The Carrollian limit of a field theory is always composed of an electric case and a magnetic case, usually referred to as electric and magnetic contractions in the literature. Carrollian ModMax theory is no different. It was expected that both the electric contraction and the magnetic contraction were covariant under the action of a same group, albeit the action itself may differ and, indeed, that was the case.

As can be seen from the generators and as was shown in the poster, both limits admit the same kind of symmetries but slightly differ on how they are expressed in the field parts. As can be noted by looking at their respective generators:

Generators for the magnetic case

PA=∂∂xAH=∂∂sJA=ϵABC(xB∂∂xC+EB∂∂EC+BB∂∂BC)Tabc=xaybzc∂∂s−ϵIJK∂xaybzc∂xIBJ∂∂EKD=xA∂∂xA+BA∂∂BAQ=s∂∂s−BA∂∂BASA=2xA(xB∂∂xB+s∂∂s)−xBxB∂∂xA−4xAEJ∂∂EJ+2xJ(EA∂∂EJ−EJ∂∂EA)−2sϵAJKBJ∂∂EK−4xABJ∂∂BJ+2xJ(BA∂∂BJ−BJ∂∂BA)W=EA∂∂EA+BA∂∂BAU=−BA∂∂EA\begin{align} \mathcal{P}_{A}=&\frac{\partial}{\partial x^{A}}\\ \mathcal{H}=&\frac{\partial}{\partial s}\\ \mathcal{J}_{A}=&\epsilon_{ABC}\left(x^{B}\frac{\partial}{\partial x^{C}}+E^{B}\frac{\partial }{\partial E^{C}}+B^{B}\frac{\partial}{\partial B^{C}}\right)\\ \mathcal{T}_{abc}=&x^{a}y^{b}z^{c}\frac{\partial}{\partial s}-\epsilon_{IJK}\frac{\partial x^{a}y^{b}z^{c}}{\partial x^{I}}B^{J}\frac{\partial}{\partial E^{K}}\\ \mathcal{D}=&x^{A}\frac{\partial}{\partial x^{A}}+B^{A}\frac{\partial}{\partial B^{A}}\\ \mathcal{Q}=&s\frac{\partial}{\partial s}-B^{A}\frac{\partial}{\partial B^{A}}\\ \notag \mathcal{S}_{A}=&2x_{A}\left(x^{B}\frac{\partial}{\partial x^{B}}+s\frac{\partial}{\partial s}\right)-x_{B}x^{B}\frac{\partial}{\partial x^{A}}\\ \notag &-4x_{A}E_{J}\frac{\partial}{\partial E^{J}}+2x^{J}\left(E_{A}\frac{\partial}{\partial E_{J}}-E_{J}\frac{\partial}{\partial E^{A}}\right)-2s\:\epsilon_{AJK}B^{J}\frac{\partial}{\partial E_{K}}\\ &-4x_{A}B_{J}\frac{\partial}{\partial B_{J}}+2x^{J}\left(B_{A}\frac{\partial}{\partial B_{J}}-B_{J}\frac{\partial}{\partial B^{A}}\right)\\ \mathcal{W}=&E^{A}\frac{\partial}{\partial E^{A}}+B^{A}\frac{\partial}{\partial B^{A}}\\ \mathcal{U}=&-B^{A}\frac{\partial}{\partial E^{A}} \end{align}

Generators for the electric case

PA=∂∂xAH=∂∂sJI=ϵIJK(xJ∂∂xK+EJ∂∂EK+BJ∂∂BK)Tabc=xaybzc∂∂s+ϵIJK∂(xaybzc)∂xIEJ∂∂BKD=xI∂∂xI−BA∂∂BAQ=s∂∂s+BA∂∂BASA=2xA(xB∂∂xB+s∂∂s)−xBxB∂∂xA−4xAEJ∂∂EJ+2xJ(EA∂∂EJ−EJ∂∂EA)−4xABJ∂∂BJ+2xJ(BA∂∂BJ−BJ∂∂BA)+2sϵAJKEJ∂∂BKW=EA∂∂EA+BA∂∂BAU=EA∂∂BA\begin{align} \mathcal{P}_{A}=&\frac{\partial}{\partial x^{A}}\\ \mathcal{H}=&\frac{\partial}{\partial s}\\ \mathcal{J}_{I}=&\epsilon_{IJK}\left(x^{J}\frac{\partial}{\partial x^{K}}+E^{J}\frac{\partial}{\partial E_{K}}+B^{J}\frac{\partial}{\partial B_{K}}\right)\\ \mathcal{T}_{abc}=&x^{a}y^{b}z^{c}\frac{\partial}{\partial s}+\epsilon_{IJK}\frac{\partial \left(x^{a}y^{b}z^{c}\right)}{\partial x^{I}}E^{J}\frac{\partial}{\partial B_{K}}\\ \mathcal{D}=&x^{I}\frac{\partial}{\partial x^{I}}-B_{A}\frac{\partial}{\partial B_{A}}\\ \mathcal{Q}=&s\frac{\partial}{\partial s}+B_{A}\frac{\partial}{\partial B_{A}}\\ \notag \mathcal{S}_{A}=&2x_{A}\left(x^{B}\frac{\partial}{\partial x^{B}}+s\frac{\partial}{\partial s}\right)-x_{B}x^{B}\frac{\partial}{\partial x^{A}}\\ \notag &-4x_{A}E_{J}\frac{\partial}{\partial E^{J}}+2x^{J}\left(E_{A}\frac{\partial}{\partial E_{J}}-E_{J}\frac{\partial}{\partial E^{A}}\right)\\ &-4x_{A}B_{J}\frac{\partial}{\partial B_{J}}+2x^{J}\left(B_{A}\frac{\partial}{\partial B_{J}}-B_{J}\frac{\partial}{\partial B^{A}}\right)+2s\:\epsilon_{AJK}E^{J}\frac{\partial}{\partial B_{K}}\\ \mathcal{W}=&E^{A}\frac{\partial}{\partial E^{A}}+B^{A}\frac{\partial}{\partial B^{A}}\\ \mathcal{U}=&E^{A}\frac{\partial}{\partial B^{A}} \end{align}

By virtue of the differences in the generators shown above, the flows that characterize their respective group actions differ, as will be shown in what follows.

Symmetries of the Carrollian electric limit of ModMax

The finite symmetry transformations of the Carrollian electric limit of ModMax theory were found by constructing the flows hXh^{\mathcal{X}} associated with each generator X\mathcal{X}. These flows are as follows:

hPA(λ,s,x,E,B)=(s,x+iA(λ),E,B)hH(λ,s,x,E,B)=(s+λ,x,E,B)hJA(λ,s,x,E,B)=(s,RA(λ)x,RA(λ)E,RA(λ)B)hTabc(λ,s,x,E,B)=(s+λxaybzc,x,E−λ∇(xaybzc)×B,B)hD(λ,s,x,E,B)=(s,eλx,E,eλB)hQ(λ,s,x,E,B)=(eλs,x,E,e−λB)hSA(λ,s,x,E,B)=(ωA(λ)s,ωA(λ)(x−iA(λ)x⋅x),TA(λ)E+OA(λ)B,TA(λ)B)hW(λ,s,x,E,B)=(s,x,eλE,eλB)hU(λ,s,x,E,B)=(s,x,eλE,B+λE),\begin{align} h^{\mathcal{P}_{A}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(s,\mathbf{x}+i_{A}(\lambda),\bm{E},\bm{B}\right)\\ h^{\mathcal{H}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(s+\lambda,\mathbf{x},\bm{E},\bm{B}\right)\\ h^{\mathcal{J}_{A}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(s,R_{A}(\lambda)\mathbf{x},R_{A}(\lambda)\bm{E},R_{A}(\lambda)\bm{B}\right)\\ h^{\mathcal{T}_{abc}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(s+\lambda x^{a}y^{b}z^{c},\mathbf{x},\bm{E}-\lambda\nabla\left(x^{a}y^{b}z^{c}\right)\times\bm{B},\bm{B}\right)\\ h^{\mathcal{D}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(s,e^{\lambda}\mathbf{x},\bm{E},e^{\lambda}\bm{B}\right)\\ h^{\mathcal{Q}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(e^{\lambda}s,\mathbf{x},\bm{E},e^{-\lambda}\bm{B}\right)\\ h^{\mathcal{S}_{A}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(\omega_{A}(\lambda)s,\omega_{A}(\lambda)\left(\mathbf{x}-i_{A}(\lambda)\mathbf{x}\cdot\mathbf{x}\right),\mathbb{T}_{A}(\lambda)\bm{E}+\mathbb{O}_{A}(\lambda)\bm{B},\mathbb{T}_{A}(\lambda)\bm{B}\right)\\ h^{\mathcal{W}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(s,\mathbf{x},e^{\lambda}\bm{E},e^{\lambda}\bm{B}\right)\\ h^{\mathcal{U}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(s,\mathbf{x},e^{\lambda}\bm{E},\bm{B}+\lambda\bm{E}\right), \end{align}

where a,b,c∈N0a,b,c\in\mathbb{N}_{0}, A∈{1,2,3}A\in\{1,2,3\}, iA:R⟶R3i_{A}:\mathbb{R}\longrightarrow\mathbb{R}^{3} is the canonical injection map to the AA-th coordinate, this is i1(λ)=(λ,0,0)i_{1}(\lambda)=(\lambda,0,0), i2(λ)=(0,λ,0)i_{2}(\lambda)=(0,\lambda,0) and i3(λ)=(0,0,λ)i_{3}(\lambda)=(0,0,\lambda); ωA(λ)\omega_{A}(\lambda) are factors given by

ωx(λ)=x2+y2+z2(x−λ(x2+y2+z2))2+y2+z2ωy(λ)=x2+y2+z2x2+(y−λ(x2+y2+z2))2+z2ωz(λ)=x2+y2+z2x2+y2+(z−λ(x2+y2+z2))2,\begin{align} \omega_{x}(\lambda)&=\frac{x^{2}+y^{2}+z^{2}}{\left(x-\lambda\left(x^{2}+y^{2}+z^{2}\right)\right)^{2}+y^{2}+z^{2}}\\ \omega_{y}(\lambda)&=\frac{x^{2}+y^{2}+z^{2}}{x^{2}+\left(y-\lambda\left(x^{2}+y^{2}+z^{2}\right)\right)^{2}+z^{2}}\\ \omega_{z}(\lambda)&=\frac{x^{2}+y^{2}+z^{2}}{x^{2}+y^{2}+\left(z-\lambda\left(x^{2}+y^{2}+z^{2}\right)\right)^{2}}, \end{align}

we also have defined, for mere convenience, the factors

Ωx(λ)=(λx−1)2+λ2(y2+z2)Ωy(λ)=(λy−1)2+λ2(x2+z2)Ωz(λ)=(λz−1)2+λ2(x2+y2);\begin{align} \Omega_{x}(\lambda)&=\left(\lambda x-1\right)^{2}+\lambda^{2}\left(y^{2}+z^{2}\right)\\ \Omega_{y}(\lambda)&=\left(\lambda y-1\right)^{2}+\lambda^{2}\left(x^{2}+z^{2}\right)\\ \Omega_{z}(\lambda)&=\left(\lambda z-1\right)^{2}+\lambda^{2}\left(x^{2}+y^{2}\right); \end{align}

sing these, we can write the two families of matrices that characterize the action of special conformal Carrollian transformations act on the electric and magnetic field. The first family of matrices is TA(λ)\mathbb{T}_{A}(\lambda), where each one is given by

Tx(λ)=Ωx(λ)(λ(λx2−2x−λ(y2+z2))+12λy(λx−1)2λz(λx−1)−2λy(λx−1)λ2(x2−y2+z2)−2λx+1−2λ2yz−2λz(λx−1)−2λ2yzλ2(x2+y2−z2)−2λx+1)Ty(λ)=Ωy(λ)(λ2(−x2+y2+z2)−2λy+1−2λx(λy−1)−2λ2xz2λx(λy−1)λ(−λ(x2+z2)+λy2−2y)+12λz(λy−1)−2λ2xz−2λz(λy−1)λ2(x2+y2−z2)−2λy+1)Tz(λ)=Ωz(λ)(λ2(−x2+y2+z2)−2λz+1−2λ2xy−2λx(λz−1)−2λ2xyλ2(x2−y2+z2)−2λz+1−2λy(λz−1)2λx(λz−1)2λy(λz−1)λ(z(λz−2)−λ(x2+y2))+1),\begin{align} \mathbb{T}_{x}(\lambda)&=\Omega_{x}(\lambda)\begin{pmatrix} \lambda \left(\lambda x^2-2 x-\lambda \left(y^2+z^2\right)\right)+1 & 2 \lambda y (\lambda x-1) & 2 \lambda z (\lambda x-1) \\ -2 \lambda y (\lambda x-1) & \lambda ^2 \left(x^2-y^2+z^2\right)-2 \lambda x+1 & -2 \lambda ^2 y z \\ -2 \lambda z (\lambda x-1) & -2 \lambda ^2 y z & \lambda ^2 \left(x^2+y^2-z^2\right)-2 \lambda x+1 \\ \end{pmatrix}\\ \mathbb{T}_{y}(\lambda)&=\Omega_{y}(\lambda)\begin{pmatrix} \lambda ^2 \left(-x^2+y^2+z^2\right)-2 \lambda y+1 & -2 \lambda x (\lambda y-1) & -2 \lambda ^2 x z \\ 2 \lambda x (\lambda y-1) & \lambda \left(-\lambda \left(x^2+z^2\right)+\lambda y^2-2 y\right)+1 & 2 \lambda z (\lambda y-1) \\ -2 \lambda ^2 x z & -2 \lambda z (\lambda y-1) & \lambda ^2 \left(x^2+y^2-z^2\right)-2 \lambda y+1 \\ \end{pmatrix}\\ \mathbb{T}_{z}(\lambda)&=\Omega_{z}(\lambda)\begin{pmatrix} \lambda ^2 \left(-x^2+y^2+z^2\right)-2 \lambda z+1 & -2 \lambda ^2 x y & -2 \lambda x (\lambda z-1) \\ -2 \lambda ^2 x y & \lambda ^2 \left(x^2-y^2+z^2\right)-2 \lambda z+1 & -2 \lambda y (\lambda z-1) \\ 2 \lambda x (\lambda z-1) & 2 \lambda y (\lambda z-1) & \lambda \left(z (\lambda z-2)-\lambda \left(x^2+y^2\right)\right)+1 \\ \end{pmatrix}, \end{align}

the second family of matrices is OA(λ)\mathbb{O}_{A}(\lambda), with

Ox(λ)=Ωx(λ)(02λ2sz−2λ2sy2λ2sz0−2λs(λx−1)−2λ2sy2λs(λx−1)0)Oy(λ)=Ωy(λ)(0−2λ2sz2λs(λy−1)−2λ2sz02λ2sx−2λs(λy−1)2λ2sx0)Oz(λ)=Ωz(λ)(0−2λs(λz−1)2λ2sy2λs(λz−1)0−2λ2sx2λ2sy−2λ2sx0).\begin{align} \mathbb{O}_{x}(\lambda)&=\Omega_{x}(\lambda)\begin{pmatrix} 0 & 2 \lambda ^2 s z & -2 \lambda ^2 s y \\ 2 \lambda ^2 s z & 0 & -2 \lambda s (\lambda x-1) \\ -2 \lambda ^2 s y & 2 \lambda s (\lambda x-1) & 0 \\ \end{pmatrix}\\ \mathbb{O}_{y}(\lambda)&=\Omega_{y}(\lambda)\begin{pmatrix} 0 & -2 \lambda ^2 s z & 2 \lambda s (\lambda y-1) \\ -2 \lambda ^2 s z & 0 & 2 \lambda ^2 s x \\ -2 \lambda s (\lambda y-1) & 2 \lambda ^2 s x & 0 \\ \end{pmatrix}\\ \mathbb{O}_{z}(\lambda)&=\Omega_{z}(\lambda)\begin{pmatrix} 0 & -2 \lambda s (\lambda z-1) & 2 \lambda ^2 s y \\ 2 \lambda s (\lambda z-1) & 0 & -2 \lambda ^2 s x \\ 2 \lambda ^2 s y & -2 \lambda ^2 s x & 0 \\ \end{pmatrix}. \end{align}

These matrices have two important properties that were relevant to proving that special conformal carrollian transformations are a symmetry of the Carrollian magnetic limit of ModMax. Said properties are relative to the R3\mathbb{R}^{3}-inner product

(TA(λ)a)⋅(TA(λ)b)=ΩA(λ)4a⋅b(TA(λ)a)⋅(OA(λ)b)=0.\begin{align} \left(\mathbb{T}_{A}(\lambda)\bm{a}\right)\cdot\left(\mathbb{T}_{A}(\lambda)\bm{b}\right)&=\Omega_{A}(\lambda)^{4}\bm{a}\cdot\bm{b} & \left(\mathbb{T}_{A}(\lambda)\bm{a}\right)\cdot\left(\mathbb{O}_{A}(\lambda)\bm{b}\right)&=0. \end{align}

Finally, remark that the generators Tabc\mathcal{T}_{abc} generate translations in Carrollian time ss by polynomials. If all values of a,b,ca,b,c are considered we have

f(x,y,z)=∑a,b,c∈N0λabcxaybzc,\begin{align} f(x,y,z)=\sum_{a,b,c\in\mathbb{N}_{0}}\lambda_{abc}x^{a}y^{b}z^{c}, \end{align}

and therefore we can summarize (22) as

hT:C∞(R3)×Em⟶Em(f,s,x,E,B)⟶hT(f,s,x,E,B):=(s+f(x,y,z),x,E−∇f×B,B),\begin{align} h^{\mathcal{T}}:C^{\infty}\left(\mathbb{R}^{3}\right)\times\mathcal{E}_{m}&\longrightarrow\mathcal{E}_{m}\\ \left(f,s,\mathbf{x},\bm{E},\bm{B}\right)&\longrightarrow h^{\mathcal{T}}\left(f,s,\mathbf{x},\bm{E},\bm{B}\right):=\left(s+f(x,y,z),\mathbf{x},\bm{E}-\nabla f\times\bm{B},\bm{B}\right), \end{align}

which corresponds to the infinite-dimensional sector of these symmetries.

Symmetries of the Carrollian magnetic limit of ModMax

Finding the finite symmetry transformations of the Carrollian magnetic limit of ModMax direcly proved computationally demandant and difficult, so instead we employed a two-step process. First, we constructed the flow for each symmetry generator of the Carrollian magnetic limit of Maxwell theory, which are:

hPA(λ,s,x,E,B)=(s,x+iA(λ),E,B)hH(λ,s,x,E,B)=(s+λ,x,E,B)hJA(λ,s,x,E,B)=(s,RA(λ)x,RA(λ)E,RA(λ)B)hTabc(λ,s,x,E,B)=(s+λxaybzc,x,E,B+λ∇(xaybzc)×E)hD(λ,s,x,E,B)=(s,eλx,E,e−λB)hQ(λ,s,x,E,B)=(eλs,x,E,eλB)hSA(λ,s,x,E,B)=(ωA(λ)s,ωA(λ)(x−iA(λ)x⋅x),TA(λ)E+OA(λ)B,TA(λ)B)hW(λ,s,x,E,B)=(s,x,eλE,eλB)hU(λ,s,x,E,B)=(s,x,E−λB,B).\begin{align} h^{\mathcal{P}_{A}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(s,\mathbf{x}+i_{A}(\lambda),\bm{E},\bm{B}\right)\\ h^{\mathcal{H}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(s+\lambda,\mathbf{x},\bm{E},\bm{B}\right)\\ h^{\mathcal{J}_{A}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(s,R_{A}(\lambda)\mathbf{x},R_{A}(\lambda)\bm{E},R_{A}(\lambda)\bm{B}\right)\\ h^{\mathcal{T}_{abc}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(s+\lambda x^{a}y^{b}z^{c},\mathbf{x},\bm{E},\bm{B}+\lambda\nabla\left(x^{a}y^{b}z^{c}\right)\times\bm{E}\right)\\ h^{\mathcal{D}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(s,e^{\lambda}\mathbf{x},\bm{E},e^{-\lambda}\bm{B}\right)\\ h^{\mathcal{Q}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(e^{\lambda}s,\mathbf{x},\bm{E},e^{\lambda}\bm{B}\right)\\ h^{\mathcal{S}_{A}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(\omega_{A}(\lambda)s,\omega_{A}(\lambda)\left(\mathbf{x}-i_{A}(\lambda)\mathbf{x}\cdot\mathbf{x}\right),\mathbb{T}_{A}(\lambda)\bm{E}+\mathbb{O}_{A}(\lambda)\bm{B},\mathbb{T}_{A}(\lambda)\bm{B}\right)\\ h^{\mathcal{W}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(s,\mathbf{x},e^{\lambda}\bm{E},e^{\lambda}\bm{B}\right)\\ h^{\mathcal{U}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)=&\left(s,\mathbf{x},\bm{E}-\lambda\bm{B},\bm{B}\right). \end{align}

Second, we proved the symmetries of both theories coincide. The workaround we found was inspired by the formula of the canonical momenta conjugate to the vector potential A\bm{A} of this theory. This allowed us to find a map between the Carrollian magnetic limit of Maxwell theory and that of ModMax, given by

E=Em+2eγsinh⁡γBm⋅EmBm2BmB=Bm,\begin{align} \mathfrak{E}&=\bm{E_{m}}+2e^{\gamma}\sinh\gamma\frac{\bm{B_{m}}\cdot\bm{E_{m}}}{B_{m}^{2}}\bm{B_{m}} & \mathfrak{B}&=\bm{B_{m}}, \end{align}

and with inverse given by

Em=E−2e−γsinh⁡γE⋅BB2BBm=B.\begin{align} \bm{E_{m}}&=\mathfrak{E}-2e^{-\gamma}\sinh\gamma\:\frac{\mathfrak{E}\cdot\mathfrak{B}}{\mathfrak{B}^{2}}\mathfrak{B} & \bm{B_{m}}&=\mathfrak{B}. \end{align}

Using this map, it was possible to show that the symmetries of both theories coincide.

Restriction to the space-time sector

It can be seen from looking at the transformations in both cases that the part pertaining to the space-time sector is the same. This is because they correspond to the action of the same space-time symmetries, with distinctions only in how they act on the electric and magnetic fields.

Even if that much is clear, it is always useful to have an explicit way of writting it. Recall that each generator is a vector field in the tangent space TET\mathcal{E}, where (E,π,C3+1)\left(\mathcal{E},\pi,C^{3+1}\right) is a fiber bundle with projection map

π:E⟶C3+1(s,x,E,B)⟼π(s,x,E,B)=(s,x).\begin{align} \pi:\mathcal{E}&\longrightarrow C^{3+1}\\ \left(s,\mathbf{x},\bm{E},\bm{B}\right)&\longmapsto\pi\left(s,\mathbf{x},\bm{E},\bm{B}\right)=\left(s,\mathbf{x}\right). \end{align}

Then the restriction to the spatial part corresponds to simply the composition of the flows with the projection map

π(hPA(λ,s,x,E,B))=(s,x+iA(λ))π(hH(λ,s,x,E,B))=(s+λ,x,E,B)π(hJA(λ,s,x,E,B))=(s,RA(λ)x)π(hTabc(λ,s,x,E,B))=(s+λxaybzc,x)π(hD(λ,s,x,E,B))=(s,eλx)π(hQ(λ,s,x,E,B))=(eλs,x)π(hSA(λ,s,x,E,B))=(ωA(λ)s,ωA(λ)(x−iA(λ)x⋅x)).\begin{align} \pi\left(h^{\mathcal{P}_{A}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)\right)=&\left(s,\mathbf{x}+i_{A}(\lambda)\right)\\ \pi\left(h^{\mathcal{H}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)\right)=&\left(s+\lambda,\mathbf{x},\bm{E},\bm{B}\right)\\ \pi\left(h^{\mathcal{J}_{A}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)\right)=&\left(s,R_{A}(\lambda)\mathbf{x}\right)\\ \pi\left(h^{\mathcal{T}_{abc}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)\right)=&\left(s+\lambda x^{a}y^{b}z^{c},\mathbf{x}\right)\\ \pi\left(h^{\mathcal{D}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)\right)=&\left(s,e^{\lambda}\mathbf{x}\right)\\ \pi\left(h^{\mathcal{Q}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)\right)=&\left(e^{\lambda}s,\mathbf{x}\right)\\ \pi\left(h^{\mathcal{S}_{A}}\left(\lambda,s,\mathbf{x},\bm{E},\bm{B}\right)\right)=&\left(\omega_{A}(\lambda)s,\omega_{A}(\lambda)\left(\mathbf{x}-i_{A}(\lambda)\mathbf{x}\cdot\mathbf{x}\right)\right). \end{align}