Variational realization of completed relations

Convention 7.3 (Two regimes for $r$, and the correct chain rule).

By Convention 2.11, exactly one of the following applies to each component $r_\alpha$, and the model declares which.

(a) $r_\alpha$ is a functional of the fields. Paper 3 permits an arbitrary index set $A$, and nothing in Sections 5–6 requires more. A differentiable action, however, needs additional analytic structure, and we impose it only where it is used. We therefore require that $S_{\mathrm{rel}}$ depend on the relation coordinates through a distinguished dynamical subfamily $$A_{\mathrm{dyn}}\subseteq A, \qquad A_{\mathrm{dyn}} \text{ finite or countable},\tag{66}$$ so that $S_{\mathrm{rel}}=S_{\mathrm{rel}}[g,A,\Phi;(r_\alpha)_{\alpha\in A_{\mathrm{dyn}}}]$ and $\partial S_{\mathrm{rel}}/\partial r_\alpha=0$ for $\alpha\notin A_{\mathrm{dyn}}$. When $A_{\mathrm{dyn}}$ is countably infinite we require in addition that the derivative series in (68) below converge unconditionally in the distributional-dual topology of the relevant space of kernels; when $A_{\mathrm{dyn}}$ is finite this is vacuous. The remaining relation coordinates, indexed by $A\setminus A_{\mathrm{dyn}}$, continue to serve as completion coordinates in the sense of Sections 5–6; they simply do not enter the action.

Now suppose $r_\alpha=r_\alpha[g,A,\Phi]$, $\alpha\in A_{\mathrm{dyn}}$, takes values in a Banach (or finite-dimensional) space $V_\alpha$ and is Fréchet differentiable on the declared domain. Then $r_\alpha$ is in general nonlocal: a Wilson observable integrates around a loop, an integral invariant integrates over $M$, a spectral quantity depends on the fields through a global eigenvalue problem. Its variation is accordingly given not by a pointwise partial derivative but by the pairing of distributional kernels with the variations: $$\delta r_\alpha =\int_M\!\Bigl[ \frac{\delta r_\alpha}{\delta g^{\mu\nu}(y)}\,\delta g^{\mu\nu}(y) +\Bigl\langle \frac{\delta r_\alpha}{\delta A_\mu(y)},\,\delta A_\mu(y)\Bigr\rangle +\Bigl\langle\!\Bigl\langle \frac{\delta r_\alpha}{\delta \Phi(y)},\,\delta\Phi(y) \Bigr\rangle\!\Bigr\rangle \Bigr]\sqrt{-g(y)}\;\mathrm{d}^4y ,\tag{67}$$ where $\delta r_\alpha/\delta g^{\mu\nu}(y)$ etc. denote the $V_\alpha$-valued distributional kernels representing the derivative, $\langle\cdot,\cdot\rangle$ is the $\operatorname{Ad}$-invariant pairing on $\mathfrak{g}$, and $\langle\!\langle\cdot,\cdot\rangle\!\rangle$ is the duality pairing between $W^{*}$ and $W$ discussed in Section 7.3. (For a Wilson loop $r_\alpha$ the kernel $\delta r_\alpha/\delta A_\mu(y)$ is supported on the loop and is a distribution, not a function; (67) is exactly the statement that must be used, and a pointwise $\partial r_\alpha/\partial A_\mu(y)$ does not exist.)

Then, writing $\partial S_{\mathrm{rel}}/\partial r_\alpha\in V_\alpha^{*}$ for the derivative of $S_{\mathrm{rel}}$ with respect to its $V_\alpha$-argument (a bounded linear functional, and an ordinary partial derivative only when $V_\alpha$ is finite-dimensional), the chain rule reads $$\frac{\delta S_{\mathrm{rel}}}{\delta g^{\mu\nu}(y)} =\left.\frac{\delta S_{\mathrm{rel}}}{\delta g^{\mu\nu}(y)}\right|_{r\;\mathrm{fixed}} +\sum_{\alpha\in A_{\mathrm{dyn}}} \Bigl\langle \frac{\partial S_{\mathrm{rel}}}{\partial r_\alpha},\; \frac{\delta r_\alpha}{\delta g^{\mu\nu}(y)}\Bigr\rangle_{V_\alpha^*\!,V_\alpha} ,\tag{68}$$ and identically with $g^{\mu\nu}(y)$ replaced by $A_\mu(y)$ or by the matter components. The sum is over the dynamical subfamily (66), and converges by the unconditional-convergence hypothesis imposed there. Only in the special case where $V_\alpha=\mathbb{R}$ and $r_\alpha$ is a local scalar field does (68) reduce to an ordinary pointwise partial derivative; in general it does not.

(b) $r_\alpha$ labels a fixed topological or boundary sector. Then $\delta r_\alpha=0$ under all variations within the sector, this is what “fixed sector” means, and $r_\alpha$ enters the field equations only as a parameter. The second term of (68) is absent.

In neither case is $r_\alpha$ varied as an independent dynamical field. A model wishing to promote $r_\alpha$ to a field must adjoin it to the configuration tuple, enlarge $\mathcal{C}_{\mathrm{geo}}$, and supply its own kinetic term; that is a different model, and its Euler–Lagrange equation is an additional equation not derived here.

Convention 7.4 (Matter variation and the dual pairing).

The representation $\rho\colon G\to\mathrm{GL}(W)$ is not assumed unitary (Convention 2.1), and $W$ may be complex. Accordingly the matter Euler operator is valued in the dual, and the variational pairing must be specified. We adopt the following, and use it consistently.

(i) Real independent components. Suppose $\Phi$ denotes a collection of real independent field components. This is always achievable: decompose a complex field into its real and imaginary parts, or a section of a complex bundle into its underlying real bundle. Then $\delta S/\delta\Phi$ is valued in the real dual bundle $P\times_{\rho^{*}}W^{*}$, and the pairing $$\langle\!\langle\cdot,\cdot\rangle\!\rangle\colon W^{*}\times W\longrightarrow\mathbb{R}$$ is the canonical duality pairing. This pairing is $G$-invariant for the contragredient action $\rho^{*}(h)=\rho(h^{-1})^{\mathsf{T}}$, and requires no metric on $W$. There is then a single Euler equation $\mathcal{E}_\Phi=0$.

(ii) Complex fields treated holomorphically. If instead $\Phi$ is treated as a complex field with $\Phi$ and $\bar\Phi$ varied independently, the usual convention for a charged scalar or a Dirac spinor, then there are two Euler equations, $$\mathcal{E}_{\Phi}:=\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta \bar\Phi}=0 \qquad\text{and}\qquad \overline{\mathcal{E}_{\Phi}}:=\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta \Phi}=0 ,\tag{69}$$ which are conjugate to one another when the action is real, and the pairing used in the variation is the sesquilinear one furnished by a $\rho$-invariant Hermitian form on $W$, which exists precisely when $\rho$ is unitary. If $\rho$ is not unitary, no such Hermitian form is available and one must revert to convention (i).

For definiteness, and to avoid carrying two conjugate equations through every formula, in the statements below $\Phi$ denotes real independent field components in the sense of (i), and $\mathcal{E}_\Phi$ is the corresponding single dual-valued Euler operator. Every result restates verbatim in convention (ii) upon replacing $\mathcal{E}_{\Phi}=0$ by the conjugate pair (69) and $\langle\!\langle\cdot,\cdot\rangle\!\rangle$ by the invariant Hermitian pairing.