mathematical physics - Is there a "covariant derivative" for conformal transformation?

A primary field is defined by its behavior under a conformal transformation $x\rightarrow x'(x)$: $$\phi(x)\rightarrow\phi'(x')=\left|\frac{\partial x'}{\partial x}\right|^{-h}\phi(x)$$

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?

Answer

I) Here we discuss the problem of defining a connection on a conformal manifold $M$. We start with a conformal class $[g_{\mu\nu}]$ of globally$^{1}$ defined metrics

$$\tag{1} g^{\prime}_{\mu\nu}~=~\Omega^2 g_{\mu\nu}$$

given by Weyl transformations/rescalings. Under mild assumption about the manifold $M$ (para-compactness), we may assume that there exists a conformal class $[A_{\mu}]$ of globally defined co-vectors/one-forms connected via Weyl transformations as

$$\tag{2} A^{\prime}_{\mu}~=~A_{\mu} + \partial_{\mu}\ln(\Omega^2).$$

In particular it is implicitly understood that a Weyl transformation [of a pair $(g_{\mu\nu},A_{\mu})$ of representatives] act in tandem/is synchronized with the same globally defined function $\Omega$ 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 $(g_{\mu\nu},A_{\mu})$ transforms as covariant tensors

$$\tag{3} g_{\mu\nu}~=~ \frac{\partial x^{\prime \rho}}{\partial x^{\mu}} g^{\prime}_{\rho\sigma}\frac{\partial x^{\prime \sigma}}{\partial x^{\nu}},$$

$$\tag{4} A_{\mu}~=~ \frac{\partial x^{\prime \nu}}{\partial x^{\mu}} A^{\prime}_{\nu}.$$

under general coordinate transformations

$$\tag{5} x^{\mu} ~\longrightarrow~ x^{\prime \nu}~= ~f^{\nu}(x).$$

III) We next introduce the unique torsionfree tangent-space Weyl connection $\nabla$ with corresponding Christoffel symbols $\Gamma^{\lambda}_{\mu\nu}$ that covariantly preserves the metric in the following sense:

$$\tag{6} (\nabla_{\lambda}-A_{\lambda})g_{\mu\nu}~=~0.$$

The Weyl connection $\nabla$ and its Christoffel symbols $\Gamma^{\lambda}_{\mu\nu}$ are independent of the pair $(g_{\mu\nu},A_{\mu})$ of representatives within the conformal class $[(g_{\mu\nu},A_{\mu})]$. (But the construction depends of course on the conformal class $[(g_{\mu\nu},A_{\mu})]$.) In other words, the Weyl Christoffel symbols are invariant under Weyl transformations

$$\tag{7} \Gamma^{\prime\lambda}_{\mu\nu}~=~\Gamma^{\lambda}_{\mu\nu}.$$

The lowered Weyl Christoffel symbols are uniquely given by

$$\Gamma_{\lambda,\mu\nu}~=~g_{\lambda\rho} \Gamma^{\rho}_{\mu\nu}$$ $$~=~\frac{1}{2}\left((\partial_{\mu}-A_{\mu})g_{\nu\lambda} +(\partial_{\nu}-A_{\nu})g_{\mu\lambda}-(\partial_{\lambda}-A_{\lambda})g_{\mu\nu} \right)$$ $$\tag{8} ~=~\Gamma^{(g)}_{\lambda,\mu\nu}+\frac{1}{2}\left(A_{\mu}g_{\nu\lambda}-A_{\nu}g_{\mu\lambda}+A_{\lambda}g_{\mu\nu} \right),$$

where $\Gamma^{(g)}_{\lambda,\mu\nu}$ denote the lowered Levi-Civita Christoffel symbols for the representative $g_{\mu\nu}$. The lowered Weyl Christoffel symbols $\Gamma_{\lambda,\mu\nu}$ scale under Weyl transformations as

$$\tag{9} \Gamma^{\prime}_{\lambda,\mu\nu}~=~\Omega^2\Gamma_{\lambda,\mu\nu}.$$

The corresponding determinant bundle has a Weyl connection given by

$$\tag{10} \Gamma_{\lambda}~=~\Gamma^{\mu}_{\lambda\mu}~=~(\partial_{\lambda}-A_{\lambda})\ln \sqrt{\det(g_{\mu\nu})}.$$

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.