Pairwise Helicity in Higher Dimensions

A foundational assumption in most discussions of the $S$-matrix in relativistic quantum field theory is that asymptotic states should factorize as tensor products of single-particle states, as classified by Wigner [1]. However, this assumption breaks down for the scattering of particles with both electric and magnetic charge, for which the angular momentum in the electromagnetic field remains finite even at asymptotic separation [2]. This effect is codified by the notion of pairwise helicity, or the charge under the pairwise little group, defined as the subgroup of the Lorentz group that leaves the momenta of a given pair of particles invariant [3, 4]. This group has nontrivial representations on certain multiparticle states, leading to fundamentally new multiparticle representations of the Poincaré group in four dimensions.

The pairwise little group naturally generalizes to the “$n$-particle” little group for $n>2$, which becomes nontrivial in sufficiently high dimension. One might wonder whether such a group can be represented nontrivially in any physical theory. While this compelling question remains open, we show in this paper that there does exist a physically manifest generalization of the pairwise little group to higher dimensions: the pairwise little group of mutually nonlocal branes.

To motivate this construction, we first recall how the pairwise helicity of four-dimensional multiparticle states arises dynamically from the electromagnetic angular momentum of dyons. While irreducible single-particle representations of the Poincaré group are induced by representations of the corresponding little group, representations of the pairwise little group induce multiparticle representations that go beyond tensor products of single-particle representations.

We next compute the angular momentum in the electromagnetic field between mutually nonlocal branes of arbitrary (complementary) dimensions. It admits a simple expression in terms of the Dirac-Schwinger-Zwanziger (DSZ) pairing [5, 6, 7] within $p$-form electrodynamics. To provide a framework for this higher-dimensional analogue of pairwise helicity, we define representations of the Lorentz group that describe infinitely extended, isotropic branes. (For reasons that will become clear, such representations do not generally lift to representations of the Poincaré group.) A $(p-1)$-brane representation of the Lorentz group in $d$ spacetime dimensions is induced by a representation of the corresponding little group, which contains the group of transverse rotations $SO(d-p)$. Multi-brane representations can further carry nontrivial quantum numbers under little groups associated with pairs of distinct branes. Such pairwise little groups include rotations in the directions mutually transverse to both branes. When this subgroup of mutually transverse rotations is $SO(2)$, we refer to the associated charge as the pairwise helicity.

Finally, we offer some thoughts on the higher “$n$-brane” little groups that could in principle induce more general multi-brane representations of the Lorentz group.

Our analysis of $(p-1)$-branes in $d$ dimensions builds on the familiar case $d=4$, $p=1$. Recall that particles (0-branes) furnish unitary irreducible representations of the Poincaré group. Such representations can be classified by their mass and their representation with respect to the little group, the subgroup of the Lorentz group that leaves some reference momentum $k$ of that mass invariant [1]. A general Lorentz transformation $\Lambda$ acts on a single-particle momentum eigenstate $|p;\sigma\rangle$ as

where $W=W(\Lambda,p,k)$ is a little group transformation, $D$ is a unitary representation of $W$, and the single-particle quantum number $\sigma$ labels a little group representation. A particle representation of the Poincaré group (and in particular, of the Lorentz group) is thus induced by a unitary irreducible representation of the little group.

On the other hand, multiparticle representations of the Poincaré group may carry additional quantum numbers that characterize the Lorentz transformation properties of multiparticle states relative to tensor products of one-particle states [3, 4]. Given a generic pair of reference four-momenta (either massive or massless), the pairwise little group is defined as the $SO(2)$ subgroup of the Lorentz group that preserves both. Since three-particle and higher little groups are trivial in four dimensions, a general $n$-particle state takes the form

where $p_{i}$ is the momentum of particle $i$, the $\sigma_{i}$ are single-particle quantum numbers (spins or helicities), and the $\binom{n}{2}$ quantized pairwise helicities $q_{ij}$ are associated with pairs of particles $i,j$. Under a Lorentz transformation $\Lambda$, these states may transform with extra phases arising from nontrivial pairwise little group representations:

The angles $\phi_{ij}=\phi(\Lambda,p_{i},p_{j})$ parametrize pairwise little group transformations, while the $D$’s are unitary representations of one-particle little group transformations.

There is precisely one setting in which nontrivial pairwise helicities are known to arise: the dynamics of electric and magnetic charges. If particles $i,j$ are dyons of charges $(e_{i},g_{i})$ and $(e_{j},g_{j})$, then $q_{ij}$ is simply the quantized angular momentum $\frac{1}{4\pi}(e_{i}g_{j}-e_{j}g_{i})$ in their mutual electromagnetic field. This is easiest to see in a Lorentz frame where their spatial momenta are equal and opposite. Indeed, for a spinless two-dyon state, $q_{ij}$ is the charge under rotations about the axis of the two asymptotically separated dyons. A careful derivation of this result for non-static sources is given in [2].

The above discussion makes clear that pairwise helicity in four dimensions is a reflection of the mutual nonlocality of the underlying electromagnetic sources.

A natural starting point for generalizing pairwise helicity to higher dimensions is to consider $p$-form electrodynamics in $d$-dimensional Minkowski space. In this theory, an electrically charged $(p-1)$-brane couples directly to the $p$-form gauge potential $A$. On the other hand, a magnetic $(d-p-3)$-brane [8, 9, 10] can be thought of as coupling to $A$ through a $(d-p-1)$-form current $G$ localized to the worldvolume of a $(d-p-2)$-dimensional Dirac brane (or generalized Dirac string [5]) that it bounds. The total $(p+1)$-form field strength is then given by $F=dA+\ast G$.

Let us label the time coordinate of $\mathbb{R}^{d-1,1}$ by $x_{0}$ and the spatial coordinates by

Suppose we have an electric $(p-1)$-brane lying along $\vec{y}$ and a magnetic $(d-p-3)$-brane lying along $\vec{z}$. For a magnetic brane (“$i$”) of charge density $g_{i}$ and an electric brane (“$j$”) of charge density $e_{j}$ in a configuration such as that of (4), we define their pairwise helicity to be

Note that the branes can be dyonic when $d=2(p+1)$, in which case we additionally assign the magnetic brane an electric charge density $e_{i}$ and the electric brane a magnetic charge density $g_{j}$.

The quantity $q_{ij}$ in (5) is quantized as a half-integer. This statement is precisely the Dirac quantization condition when $d\neq 2(p+1)$ [8, 9, 10, 11] and the more refined DSZ quantization condition when $d=2(p+1)$ [12, 13, 14, 15, 16]. Indeed, the trajectories of electric and magnetic branes can link in $(d-1)$-dimensional space. Hence amplitudes involving these objects incur an Aharonov-Bohm phase, and quantum-mechanical consistency of the theory places constraints on the allowed electric and magnetic charges.

More directly, however, $q_{ij}$ is nothing other than the angular momentum in the electromagnetic field of branes $i$ and $j$. To see this, we focus on the spatial directions $\mathbb{R}^{d-1}$ in (4). We take brane $i$ (which fills the $\vec{z}$-directions) to lie at the origin in $\vec{x}$, while we take brane $j$ (which fills the $\vec{y}$-directions) to lie at the point $\smash{\vec{R}}=(0,0,R_{3})$ in $\vec{x}$. The angular momentum corresponding to rotations in the $x_{1},x_{2}$ directions is

where $T^{0}{}_{i}=\frac{1}{p!}F^{0}{}_{i_{1}\cdots i_{p}}F_{i}{}^{i_{1}\cdots i_{p}}$ is the linear momentum density of the gauge field. We can write the electric and magnetic fields sourced by the branes as differential forms $E$ and $B$ on $\mathbb{R}^{d-1}$, with components $E^{i_{1}\cdots i_{p}}=F^{0i_{1}\cdots i_{p}}$ and $B^{i_{1}\cdots i_{d-p-2}}=(-1)^{p}(\ast F)^{0i_{1}\cdots i_{d-p-2}}$. They satisfy

where $\smash{\widehat{\Sigma}_{i,j}}$ denote the Poincaré duals of the brane submanifolds $\Sigma_{i,j}\subset\mathbb{R}^{d-1}$ (the operations $d$, $\ast$, and $\widehat{\phantom{m}}$ in (7) are all defined with respect to $\mathbb{R}^{d-1}$). Using (7), we find:

which exactly reproduces (5). When $d=2(p+1)$, the $e_{i}g_{j}$ term in (5) follows from a symmetry argument. Crucially, the result (8) is independent of the separation $|\smash{\vec{R}}|=|R_{3}|$, so it holds equally well for branes that are asymptotically separated in the $x_{3}$-direction.¹¹1Our mathematical conventions, details on the derivation of (8), and additional context on $p$-form electrodynamics can be found in Appendices B, C, and D.

Having explained the dynamical meaning of $q_{12}$, we would like to show that it deserves the moniker “pairwise helicity,” i.e., to interpret it as an $SO(2)$ pairwise little group charge. We propose that states of multiple branes can transform with nontrivial pairwise helicities when they contain pairs of mutually nonlocal branes with $SO(2)$ pairwise little groups. To formalize this statement, we now suggest a framework in which to understand pairwise helicity more systematically.

It is interesting to ask whether branes, like particles, can be described as representations of an appropriate group of spacetime symmetries. Since branes have infinitely many more degrees of freedom than particles, making such a question tractable requires certain simplifying assumptions. For this reason, we concentrate on $(p-1)$-branes of positive tension in $d$ spacetime dimensions that are completely isotropic [17]. The tangent space to a point on the brane worldvolume is a $p$-dimensional linear subspace of Minkowski space that contains a timelike direction. This “brane subspace” is characterized by a decomposition

where $\Pi$ and $\Pi^{\perp}$ are orthogonal projection operators onto the parallel and transverse directions. They have rank $p$ and $d-p$, respectively. The projection tensor $\Pi$ is manifestly invariant under internal $SO(p-1,1)$ Lorentz transformations, i.e., independent of the choice of basis for the brane subspace.

In a given $SO(d-1,1)$ Lorentz frame, the relativistic $d$-velocity of the brane is the normalized timelike vector in this subspace given by

We use “mostly minus” signature, so that $u_{\mu}u^{\mu}=1$. The spatial orientation of the brane is then characterized by a spatial projection tensor $\Xi$ of rank $p-1$:

In any Lorentz frame, the brane’s $d$-velocity is parallel to its worldvolume directions ($\Pi^{\mu}{}_{\nu}u^{\nu}=u^{\mu}$) but orthogonal to its $p-1$ purely spatial directions ($\Xi^{\mu}{}_{\nu}u^{\nu}=0$).

Note that, despite our notation, $u$ and $\Xi$ do not generally transform as Lorentz tensors. On the other hand, because the brane orientation $\Pi$ transforms as a symmetric two-index Lorentz tensor, we can use it to construct representations of the Lorentz group $SO(d-1,1)$. First, we observe that the subgroup of the Lorentz group that leaves $\Pi$ invariant is $SO(p-1,1)_{\parallel}\times SO(d-p)_{\perp}$, corresponding to rotations or boosts in the directions parallel ($\parallel$) and transverse ($\perp$) to the brane. Indeed, we can always choose a Lorentz frame in which the worldvolume projection tensor takes the standard form

This can be achieved by boosting to the brane’s rest frame, where $\tilde{u}^{\mu}=(1,0,\ldots,0)$, and performing an appropriate spatial rotation. For simplicity, we consider only brane representations of the Lorentz group in which the noncompact $SO(p-1,1)_{\parallel}$ factor is represented trivially. We refer to the compact $SO(d-p)_{\perp}$ factor as the single-brane little group, dropping the subscript $\perp$ from here on. To specify a representation of the $SO(d-p)$ little group, we introduce a collective label $\sigma$ that includes all necessary Casimirs. We call $\sigma$ the “spin” under $SO(d-p)$.

Following Wigner, we introduce a collection of single-brane states $|\Pi;\sigma\rangle$ that transform in an infinite-dimensional unitary representation of the $d$-dimensional Lorentz group according to

where $W=W(\Lambda,\Pi,\tilde{\Pi})$ is a little group transformation. (Here, we write $\Pi$ for the matrix $\Pi^{\mu}{}_{\nu}$ without indices.) These brane states, unlike particle states, are not constructed as eigenstates of the translation generators of the Poincaré algebra. Indeed, while the $d$-momentum per unit volume is given by $p^{\mu}=Tu^{\mu}$ in terms of the brane tension $T$ (with mass dimension $[T]=p$), $p^{\mu}$ does not correspond to the integrated Poincaré charge $P^{\mu}$ (which diverges in states of infinite branes), nor does it transform as a Lorentz vector.²²2Further details on this construction and its subsequent generalization to multiple branes are given in Appendix A.

In the case of a particle ($p=1$), we have $T=m>0$ and $\Pi^{\mu\nu}=u^{\mu}u^{\nu}$. Hence $\Pi$ and $u$ contain precisely the same information, under the assumption that the timelike component of $u$ is positive. In this case, (10) becomes a tautology and $u$ transforms as a genuine Lorentz vector. The little group reduces to the expected $SO(d-1)$ for a massive particle. In the opposite case of a space-filling brane ($p=d$), we have $\Pi^{\mu\nu}=\eta^{\mu\nu}$ in any frame, and $u$ transforms as a Lorentz scalar.

We now illustrate how the definitions of pairwise little group and pairwise helicity generalize to a state of two branes relevant to the calculation of the preceding section. We then turn to multi-brane states in generality.

Consider an electric $(p-1)$-brane and a magnetic $(d-p-3)$-brane that span orthogonal directions in space, as in (4). In other words, we suppose that there exists a frame in which their spatial projection tensors $\Xi_{1}$ and $\Xi_{2}$ take the standard forms

Suppose also that the branes have generic (spatially transverse) $d$-velocities $u_{1}$, $u_{2}$. Via an $SO(3,1)\subset SO(d-1,1)$ Lorentz transformation acting only on $(x_{0},\vec{x})$, which preserves the spatial projection tensors (14)–(15), any $u_{1}$, $u_{2}$ may be brought to the standard forms

where $(u_{i}^{0})^{2}-u_{c}^{2}-(\vec{u}_{i})^{2}=1$ for $i=1,2$. This frame makes manifest that there exists an $SO(2)$ subgroup of the Lorentz group that preserves $\tilde{\Xi}_{1}$, $\tilde{\Xi}_{2}$, and any two generic transverse $d$-velocities simultaneously.³³3Although the full pairwise little group of two branes in the configuration (14)–(17) is $SO(2)\times SO(p-2)\times SO(d-p-4)$, we focus on the $SO(2)$ subgroup of external rotations because it will turn out to admit a transparent physical interpretation. When the $\vec{x}$-components of their velocities are equal and opposite along the $x_{3}$-axis as in (16)–(17), the $SO(2)$ pairwise little group of these two branes consists of rotations $R_{12}$ in the plane of $x_{1},x_{2}$.

We can now define a two-brane representation of the Lorentz group with respect to the reference projectors $(\tilde{\Pi}_{i})^{\mu\nu}=(\tilde{u}_{i})^{\mu}(\tilde{u}_{i})^{\nu}+(\tilde{\Xi}_{i})^{\mu\nu}$ defined by (14)–(17). This representation is carried by states $|\Pi_{1},\Pi_{2};\sigma_{1},\sigma_{2};\sigma_{12}\rangle$, where the $\sigma_{i}$ label single-brane little group representations and $\sigma_{12}\equiv q_{12}$ labels a representation of the aforementioned $SO(2)$ pairwise little group. These states transform according to

where the $D$’s are unitary representations of single-brane little group transformations. As any pairwise little group transformation takes the form $W_{12}=R_{12}(\phi_{12})$ for some angle $\phi_{12}=\phi(\Lambda,\Pi_{1},\Pi_{2})$, we have written $D(W_{12})=e^{iq_{12}\phi_{12}}$. The $SO(2)$ charge $q_{12}$, or the pairwise helicity, can thus be identified with the electromagnetic angular momentum of these mutually nonlocal branes (which are asymptotically separated along the $x_{3}$-axis). Furthermore, we have seen that $q_{12}$ satisfies a quantization condition that gives rise to a bona fide representation of $SO(2)$.

Finally, we define general states of asymptotically separated, non-intersecting branes in $\mathbb{R}^{d-1,1}$, in the spirit of the four-dimensional multiparticle states of [3, 4]. Such states may have quantum numbers that are not captured by tensor products of single-brane states.

We define a general $N$-brane Lorentz representation by a set of reference projection tensors $\{\tilde{\Pi}_{i}\,|\,i=1,\ldots,N\}$ and a set of representations $\{\sigma_{\ell}\,|\,\ell\in\mathcal{L}\}$, where $\mathcal{L}$ indexes all possible little groups $G_{\ell}$, which are subgroups of the Lorentz group that stabilize the reference data. Explicitly, we take $\mathcal{L}$ to consist of all $2^{N}-1$ nontrivial subsets of $\{1,\ldots,N\}$. For each $\ell\in\mathcal{L}$, we take $G_{\ell}$ to consist of those Lorentz transformations $\lambda\in SO(d-1,1)$ for which

Every state in this representation is defined in terms of some canonical Lorentz transformations relating it to the reference state $|\tilde{\Pi}_{1},\ldots,\tilde{\Pi}_{N};\{\sigma_{\ell}\,|\,\ell\in\mathcal{L}\}\rangle$, and an arbitrary state transforms as

where each $W_{\ell}\in G_{\ell}$ is a little group transformation that depends on $\{\Pi_{i}\,|\,i\in\ell\}$ as well as on $\Lambda$. Those $N$-brane representations that are induced by specifying nontrivial representations $\sigma_{\ell}$ only for $|\ell|=1$ are tensor products of single-brane representations. In principle, however, we are free to specify a nontrivial representation of the entire little group $\prod_{\ell\in\mathcal{L}}G_{\ell}$. We have shown that the induced representations arising from nontrivial $\sigma_{\ell}$ for $|\ell|=2$ have physical relevance, as was known to be the case even for particles [4].

In $d=4$, all higher little groups beyond the pairwise ones are generically trivial. This is no longer true in higher dimensions. Therefore, it is natural to ask whether the higher little groups with $|\ell|>2$ have any significance in $d>4$. We content ourselves with addressing a simpler question: given a set of reference data for $n$ branes, what is the $n$-brane little group (for any $n=|\ell|$)?

To approach this question, it is simplest to begin with the case of particles. We define the $n$-particle little group as the subgroup of the $d$-dimensional Lorentz group that leaves invariant an arbitrary set of $n$ momenta (or velocities). For generic momenta, this group is $SO(d-n)$. The reasoning is simple: given $n$ generic (i.e., linearly independent) $d$-vectors, this $SO(d-n)$ preserves the orthogonal complement of their span.

Now consider $n$ branes with projection tensors $\Pi_{i}$ for $i=1,\ldots,n$. Let $V_{i}$ denote the corresponding linear subspaces of Minkowski space $\mathbb{R}^{d-1,1}$, i.e., the images of the $\Pi_{i}$. Lorentz transformations that preserve a given subspace of Minkowski space act purely within the subspace or purely within its orthogonal complement. Therefore, those Lorentz transformations acting within any of the subspaces

for $(i_{1},\ldots,i_{n})\in\{0,1\}^{n}$, where $V_{i}^{0}\equiv V_{i}$ and $V_{i}^{1}\equiv V_{i}^{\perp}$, preserve all subspaces $V_{i}$ simultaneously. The $n$-brane little group is thus at least as large as

where the group $SO(m)$ is understood to be trivial for integers $m\leq 1$.⁴⁴4Note that the subspace (21) can contain a timelike direction only when $(i_{1},\ldots,i_{n})=(0,\ldots,0)$. If it does, then the signature of the corresponding $SO$ factor in (22) should be adjusted accordingly. For our branes of interest, we would expect the corresponding $SO$ factor to be represented trivially, since it consists only of Lorentz transformations internal to all branes. The actual little group may in fact be larger (in particular, we have ignored possible discrete subgroups).

Let us see what the formula (22) gives in special cases. If all $n$ branes are particles with generic $d$-momenta, then none of the momenta are collinear. Hence the only factor that contributes to (22) is that with $(i_{1},\ldots,i_{n})=(1,\ldots,1)$, giving back the $n$-particle little group

where we have used the fact that the intersection of $n$ generic hyperplanes ($(d-1)$-dimensional subspaces) has dimension $d-n$. In the other extreme case that all $n$ branes are space-filling, we get simply

For a single $(p-1)$-brane ($n=1$) spanning the subspace $V$, we get the expected little group

including the internal Lorentz transformations that we take to be represented trivially. Finally, consider a $(p_{1}-1)$-brane and a $(p_{2}-1)$-brane. Generically, the intersection of $d_{1}$- and $d_{2}$-dimensional linear subspaces has dimension $\max(d_{1}+d_{2}-d,0)$. So for generic configurations of these two branes, the corresponding pairwise little group always contains an $SO(d-p_{1}-p_{2})$ factor.⁵⁵5Note that the specific two-brane configuration in (14)–(17) was chosen to simplify the angular momentum analysis for $p_{1}=d\linebreak[1]-\linebreak[1]p_{2}\linebreak[1]-2\equiv p$, so the corresponding pairwise little group is larger than would be expected generically.

We have not presented an exhaustive classification of multi-brane representations of the Lorentz group. While the subgroup of $SO(d-1,1)$ that preserves any given set of brane subspaces always contains a subgroup isomorphic to (22), we have not attempted to find the full stabilizer subgroup or to catalogue all such groups that could arise. In addition, we have excluded non-generic “tensionless” brane representations with null (lightlike) $d$-velocities.

In this paper, we have constructed representations of the Lorentz group that describe ideal branes whose worldvolume directions span linear subspaces of Minkowski spacetime. We have illustrated how quantum numbers associated to pairwise little groups of multi-brane representations can arise from the coupling of branes to abelian gauge fields in higher dimensions.

This study raises many more questions than answers. We highlight some of these questions below.

I thank Csaba Csáki, Ofri Telem, and John Terning for initial collaboration. I thank Jacques Distler, Dan Freed, Vadim Kaplunovsky, Andreas Karch, Liam McAllister, and Aaron Zimmerman for helpful discussions. Finally, I thank Steven Weinberg [26] and Jacques Distler [27] for providing the intellectual impetus to contemplate representations of the Lorentz group in higher dimensions. This work was supported in part by the NSF grant PHY-1914679.