Some Remarks on Relational Nature of Gauge Symmetry
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Some remarks on relational nature of gauge symmetry M. M. Amaral∗ UERJ − Universidade do Estado do Rio de Janeiro Instituto de F´ısica − Departamento de F´ısica Te´orica Rua S~aoFrancisco Xavier 524, 20550-013 Maracan~a,Rio de Janeiro, RJ, Brasil. August 29, 2014 Abstract We review the gauge theories on the relational point of view. With this new insight we approach the Yang Mills theories and the problem of conﬁnement. We point out that we can intuit the relational gauge nature from the Gribov mechanism in the study of conﬁnement in strong coupled gauge theories. ∗[email protected] 1 Introduction In contemporary physics the understanding of physical systems in terms of relations between their funda- mental entities has been addressed in diﬀerent contexts. An important example comes from the founda- tions of quantum mechanics, the relational quantum mechanics [1] that defends the thesis of the rejection of absolute state of a system in favor of a weaker notion of state and values of physical quantities related to something else. Another is to try to understand or interpret the concept of gauge in relational terms [2] (and see [3] for some discussions), by reviewing the usual interpretation of gauge as mathematical redundancy. This discussion appears in the case of electromagnetism as a view of thinking the Maxwell potential, a variable non-gauge-invariant, as more fundamental and not its derivative. See [4] to an approach of electromagnetism in these terms. The quantum and the gauge, two important foundations of our most successful theories. It is relevant that they point in the same direction, an even greater foundation at the core of nature, the relations itself should be more important than the individuals themselves. We will concentrate on the gauge that has electromagnetism as the great paradigm, and above references give the tone of the approach we want to address the relational term here. Gauge symmetries are omnipresent and fundamental in contemporary theoretical physics because all four fundamental interactions, electromagnetism, the strong and weak nuclear forces, gravitation through general relativity, and several models of uniﬁcation, such as string theories are described by theories of this type. Successful examples of gauge theories are quantum electrodynamics (QED) and quantum chromodynamics (QCD) [5] that are part of the Standard Model of particle physics. In light of this we ask ourselves about the true nature of what the gauge is, and we seek to exploit fully its consequences. We also emphasize that interesting philosophical discussions have recently appeared in the literature, see [2] and references therein. In this paper we begin by reviewing an example of dynamic of a classical gauge system shown in Rovelli's paper \Why Gauge?" (RWG) [2], recalling their interpretation and getting new insight into the nature of the gauge, and we use these ideas to address the QED and Yang Mills theories (YM) [5], and in particular the QCD. Then we will analyze some aspects of quantization of such gauge theories in this point of view. Finally we discuss relational nature of gauge symmetry in conﬁned phase of QCD. The problem of conﬁnement of quarks and gluons [6] is a challenging issue, which is relevant in the context of general investigations of strongly coupled gauge theories. From the an analytical perspective, a possible approach to the problem of conﬁnement in YM theories, comes from the analysis of the Gribov copies [7]. This is known as Gribov problem, and the model of Gribov{Zwanziger (GZ) [8, 9] has an important role. We will interpret the Gribov problem (or more generally, Gribov mechanism) in these relational terms, ie, we will see that gauge in the strongly coupled gauge theories (which naturally suggests more interaction, coupling and relation) reveals a very particular structure of our world, one relational structure between physical quantities in the sense that each individual or system can be thought of or described in terms of their relations with other individuals or systems and that this type of relation is more fundamental. Indicate this relational structure and pointing out that it can also be obtained and generalized from the Gribov mechanism, we consider the step beyond the considerations of the RWG. 2 Notes on relational classical gauge Let's review a simple prototype of a gauge system, used in the RWG, to help to visualize the relational character involving gauge theories. We will see that this dynamic system already shows the relational 1 feature that later we will argue that is present in the successful gauge theories QED and QCD, ie, general property that has to do with gauge itself. Consider then a system S1 formed by a set of N variables xn(t) (with n = 1:::N) whose dynamics is governed by the Lagrangian: N−1 1 X L = (_x − x_ )2 (1) 1 2 n+1 n n=1 and the equations of motion are invariant under the gauge transformations 0 xn ! xn = xn + λ (2) for an arbitrary function λ(t). The evolution of xn(t) is undetermined by the equations of motion. A complete set of gauge invariant quantities is given by an = xn+1 − xn (3) with n = 1:::N − 1. And similarly consider a system S2 consisting by a set of M variables yn(t) (com n = 1:::M) whose dynamics is governed by the Lagrangian: M−1 1 X L = (_y − y_ )2 (4) 2 2 n+1 n n=1 and the equations of motion are invariant under the gauge transformations 0 0 yn ! yn = yn + λ (5) 0 for an arbitrary function λ (t). As before, the evolution of yn(t) is undetermined by the equations of motion. A complete set of gauge invariant quantities is given by bn = yn+1 − yn (6) with n = 1:::N − 1. Now, consider the coupled system deﬁned by the variables xn(t), yn(t), and the Lagrangian L = L1 + L2 + L12 (7) where 1 L = (_y − x_ )2 (8) 12 2 1 N The coupled system is invariant under the gauge transformations (2) and (5) if λ0(t) = λ(t): Its observable gauge invariant are more than those of the individual systems: they are given by the gauge invariant observables of the ﬁrst system, the observables of the second, plus a new one: c = y1=xN (9) which depends on gauge variables of both systems. An important conclusion here (of RWG) is that gauge non invariant variables store potential of relation with other systems, like a measurement apparatus for example. In this sense it is a \partial observable" [21]. 2 In addition to the original considerations of the paper in question we can observe and intuit from this example that the essential information of a system of gauge, S1 for example, can be taken from some kind of sum of interrelations of the system itself (an) more the information coming from the coupling with one or more other systems, in this case, the gauge invariant observable c that captures the information of the two coupling systems. Thus we can anticipate a conclusion that can also be obtained from the main gauge theories, which we will see below, that these results may be interpreted heuristically for a gauge system, S1, as: X S1 = (relations(internal + whole)) : (10) That is, for a system Si, we can represent in the following way: X Si = SiOijSj ; (11) j where Oij is an object that relates the Si, Sj. Let's now do the same study for QED, the great paradigm of a gauge theory. Our system S1 now involves the gauge ﬁeld Aµ (photon) with 1 L = − F µνF ; (12) 1 4 µν where Fµν = @µAν − @νAµ : (13) Fµν and then L1 are invariant under the gauge transformations (symmetries): 0 Aµ ! Aµ = Aµ + @µα (14) The system S2 in QED involves the Dirac ﬁeld for the electron or positron ( ; ¯) and: µ L2 = i ¯γ @µ : (15) with gauge invariance: 0 ! 0 = eiα : (16) The coupled system is: L = L1 + L2 + L12 (17) where µ L12 = −e ¯γµA (18) The coupled system is invariant under the gauge transformations (14) and (16) if α0(x) = −eα(x): This is an abelian gauge symmetry with the gauge group U(1), therefore the QED is a U(1) gauge theory. Note that the interaction term L12 is essential for the full description of the system S1 and in the framework of quantum ﬁeld theories it is calculated by the well known tools of perturbation theory [5]. That is, in a gauge system of the type S1, the relational information with the whole, in the case the system S2 with which it is coupled, coming through perturbative corrections where the gauge non invariant variable plays important role in according to interpretation (10). We will address the issue of perturbative expansion in quantum version in the next section. 3 We now generalize the analogy to nonabelian gauge theories [5]. We have then SU(N) gauge theories. In this case we have richer theory with more subsystems that previous cases. Now considering only the a bosonic term of QED we have more than one system involved, the gauge ﬁeld is now Aµ (in the adjoint 2 a representation of SU(N)), with a = 1 :::N − 1. Thus the analogue to (12) is the set of systems Aµ and their interactions, described by the Yang Mills Lagrangian: 1 L = − F aµνF a ; (19) YM 4 µν with a a a abc b c Fµν = @µAν − @νAµ + gf AµAν ; (20) and [T a;T b] = if abcT c ; (21) where the matrices T a are called generators of the group deﬁned in the appropriate representation, the constants of proportionality f abc are the structure constants and g is the coupling constant.