Symmetries in Physics
Symmetry has been a guiding principle for physics ever since its modern conception in the work of Issac Newton[1], which eventually led to the reformulation of classical mechanics by Lagrange[2,3] and Hamilton[4]. In the context of dynamical physical systems, symmetries are transformations of the degrees of freedom such that either the physical configuration has a well-defined transformation or it remains the same (i.e. is invariant). Accordingly, symmetries come in two broad types: (1) Genuine symmetries; (2) Gauge symmetries. Each type of such transformations forms a group.[a]
As the name suggests, the second type of symmetries are not really symmetries but have to be considered as redundancies in the description of the physical systems. As such, it might be a misnomer to call them symmetries and a better name that is usually used is gauge redundancies, which still form a group. Furthermore, genuine symmetries can be further decomposed into spacetime symmetries and global symmetries. For example, the group of symmetries of the four-dimensional Minkowski spacetime is the Poincaré group while the global symmetry group of say quantum chromodynamics, the theory describing the strong interactions, includes the $\text{SU}(2)\times\text{SU}(2)$ group of chiral symmetries.
A physical system is described by a four-tuple $(D,L,G,H)$, where $D$ collectively denotes the physical degrees of freedom of the system together with their physical properties, $L$ is called the Lagrangian describing the dynamics of the system, $G$ and $H$ are the groups of genuine symmetries and redundancies of the system. As an example, one can consider a particle moving on the surface of a sphere. Here, $S$ is just the particle, its mass, and its location with respect to say the centre of the sphere, $G$ is the symmetry group of the sphere, which is the orthogonal group $\text{O}(3)$, and $H$ is trivial since there are no gauge redundancies.
Noether's Theorem
The importance of symmetry in modern physics has been conceived in the foundational work of Emmy Noether[5]. In this viewpoint, the presence of symmetries is associated with the existence of conservation laws. For example, consider a particle moving in one dimension. The degree of freedom is the particle's location. Quantitatively, this degree of freedom is described by its location $x$ and the aim is to describe $x$ as a function of time $t$, i.e. $x=x(t)$. To proceed further, we need to give the Lagrangian $L$, which is given by the difference between the kinetic $K$ and potential $U$ energies, i.e. $L=K-U$. For a particle of mass $m$, $K=\frac{1}{2}m(d x/d t)^2:=\frac{1}{2}m\dot{x}^2$, with $\dot{x}$ is the velocity of the particle, and $U$ is a generic function of $x$ and $t$, i.e. $U=U(x,t)$. The dynamics of the system can then be described by a solution to the Euler-Lagrange equations of motion
\begin{equation}
\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{x}}\right)-\frac{\partial L}{\partial x}=0,
\end{equation}subject to a boundary condition. If we assume that $U(x,t)$ does not depend on $x$ (i.e. the symmetry of translation of the system along $x$), then $\frac{\partial L}{\partial x}=0$, and we end up with
\begin{equation}
\frac{d }{d t}(m\dot{x})=0.
\end{equation}This means that the quantity $m\dot{x}$, called the momentum of the particle, is independent of time, and hence is conserved. On the other hand, by computing $\frac{d}{d t}L$ and using the Euler-Largrange equations, we have
\begin{equation}
\frac{d}{d t}\left(\dot{x}\frac{\partial L}{\partial\dot{x}}-L\right)=-\frac{\partial L}{\partial t}.
\end{equation}The only explicit time-dependence in $L$ can enter through the potential $U(x,t)$. Therefore, assuming $U=U(x)$ (i.e. the symmetry of translation of the system along $t$), we see that $\dot{x}\frac{\partial L}{\partial\dot{x}}-L$ is a conserved quantity. It is easy to see that it is just $K+U$, the total energy of the system.
Symmetries and Quantum Mechanics
As quantum mechanics describes the behaviour of physical systems at microscopic scales, the role of symmetries is even more prominent. As a system $S$ has a symmetry $G$, the degrees of freedom of the system are fixed by determining how they transform under $G$. On the other hand, as $H$ is describing the redundancies of the system, only invariant quantities under a transformation by elements of $H$ are physically relevant. This intuition can be make quantitative by the notion of the representation theory of $G$ and $H$. Basically, this means that our degrees of freedom belong to an abstract space and elements of $G$ and $H$ are represented as transformations acting on that abstract space. For example, in the case of quantum mechanics, this abstract space is the Hilbert space of the system consisting of possible quantum states. Then, the degrees of freedom are characterised by how they transform under $G$ and $H$. The hallmark of this construction is Wigner's Theorem, according to which symmetries of a quantum-mechanical system can be represented as unitary or anti-unitary operators acting on the Hilbert space of the theory[6,7,8,9]. To find a proper quantisation of a physical system, one thus needs to know how to classify (or label) representations of $G$, as they provide the label for degrees of freedom of the system.
Symmetries in the Absence of Gravity
In the absence of gravity, i.e. when the special theory of relativity[10,11] is sufficient for the description of physical systems, the spacetime symmetries are encoded in the four-dimensional Poincaré group. According to what we have said so far, the degrees of freedom are thus labelled by how they transform under this group. It turns out that the degrees of freedom are labeled by two quantities associated with the Lie algebra of the Poincaré group: the mass and their spin, which in turn are determined by the two Casimir elements of the Poincaré algebra. On top of these spacetime symmetries, one can introduce various global and gauge symmetries as dictated by the observation. It turns out that this characterisation of physical degrees of freedom provides the required groundwork for the quantisation of physical theories describing observable physics in the absence of gravitational interactions[12,13,14].
Let us elaborate a bit on the mathematical structure of symmetry group in this case. In the absence of gravity, for a physical system compatible with special relativity, the physical processes happen in Minkowski spacetime $M$. Such a system should thus be invariant under isometries of $M$ since it should not matter where on $M$ we put the physical system as long as we do not change the grids by which we do the measurements: these isometries are translations in space and in time, orthogonal spacetime transformations involving rotation in spatial directions and boosts which involve both space and time transformations. The latter types together are called Lorentz transformations. Translations, denoted as $\mathbb{R}^{1,3}$, and Lorentz transformations, denoted as $\text{Lor}(M)$ form the Poincaré group $\text{Poin}(M)$ which is the isometry group of physical spacetime $M$ in the absence of gravity. It turns out that there is a mixture between translations and Lorentz transformations, i.e. if one does first a translation and then a Lorentz transformation, the result is not the same as first doing the Lorentz transformation and then the translation. In mathematical language, this is expressed by saying that Lorentz transformations is acting non-trivially on translations. This leads to the notion of semi-direct product. In this case,
\begin{equation}
\text{Poin}(M)=\text{Lor}(M)\ltimes\mathbb{R}^{1,3},
\end{equation}where the symbol $\ltimes$ emphasises that there is a mixture between $\text{Lor}(M)$ and $\mathbb{R}^{1,3}$, or more precisely, an action of $\text{Lor}(M)$ on $\mathbb{R}^{1,3}$.
As we will see below, even in the presence of gravity, we would have a semi-direct product structure $G=K\ltimes N$, for some group $K$ and $N$, and $\ltimes$ means that $K$ acts on $N$ in some way. In particular, the Cartesian product $K\times N$ is a particular example of this construction where the action is trivial since there is no mixture between transformations by $K$ and those by $N$ in that case.
The Biggest Elephant in the Room, Quantum Gravity
The most prominent challenge of modern theoretical physics is the formulation of the quantum theory of gravitational interactions, a puzzle which more than a century after the formulation of the accepted theory of classical gravity by Einstein[15] is persisting very hard against any final resolution. In their ground-breaking work on the foundation of canonical formulation of quantum field theory in the context of quantum electrodynamics[16,17], Pauli and Heisenberg claimed[16][b]
One should mention that a quantisation of the gravitational field, which appears to be necessary for physical reasons, may be carried out without any new difficulties by means of a formalism wholly analogous to that applied here.
This project was pushed forward to some extent by Rosenfeld[19,20]. However, it turned out to be significantly more involved than it was originally conceived by Pauli and Heisenberg. As later was realised in the pioneering work of Matvei Petrovich Bronstein, the non-linear nature of gravitational interactions makes the Pauli-Heisenberg quantisation scheme hard to properly implement[21,22]. As he notes
The elimination of the logical inconsistencies connected with this result requires a radical reconstruction of the theory, and in particular, the rejection of a [pseudo-]Riemannian geometry dealing, as we have seen here, with values unobservable in principle, and perhaps also rejection of our ordinary concepts of space and time, modifying them by some much deeper and nonevident concepts. Wer's nicht glaubt, bezahlt einen Taler (Let him who doubts it pay a Thaler).
Shortly after Bronstein's work, Jacques Solomon emphasised[23]
In the case when the gravitational field is not weak, the very method of quantisation based on the superposition principle fails, so that it is no longer possible to apply a relation such as [the equation setting a lower limit on the measurability of the linearised field strength] in an unambiguous way ... Such considerations are of a sort to put seriously in doubt the possibility of reconciling the present formalism of field quantisation with the non-linear theory of gravitation.
It should be noted that the non-linearity is not the only issue with the quantisation of gravity, as non-Abelian gauge theories are also non-linear but can be quantised. One of the most challenging aspects is the issue of non-renormalisability of Einstein's theory[24,25,26], a problem that has been successfully resolved for non-Abelian gauge theories[27,28,29,30] and is considered as one of the greatest triumphs of 20th-century physics. This basically means it is not possible to extract observable quantities with finite values from the quantised general relativity. On a different front, another struggle with this multi-headed monster comes from its connection to various outstanding questions in mathematics like the classification of (topological or differentiable) four-manifolds, problems that may not even be solvable, or having a handle on the space of Lorentzian metrics on a four-manifold. Yet another demand from such a theory is that it should provide a totally new perspective on many deep questions like the origin of time, its emergence, and also its macroscopic direction (i.e. its forwardness). Although some ideas put some doubts on the necessity for the quantisation of gravity[31,32,33,34], the logical follow-up to the known physics and the problems which involve the simultaneous presence of gravity and quantum mechanics have motivated physicists to face this extraordinary challenge. It should be emphasised that the problem of quantum gravity is not necessarily related to the unification of fundamental interactions, an idea pursued in string theory.
The Role of Symmetries in Classical and Quantum Gravity
Proceeding in the same spirit of the non-gravitational physics explained above, one may wonder what happens in the presence of gravity. The first puzzle is to identify the proper degrees of freedom to be quantised. As we have seen above, in the absence of gravity, these degrees of freedom are labelled by a symmetry group. This will lead us to the question of identifying the proper notion of symmetry group in the presence of gravity.
The general theory of relativity, the theory describing classical gravity, has a big group of redundancies, the group of arbitrary coordinate transformations of spacetime, forming its diffeomorphism group. By analogy with the case of Minkowski spacetime and its isometry group, i.e. the Poincaré group, one might be tempted to consider the isometries of a fixed metric on spacetime as a symmetry group in gravity. However, in quantum gravity, one is dealing with all possible metrics on a spacetime, therefore one cannot consider the isometries of a given metric to define a notion of symmetry group in gravity. On the other hand, spacetimes which are compact are not usually interesting as they contain closed time-like curves. Noncompact spacetimes are difficult to deal with due to various mathematical subtleties. A method to work with non-compact spacetimes is to consider their conformal completion by adding certain extra points at infinity, a method invented by Penrose[35,36]. These conformal completions have boundaries, and the presence of boundaries forces the impositions of (asymptotic) structures. By fixing such a structure, some of the diffeomorphisms that preserve this structure become physical (asymptotic) symmetries acting on the (asymptotic) phase space of the theory. It turns out that by certain mild assumptions, the set of all such transformations forms an infinite-dimensional group. The explicit form of this group depends on the universal structure, and the presence or absence of cosmological constant and its sign.
The Concept of Spin and Symmetry Groups in Gravity
The first and most famous example of symmetry groups in the absence of cosmological constant (i.e. for asymptotically-flat spacetimes) appeared in the famous work of Bondi, Van der Burg, and Metzner and also Sachs, the so-called BMS group[37,38]. To introduce this group, we note that the topology of the boundary of the conformal completion of asymptotically-flat spacetimes is fixed to be $S^2\times\mathbb{R}$, where $S^2$ is the 2-dimensional sphere, and $\mathbb{R}$ is the time direction. Then, the BMS group has the following form
\begin{equation}
\text{BMS}:=\text{Lor}(S^2)\ltimes C^\infty(S^2).
\end{equation} $\text{Lor}(S^2)$ denotes the Lorentz group, realized as the group of conformal Killing vectors of $S^2$, and $C^\infty(S^2)$ denotes the space of smooth functions on $S^2$. Therefore, one might hope by studying the representation theory of this group, the correct degrees of freedom for gravity can be identified. However, there are two major obstacles: (1) the BMS group, unlike the case of Poincaré group, is infinite-dimensional and its representation theory is significantly more complicated[39,40,41,42,43,44,45]; (2) Even if there is no issue with the representations theory, another major issue, as pointed out by Sachs[38], is that there is no notion of spin associated with this group.
Let us be a bit more precise about what we mean by the notion of spin. As the structure of the BMS group shows, the symmetry groups that appear in the context of gravity have the structure of a semi-direct product $G=K\ltimes N$. If we denote the corresponding Lie algebra as $\mathfrak{g}=\mathfrak{k}\oplus_{\text{ad}} \mathfrak{n}$, where $\text{ad}$ denotes the adjoint action of $\mathfrak{k}$ on $\mathfrak{n}$, then one can define the so-called isotropy or little algebra of an element $n\in\mathfrak{n}$ as follows
\begin{equation}
\text{lit}(n):=\big\{X\in\mathfrak{k}\,|\,\text{ad}_Xn=0\big\}.
\end{equation} For generic values, the isotropy subalgebra of two distinct points $n$ and $n'$ are isomorphic. Now, what we will call spin $s$ has two properties: (1) it generates $\text{lit}(n)$, and (2) it is invariant under translations by elements of $\mathfrak{n}$, i.e. $[s,\mathfrak{n}]=0$, where $[\cdot,\cdot]$ denotes the Lie bracket of $\mathfrak{g}$. The most familiar example of spin is for the Poincaré algebra. In that case, the so-called Pauli-Lubański (pseudo-)vector generates the isotropy subalgebra of the Poincaré algebra and also is invariant under translations of the Minkowski spacetime[47,48]. Therefore, if we would like to generalise the notion of spin in the presence of gravity, our task is to first identify the correct symmetry group in gravity.
As we stated above, the notion of spin does not exist for the BMS algebra, essentially due to the fact that the isotropy group is discrete and there is no Lie algebra associated with it. On the other hand, there should be a notion of spin in the presence of gravity that when the gravity is switched off just gives us back the notion of spin in the absence of gravity. The later notion of spin is well-defined and it is essential for the formulation of quantum field theory. As we cannot associate a good notion of spin with the BMS group, one possible resolution is to consider alternative symmetry groups in gravity, by demanding a different asymptotic structure. It turns out essential to enlarge the BMS group to the so-called generalised BMS (GBMS) group to account for certain physical corrections to the scattering processes[48,49]. This group has the following form[c]
\begin{equation}
\text{GBMS}:=\text{Diff}(S^2)\ltimes C^\infty(S^2),
\end{equation} where $\text{Diff}(S^2)$ is the diffeomorphism group of $S^2$. In physical terminology, the elements of $\text{Diff}(S^2)$ are called super-rotations, and the elements of $C^\infty(S^2)$ are called super-translations, i.e. translations along $\mathbb{R}$ in $\mathbb{R}\times S^2$ that depend on the points of $S^2$. This group, being a far bigger group than the BMS group, is still infinite-dimensional and the study of its representation theory would be quite complicated. Therefore, one would look for an alternative route to study this group and the possible notion of spin in gravity.
The Method of Coadjoint Orbits
In the canonical formulation[4], a classical dynamical system is studied through its phase space which, by definition, is the space of solutions to the Euler-Lagrange equations of motion. An important feature of this space is that it is naturally a symplectic manifold and hence can be quantised, i.e. a Hilbert space can be assigned to it. On the other hand, the Hilbert space of a dynamical system is the representation space of its symmetry group $G$ and also carries the trivial representation of the group of gauge transformations $H$. Therefore, the broad question is how to construct the representation space of a symmetry group $G$ of a physical system. One powerful method, invented by Alexandre Kirillov[50,51,52], is the method of coadjoint orbits. According to this method, a representation space can be associated with each coadjoint orbit of a group $G$, i.e. the orbit of coadjoint action of $G$ on $\mathfrak{g}^*$, where $\mathfrak{g}^*$ is the dual of $\mathfrak{g}=\text{Lie}(G)$, the Lie algebra of $G$. Each such orbit is a symplectic manifold and hence can be quantised using various methods such as geometric quantisation or the quantisation via branes[53,54].
As GBMS acts as symmetries on the phase space of a gravitational system (i.e. an asymptotically-flat spacetime), one can construct the associated Hilbert space by first classifying the coadjoint orbits of GBMS. Once this is done, the phase space can be realised as a certain decomposition of the symplectic manifolds constructed out of coadjoint orbits. The first part of this program is achieved in[55]. An important part of the classification of coadjoint orbits of GBMS is the construction of certain functions on the orbits, the so-called Casimir function(al)s, which are invariant under the coadjoint action. The definition of these function(al)s depends on a quantity that, in the context of fluid dynamics, is called vorticity. To make this a bit more quantitative, first notice that $\mathfrak{gbms}^*$, the dual of the Lie algebra $\mathfrak{gbms}:=\text{Lie}(\text{GBMS})$, is the set of all pairs $(j,m)$, where $j\in\mathfrak{diff}(S^2)^*$ and $m\in C^\infty(S^2)^*$, with $\mathfrak{diff}(S^2)^*$ is the dual of $\mathfrak{diff}(S^2)=\text{Lie}(\text{Diff}(S^2))$ and $C^\infty(S^2)^*$ denotes the dual of $C^\infty(S^2)$. $j$ and $m$ are a one-form and a function on $S^2$, respectively. In terms of these quantities, the vorticity is a two-form defined as
\begin{equation}
w:=m^{-\frac{2}{3}}\,d_{S^2}(m^{-\frac{2}{3}}j),
\end{equation} where $d_{S^2}=\sum_{a=1}^2 d\sigma^a\partial_{\sigma^a}$ is the differential on $S^2$ written in local coordinates $\sigma^a$ on $S^2$. The intriguing feature of this quantity is that it is invariant under the action of supertranslations (i.e. under the coadjoint actions of elements in $C^\infty(S^2)$) and furthermore, they transform under superrotations by pullback. These desired properties can be used then to construct Casimir function(al)s on coadjoint orbits[55]. It turns out that the celebrated No-Hair Theorem of the Kerr black hole[56,57,58] can be entirely restated in terms of these Casimirs; In particular, the mass and angular momentum of the Kerr black hole can be computed in terms of these Casimirs[55].
And Finally, the Spin
Our main objective was to define the spin for the symmetry algebra $\mathfrak{gbms}$. The first point to notice is that as we are working with coalgebra $\mathfrak{gbms}^*$ (and not the Lie algebra $\mathfrak{gbms}$ itself), we need to modify the definition of the isotropy subalgebra $\text{lit}(n)$ given above. For $m\in C^\infty(S^2)^*$, this is given by\begin{equation}
\text{lit}^*(m):=\big\{X\in\mathfrak{diff}(S^2)\,|\,\text{ad}^*_Xm=0\big\},
\end{equation}where $\text{ad}^*_X$ is the coadjoint action by a vector field $X$ on $S^2$. Then, one can define the following quantity, smeared by a function $\phi$ on the sphere, from the vorticity\begin{equation}
S[\phi]:=\int_{S^2}\phi\,m^{\frac{2}{3}}w.
\end{equation}This, as a quantity on $\mathfrak{gbms}^*$, has two properties: (1) it is invariant under supertranslations, and (2) it generates the isotropy subalgebra of $\mathfrak{gbms}$, which is given by\begin{equation}
\text{lit}^*(m)=\mathfrak{sdiff}_m(S^2)\oplus C^\infty(S^2),
\end{equation} where $\mathfrak{sdiff}_m(S^2)$ is the subalgebra of $\mathfrak{diff}(S^2)$ that preserves an $m$-dependent area form. The fact that $S[\phi]$ generates the action of the isotropy subalgebra $ \text{lit}^*(m)$ on $\mathfrak{gbms}^*$ is the following statement\begin{equation}
\{S[\phi],S[\phi']\}_{\mathfrak{gbms}^*}=S[\{\phi,\phi'\}_{S^2,m}],
\end{equation}where $\{\cdot,\cdot\}_{\mathfrak{gbms}^*}$ is the Lie-Poisson structure on $\mathfrak{gbms}^*$ and $\{\phi,\phi'\}_{S^2,m}$ is a certain $m$-dependent Poisson bracket on $S^2$. Therefore, $S[\phi]$ is the proposal for spin degrees of freedom of gravity in an asymptotically-flat spacetime. However, one last question remains!
Moment Map
One may be curious that all of this is about algebraic aspects of the $\mathfrak{gbms}$ algebra, so where is the physics? The action of $\mathfrak{gbms}$ on a certain phase space of gravity $\Gamma$ is canonical, and hence a moment map $\mu:\Gamma\to\mathfrak{gbms}^*$ can be assigned to it. This moment map, which is explicitly constructed in[55], as a smooth map can be used to construct quantities on $\Gamma$ from those on $\mathfrak{gbms}^*$ by the pullback operation. Therefore, one can translate statements about quantities on $\mathfrak{gbms}^*$ to the phase space $\Gamma$. For example, one can define a quantity on $\Gamma$\begin{equation}
\mathcal{S}[\phi]:=\mu^*S[\phi],
\end{equation}which by construction satisfies\begin{equation}
\{\mathcal{S}[\phi],\mathcal{S}[\phi']\}_{\Gamma}=\mathcal{S}[\{\phi,\phi'\}_{S^2,m}],
\end{equation} with $\{\cdot,\cdot\}_{\Gamma}$ being the canonical Poisson bracket on $\Gamma$. This relation is nothing but the statement that $\mathcal{S}[\phi]$ implements the action of $\mathfrak{sdiff}_m(S^2)$ on the phase space $\Gamma$. Hence, it can be interpreted as the gravitational spin, as claimed before.
The Road Ahead
There are many things left to be done. Some of the most exciting ones are:
- One can study the coadjoint orbits of the Poincaré algebra. The result is that there are two Casimir functions, which upon quantisation give the usual Casimir elements of the Poincaré algebra, hence the labels for elementary particles. By the above result, one can now ask the similar question about the generalised BMS group: What are the quantisation of these coadjoint orbits? And what is the role of spin in this quantisation?
- How to express the phase space in terms of the coadjoint orbits? More precisely, can phase space be expressed as the inverse image of coadjoint orbits under the moment map constructed in[55]?
- How to compute the entropy of black holes in asymptotically-flat spacetimes in terms of the Casimirs of asymptotic-symmetry algebra? Is it even possible or meaningful?
- It has been shown in[55] that the embedding of a copy of Lorentz algebra in $\mathfrak{gbms}$ can be understood using a symmetry-breaking mechanism. How would this affect the formulation of scattering processes in asymptotically-flat spacetimes?
- Classical symmetries could be anomalous at the quantum level, which means that due to some quantum-mechanical effect, the symmetry group breaks into a subgroup, possibly the trivial one. A fascinating question is the possibility of the survival of asymptotic symmetries of gravity at the quantum level. The existence of anomalies would have important physical implications.
As explained above, the answers to these questions pave the way for understanding the Hilbert space of quantum gravity in asymptotically-flat spacetimes, a long-standing dream! Even that would not be the end of the story as for cosmological questions one must include the positively-valued cosmological constant, an endeavour which by itself could be a life-long project!
Seyed Faroogh Moosavian is a postdoctoral research associate in mathematical physics at Oxford Mathematical Institute.
Endnotes
[a] In the following, and for simplicity, the modern and generalised conception of symmetry in terms of more exotic mathematical structures like higher groups and fusion categories is totally avoided.
[b] All the quotes are taken from the article by Stachel[18].
[c] The action of $\text{Diff}(S^2)$ on $C^\infty(S^2)$ involves a certain density weight, which has been omitted in the notation.
References
[1] Newton, I. (1687). Philosophiae Naturalis Principia Mathematica; (p. 581 pp). New-York : Published by Daniel Adee.
[2] Lagrange, J.-L. (1811). Mécanique Analytique, Vol. 1 (2nd ed.). Paris, Ve Courcier.
[3] Lagrange, J.-L. (1815). Mécanique Analytique, Vol. 2 (2nd ed.). Paris, Ve Courcier.
[4] Hamilton, W. R. (1833). On a General Method of Expressing the Paths of Light and of the Planets by the Coefficients of a Characteristic Function. Dublin Uni. Rev. Q. Mag., 1, 795–826.
[5] Noether, E. (1918). Invariante Variationsprobleme [Invariant Variation Problems]. Nachr. Ges. Wiss. Gottingen, Math.-Phys. Kl., 1918, 235–257 [arXiv:physics/0503066].
[6] Wigner, E. (1931). Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren. Vieweg+Teubner Verlag.
[7] Wigner, E. (1959). Group Theory: And Its Application to the Quantum Mechanics of Atomic Spectra. Elsevier.
[8] Bargmann, V. (1964). Note on Wigner’s Theorem on Symmetry Operations. J. Math. Phys., 5, 862–868.
[9] Uhlhorn, U. (1963). Representation of Symmetry Transformations in Quantum Mechanics. Ark. Fys., 23.
[10] Einstein, A. (1905). Zur Elektrodynamik Bewegter Körper [On the Electrodynamics of Moving Bodies]. Ann. Phys., 322(10), 891–921.
[11] Einstein, A. (1905). Ist die Trägheit Eines Körpers von Seinem Energieinhalt Abhängig? [Does the Inertia of a Body Depend Upon Its Energy Content?]. Ann. Phys., 323(13), 639–641.
[12] Glashow, S. L. (1961). Partial Symmetries of Weak Interactions. Nucl. Phys., 22, 579–588.
[13] Weinberg, S. (1967). A Model of Leptons. Phys. Rev. Lett., 19, 1264–1266.
[14] Salam, A. (1968). Weak and Electromagnetic Interactions. In N. Svartholm (Ed.), Physical Formulations: Elementary Particle Theory. Relativistic Groups and Analyticity. Proceedings of the Eighth Nobel Symposium, Aspenäsgården, Lerum, Sweden, May 1968. (pp. 367–377).
[15] Einstein, A. (1915). Die Feldgleichungen der Gravitation [The Field Equations of Gravitation]. Sitzungsber. Kgl. Preuss. Akad. Wiss., 844–847.
[16] Heisenberg, W., & Pauli, W. (1929). Zur Quantendynamik der Wellenfelder [On the Quantum Dynamics of Wave Fields]. Z. Phys., 56(1–2), 1–61.
[17] Heisenberg, W., & Pauli, W. (1930). Zur Quantentheorie der Wellenfelder. II [On the Quantum Dynamics of Wave Fields II]. Z. Phys., 59(3–4), 168–190.
[18] Stachel, J. (1999). The Early History of Quantum Gravity (1916–1940). In B. R. Iyer & B. Bhawal (Eds.), Black Holes, Gravitational Radiation and the Universe - Essays in Honor of C.V. Vishveshwara (pp. 525–534). Springer Netherlands.
[19] Rosenfeld, L. (1930). Zur Quantelung der Wellenfelder [On the Quantization of Wave Fields]. Ann. Phys., 397(1), 113–152.
[20] Rosenfeld, L. (1930). Über die Gravitationswirkungen des Lichtes [On the Gravitational Effects of Light]. Z. Phys., 65(9–10), 589–599.
[21] Bronstein, M. P. (1936b). Quantentheorie Schwacher Gravitationsfelder [Quantum Theory of Weak Gravitational Field]. Phys. Z. Sowjetunion, 9(2–3), 140–157.
[22] Bronstein, M. P. (1936a). Kvantovanie Gravitatsionnykh Voln [Quantization of Gravitational Waves]. Zh. Eksp. Theor. Fiz, 6, 195–236.
[23] Solomon, J. (1938). Gravitation et Quanta [Gravitation and Quanta]. J. Phys. Radium, 9(11), 479–485.
[24] ’t Hooft, G., & Veltman, M. J. G. (1974). One-Loop Divergencies in the Theory of Gravitation. Ann. Inst. H. Poincare A Phys. Theor., 20(1), 69–94.
[25] Goroff, M. H., & Sagnotti, A. (1985). Quantum Gravity at Two Loops. Phys. Lett. B, 160, 81–86.
[26] van de Ven, A. E. M. (1992). Two-Loop Quantum Gravity. Nucl. Phys. B, 378, 309–366.
[27] Veltman, M. J. G. (1968). Perturbation Theory of Massive Yang–Mills Fields. Nucl. Phys. B, 7, 637–650.
[28] ’t Hooft, G. (1971). Renormalization of Massless Yang–Mills Fields. Nucl. Phys. B, 33, 173–199.
[29] ’t Hooft, G. (1971). Renormalizable Lagrangians for Massive Yang–Mills Fields. Nucl. Phys. B, 35, 167–188.
[30] ’t Hooft, G., & Veltman, M. J. G. (1972). Regularization and Renormalization of Gauge Fields. Nucl. Phys. B, 44, 189–213.
[31] van Dantzig, D. (1937). Some Possibilities of the Future Development of the Notions of Space and Time. Erkenntnis, 7, 142–146.
[32] van Dantzig, D. (1956). On the Relation Between Geometry and Physics and the Concept of Space-Time. Fünfzig Jahre Relativitätstheorie. Verhandlungen, Bern, 11–16 Juli 1955: Jubilee of Relativity Theory. Proceedings Volume 4 of Helvetica Physica Acta: Supplementum, 48–53.
[33] Jacobson, T. (1995). Thermodynamics of Space-Time: The Einstein Equation of State. Phys. Rev. Lett., 75, 1260–1263 [arXiv:gr-qc/9504004].
[34] Verlinde, E. P. (2011). On the Origin of Gravity and the Laws of Newton. J. High Energy Phys., 4, 029 [arXiv:1001.0785].
[35] Penrose, Roger. (1963). Asymptotic Properties of Fields and Space-Times. Phys. Rev. Lett., 10, 66–68.
[36] Penrose, R. (1965). Zero Rest-Mass Fields Including Gravitation: Asymptotic Behavior. Proc. Roy. Soc. Lond. A, 284, 159.
[37] Bondi, H., Van der Burg, M. G. J., & Metzner, A. W. K. (1962). Gravitational Waves in General Relativity. VII. Waves from Axisymmetric Isolated Systems. Proc. Roy. Soc. Lond. A, 269, 21–52.
[38] Sachs, R. (1962). Asymptotic Symmetries in Gravitational Theory. Phys. Rev., 128(6), 2851–2864.
[39] McCarthy, P. J. (1972). Representations of the Bondi-Metzner-Sachs Group I. Determination of the Representations. Proc. R. Soc. Lond. A. Math. Phys. Sci., 330(1583), 517–535.
[40] McCarthy, P. J. (1973). Representations of the Bondi-Metzner-Sachs Group II. Properties and Classification of the Representations. Proc. R. Soc. Lond. A. Math. Phys. Sci., 333(1594), 317–336.
[41] McCarthy, P. J. (1973). Representations of the Bondi-Metzner-Sachs Group III. Poincaré Spin Multiplicities and Irreducibility. Proc. R. Soc. Lond. A. Math. Phys. Sci., 335(1602), 301–311.
[42] Crampin, M., & McCarthy, P. J. (1974). Physical Significance of the Topology of the Bondi-Metzner-Sachs Group. Phys. Rev. Lett, 33(9), 547–550.
[43] McCarthy, P. J. (1975). The Bondi-Metzner-Sachs Group in the Nuclear Topology. Proc. R. Soc. Lond. A. Math. Phys. Sci., 343(1635), 489–523.
[44] Crampin, M., & McCarthy, P. J. (1976). Representations of the Bondi-Metzner-Sachs Group IV. Cantoni Representations are Induced. Proc. R. Soc. Lond. A. Math. Phys. Sci., 351(1664), 55–70.
[45] McCarthy, P. J. (1978). Lifting of Projective Representations of the Bondi-Metzner-Sachs Group. Proc. R. Soc. Lond. A. Math. Phys. Sci., 358(1693), 141–171.
[46] Lubański, J. K. (1942a). Sur la Théorie des Particules Élémentaires de Spin Quelconque. I [On the Theory of Elementary Particles with any Spin I]. Physica, 9(3), 310–324.
[47] Lubański, J. K. (1942b). Sur la Théorie des Particules Élémentaires de Spin Quelconque. II [On the Theory of Elementary Particles with any Spin II]. Physica, 9(3), 325–338.
[48] Campiglia, M., & Laddha, A. (2014). Asymptotic Symmetries and Subleading Soft Graviton Theorem. Phys. Rev. D, 90(12), 124028 [arXiv:1502.02318].
[49] Campiglia, M., & Laddha, A. (2015). New Symmetries for the Gravitational S-Matrix. J. High Energy Phys., 4, 076 [arXiv:1408.2228].
[50] Kirillov, A A. (1962). Unitary Representations of Nilpotent Lie Groups. Russ. Math. Surv., 17(4), 53–104.
[51] Kirillov, A. A. (1999). Merits and Demerits of the Orbit Method. Bull. Amer. Math. Soc., 36(4), 433–489.
[52] Kirillov, A. (2004). Lectures on the Orbit Method (p. 408pp). American Mathematical Society.
[53] Gukov, S., & Witten, E. (2009). Branes and Quantization. Adv. Theor. Math. Phys., 13(5), 1445–1518 [arXiv:0809.0305].
[54] Gaiotto, D., & Witten, E. (2021). Probing Quantization via Branes. Surveys Diff. Geom., 24(1), 293–402 [arXiv:2107.12251].
[55] Freidel, L., Moosavian, S. F., & Pranzetti, D. (2024). On the Definition of the Spin Charge in Asymptotically-Flat Spacetimes. Class. Quant. Grav., 42(1), 015011 [arXiv:2403.19547].
[56] Israel, W. (1967). Event Horizons in Static Vacuum Space-Times. Phys. Rev., 164(5), 1776–1779.
[57] Carter, B. (1971). Axisymmetric Black Hole Has Only Two Degrees of Freedom. Phys. Rev. Lett., 26, 331–333.
[58] Robinson, D. C. (1975). Uniqueness of the Kerr Black Hole. Phys. Rev. Lett., 34, 905–906.