Ashtekar variables

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Abhay Ashtekar (2015), Scholarpedia, 10(6):32900. doi:10.4249/scholarpedia.32900 revision #186565 [link to/cite this article]
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In the spirit of Scholarpedia, this invited article is addressed to students and younger researchers. It provides the motivation and background material, a summary of the main physical ideas, mathematical structures and results, and an outline of applications of the connection variables for general relativity. These variables underlie both the canonical/Hamiltonian and the spinfoam/path integral approaches in loop quantum gravity.



This article describes a new formulation of general relativity, introduced in the mid 1980s (Ashtekar A, 1986; Ashtekar A, 1987; Ashtekar A, 1991). The main motivation was to launch a non-perturbative, background independent approach to unify principles of general relativity with those of quantum physics. The approach has since come to be known as Loop Quantum Gravity (LQG) and is being pursued using both Hamiltonian methods (Canonical LQG) and path integral methods (Spinfoams). Details of this program can be found, e.g., in an introductory text addressed to undergraduates (Gambini R, Pullin J, 2012), a more advanced review article (Ashtekar A, Lewandowski J, 2004), and monographs (Rovelli C, 2004; Thiemann T, 2007; Rovelli C, Vidotto F, 2014). This reformulation of general relativity has also had some applications to black hole physics (Ashtekar A, Krishnan B, 2004; Ashtekar A, Krasnov K, 1999; Ashtekar A, Reuter M, Rovelli C, 2015), numerical relativity (Yoneda G, Shinkai H-A, 1999; Yoneda G, Shinkai H-A, 1999; Yoneda G, Shinkai H-A, 2000), and cosmology (Ashtekar A, Singh P, 2011; Bethke L, Magueijo J, 2011; Ashtekar A, Reuter M, Rovelli C, 2015). It is closely related to developments in Twistor theory, particularly Penrose's non-linear graviton (Penrose R, 1976) and more recent advances in calculations of scattering amplitudes using twistorial techniques (Adamo T, Casali E, Skinner D, 2014; Arkani-Hamed N, Trnka J, 2013).1

The article is organized as follows. Section 2 discusses the motivation behind this reformulation and explains some of the key ideas in general terms. This non-technical introduction is addressed to readers interested only in the underlying ideas and the global context. Section 3 presents a brief but self-contained summary of the reformulation. The level of this discussion is significantly more technical; in particular it assumes familiarity with basic differential geometry and mathematical underpinnings of general relativity. Section 4 provides a short summary and two directions for future research. There is a fascinating but largely unknown historical episode involving Albert Einstein and Erwin Schrödinger that is closely related to this reformulation. It is described in Appendix A.

Motivation and Key ideas

Smooth Lorentzian metrics $g_{ab}$ (with signature $-,+,+,+$) constitute the basic mathematical variables of general relativity. $g_{ab}$ endows the space-time manifold with a pseudo-Riemannian geometry and the gravitational field is encoded in its curvature. Key predictions of general relativity which have revolutionized our understanding of the universe can be traced back to the fact that the metric $g_{ab}$ ---and hence space-time geometry--- is a dynamical entity in this theory. For example, it is this feature that makes it possible for the universe to expand. It is this feature that lets ripples of curvature propagate, providing us with gravitational waves. In the strong curvature regions, it is this feature that warps the space-time geometry so much that even light is trapped, leading to the formation of black holes.

Let me use an analogy with particle mechanics to further explain the fundamental role played by metrics in general relativity. Just as position serves as the configuration variable for a particle, the positive, definite three dimensional metric of space, $q_{ab}$, can be taken to be the configuration variable in general relativity. Given the initial position and velocity of a particle, Newton's laws provide us with its trajectory in the position space. Similarly, given a three dimensional metric $q_{ab}$ and its time derivative $K_{ab}$ at an initial instant, Einstein's equations provide us with a four dimensional space-time, or, a trajectory in the infinite dimensional space of 3-geometries.2 This superspace of 3-metrics is thus the arena for describing dynamics in general relativity. Thus, general relativity can be regarded as the dynamical theory of 3-metrics $q_{ab}$ and was therefore labeled geometrodynamics by John Wheeler (Wheeler J A, 1964).

However, this central role played by the metric also sets general relativity apart from all other fundamental forces of Nature. For, theories of electro-weak and strong interactions are also geometric, but the basic dynamical variable in these theories is a (matrix-valued) vector potential, or a connection. Thus, these theories describe connection-dynamics (in the fixed space-time geometry of Minkowski space). The connections enable one to parallel-transport geometrical/physical entities along curves. In electrodynamics, the entity is a charged particle such as an electron; in chromodynamics, it is a particle with internal color, such as a quark. Generally, if we move the object around a closed loop in an external gauge field, its state does not return to its initial value but is rotated by a non-trivial unitary matrix. The unitary matrix is a measure of the flux of the field strength ---i.e. the curvature of the connection--- across a surface bounded by the loop. In the case of electrodynamics, the connection $A_{a}$ is a 1-form on 4-dimensional Minkowski space-time that takes values in the Lie algebra of ${\rm U(1)}$, and the curvature is the field strength $F_{ab}$. In chromodynamics, the connection $A_{a}^{i}$ is a 1-form that takes values in the Lie algebra of the non-Abelian group ${\rm SU(3)}$ and its curvature $F_{ab}^{i}$ encapsulates the strong force. The Lagrangian governing dynamics of all these connections dynamics is just quadratic in curvature.

But the space-time metric of general relativity also determines a specific connection named after Levi-Civita. It enables one to parallel transport vectors in the tangent space of a point to those at another point in presence of a gravitational field encoded in the space-time metric. Furthermore, the Einstein-Hilbert Lagrangian also features the curvature of this connection. Therefore, it is natural to ask if one can recast general relativity as connection-dynamics thereby bringing it closer to the theories of other basic forces of Nature.

Early attempts along these lines were motivated by considerations of unifying general relativity with electrodynamics, dating back long before the advent of Yang Mills theories (that govern the electro-weak and strong interactions). The earliest of these investigations were by Einstein, Eddington and Schrödinger, where only the Levi-Civita type connection was used as the basic dynamical variable (see, e.g., (Schrödinger E, 1947; Schrödinger E, 1948)). But the equations became complicated and physics remained opaque. The fascinating episode involving Einstein and Schrödinger I referred to in section 1, occurred in this phase. It is surprising that this episode is not widely known today but it appears to have cast a long shadow on the subject, dampening for quite some time subsequent attempts to construct theories of relativistic gravity based connection-dynamics.

The attempts were revived decades later, after the advent of geometrical formulations of Yang Mills theories, along two directions. In the first, by mimicking Yang Mills theory closely, one was led to new theories of gravity (see, e.g., (Yang C N, 1974; Mielke E W, Maggiolo A A R, 2004)), which, it was hoped would be better suited for incorporating quantum aspects of gravity. However, these ideas have lost their original appeal because, on the one hand, general relativity has continued to meet with impressive observational successes, and, on the other hand, the original hopes that the new theories would readily lead to satisfactory quantum gravity completions of general relativity were not realized. The second direction involved recasting general relativity itself as a theory of connections. Here, the motivation came from ‘Palatini type’ actions for general relativity in which both the metric and the connection are regarded as independent dynamical variables. The relation between them which says that the metric serves as a ‘potential’ for the connection now emerges as a field equation. This framework was extended using in part the strategies employed in the early attempts described above. It was shown that general relativity coupled with electrodynamics could be treated as a gauge theory in the modern sense of the term, with only the electromagnetic and gravitational connections as basic variables (see, e.g., (Ferraris M, Kijowski J, 1981; Ferraris M, Kijowski J, 1982)). The space-time metric did not appear in the action but emerged as a ‘derived’ quantity. However, when one carries out the Legendre transform to pass to the Hamiltonian theory, one essentially recovers the same phase space as geometrodynamics. Einstein equations again provide dynamical trajectories on the superspace of metrics. Consequently one cannot import techniques from gauge theories in the passage to quantum theory.

This status quo changed in the mid-1980s (Ashtekar A, 1986; Ashtekar A, 1987). The new twist was motivated by entirely different considerations, rooted in the appreciation of the role played by chirality ---i.e. self-duality (or anti self-duality)--- in simplifying Einstein dynamics. The simplification was made manifest by apparently diverse results: Penrose's twistorial construction of the ‘non-linear graviton’ (Penrose R, 1976; Penrose R, Rindler W, 1987); Newman's construction of ‘H-spaces’ (Ko M, Ludvigsen M, Newman E T, Tod K P, 1981); and results on ‘asymptotic quantization’ (Ashtekar A, 1981; Ashtekar A, 1981; Ashtekar A, 1981; Ashtekar A, 1987) based on the structure of the radiative degrees of freedom of the gravitational field in exact general relativity. The first two of these constructions provided a precise sense in which the chiral sectors of general relativity are ‘exactly integrable’, while the third showed that chirality considerably simplifies the description of the asymptotic quantum states of the gravitational field in the exact, fully non-linear general relativity, bringing them closer to the Fock spaces of gravitons used in perturbative treatments of quantum gravity.

In the new formulation (Ashtekar A, 1986; Ashtekar A, 1987) the connections of interest parallel transport chiral spinors ---rather than vectors--- in a gravitational field. The use of the spin-connection was motivated by certain simplifications in the Hamiltonian formulation of general relativity (Sen A, 1982; Ashtekar A, Horowitz G T, 1982). These mathematical objects have direct physical content because the left handed fermions of the standard model of particle physics are represented by chiral spinors. The new framework is a generalization of Palatini's in that it uses orthonormal tetrads ---i.e., ‘square roots of metrics’--- in place of metrics, and spin connections in place of the Levi Civita connections. Then, in the Hamiltonian framework, the phase space is the same as in the ${\rm SU(2)}$ Yang-Mills theory. Thus, the dynamics of general relativity is now represented by trajectories in the infinite dimensional superspace, but now of connections rather than of metrics. But, as discussed in detail in section 3, the equations that govern the dynamics of the gravitational field on this common phase space are very different from those governing the Yang-Mills dynamics. In the Yang-Mills case, the space-time geometry is fixed once and for all and equations make use of the background Minkowski metric. On the phase space of general relativity, by contrast, there is no background geometry and so all equations have to be written entirely in terms of (spin) connections and their canonically conjugate momenta. However, the dynamical trajectories can be projected down to the configuration space, which is now the infinite dimensional superspace of ${\rm SU(2)}$ connections, exactly as in the Yang-Mills theory. In this sense, the kinematical arena for all interactions ---including gravitational--- is now a connection superspace. Furthermore, in the case of general relativity, the projected trajectories can be interpreted as geodesics on the connection superspace, with respect to the ‘super-metric’ that features in the so called Hamiltonian constraint. In this sense, the new setting brings out the underlying simplicity of the dynamics of general relativity.3 This simplicity can also be seen directly in the form of equations. In contrast to geometrodynamics, where the equations involve rather complicated non-polynomial functions of the spatial metric, in the new connection-dynamics all equations are low order polynomials in the basic phase space variables. Finally, because the phase space is the same as in Yang-Mills theory, now it is possible to import into gravity certain powerful mathematical techniques from Yang-Mills theories. All these features play an important role in the emergence of quantum geometry. In LQG, this specific quantum geometry is crucial both in the Hamiltonian theory and spinfoams, and in applications of these theories to black holes and cosmology.

This concludes the broad-brush overview of the underlying ideas.

General relativity as a dynamical theory of spin-connections

Let us begin with a quote from Einstein's 1946 biographical notes on how to formulate relativistic theories of gravity (Einstein A, 1973):

The major question for anyone doing research in this field is: Of which mathematical type are the variables… which permit the expression of physical properties of space… Only after that, which equations are satisfied by these variables?

Recall that before the advent of Yang-Mills theories, Einstein, Schrödinger, Palatini and others focused on the Levi-Civita connections that enable one to parallel transport space-time vectors along curves. However, in Yang-Mills theories, the parallel transported objects have ‘internal indices’. As explained in section 2, in the gravitational context, the natural objects ‘with internal indices’ are spinors. Thus, in the approach described in this article, the answer to the first major question of Einstein's is: spin-connections. As for the second question, in approaches motivated directly by non-Abelian gauge theories (Yang C N, 1974; Mielke E W, Maggiolo A A R, 2004), one mimics the Yang-Mills strategy and introduces actions that are quadratic in the curvature of the connections. Then the field equations turn out to be of order four (or higher) in the metric. In the present approach, by contrast, one retains general relativity. Thus, the equations satisfied by the spin-connections and their conjugate momenta will be equivalent to Einstein's, which contain only second derivatives of the metric.

Preliminaries: 2+1 dimensional General Relativity

It is instructive to begin with 2+1 dimensions, first because many of the essential conceptual aspects of connection dynamics are more transparent there, and second, because the discussion brings out the key difficulties that arise in 3+1 dimensions. Indeed, even today, new ideas for background independent, non-perturbative quantum gravity are often first tried in 2+1 dimensions. Now, the 2+1 dimensional Lorentz group ${\rm SO(1,2)}$ ---or its double cover, $SU(1,1)$--- serves as the internal gauge group. It acts on orthonormal co-frames $e_{a}^{I}$, where the index $a$ refers to the 1-form nature of the co-frame and the index $I$ takes values in the Lie algebra ${\rm so(1,2)}$ of ${\rm SO(1,2)}$. The space-time metric is given by $g_{ab} = e_{a}^{I}e_{b}^{J} \eta_{IJ}$, where $\eta_{IJ}$, the Cartan-Killing metric on ${\rm so(1,2)}$, has signature $-,+,+$. Note that $\eta_{IJ}$ constitutes the kinematic structure of the theory, fixed once and for all, in the ‘internal space’. It enables one to freely raise and lower the internal indices. The co-frames $e_{a}^{I}$, on the other hand are dynamical variables that directly determine the space-time metric $g_{ab}$ which is guaranteed to have signature $-,+,+$ simply from the signature of $\eta_{IJ}$. The transition from the metric $g_{ab}$ to its ‘square-root’ $e_{a}^{I}$ becomes essential for incorporation of spinors in the theory. Physically, of course, spinors serve as fundamental matter fields. However, interestingly, even if one restricts oneself to the purely bosonic sector, spinors provide a more transparent way to establish a number of mathematical results, the most notable being the positive energy theorem in 2+1 dimensions (Ashtekar A, Varadarajan M, 1994) (that follows the key ideas introduced by Sen, Nester, Sparling and Witten in 3+1 dimensions (Witten E, 1981)).

In the Palatini formulation, the fundamental dynamical variables are the co-frames $e_{a}^{I}$ and a connection 1-form $A_{a}^{IJ}$ that takes values in the Lie algebra of ${\rm so(1,2)}$. For pure gravity, the action is given by \[ \tag{1} S_{P}^{(2+1)} (e, A) = \frac{1}{8\pi G_{3}}\int_{M} {\rm d}^{3}x\, \epsilon_{IJK} \tilde{\eta}^{abc}e^{I}_{a} F^{JK}_{bc} \] where $\tilde{\eta}^{abc}$ is the (metric independent) space-time Levi-Civita tensor density of weight 1 and $F_{ab}^{IJ} = 2 \partial_{[a}A_{b]}^{IJ} + A_{a}{}^{IK}A_{bKJ}$ is the curvature 2-form of $A_{a}^{IJ}$. When one carries out a Legendre transform of this action, one is led to the Hamiltonian framework. The pull-back of $A_{a}^{IJ}\epsilon_{IJ}{}^{K}$ to a spatial 2-manifold, which for simplicity we denote by $A_{a}{}^{K}$, turns out to be the configuration variable. The conjugate momentum is the ‘electric field’ $\tilde{E}^{a}_{I} = \tilde{\eta}^{ab}e_{b\,I}$, the dual of the pull-back of the co-triad to the spatial 2-manifold. Thus, now the configuration space is simply the space of connections and the (spatial) metric is ‘derived’ from the canonically conjugate momentum $\tilde{E}^{a}_{I}$.4 Recall that in general relativity, some of Einstein's equations constrain initial data and the remaining equations describe dynamics. Furthermore, because there are no background fields, in the phase space framework, the Hamiltonian generating dynamics is a linear combination of constraints. In the metric variables, constraints are rather complicated, non-polynomial functions of the 2-metric and its canonically conjugate momentum. In connection-dynamics under considerations, by contrast, the constraints are very simple: \[ \tag{2} \mathcal{D}_{a} \tilde{E}^{a}_{I} =0, \qquad \text{and} \qquad F^{IJ}_{ab} = 0, \] where $\mathcal{D}$ is the gauge covariant derivative defined by the connection $A_{a}^{IJ}$ and $F_{ab}^{IJ}$ its curvature and $\tilde{E}^{a}_{I} = \tilde{\eta}^{ab}e_{b\,I}$, the dual of the co-triad, can be regarded as the ‘electric field’ in the Yang-Mills terminology. (Recall that the lower case indices now refer to the 2-dimensional tangent space of the spatial manifold, where all fields in the Hamiltonian theory live.) The first constraint is just the familiar Gauss law while the second says that the field strength of the pulled-back connection vanishes. Since the Hamiltonian generating dynamics is just a linear combination of these constraints, the dynamical equations are also low order polynomials in the connection variables although they are equivalent to the standard Einstein's equations which are non-polynomial in the metric variables. (For further details, see, e.g., (Ashtekar A, 1991; Ashtekar A, 1995).)

3+1 dimensional General Relativity: The Lagrangian framework

For simplicity, I will continue to restrict myself to source-free general relativity. The connection-dynamics framework is, however, quite robust: all its basic features ---including the fact that all equations are low order polynomials in the basic canonical variables--- remain unaltered by the inclusion of a cosmological constant and coupling of gravity to Klein-Gordon fields, (classical or Grassmann-valued) Dirac fields and Yang-Mills fields with any internal gauge group (Ashtekar A, Romano J D, Tate R S, 1989; Ashtekar A, 1991). This is true both for the Lagrangian framework discussed in this sub-section and the Hamiltonian framework discussed in the next sub-section.

Let us begin by extending the underlying ideas from 2+1 to to 3+1 dimensions. Now the ${\rm SO(2,1)}$ connection is replaced by the ${\rm SO(3,1)}$ Lorentz connection ${}^{4}\omega_{a}^{IJ}$ and the co-triad $e_{a}^{I}$ by a co-tetrad $e_{a}^{I}$, where $I,J,...$ now denote the internal ${\rm so(3,1)}$ indices labeling the co-tetrads, and $a,b,..$ denote 4-dimensional space-time indices. However, because the underlying manifold $M$ is now 4-dimensional, the Palatini action contains two co-tetrads, rather than just one: \[ \tag{3} S_P(e, {}^4\omega) := {1\over 16\pi G}\int_{M} {\rm d}^4 x\, \epsilon_{IJKL}\tilde\eta^{abcd}e_{aI} e_{bJ}({}^4\!R_{cd}{}^{KL}), \] where $G$ is Newton's constant, $\tilde\eta^{abcd}$ is the metric independent Levi-Civita density on space-time and ${}^4R_{ab}{}^{IJ}$ is the curvature tensor of the ${\rm SO(3,1)}$ connection ${}^4\omega_a^{IJ}$. (Note that the internal indices can again be raised and lowered freely using the fixed, kinematical metric $\eta_{IJ}$ on the internal space.) Hence, when one performs the Legendre transform, the momentum $\tilde{\Pi}^a_{IJ}$ conjugate to the connection $A_a^{IJ}$ is the dual $\tilde{\eta}^{abc} \epsilon^{IJ}{}_{KL}e_b^K e_c^L$ of a product of two co-triads rather than of a single co-triad as in the 2+1 dimensional case. The theory then has an additional constraint --saying that the momentum is “decomposable” in this manner-- which spoils the first class nature of the constraint algebra. Following the Dirac procedure, one can solve for the second class constraint and obtain new canonical variables. It is in this elimination that one loses the connection 1-form altogether and is led to geometrodynamics. (For details, see (Ashtekar A, Balachandran A P, Jo S G, 1989) and chapters 3 and 4 in (Ashtekar A, 1991).)

However, these complications disappear if one requires the connection to take values only in the self dual (or, alternatively anti-self dual) part of ${\rm SO(3,1)}$. Furthermore, the resulting connection dynamics is technically significantly simpler than geometrodynamics. It is this simplicity that leads to LQG. Thus, in connection-dynamics, the answer to the first question posed by Einstein in his autobiographical notes is that (the configuration) variables of the theory should be chiral connections. I will now elaborate on this observation.

Let me first explain what I mean by self duality here. If one begins with a Lorentz connection ${}^4\omega_a^{IJ}$, the self dual connection ${}^4\!A_a^{IJ}$ is given by dualizing over the internal indices: \[ \tag{4} {}^4\!A_a^{IJ} = \frac{1}{2G}({}^4\omega_a^{IJ} - \tfrac{i}{2} \epsilon^{IJ}{}_{KL}{}^4\omega_a^{KL}), \] where $G$ is Newton's constant. (This factor has been introduced for later convenience and plays no role in this discussion of the mathematical meaning of self duality.) However, one regards the self dual connections themselves as fundamental; they are subject just to the following algebraic condition on internal indices: \[ \tag{5} \tfrac{1}{2} \epsilon^{IJ}{}_{KL} {}^4\!A_a^{KL} = i\;{}^4\!A_a^{IJ}. \] Let me emphasize that, unlike in the analysis of self dual Yang-Mills fields on a given space-time, the notion of self duality here refers to the internal rather than space-time indices: to define the duality operation, we use the kinematical internal metric $\eta_{IJ}$ (and its alternating tensor $\epsilon^{IJ}{}_{KL}$) rather than the dynamical space-time metric (to be introduced later).

The new action is obtained simply by substituting the real ${\rm SO(3,1)}$ connection ${}^4\omega_a^{IJ}$ by the self dual connection $A_a^{IJ}$ in the Palatini action (modulo overall constants): \[ \tag{6} S(e, {}^4\!A) := \frac{1}{16\pi} \int_{\rm M} {\rm d}^4 x\, \epsilon_{IJKL}\tilde\eta^{abcd} e_{aI} e_{bJ}({}^4\!F_{ab}{}^{KL}), \] where, \[ \tag{7} {}^4F_{abI}{}^J := 2 \partial_{[a} {}^4\!A_{b]I}{}^J + G{}^4A_{aI} {}^M {}^4A_{bM}{}^J - G{}^4A_{bI} {}^M{}^4A_{aM}{}^J \] is the field strength of the connection ${}^4\!A_{aI}{}^J$. Thus, $G$ plays the role of the coupling constant. Note incidentally that because of the factors of $G$, ${}^4A_{aI}{}^J$ and ${}^4F_{abI}{}^J$ do not have the usual dimensions of connections and field strength.

By setting the variation of the action with respect to ${}^4\!A_{a}{}^{IJ}$ to zero we obtain the result that ${}^4\!A_{a}{}^{IJ}$ is the self dual part of the (torsion-free) connection ${}^4\Gamma_a{}^{IJ}$ compatible with the tetrad $e_a^I$. Thus, ${}^4\!A_{a}{}^{IJ}$ is completely determined by $e_a^I$. Setting the variation with respect to $e_a^I$ to zero and substituting for the connection from the first equation of motion, we obtain the result that the space-time metric $g^{ab} =e^a{}_I e^b{}_J\eta^{IJ}$ satisfies the vacuum Einstein's equation. Thus, as far as the classical equations of motion are concerned, the self dual action (6) is completely equivalent to the Palatini action (3).

This result seems surprising at first. Indeed, since ${}^4\!A_{a}{}^{IJ}$ is the self dual part of ${}^4\omega_a^{IJ}$, it follows that the curvature ${}^4\!F_{ab} {}^{IJ}$ is the self dual part of the curvature ${}^4\!R_{ab}{}^{IJ}$. Thus, the self dual action is obtained simply by adding to the Palatini action an extra (imaginary) term. This term is not a pure divergence. How can it then happen that the equations of motion remain unaltered? This comes about as follows. First, the compatibility of the connections and the tetrads forces the “internal” self duality to be the same as the space-time self duality, whence the curvature ${}^4\!F_{abI}{}^J$ can be identified with the self dual part, on space-time indices, of the Riemann tensor of the space-time metric. Hence, the imaginary part of the field equation says that the trace of the dual of the Riemann tensor must vanish. This, however, is precisely the (first) Bianchi identity! Thus, the imaginary part of the field equation just reproduces an equation which holds in any case; classically, the two theories are equivalent. However, the extra term does change the definition of the canonical momenta in the Legendre transform --i.e., gives rise to a canonical transform on the Palatini phase space-- and this change, in turn, enables one to regard general relativity as a theory governing the dynamics of 3-connections rather than of 3-geometries. (For details, see (Ashtekar A, 1986; Ashtekar A, 1987; Ashtekar A, 1991; Ashtekar A, Lewandowski J, 2004; Rovelli C, 2004; Thiemann T, 2007; Ashtekar A, Romano J D, Tate, R S, 1989)).

Hamiltonian framework

Since in the Lorentzian signature self dual fields are necessarily complex, it is convenient to begin with complex general relativity --i.e. by considering complex, Ricci-flat metrics $g_{ab}$ on a real 4-manifold $M$-- and take the “real section” of the resulting phase-space at the end. Let $e_a^I$ then be a complex co-tetrad on $M$ and ${}^4\!A_{a}{}^{IJ}$ a self dual ${\rm SO(3,1)}$ connection, and let the action be given by (6). Let us assume that the space-time manifold $M$ has the topology $\Sigma\times\mathbb{R}$ and carry out the Legendre transform. This procedure is remarkably straightforward especially when compared to geometrodynamics. The resulting canonical variables are then complex fields on a (“spatial”) 3-manifold $\Sigma$. To begin with, the configuration variable turns out to be a 1-form $A_a^{IJ}$ on $\Sigma$ which takes values in the self dual part of the (complexified) ${\rm SO(3,1)}$ Lie-algebra and its canonical momentum $\tilde{E}_a^{IJ}$ is a self dual vector density which takes values also in the self dual part of the ${\rm SO(3,1)}$ Lie algebra. (Thus, in the Hamiltonian framework, the lower case latin indices refer to the spatial 3-manifold $\Sigma$.) The key improvement over the Palatini framework is that there are no additional constraints on the algebraic form of the momentum (Ashtekar A, 1991; Ashtekar A, 1989). Hence, all constraints are now first class and the analysis retains its simplicity. For technical convenience, one can set up, once and for all, an isomorphism between the self dual sub-algebra of the Lie algebra of ${\rm SO(3,1)}$ and the Lie algebra of ${\rm SO(3)}$. When this is done, we can take our configuration variable to be a complex, ${\rm SO(3)}$-valued connection $A_a^i$ and its canonical momentum, a complex spatial triad $\tilde{E}_i^a$ with density weight one, where ‘$a$’ is the manifold index and ‘$i$’ is the triad or the ${\rm SO(3)}$ internal index.

The (only non-vanishing) fundamental Poisson brackets are: \[ \tag{8} \{\tilde{E}^a{}_i(x),\,A_b{}^j(y)\}=-i\delta^a{}_b \delta_i{}^j\delta^3(x,y). \] The geometrical interpretation of these canonical variables is as follows. As we saw above, in any solution to the field equations, ${}^4\!A_{a}{}^{IJ}$ turns out to be the self dual apart of the spin-connection defined by the tetrad, whence $A_a^i$ has the interpretation of being a potential for the self dual part of the Weyl curvature. $\tilde{E}_i^a$ can be thought of as a “square-root” of the 3-metric (times its determinant) on $\Sigma$. More precisely, the relation of these variables to the familiar geometrodynamical variables, the 3-metric $q_{ab}$ and the extrinsic curvature $K_{ab}$ on $\Sigma$, is as follows: \[ \tag{9} GA_a{}^i = \Gamma_a{}^i - i K_a{}^i \quad {\rm and} \quad \tilde{E}^a{}_i \tilde{E}^{bi} = (q) q^{ab} \] where, as before, $G$ is Newton's constant, $\Gamma_a{}^i$ is the spin-connection determined by the triad, $K_a{}^i$ is obtained by transforming the space index ‘$b$’ of the extrinsic curvature $K_{ab}$ into an internal index by the triad $E^a_i := (1/\sqrt{q})\tilde{E}^a_i$, and $q$ is the determinant of $q_{ab}$. Note, however, that, as far as the mathematical structure is concerned, we can also think of $A_a^i$ as a (complex) ${\rm so(3)}$-Yang-Mills connection and $\tilde{E}_i^a$ as its conjugate electric field. Thus, the phase space has a dual interpretation. It is this fact that enables one to import into general relativity and quantum gravity ideas from Yang-Mills theory and quantum chromodynamics and may, ultimately, lead to a unified mathematical framework underlying the quantum description of all fundamental interactions. In what follows, we shall alternate between the interpretation of $\tilde{E}_i^a$ as a triad and as the electric field canonically conjugate to the connection $A_i^a$.

Since the configuration variable $A_a^i$ has nine components per space point and since the gravitational field has only two degrees of freedom, we expect seven first class constraints. This expectation is indeed correct. The constraints are given by: \[ \tag{10} \begin{split} {\cal G}_i(A,\tilde{E}) &:= \mathcal{D}_a \tilde E^a{}_i=0 \\ {\cal V}_a (A,\tilde{E}) &:= \tilde{E}^b{}_i\, F_{ab}{}^i\equiv \mathrm{tr}\, E\times B =0 \\ {\cal S} (A,\tilde{E}) &:= \epsilon^{ijk}\tilde E^a{}_i\,\tilde E^b{}_j\,F_{abk} \equiv \mathrm{tr}\, E\times E\cdot B =0, \end{split} \] where $F_{ab}{}^i:=2\partial_{[a} A_{b]}{}^i + G\epsilon^{ijk} A_{a{}j}A_{b{}k}$ is the field strength constructed from $A_a^i$, $B$ stands for the magnetic field $\tilde\eta^{abc}F_{bc}^i$, constructed from $F_{ab}^i$, and $\mathrm{tr}$ refers to the standard trace operation in the fundamental representation of ${\rm SO(3)}$. Note that all these equations are simple polynomials in the basic variables; the worst term occurs in the last constraint and is only quadratic in each of $\tilde{E}_i^a$ and $A_a^i$ . The three equations are called, respectively, the Gauss constraint, the vector constraint and the scalar constraint. The first, Gauss law, arises because we are now dealing with triads rather than metrics. It simply tells us that the internal ${\rm SO(3)}$ triad rotations are “pure gauge”. Modulo these internal rotations, the vector constraint generates spatial diffeomorphisms on $\Sigma$ while the scalar constraint is responsible for diffeomorphisms in the “time-like directions”. Thus, the overall situation is the same as in triad geometrodynamics.

From geometrical considerations we know that the “kinematical gauge group” of the theory is the semi-direct product of the group of local triad rotations with that of spatial diffeomorphisms on $\Sigma$. This group has a natural action on the canonical variables $A_a^i$ and $\tilde{E}_i^a$ and thus admits a natural lift to the phase-space. This is precisely the group formed by the canonical transformations generated by the Gauss and the vector constraints. Thus, six of the seven constraints admit a simple geometrical interpretation. What about the scalar constraint? Note that, being quadratic in momenta, it is of the form $G^{\alpha\beta} p_\alpha p_\beta=0$ on a generic phase space, where, the connection supermetric $\epsilon^{ijk}F_{ab k}$ plays the role of $G^{\alpha\beta}$ and the momenta $\tilde{E}^{a}{}_{i}$ of $P_{\alpha}$. Consequently, the motions generated by the scalar constraint in the phase space correspond precisely to the null geodesics of the “connection supermetric”. As in geometrodynamics, the space-time interpretation of these canonical transformations is that they correspond to “multi-fingered” time-evolution. Thus, we now have an attractive representation of the Einstein evolution as a null geodesic motion in the (connection) configuration space.5 If $\Sigma$ is spatially compact, the Hamiltonian is given just by a linear combination of constraints. In the asymptotically flat situation, on the other hand, constraints generate only those diffeomorphisms which are asymptotically identity. To obtain the generators of space and time translations, one has to add suitable boundary terms. In a 3+1 framework, these translations are coded in a lapse-shift pair. The lapse --which tends to a constant value at infinity-- tells us how much of a time translation we are making while the shift --which approaches a constant vector field at infinity-- tells us the amount of space-translation being made. Given a lapse6 $\underset{\sim}{N}$ and a shift $N^a$, the Hamiltonian is given by: \[ \tag{11} \begin{split} H(A,\tilde E) &= i \int_{\Sigma} {\rm d}^3 x \, (N^a F_{ab}{}^i\tilde E^b{}_i -\tfrac{i}{2}\underset{\sim}{N} \epsilon^{ijk}F_{ab k} \tilde E^a{}_i \tilde E^b{}_j) \\ & \qquad - \oint_{\partial\Sigma} {\rm d}^2S_a\,(\underset{\sim}{N} \epsilon^{ijk} A_{bk} \tilde {E}^a{}_i \tilde{E}^b{}_j + 2 i N^{[a} \tilde E^{b]}{}_i A_b{}^i). \end{split} \] The dynamical equations are easily obtained since the Hamiltonian is also a low order polynomial in the canonical variables. We have \[ \tag{12} \begin{split} \dot{A}_a^i &= -i\epsilon^{ijk}\underset{\sim}{N}\tilde{E}^b_jF_{ab}{}_{k} - N^bF^i_{ab} \\ \dot{E}^a_i &= i\epsilon_i^{jk}\mathcal{D}_b(\underset{\sim}{N}\tilde{E}^a_j \tilde{E}^b_k) -2\mathcal{D}_b(N^{[a}\tilde{E}^{b]i}) \end{split} \] Again, relative to their analogs in geometrodynamics, these equations are significantly simpler.

So far, we have discussed complex general relativity. To recover the Lorentzian theory, we must now impose reality conditions, i.e., restrict ourselves to the real, Lorentzian section of the phase-space. Let me explain this point by means of an example. Consider a simple harmonic oscillator. One may, if one so wishes, begin by considering a complex phase-space spanned by two complex co-ordinates $q$ and $p$ and introduce a new complex co-ordinate $z= q - ip$. ($q$ and $p$ are analogous to the triad $\tilde{E}_i^a$ and the extrinsic curvature $K_a{}^i$, while $z$ is analogous to $A_a^i$.) One can use $q$ and $z$ as the canonically conjugate pair, express the Hamiltonian in terms of them and discuss dynamics. Finally, the real phase-space of the simple harmonic oscillator may be recovered by restricting attention to those points at which $q$ is real and $ip = q-z$ is pure imaginary (or, alternatively, $\dot{q}$ is also real.) In the present phase-space formulation of general relativity, the situation is analogous. In terms of the familiar geometrodynamic variables, the reality conditions are simply that the 3-metric be real and the extrinsic curvature --the time derivative of the 3-metric-- be real. If these conditions are satisfied initially, they continue to hold under time-evolution. In terms of the present canonical variables, these become: i) the 3-metric $\tilde E^a{}_i \tilde E^{bi}$ (with density weight 2 be real, and, ii) its Poisson bracket with the Hamiltonian $H$ be real, i.e., \[ \tag{13} \begin{split} (\tilde{E}^a{}_i \tilde{E}^{bi})^\star &= \tilde{E}^a{}_i \tilde{E}^{bi} \\ \big(\epsilon^{ijk}\tilde{E}^{(a}{}_i \mathcal{D}_c(\tilde{E}^{b)}{}_k\tilde{E}^c{}_j ) \big)^\star &= - \epsilon^{ijk}\tilde E^{(a}{}_i \mathcal{D}_c(\tilde{E}^{b)}{}_k\tilde{E}^c{}_j), \end{split} \] where $\star$ denotes complex-conjugation. (Note, incidentally, that in Euclidean relativity, these conditions can be further simplified since self dual connections are now real: The reality conditions require only that we restrict ourselves to real triads and real connections.) As far as the classical theory is concerned, we could have restricted to the “real slice” of the phase-space right from the beginning. In quantum theory, on the other hand, it may be simpler to first consider the complex theory, solve the constraint equations and then impose the reality conditions as suitable Hermitian-adjointness relations. Thus, the quantum reality conditions would be restrictions on the choice of the inner-product on physical states.

Could we have arrived at the phase-space description of real general relativity in terms of ($A_a^i$, $\tilde{E}_i^a$) without having to first complexify the theory? The answer is in the affirmative. This is in fact how the new canonical variables were first introduced (Ashtekar A, 1986; Ashtekar A, 1987). The idea is to begin with the standard Palatini action for real tetrads and real Lorentz-connections, perform the Legendre transform and obtain the phase-space of real relativity à la Arnowitt, Deser and Misner. The basic canonical variables in this description can be taken to be the density weighted triads $\tilde{E}_i^a$ and their canonical conjugate momenta $\pi_a^i$. The interpretation of $\pi_a^i$ is as follows: In any solution to the field equations, i.e., “on shell,” $K_{ab}:= \pi_{(a}^i E_{b)i}$ turns out to be the extrinsic curvature. Up to this point, all fields in question are real. On this real phase space, one can make a (complex) canonical transformation to pass to the new variables: $(\tilde{E}^a_i, \pi_a^i)\to (\tilde{E}^a_i , GA_a^i := \Gamma_a^i - i \pi_a^i \equiv (\delta F/\delta\tilde{E}^a_i) - i \pi_a^i)$, where the generating function $F(\tilde{E})$ is given by: $F(\tilde{E}) = \int_{\Sigma} {\rm d}^3x \tilde{E}^a_i \Gamma_a^i$, and where $\Gamma_a^i$ are the spin-coefficients determined by the triad $\tilde{E}_i^a$. Thus, $A_a^i$ is now just a complex coordinate on the traditional, real phase space. This procedure is completely analogous to the one which lets us pass from the canonical coordinates $(q,p)$ on the phase space of the harmonic oscillator to another set of canonical coordinates $(q, z = dF/dq - ip)$, with $F(q) = \frac{1}{2}q^2$, and makes the analogy mentioned above transparent. Finally, the second of the reality conditions, (4.2.9), can now be re-expressed as the requirement that $GA_a^i - \Gamma_a^i$ be purely imaginary, which follows immediately from the expression of $A_a^i$ in terms of the real canonical variables $(\tilde{E}^a_i, K_a^i)$.

I will conclude this sub-section with a few remarks.

  • In broad terms the Hamiltonian framework developed above is quite similar to that for the 2+1 theory: in both cases, the configuration variable is a connection, the momentum can be regarded as a “square root” of the metric, and the constraints, the Hamiltonian and the equations of motion are all low order polynomials in these basic variables. However, there are a number of key differences. The first is that the connection in the 3+1 theory is the self dual ${\rm SO(3,1)}$ spin-connection while that in the 2+1 theory is the real ${\rm SO(2,1)}$ spin-connection. More importantly, while in the 2+1-theory the connection is constrained to be flat a simple counting argument shows that there is no such implication in the 3+1 theory. This is the reason behind the main difference between the two theories: Unlike in the 2+1 case, the 3+1 theory has local degrees of freedom and hence gravitons.
  • A key feature of this framework is that all equations of the theory --the constraints, the Hamiltonian and hence the evolution equations and the reality conditions-- are simple polynomials in the basic variables $\tilde{E}_i^a$ and $A_a^i$. This is in striking contrast to the ADM framework where the constraints and the evolution equations are non-polynomial in the basic canonical variables. An interesting --and potentially, powerful-- consequence of this simplicity is the availability of a nice algorithm to obtain the “generic” solution to the vector and the scalar constraint (Capovilla R, Dell J, Jacobson T, 1989). Choose any connection $A_a^i$ such that its magnetic field $\tilde B^a{}_i := \tilde\eta^{abc} F_{bci}$, regarded as a matrix, is non-degenerate. A “generic” connection $A_a^i$ will satisfy this condition; it is not too restrictive an assumption. Now, we can expand out $\tilde{E}_i^a$ as $\tilde{E}_i^a = M_i{}^j \tilde B^a{}_j$ for some matrix $M_i{}^j$. The pair ($A_a^i$ , $\tilde{E}_i^a$ ) then satisfies the vector and the scalar constraints if and only if $M_i{}^j$ is of the form $M_i{}^j = [\phi^2 -\frac{1}{2} \mathrm{tr} \phi^2 ]_i{}^j$, where $\phi_i{}^j$ is an arbitrary trace-free, symmetric field on $\Sigma$. Thus, as far as these four constraints are concerned, the “free data” consists of $A_a^i$ and $\phi_i{}^j$.
  • The phase-space of general relativity is now identical to that of complex-valued Yang-Mills fields (with internal group ${\rm SO(3)}$). Furthermore, one of the constraint equations is precisely the Gauss law that one encounters on the Yang-Mills phase-space. Thus, we have a natural embedding of the constraint surface of Einstein's theory into that of Yang-Mills theory: Every initial datum ($A_a^i$ , $\tilde{E}_i^a$) for Einstein's theory is also an initial datum for Yang-Mills theory which happens to satisfy, in addition to the Gauss law, a scalar and a vector constraint. From the standpoint of Yang-Mills theory, the additional constraints are the simplest diffeomorphism and gauge invariant expressions one can write down in absence of a background structure such as a metric. Note that the degrees of freedom match: the Yang-Mills field has $2$(helicity) $\times 3$ (internal) $= 6$ degrees and the imposition of four additional first-class constraints leaves us with $6-4 =2$ degrees of freedom of Einstein's theory. I want to emphasize, however, that in spite of this close relation of the two initial value problems, the Hamiltonians (and thus the dynamics) of the two theories are very different. Nonetheless, the similarity that does exist can be exploited to obtain interesting results relating the two theories. For example, there are interesting and surprising relations between instantons in the two theories. (See, e.g. (Samuel J, 2000)).
  • Since all equations are polynomial in $A_a^i$ and $\tilde{E}_i^a$ they continue to be meaningful even when the triad (i.e. the “electric field”) $\tilde{E}_i^a$ becomes degenerate or even vanishes. As in the 2+1 theory, this feature enables one to use in quantum theory a representation in which states are functionals of $A_a^i$ and $\hat{E}^a_i$ is represented simply by a functional derivative with respect to $A_a^i$, thereby shifting the emphasis from triads to connections. In fact, Capovilla, Dell and Jacobson (Capovilla R, Dell J, Jacobson T, 1989; Capovilla R, Dell J, Jacobson T, Mason L, 1991) have introduced a Lagrangian framework which reproduces the Hamiltonian description discussed above but which never even introduces a space-time metric or tetrads! This formulation of “general relativity without metric” lends strong support to the viewpoint that the traditional emphasis on metric-dynamics, however convenient in classical physics, is not indispensable.

This completes the discussion of the Hamiltonian description of general relativity which casts it as a theory of self dual connections. We have transformed triads from configuration to momentum variables and found that self dual connections serve as especially convenient configuration variables. In effect, relative to the Arnowitt-Deser-Misner geometrodynamical description, we are looking at the theory “upside down” or “inside out”. And this unconventional way of looking reveals that the theory has a number of unexpected and, potentially, profound features: it is much closer to gauge theories (particularly the topological ones) than was previously imagined; its constraints are the simplest background independent expressions one can write down on the phase space of a gauge theory; its dynamics has a simple geometrical interpretation on the space of connections; etc. It opens new doors particularly for the task of quantizing the theory. We are led to shift emphasis from metrics and distances to connections and holonomies and this, in turn suggests fresh approaches to unifying the mathematical framework underlying the four basic interactions (see, e.g., (Peldán P, 1993)).

Real connection-variables

Because of the topic assigned to me by the Editors, so far I have focused on chiral connection-variables. As we saw, in the classical theory, they provide a viable reformulation of general relativity and, by bringing the theory closer to the successful gauge theories describing other interactions, they provide new tools for the passage to quantum theory. However, since a chiral connection is complex-valued in the Lorentzian signature, the holonomies it defines take values in a non-compact subgroup of ${\rm SL(2, C)}$ generated by the self-dual sub-space of its Lie-algebra. This creates a major obstacle in developing a well-defined integration theory ---that respects gauge and diffeomorphism invariance--- on the infinite dimensional space of these connections. Without this integration theory, we cannot construct the Hilbert space of quantum states, introduce physically interesting operators thereon, and analyze properties of these operators. Since the difficulty stems from the non-compact nature of the subgroup ${\rm SL(2, C)_{sd}}$ of ${\rm SL(2, C)}$ in which holonomies take values, a natural strategy is to perform a ‘Wick transform’ in the internal space that sends ${\rm SL(2, C)_{sd}}$ to an ${\rm SU(2)}$ subgroup of ${\rm SL(2, C)}$ (which is compact). This strategy has been adopted in most of the mainstream work in LQG since mid-1990s. Concretely, the desired Wick transform is performed by sending self dual connections $A_{a}^{i}$ to connections ${}^{\gamma}\!A_a{}^i$, simply by replacing $i$ in (9) with a real parameter $\gamma$, called the Barbero-Immirzi parameter (Immirzi G, 1997; Barbero F, 1995) (which is assumed to be positive without loss of generality): \[ \tag{14} G\, {}^{\gamma}\!A_a^i := \Gamma_a{}^i - \gamma K_a{}^i. \] While the subgroup of ${\rm SL(2, C)}$ one thus obtains depends on the choice of $\gamma$, it is always an ${\rm SU(2)}$ subgroup, whence the integration theory is insensitive to the specific choice of $\gamma$. Note that this ‘Wick transform’ is performed on the internal space, where it is well-defined also in curved space-times; it is distinct from the standard space-time Wick transform performed in Minkowskian quantum field theories which does not have a well-defined extension to general curved space-times. Nonetheless the basic motivation is the same as in Minkowskian quantum field theories: one can regard it as a method of regularizing the functional integrals that are ill-defined in the Lorentzian sector.

After this passage, it was possible to develop a rigorous integration theory as well as introduce notions from geometry on the (infinite dimensional) configuration space of connections ${}^{\gamma}\!A_{a}^{i}$ and systematically develop a specific quantum theory of Riemannian geometry (Ashtekar A, Lewandowski L, 1994; Ashtekar A, Lewandowski L, 1995; Ashtekar A, Lewandowski L, 1997; Ashtekar A, Lewandowski L, 1997). The construction of the Hilbert space of quantum states and the definition of the elementary (holonomy and flux) operators is insensitive to the choice of $\gamma$. However, now $\gamma$ enters in the relation between the momentum ${}^{\gamma}\!\Pi^{a}_{i}$ conjugate to ${}^{\gamma}\!A_{a}^{i}$ and the orthonormal triad $\tilde{E}^{a}_{i}$. As a result it also enters the expressions of various geometric operators.

However, the qualitative features of quantum geometry do not depend on the specific choice of $\gamma$. Just as the flux of the magnetic field is quantized in a type II superconductor, the flux of the ‘electric field’ $\tilde{E}^{a}_{i}$ is quantized in the quantum Riemannian geometry. Since the electric field also serves as a (density weighted triad) in the classical theory, determining the spatial metric $q_{ab}$, the quantum Riemannian geometry now acquires an interesting and very non-trivial discreteness. More precisely, the eigenvalues of geometric operators such as areas of 2-surfaces or volumes of 3-dimensional regions are discrete. This discreteness has non-trivial consequences on quantum dynamics. In particular, in cosmological models, quantum geometry creates a brand new repulsive force which is negligible under normal circumstances but rises quickly in the Planck regime and overwhelms the classical attraction. In intuitive terms, under normal circumstances, general relativity provides an excellent approximation to quantum dynamics. But if this dynamics drives a curvature scalar to the Planck regime, quantum geometry effects become prominent and ‘dilute’ the curvature scalar, preventing the formation of a strong curvature singularity. (For details, see (Ashtekar A, Singh P, 2011) and references therein.) Similarly, in the path integral approach, one is now naturally led to sum over the specific, discrete quantum geometries provided by the detailed LQG framework. As a result, one can express the transition amplitudes as a sum, each term in which is ultraviolet finite. (For details, see (Ashtekar A, Reuter M, Rovelli C, 2015) and references therein). Thus, the connection-dynamics formulation of general relativity leads one along new paths that combine techniques from gauge theory and the underlying diffeomorphism invariance. This combination has led to unforeseen results representing concrete advances through LQG.

However, the ‘internal Wick transform’ strategy has two limitations. First, the form of the constraints (and evolution equations) is now considerably more complicated already in the classical theory. These complications seemed so formidable that while the idea of moving away from chiral connections was considered, it was not pursued initially. Almost a decade after the introduction of chiral connections, the strategy was worked out in complete detail by Barbero (Barbero F, 1995) at the classical level. Soon thereafter, Thiemann (Thiemann T, 1996; Thiemann T, 1998; Thiemann T, 1998; Thiemann T, 1998; Thiemann T, 2007) introduced several astute techniques to handle the complications in the canonical approach within LQG. It is only then that the strategy became mainstream. A second limitation of the strategy is that while the connection ${}^{\gamma}\!A_a^i$ is well-defined on the spatial 3-manifold and continues to have a simple relation (14) to the ADM variables, it does not have a natural 4-dimensional geometrical interpretation even in solutions to the field equations (Samuel J, 2000). As I illustrated above, significant advances have occurred in spite of these limitations. Still the situation could be improved significantly by seriously pursuing the viewpoint that the passage to ${}^{\gamma}\!A_a^i$ is only a mathematical construct and all questions should be phrased and all answers be given using the Lorentzian chiral connection. This idea is acquiring momentum over the last few years (see, e.g., (Geiller M, Noui K, 2014; Wieland W M, 2012; Wieland W M, 2015) but we are still far from a complete picture.


The connection-dynamics formulation of general relativity provide a fresh perspective on the deep underlying simplicity of Einstein's equations. As I emphasized in the beginning of this article, the central feature of general relativity is that gravity is encoded in the very geometry of space-time. Therefore, a quantum theory of gravity should have at its core a quantum theory of geometry. The connection-dynamics formulation opens new vistas to construct this theory. Specifically, it provides brand new tools for this purpose: holonomies defined by the gravitational connection (i.e., Wilson loops), and quantization of the flux of the conjugate electric field across 2-surfaces. These tools have led to a rich Riemannian quantum geometry (Ashtekar A, Lewandowski J, 2004; Rovelli C, 2004; Thiemann T, 2007). The novel aspects associated with the fundamental discreteness of this geometry have already led to some unforeseen consequences. These include the resolution of strong curvature singularities in a variety of cosmological models (Ashtekar A, Singh P, 2011; Singh P, Singh P, 2009); new insights into the microstructure of the geometry of quantum horizons (Ashtekar A, Krishnan B, 2004; Ashtekar A, Krasnov K, 1999; Ashtekar A, Reuter M, Rovelli C, 2015); and, a derivation of the graviton propagator in the background independent, non-perturbative setting of spinfoams (Ashtekar A, Reuter M, Rovelli C, 2015; Bianchi E, Magliaro E, Perini C, 2009; Bianchi E, Magliaro E, Perini C, 2012).

The formulation also provides a small generalization of Einstein's theory: Since the equations are polynomial in the canonical variables, they do not break down if the (density weighted) triad $\tilde{E}^{a}_{i}$ were to become degenerate or even vanish. Consequently, unlike the Arnowitt-Deser-Misner formalism, evolution remains well-defined even when the 3-metric becomes degenerate or even vanishes at some points during evolution. This extension of general relativity was studied in some detail (Bengtsson I, Jacobson T, 1997; Bengtsson I, Jacobson T, 1998), and in particular, the causal structure of these generalized solutions has been analyzed (Matschull H J, 1996). These investigations may well be useful in the future analysis of various phases of quantum gravity. Another tantalizing aspect of the connection formulation is that is also leads to a Hamiltonian formulation of general relativity in terms of fields (with density weight 1) on the spatial 3-manifold which have only internal indices (Ashtekar A, Henderson A, Sloan D, 2009). This formulation is particularly well-suited to analyze the behavior of the gravitational field as one approaches space-like singularities. Indeed, there is now a specific formulation of the Belinskii-Khalatnikov-Lifshits (BKL) conjecture in terms of these variables provided by the (self-dual) connection dynamics and numerical simulations have been performed using these variables, exhibiting the conjectured BKL behavior (Ashtekar A, Henderson A, Sloan D, 2009; Ashtekar A, Henderson A, Sloan D, 2011). While other formulations of the BKL conjecture are motivated primarily by considerations involving differential equations, this formulation comes with a Hamiltonian framework and is therefore well suited for the analysis of the fate of generic space-like singularities in LQG. The extension to include degenerate metrics and the Hamiltonian framework involving only fields with internal indices have a strong potential to shed new light on the fate of generic, space-like singularities of general relativity but have remained largely unexplored so far.

Finally, the connection dynamics formulation of general relativity, discussed here, has been generalized in several ways: generalizations of Einstein dynamics in four dimensions through interesting deformations of the constraint algebra (Krasnov K, 2007); general relativity in higher space-time dimensions (Bodendorfer N, Thiemann T, Thurn A, 2013) ; and, inclusion of supersymmetries (Bodendorfer N, Thiemann T, Thurn A, 2012). There is considerable ongoing research in these interesting directions. However, these developments fall beyond the charge assigned to me by the Editors.

Appendix A: Connection-dynamics: A historical anecdote

I will now describe the early attempt by Einstein and Schrödinger, mentioned in section 1, to formulate theories of relativistic gravity in terms of connections rather than space-time metrics.7 The episode is of interest not only for its scientific content but also for sociology of science.

In the 1940s, both men were working on unified field theories. They were intellectually very close. Indeed, Einstein wrote to Schrödinger saying that he was perhaps the only one who was not ‘wearing blinkers’ in regard to fundamental questions in science and Schrödinger credited Einstein for inspiration behind his own work that led to the Schrödinger equation. Einstein was at the Institute of advanced Study in Princeton and Schrödinger at the Institute for Advanced Study in Dublin. During the years 1946-47, they frequently exchanged ideas on unified field theory and, in particular, on the issue of whether connections should be regarded as fundamental in place of space-time metrics. In fact the dates on their letters often show that the correspondence was going back and forth with astonishing speed. It reveals how quickly they understood the technical material the other hand sent, how they sometimes hesitated and how they teased each other. Here are a few quotes:

The whole thing is going through my head like a millwheel: To take $\Gamma$ [the connection] alone as the primitive variable or the $g$'s [metrics] and $\Gamma$'s ? ...

---Schrödinger, May 1st, 1946.

How well I understand your hesitating attitude! I must confess to you that inwardly I am not so certain ... We have squandered a lot of time on this thing, and the results look like a gift from devil's grandmother.

---Einstein, May 20th, 1946

Einstein was expressing doubts about using the Levi-Civita connection alone as the starting point which he had advocated at one time. Schrödinger wrote back that he laughed very hard at the phrase ‘devil's grandmother’. In another letter, Einstein called Schrödinger ‘a clever rascal’. Schrödinger was delighted: ‘No letter of nobility from emperor or king ... could to me greater honor...’. This continued all through 1946.

Then, in the beginning of 1947, Schrödinger thought he had made a breakthrough.8 He wrote to Einstein:

Today, I can report on a real advance. May be you will grumble frightfully for you have explained recently why you don't approve of my method. But very soon, you will agree with me...

---Schrödinger, January 26th, 1947

Schrödinger sincerely believed that this advance was revolutionary. In his view the ‘breakthrough’ was to drop the requirement that the (Levi-Civita) connection be symmetric, i.e., to allow for torsion. This does not seem to be that profound an idea. But at the time Schrödinger believed it was. In discussions at Dublin, he referred to similar previous attempts by Einstein and Eddington ---in which this symmetry was assumed--- and suggested that they did not work simply because of this strong restriction:

I will give you a good simile. A man wants a steed to take a hurdle. He looks at it and says ‘Poor thing, it has four legs, it will be very difficult for him to control all four of them. ... I will teach him in successive steps. I will bind his hind legs together. He will learn to jump on his fore legs alone. That will be much simpler. Later on, perhaps, he will learn it with all four. This describes the situation perfectly. The poor thing, $\Gamma^i_{kl}$, got its hind legs bound together by the symmetry condition, $\Gamma^i_{kl} = \Gamma^i_{lk}$, taking away 24 of its 64 degrees of freedom. The effect was, it could not jump and it was put away as good for nothing.

The paper was presented to the Royal Irish academy on January 27th, the day after he wrote to Einstein. The Irish prime minister (the Taoiseach) Eamon de Valera, a mathematician, and a number of newspaper reporters were present. Privately, Schrödinger spoke of a second Nobel prize. The next day, the following headlines appeared:

Twenty persons heard and saw history being made in the world of physics. ... The Taoiseach was in the group of professors and students. .. [To a question from the reporter] Professor Schrödinger replied “This is the generalization. Now the Einstein theory becomes simply a special case ...”

---Irish Press, January 28th, 1947

Not surprisingly, the headlines were picked up by New York Times which obtained photocopies of Schrödinger's paper and sent them to prominent physicists --including of course Einstein-- for comments. As Walter Moore, Schrödinger's biographer puts it, Einstein could hardly believe that such grandiose claims had been made based on a what was at best a small advance in an area of work that they both had been pursuing for some time along parallel lines. He prepared a carefully worded response to the request from New York Times:

... It seems undesirable to me to present such preliminary attempts to the public in any form. It is even worse when an impression is created that one is dealing with definite discoveries concerning physical reality. Such communiqués given in sensational terms give the lay public misleading ideas about the character of research. The reader gets the impression that every five minutes there is a revolution in Science, somewhat like a coup d’état in some of the smaller unstable republics. ...

Einstein's comments were also carried by the international press. On seeing them, Schrödinger felt deeply chastened and wrote a letter of apology to Einstein. Unfortunately, as an excuse for his excessive claims, he said he had to ‘indulge in a little hot air in my present somewhat precarious [financial] situation’. It seems likely that this explanation only worsened the situation. Einstein never replied. He also stopped scientific communication with Schrödinger for three years.

The episode must have been shocking to those few who were exploring general relativity and unified field theories at the time. Could it be that this episode effectively buried the desire to follow-up on connection formulations of general relativity until an entirely new generation of physicists who were blissfully unaware of this episode came on the scene?

Acknowledgments In the work summarized here, I profited a great deal from discussions with a large number of colleagues. I would especially like to thank Amitabha Sen, Gary Horowitz, Joe Romano, Ranjeet Tate, Jerzy Lewandowski, Joseph Samuel, Carlo Rovelli and Lee Smolin. This work was supported in part by the NSF grant PHY-1205388 and the Eberly research funds of Penn state.


1 At this point, I should emphasize that the bibliography provided in this article is very far from being exhaustive; these references only provide a point of entry for various topics. For more exhaustive lists, the reader can use the one compiled in (Beetle C, Corichi A, 1997) for papers written in the first decade of LQG, and references given in (Ashtekar A, Lewandowski J, 2004; Rovelli C, 2004; Thiemann T, 2007; Rovelli C, Vidotto F, 2014; Ashtekar A, Reuter M, Rovelli C, 2015) for later developments.
2 Actually, only six of the ten Einstein's equations provide the evolution equations. The other four do not involve time-derivatives at all and are thus constraints on the initial values of $q_{ab}$ and its time derivative $K_{ab}$, the extrinsic curvature. However, if the constraint equations are satisfied initially, they continue to be satisfied at all times.
3 This simplicity is not manifest in geometrodynamics where the dynamical trajectories on the superspace of metrics are not geodesics but rather represent a motion in a potential that is, moreover, non-polynomial in the basic configuration variable.
4 Note that the conjugate momentum is a vector density, and can thus be thought of as the ‘electric field’ in the Yang-Mills terminology. We could have worked with the 1-forms $e_{a}^{I}$ without dualization through $\tilde{\eta}^{ab}$. Here, and also in the 3+1 theory, the dualization is carried out only to bring out the similarity with the traditional Hamiltonian formulations of Yang-Mills theories.
5 At first sight, it may appear that this interpretation requires $G^{\alpha\beta}$ to be non-degenerate since it is only in this case that one can compute the connection compatible with $G^{\alpha\beta}$ unambiguously and speak of null geodesics. However, in the degenerate case, there exists a natural generalization of this notion of null geodesics which is routinely used in Hamiltonian mechanics.
6 In this framework, the lapse naturally arises as a scalar density $\underset{\sim}{N}$ of weight $-1$. It is $\underset{\sim}{N}$ that is the basic, metric independent field. The “geometric” lapse function $N$ is metric dependent and given by $N=\sqrt{q}\underset{\sim}{N}$. Note also that, unlike in geometrodynamics, Newton's constant never appears explicitly in the expressions of constraints, Hamiltonians, or equations of motion; it features only through the expression for $F_{ab}{}^i$ in terms of the connection.
7 For further details and more complete quotes, see e.g. Chapter 11 in (Moore W, 1999).
8 He used only a non-symmetric Levi-Civita connection $\Gamma^i_{kl}$ on space-time as the basic variable and regarded the square-root of the Ricci tensor $R_{ik}$ of the connection as the Lagrangian density. Space-time metric did not even feature in the main equations. The theory was to naturally unify gravitation with electromagnetism.


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