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Non-abelian theories and generalized Dirac variables
Within the literature of Quantum ChromoDynamics, hadronic Physics and more generally in the context of non-abelian gauge theories, there have been attempts to generalize the construction of Dirac. Not surpris-ingly, we nd instances of (2.6) which are then naturally called generalized Dirac variables. In (Lavelle and McMullan, 1997) e.g , which touches upon the problem of de ning gauge invariant coloured states for the quarks in QCD and combining them in hadronic bound states, the term generalized Dirac vari-able is not used but the will to extend his idea is well assumed. Moreover by an interesting terminological coincidence, if ψ is a quark, the gauge-invariant quantity de ned by ψphys = h 1ψ (equation [5.2] of their paper) is called a “dressed quark”. And the eld h transforming according to hU = U 1h, for U 2 SU (3) (equation [5.1]), is called it a “dressing”. The composite eld, or generalized Dirac variable, associated to the SU (3)-gauge eld A is de ned as, (Aphys)i = h 1Aih + h 1@ih (equation [5.5]), with i a spatial index. Notice how gauge-invariance and observability are tied (if not identi ed) by the subscript “phys” used to denote the composite elds. In accordance with Dirac’s remark cited above, and by virtue of its explicit construction as a (still non-local) function of the gauge potential, the dressing eld h is interpreted as a gluon sea surrounding the bare quark ψ and the bare gluon A. This is announced from the introduction of the paper: “One views dressings as surrounding the charged particles with a cloud of gauge eld”. The aim of the authors is then stressed: “[…] such dressed quarks may be combined to form colourless hadrons in the way commonly done in the constituent quark model. […] gluons may also be dressed”. A most salient point of this paper is the will to draw a link between “dressings” and gauge xing. Never-theless, on account of our closing discussion in 2.2.1, we are bound to dispute this will. However the related nice suggestion of a connection between Gribov ambiguity and quark con nement might be left untouched. A discussion of this interpretive issue is proposed in appendix A.1.
The paper (Lorcé, 2013b), which reviews and addresses the problem of the proton spin decomposition in terms of gauge-invariant contributions with clear partonic interpretation, is an example of an approach whose link with generalized Dirac variables went at rst unnoticed. Though, a careful analysis shows that the rst part of the paper can be entirely founded on the dressing eld method. Nevertheless the construction presented there is interpreted as a speci c gauge tranformation, a viewpoint that we must again dispute on account of 2.2.1. I propose the full analysis in appendix A.2. The authors of this paper and of (Fournel et al., 2014) came to a short correspondence (iniated by the former) on the contact points of both works. As a mark of fruitful exchanges, the dressing eld and the com-posite elds/Dirac variables are more clearly identi ed in the subsequent paper (Lorcé, 2013a). In there, the transformation law of the matrix valued eld Upure, UHpure (x ) = U 1 (x )Upure (x ) (slight correction of equation [22] of Lorcé’s paper), identi es it with a dressing eld. Moreover the elds ϕ (x ) = Upure1 (x )ϕ (x ) (equation [44]) and Aµ (x ) = Upure1 (x ) Aµ (x ) + i @µ Upure (x ) (equation [45]) are instancesDof (2.6).
The remarks made by thef author aregrelevant : “[…] despite appearances, eqs [44] and [45] are not gauge transformations. In practive, the matrices Upure (x ) can be expressed in terms of the gauge elds Aµ (x ) (Lorcé, 2013c), and can be thougth of as dressing elds”. In view of this clear statement the next sentence is quite surprising: “From a geometrical point of view Upure (x ) simply determines a reference con guration in the internal space. The gauge-invariant eld ϕ (x ) then represents “physical” deviations from this reference con-equations [44] and [45] are not gauge transformations, as we plainly guration”. We have to object to this. If agree on, then the dressing Upure does not belong to the gauge group and does not determine a point (“ref-erence con guration”, or “abstract reference frame” as we termed it) in the ber (“internal space”) of the underlying bundle. The transformation of the Lagrangian is considered in the last part of the paper and we nd the key words: “[…] one can switch between gauge-covariant and invariant canonical formalism by a mere change of variables”. All this is repeated and concisely synthesized, with the same misinterpretation though, on p52-53 of the extensive review (Leader and Lorcé, 2014) on the problem of the proton spin decomposition. We should notice two important facts. First, while in (Dirac, 1955), (Lavelle and McMullan, 1997) and (Lorcé, 2013b) the dressing elds are constructed as non-local functions of the gauge potential A, this was not the case in the two simple toy models presented above where the dressings were local functions of an auxiliary eld. But this is not an artefact of over-simplistic models. From now on we will see physically substantial examples (to say the least) where the dressing is not a function of the gauge potential or, if it is, it is still local.
Secondly, in the aforementioned works the question of the loss of the manifest Lorentz covariance of the composite elds is of constant worry and much energy is deployed in order to settle this question in each speci c construction. Notice that due to our di erential geometric, thus intrinsic, formulation of the dressing method, the question of the Lorentz covariance and even of the general covariance of our composite elds never arises. Once again, from now on the examples we consider are free of such concerns.
One last remark. All constructions above were instances of a dressing with value in H , the full struc-ture group, therefore instances of complete neutralization of gauge symmetry and full geometrization. The composite elds/generalized Dirac variables had no residual gauge freedom and belonged to the natural ge-ometry of the space-time base manifold M . An interesting case of partial neutralization, thus of residual gauge freedom, is presented below. A more complicated illustration is to be worked out in the next section. For the moment let us go to our two most relevant examples, the Electroweak sector of the Standard Model and General Relativity.
Dressing fields in the Cartan-Möbius geometry
As far as Physics is concerned, one ought to be interested in conformal geometry since conformal transforma-tions of a Lorentz manifold preserve the class of null geodesics. In other words the causal structure of space-time is preserved by conformal transformations. As a consequence, conformal symmetry is a fundamental symmetry of any relativistic theory of massless elds: Maxwell’s theory, General Relativity in vacuum and both of these interactions coupled to each other and/or to massless scalar or fermion elds give conformally invariant theories. In high energy particle physics, that is in accelerators, in violent astrophysical phenomena or in early cosmology, all situations where the masses of the particles are negligible compared to their total energy, the conformal symmetry may be considered as an approximate symmetry.
From the viewpoint of Mathematics, conformal geometry is now classically approached through the jet formalism. It also has a natural Cartan geometric structure. Indeed the bundle of any Cartan geometry is a reduction of an adequate higher-order structure, and is studied from both perspectives in (Ogiue, 1967) and (Kobayashi, 1972).20 Nevertheless, the jet formalism is intricate to use. The mere jet multiplication, which is the partial derivative of the composition of functions, is already complex at second order and becomes formidable at higher order. Moreover the jets usually do not allow torsion. So, following (Sharpe, 1996) we will use a matrix formalism which allows to focus on the Cartan geometric aspect, allows torsion, is more handy and better suited to apply the dressing eld method. We will occasionally stress the equivalence between the two formalisms at chosen moments, by reference to the mentioned classic papers on the jet approach. Conformal geometry seen as Cartan geometry has been discussed in 1.2, the exposure followed (Sharpe, 1996).
The symmetry of the gauge fixed e ective Lagrangian
As we’ve seen several times, the gauge symmetry poses a problem for anyone who wants to apply the path integral algorithm for quantization. We’ve also mentionned the fact that an obvious move is to select a single representative in each gauge orbit by gauge xing. This was the celebrated contribution of (Faddeev and Popov, 1967) to nd a clever general way to do so. There is no need here to be involved in the technical details so I just sketch the idea, following (Bertlmann, 1996). The path integral Z = A F dAdφ eiS (A; φ ) , where S (A;φ) = M LYM + Lmatter is the action of the classical γ symmetry. A gauge xing condition χ(A )=0 gauge eld theory, is ill-deR ned. It diverges due to the gauge δ R(Aγ) δχ(Aγ) is then chosen and is inserted in Z as the identity R dγ det χ δ χ(Aγ) = 1, where det is δ γ δ γ the Faddeev-Popov determinant. Then the gauge transformation γ 0 = γ 1 is performed on Z and due to the invariance of the measure and of S (A;φ) it gives,1 Z = ZA F dγdAdφ det δγ δ χ (A) eiS (A;φ ) : δχ(A).
The integrand does not explicitly depend on the gauge group element γ so integration over it factorizes and produces and in nite normalization that can be conventionally removed. The Faddeev-Popov determinant is shown to be the determinant of a di erential operator (depending on the chosen gauge xing condition χ ) and the next step is to express it a Gaussian integral over the g-valued Grassmann variables v and v¯, the so-called Faddeev-Popov ghost and antighost, so that nally one obtains, Z = Z( A F ) =G dAdφdvdv¯ eiS (A;φ;v;v¯) ; with S (A;φ;v;v¯) = ZM Le (A;φ;v;v¯).
Table of contents :
Introduction
1 Geometry of fundamental interactions
1.1 The geometry of gauge elds
1.1.1 The basics of bundle geometry
1.1.2 Connection & Curvature
1.1.3 Lagrangian
1.2 Cartan geometry and Cartan connections: a language for Gravitation
1.2.1 A language for classical gravitation
1.2.2 Global denition of a Cartan geometry
1.2.3 Reductive geometries and gravity
1.3 Downsides of gauge symmetry
1.3.1 Problem with gauge theories?
1.3.2 What standard solutions?
1.3.3 A new approach
2 The dressing eld
2.1 Gauge theories: a recipe
2.1.1 What is a gauge theory?
2.1.2 Examples
2.2 Dressing eld
2.2.1 Easy propositions
2.2.2 Dressing eld and residual gauge freedom
2.3 Applications to Physics
2.3.1 Abelian theories and Dirac variables
2.3.2 Non-abelian theories and generalized Dirac variables
2.4 Application to geometry
2.4.1 The dressing and higher-order G-structures
2.4.2 Dressing elds in the Cartan-Möbius geometry
3 Dressing eld and BRS formalism
3.1 The BRS approach, an outline
3.1.1 The symmetry of the gauge xed eective Lagrangian
3.1.2 Cohomological viewpoint
3.2 The dressing eld method in the BRS formalism
3.2.1 Modifying the BRS algebra
3.2.2 Reduced BRS algebra and higher-order G-structures
3.3 Applications
3.3.1 General Relativity
3.3.2 Cartan-Möbius geometry
3.4 Extended BRS algebra: innitesimal dieomorphisms
3.4.1 Translations and local dieomorphisms
3.4.2 Examples
4 Dressing eld and anomalies
4.1 Anomalies in Quantum Field Theory
4.1.1 What is an anomaly?
4.1.2 BRS characterization of anomalies
4.1.3 The Stora-Zumino chain of descent equations and the consistent anomaly
4.2 The place of the dressing eld
4.2.1 Four possibilities
4.2.2 Cases (I) and (I’)
4.2.3 Cases (II) and (II’)
4.3 An attempt toward the Weyl anomaly
4.3.1 The Weyl anomaly
4.3.2 A candidate polynomial
4.3.3 The dressing eld method and the Weyl anomaly
Conclusion
A Papers analysis
A.1 A paper by M. Lavelle & D. McMullan
A.2 A paper by C. Lorcé
A.3 Papers by N. Boulanger
A.3.1 First paper
A.3.2 Second paper
B A minimal dressing eld for General Relativity
C Some Detailed Calculations
C.1 Calculation of the entries of 0
C.2 The remaining symmetries of the nal composite elds
C.2.1 Coordinate changes for $0 and 0
C.2.2 Residual Weyl gauge symmetry of $0 and 0
D Published papers/ in preparation
D.1 Gauge invariant composite elds out of connections with examples
D.2 Gauge eld theories: various mathematical approaches
D.3 Weyl residual gauge freedom out of conformal geometry, with a new BRS tool
D.4 Nucleon spin decomposition and dierential geometry
