Prepared for submission to JHEP Conserved and non-conserved Noether currents from the quantum effective action Stefan Floerchinger1 and Eduardo Grossi2 1Institut f¨ur Theoretische Physik Universit¨at Heidelberg, Philosophenweg 16, D-69120 Heidelberg 2Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA E-mail: [email protected], [email protected] Abstract: The quantum effective action yields equations of motion and correlation functions including all quantum corrections. We discuss here how it encodes also Noether currents at the full quantum level. This holds both for covariantly conserved currents associated to real symmetries that leave the action invariant as well as for non-conserved Noether currents associated to extended symmetry transformations which change the action, but in a specific way. We discuss then in particular symmetries and extended symmetries associated to space-time geometry for relativistic quantum field theories. These encompass local dilatations or Weyl gauge transformation, local Lorentz transformations and local shear transformations. Together they constitute the symmetry group of the frame bundle GL(d). The corresponding non-conserved Noether currents are the dilatation or Weyl current, the spin current and the shear current for which divergence-type equations of motion are obtained from the quantum effective action. arXiv:2102.11098v1 [hep-th] 22 Feb 2021 Contents 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 formu- lation 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 micro- scopic 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 cor- responding 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 polar- ization 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, be- cause 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