A primary field is defined by its behavior under a conformal transformation x→x′(x): ϕ(x)→ϕ′(x′)=|∂x′∂x|−hϕ(x)
It's fairly easy to see that the gradient of the field doesn't have this nice property under the same transformation, it gets a non homogeneous term. Still, is it possible to construct a derivative that would behave nicely under conformal mappings and give the usual derivative for Lorentz transformations? By adding a "connection" similarly as what is done in general relativity or gauge theories. And if not, why?
I) Here we discuss the problem of defining a connection on a conformal manifold M. We start with a conformal class [gμν] of globally1 defined metrics
g′μν = Ω2gμν
given by Weyl transformations/rescalings. Under mild assumption about the manifold M (para-compactness), we may assume that there exists a conformal class [Aμ] of globally defined co-vectors/one-forms connected via Weyl transformations as
A′μ = Aμ+∂μln(Ω2).
In particular it is implicitly understood that a Weyl transformation [of a pair (gμν,Aμ) of representatives] act in tandem/is synchronized with the same globally defined function Ω in eqs. (1) and (2) simultaneously.
II) Besides Weyl transformations, we can act (in the active picture) with diffeomorphisms. Locally, in the passive picture, the pair (gμν,Aμ) transforms as covariant tensors
gμν = ∂x′ρ∂xμg′ρσ∂x′σ∂xν,
Aμ = ∂x′ν∂xμA′ν.
under general coordinate transformations
xμ ⟶ x′ν = fν(x).
III) We next introduce the unique torsionfree tangent-space Weyl connection ∇ with corresponding Christoffel symbols Γλμν that covariantly preserves the metric in the following sense:
(∇λ−Aλ)gμν = 0.
The Weyl connection ∇ and its Christoffel symbols Γλμν are independent of the pair (gμν,Aμ) of representatives within the conformal class [(gμν,Aμ)]. (But the construction depends of course on the conformal class [(gμν,Aμ)].) In other words, the Weyl Christoffel symbols are invariant under Weyl transformations
Γ′λμν = Γλμν.
The lowered Weyl Christoffel symbols are uniquely given by
Γλ,μν = gλρΓρμν = 12((∂μ−Aμ)gνλ+(∂ν−Aν)gμλ−(∂λ−Aλ)gμν) = Γ(g)λ,μν+12(Aμgνλ−Aνgμλ+Aλgμν),
where Γ(g)λ,μν denote the lowered Levi-Civita Christoffel symbols for the representative gμν. The lowered Weyl Christoffel symbols Γλ,μν scale under Weyl transformations as
Γ′λ,μν = Ω2Γλ,μν.
The corresponding determinant bundle has a Weyl connection given by
Γλ = Γμλμ = (∂λ−Aλ)lndet
IV) Let us next define a conformal class [\rho] of a density \rho of weights (w,h), who scales under Weyl transformations as
\tag{11} \rho^{\prime}~=~ \Omega^w\rho
with Weyl weight w, and as a density
\tag{12} \rho^{\prime}~=~\frac{\rho}{J^h}
of weight h under general coordinate transformations (5). Here
\tag{13} J ~:=~\det(\frac{\partial x^{\prime \nu}}{\partial x^{\mu}})
is the Jacobian.
Example: The determinant \det(g_{\mu\nu}) is a density with h=2 and w=2d, where d is the dimension of the manifold M.
V) The concept of (conformal classes of) densities \rho of weights (w,h) can be generalized to (conformal classes of) tensor densities T^{\mu_1\ldots\mu_m}_{\nu_1\ldots\nu_n} of weights (w,h) in a straightforward manner. For instance, a vector density of weights (w,h) transforms as
\tag{14} \xi^{\prime \mu}~=~ \frac{1}{J^h}\frac{\partial x^{\prime \mu}}{\partial x^{\nu}} \xi^{\nu}
under general coordinate transformations (5), and scales as
\tag{15} \xi^{\prime \mu}~=~\Omega^w \xi^{\mu}
under Weyl transformations. Similarly, a co-vector density of weights (w,h) transforms as
\tag{16} \eta^{\prime}_{\mu}~=~ \frac{1}{J^h}\frac{\partial x^{\nu}}{\partial x^{\prime \mu}} \eta_{\nu}
under general coordinate transformations (5), and scales as
\tag{17} \eta^{\prime}_{\mu}~=~\Omega^w \eta_{\mu}
under Weyl transformations. And so forth for arbitrary tensor densities T^{\mu_1\ldots\mu_m}_{\nu_1\ldots\nu_n}.
Example: The metric g_{\mu\nu} is a tensor density with h=0 and w=2. The one-form A_{\mu} is not a tensor density, cf. eq. (2).
VI) Finally, one can discuss the definition of covariantly conserved (conformal classes of) tensor densities T^{\mu_1\ldots\mu_m}_{\nu_1\ldots\nu_n}. A density \rho of weights (w,h) is covariantly conserved if
\tag{18} (\nabla_{\lambda}-\frac{w}{2}A_{\lambda})\rho~\equiv~ (\partial_{\lambda}-h \Gamma_{\lambda}-\frac{w}{2}A_{\lambda})\rho~=~0.
A vector density of weights (w,h) is covariantly conserved if
\tag{19} (\nabla_{\lambda}-\frac{w}{2}A_{\lambda})\xi^{\mu}~\equiv~ (\partial_{\lambda}-h \Gamma_{\lambda}-\frac{w}{2}A_{\lambda})\xi^{\mu}+\Gamma_{\lambda\nu}^{\mu}\xi^{\nu} ~=~0.
A co-vector density of weights (w,h) is covariantly conserved if
\tag{20}(\nabla_{\lambda}-\frac{w}{2}A_{\lambda})\eta_{\mu}~\equiv~ (\partial_{\lambda}-h \Gamma_{\lambda}-\frac{w}{2}A_{\lambda})\eta_{\mu}-\Gamma_{\lambda\mu}^{\nu}\eta_{\nu} ~=~0.
In particular, if T^{\mu_1\ldots\mu_m}_{\nu_1\ldots\nu_n} is a tensor density of weights (w,h), then the covariant derivative (\nabla_{\lambda}-\frac{w}{2}A_{\lambda})T^{\mu_1\ldots\mu_m}_{\nu_1\ldots\nu_n} is also a tensor density of weights (w,h).
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^{1} We ignore for simplicity the concept of locally defined conformal classes.
