Helicity bundles, rotations and Berry curvature

For $u,v\in {\mathbb{C}}^3$, extend the real Euclidean dot and right-handed cross products complex-bilinearly, with coordinate formulas $u\cdot v:={\sum }_{j=1}^3{u}_j{v}_j,u\times v:=(u_2{v}_3-u_3{v}_2,u_3{v}_1-u_1{v}_3,u_1{v}_2-u_2{v}_1)\mathrm{.}$ The dot product in these formulas uses no complex conjugation; it differs from the Hermitian product $⟨u,v{⟩}_H:={\sum }_{j=1}^3\overline{u_j}{v}_j$. Write $\mathrm{\parallel }z{\mathrm{\parallel }}_H:=\sqrt{⟨z,z{⟩}_H}$ for the associated nonnegative norm on ${\mathbb{C}}^3$.

Let $S^2:=\{n\in {\mathbb{R}}^3:n\cdot n=1\}$ with the smooth structure given by the six graph charts solving $n_j=\pm \sqrt{1-{\sum }_{k\mathrm{\ne }j}{n}_k^2}$ on the hemispheres $\pm n_j>0$. Orient it outward using the ordered spatial basis $x^1,x^2,x^3$. Its tangent plane is $T_n{S}^2=\{X\in {\mathbb{R}}^3:n\cdot X=0\}$. Physically, $n=k\mathrm{/}\mathrm{\mid }k\mathrm{\mid }$ for a nonzero spatial wavevector $k$ in the SI Fourier description, where $k$ has units ${\mathrm{m}}^{-1}$; hence $n$ is dimensionless and the radial magnitude is discarded. For $\sigma \in \{+1,-1\}$, set $C_{n}z:=in\times z$, $P_T(n)z:=z-n(n\cdot z)$, and $P_{\sigma }(n):=\frac{1}{2}(P_T(n)+\sigma C_n)$ on ${\mathbb{C}}^3$. Define the helicity fibre $L_{\sigma }(n):=\mathrm{range}{P}_{\sigma }(n)$ and the total space $L_{\sigma }:=\{(n,z):n\in S^2,z\in L_{\sigma }(n)\}$, projected to $S^2$ by $(n,z)\mapsto n$.

For an open set $U\subseteq S^2$, a unit frame of $L_{\sigma }$ over $U$ is a smooth map $v:U\to {\mathbb{C}}^3$ such that $v(n)\in L_{\sigma }(n)$ and $⟨v(n),v(n){⟩}_H=1$ for every $n\in U$. Since each fibre is a complex line, this vector spans that fibre. A local unit frame is such a map on an open neighborhood in $S^2$; a global unit frame is one defined on all of $S^2$.

Fibrewise addition and scalar multiplication are inherited from ${\mathbb{C}}^3$. A unit frame $v$ gives the local trivialization $(n,a)\mapsto (n,av(n))$ from $U\times \mathbb{C}$ to $L_{\sigma }{\mathrm{\mid }}_U$, with inverse $(n,z)\mapsto (n,⟨v(n),z{⟩}_H)$. The construction in the Proof supplies such a frame near every $n$, so $L_{\sigma }$ is a smooth rank-one complex vector subbundle of the product bundle $S^2\times {\mathbb{C}}^3\to S^2$, whose projection is $(n,z)\mapsto n$.

On a spherical chart with $0<\theta <\pi$ and a smooth longitude branch $\varphi$, write $n=(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$, $e_{\theta }=(\cos \theta \cos \varphi ,\cos \theta \sin \varphi ,-\sin \theta )$, and $e_{\varphi }=(-\sin \varphi ,\cos \varphi ,0)$. The local unit helicity frame is $v_{\sigma }:=(e_{\theta }+i\sigma e_{\varphi })\mathrm{/}\sqrt{2}$. For any unit frame $v$ over an open set $U$, write $v^{†}dv:={\sum }_{j=1}^3\overline{v_j}d{v}_j$, where $d{v}_j$ is the exterior derivative of its $j$th component. Define its Berry connection one-form by $A:=i{v}^{†}dv$. A smooth phase change $v^{\mathrm{\prime }}=e^{i\chi }v$, with real smooth $\chi$, changes this local form to $A^{\mathrm{\prime }}=A-d\chi$. The local Berry curvature is the real two-form $F:=dA$ on $U$; no global gluing assertion is made here.

Exact helicity projectors in vacuum

Classical transverse Fourier amplitudes

Lorentzian Hodge star and periods

The cross product in ${\mathbb{R}}^3$

The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$

Oriented smooth manifolds and oriented charts

A smooth differential $k$-form

The wedge product of differential forms

The exterior derivative by the invariant vector-field formula

Fix $\sigma \in \{+1,-1\}$ and let $L_{\sigma }(n)$ be the range of the helicity projector $P_{\sigma }(n)$ over the outward-oriented unit sphere $S^2$. On an angular chart with $0<\theta <\pi$ and a smooth longitude branch $\varphi$, use $v_{\sigma }=(e_{\theta }+i\sigma e_{\varphi })\mathrm{/}\sqrt{2}$ and the connection convention $A_{\sigma }=i{v}_{\sigma }^{†}d{v}_{\sigma }$. The spherical frame has Berry one-form $A_{\sigma }=\sigma \cos \theta d\varphi$ and curvature $F_{\sigma }=d{A}_{\sigma }=-\sigma \sin \theta d\theta \wedge d\varphi$. If ${\omega }_{S^2,n}(X,Y):=n\cdot (X\times Y)$ is the outward area form for $X,Y\in T_n{S}^2$, then this curvature extends smoothly to the whole sphere as $F_{\sigma }=-\sigma {\omega }_{S^2}$. Write ${\int }_{S^2}$ for the outward-oriented integral, evaluated as the sum over the eight radially projected octant triangles described in the Proof. Its total flux is ${\int }_{S^2}{F}_{\sigma }=-4\pi \sigma$, so the flux divided by $2\pi$ is $-2\sigma$ in this convention. The line bundle has no globally smooth unit frame.

Helicity eigenlines over the direction sphere

The exterior derivative squares to zero

Smooth singular chain and cochain complexes

Integral of a form over a smooth singular simplex

Stokes theorem for smooth singular chains

On the overlap $S^2\setminus \{+e_3,-e_3\}$ the longitude phase is single-valued after exponentiation because $\sigma =\pm 1$. The two local frame choices $v_N=e^{i\sigma \varphi }{v}_{\sigma }$ and $v_S=e^{-i\sigma \varphi }{v}_{\sigma }$ obey the displayed transition. The arrow records only this change of frame; the proof above establishes the curvature and obstruction without relying on the diagram.

For $u,v\in {\mathbb{C}}^3$, the dot and cross products are the complex-bilinear extensions with coordinate formulas $u\cdot v={\sum }_{j=1}^3{u}_j{v}_j$ and $u\times v=(u_2{v}_3-u_3{v}_2,u_3{v}_1-u_1{v}_3,u_1{v}_2-u_2{v}_1)$, as defined in the helicity line-bundle item.

Fix $n\in S^2$ and $\sigma \in \{+1,-1\}$. Put $U(1):=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$. For $n\in {\mathbb{R}}^3$ and $z\in {\mathbb{C}}^3$, the dot in the rotation formula is $n\cdot z={\sum }_{j=1}^3{n}_j{z}_j$, the complex-bilinear dot product defined in the earlier helicity line-bundle definition. For the dimensionless angle $\alpha \in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$, define the proper active rotation fixing $n$ by $R_{n,\alpha }z=(n\cdot z)n+\cos \alpha (z-(n\cdot z)n)+\sin \alpha (n\times z)$ on ${\mathbb{C}}^3$, with the complex-linear extension of the real rotation action, and set $SO(2{)}_n:=\{R_{n,\alpha }:\alpha \in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})\}$. It acts on the whole helicity fibre $L_{\sigma }(n)$ by $R_{n,\alpha }z=e^{-i\sigma \alpha }z$, so the fibre carries the character ${\rho }_{\sigma }(R_{n,\alpha })=e^{-i\sigma \alpha }$. For a piecewise $C^1$ closed direction path $\gamma :[0,1]\to S^2$, choose a finite subdivision $0=t_0<\cdots <t_m=1$ on whose closed subintervals $\gamma$ is $C^1$. For each segment choose an open set $U_j\subseteq S^2$ containing its image and a unit frame $v_j:U_j\to {\mathbb{C}}^3$ as defined in the helicity line-bundle definition; write $A_j:=i{\sum }_{\mathrm{\ell }=1}^3\overline{(v_j{)}_{\mathrm{\ell }}}d(v_j{)}_{\mathrm{\ell }}$. For a one-form $A_j$ on $U_j$, define its oriented curve integral on this segment by ${\int }_{\gamma \mathrm{\mid }[t_{j-1},t_j]}{A}_j:={\int }_{t_{j-1}}^{t_j}(A_j{)}_{\gamma (t)}(\dot{\gamma }(t))dt,$ the ordinary Riemann integral of the continuous integrand on that segment. For a $C^1$ section $s(t)\in L_{\sigma }(\gamma (t))$, define its projected derivative by ${\mathrm{\nabla }}_{\dot{\gamma }}s(t):=P_{\sigma }(\gamma (t))\dot{s}(t)$, with the derivative taken componentwise in ${\mathbb{C}}^3$; parallel transport is the solution of ${\mathrm{\nabla }}_{\dot{\gamma }}s=0$ on each segment. If $v_{j+1}(\gamma (t_j))=g_j{v}_j(\gamma (t_j))$ at each internal join and $v_m(\gamma (1))=g_m{v}_1(\gamma (0))$, where $g_j,g_m\in U(1)$, then its holonomy, the scalar multiplying an initial fibre vector after transport around $\gamma$, is ${\mathrm{H}\mathrm{o}\mathrm{l}}_{\gamma }=({\prod }_{j=1}^m\exp (i{\int }_{\gamma \mathrm{\mid }[t_{j-1},t_j]}{A}_j))\left({\prod }_{j=1}^{m-1}{g}_j^{-1}\right)g_m$. In one unit frame defined on a neighborhood of the whole loop, write ${\oint }_{\gamma }A:={\sum }_{j=1}^m{\int }_{\gamma \mathrm{\mid }[t_{j-1},t_j]}A$; then the formula reduces to ${\mathrm{H}\mathrm{o}\mathrm{l}}_{\gamma }=\exp (i{\oint }_{\gamma }A)$. If in addition $\gamma$ is piecewise smooth in the sense that each rescaled segment ${\gamma }_j(u):=\gamma (t_{j-1}+(t_j-t_{j-1})u)$, $0\le u\le 1$, is a smooth singular $1$-simplex, define the finite smooth singular $1$-chain $c_{\gamma }:={\sum }_{j=1}^m{\gamma }_j$ with its positive path orientation. If a finite oriented smooth singular $2$-chain $\mathrm{\Sigma }$ has image in one open frame domain $U$ and $\mathrm{\partial }\mathrm{\Sigma }=c_{\gamma }$, then ${\mathrm{H}\mathrm{o}\mathrm{l}}_{\gamma }=\exp (i{\int }_{\mathrm{\Sigma }}{F}_{\sigma })$.

The piecewise $C^1$ subdivision is understood with one-sided derivatives continuous up to each subinterval endpoint. In the final clause, each rescaled path segment must extend to a smooth map from an open neighborhood of its affine interval into $S^2$; this is the neighborhood-extension condition for a smooth singular $1$-simplex. The open frame domain $U$ is an open subset of $S^2$ carrying one smooth unit frame on all of $U$, and the image of $\mathrm{\Sigma }$ lies in $U$.

Helicity eigenlines over the direction sphere

Exact helicity projectors in vacuum

Curvature and the obstruction to a global polarization basis

Stokes theorem for smooth singular chains