Classical complex-scalar connection derivative and action

Let $(M,g,N,o)$ be an oriented time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$ as in Oriented Lorentzian fields, observers and coordinate conventions. Set $U(1):=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ with multiplication. A smooth right principal $U(1)$ bundle is a smooth manifold $P$ with a smooth surjective submersion $\pi :P\to M$, a smooth right $U(1)$ action, and smooth local equivariant product charts ${\pi }^{-1}(U_i)\cong U_i\times U(1)$; the action is free and transitive on each fibre. This is the smooth version of the right-action convention in Principal g bundle and associated fiber bundle. A principal connection on $P$ is a smooth distribution $H\subseteq TP$ with $T_{p}P=\ker (d{\pi }_p)\oplus H_p$ and $d{R}_z(H_p)=H_{pz}$. Equivalently, it is a smooth real one-form $\omega$ on $P$ that is right-invariant and satisfies $\omega ({\zeta }_P)=1$, where ${\zeta }_P(p):={\frac{d}{dt}\mid }_{0}p\cdot e^{it}$; then $H=\ker \omega$. Choose a fixed smooth right principal $U(1)$ bundle with a principal connection $\omega$. In a smooth trivialization ${\tau }_i:P{\mathrm{\mid }}_{U_i}\to U_i\times U(1)$, set ${\sigma }_i(x):={\tau }_i^{-1}(x,1)$ and $a_i:={\sigma }_i^{\ast }\omega$. Choose the normalized local data $(U_i,g_{ij},a_i;{\kappa }_0)$ as in Local potentials and smooth circle-bundle connection data; the potentials used below are their rescalings $A_i:={\kappa }_0{a}_i$. On any overlap neighborhood admitting a smooth real phase lift $f_{ij}$, write $g_{ij}=e^{i{f}_{ij}}$. Use the dual representation ${\rho }_k(z):=z^{-k}$ and define charged scalar representatives by the global transition rule ${\phi }_j=g_{ij}^k{\phi }_i$, equivalently $e^{ik{f}_{ij}}{\phi }_i$ on each phase-lift neighborhood; the connection rule there is

$A_j=A_i+{\kappa }_{0}d{f}_{ij}\mathrm{.}$

Here ${\kappa }_0>0$ is the chosen circle normalization with $[{\kappa }_0]=\mathrm{W}\mathrm{b}$, $k\in \mathbb{Z}$ is the representation weight, and $\lambda :=k\mathrm{/}{\kappa }_0$ has units ${\mathrm{W}\mathrm{b}}^{-1}$. Because $g_{ij}{g}_{jk}=g_{ik}$ and $k$ is an integer, the maps $g_{ij}^k$ satisfy the scalar transition cocycle. A charged scalar is a family of smooth local functions ${\phi }_i$ satisfying this rule; all phase-lift calculations below are local, and no global phase lift is assumed. The constant $c>0$ is the length-to-time conversion parameter of Oriented Lorentzian fields, observers and coordinate conventions, with units $\mathrm{m}{\mathrm{s}}^{-1}$. Define $F{\mathrm{\mid }}_{U_i}:=d{A}_i$; the transition law gives $d{A}_j=d{A}_i$, so $F$ is a global curvature two-form. In a global real-potential model on a trivial bundle, one may instead take a global $A$ and any real $\lambda$; no charge quantization is imposed there.

In a local trivialization define the gauge-covariant derivative and its lower-index one-form by

$P_a:=(D\phi {)}_a:={\mathrm{\partial }}_a{\phi }_i-i\lambda A_{i,a}{\phi }_i\mathrm{.}$

The transition laws give $P_{j,a}=e^{ik{f}_{ij}}{P}_{i,a}$, since $\lambda {\kappa }_0=k$. For any charged covector $Q_b$ with the same phase transition, define the gauge- and metric-covariant derivative by $D_a{Q}_b:={\mathrm{\nabla }}_a{Q}_b-i\lambda A_a{Q}_b$ in a local trivialization; for a charged scalar $\eta$, set $D_a\eta :={\mathrm{\partial }}_a\eta -i\lambda A_a\eta$. Conjugate fields use the opposite gauge sign. Thus $D_a{P}_b$ is a well-defined charged covariant two-tensor, and $D_a{P}^a:=g^{ab}{D}_a{P}_b$. Consequently $\mathrm{\mid }\phi {\mathrm{\mid }}^2$ and $g^{ab}\overline{P_a}{P}_b$ are global scalar fields. Let $V:\mathcal{U}\to \mathbb{R}$ be smooth on an open set $\mathcal{U}\subseteq \mathbb{R}$ containing the range of $\mathrm{\mid }\phi {\mathrm{\mid }}^2$, write $V^{\mathrm{\prime }}$ for its ordinary real derivative, and choose a positive normalization constant $K_{\phi }>0$.

The kinetic scalar $g^{ab}\overline{P_a}{P}_b$ is real because its complex conjugate equals itself by symmetry of $g^{ab}$. Put

$L_{\phi }:=g^{ab}\overline{P_a}{P}_b+V(\mathrm{\mid }\phi {\mathrm{\mid }}^2),S_{\phi }:=-\frac{K_{\phi }}{2c}{\int }_M{L}_{\phi }{\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$

For SI dimensional bookkeeping in length coordinates, take $[\phi ]=1$, $[A_a]=\mathrm{W}\mathrm{b}{\mathrm{m}}^{-1}$, $[P_a]={\mathrm{m}}^{-1}$, $[V]={\mathrm{m}}^{-2}$, and $[K_{\phi }]=\mathrm{J}{\mathrm{m}}^{-1}$; then $S_{\phi }$ has units $\mathrm{J}\mathrm{s}$. These are formal dimensional assignments, not measured values or physical postulates. On noncompact $M$, the integral is used when finite; otherwise only differences between configurations agreeing outside a compact set are defined, so every variation below is a compactly supported integral.

Allowed first variations are compactly supported smooth complex-section variations $\delta \phi$, compactly supported real global one-form variations $\alpha =\delta A$ added to every local $A_i$ (so the same bundle transition functions are retained), and compactly supported symmetric inverse-metric variations $q^{ab}=\delta g^{ab}$. The formula defines a classical mathematical functional; it asserts no quantum interpretation, value of $\mathrm{\hslash }$, or material-response model.

• The seven typed foundational replacement carriers declared in deps are now authored on emg-mg02; no independent acceptance is asserted: Finite-real supremum and infimum domain, Real metric and pseudometric convention, Metric nonnegativity from the three axioms, Explicit rational and real power regimes, Sequence-limit criterion with explicit choice use, Boundaryless domain for geometric completeness, and Set-sized objects and definable-class schemas.
• The published Lorentzian supplier carries the stated ${\mathrm{A}\mathrm{C}}_{\omega }$ convention through its tangent-bundle setup; the local patch calculation here makes no further choice. No full axiom of choice is used. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))
• In a nontrivial circle sector the integer weight condition $\lambda {\kappa }_0\in \mathbb{Z}$ is the representation condition selected for this bundle. In a trivial global real-potential sector $\lambda$ may be any real number; the two cases are not identified.