Sunday, January 22, 2006

Lecture 4 Smolin on Background Independence
Lecture 4
Nonlocality of the classical GR observables

Smolin writes:

"4.1 The problem of time and related issues
As I emphasized at the beginning of this section, a truly fundamental theory cannot be formulated in terms of boundaries or asymptotic conditions. This, together with diffeomorphism invariance, implies that the hamiltonian is a linear combination of constraints 11. (11 This is reviewed in [38, 16].)
This is no problem for defining and solving the evolution equations, but it does lead to subtleties in the question of what is an observable. One important consequence is that one cannot define the physical observables of the theory without solving the dynamics. In other words, as Stachel emphasizes, there is no kinematics without dynamics. This is because all observables are relational, in that they describe relations between physical degrees of freedom. You cannot just ask what is happening at a manifold point, or an event, labeled by some coordinate, and assume you are asking a physically meaningful question. The problem is that because of diffeomorphism invariance, points are not physically meaningful without a specification of how a point or event is to be identified by the values of some physical degrees of freedom. As a result, even observables that refer to local points or regions of physical spacetime are non-local in the sense that as functions of initial data they depend on data in the whole initial slice.

As a result, the physical interpretation of classical general relativity is more subtle than is usually appreciated. In fact, most of what we think we understand naively about how to interpret classical GR applies only to special solutions with symmetries, where we use the symmetries to define special coordinates. These methods do not apply to generic solutions, which have no symmetries. It is possible to give a physical interpretation to the generic solutions of the theory, but only by taking into account the issues raised by the facts that all physical observables must be diffeomorphism invariant, and the related fact that the hamiltonian is a sum of constraints [38].

We see here a reflection of Leibniz’s principles, in that the interpretation that must be given to generic solutions, without symmetries, is completely different from that given to the measure zero of solutions with symmetries.
One can actually argue something stronger [16]: Suppose that one could transform general relativity into a form in which one expressed the dynamics directly in terms of physical observables. That is, observables, which commute with all the constraints, but still measure local degrees of freedom. Then the solutions with symmetries might just disappear. This is because, being diffeomorphism invariant, such observables can distinguish points only by their having different values of fields. Such observables must degenerate when one attempts to apply them to solutions with symmetries. Thus, expressed in terms of generic physical observables, there may be no symmetric solutions. If this is true this would be a direct realization of the identity of the indiscernible in classical general relativity.

Thus, even at the classical level, there is a distinction between background independent and background dependent approaches to the physical interpretation. If one is interested only in observables for particles moving within a given spacetime, one can use a construction that regards that spacetime as fixed. But if one wants to discuss observables of the gravitational field itself, one cannot use background dependent methods, for those depend on fixing the gravitational degrees of freedom to one solution. To discuss how observables vary as we vary the solution to the Einstein equations we need functions of the phase space variables that make sense for all solutions. Then one must work on the full space of solutions, either in configuration space or phase space.
One can see this with the issue of time. If by time you mean time experienced by observers following worldlines in a given spacetime, then we can work within that space-time. For example, in a given spacetime time can be defined in terms of the causal structure. But if one wants to discuss time in the context in which the gravitational degrees of freedom are evolving, then one cannot work within a given spacetime. One constructs instead a notion of time on the infinite dimensional phase or configuration space of the gravitational field itself. Thus, at the classical level, there are clear solutions to the problems of what is time and what is an observable in general relativity.

Any quantum theory of gravity must address the same issues. Unfortunately, background dependent approaches to quantization evade these issues, because they take for granted that one can use the special symmetries of the non-dynamical backgrounds to define physical observables. To usefully address issues such as the problem of time, or the construction of physical observables, in a context that includes the quantum dynamics of the spacetime itself, one must work in a background independent formulation.

However, while the problem of time has been addressed in the context of background independent approaches to quantum gravity, the problem has not been definitively solved. The issue is controversial and there is strong disagreement among experts. Some believe the problem is solved, at least in principle, by the application of the same insights that lead to its solution in classical general relativity [38]. Others believe that new ideas are needed [16]. While I will not dwell on it here, the reader should be aware that the problem of time is a key challenge that any complete background independent quantum theory of gravity must solve.

5 Relationalism and the search for the quantum theory of gravity
Let us begin by noting that conventional quantum theories are background dependent theories. The background structures for a quantum theory include space and time, either Newtonian or in the case of QFT, some fixed, background spacetime. There are additional background structures connected with quantum mechanics, such as the inner product. It is also significant that the background structures in quantum mechanics are connected to the background space and time. For example, the inner product codes probability conservation, in a given background time coordinate.

Thus, when we attempt to unify quantum theory with general relativity we have to face the question of whether the resulting theory is to be background dependent or not.

There are two kinds of approaches, which take the two possible answers yes and no. These are called background independent and background dependent approaches.
Background dependent approaches study quantum theory on a background of a fixed classical spacetime. These can be quantum theories of gravity in a limited sense in which they study the quantization of gravitational waves defined as moving (to some order of approximation) on a fixed background spacetime. One splits the metric into two pieces

gab= bab+ hab (3)

where bab is the background metric, a fixed solution to the Einstein equations, and hab is a perturbation of that solution. In a background dependent approach one quantizes hab using structures that depend on the prior specification of bab, as if hab were an ordinary quantum field, or some substitute such as a string.

Background dependentapproaches include

perturbative quantum general relativity

string theory.

Perturbative quantum general relativity does not lead to a good theory, nor are the problems cured by modifying the theory so as to add supersymmetry or other terms to the field equations. It is hard to imagine a set of better motivated conjectures than those that drove interest in string theory. Had string theory succeeded as a background dependent theory, it would have served as a counter-argument to the thesis of this essay 12. Conversely, given that the problems string theory faces seem deeply rooted in the structure of the theory, it may be worthwhile to examine the alternative, which is background independent theories.
in recent years there has been healthy development of a number of different background independent approaches to quantum gravity. These include,

Causal sets

Loop quantum gravity (or spin foam models)

Dynamical triangulations models.

Certain approaches to non-commutative geometry [23].

A number of approaches that posit a fundamental discrete quantum theory from which classical spacetime is conjectured to emerge at low energies [22].

Attempts to formulate string theory as a background independent theory.

I will briefly describe the first three. These are well enough understood to illustrate both the strengths of the relational view for quantum gravity and the hard issues that any such approach must overcome.
12A more detailed summary of the results achieved in string theory and other approaches to quantum gravity, together with a list of problems that remain unsolved is given in [2].

5.1 The causal set theory
To describe the causal set model we need the definition of a causal set.
A causal set is a partially ordered set such that the intersection of the past and future of any pair of events is a finite set. The elements of the causal set are taken to be physical events and their partial ordering is taken to code the relation of physical causation.
The basic premises ofthe causalsetmodelare[24]
1) A history of the universe consists of nothing but a causal set. That is, the fundamental events have no properties except their mutual causal relations 13.
2) The quantum dynamics is defined by assigning to each history a complex number which is to be its quantum amplitude 14.
The motivation for the causal set hypothesis comes from the expectation that the geometry of spacetime becomes discrete at the Planck scale. This leads one to expect that, given any classical spacetime {M,gab}, one will be able to define a causal set C which approximates it … The inverse problem for causal sets: Given a classical spacetime {M,gab,f} , it is easy to define a causal set C which approximates it in the sense just defined. But almost no causal set C approximates a low dimensional manifold in this sense. Moreover, we do not have a characterization, expressed only in terms of the relations in a causal set, C, which would allow us to pick out those causal sets that do approximation spacetimes. We can only do this by first constructing classical spacetimes, and then extracting from them a causal set that approximates them. Moreover no dynamical principle has been discovered which would generate causal sets C that either directly approximate low dimensional classical spacetimes, or have coarse grainings or approximations that do so. … This is an example of a more general class of problems, which stems from the fact that combinatorially defined discrete structures are very different from continuous manifolds. A very general combinatorial structure is a graph. The possibility of a correspondence between a graph and a smooth geometry is based on two definitions.

Definition: The metric on a graph is defined by g(j, k) for two nodes k and j is the minimal number of steps to walk from j to k along the graph.

Definition: A graph •is said to approximate a manifold and metric {M, gab} if there is an embedding of the nodes into points of {M, gab} such that the graph distance g(j, k)is equal to the metric distance between the images of the nodes in {M, gab}.

It is easy to see that the following issue confronts us.

Inverse problem for graphs. Given any {M, gab }it is easy to construct a graph • that approximates it. But, assuming only that the dimension is much less than the number of nodes, for almost no graphs do there exist low dimensional smooth geometries that they approximate.

Because of the inverse problem, it is fair to say that the causal set program has unfortunately so far failed to lead to a good physical theory of quantum gravity, but it is useful to review the logic employed:

Logic of the causal set program:

GR is relational, and the fundamental relations are causal relations.

But GR is continuous and it is also non-quantum mechanical

We expect that a quantum theory of spacetime should tell us the set of physical events is discrete.

Therefore a quantum spacetime history should consist of a set of events, which is a discrete causal structure.

Moreover, the causal structure is sufficient to define the physical classical spacetime, so it should be sufficient to describe a fundamental quantum history.

But this program so far fails because of the inverse problem.

Given the seriousness of the inverse problem, it is possible to imagine that the solution is that there are more fundamental relations, besides those of causality.It should be said that this direction is resisted by some proponents of causal sets, who are rather “purist” in their belief that the relation of causality is sufficient to constitute all of physics. But a possible answer to this question is given by another program, loop quantum gravity, where causal relations are local changes in relational structures that describe the quantum geometry of space[20].

5.2 Loop quantum gravity
Loop quantum gravity was initiated in 1986 and is by now a well-developed research program, with on the order of 100 practitioners. There is now a long list of results, many of them rigorous. Here I will briefly summarize the key results that bear on the issue of relational space and time15.

5.2.1 Basic results of loop quantum gravity
Loop quantum gravity is based on the following observation, introduced by Sen and Ashtekar for general relativity and extended to a large class of theories including general relativity andsupergravity in spacetime dimensions three and higher 16.

General relativity and supergravity, in any spacetime dimension greater than or equalto 2 + 1, can be rewritten as gauge theories, such that the configuration space is the space of a connection field, Aa, on a spatial manifold S•. The metric information is contained in the conjugate momenta. The gauge symmetry includes the diffeomorphisms of a spacetime manifold, usually taken to be •SxR. The dynamics takes a simple form that can be understood as a constrained topological field theory. This means that the action contains one term, which is a certain topological field theory called BF theory, plus another term which generates a quadratic constraint.

Consider such a classical gravitational theory, T, whose histories are described as diffeomorphism equivalence class of connections and fields, {M,Aa,f}. To define the action principle one must assume that the topology, dimension and differential structure of spacelike surfaces, S, are fixed.

The following results have then been proven [7]:
1. The quantization of T results in a unique Hilbert space, H of diffeomorphism in-variant states. There is a recent uniqueness theorem [25], which guarantees that for dimension of • two or greater, there is a unique quantization of a gauge field such that

i)the Wilson loops are represented by operators that create normalizable states,

ii)its algebra with the operator that measures “electric” field •ux is represented faithfully and

iii)the diffeomorphisms of • are unitarily implemented without anomaly.
This unique Hilbert space has a beautiful description. There is a orthonormal basis of H whose elements are in one to one correspondence with the embeddings of certain labeled graphs • in •.(The label set varies depending on the dimension, matter fields, and with supersymmetry.)

Because H carries a unitary representation of Diff(•) it is possible rigorously to mod out by the action of diffeomorphisms and construct a Hilbert space, H diffof spatially diffeomorphism invariant states. This has a normalizable basis in one to one correspondence with the diffeomorphism classes of the embeddings in •ofthe labeled graphs. This is a very satisfactory description from the point of view of relationalism. There is no more relational structure than a graph, as two nodes are distinguished only by their pattern of connections to the rest of the graph. The labels come from the theory representations of a group or algebra A. The edges are labeled by representations of A, which describe properties shared between the nodes they connect. The labels on nodes are invariants of A, which likewise describe properties shared by the representations on edges incident on those nodes.

Because there is a background topology, there is additional information coded in how the edges of the graph knot and link each other. Given the choice of background topology, this information is also purely relational.
A quantum history is defined by a series of local moves on graphs that take the initial state to the final state[20]. The set of local moves in each history define a causalset.

Hence, the events of the causal set arise from local changes in another set of relations, that which codes the quantum geometry of a spatial slice. The structure that merges the relational structure of graphs with that of causal sets is now called a causal spin foam.

The amplitudes for local moves that follow from the quantization of the Einstein equations are known in closed form. The sums over those amplitudes are known to be ultraviolet finite. Similarly, the quantum Einstein equations in the Hamiltonian form have been implemented by exact operator equations on the states.

In the case of a spin foam model for 2+1 gravity coupled to massive particles, it has been shown in detail that the theory can be re-summed, yielding an effective field theory on a non-commutative spacetime [21]. This provides an explicit demonstration of how physics in classical spacetime can emerge from a non-trivial background independent quantum theory of gravity. The resulting effective field theory has in addition deformed Poincare symmetry, which confirms, in this case, the general conjecture that the low energy limit of loop quantum gravity has deformed Poincare symmetry[7].

The quantum spacetime is discrete in that each node of the graph corresponds to a finite quantum of spatial volume. The operators that correspond to volumes, areas and lengths are finite,and have discrete spectra with finite non-zero minimal values. Hence a graph with a•finite number of nodes and edges defines a region of space with •finite volume and area.

There are a number of robust predictions concerning subjects like black hole entropy. Evidence has recently been found that both cosmological and black hole singularities bounce, so the evolution of the universe continues through apparent classical cosmological singularities.

There are explicit constructions of semiclassical states, coarse grained measurements of which reproduce classical geometries. Excitations of these states, with wavelengths long in Planck units, relative to those classical geometries, have been shown to reproduce the physics of quantum fields and linearized gravitational waves on those backgrounds.

5.2.2 Open problems of loop quantum gravity:
There are of course many, in spite ofthe fact that the theory is well defined.

Classical limit problem: Find the ground state of the theory and show that it is a semiclassical state, excitations of which quantum fieldtheory and classical GR.

Do science problem: By studying the excitations of semiclassical states, make predictions for doable experiments that can test the theory up or down.

Remove the remaining background dependence problem: The results so far defined depend on the fact that the dimension and topology of the spatial manifold, •, is fixed, so that the graphs are embedded in •. This helps by lessoning the inverse problem. Can this be removed-and the inverse problem solved-so that all the structure that was background for previous theories, including dimension and topology, is explained as following from solutions to a relational theory17?

We note that in some formulations of spin foam models, the dependence on a fixed background topology is dropped, so that the states and histories are defined as pure combinatorial structures. But this makes the problem of recovering classical general relativity from the low energy limit more complicated.

The problem of time: The different proposals that have been made to resolve the problem of time in quantum gravity and cosmology can all be studied in detail in loop quantum gravity and related cosmological models. While there are some interesting results, the opinion of this author is that the problem remains open.

These are hard problems, and remain unsolved, but some progress is being made on all of them. It is important to mention that there are real possibilities for experimental tests ofthe theory. This is because the discrete structure of space and time implies modifications in the usual relations between energy and momenta

E^2= p^2 + m^2+ lpE3 + ... (4)

This turns out to have implications for experiments currently underway, having to do with ultra high-energy cosmic rays and gamma ray bursts, amongst others 18. Loop quantum gravity appears to make predictions for these experiments[41].

5.2.3 Lessons from loop quantum gravity for the relational program
So far as the relational/absolute debate is concerned, loop quantum gravity teaches us several lessons:

So long as we keep as background those aspects of space and time that are background for classical GR, (the topology, dimension and differential structure), we can find a quantum mechanical description of the metric and fields. Thus LQG is partly relational, in exactly the same way that GR is partly relational.

Loop quantum gravity does give us a detailed description of quantum spatial and spacetime geometry. There are many encouraging results, such as finiteness, and the derivation of an explicit language of states, histories, and observables for general background independent theories of quantum gravity. It is possible to do nontrivial computations to study the dynamics of quantum spacetime, and applications to physical problems such as black holes and cosmology yield results that are sensible and, in some cases, testable. It is very satisfying that the description of quantum geometry and quantum histories are formulated using beautiful relational structures such as graphs and causal sets.

This description is flexible and can accommodate different hypotheses as to the dimension of spacetime, matter couplings, symmetries and supersymmetries.

There do remain hard open problems having to do with how a classical spacetime is to emerge from a purely background independent description. A related challenge is to convincingly resolve the problem of time. Nevertheless, significant progress is being made on these problems [21], and it even appears to be possible to derive predictions for experiment by expanding around certain semiclassical states[41].

The main barrier to making an entirely relational theory of quantum spacetime appears to be the inverse problem.

5.3 Causal dynamical triangulation models
These are models for quantum gravity, based on a very simple construction [26]-[32]. A quantum spacetime is represented by a combinatorial structure, which consists of a large number N of d dimensional simplexes (triangles for two dimensions, tetrahedra for three etc.) glued together to form a discrete approximation to a spacetime. Each such discrete spacetime is given an amplitude, which is gotten from a discrete approximation to the action for general relativity. Additional conditions are imposed, which guarantee that the resulting structure is the triangulation of some smooth manifold (otherwise there is a severe inverse problem.) For simplicity the edge lengths are taken to be all equal to a fundamental scale, which is considered a short distance cutoff 19 One defines the quantum theory of gravity by a discrete form of the sum over histories path integral, in which one sums over all such discrete quantum spacetimes, each weighed by its amplitude.

These models were originally studied as an approach to Euclidean quantum gravity (that is the path integral sums over spacetimes with Euclidean signature, rather than the Lorentzian signature of physical spacetime.) In these models the topology is not fixed, so one has a model of quantum gravity in which one can investigate the consequences of removing topology from the background structure and making it dynamical [26].

More recently, a class ofmodels have been studied corresponding to Lorentzian quantum gravity. In these cases additional conditions are fixed, corresponding to the existence of a global time slicing, which restricts the topology to be of the form of SXR, where S•is a fixed spatialtopology [27]20.
Some of the results relevant for the debate on relationalism include,

In the Euclidean case, for spacetime dimensions d > 2, the sum over topologies cannot be controlled. The path integral is, depending on the parameters of the action chosen, unstable to the formation of either an uncontrolled spawning of “baby universes”, or to a crunch down to degenerate triangulations. Neither converges to allow a coarse grained approximation in terms of smooth manifolds of any dimension.

In the Lorentzian case, when the simplices have spacetime dimension d = 2,3,4, where the topology is fixed and the formation of baby universes suppressed, there is evidence for convergence to a description of physics in manifolds whose macroscopic dimension i the same as the microscopic dimension. For the case of d = 4 there results are recent and highly significant [27, 28]. In particular, there is now detailed numerical evidence for the emergence of 3+1 dimensional classical spacetime at large distances from a background independent quantum theory of gravity[28].

The measure of the path integral is chosen so that each triangulation corresponds to a diffeomorphism class {M,gab,f}. The physical observables such as correlation functions measured by averaging over the triangulations correspond to diffeomorphism invariant relational observables in spacetime.
19 There is a different, but related approach, called Regge calculus, in which the triangulations are fixed while the edge lengths are varied.
20 The condition ofa •fixed global time slicing can be relaxed to some extent[29]

These results are highly significant for quantum gravity. It follows that earlier conjectures about the possibility of defining quantum gravity through the Euclidean path integral cannot be realized. The sum has to be done over Lorentzian spacetimes to have a hope of converging to physics that has a coarse grained description in smooth space-times. Further, earlier conjectures about summing over topologies in the path integral also cannot be realized.

As far as relationalism is concerned we reach a similar conclusion to that of loop quantum gravity. There is evidence for the existence of the quantum theory when structures including topology, dimension and signature are fixed, as part of the background structure,just as they are in classical general relativity. When this is done one has a completely relational description of the dynamics of a discrete version of metric and fields. Furthermore, in the context of each research program there has recently been reported a detailed study showing of how classical spacetime emerges from an initially discrete, background independent theory. This is an analytic result in the case of spin foam models in 2+1 dimensions, with matter [21], and numerical results in 3+1 dimensions in the causal dynamical triangulations case[27, 28]. This is very encouraging, given that the problem of how classical spacetime emerges is the most challenging problem facing background independent approaches to quantum gravity.”

Comment 9: In my theory this is easy.

L = Lp{(dtheta)(phi) – (theta)(dphi)} = curved tetrad 1-form

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