkowski a a form ηab such that the frame field eµ is just the tetrad. Moreover, keeping Ωµ b in (4.113) fixed implies then through (4.102) to keep also the Weyl gauge field Bµ fixed. µ b Similarly, the field S a must be the the hypermomentum current [43, 44, 46, 47] introduced already a in eq. (4.38), now in the unimodular frame. This is because keeping the frame field eµ andg âb fixed means a ρ a ν to keep also the coordinate metric gµν fixed and according to (4.108) one has then δΩµ b = δΓµν eρ e b. Finally, Uâb must be the trace-less part of the field U µν(x) introduced in (4.38), now in the unimodular frame, Uâb = U µν (1/d)gµν U ρ e ae b. (4.114) − ρ µ ν a a This is because keeping the frame field eµ and the connection Ω µ b fixed but varying the metricg âb is like ρ µν keeping the connection Γµ σ fixed and varying only the metric such that g δgµν = 0. If instead the variation is done such that (4.108) is obeyed at fixed (vanishing) contorsion and non- metricity we write 1 δΓ= ddx√g T µ (x)δe a(x)+ Tâb(x)δgˆ (x) . (4.115) a µ 2 ab Z µ Here T a must be the symmetric energy-momentum tensor as follows from comparison with (4.76) for a δgâb = 0. Moreover, when keeping the frame field fixed, δeµ = 0, we can compare to (4.7) and find that

Tˆab = T µν (1/d)gµν T ρ e ae b, (4.116) − ρ µ ν must be the trace-free part of the symmetric energy-momentum tensor in the unimodular frame. By using (4.108) and eq. (4.5) together with the variation in (4.113) and comparing to (4.115) we ﬁnd the following relation

µν µν 1 ρµν µνρ µρν ρνµ νµρ νρµ T =T + ρ [S + S S S + S S ] , 4∇ − − − (4.117) µν 1 ρµν µνρ νµρ =T + ρ [S + S + S ] . 2∇ In the second line we have used eq. (4.40) and Sρµν is the spin current. We recognize the Belinfante- Rosenfeld relation between symmetric and canonical energy-momentum tensor as seen before in eq. (4.77). The anti-symmetric part of (4.117) gives eq. (4.83). Similarly we ﬁnd using (4.41)

µν µν 1 ρµν Tˆ (x)=Uˆ (x)+ ρQ . (4.118) 2∇ This is actually eq. (4.47) obtained previously.

– 29 – Local SL(d) transformations. In a next step let us consider local SL(d) transformations. When both a the frame field eµ and the unimodular metricg âb are being transformed the coordinate frame metric gµν in (4.87) is invariant. Accordingly also the effective action must be invariant at stationary matter fields. This is indeed the case when T µν as defined by (4.115) is symmetric and when Tˆµν is its trace-less part.

Consequences of Weyl gauge transformations. Under Weyl gauge transformations the frame fields transforms as in (4.91) and the general linear connection as in (4.105). The change in the effective action (4.113) must equal (4.49). We find thus

µ 1 µ a µ T ∂µ √gS = U . (4.119) µ − √g a µ Using (4.39) and (4.77) this can be brought to the form (4.46) as it should be. In summary, in the unimodular frame one can see nicely the full transformation group of the frame bundle GL(d). The associated (non-conserved) Noether currents are the spin current Sµρσ, the Weyl current W µ and the shear current Qµρσ. The corresponding divergence-type relations are given by eq. (4.83), eq. (4.46) and eq. (4.47).

5 Example: scalar ﬁeld theory with non-minimal coupling

In this section we will discuss an example for an effective action and the resulting construction of the different tensor fields defined in section 4.2. The example is illustrative and rather simple. We take the effective action for a single real scalar field to be

d 1 µν 1 2 Γ= d x√g g ∂µϕ∂ν ϕ U(ϕ) ξRϕ . (5.1) −2 − − 2 Z σν σν ρ Here, U(ϕ) is the effective potential, R = g Rσν = g R σρν is the Ricci scalar and ξ denotes its non- minimal coupling to the scalar field ϕ. For clarity let us note that the Einstein-Hilbert action would be in d our conventions SEH = d x√gR/(16πGN). From dimensional analysis the scaling dimension of ϕ follows as ∆ϕ = (d 2)/2. − R The non-minimal coupling has the value ξ = (d 2)/(4d 4) when the action (5.1) has a conformal − − symmetry. This value can also be seen as a renormalization group fixed point. On the other side, ξ =0 is not a renormalization group fixed point and thus ξ is generated by quantum fluctuations even if it should be absent in the microscopic action. Let us note that as an effective action (5.1) should be seen as an approximation. In particular one can expect that quantum fluctuation induce more complex kinetic terms, higher order derivatives, more involved couplings to the curvature tensor, as well as non-local terms. Nevertheless, we can use the model in (5.1) for some illustrations. Let us first consider (5.1) in the context of strictly Riemannian geometry and determine the energy- momentum tensor according to eq. (4.7). This leads to the so-called improved energy-momentum tensor [65],

µν µ ν µν 1 ρσ 1 2 µν 2 µν ρ 2 µ ν 2 T =∂ ϕ∂ ϕ g g ∂ρϕ∂σϕ + U(ϕ)+ ξRϕ + ξ R ϕ + g ρϕ ϕ . (5.2) − 2 2 ∇ ∇ −∇ ∇

– 30 – Let us now extend eq. (5.1) to a geometry with general aﬃne connection as discussed in section 4.2. This amounts to taking the connection as independent of the metric or alternatively to introduce contorsion, the Weyl gauge ﬁeld and proper non-metricity. The action (5.1) becomes

d 1 µν 1 2 Γ= d x√g g µϕ νϕ U(ϕ) ξRϕ , (5.3) −2 ∇ ∇ − − 2 Z with the co-covariant derivative acting on a scalar ﬁeld like

d 2 µϕ = (∂µ ∆ϕBµ)ϕ = ∂µ − Bµ ϕ. (5.4) ∇ − − 2 The Ricci scalar R is given in eq. (4.34) and its variation in (4.36). We recall also the connection between the Weyl gauge ﬁeld and the connection in (4.28). Similarly one ﬁnds from (4.30)

1 ρ 1 ρσ δBµ = δΓ g µδgρσ . (5.5) d µ ρ − 2 ∇ µν µ σ The (non-conserved) tensor U and the hypermomentum tensor S ρ are deﬁned through eq. (4.38). The variation of the action with respect to the metric at ﬁxed connection yields (evaluated at vanishing contorsion and non-metricity)

µν µ ν µν 1 ρσ 1 2 µν 2 d 2 µν ρ U =∂ ϕ∂ ϕ g g ∂ρϕ∂σϕ + U(ϕ)+ ξRϕ + ξR ϕ + − g ρ(ϕ∂ ϕ). (5.6) − 2 2 2d ∇ Similarly the variation with respect to the connection at ﬁxed metric yields the hypermomentum current,

µ σ d 2 σ µ 2 µσ 2 µ σ 2 S = − δ ∂ ϕ ξg ∂ρϕ + ξδ ∂ ϕ . (5.7) ρ − 2d ρ − ρ The spin tensor can be obtained from this trough eq. (4.40), 1 Sµρσ = (S µρσ S µσρ)= ξgµσ∂ρϕ2 + ξgµρ∂σϕ2. (5.8) 2 − − The shear and dilation current follow through eq. (4.41). We ﬁnd for the dilatation or Weyl current

2d 2 d 2 W µ = ξ − − ∂µϕ2, (5.9) d − 2d and for the shear current 2 Qµρσ = ξgµσ∂ρϕ2 ξgµρ∂σϕ2 + ξ gρσ∂µϕ2. (5.10) − − d We note that the dilatation current vanishes for the conformal choice ξ = (d 2)/(4d 4). We also note − − that the shear tensor is in general non-zero. However, it is proportional to gradients and should therefore vanish in many equilibrium situations.

– 31 – 6 Conclusions

We have developed here a formalism to determine expectation values as well as correlation functions of Noether currents from the quantum effective action. These contain immediately all corrections due to quantum fluctuations. Technically, the method works with external gauge fields on which the quantum effective action depends in addition to expectation values of matter fields. The method is very versatile and can be used in particular for the real, conserved Noether currents associated to global symmetries of the quantum effective action. However, it can also be used for a class of transformations that have been called “extended symmetries” [40–42], under which the action is not invariant but changes in a specific way. More precisely, this change must be proportional to a term that is actually known at the macroscopic level of the quantum effective action in order for the transformation to be useful in practise. Associated to such “extended symmetries” one finds non-conserved “Noether currents”. Their equation of motion has a form similar to a covariant conservation law but with a non-vanishing known term on the right hand side. After a general discussion of this construction we turned to applications of these ideas in the context of space-time geometry. First, the symmetry under general coordinate transformations leads as usual to the covariant conservation of the symmetric energy-momentum tensor. More interesting are further transformations corresponding to changes of basis in the frame- and spin bundle. In particular we discuss local internal Lorentz transformations (including rotations and boosts), local dilatations or Weyl transformations, and local shear transformations. The associated currents are the spin current, the dilatation or Weyl current and the shear current. Together they form a rank-three tensor known as hypermomentum current [43, 44, 46, 47]. The latter can also be understood as the (non-conserved) Noether current associated to GL(d) transformations in the frame bundle. It is only under special circumstances that real conservation laws arise. For example, the Weyl current is conserved in the presence of a conformal symmetry (but then typically vanishes). Or the spin current is conserved when the canonical energy-momentum tensor is symmetric. (Our formalism yields an expression for the canonical energy-momentum tensor in terms of a variation of the effective action.) The shear current is usually not conserved, except when it vanishes. An interesting application of the insights gained here might concern relativistic fluid dynamics. While the usual formulation builds up on the covariant conservation law for the energy-momentum tensor, additional equation of motion are available in our formalism, and their connection to the quantum effective action is now understood. It is very interesting that for a given quantum effective action the currents themselves are known, as well as their correlation functions, at least in principle. On the other side, the state dependence of the quantum effective action might be carried to rather good approximation by the (fluid dynamic) degrees of freedom of the energy-momentum tensor, by the matter field expectation values, and additionally by the components of the hypermomentum current. This could lead to a rather powerful formalism for quantum field dynamics out-of-equilibrium. Understanding how the components of the hypermomentum tensor evolve might be of interest for many situations in non-equilibrium quantum field theory, such as in condensed matter physics, heavy ion collisions, or cosmology. We believe that in particular the dilatation current and the shear current are interesting because their divergence-type equation of motion could give the evolution equations for the non-equilibrium degrees of freedom related to bulk and shear viscous dissipation. Moreover, the structure of the equations of motion is such that these equations could actually be causal in the relativistic sense as explained in ref. [21].

– 32 – In these regards, our equations are similar to the equations of motion for the so-called divergence-type theories of relativistic fluid dynamics. We plan to investigate these matters in more detail in a forthcoming publication. Technically we obtain equations of motion for the components of the hypermomentum tensor by varying the affine connection independent of the metric, tetrad or frame field. Conceptually this amounts to varying the non-Riemannian parts of space-time geometry, specifically contorsion, the Weyl gauge field and proper non-metricity. It is important to note here that for us the non-Riemannian geometry is a purely calculational device. After the variations are done we evaluate all expressions at vanishing contorsion and non-metricity, i. e. in the (pseudo) Riemannian geometry of general relativity. However, from the point of view of theories of modified gravity (beyond Einsteins theory of general relativity), our findings may also be of interest. Specifically, it has been argued that the spin current, dilatation current and shear current are natural source terms to appear in such extended theories of gravity [43–57]. It is therefore useful to understand well under which circumstances they are non-zero. In the present paper we have focused entirely on (quantum) field theory. However, in light of our findings it might also be interesting to revisit actions for particles (or strings or branes), and to investigate wether and how contributions to the shear current, Weyl current and spin current would arise from variations in geometry there. On the example of a scalar field theory with non-minimal coupling to gravity we have shown that all components of the hypermomentum tensor can be non-vanishing. However, they are proportional to gradients and might therefore vanish in many equilibrium situations. We believe that quantum fluctuations indeed induce typically a non-vanishing shear and dilatation current in non-equilibrium situations. This point can be investigated further, for example with the functional renormalization group [3–6], and we plan to do so.

Acknowledgments

This work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster), SFB 1225 (ISOQUANT) as well as FL 736/3-1, and by U.S. Department of Energy, Oﬃce of Science, Oﬃce of Nuclear Physics, grants Nos. DE-FG-02-08ER41450.

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– 36 –