**1. Introduction**

A formulation of dressed photons in quantum field theory is given by the Clebsch dual variable, motivated by fluid dynamics [1–3]. The Clebsch parametrization of the rotational model of the velocity field *<sup>U</sup>μ* is formulated of the form *<sup>U</sup>μ* = *λ*∇*μφ* with two scalar fields *λ*, *φ*. We define the covariant vectors *Cμ* = ∇*μφ* and *Lμ* = <sup>∇</sup>*μλ*, and the bi-vector *<sup>S</sup>μν* = *<sup>C</sup>μLν* − *<sup>L</sup>μC<sup>ν</sup>*. The energy–momentum tensor is defined by *T* ˆ *ν μ* = <sup>−</sup>*SμσSνσ*. It is shown

$$
\hat{T}^{\nu}\_{\mu} = \rho \mathbf{C}\_{\mu} \mathbf{C}^{\nu} \tag{1}
$$

by a simple computation [1].

Our main concern is this last Equation (1). This looks like Veronese embedding in projective geometry. In this paper, we introduce the model in arbitrary dimension and describe the symmetry of this model. Most of the material comes from the modern treatment of classical invariant theory [4,5]. Especially, the quadratic map arising in reductive dual pair [6,7] is used as one of the key ingredients in this paper to construct geometric objects describing the symmetry. This enables us to give another explanation of the last Equation (1) on *T* ˆ .

Physical study of dressed photons, including experiments and related applications, called dressed photon phenomenon, has already been summarized in our previous paper [8]. This paper serves as a complementary observation on symmetry of theoretical foundations of dressed photon Equation [1], which would be expected as is in classical electromagnetism. We conclude that the symmetry is well described in terms of compact homogeneous space, such as Grassmann manifolds and flag manifolds, as well as pre-homogeneous vector spaces, which is not a homogeneous space, but still has a large symmetry. It is also significant that a part of discussion is not restricted to a specific dimension, so that half of them are formulated in arbitrary dimension.

The construction of this paper is as follows: In Section 2, we work over the complex number field C, and do si in arbitrary dimensions *n* ≥ *r* ≥ *s*. In Section 3, we consider the special case *n* = 4,*r* = 2,*s* = 1 with the real number field R. The symmetry and invariants are mostly the same for C and for R; however, there is a subtle and rather complicated problem on connected components over R. In order to concentrate this complication for R, the common features of the model are discussed over C, and the different point is separately treated in Section 3.

**Citation:** Ochiai, H. Symmetry of Dressed Photon. *Symmetry* **2021**, *13*, 1283. https://doi.org/10.3390/ sym13071283

Academic Editor: Ignatios Antoniadis

Received: 1 July 2021 Accepted: 14 July 2021 Published: 16 July 2021

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**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

### **2. The Model over the Complex Numbers**

*2.1. Symmetry in Arbitrary Dimension*

Let *<sup>M</sup>*(*<sup>n</sup>*,*r*, C), Sym(*<sup>n</sup>*, C), Alt(*<sup>n</sup>*, C) be the set of *n* by *r* matrices, symmetric matrices, and skew-symmetric matrices with complex entries. We denote by *<sup>M</sup>*(*<sup>n</sup>*,*r*, <sup>C</sup>)rk≤*i* the subset consisting of matrices of rank at most *i*. The transpose of a matrix *X* is denoted by *XT*. Classical invariant theory gives the following maps:

Let *J* ∈ Alt(*<sup>r</sup>*, C)rk=*r*. We define the map

$$\mathbb{S}: \mathcal{M}(n, r, \mathbb{C}) \longrightarrow \text{Alt}(n, \mathbb{C}) \quad \text{by} \quad X \mapsto XfX^{\top}.$$

If *r* ≥ *n*, then this map is surjective. If *r* < *n*, then the image of this map is Alt(*<sup>n</sup>*, <sup>C</sup>)rk≤*<sup>r</sup>*. This map is *GL*(*<sup>n</sup>*, C) × *Sp*(*<sup>r</sup>*, C)-equivariant, in the sense that S(*lXh*) = *l* S(*X*)*l<sup>T</sup>* for any *l* ∈ *GL*(*<sup>n</sup>*, C) and *h* ∈ *Sp*(*<sup>r</sup>*, C), where the symplectic group attached to *J* is defined by *Sp*(*<sup>r</sup>*, C) = *Sp*(*J*, C) = {*h* ∈ *<sup>M</sup>*(*<sup>r</sup>*, C) | *hJh<sup>T</sup>* = *J*}.

Let *g* ∈ Sym(*<sup>n</sup>*, C)rk=*n*. We define the map

$$\mathbb{G}: \mathsf{M}(n, r, \mathbb{C}) \longrightarrow \mathrm{Sym}(r, \mathbb{C}) \quad \text{by} \quad X \mapsto X^T \mathcal{g}X.$$

If *r* ≥ *n*, then this map is surjective. If *r* < *n*, then the image of this map is Sym(*<sup>r</sup>*, <sup>C</sup>)rk≤*<sup>n</sup>*. This map is *<sup>O</sup>*(*<sup>n</sup>*, C) × *GL*(*<sup>r</sup>*, C)-equivariant, in the sense that G(*lXh*) = *hT*G(*X*)*h* for any *l* ∈ *<sup>O</sup>*(*<sup>n</sup>*, C) and *h* ∈ *GL*(*<sup>r</sup>*, C), where the orthogonal group attached to *g* is defined by *<sup>O</sup>*(*<sup>n</sup>*, C) = *<sup>O</sup>*(*g*, C) = {*l* ∈ *<sup>M</sup>*(*<sup>n</sup>*, C) | *lTgl* = *g*}. Especially, put *r* = *n* and restrict the domain, we define

$$\mathbb{T}: \operatorname{Alt}(n, \mathbb{C}) \longrightarrow \operatorname{Sym}(n, \mathbb{C}) \quad \text{by} \quad X \mapsto X \operatorname{g} X^T = -X \operatorname{g} X = X^T \operatorname{g} X.$$

This is *<sup>O</sup>*(*<sup>n</sup>*, C)-equivariant: T(*lXl<sup>T</sup>*) = *l* <sup>T</sup>(*X*)*lT*.

From now on, we assume that *n* ≥ *r* ≥ *s*. Each *GL*(*<sup>r</sup>*, C)-orbit on Sym(*<sup>r</sup>*, C) is parametrized by the rank. The closure relation of orbits is linear, so that the closure of Sym(*<sup>r</sup>*, C)rk=*<sup>s</sup>* is Sym(*<sup>r</sup>*, <sup>C</sup>)rk≤*<sup>s</sup>*. We define

$$Y(\mathbb{C}) = M(n, r, \mathbb{C})\_{\text{rk} \star r} \cap \mathbb{G}^{-1}(\text{Sym}(r, \mathbb{C})\_{\text{rk} \le s})$$

Our main target is the description of the image of *Y*(C) by the map T ◦ S:

$$\text{Sym}(r,\mathbb{C})\_{\text{rk}\leq\_{\text{s}}} \xleftarrow{\text{G}} M(n,r,\mathbb{C})\_{\text{rk}=r} \xrightarrow{\text{G}} \text{Alt}(n,\mathbb{C}) \xrightarrow{\text{T}} \text{Sym}(n,\mathbb{C}).\tag{2}$$

In order to state the main result, we introduce several auxiliary spaces and maps. We fix *g* ∈ Sym(*<sup>s</sup>*, C)rk=*s*. We define the maps

$$\mathbb{V}: \mathcal{M}(n, \mathbf{s}, \mathbb{C}) \longrightarrow \text{Sym}(n, \mathbb{C}) \quad \text{by} \quad \mathbb{V}(X) = X \mathbb{g}' X^T.$$

$$\mathbb{V}': \mathcal{M}(r, \mathbf{s}, \mathbb{C}) \longrightarrow \text{Sym}(r, \mathbb{C}) \quad \text{by} \quad \mathbb{V}'(X') = X' \mathbb{g}' X'^T.$$

Note that these maps are similar to G, but transposed. Especially, the orthogonal group *<sup>O</sup>*(*g*, C) acts transitively on each fiber of an element of Sym(*<sup>r</sup>*, C)rk=*s*.

We define *Z*(C) to be the fiber product of the map G : *Y*(C) → Sym(*<sup>r</sup>*, <sup>C</sup>)rk≤*s* and V : *<sup>M</sup>*(*<sup>r</sup>*,*s*, C)rk=*<sup>s</sup>* → Sym(*<sup>r</sup>*, <sup>C</sup>)rk≤*s*:

$$\begin{split} Z(\mathbb{C}) &= \mathcal{Y}(\mathbb{C}) \times\_{\text{Sym}(r,\mathbb{C})\_{\text{rk}\leq s}} M(r, s, \mathbb{C})\_{\text{rk}=s} \\ &= \{ (X, X') \in M(n, r, \mathbb{C}) \times M(r, s, \mathbb{C}) \mid \text{rk}(X) = r, \text{rk}(X') = s, X^T \text{g}X = X' \text{g}'X'^T \} .\end{split}$$

We have the commutative diagram

*<sup>M</sup>*(*<sup>n</sup>*,*r*, C) ←− *Y*(C) V ←−−−− *Z*(C) G ⏐⏐⏐ G ⏐⏐⏐ G ˜ ⏐⏐⏐ Sym(*<sup>r</sup>*, C) ←− Sym(*<sup>r</sup>*, <sup>C</sup>)rk≤*s* V ←−−−− *<sup>M</sup>*(*<sup>r</sup>*,*s*, C)rk=*<sup>s</sup>* (3)

where the right square is Cartesian.

The map T ◦ S does not factor through the map V. However, when we lift the map from *Y*(C) to *<sup>Z</sup>*(C), the map factor through V. To be more precise, we have the following:

$$\text{Theorem 1. } (\mathbb{T} \circ \mathbb{B})(X) = (\mathbb{V} \circ \phi)(X, X') \text{ for all } (X, X') \in Z(\mathbb{C}), \text{ where we define}$$

$$\phi: M(\mathfrak{n}, r, \mathbb{C})\_{\text{rk} \gets r} \times M(r, \mathfrak{s}, \mathbb{C})\_{\text{rk} \gets s} \longrightarrow M(\mathfrak{n}, \mathfrak{s}, \mathbb{C})\_{\text{rk} \gets s} \quad \text{by} \quad (X, X') \mapsto XIX' \tag{4}$$

$$\begin{array}{l} \textbf{Proof.} \ (\mathbb{T}\circ\mathbb{S})(X) = \mathbb{T}(X\!\!\!/X^{\top}) = (X\!\!\!\!/X^{\top})\!\!\!/ (X\!\!\!\!/X^{\top})^{\top} = X\!\!\!\!\!\!/\mathbb{G}(X)\!\!\!\!/^{\top}\mathbb{X}^{\top} \\\ = X\!\!\!\!/\mathbb{V}^{\prime}(X^{\prime})\!\!\!/^{\top}\mathbb{X}^{\top} = (X\!\!\!\!/X^{\prime})\!\!\!\!/^{\ast}(X\!\!\!\!/X^{\prime})^{\top} = \mathbb{V}(X\!\!\!\!\!/X^{\prime}) = (\mathbb{V}\circ\phi)(X,X^{\prime}). \end{array}$$

This theorem is illustrated as the following commutative diagram:

$$\begin{array}{ccccc} Z(\mathbb{C}) & \longrightarrow & M(n,r,\mathbb{C})\_{\mathbb{r}\mathbb{k}=r} \times M(r,s,\mathbb{C})\_{\mathbb{r}\mathbb{k}=s} & \xrightarrow{\phi} & M(n,s,\mathbb{C})\_{\mathbb{r}\mathbb{k}=s} \\ \mathbb{V}^{\mathbb{v}} & & & & \Big|\mathbb{V} \\ \mathbb{V}(\mathbb{C}) & \longrightarrow & M(n,r,\mathbb{C})\_{\mathbb{r}\mathbb{k}=r} & \xrightarrow{\mathbb{S}} & \text{Alt}(n,\mathbb{C}) & \xrightarrow{\mathbb{T}} & \text{Sym}(n,\mathbb{C}) \end{array} \tag{5}$$

Note that the maps S, G,T, V, V are common in classical invariant theory and theory of reductive dual pair, though the space *Y*(C) and *Z*(C) is unique in our setting.
