In that context, in [40] Dirac says of the dressed fermion field ${\psi }^{u[A]}≔u[A{]}^{-1}\psi$ that, upon quantization, it is “the operator of creation of an electron together with its Coulomb field, or possibly the operator of absorption of a positron together with its Coulomb field. It is to be contrasted with the operator $\psi$, which gives the creation or absorption of a bare particle. A theory that works entirely with gauge-invariant operators has its electrons and positrons always accompanied by Coulomb fields around them, which is very reasonable from the physical point of view.” Emphasis is his.

Kibble wrote: “It is perfectly possible to describe [the model] without ever introducing the notion of symmetry breaking, merely by writing down the Lagrangian (66) [the dressed one $L({\varphi }^u)$, Eq. (4)]. Indeed if the physical world were really described by this model, it is (66) rather than (64) [the bare Lagrangian $L(\varphi )$] to which we should be led by experiment.” Kibble was publishing just eight months before the famous paper by Weinberg “A model of leptons” [47] containing the electroweak theory.

Remark that here $u[\tilde{A}]=\overline{e}:M\to GL(4)\supset SO(1,3)$, i.e., the dressing field takes value in a group $G\supset H$, yet still has the defining $\mathcal{H}$-transformation property. This more general situation is fully accounted for by the DFM [28, 57, 58], of which a slightly simplified version was given in Section 3a. Early on, [57] section 4.3 discussed the possibility of extracting an $SO$-valued dressing field from the $GL$-valued tetrad (via an extension of the Schweinler-Wigner orthogonalization procedure [59]). See also Appendix B of [58] for details, as well as footnote 12 in Francois2018, and section 4.3.2. of [60]. Such a “minimal dressing” extraction from the tetrad is the basis of a recent quantum gravity approach by Thiemann [61], and called there the “triangular gauge”; an unfortunate choice of terminology given the clear conceptual and mathematical distinction between dressing and gauge-fixing [35].

The dressing field $(\upsilon ,u)$ can be seen as the local version of a field-dependent dressing field $\mathit{u}$ on a principal fiber bundle $P$ over $M$. The DFM procedure in this case allows to define the relational, physical ‘enriched’ spacetime as a dressed bundle JTF-Ravera2024c,BFR-fbsubst,JTF-Ravera2024gRGFT.

The origin of the latter being indeed the group ${\mathrm{Aut}}_v(P)$ of vertical automorphisms of the principal fiber bundle $P$ over $M$, whose geometry underlies the kinematics of GFT. These automorphisms of $P$ induce, by definition, the identity transformation ${\mathrm{id}}_M$ of $M$. They form a (normal) subgroup of the full group $\mathrm{Aut}(P)$ of automorphisms of $P$, which induces diffeomorphisms $\mathrm{Diff}(M)$ of $M$. The geometry of principal bundles is the unifying geometric framework for gRGFT. See [24, 28].

DeWitt [68] would say that the dressing field $\upsilon [\varphi ]$ provides what he calls “intrinsic coordinates”, whereby one may build invariants: in our language, the dressed fields ${\varphi }^{\upsilon }≔{\upsilon }^{*}\varphi$. To quote him more extensively, he argues that “what appears to be needed, if one insists on maintaining manifest covariance [we would rather say, $\mathrm{Diff}(M)$-invariance], is a mean of constructing local invariants. A possible procedure is to introduce four independent scalars ${\zeta }^A$, $A=0$, 1, 2, 3 [yielding a dressing field for $\mathrm{Diff}(M)$ as defined in Eq. (6)], formed out of the metric and its derivative [i.e., a dressing field $\upsilon [g]$] and then to use these to define an intrinsic coordinate system.” His emphasis. DeWitt cites Komar and Bergmann [64, 65] as doing just that, but noting possible difficulties with using the metric; to avoid them he continues “[...] we shall introduce directly into the discussion an additional physical system. This system will serve to furnish us with a reasonably full-proof set of intrinsic coordinates [i.e., a dressing field $\upsilon [\varphi ]$] while at the same time forming a combined physical system with the gravitational field.” The emphasis here is ours, and highlights the relational character of the physical system to be analyzed. He continues “In principle, any additional system which provides a ‘useful’ set of four scalars will do.”—see indeed Section 4b for an illustration—adding “It might be supposed that [the additional system] has merely a technical utility, constituting an otherwise foreign element in the discussion. Such is by no means the case. The role played by the medium in providing a physical coordinate system proves to be a fundamental one [...]” (our emphasis again), with which we fully agree. Echoing (perhaps unwittingly) Einstein’s point-coincidence argument described earlier, DeWitt indeed insists that “A [...] medium of some kind is needed [...] in order to give an operational meaning to the concept of ‘space-time geometry’ in the first place.” We thank a referee for pointing out this reference by DeWitt.

Remark that, seeing $G(\upsilon ):M^{\upsilon }\to GL(n)$, this transformation closely resembles that of an ‘internal’ dressing (2).

We may e.g., quote DeWitt [68] again; if “[i]n principle, any additional system which provides a ‘useful’ set of four scalars [dressing $\upsilon [\varphi ]$] will do. Actually, we shall choose the most intuitively obvious system possible, namely, a stiff elastic medium carrying a framework of clocks.” whose “physical constitution” is considered “only phenomenologically”.

The scalars could be taken to be, if not as the components of the 4-velocity of the fluid particles ${\phi }^a=u^a$ (as determined by any arbitrary frame field, i.e., a section of the frame bundle $LM$ of $M$), at least as a (sub)set of scalars in terms of which the fluid 4-velocity is expressed, $u^a=u^a(\phi )$, as is done in the so-called “velocity-potential” representations. A typical choice of effective Lagrangian is $L_{\mathrm{matter}}(g,\phi )=\rho {\mathrm{vol}}_g$, with ${\mathrm{vol}}_g$ the volume form induced by the metric field $g$ and $\rho =\rho (\phi ,\dots )$ the rest energy density of the fluid expressed as a function of the scalars $\phi$ and possibly of other thermodynamical parameters (entropy per baryon, chemical potential, etc.). The field equations for the scalars are equivalent to particle (baryon) number conservation and covariant conservation of the stress-energy tensor, $\nabla \mathsf{T}(g.\phi )=0$. Such Lagrangian description was pioneered notably by Taub [75, 76], Schutz [77], as well as Carter [78], Kijowski and Tulczyjew [79]—see also Brown [80, 81]—and is an integral part of the field of relativistic fluid dynamics and numerical relativity [82, 83]. In this field, it is also often the case that the fluid distribution is described by a set of scalars fields labelling fluid particles and called “Lagrangian coordinate fields“, or yet “comoving coordinates” (standard, unphysical, coordinates on $M$ being called “Eulerian coordinates”); these are clearly fit for our purpose (e.g., DeWitt [68] uses just this viewpoint). Velocity-potentials, or a subset thereof, are sometimes used as Lagrangian coordinates. Keen readers may detect in this literature many instances of the DFM philosophy; e.g., the reference manifold $N$ (that is $R^4$ in the case at hand), the source space of the dressing field $\upsilon$, generalizes what is variously called “material space” [79], “fluid space” [80], and “matter space” [81, 82]—or yet “fleet” (of fluid particles) [81]. Also, dressed fields ${\varphi }^{\upsilon }$ extend what Carter calls “material tensors” [78], while $g^{\upsilon }$ relates e.g., to the “matter space/fleet metric” of [81].

As stressed already above, the dressed field equations have the same functional expression as the bare ones. In components, they are related by $$\begin{gathered} {\mathsf{G}}_{ab}^{\upsilon }+\mathrm{\Lambda }{g}_{ab}^{\upsilon }-\kappa {\mathsf{T}}_{ab}^{\upsilon } \\ ={G(\upsilon {)}_a}^{\mu }({\mathsf{G}}_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }-\kappa {\mathsf{T}}_{\mu \nu }){G(\upsilon {)}^{\nu }}_b=0. \end{gathered}$$Which superficially resembles the general-covariance of Einstein’s equation, yet is conceptually distinct.