Structural Aspects of Quantum Field Theory and Noncommutative Geometry

Grensing, Gerhard

doi:10.1142/8771

Cited by 12 publications

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“…After recalling the definition of the canonical EMT [15,19,22], we consider the example of pure YM theories (while referring whenever necessary to the summary of non-Abelian gauge theories given in appendix A). Thereafter, we discuss the general properties of the EMT (local conservation law, symmetry, gauge invariance, tracelessness) as well as the addition of superpotential terms to the canonical EMT: The aim of these additions is to "improve" the characteristics of the canonical EMT, so that it becomes symmetric and gauge invariant (and traceless for scale invariant Lagrangians) if it does not have these properties.…”

Section: Emt's For Gauge Theories In Minkowski Spacementioning

confidence: 99%

“…(In particular one recovers the canonical EMT if this one is already symmetric. We note that Belinfante's symmetric EMT admits a natural geometric formulation if gravity is viewed as a gauge theory of the Poincaré group, see references [15,43].) One can also recover the traceless Callan-Coleman-Jackiw EMT in the flat space limit of a conformally invariant field theory on curved space.…”

Section: Definition Of the Physical Emtmentioning

confidence: 99%

“…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 .…”

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

et al. 2016

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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...

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Section: Emt's For Gauge Theories In Minkowski Spacementioning

confidence: 99%

Section: Definition Of the Physical Emtmentioning

confidence: 99%

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confidence: 99%

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

et al. 2016

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“…If we consider the cutoff M to be proportional to the Planck mass M P l ≈ 10 18 GeV, then we have a huge discrepancy of almost 120 orders of magnitude between the vacuum energy of the quantum field theory and the observed cosmological constant ρ ob vac ≈ 10 −47 GeV 4 , leading to the "worst prediction of theoretical physics". However, this is just an estimate and no physical quantity can be cut-off dependent, since this dependance should be absorbed in the renormalized cosmological costant [83]. Since there are indications that a change in the UV behavior of the kinetic operator could drastically affect the structure of the vacuum energy [8,27,28], we will be interested in investigating κ-Poincaré invariant field theories because they are characterized by kinetic operators which exhibit significant modifications in the UV regime.…”

Section: Vacuum Energy and Cosmological Constantmentioning

confidence: 99%

“…The novelty that the noncommutativity brought to this problem is the fact that the UV behavior of the theory is changed, resulting in a different behavior, that is change in the level of divergence of the vacuum energy. This, of course, is just a starting point and one should proceed with renormalization of the vacuum energy in order to absorb the cut-off dependance into the counterterms [83]. This is an adventure of its own, since the quantum corrections are non trivial [46,47] and furthermore the counterterm should be steaming from the cosmological constant part of the gravitational action.…”

Section: Final Remarksmentioning

confidence: 99%