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II. FROM EINSTEIN-HAMILTON-JACOBI TO family of solutions S[q, q0] parametrized by a 3-metric EINSTEIN-SCHRODINGER¨ 0 qab(~x). Then we can define the momenta

0 In the following I make no attempt at a full review ab 0 δS[q, q ] p0 [q, q ](~x)= 0 (7) of the vast literature where the WdW equation has been δqab(~x) discussed, utilized, and has left its mark. Instead, I focus on the main issues the equation has raised, on what we which are thus nonlocal functions of q and q0. For any 0 have learned from it, and on the possibility it has opened choice of q and p0 satisfying (4) and (5), there exists a for describing quantum spacetime. spacetime which is a solution of Einstein’s equations and I start by illustrating the path leading to the WdW for any such solution, any field q satisfying equation via Peres’ Hamilton-Jacobi formulation of gen- ab 0 ab eral relativity [1], in a bit more detail than above. This p0 [q, q ](~x)= p0 (~x). (8) is relevant not only for history, but especially because is the metric of a spacelike 3d surface imbedded in this it clarifies the meaning of the WdW equation, shedding spacetime. Since the q0 and p are related by equa- light on the confusion it raised. Some difficulties of the 0 tions (4) and (5), the last equation does not determine q WdW equation are more apparent than real. They have uniquely: the different solutions correspond to different nothing to do with quantum mechanics; they stem from spacelike surfaces in spacetime, and different coordinati- the application of the Hamilton-Jacobi language to a gen- zation of the same; in fact, all of them. Therefore the erally covariant context. solution to the Einstein-Hamilton-Jacobi system (1)-(6) Einstein wrote general relativity in terms of the provides in principle the full solution of the Einstein’s Lorentzian metric g (~x, t), where µ,ν = 0, ..., 3 are µν equations. These results follow from a simple generaliza- spacetime indices and (~x, t) are coordinates on space- tion of standard Hamilton-Jacobi theory. time. Peres’ starting point was the Arnowit-Deser- This was the starting point of Bryce DeWitt. Now re- Misner change of variables [5] call that in his milestone 1926 article [7], Schr¨odinger in- oo qab = gab,Na = gao, N = √ g . (3) troduced (what is called today) the Schr¨odinger equation − for the hydrogen atom by taking the Hamilton-Jacobi a ab where a,b =1, 2, 3. N and N = q Nb are the “Lapse” equation of an electron in a Coulomb potential: ab and “Shift” functions, and q is the inverse of qab. The interest of these variables is that the Lagrangian does ∂S(~x, t) 1 ∂S(~x, t) ∂S(~x, t) e2 + + = 0 (9) not depend on the time derivatives of Lapse and Shift. ∂t 2m ∂~x · ∂~x ~x Therefore the only true dynamical variable is the three | | metric qab(~x), the Riemannian metric of a t = constant and replacing derivatives with ( i~ times) derivative op- surface. The canonical hamiltonian vanishes, as is always erators: − the case for systems invariant under re-parametrization ∂ ~2 ∂ ∂ e2 of the lagrangian evolution parameter, and the dynamics i~ + ψ(~x, t)=0. (10) − ∂t − 2m ∂~x · ∂~x ~x  is coded by the two ADM constraints | |

1 ac bd In a shortly subsequent article [8], Schr¨odinger offers a qabqcd qacqbd p p det q R[q] = 0 (4)  − 2  − rationale for this procedure: the eikonal approximation of a wave equation, which defines the geometrical optic and approximation where wave packets follow definite trajec- tories, can be obtained by the opposite procedure. If ab Da p =0, (5) we interpret classical mechanics as the eikonal approx- imation to a wave mechanics, we can guess the wave on the three metric q and its conjugate momentum p. equation by this procedure. Given the immense suc- (Da is the covariant derivative of the 3-metric.) cess of Schr¨odinger’s leap, trying the same strategy for The dynamics can be written in Hamilton-Jacobi form gravity is obviously tempting. This is what led DeWitt by introducing the Hamilton-Jacobi function S[q], which and Wheeler to equation (2). Next to it, the second is a functional of qab(~x) and demanding that this sat- Hamilton-Jacobi equation, following from (5), remains ab isfies the two equations above with p replaced by the unaltered: functional derivative δS[q]/δqab. The first equation gives (1), while the second can be rather easily shown [6] to be δ Da Ψ[q]=0, (11) equivalent to the request δqab

S[q]= S[˜q] (6) and, as before, can be simply shown to be equivalent to the requirement that Ψ[q] = Ψ[q′] if q and q′ are related for any two 3d metrics q andq ˜ related a 3d change of by a 3d coordinate transformation. That is, the wave coordinates. Solving (1) and (6) amounts to solve the function is only a function of the “3-geometry”, namely Einstein equations. To see how, say we have found a the equivalence class of metrics under a diffeomorphism, 3 and not of the specific coordinate dependent form of the same, will then differ. Given the appropriate initial data, qab(~x) tensor. general relativity allows us to compute the value of T1 The Schr¨odinger equation (10) gives, in a sense, the when the second clock reads T2, or viceversa. Does this full dynamics of the electron in the hydrogen atom. Sim- describe the evolution of T1 in the “time” T2, or, rather, ilarly, one expects the WdW equation (2), properly un- the evolution of T2 in the “time” T1? derstood and properly defined, to give the full dynamics The question is clearly pointless: general relativity of quantum gravity. describes the relative evolution of the two variables T1 and T2, both given by (12) but computed along different worldliness. The two “times” T1 and T2 are on the same III. PHYSICS WITHOUT BACKGROUND TIME footing. The example shows that general relativity de- scribes the relative evolution of variable quantities with The immediately puzzling aspect of the WdW equa- respect to one another, and not the absolute evolution of tion, and the one that has raised the largest confusion, is variables in time. the absence of a time variable in the equation. This has Mathematically this is realized by parametrizing the been often wrongly attributed to some mysterious quan- motions. Instead of using T1(T2) or T2(T1) to de- tum disappearance of time. But things are simpler: the scribe evolution, the theory uses the parametric form disappearance of the time variable is already a feature of T1(t),T2(t), where t is an arbitrary parameter, which can the classical Hamilton-Jacobi formulation of general rel- be chosen freely. The coordinates (t, ~x) in the argument ativity. It has nothing to do with quantum mechanics. It of the gravitational field gµν (~x, t) are arbitrary parame- it is only a consequence of the peculiar manner in which ters of this sort. evolution is described in general relativity. This manner of describing evolution is more general In Newtonian and special relativistic physics the time than giving the evolution in a preferred time parame- variable represent the reading of a clock. It is therefore a ter. The formal structure of dynamics can be general- quantity to which we associate a well determined proce- ized to this wider context. This was early recognized by dure of measurement. Not so in general relativity, where Dirac [10] and the corresponding generalized formulation the reading of a clock is not given by the time variable t, of mechanics has been discussed by many authors in sev- but is instead expressed by a line integral depending on eral variants (see for instance [11, 12]), often under the the gravitational field, computed along the clock’s world- deceptively restrictive denomination of “constrained sys- line γ, tem dynamics”. “Constrained system dynamics” is not the dynamics of special systems that have constraints: it µ ν T = gµν dx dx . (12) is a generalization of dynamics which avoids the need of Z γ p picking one of the variables and treating it as the special, independent, evolution parameter. In the correspond- The coordinate t in the argument of gµν (~x, t) which is ing generalization of Hamilton-Jacobi theory, parameter the evolution parameter of the Lagrangian and Hamil- time does not appear at all, because the Hamilton-Jacobi tonian formalisms has no direct physical meaning, and theory gives the relation between observable variables di- can be changed freely. Such change in the manner evolu- rectly. tion is described is is not a minor step. Einstein’s wrote Thus, the reason that the coordinate time variable t that the biggest difficulty he had to overcome in order to does not show up in the WdW equation is not at all find general relativity was to understand “the meaning 1 mysterious after all. It is the same reason for which co- of the coordinates”. The physical prediction of general ordinates do not show up in the physical predictions of relativity, which can be directly tested against experience classical general relativity: they have no physical mean- are not given by the evolution of physical quantities in ing, and the theory can well do without. the coordinate t, but, rather, in the relative evolution of In particular, the absence of t in the WdW equation physical quantities with respect to one another. This is does not imply at all that the theory describes a frozen why using the time variable t is not required for making world, as unfortunately often suggested. One can pick sense of general relativity. a function of the gravitational field, or, more realisti- To clarify this crucial point, consider two clocks on cally, couple a simple system to the gravitational field, the surface of the Earth; imagine one of them is thrown and use it as a physical clock, to coordinatize evolution upward, and then falls back down near the first. The in a physically relevant manner. A common strategy in reading of the two clocks, say T and T , initially the 1 2 quantum cosmology, for instance, is to include a scalar field φ(~x, t) in the system studied, take the approximation where φ(~x, t) is constant in space φ(~x, t)= φ(t) and give 1 it a simple dynamics, such as a linear growth in proper “Why were a further seven years required for setting up the gen- eral theory of relativity? The principal reason is that one does time. Then the value of φ can be taken as a “clock” –it not free oneself so easily from the conception that an immedi- coordinatizes trajectories of the system– and the WdW ate physical significance must be attributed to the coordinates.” wave function Ψ[q, φ] can be interpreted as describing the Albert Einstein, in [9]. evolution of Ψ[q] in the physical variable φ. 4

The full structure of quantum mechanics in the absence in terms of the transition amplitudes (14). In practice, of a preferred time variable has been studied by several setting up a concrete realization of this framework is com- authors. Reference [12] is my own favorite version. It can plicated. To see where we are today along the path of be synthesized as follows. The variables that can inter- making sense of the quantum theory formally defined by act with an external system (“partial observables”) are the WdW equation, and to describe the legacy of the represented by self-adjoint operators An in an auxiliary equation, I now turn to some specific current approach Hilbert space where the WdW operator C is defined. to quantum gravity. The WdW equationK

CΨ=0 (13) IV. WDW IN STRINGS defines a linear subspace , and we call P : H H → H the orthogonal projector. If zero lies in the continuous String theory developed starting from conventional spectrum of C, then is not a proper subspace of . It H K quantum field theory and got in touch with general rela- is a generalized subspace namely a linear subspace of a tivity only later. Even in dealing with gravity, string the- suitable completion of in a weak norm. In this case, ory was for sometime confined to perturbations around still inherits a scalar productK from : this can be definedH K Minkowski space, where the characteristic features of using various techniques, such as spectral decomposition general relativity are not prominent. For this reason, the of , or group averaging. P : is still defined, it is specific issues raised by the fact that spacetime is dy- K 2 H → H not a projector, as P diverges, but we still have CP = namical have not played a major role in the first phases P C =0. If An is a complete set of commuting operators of development. But major issues cannot remain long (in the sense of Dirac) and an a basis diagonalizing | i hidden, and at some point string theory has begun to them, then face dynamical aspects of spacetime. In recent years, the realization that in the world there is no preferred W (a′; a)= a′ P an (14) h n| | i spacetime has impacted string theory substantial and the string community has even gone to the opposite extreme, is the amplitude for measuring a after a has been n′ n largely embracing more or less precise holographic ideas, measured, from which transition{ probabilities} { } can be de- where bulk spacetime is not anymore a primary ingredi- fined by properly normalizing. ent. In this context the WdW equation has reappeared This formalism is a generalization of standard quantum in various ways in the context of the theory. mechanics. It reduces to the usual case if For instance, in AdS/CFT setting, a constant radial ∂ coordinate surface can be seen as playing the role of the C = i~ + H (15) ∂t ADM constant time surface, the quantisation of the cor- responding constraint of the bulk gravity theory gives a where H is a standard Hamiltonian. In this case, if q,t { } WdW equation and its Hamilton-Jacobi limit turns out is a complete set of quantum numbers, then to admit an interpretation as a renormalization group equation for the boundary CFT [13, 14]. W (q′,t′; q,t)= t′, q′ P a,t (16) h | | i In fact, the full AdS/CFT correspondence can be seen turns out to be the standard propagator as a realization of a WdW framework: the correspond critically relies on the ADM Hamiltonian being a pure i Ht boundary term, since its “bulk” part is pure gauge. See W (q′,t′; q,t)= q′ e− ~ q (17) h | | i [15] and, for a direct attempt to derive the AdS/CFT which has the entire information about the quantum dy- correspondence from the WdW equation, [16]. See also namics of the system. Thus, in general the WdW equa- the discussion on dS/CFT, where the “wave function of tion is just a generalization of the Schr¨odinger equation, the universe” seen as a functional of 3-metrics, just as in to the case where a preferred time variable is not singled canonical general relativity plays a major role [17]. out. Equation (2) is the concrete form that (13) takes There is a more direct analog to the WdW equation when the dynamical system is the gravitational field. in the foundation of string theory. In its “first quantisa- The first merit of the equation written by Wheeler and tion”, string theory can be viewed as a two-dimensional DeWitt in 1965 is therefore that it has opened the way to field theory on the world-sheet. The action can be taken the generalization of quantum theory needed for under- to be the Polyakov action, which is generally covariant standing the quantum properties of our general covariant on the world-sheet, and therefore the dynamics is entirely world. determined by the constraints as in the case of general The world in which we live is, as far as we understand, relativity. In fact, the two general relativity constraints well described by a general covariant theory. In principle, (4) and (5) have a direct analog in string theory as the any quantity that we can observe and measure around us Virasoro constraints can be represented by an operator An on , and all dy- namical relations predicted by physics canK be expressed Π2 X 2 = 0 (18) − |∇ | 5 and be constructed, in the continuum, as follows. Choose a 3 loop γ = S1 R , namely a closed line in space and X Π=0 (19) → ∇ · consider the trace of the holonomy of A along this loop, namely the quantity where X are the coordinates of the string and Π its mo- mentum. As in general relativity, the second equation H A implements the invariance under 1-dimensional spatial Ψγ [A]= TrPe γ (22) change of coordinates on the world sheet, while the first is a “hamiltonian constraint” which codes its dynamics. The left- and right-moving null combinations (associated where P indicates the standard path ordered exponentia- with x t) are the left- and right-moving Virasoro alge- tion. It turns out that Ψγ[A] is a solution of the Ashtekar- bras. However,± the standard quantization of the string is WdW equation (21), if the loop has no self-intersection (if obtained in a manner different from the way the WdW γ is injective) [20, 21]. A simplified derivation of this re- is derived. The two equations combined in right and left sult is to observe that the functional derivative of a holon- null combinations and Fourier transformed give the Vi- omy vanishes in the space points ~x outside the loop and a a rasoro operators Ln, of which only the n 1 components is otherwise proportional to the tangentγ ˙ = dγ (s)/ds are imposed as operator equations on the≥ quantum states. to the loop: The expectation value of the Ln for negative n vanishes on physical states, so the full constraint is still recovered δΨγ a 3 R A in the classical limit (in a way similar to Gupta-Bleuler). = ds γ˙ (s) Aa(γ(s)) δ (γ(s), ~x) Pe γ (23) δAa(~x) Iγ In addition, states are required to be eigenstates of L0, with eigenvalue given by the central charge of the corre- sponding conformal field theory. where a trace is understood and the path order integral The reason for these choices is that the theory is re- starts at the loop point ~x. The left hand side of (21), is a b quired to make sense in the target space and respect its therefore proportional to Fab γ˙ γ˙ , which vanishes be- Poincar´einvariance. But imposing all the constraints cause of the antisymmetry of Fab if the loop has a no Ln = 0 would be inconsistent, because of the non trivial- self-intersection. If it has intersections, in the intersec- ity of the Ln’s algebra: a warning about naive treatment tion point there are two different tangents and mixed of the hamiltonian constraints that must be kept in mind terms do not cancel. Thus, loop states Ψγ without inter- also in the case of general relativity. sections are exact solutions of the WdW equation. Non-intersecting loop-states alone do not describe a V. WDW IN LOOPS realistic quantum space, because they are eigenstates of the volume V = det qab with vanishing eigenvalue. (In- 2 a b c a b c deed, V ǫabcE E E ǫabcγ˙ γ˙ γ˙ = 0). Therefore The line of research where the WdW equation has had intersections∼ play a role in∼ the theory [22]. But acting the largest impact is Loop Gravity (LQG), currently the on a loop state with intersections, the Ashtekar-WdW most active attempts to develop a specific theory of quan- operator acts non trivially only at the intersection point. tum gravity addressing the possibility of quantum super- This is the basis fact underpinning LQC. position of spacetimes. LQG, indeed, is born from a partial solution of the The loop representation of quantum general relativity WdW equation, and its structure is based on this partial can formally be obtained by moving from the connection basis Ψ[A]= A Ψ to a basis formed by loop states with solution. General relativity can be rewritten in terms h | i of variables different from the metric used by Einstein. intersections. An orthonormal basis of such loop-based These, developed by Abhay Ashtekar in the late eighties, states with intersection is given by the spin network basis are the variables of an SU(2) gauge theory, namely (in [23], which provides today the standard basis on which the theory is defined. In this basis, the WdW operator the hamiltonian framework) an SU(2) connection Aa and its“electric field” conjugate momentum Ea [18]. Rewrit- acts only at intersections, which are called the “nodes” ten in terms of these variables, the hamiltonian constraint of the network. A rigorous and well-defined definition of of general relativity reads the WdW operator in this representation has been given by Thomas Thiemann, and is constructed and studied in a b C = FabE E = 0 (20) detail in his book [24]. Simplified versions of this operator are heavily used in and the WdW equation takes the simpler form the field of Loop Quantum Cosmology, namely the ap- δ δ plication of loop-gravity results to quantum cosmology F Ψ[A]=0. (21) ab δA δA [25]. Loop quantum cosmology is producing some tenta- a b tive preliminary predictions about possible early universe Remarkably, we know a large number of solutions of this quantum gravity effects on the CMB (see for example equation. These were first discovered using a lattice dis- [26]). If these were verified, the result would be of major cretization by Ted Jacobson and Lee Smolin [19], and can importance and WdW would have been a key ingredient. 6

VI. WDW AND PATH INTEGRALS The oscillation between canonical and covariant meth- ods is well known in fundamental theoretical physics, and Since its earliest days [27], the inspiration for the is not peculiar of quantum gravity. The two approaches search for a good quantum theory of quantum gravity have complementary strengths: the hamiltonian theory has oscillated between the canonical WdW framework captures aspects that are easily overlook ed in the covari- and the covariant framework provided by a “path inte- ant language, especially in the quantum context, while gral over geometries” the lagrangian framework allows symmetries to remain manifest, is physically far more transparent, and leads to more straightforward calculations techniques. Z = D[g] ei R √gR[g]. (24) Z In the loop context, the difficulties of dealing with the It is hard to give this integral a mathematical sense, WdW equation have pushed a good part of the com- or to use it for computing transition amplitudes within munity to adopt alternative, covariant methods for com- some approximation scheme, but the formal expression puting the transition amplitudes (14), which makes use (24) has provided an intuitive guidance for constructing of the so called “spinfoam” techniques [31], a sum-over- the theory. Formally, the path integral is related to the paths technique of computing amplitudes between spin WdW equation, in same manner in which the Feynman network states which can be seen as a well-defined ver- path integral that defines the propagator a non relativis- sion of equation (24). tic particle is a solution of the Schr¨odinger equation. This relation has taken a particularly intriguing form in the After all, the initial unhappiness of Bryce DeWitt context of the Euclidean quantum gravity program, de- with an equation entangled within the complexities of veloped by Hawking and his collaborators [28], where the the Hamiltonian formalism and having the bad man- wave functional ners of breaking the manifest covariance between space and time, were not unmotivated. The WdW equation is R √gR[g] not necessarily the best manner for actually defining the Ψ[q] D[g] e− (25) Z∂g=q quantum dynamics and computing transition amplitudes in quantum gravity. can is shown by some formal manipulations to be a so- lution of the WdW equation. Here the integration is But it remains the equation that has opened the world over euclidean 4-metrics, inducing the 3-metric q on a 3d of background independent quantum gravity, a unique boundary. The construction is at the basis of beautiful source of inspiration, and a powerful conceptual tool that ideas such as the Hartle-Hawking “no boundary” defini- has forced us to understand how to actually make sense tion of a wave function for cosmology [29, 30]. of a quantum theory of space and time.

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