Symmetry of Dressed Photon

: Motivated by describing the symmetry of a theoretical model of dressed photons, we introduce several spaces with Lie group actions and the morphisms between them depending on three integer parameters n ≥ r ≥ s on dimensions. We discuss the symmetry on these spaces using classical invariant theory, orbit decomposition of prehomogeneous vector spaces, and compact reductive homogeneous space such as Grassmann manifold and ﬂag variety. Finally, we go back to the original dressed photon with n = 4, r = 2, s = 1.


Introduction
A formulation of dressed photons in quantum field theory is given by the Clebsch dual variable, motivated by fluid dynamics [1][2][3]. The Clebsch parametrization of the rotational model of the velocity field U µ is formulated of the form U µ = λ∇ µ φ with two scalar fields λ, φ. We define the covariant vectors C µ = ∇ µ φ and L µ = ∇ µ λ, and the bi-vector S µν = C µ L ν − L µ C ν . The energy-momentum tensor is defined byT ν µ = −S µσ S νσ . It is shownT ν µ = ρC µ C ν (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) onT.
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.
If r ≥ n, then this map is surjective. If r < n, then the image of this map is Alt(n, C) rk≤r . This map is GL(n, C) × Sp(r, C)-equivariant, in the sense that S(lXh) = l S(X) l T for any l ∈ GL(n, C) and h ∈ Sp(r, C), where the symplectic group attached to J is defined by Let g ∈ Sym(n, C) rk=n . We define the map G : M(n, r, C) −→ Sym(r, C) by X → X T gX.
If r ≥ n, then this map is surjective. If r < n, then the image of this map is Sym(r, C) rk≤n .
This map is O(n, C) × GL(r, C)-equivariant, in the sense that G(lXh) = h T G(X)h for any l ∈ O(n, C) and h ∈ GL(r, C), where the orthogonal group attached to g is defined by O(n, C) = O(g, C) = {l ∈ M(n, C) | l T gl = g}. Especially, put r = n and restrict the domain, we define This is O(n, C)-equivariant: T(lXl T ) = l T(X) l T . From now on, we assume that n ≥ r ≥ s. Each GL(r, C)-orbit on Sym(r, C) is parametrized by the rank. The closure relation of orbits is linear, so that the closure of Sym(r, C) rk=s is Sym(r, C) rk≤s . We define Y(C) = M(n, r, C) rk=r ∩ G −1 (Sym(r, C) rk≤s ) Our main target is the description of the image of Y(C) by the map T • S: In order to state the main result, we introduce several auxiliary spaces and maps. We fix g ∈ Sym(s, C) rk=s . We define the maps Note that these maps are similar to G, but transposed. Especially, the orthogonal group O(g , C) acts transitively on each fiber of an element of Sym(r, C) rk=s .
We define Z(C) to be the fiber product of the map G : Y(C) → Sym(r, C) rk≤s and V : M(r, s, C) rk=s → Sym(r, C) rk≤s : We have the commutative diagram 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 Z(C), the map factor through V. To be more precise, we have the following: This theorem is illustrated as the following commutative diagram: 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.

Grassmann and Flag Manifold
We will show that the map φ introduced in Theorem 1 has an interpretation in the projective setting. We still assume n ≥ r ≥ s. The Grassmann manifold Grass(n, r, C) is the set of r-dimensional subspace of C n . This is identified with M(n, r, C) rk=r /GL(r, C) ∼ = Grass(n, r, C).
Every r-dimensional subspace of C n is spanned by r linear independent column vectors in C n .
The flag manifold Flag(n; k 1 , . . . , k m , C) is the set of flags of type (k 1 , k 2 , . . . , k m ), which is defined to be a sequence of subspaces . . . , m). Grassmann manifold is a special case of flag manifolds with m = 1. On the other hand, a flag variety is regarded as the incidence variety of the product of Grassmann manifolds. For example, Flag(n; k 1 , We have an isomorphism (M(n, r, C) rk=r × M(r, s, C) rk=s )/(GL(r, C) × GL(s, C)) ∼ = Flag(n; s, r, C).
In the following commutative diagram, each space in the upper line, which arises in Theorem 1, is a locally closed subset of an affine space, while each space in the lower line is a projective variety.
The maps in the lower line are given by V 2 ← (V 1 , V 2 ) → V 1 . This double fibration is often used in Radon transform and Heck correspondence [9].

The Model over Real Numbers
We now consider the special case n = 4, r = 2, s = 1, and replace C by R. Let J = 0 1 −1 0 be the standard non-degenerate skew-symmetric matrix. Note that J T = −J and det J = 1. Let g be the diagonal matrix with diagonal entries (1, −1, −1, −1). Finally, we put g = 1.
Most of the story in the previous section does hold over the real number field R as well. However, the disconnectedness makes things complicated. For example, although the map V : M(2, 1, C) −→ Sym(2, C) rk≤1 given by V (X ) = X X T is surjective, the map In order to improve this defect, we introduce a non-zero scalar multiplication so that we modify the map V by V 2 given below (7).

Quadratic Polynomial
Let us consider the matrix X = (C, L) = which realizes the definition of S µν . The map S is GL(4, R) × SL(2, R)-equivariant, where we remark the accidental isomorphism of lower rank groups: The action of GL(4, R) on Alt(4, R) is prehomogeneous [10]. The image Alt(4, R) rk≤2 is the complement of the open GL(4, R)-orbit Alt(4, R) rk=4 , and its defining equation is given by the basic relative invariant, Pfaffian Pf(S) = S 01 S 23 + S 02 S 31 + S 03 S 12 .

Symmetry Breaking
We restrict the general linear group GL(4, R) to the subgrouop O(1, 3). Let Gram matrix with respect to this metric is given by the map where Sym(n, R) is the set of real symmetric matrices of size n. The map G is O(1, 3) × GL(2, R)-equivariant: and an analogue of Veronese map is defined by The fiber product of two maps G : Y(R) −→ Sym(2, R) rk≤1 ,C → G(C), then we obtain a real counterpart of (3):

Grassmann and Flag Manifold
The flag manifold is realized as an incidence variety of the product of two Grassmann manifold: For (X, v) ∈ M(4, 2, R) rk=2 × S 1 , two column vectors of X spans a two-dimensional subspace V 2 , and a column vector X Jv generate a one-dimensional subspace V 1 in V 2 . The map Grass(4, 2, R) ←− Flag(4; 1, 1, 2, R) −→ Grass(4, 1, R) is the double fibration.

The Interpretation of the Off-Shell Condition
The vectors C and L in Clebsch parametrization should satisfy the following off-shell conditions [2]: We putR := 0 0 0 −ρ . Then, the condition (8) is written as G(X) =R.
In particular, in the case v = 0 1 ∈ S 1 , we compute the maps V 2 , Φ and V 4 : This coincides with the result in [2]. An unnatural J in the definition of Φ is for the sake of compatibility with the existing formula.
The group GL(2, R) acts on Sym(2, R) rk=1 and the stabilizer atR is a Borel subgroup Then, G : Y(R) −→ Sym(2, R) rk=1 is a GL(2, R)-equivariant bundle. We regard the off-shell condition specifies a fiber of this bundle. A symmetry is hidden in the horizontal direction of this bundle, the group action of GL(2, R). Of course, form the Clebsch parametrization point of view, the role of C and L is not the same; the off-shell condition specifies the special isotropic direction for C: the choice of this direction is controlled by the homogeneous space GL(2, R)/B.

Discussion
We describe the symmetry of equations of dressed photon in a general manner. The tensor S is understood as an affine version of Plücker coordinates of Grassmann manifold Grass(4, 2, R). The splitting expression of the tensorT is related with an affine version of flag manifold Flag(4; 1, 2, R). We find the off-shell condition (8) chooses the special fiber of the homogeneous bundle. This mathematical interpretation of the choice may have a physical interpretation, especially in the context of Clebsch variables, however, which must be a future work. We also remark that the existence of the symmetry in arbitrary dimension suggests a feedback from the theory of dressed photon to the theory of reductive dual pairs on the pullback of nilpotent orbits [6], which is also a topic of future study.