uantum gravity

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uantum Gravity

First published Mon 26 Dec, 2005

Quantum Gravity: A physical theory describing the gravitational interactions of matterand energy in which matter and energy are described by quantum theory. In most, but not

all, theories of quantum gravity, gravity is also quantized. Since the contemporary theory

of gravity, general relativity, describes gravitation as the curvature of spacetime by matterand energy, a quantization of gravity implies some sort of quantization of spacetime

itself. Insofar as all extant physical theories rely on a classical spacetime background, this

presents profound methodological and ontological challenges for the philosopher and the

physicist.

1. Introduction

2. Gravity Meets Quantum Theory

3. Methodologyo 3.1 Theory

o 3.2 Experiment

4. Theoretical Frameworks

o 4.1 String theory

o 4.2 Canonical and loop quantum gravity

4.2.1 Geometric variables 4.2.2 Problem of time

4.2.3 Ashtekar, loop, and other variables

o 4.3 Other approaches

5. Philosophical Issues

o 5.1 Timeo 5.2 Ontology

o 5.3 Status of quantum theory

o 5.4 Methodology

6. Conclusion

Bibliography

Other Internet Resources

Related Entries

1. Introduction

Dutch artist M.C. Escher's elegant pictorial paradoxes are prized by many, not least by

philosophers, physicists, and mathematicians. Some of his work, for exampleAscendingand Descending, relies on optical illusion to depict what is actually an impossiblesituation. Other works are paradoxical in the broad sense, but notimpossible:Relativity

depicts a coherent arrangement of objects, albeit an arrangement in which the force of

gravity operates in an unfamiliar fashion. (See the Other Internet Resources section
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below for images.) Quantum gravity itself may be like this: an unfamiliar yet coherent

arrangement of familiar elements. Or it may be likeAscending and Descending, an

impossible construction which looks sensible in its local details but does not fit togetherinto a coherent whole.

Quantum gravity primarily refers to an area of research, rather than a particular theoryofquantum gravity. Several approaches exist, none of them entirely successful to date.

Thus the philosopher's task, if indeed she has one, is different from what it is whendealing with a more-or-less settled body of theory such as classical Newtonian

mechanics, general relativity, or quantum mechanics. In such cases, one typically

proceeds by assuming the validity of the theory or theoretical framework and drawing theontological and perhaps epistemological consequences of the theory, trying to understand

what it is that the theory is telling us about the nature of space, time, matter, causation,

and so on. Theories of quantum gravity, on the other hand, are bedeviled by a host oftechnical and conceptual problems, questions, and issues which make them unsuited to

this approach. However, philosophers who have a taste for a broader and more open-

ended form of inquiry will find much to think about.

2. Gravity Meets Quantum Theory

The difficulties in reconciling quantum theory and gravity into some form of quantum

gravity come from theprima facie incompatibility of general relativity, Einstein's

relativistic theory of gravitation, with quantum field theory, the framework for the

description of the other three forces (electromagnetism and the strong and weak nuclearinteractions). Whence the incompatibility? General relativity is described by Einstein's

equations, which amount to constraints on the curvature of spacetime (the Einstein tensor

on the left-hand side) due to the presence of mass and other forms of energy, such as

electromagnetic radiation (the stress-energy-momentum tensor on the right-hand side).(See John Baez's webpages inOther Internet Resources for an excellent introduction.) In

doing so, they manage to encompass traditional, Newtonian gravitational phenomenasuch as the mutual attraction of two or more massive objects, while also predicting new

phenomena such as the bending of light by these objects (which has been observed) and

the existence of gravitational radiation (which has to date only been indirectly observedvia the decrease in the period of binary pulsars). (For the latter observation, see the 1993

Physics Nobel Prize presentation speech by Carl Nordling .)

In general relativity, mass and energy are treated in a purely classical manner, where

classical means that physical quantities such as the strengths and directions of various

fields and the positions and velocities of particles have definite values. These quantitiesare represented by tensor fields, sets of (real) numbers associated with each spacetime

point. For example, the stress, energy, and momentum Tab(x,t) of the electromagnetic

field at some point (x,t), are functions of the three componentsEi,Ej,Ek,Bi,Bj,Bk of theelectric and magnetic fields Eand B at that point. These quantities in turn determine, via

Einstein's equations, an aspect of the curvature of spacetime, a set of numbers Gab(x,t)

which is in turn a function of the spacetime metric. The metricgab(x,t) is a set of numbersassociated with each point which gives the distance to neighboring points. At the end of
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the day, a model of the world according to general relativity consists of a spacetime

manifold with a metric, the curvature of which is constrained by the stress-energy-

momentum of the matter distribution. All physical quantities the value of thex-component of the electric field at some point, the scalar curvature of spacetime at some

point have definite values, given by real (as opposed to complex or imaginary)

numbers. Thus general relativity is a classical theory in the sense given above.

The problem is that our fundamental theories of matter and energy, the theoriesdescribing the interactions of various particles via the electromagnetic force and the

strong and weak nuclear forces, are all quantum theories. In quantum theories, these

physical quantities do not in general have definite values. For example, in quantummechanics, the position of an electron may be specified with arbitrarily high accuracy

only at the cost of a loss of specificity in the description of its momentum, hence its

velocity. At the same time, in the quantum theory of the electromagnetic field known asquantum electrodynamics (QED), the electric and magnetic fields associated with the

electron suffer an associated uncertainty. In general, physical quantities are described by

a quantum state which gives a probability distribution over many different values, andincreased specificity (narrowing of the distribution) of one property (e.g., position,electric field) gives rise to decreased specificity of its canonically conjugate property

(e.g., momentum, magnetic field). This is an expression of Heisenberg'sUncertainty

Principle.

On the surface, the incompatibility between general relativity and quantum theory might

seem rather trivial. Why not just follow the model of QED and quantize the gravitational

field, similar to the way in which the electromagnetic field was quantized? Just as we

associate a quantum state of the electromagnetic field with the quantum state ofelectrically charged matter, we should, one might think, similarly just associate a

quantum state of the gravitational field with the quantum state of both charged anduncharged matter. This is more or less the path that has been taken, but it encountersextraordinary difficulties. Some physicists consider these to be technical difficulties,

having to do with the non-renormalizability of the gravitational interaction and the

consequent failure of the perturbative methods which have proven effective in ordinaryquantum field theories. However, these technical problems are closely related to a set of

daunting conceptualdifficulties, of interest to both physicists and philosophers.

The conceptual difficulties basically follow from the nature of the gravitational

interaction, in particular the equivalence of gravitational and inertial mass, which allowsone to represent gravity as a property of spacetime itself, rather than as a field

propagating in a (passive) spacetime background. When one attempts to quantize gravity,

one is subjecting some of the properties of spacetime to quantum fluctuations. Butordinary quantum theory presupposes a well-defined classical background against which

to define these fluctuations (Weinstein, 2001a, b), and so one runs into trouble not only in

giving a mathematical characterization of the quantization procedure (how to take into

account these fluctuations in the effective spacetime structure?) but also in giving aconceptual and physical account of the theory that results, should one succeed. We will

look in more detail at how these conceptual problems arise in two different research
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programs below. But first, we will talk a bit about some general methodological issues

which haunt the field.

3. Methodology

Research in quantum gravity has a rather peculiar flavor, owing to both the technical andconceptual difficulty of the field and the remoteness from experiment. Thus conventional

notions of the relation between theory and experiment have a tenuous foothold, at best.

3.1 Theory

As remarked in the introduction, there is no single, generally agreed-upon body of theory

in quantum gravity. The majority of the physicists working in the field work on stringtheory, an ambitious program which aims at providing a unified theory of all interactions.

A non-negligible minority work on what is now called loop quantum gravity, the goal of

which is simply to provide a quantum theory of the gravitational interaction. There is also

significant work in other areas. [Good recent reviews of the theoretical landscape includeCarlip 2001 and Smolin 2001 (Other Internet Resources section below), 2003.] But there

is no real consensus, for two reasons.

The first reason is that it is extremely difficult to make any concrete predictions in thesetheories. String theory, in particular, is plagued by a lack of predictions because of the

tremendous number of distinct ground or vacuum states in the theory, with an absence of

guiding principles for singling out the physically significant ones. Though the stringcommunity prides itself on the dearth of free parameters in the theory (in contrast to the

nineteen or so free parameters found in the standard model of particle physics), the

problem arguably resurfaces in the huge number of vacua associated with different

compactifications of the nine space dimensions to the three we observe. Attempts toexplain why we live in the particular vacuum that we do have recently given rise to

appeals to the infamous anthropic principle (Susskind, 2003 [CHECK THE DATE]),

whereby the existence of humans is invoked to, in some sense, explain the fact that wefind ourselves in a particular world.

Loop quantum gravity is less plagued by a lack of predictions, and indeed it is often

claimed that the discreteness of area and volume are concrete predictions of the theory.

Proponents of this approach argue that this makes the theory more susceptible tofalsification, thus more scientific (in the sense of Popper; see the entry on Karl Popper)

than string theory. However, it is still quite unclear, in practice and even in principle, how

one might actually observe these quantities.

3.2 Experiment

The second reason for the absence of consensus is that there are no experiments in

quantum gravity, and little in the way of observations that might qualify as direct orindirect data or evidence. This stems in part from the lack of theoreticalpredictions, since

it is difficult to design an observational test of a theory if one does not know where to
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look or what to look at. But it also stems from the fact that most theories of quantum

gravity appear to predict departures from classical relativity only at energy scales on the

order of 1019 GeV. (By way of comparison, the proton-proton collisions at Fermilab havean energy on the order of 103 GeV.) Whereas research in particle physics proceeds in

large part by examining the data collected in large particle accelerators, accelerators

which are able to smash particles together at sufficiently high energies to probe theproperties of atomic nuclei, gravity is so weak that there is no way to do a comparable

experiment that would reveal properties at the energy scales at which quantum

gravitational effects are expected to be important.

Though progress is being made in trying to at least draw observational consequences ofloop quantum gravity, a theory of quantum gravity which arguably does make predictions

(Amelino-Camelia, 2003, in the Other Internet Resources section below; D. Mattingly,

2005), it is remarkable that the most notable test of quantum theories of gravityimposed by the community to date involves a phenomenon which has never been

observed, the so-called Hawking radiation from black holes. Based on earlier work of

Bekenstein (1973) and others, Hawking (1974) predicted that black holes would radiateenergy, and would do so in proportion to their gravitational temperature, which was inturn understood to be proportional to their mass, angular momentum, and charge.

Associated with this temperature is an entropy (see the entry on the philosophy of

statistical mechanics), and one would expect a theory of quantum gravity to allow one tocalculate the entropy associated with a black hole of given mass, angular momentum, and

charge, the entropy corresponding to the number of quantum states of the gravitational

field having the same mass, charge, and angular momentum. (See Unruh (2001) andreferences therein.)

In their own ways, string theory and loop quantum gravity have both passed the test of

predicting an entropy for black holes which accords with Hawking's calculation. Stringtheory gets the number right for a not-particularly physically realistic subset of blackholes called near-extremal black holes, while loop quantum gravity gets it right for

generic black holes, but only up to an overall constant.Ifthe Hawking effect is real, then

this consonance could be counted as evidence in favor of either or both theories.

It should be noted, finally, that to date neither of the main research programs has beenshown to give rise to the world we see at low energies. Indeed, it is a major challenge of

loop quantum gravity to show that it has general relativity as a low-energy limit, and a

major challenge of string theory to show that it has the standard model of particle physicsplus general relativity as a low-energy limit.

The absence of relevant experiments in quantum gravity is a peculiarity which has drawn

little attention to date from philosophers of science (an exception is Butterfield & Isham,

2001). However, it would seem to be fertile ground for philosophical analysis, in that itraises the interesting question of how a science should and does proceed in the absence of

data.

4. Theoretical Frameworks
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4.1 String Theory

Known variously as string theory, superstring theory, and M-theory, this program has its

roots, indirectly, in the observation, dating back to at least the 1950s, that classical

general relativity looks in many ways like the theory of a massless spin-two field

propagating on the flat Minkowski spacetime of special relativity. [See Rovelli 2001b(Other Internet Resources section below), and 2006 for a capsule history, and Greene

2000 for a popular account.] This observation led to early attempts to formulate a

quantum theory of gravity by quantizing this spin-two theory. However, it turned outthat the theory is not perturbatively renormalizable, meaning that there are ineliminable

infinities. Attempts to modify the classical theory to eliminate this problem led to a

different problem, non-unitarity, and so this general approach was moribund until themid-1970s, when it was discovered that a theory of one-dimensional strings developed

around 1970 to account for the strong interaction, actually provided a framework for a

unified theory which included gravity, because one of the modes of oscillation of thestring corresponded to a massless spin-two particle (the graviton).

The original and still prominent idea behind string theory was to replace the point

particles of ordinary quantum field theory (particles like photons, electrons, etc) with

one-dimensional extended objects called strings. (See Weingard, 2001 and Witten, 2001for overviews of the conceptual framework.) In the early development of the theory, it

was recognized that construction of a consistent quantum theory of strings required that

the strings live in a larger number of spatial dimensions than the observed three;eventually, most string theories came to be formulated in nine space dimensions and one

time dimension. Strings can be open or closed, and have a characteristic tension and

hence vibrational spectrum. The various modes of vibration correspond to various

particles, one of which is the graviton. The resulting theories have the advantage of being

perturbatively renormalizable, at least to second order. This means that perturbativecalculations are at least mathematically tractable. Since perturbation theory is an almost

indispensable tool for physicists, this is deemed a good thing.

String theory has undergone several mini-revolutions over the last several years, one of

which involved the discovery of various duality relations, mathematical transformations

connecting, in this case, what appeared to be mathematically distinct string theories

type I, type IIA, type IIB, HE and HO to one another and to eleven-dimensionalsupergravity (a particle theory). The discovery of these connections led to the conjecture

that all of the string theories are really aspects of a single underlying theory, which was

given the name M-theory (though M-theory is also used more specifically to describe

the unknown theory of which eleven-dimensional supergravity is the low energy limit).The rationale is that what looks like one theory at strong coupling (high energy

description) looks like another theory at weak coupling (lower energy, more tractabledescription), and that if all the theories are related to one another, then they must all be

aspects of some more fundamental theory. Though attempts have been made, there has

been no successful formulation of this theory: its very existence, much less its nature, is

still largely a matter of conjecture.


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4.2 Canonical and Loop Quantum Gravity

Whereas string theory views the curved spacetime of general relativity as an effective

modification of a flat (or other fixed) background geometry by a massless spin-two field,

the canonical quantum gravity program treats the spacetime metric itself as a kind of

field, and attempts to quantize it directly.

Technically, most work in this camp proceeds by writing down general relativity in so-

called canonical or Hamiltonian form, since there is a more-or-less clearcut way to

quantize theories once they are put in this form (Kuchar, 1993; Belot & Earman, 2001).In a canonical description, one chooses a particular set of configuration variablesxi and

canonically conjugate momentum variablespi which describe the state of a system at

some time. Then, one obtains the time-evolution of these variables from the HamiltonianH(xi,pi). Quantization proceeds by treating the configuration and momentum variables as

operators on a quantum state space (a Hilbert space) obeying certain commutation

relations analogous to the classical Poisson-bracket relations, which effectively encode

the quantum fuzziness associated with Heisenberg's uncertainty principle.

Although advocates of the canonical approach often accuse string theorists of relying too

heavily on classical background spacetime, the canonical approach does something which

is arguably quite similar, in that one begins with a theory that conceives time-evolution in

terms of evolving some data given on a spacelike surface, and then quantizing the theory.The problem is that if spacetime is quantized, this assumption does not make sense in

anything but an approximate way. This issue in particular is decidedly neglected in both

the physical and philosophical literature (but see Isham (1993)), and there is more thatmight be said.

4.2.1 Geometric variables

Early attempts at quantizing general relativity by Dirac, Wheeler, DeWitt and others in

the 1950s and 1960s worked with a seemingly natural choice for configuration variables,namely geometric variablesgij corresponding to the various components of the three-

metric describing the intrinsic geometry of the given spatial slice of spacetime. One can

think about arriving at this via an arbitrary slicing of a 4-dimensional block universe by

3-dimensional spacelike hypersurfaces. The conjugate momenta ij then effectivelyencode the time rate-of-change of the metric, which, from the 4-dimensional perspective,

is directly related to the extrinsic curvature of the slice (meaning the curvature relative to

the spacetime in which the slice is embedded). This approach is known as

geometrodynamics.

In these geometric variables, as in any other canonical formulation of general relativity,

one is faced with constraints, which encode the fact that the canonical variables cannot be

specified independently. A familiar example of a constraint is Gauss's law from ordinaryelectromagnetism, which states that, in the absence of charges, E(x) = 0 at every point

x. It means that the three components of the electric field at every point must be chosen

so as to satisfy this constraint, which in turn means that there are only two true degrees


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of freedom possessed by the electric field at any given point in space. (Specifying two

components of the electric field at every point dictates the third component.)

The constraints in electromagnetism may be viewed as stemming from the U(1) gaugeinvariance of Maxwell's theory, while the constraints of general relativity stem from the

diffeomorphism invariance of the theory.Diffeomorphism invariance means, informally,that one can take a solution of Einstein's equations and drag it (meaning the metric and

the matter fields) around on the spacetime manifold and obtain a mathematically distinctbut physically equivalent solution. The three supermomentum constraints in the

canonical theory reflect the freedom to drag the metric and matter fields around in

various directions on a given three-dimensional spacelike hypersurface, while the super-Hamiltonian constraint reflects the freedom to drag the fields in the time direction, and

so to the next hypersurface. (Each constraint applies at each point of the given

spacelike hypersurface, so that there are actually 4 3 constraints, four for each point.)In the classical (unquantized) canonical formulation of general relativity, the constraints

do not pose any particular conceptual problems. One effectively chooses a background

space and time (via a choice of the lapse and shift functions) on the fly, and one can beconfident that the spacetime that results is independent of the particular choice.Effectively, different choices of these functions give rise to different choices of

background against which to evolve the foreground. However, the constraints pose a

serious problem when one moves to quantum theory.

4.2.2 Problem of time

All approaches to canonical quantum gravity face the so-called problem of time in one

form or another (Kucha (1992) and Isham (1993) are excellent reviews). The problem

stems from the fact that in preserving the diffeomorphism-invariance of general relativity

depriving the coordinates of the background manifold of any physical meaning theslices of spacetime one is considering inevitably include time, just as they include

space. In the canonical formulation, the diffeomorphism invariance is reflected in the

constraints, and the inclusion of what would ordinarily be a time variable in the data isreflected in the existence of the super-Hamiltonian constraint. The difficulties presented

by this constraint constitute the problem of time.

Attempts to quantize general relativity in the canonical framework proceed by turning thecanonical variables into operators on an appropriate state space (e.g., the space of square-

integrable functions over three-metrics), and dealing somehow with the constraints.

When quantizing a theory with constraints, there are two possible approaches. The

approach usually adopted in gauge theories is to deal with the constraints beforequantization, so that only true degrees of freedom are promoted to operators when

passing to the quantum theory. There are a variety of ways of doing this so-called gauge

fixing, but they all involve removing the extra degrees of freedom by imposing somespecial conditions. In general relativity, fixing a gauge is tantamount to specifying a

particular coordinate system with respect to which the physical data is described

(spatial coordinates) and with respect to which it evolves (time coordinate). This isdifficult already at the classical level, since the utility and, moreover, the very tractability
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of any particular gauge generally depends on the properties of the solution to the

equations, which of course is what one is trying to find in the first place. But in the

quantum theory, one is faced with the additional concern that the resulting theory maywell not be independent of the choice of gauge. This is closely related to the problem of

identifying true, gauge-invariant observables in the classical theory (Torre 2005, in the

Other Internet Resources section).

The preferred approach in canonical quantum gravity is to impose the constraints afterquantizing. In this constraint quantization approach, due to Dirac, one treats the

constraints themselves as operatorsA, and demands that physical states be those

which are solutions to the resulting equationsA = 0. The problem of time is associatedwith the super-Hamiltonian constraint. The super-HamiltonianHis responsible for

describing time-evolution in the classical theory, yet its counterpart in the constraint-

quantized theory,H = 0, wouldprima facie seem to indicate that the true physicalstates of the system do not evolve at all. Trying to understand how, and in what sense, the

quantum theory describes the time-evolution of something, be it states or observables, is

the essence of the problem of time.

4.2.3 Ashtekar, loop, and other variables

In geometrodynamics, all of the constraint equations are difficult to solve (though thesuper-Hamiltonian constraint, known as the Wheeler-DeWitt equation, is especially

difficult), even in the absence of particular boundary conditions. Lacking solutions, one

does not have a grip on what the true, physical states of the theory are, and one cannothope to make much in the way of predictions. The difficulties associated with geometric

variables are addressed by the program initiated by Ashtekar and developed by his

collaborators (for a review and further references see Rovelli 2001b (Other Internet

Resources), 2001a, 2004). Ashtekar used a different set of variables, a complexifiedconnection (rather than a three-metric) and its canonical conjugate, which made it

simpler to solve the constraints. The program underwent further refinements with the

introduction of the loop transform, and further refinements still when it was understoodthat equivalence classes of loops could be identified with spin networks. (See Smolin

(2001, 2004) for a popular introduction.)

4.3 Other Approaches

There are many other approaches to quantum gravity as well. Some (e.g., Huggett 2001,

Wthrich 2004 (Other Internet Resources section); J. Mattingly 2005) have argued that

semiclassical gravity, a theory in which matter is quantized but spacetime is classical, is aviable alternative. Other approaches include twistor theory (currently enjoying a revival

in conjunction with string theory), Bohmian approaches (Goldstein & Teufel, 2001),causal sets (see Sorkin 2003, in the Other Internet Resources section) in which the

universe is described as a set of discrete events along with a stipulation of their causal

relations, and other discrete approaches (see Loll, 1998). Also of interest are arguments tothe effect that gravity itself may play a role in quantum state reduction (Christian, 2001;

Penrose, 2001).


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5. Philosophical Issues

Quantum gravity raises a number of difficult philosophical questions. To date, it is the

ontological aspects of quantum gravity that have attracted the most interest fromphilosophers, and it is these we will discuss in the first three sections below. In the final

section, though, we will briefly discuss the methodological and epistemological issueswhich arise.

First, however, let us discuss the extent to which ontological questions are tied to aparticular theoretical framework. In its current stage of development, string theory

unfortunately provides little indication of the more fundamental nature of space, time,

and matter. Despite the consideration of ever more exotic objects strings, p-branes, D-branes, etc. these objects are still understood as propagating in a background

spacetime. Since string theory is supposed to describe the emergence of classical

spacetime from some underlying quantum structure, these objects are not to be regardedas truly fundamental. Rather, their status in string theory is analogous to the status of

particles in quantum field theory (Witten, 2001), which is to say that they are relevantdescriptions of the fundamental physics only in situations in which there is a background

spacetime with appropriate symmetries.

The duality relations between the various string theories suggest that they are all

perturbative expansions of some more fundamental, non-perturbative theory known as

M-theory (Polchinski, 2002, see the Other Internet Resources section below). This,

presumably, is the most fundamental level, and understanding the theoretical frameworkat that level is central to understanding the underlying ontology of the theory. Matrix

theory is an attempt to do just this, to provide a mathematical formulation of M-theory,

but it remains highly speculative. Thus although string theory purports to be a

fundamental theory, the ontological implications of the theory are still obscure.

Canonical quantum gravity, in its loop formulation or otherwise, has to date been of

greater interest to philosophers because it appears to confront fundamental questions in a

way that string theory, at least in its perturbative guise, does not. Whereas perturbativestring theory treats spacetime in an essentially classical way, canonical quantum gravity

treats it as quantum-mechanical, at least to the extent of treating the geometric structure

(as opposed to, say, the topological or differential structure) as quantum-mechanical.

5.1 Time

As noted inSection 4.2.2 above, the treatment of time presents special difficulties incanonical quantum gravity. These difficulties are connected with the special role time

plays in physics, and in quantum theory in particular. Physical laws are, in general, laws

of motion, of change from one time to another. They represent change in the form ofdifferential equations for the evolution of, as the case may be, classical or quantum states;

the state represents the way the system is at some time, and the laws allow one to predict

how it will be in the future (or retrodict how it was in the past).
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It is not surprising, then, that a theory of quantum spacetime would have a problem of

time, because there is no classical time against which to evolve the state. The problem

is not so much that the spacetime is dynamical; there is no problem of time in classicalgeneral relativity. Rather, the problem is roughly that in quantizing the structure of

spacetime itself, the notion of a quantum state, representing the structure of spacetime at

some instant, and the notion of the evolution of the state, do not get any traction, sincethere are no real instants. (In some approaches to canonical gravity, one fixes a time

before quantizing, and quantizes the spatial portions of the metric only. This approach is

not without its problems, however; see Isham (1993) for discussion and furtherreferences.)

One can ask whether the problem of time arising from the canonical program tells us

something deep and important about the nature of time. Julian Barbour (2001a,b), for

one, thinks that it tells us that time is illusory (see also Earman (2002) in this connection).It is argued that the fact that quantum states do not evolve under the super-Hamiltonian

means that there is no change. However, it can also be argued (Weinstein, 1999a,b) that

the super-Hamiltonian itself should not be expected to generate time-evolution; rather,one or more true Hamiltonians should play this role. (See Butterfield & Isham (1999)and Rovelli (2006) for further discussion.)

5.2 Ontology

The problem of time is closely connected with a general puzzle about the ontology

associated with quantum spacetime. Quantum theory in general resists any

straightforward ontological reading, and this goes double for quantum gravity. Inquantum mechanics, one has particles, albeit with indefinite properties. In quantum field

theory, one again has particles (at least in suitably symmetric spacetimes), but these are

secondary to the fields, which again are things, albeit with indefinite properties. On theface of it, the only difference in quantum gravity is that spacetime itself becomes a kind

of quantum field, and one would perhaps be inclined to say that the properties of

spacetime become indefinite. But space and time traditionally play important roles inindividuating objects and their propertiesin fact a field is in some sense a set of

properties of spacetime points and so the quantization of such raises real problems for

ontology.

In the loop quantum gravity program, the area and volume operators have discrete

spectra. Thus, like spins, they can only take certain values. This suggests (but does not

imply) that space itself has a discrete nature, and perhaps time as well (depending on how

one resolves the problem of time). This in turn suggests that space does not have thestructure of a differential manifold, but rather that it only approximates such a manifoldon large scales, or at low energies.

5.3 Status of quantum theory

Whether or not spacetime is discrete, the quantization of spacetime entails that our

ordinary notion of the physical world, that of matter distributed in space and time, is at


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best an approximation. This in turn implies that ordinary quantum theory, in which one

calculates probabilities for events to occur in a given world, is inadequate as a

fundamental theory. As suggested in the Introduction, this may present us with a viciouscircle. At the very least, one must almost certainly generalize the framework of quantum

theory. This is an important driving force behind much of the effort in quantum

cosmology to provide a well-defined version of themany-worlds orrelative-stateinterpretations. Much work in this area has adopted the so-called decoherent histories or

consistent histories formalism, whereby quantum theories are understood to make

probabilistic predictions about entire (coarse-grained) histories. Almost all of this workto date construes histories to be histories of spatiotemporal events, and thus presupposes a

background spacetime; however, the incorporation of a dynamical, quantized spacetime

clearly drives much of the cosmology-inspired work in this area.

More generally, one might step outside the framework of canonical, loop quantumgravity, and ask why one should only quantize the metric. As pointed out by Isham

(1994, 2002), it may well be that the extension of quantum theory to general relativity

requires one to quantize, in some sense, not only the metric but also the underlyingdifferential structure and topology. This is somewhat unnatural from the standpoint whereone begins with classical, canonical general relativity and proceeds to quantize (since

the topological structure, unlike the metric structure, is not represented by a classical

variable). But one might well think that one should start with the more fundamental,quantum theory, and then investigate under which circumstances one gets something that

looks like a classical spacetime.

5.4 Methodology

The nature of the enterprise, in particular its seeming remoteness from experiment, gives

rise to significant methodological and epistemological questions as well, focusing on theproblem of how to construct or discover a scientific theory for phenomena which are so

remote from observation. Are beauty and consistency either necessary or sufficient? The

pronounced split between the string theory community and the loop quantum gravitycommunity has nothing to do with empirical success or lack thereof, but much to do with

factors which might normally play a role only on the periphery of the scientific

enterprise. The history of the enterprise of quantum gravity might well be worth

historico-philosophical scrutiny, much as the history of cosmology has been, cosmologyhaving also been rather data-starved until recently. (See Kragh (1999) for an excellent

account of the big-bang/steady-state controversy.)

6. Conclusion

In the author's opinion, it is unlikely that a final theory of quantum gravity if indeedthere is one will look much like any of the current candidate theories, be they string

theory, canonical gravity, or other approaches. However, the philosophical and

conceptual study of quantum gravity is useful insofar as it prompts one to considerquestions which are surely raised by this almost quixotic undertaking, questions which
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will ultimately require some unknown combination of philosophical, physical, and

mathematical insight to answer.

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