The energy–momentum tensor(s) in classical gauge theories

Abstract

We give an introduction to, and review of, the energy-momentum tensors in classical gauge field theories in Minkowski space, and to some extent also in curved space-time. For the canonical energy-momentum tensor of non-Abelian gauge fields and of matter fields coupled to such fields, we present a new and simple improvement procedure based on gauge invariance for constructing a gauge invariant, symmetric energy-momentum tensor. The relationship with the Einstein-Hilbert tensor following from the coupling to a gravitational field is also discussed. Plan of the paper:In the present note we focus, for classical YM (Yang-Mills) theories in ndimensional Minkowski space, on the EMT (energy-momentum tensor, also referred to as stressenergy tensor or stress tensor for short): the components T µν of the EMT can be interpreted as follows, e.g. see reference [12]. T 00 represents the energy density, T 0i the i-momentum density (or energy flux density) and T ij the i-momentum flux density in the j-direction. Regarding the field theoretical system as a collection of particles, we can also interpret T ii ≡ p as the pressure and (T ij ) with i = j as the shear stress. In particular, we will study here the so-called canonical EMT whose components T µν can represent the conserved current densities which are associated (by virtue of Noether's first theorem) to the invariance of the action under spacetime translations. The corresponding conserved charges P ν ≡ R n−1 d n−1 x T 0ν can define the total energy and momentum of the physical system.As is well known (for instance for Maxwell's theory), the tensor T µν can is neither symmetric nor gauge invariant in general and thereby needs to be "improved". This is traditionally realized by the "symmetrization procedure of Belinfante" [13,14] which relies on the spin angular momentum density, but this method does not work straightforwardly in the case where matter fields are minimally coupled to a gauge field [15]. After a short introduction to the subject and problematics in subsection 2.1, we will show in subsection 2.2 that the improvement can be realized in a simple manner for pure gauge fields or for interacting gauge and matter fields by taking into account the local gauge invariance 1 . A conceptually quite different approach consists of coupling the gauge and matter fields to gravity and deducing the so-called Einstein-Hilbert EMT in Minkowski space from the metric EMT on curved space. This approach is outlined for YM theories in section 3 and it is readily shown that the different results in Minkowski space coincide with each other. Concerning the latter point we should mention that more general and abstract approaches have been considered in the literature (we refer in particular to the systematic study [17] based on the earlier work [18]), but we hope that the elementary discussion of the different aspects presented here is useful both as a short introduction to, and as an overview of, the subject. While our paper is devoted to the classical theory, we conclude with some remarks...

Field Theory with Coordinate Dependent Noncommutativity

“…For real-valued fields φ we thus recover the well-known result which has been obtained by other arguments in the literature [83]. In the commutative limit, the expression T free,θ=const reduces to the familiar result from Minkowski space [20].…”

“…The local conservation law ∂ µ j µ = 0 (which holds for any solution of the equation of motion) now expresses the conservation of electric charge. In gauge field theories, multiplication of the functional derivative δS/δA µ with the field strength tensor F µν readily yields the physical, gauge invariant EMT T µν of the gauge field (A µ ) whereas the standard Noether procedure leads to the canonical, non-gauge invariant EMT T µν can (which has to be improved by the Belinfante or other procedures [20]).…”

“…With the description of gravity in mind, the formulation of noncommutative field theories (and in particular of gauge theories) on generic symplectic manifolds with curvature and/or torsion has been addressed by various authors using diverse approaches, e.g., see [3,4,5,11,12,24,28,29,31,32,33,43,44,46,51,59,63,64,81,94,104,105,106] as well as [86,95] for some nice introductions and overviews of the literature up to the year 2010. In relationship with the main subject of the present work (in particular the conservation laws for field theories on flat noncommutative space-time) we note that it should also be possible to obtain the energymomentum tensor (EMT) of matter fields in flat space-time by coupling these fields to a metric tensor field: the EMT is then given by the flat space limit of the curved space EMT defined as the variational derivative of the matter field action with respect to the metric tensor (see [20] and references therein for a justification of this procedure). Here we outline the approach to curved noncommutative space which was recently put forward by M. Dobrski [44] who discussed the case of pure gravity following a series of related works by the same author, notably [43]: this formulation appears to fit nicely with the one that we considered here for flat noncommutative space-time.…”

Forces inside hadrons: Pressure, surface tension, mechanical radius, and all that

“…Form factors of the non-symmetric canonical EMT were discussed in [49]. For a discussion of classical aspects of the EMT we refer to [50,51]. Let us remark that if the symmetries of QCD, parity and time reversal, are relaxed, then more form factors are possible.…”

Anomaly of a gauge theory under rescaling of the fields

“…(8) looks complicated and unhelpful. However in [8] a more amenable version for a general gauge theory was introduced in the form:…”