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date: 22 November 2017

# Time in Quantum Gravity

## Abstract and Keywords

This chapter notes that quantum gravity places the concept of time on a new level. In the absence of experimental hints, mathematical and conceptual issues must be chosen as the guides in the search for such a theory. Just as reconceiving classical notions of time was key for Einstein, in his discovery of special relativity, so too many believe that time will again hold the clue for theoretical advancement, but this time with quantum gravity. The chapter details the challenge of reconciling quantum theory with relativity, concentrating especially on why time in particular causes trouble. It describes a result in canonical quantum gravity which is possibly of signal importance, namely, that fundamentally there is no time at all, and discusses the problem of time, quantization, semiclassical time, loop quantum gravity, and string theory.

# 1. Introduction

Time is a fundamental concept in all physical theories. Because it enters the dynamical laws, changing these laws is inextricably linked with a change in the notion of time.

Modern science started with the advent of Newtonian mechanics. In a certain sense, Newton ‘invented’ time in order to formulate his laws. Like space, time in Newtonian physics is absolute, that is, it is externally given and unaffected by any material agency. This reflects the ontological idea that the world evolves in time in an objective sense. It is possible to introduce a Newtonian picture of four‐dimensional spacetime, which can be foliated into three‐dimensional hypersurfaces of Euclidean geometry in an absolute sense. There is an absolute notion of simultaneity: It is objectively clear whether two points in spacetime are simultaneous or not. As pointed out by Ludwig Lange in 1885, a central property of Newtonian spacetime is its affine structure: straight timelike worldlines are distinguished because they describe the free (inertial) motion of objects, cf. Ehlers (1973).

The notions of absolute space and time were criticized early on, by Leibniz, Huy-gens, and Berkeley, and in the nineteenth century by Ernst Mach, with the argument that they are unobservable. These authors favoured a picture of the world in which all motion is relative. They were, however, not able to construct a viable alternative dynamics. Models for relative motion in mechanics were only developed in the twentieth century (Barbour 1986).

Albert Einstein, in 1905, recognized that the notion of an absolute simultaneity is empirically unfounded. This led him to his special theory of relativity. While its causal structure has changed, spacetime is still absolute in the sense of being non‐dynamical: it acts on matter and fields, but is itself not acted upon; it is only an arena for the dynamics.

(p. 664) Spacetime becomes dynamical only with the advent of general relativity in 1915. Its geometry is a manifestation of the gravitational field, and gravity is dynamical. There is now a complicated non‐linear interaction between gravity and non‐gravitational fields, as encoded in the Einstein field equations. The running of a clock depends on its position in spacetime, and the clock acts back on spacetime due to its mass. This back action is very natural because ‘it is contrary to the scientific mode of understanding to postulate a thing that acts, but which cannot be acted upon’ (Einstein 1922).

On the other hand, quantum mechanics retains the Newtonian concept of absolute time. The time parameter t in the Schrödinger equation is non‐dynamical and not transformed into an operator. The wave function evolves in time and gives probability amplitudes and transition rates for physical entities with respect to external time; the total probability is conserved in time.

The situation does not change much in special‐relativistic quantum field theory. Quantum fields evolve on the externally given, rigid Minkowski spacetime of special relativity. The Standard Model of strong and electroweak interactions is formulated on this background.

This is no longer possible for gravity. Since it entails dynamical degrees of freedom for spacetime, the latter can no longer serve as a background. One thus has to look for a background‐independent quantum theory. If one sticks to the universality of quantum theory—and there is no evidence to the contrary—gravity and spacetime must be quantized, too, leading to a theory of quantum gravity. It is obvious that this will have drastic consequences for the concept of time. The clash between the external time in quantum theory and the dynamical time in general relativity is one aspect of the problem of time in quantum gravity, on which I shall elaborate in section 3.

In the following, I shall first motivate why a theory of quantum gravity is needed and give a brief overview of the existing approaches (section 2). I shall then discuss the problem of time in more detail (section 3). Sections 4 and 5 are devoted to two methods towards its solution: the choice of time before and after quantization. Section 6 will deal with the recovery of the old, ‘external’ time of quantum theory as an approximate notion in an appropriate semiclassical limit. In section 7, I shall briefly discuss the concept of time in loop quantum gravity and string theory, and section 8 is devoted to the arrow of time in quantum gravity. I shall end with a brief Conclusion in section 9.

I want to end this introduction with a quote by Einstein. In his foreword to the book Concepts of Space by Max Jammer, he writes

It was a hard struggle to gain the concept of independent and absolute space which is indispensible for the theoretical development. And it has not been a smaller effort to overcome this concept later on, a process which probably has not yet come to an end.1

(p. 665) Although he writes here only about space, the same holds for time. The ‘process which probably has not yet come to an end’ can perhaps be interpreted as referring to a future fundamental theory such as quantum gravity will be.

The literature on this subject is enormous. I have therefore cited only a few original articles. More details and a guide to the literature can be found, for example, in Butter‐field and Isham (1999), Isham (1993), Kiefer (2007), Kuchař (1992), and Rovelli (2004).

# 2. Why Quantum Gravity?

What are the main motivations for developing a quantum theory of gravity? Since there are currently no experimental hints, the main reasons are conceptual.

Within general relativity, one can prove singularity theorems which show that the theory is incomplete: under very general conditions, singularities are unavoidable, cf Hawking and Penrose (1996). Singularities are borders of spacetime beyond which geodesics cannot be extended; the proper time of any observer comes to an end. Such singularities can be rather mild, that is, of a topological nature, but they can also have diverging curvatures and energy densities. In fact, the latter situation seems to be realized in two important physical cases: the Big Bang and black holes. The presence of the cosmic microwave background (CMB) radiation indicates that a Big Bang has happened in the past. Curvature singularities also seem to lurk inside the event horizon of black holes. One thus needs a more comprehensive theory to understand these situations, and the general expectation is that a quantum theory of gravity is needed, in analogy to quantum mechanics in which the classical instability of atoms has disappeared. The origin of our universe cannot be described without such a theory, so cosmology remains incomplete without quantum gravity.

Because of its geometric nature, gravity interacts universally with all forms of energy. Since it thus interacts with the quantum fields of the Standard Model, it would seem natural that gravity itself is described by a quantum theory. It is hard to understand how one could construct a unified theory of all interactions in a hybrid classical‐quantum manner. In fact, all attempts to do so have failed up to now.

Since gravity is a manifestation of spacetime geometry, quantum gravity should make definite statements about the microscopic behaviour of spacetime. For this reason it has been speculated long ago that the inclusion of gravity can avoid the divergences that plague ordinary quantum field theories. These divergences arise from the highest momenta and thus from the smallest scales. This speculation is well motivated, and can be traced back to the background independence discussed in the Introduction. If the usual divergences have really to do with the smallest scales of the background spacetime, they should disappear together with the background.

A direct experimental test of quantum gravity is hard to perform. This is connected with the smallness of the corresponding length and time scales and the largeness of the mass (energy) scale: combining the gravitational constant, G, the speed of light, c, and the quantum of action, ℏ, into units of length, time, and mass, respectively, one arrives (p. 666) at the famous Planck units,

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To probe, for example, the Planck length with contemporary accelerators, one would have to build a machine of the size of our Milky Way! Direct observations should thus be possible mainly in astrophysics—probing the early universe and the structure of black holes.

Concerning now the attempt to construct a full quantum theory of gravity, the question arises: what are the main approaches? In brief, one can distinguish between

• Quantum general relativity: The most straightforward attempt, both conceptually and historically, is the application of ‘quantization rules’ to classical general relativity. This approach can be divided further into

• Covariant approaches: These are approaches that employ four‐dimensional covariance at some stage of the formalism. Examples include perturbation theory, effective field theories, renormalization‐group approaches, and path integral methods.

• Canonical approaches: Here one makes use of a Hamiltonian formalism and identifies appropriate canonical variables and conjugate momenta. Examples include quantum geometrodynamics and loop quantum gravity.

• String theory (M‐theory): This is the main approach to construct a unifying quantum framework of all interactions. The quantum aspect of the gravitational field only emerges in a certain limit in which the different interactions can be distinguished.

• Other fundamental approaches, such as a direct quantization of topology, or the theory of causal sets.

All these approaches, in their non‐perturbative version, are either already formulated in a background‐independent way, or have at least the ambition to aim at such a formulation. They thus all face the problem of time, to which I shall now turn in more detail.

# 3. The Problem of Time

As we have seen, time in quantum theory is a non‐dynamical quantity. The parameter t in the Schrödinger equation, (p. 667)

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is identical to Newton's absolute time. On the other hand, time in general relativity is dynamical because it is part of spacetime as described by Einstein's equations,

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Spacetime and matter are inextricably linked by these non‐linear dynamical equations, and it is conceptually impossible to introduce a non‐dynamical background into this framework.

What, then, happens with the concept of time if gravity is quantized? Although the problem of time is present in all approaches, I shall restrict myself to quantum geometrodynamics, because there the discussion can be presented in a most transparent way. Quantum geometrodynamics is the oldest version of canonical quantum gravity; its fundamental variable is the three‐dimensional metric. Some remarks on loop quantum gravity and string theory are made in section 7 below.

Quantum geometrodynamics is one of the most conservative approaches. Quantization is here performed in a spirit similar to Erwin Schrödinger's original heuristic approach to quantum mechanics. It leads to a wave equation which correctly produces the Einstein equations (5) in the semiclassical limit. To the very least, quantum geometrodynamics should provide a good approximation to any full quantum theory of gravity for not‐too‐small length scales.

Click to view larger

figure 23.1 Foliation of spacetime into spacelike hypersurfaces

In order to understand the concept of time, let us inspect in more detail the canonical formalism. Starting point is the ‘3+1 decomposition’, that is, the split of spacetime into a foliation of three‐dimensional spacelike hypersurfaces. This is schematically shown in Figure 23.1. A necessary requirement for this to work is that the spacetime

(p. 668) manifold, 𝓜, be globally hyperbolic, that is, isomorphic to the cartesian product ℝ×Σ, where denotes a three‐dimensional manifold.

It is illustrative to compare the situation with mechanics. A particle trajectory is the succession of spatial points in time. Similarly, spacetime can be understood as a generalized trajectory in the sense that it is a succession of three‐dimensional spaces. In quantum mechanics, trajectories are no longer part of the formalism; instead one has wave functions that only depend on the spatial configurations. In the same manner, upon quantization, only a wave functional depending on the three‐dimensional spaces remains. There is a difference, though. Whereas quantum mechanics still contains the absolute time t, no such time is available in canonical quantum gravity because there is no absolute time in general relativity. One should thus expect that the equations of quantum gravity are fundamentally timeless.

This is indeed what happens. Already the classical theory contains constraints, that is, equations which constrain the possibilities to choose initial data on a space‐ like hypersurface. The presence of such constraints is intimately connected with the presence of the freedom to choose the coordinates in general relativity. In the picture of Figure 23.1, this encapsulates the freedom to choose both the foliation as well as the three spatial coordinates on each space. In fact, the first four of the ten Einstein equations (5) are constraints. They are called Hamiltonian constraint (the one responsible for the freedom to choose the foliation in an arbitrary way) and momentum or diffeomorphism constraints (the three that are responsible for the choice of spatial coordinates). An analogy to these constraints in electrodynamics is Gauss law, ∇E = (4π;/c)ρ, whose presence can be traced back to the gauge invariance. Since Gauss' law is devoid of time, it is a constraint on the initial data of the theory. Quite generally, constraints generate redundancy transformations such as gauge or coordinate transformations.

It turns out that the full Hamiltonian of gravity can be written, apart from possible boundary terms, as a linear combination of these constraints. The Hamiltonian is thus itself a constraint. This is connected with the absence of absolute time. Unlike theories with a background, the gravitational Hamiltonian can no longer generate time translations. Instead it generates a change of foliation and spatial coordinate transformations; since these changes are redundancies without observable consequence, the Hamiltonian must be a constraint. It will be clear from these remarks that any background‐independent theory, not only general relativity, leads to a Hamiltonian which is a constraint.

In a certain sense, the constraints in general relativity already contain all the information of the theory. A theorem states that Einstein's equations are the unique propagation laws consistent with the constraints. To demand the validity of the constraints on each hypersurface necessarily entails the fact that all Einstein's equations must hold on the foliated spacetime. The analogy in electrodynamics states that Maxwell's equations are the unique propagation laws consistent with the Gauss constraint. Constraints and dynamical equations are inextricably mixed.

(p. 669) After quantization, the ‘trajectory’, that is, the spacetime has vanished. Only the three‐dimensional space Σ remains. Consequently, only the constraints are left: the remaining six Einstein equations have disappeared. All the information about the quantum theory is therefore contained in the quantum version of the constraints. They thus constitute the central equations of quantum gravity.

# 4. Time Before Quantization

The constraints of the classical theory are constraints on the initial data, that is, on the generalized positions and their momenta. As already mentioned, the generalized positions are the components of the three‐dimensional metric, h ab(x). The conjugate momenta, p cd (x), are a linear combination of the components of the extrinsic curvature. (The extrinsic curvature is a measure of the embedding of space into spacetime.) How should one transform these constraints into the quantum theory?

A first attempt would be to solve the constraints already on the classical level. What does this mean? It is well known that classical mechanics can be put into a parametrized form, where time t is elevated to a dynamical variable; t as well as all the q i are then supposed to depend on an arbitrary parameter τ (cf. Lanczos 1986). It turns out that the resulting formalism is invariant under arbitrary reparametrizations of τ; it is called ‘time‐reparametrization invariant’. As a consequence of this invariance, a constraint appears. It is of the form

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where p t is the momentum conjugate to t (which in this formalism is dynamical), and H is the usual Hamiltonian. Following a general prescription introduced by Dirac, the constraint (6) can be transformed into the quantum theory by turning all variables into operators and interpreting the resulting constraint as a restriction on physically allowed wave functions. Substituting p t by (ℏ/i)/∂t and H by Ĥ, one then arrives at the Schrödinger equation (4).

The question now is whether this procedure can also be performed in the much more complicated case of the constraints of general relativity. If it could, one would be able to identify a function of the canonical variables h ab(x) and p cd (x) that could play the role of an appropriate time variable already at the classical level. Note that this would not be a time coordinate on spacetime, but a function of the canonical variables which are defined solely on space.

After such an identification, the constraints would then be in a form similar to (6). One could then quantize the constraints in the same way and arrive at a functional version of the Schrödinger equation (4). This would have great advantages: one could extrapolate all the interpretational structure, such as the usual inner product and its conservation in time (unitarity) from quantum mechanics to quantum gravity.

(p. 670) There are, however, a lot of problems with such an attempt. Firstly, a transformation of the constraints into a form similar to (6) is not possible globally, that is, on the whole phase space. Secondly, even if attention is restricted to a local identification of time, there is the ‘multiple‐choice problem’: there are many choices for such a time, and the corresponding quantum theories are generically not unitarily equivalent. Thirdly, the ensuing reduced Hamiltonian depends on this time and has in general a very complicated structure (containing square roots, etc.). In addition, there is the practical task of actually transforming the constraints into a form similar to (6), a task that has been accomplished only in a few relatively simple cases (such as cylindrical gravitational waves and spherically‐symmetric black holes). Most authors do not, therefore, prefer this approach but try to identify an appropriate time, if any, after quantization.

# 5. Time After Quantization

In this alternative approach one takes the contraints as they appear and tries to transform them directly into quantum constraint equations. As we have seen, the constraints combine into the Hamiltonian of general relativity, which is then again a constraint, H = 0. (We shall neglect boundary terms.) The application of Dirac's prescription then leads to the equation

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where Ψ is the quantum gravitational wave functional, which depends on the three‐ dimensional metric h ab(x). There remain, of course, the usual problems of factor ordering and regularization.

Strictly speaking, (7) are infinitely many equations, one equation at each space point. If non‐gravitational fields are present, they will be included into the constraint, and (7) then refers to the full Hamiltonian of gravity and matter. In honour of the work by Bryce DeWitt (1967) and John Wheeler (1968), Equation (7) is called the Wheeler– DeWitt equation. It is the central equation of quantum geometrodynamics.

The Wheeler–DeWitt equation is fundamentally timeless (in the sense that a time parameter is absent); all components of the three‐dimensional metric are on equal footing. Its solutions are thus static waves. A closer inspection of its kinetic term exhibits, however, a particular and important feature: the kinetic term is of an indefinite nature. More precisely, the Wheeler–DeWitt equation is locally (that is, at each space point) of a hyperbolic structure. Unlike the Schrödinger equation, it has the form of a wave equation similarly, for example, to the Klein–Gordon equation. A wave equation is characterized by the fact that one of the variables comes with the opposite sign in the kinetic term; this variable is usually related to time. The structure of the Wheeler–DeWitt equation thus suggests the presence of an intrinsic timelike variable, in short: intrinsic time.

(p. 671) The intrinsic time is constructed from part of the three‐dimensional metric h ab(x). More precisely, it is the size (instead of the shape) of the three‐dimensional geometry. This becomes evident in models of quantum cosmology, cf. Kiefer and Sand-höfer (2008). There, only very few variables are quantized, such as the scale factor (‘radius’) of the universe and a homogeneous scalar field (representing matter). Consider, for example, a simple model of a closed Friedmann‐Lemaître universe with scale factor α, containing a massive scalar field ϕ. The Wheeler–DeWitt equation then reads (after a suitable redefinition of variables)

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where α ≡ ln α; the variable α has the advantage that its range is from −∞ to +∞ instead of 0 to ∞, which holds for α. One recognizes that the sign of the kinetic α‐term has the ‘wrong’ sign compared to the standard matter kinetic term; it is thus the size of the universe which can serve as the intrinsic time. (Additional cosmological and matter variables will all come with the same sign as the ϕ‐term.)

This new concept of time has far‐reaching consequences: the classical and the quantum model exhibit two drastically different concepts of determinism, see Figure 23.2.

Let us consider the case of a classically recollapsing universe. In the classical case (left) we have a trajectory in configuration space: although it can be parametrized in many ways, the important point is that it can be parametrized by some time parameter. Therefore, upon solving the classical equations of motion, the recollapsing part of the trajectory is the deterministic successor of the expanding part: the model universe expands, reaches a maximum point, and recollapses.

Not so for the quantum model. There is no classical trajectory and no classical time parameter, and one must take the wave equation as it stands. The wave function only distinguishes small α from large α, not earlier t from later t. There is thus no intrinsic difference between Big Bang and Big Crunch. If one wants to construct a wave packet following the classical trajectory as a narrow tube, one has to impose the presence of two packets as an initial condition at small α; if one chose only one packet, one would obtain a wave function which is spread out over configuration space and which does not resemble anything close to a narrow wave packet. (Equation (8) with m = 0 directly corresponds to the situation of Figure 23.2.)

Wave packets are of crucial importance when studying the validity of the semi‐ classical approximation. In quantum cosmology, this issue has to be discussed from the viewpoint of the Wheeler–DeWitt equation (8). If the classical model describes a recollapsing universe, one has to impose in the quantum theory onto the wave function the restriction that it go to zero for α →∞; with respect to intrinsic time, this corresponds to a ‘final condition’. Calculations show that it is then not possible to have narrow wave packets all along the classical trajectory: the packet disperses, and the references therein. This is again a consequence of the novel concept of time in quantum gravity.

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figure 23.2 The classical and the quantum theory of gravity exhibit drastically different notions of determinism

From Kiefer (2007)

(p. 672) But how do classical properties arise if wave packets necessarily disperse? The answer to this question is decoherence—the irreversible emergence of classical behaviour through the unavoidable interaction with an ubiquitous environment (Joos et al. 2003). ‘Environment’ is a general name for uncontrollable or irrelevant variables. In quantum cosmology, such degrees of freedom can be small density fluctuations or weak gravitational waves. They can act as an ‘environment’ which becomes quantum entangled with α and ϕ, causing their classical appearance. This classical appearance holds for most of the evolution of the universe. Possible exceptions are the Planck regime (small α) and the turning point of a classically recollapsing quantum universe (section 8).

# 6. Semiclassical Time

The Wheeler–DeWitt equation (7) is timeless in the sense that classical spacetime is absent and only space is present. While this holds at the most fundamental level, spacetime and with it the familiar time parameter t must emerge at an appropriate level of approximation. In particular, the functional (field‐theoretic) version of the Schrödinger equation (4) must reappear as an effective equation. Together with the t, the imaginary unit i must show up: whereas the Wheeler–DeWitt equation is a real equation, allowing for real solutions, the Schrödinger equation is complex, a feature that is of the highest importance for the interpretational structure of quantum mechanics. How can the emergence of both t and i be understood (Kiefer 2007)?

It can be seen from (8) that the Planck mass m P appears explicitly in the gravitational part of the kinetic term. (This is also true for the full Wheeler–DeWitt equation (7)). Assuming, then, that one is in a regime where the relevant masses and energies are much smaller than the Planck mass, one can make a formal expansion in inverse (p. 673) powers of the Planck mass. This is similar to what is called Born–Oppenheimer expansion in molecular physics. In this way, one arrives at the following approximate solution of (7):

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where h is an abbreviation for the three‐dimensional metric, and ϕ stands for non‐ gravitational fields. In short, the approximation scheme leads to the following results:

• S 0 obeys the Hamilton–Jacobi equation for the gravitational field and thereby defines a classical spacetime which is a solution to Einstein's equations. (This order is akin to the recovery of geometrical optics from wave optics via the eikonal equation.)

• ψ obeys an approximate (functional) Schrödinger equation,

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where H m denotes the Hamiltonian for the non‐gravitational fields ϕ. Note that the expression on the left‐hand side of (10) is a shorthand notation for an integral over space, in which ∇ stands for functional derivatives with respect to the three‐ metric. In the case of (8) one has, for example,

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Semiclassical time t is thus defined in this limit from the dynamical variables. This bears a resemblance to the notion of ephemeris time in astronomy.

• The next order of the Born–Oppenheimer scheme yields quantum gravitational correction terms proportional to the inverse Planck mass squared, $1 / m p 2 .$ The presence of such terms may in principle lead to observable effects, for example, in the anisotropy spectrum of the cosmic microwave background radiation.

The Born–Oppenheimer expansion scheme distinguishes a state of the form (9) from its complex conjugate. In fact, in a generic situation both states will decohere from each other, that is, they will become dynamically independent from each other (Joos et al. 2003). This is a type of symmetry breaking in analogy to the occurrence of parity violating states in chiral molecules. It is through this mechanism that the t and the i in the Schrödinger equation emerge.

The recovery of the Schrödinger equation (10) raises an interesting issue. It is well known that the notion of Hilbert space is connected with the conservation of probability (unitarity) and thus with the presence of an external time (with respect to which the probability is conserved). The question then arises whether the concept of a Hilbert space is still required in the full theory where no external time is present.

(p. 674) It could then be that this concept makes sense only on the semiclassical level where (10) holds. This question is not yet settled.

This idea of recovering time is also of interest in ordinary quantum mechanics. One can adopt there the idea that the fundamental equation is the stationary Schrödinger equation, not the time‐dependent one. Mott (1931), for example, considered a time‐ independent Schrödinger equation for a total system consisting of an alpha‐particle and an atom. If the state of the alpha‐particle can be described by a plane wave (corresponding in this case to high velocities), one can make an ansatz similar to (9) and derive a time‐dependent Schrödinger equation for the atom alone, for which time is defined by the state of the alpha‐particle.

# 7. Time in Loop Quantum Gravity and String Theory

So far, we have restricted our discussion to quantum geometrodynamics. I have argued that the situation encountered there should be typical for all theories of quantum gravity, at least for those which start from a classical theory devoid of an absolute background. Let us see how the situation is in two highly discussed approaches: loop quantum gravity and string theory.

Loop quantum gravity is a particular version of canonical quantum gravity (Rovelli 2004). Instead of three‐metric and extrinsic curvature, the canonical variables are now holonomies along loops and fluxes of generalized electric fields through the loops, concepts well known from gauge theories. In fact, one of the main merits of loop quantum gravity is the use of concepts akin to Yang–Mills theories. It is thus hoped that this will be helpful in the search for a unified theory. It seems that loop quantum gravity is mathematically better behaved than quantum geometrodynamics.

Since loop quantum gravity is a canonical theory, one arrives again at constraint equations of the form (7). The detailed structure of this equation is different, but its interpretation with regard to the problem of time is similar: there is no time parameter at the fundamental level. An important new feature is the emergence of a discrete structure of space: one can rigorously define geometric operators (such as an area operator) on the kinematical level (that is, before solving all constraints), which has a discrete spectrum: there thus exists a smallest quantum of area.

One would expect that this discreteness in space leaves its imprints on the time that emerges on a semiclassical level (section 6). Unfortunately, the semiclassical approximation has not yet been completed on the level of full loop quantum gravity. Some insights can, however, be gained from loop quantum cosmology (Bojowald 2005), that is, the application of loop quantum gravity to cosmology. The Wheeler–DeWitt equation (8) is there replaced by a difference equation: only discrete values of the scale factor α are allowed. For large‐enough values of α, that is, for values sufficiently (p. 675) above the Planck scale, this difference equation becomes indistinguishable from the Wheeler–DeWitt equation. The semiclassical approximation with its recovery of both the time parameter t and the imaginary unit thus proceeds as in section 6. But if viewed from the fundamental perspective of loop quantum cosmology, this semiclassical time parameter inherits a small discrete structure recognizable only at the scale of the Planck time.

String theory, on the other hand, is conceptually different from all of the above approaches. It does not involve a direct quantization of the classical theory of relativity, but instead aims at directly constructing a fundamental quantum theory of all interactions, see, for example, Zwiebach (2004). Quantum general relativity should thus only emerge in an appropriate limit in which the various interactions such as gravity are distinguishable.

String theory starts from the formulation of a string on a given background spacetime. In more recent years, one has learnt that also higher‐dimensional objects (‘branes’) have to be taken into account; they, too, propagate on the background. Eventually, spacetime itself should be constructed out of strings and branes. It seems, however, that string theory has not yet rescued itself from a background‐independent formulation. But an important step towards such a formulation could be provided by the so‐called AdS/CFT‐correspondence, cf. Horowitz (2005). This correspondence states that non‐perturbative string theory in a background spacetime which is asymptotically anti‐de Sitter (AdS) is equivalent to a conformal field theory (CFT) defined in a flat spacetime of one less dimension. In a sense, a theory containing gravity (the AdS sector) is equivalent to a theory without gravity in one less dimension (the CFT sector), cf. Maldacena (2007). In addition to the problem of time, string theory thus seems to lead to a ‘problem of space’.

The AdS/CFT‐correspondence can be interpreted as a non‐perturbative and partly background‐independent definition of string theory, since the CFT is defined non‐ perturbatively, and the background metric enters only through boundary conditions at infinity. Full background independence in the sense of canonical quantum gravity has, however, not yet been achieved. The problem of time in string theory thus still awaits its solution.

# 8. The Direction of Time

Although most of the fundamental physical laws are invariant under time reversal, there are several classes of phenomena in Nature that exhibit an arrow of time, see, for example, Zeh (2007). Because most subsystems in the universe cannot be considered as isolated, these various arrows of time all point in the same direction. The question then arises whether there exists a master arrow of time underlying all these arrows. The tentative answer is yes. Already Ludwig Boltzmann has speculated about a possible foundation of the Second Law of thermodynamics from cosmology: (p. 676) it is the huge temperature gradient between the hot stars and the cold space which provides the entropy capacity which is necessary for the entropy to increase (instead of being already at its maximum).

This is also the modern point of view, although the arguments are somewhat different. The global expansion of the universe, together with the gravitational collapse of substructures (leading to stars etc.), defines a gravitational arrow that seems to monitor the other arrows. If there is a special initial condition of low entropy in the early universe, statistical arguments can be invoked to demonstrate that the entropy of the universe will increase with increasing size. But where does such an initial condition come from, and how can the entropy of the universe be calculated?

There are several subtle issues connected with these questions. First, one does not yet know a general expression for the entropy of the gravitational field, except for the black‐hole entropy, which is given by the Bekenstein–Hawking formula. As for the general case, Roger Penrose has suggested the use of the Weyl tensor as a measure of gravitational entropy. He has also estimated from the Bekenstein–Hawking formula how unlikely our universe in fact is (Penrose 1981). Second, because the very early universe is involved, the problem has to be treated within quantum gravity. But as we have seen, there is no external time in quantum gravity – so what does the notion ‘arrow of time’ mean in a timeless theory?

The following discussion will again be based on quantum geometrodynamics, that is, on the Wheeler–DeWitt equation. It should be possible to implement an analogous reasoning in loop quantum gravity and string theory.

An important observation is that the Wheeler–DeWitt equation exhibits a fundamental asymmetry with respect to the intrinsic time introduced above. Very schematically, one can write this equation as (with a convenient choice of units)

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where again α = ln α =, and the {x i} denote inhomogeneous degrees of freedom describing perturbations of the Friedmann universe; they can describe weak gravitational waves or density perturbations. The important property of the equation is that the potential becomes small for α → − ∞ (where the classical singularities would occur), but complicated for increasing α the Wheeler–DeWitt equation thus possesses an asymmetry with respect to ‘intrinsic time’ α. One can in particular impose the simple boundary condition

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(12)

which would mean that the degrees of freedom are initially not entangled. Defining an entropy as the entanglement entropy between relevant degrees of freedom (such as α) and irrelevant degrees of freedom (such as most of the {x i}), this entropy vanishes initially but increases with increasing α because entanglement increases due to the presence of the potential. In the semiclassical limit where t is constructed from α (p. 677) (and other degrees of freedom), cf. (10), entropy increases with increasing t. This then defines the direction of time and would be the origin of the observed irreversibility in the world. The expansion of the universe would then be a tautology. Due to the increasing entanglement, the universe rapidly assumes classical properties for the relevant degrees of freedom due to decoherence.

This process has interesting consequences for a classically recollapsing universe (Kiefer and Zeh 1995). Since Big Bang and Big Crunch correspond to the same region in configuration space (α → − ∞), an initial condition for α → − ∞ would encompass both regions, cf. Figure 23.2. This would mean that the above initial condition would always correlate increasing size of the universe with increasing entropy: the arrow of time would formally reverse at the classical turning point. As it turns out, however, a reversal cannot be observed because the universe would enter a quantum phase. Further consequences concern black holes within such a universe because no horizon and no singularity would ever form. It is, of course, not clear whether this situation is actually realized in Nature. This scenario shows, however, that drastic consequences for our understanding of the universe can arise from the direct combination of two well‐established theories: general relativity and quantum theory.

# 9. Conclusion

Quantum gravity places the concept of time on a new level. In the absence of experimental hints, mathematical and conceptual issues must be chosen as the guides in the search for such a theory. The situation can be compared with Einstein's investigation into the meaning of time in 1905, which led him to develop his special theory of relativity.

We have seen that general relativity does not contain a non‐dynamical background spacetime. Upon quantization, spacetime disappears in the same way as a particle trajectory has disappeared in quantum mechanics; only space remains. There is thus no time parameter in quantum gravity. I have discussed this explicitly within the framework of quantum geometrodynamics, but the situation should be similar in all reasonable approaches. String theory, for example, leads to general relativity in an appropriate limit, and its quantization should thus lead to the absence of a time parameter, too.

In quantum geometrodynamics, a sensible notion of intrinsic time can be introduced. In simple models, it can be constructed from the size of the universe—the universe thus provides its own clock. The standard time parameter appears on a semiclassical level. In loop quantum gravity, it should exhibit a discrete structure recognizable for times of the order of the Planck time.

The origin of the arrow of time can in principle be understood from quantum gravity. Intrinsic time enters asymmetrically into the Wheeler–DeWitt equation and thus allows for a natural initial condition that leads to an entropy correlated with the size of the universe. In the semiclassical limit, where a time parameter t appears, this (p. 678) entails the Second Law of thermodynamics. Both the familiar time and its arrow can thus be understood from quantum gravity, which itself is fundamentally timeless.

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## Notes:

(1) Es hat schweren Ringens bedurft, um zu dem für die theoretische Entwicklung unentbehrlichen Begriff des selbständigen und absoluten Raumes zu gelangen. Und es hat nicht geringerer Anstrengung bedurft, um diesen Begriff nachträglich wieder zu überwinden—ein Prozeß, der wahrscheinlich noch keineswegs beendet ist.