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- Quantum Structure of Spacetime
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This is due to the extreme relative weakness of gravity, which makes it very complicated to isolate the gravitational effects experimentally. Therefore, one should keep in mind that the short-range behaviour of gravity is not particularly well understood and could potentially be subject to rather drastic modifications at the scale of single atoms. The usual assumption to simply neglect gravitational effects in particle physics therefore is rather bold based on direct experimental data—despite being very successful in view of the well-working Standard Model.
Feynman diagrams. When it was realized, that in every consistent variant of string theory there is a certain closed string state with just the right properties to identify it with the graviton—the virtual exchange particle of the gravitational interaction—interest in the old theory was renewed. And after Green and Schwarz proved in that superstring theory comes without a certain type of anomalies signalling significant instabilities and conceptual problems in any quantum theory , research in the area of string theory really took off.
String theory rejects the idea of a point particle as the fundamental constituent of the theory, which is the central concept in quantum field theory. By introducing 1-dimensional, extended objects and higher dimensional membranes in its further developments this gives rise to a natural smallest scale in form of the string length.
The average size of this string is expected to be much smaller than the smallest sizes probed experimentally, currently centimetre, and might be as small as the Planck length, which is of the order of centimetre. So when one studies string theory at low energies compared to the Planck scale, it becomes difficult to see that strings are extended objects—they behave effectively 0-dimensional, i. With this perspective in mind, point-like quantum field theory can be regarded as a sort of effective theory for strings at low energies.
This limit in conceptual smallness helps avoiding most of the described problems. It naturally smears out point-like interactions and imposes a certain cutoff scale dictated by the string length, thus yielding a consistent and unified picture of quantum theory and general relativity due to the presence of a spin-2 particle state. Naive string interaction diagrams appear as webs of tubes closed strings or ribbons open strings , and can be stretched and deformed into a great number of inequivalent Feynman diagrams of the corresponding field theory.
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Take a look at the picture: The string worldsheets a , b and c are obtained by stretching and deforming, only diagram d has a different loop order and topology. At least in theory the computation of scattering amplitudes becomes much simpler in string theory. M-theory duality web. One rather odd property of supersymmetric string theories is that it only works consistently in 10 space-time dimensions. This is obviously in contrast to the everyday experience of 3 spatial and 1 temporal dimension, thus 6 dimensions have to be accounted for somehow. For example, rolling up each additional coordinate to a circle gives a 6-dimensional torus, but one may also consider much more complicated compact 6-dimensional spaces.
Furthermore, if there are higher-dimensional objects like D-branes in the considered configuration, those may wrap around certain parts of the compactified coordinates, which dramatically changes their respective physical properties.
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In essence, shape and geometry both of the space-time and the objects propagating inside of it govern everything in string theory. Hopes to observe stringy effects in current and future accelerator experiments are—unfortunately—vague and rather speculative at the moment. But string theory itself is only the perturbative starting point of even more involved theories. However, in subsequent years it was shown that all those theories were actually connected by an intricate web of dualities and symmetries, which suggested the existence of an unique non-perturbative theory lurking somewhere in the background.
Furthermore, many of the issues facing canonical quantum gravity are also firmly rooted in conceptual difficulties facing the classical theory, which philosophers are already well acquainted with e.
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As noted in Section 3. 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 of differential 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.
The problem is not so much that the spacetime is dynamical; there is no problem of time in classical general relativity in the sense that a time variable is present. 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 for discussion and further references. 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 a,b , for one, thinks that it tells us that time is illusory see also Earman, , 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. Bradley Monton has argued that a specific version of canonical quantum gravity — that with a so-called constant mean extrinsic curvature [CMC] or fixed foliation — has the necessary resources to render presentism the view that all and only presently existing things exist a live possibility see the section on Presentism, Eternalism, and The Growing Universe Theory in the entry on time for more on presentism. Though he readily admits that CMC formulations are outmoded in the contemporary theoretical landscape, he nonetheless insists that given the lack of experimental evidence one way or the other, it stands as a viable route to quantum gravity, and therefore presentism remains as a possible theory of time that is compatible with frontier theoretical physics.
It is more of a piece of machinery that is used within a pre-existing approach namely, the canonical approach. Simply not being ruled out on experimental grounds does not thereby render an approach viable. This at least has the added benefit of being a research programme that is being actively pursued. A common claim that appears in many discussions of the problem of time especially amongst philosophers is that it is restricted to canonical formulations of general relativity, and has something to do with the Hamiltonian formalism see Rickles a, pp.
The confusion lies in the apparently very different ways that time is treated in general relativity as standardly formulated, and as it appears in a canonical, Hamiltonian formulation. In the former there is no preferred temporal frame, whereas the latter appears to demand such a frame in order to get off the ground cf. Curiel, , p.
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The canonical framework is simply a tool for constructing theories, and one that makes quantization an easier prospect. As a matter of historical fact the canonical formulation of general relativity is a completed project, and has been carried out in a variety of ways, using compact spaces and non-compact spaces, and with a range of canonical variables.
However, there is no question that general relativity is compatible with the canonical analysis of theories, and the fact that time looks a little strange in this context is simply because the formalism is attempting to capture the dynamics of general relativity. In any case, the peculiar nature of general relativity and quantum gravity, with respect to the treatment of time, resurfaces in arguably the most covariant of approaches, the Feynman path-integral approach.
In this case that central task is to compute the amplitude for going from an initial state to a final state where these states will be given in terms of boundary data on a pair of initial and final hypersurfaces. However, one cannot get around the fact that general relativity is a theory with gauge freedom, and so whenever one has diffeomorphic initial and final hypersurfaces, the propagator will be trivial. A similar confusion can be found in discussions of the related problem of defining observables in canonical general relativity.
The claim gets its traction from the fact that it is very difficult to construct observables in canonical general relativity, while apparently it is relatively straightforward in the standard Lagrangian description. See, e. Curiel cites a theorem of Torre, , to the effect that there can be no local observables in compact spacetimes, to argue that the canonical formulation is defective somehow. Again, this rests on a misunderstanding over what the canonical formalism is and how it is related to the standard spacetime formulation of general relativity. That there are no local observables is not an artefact of canonical general relativity.
The notion that observables have to be non-local in this case, relational is a generic feature that results precisely from the full spacetime diffeomorphism invariance of general relativity and is, in fact, implicit in the theorem of Torre mentioned earlier. It receives a particularly transparent description in the context of the canonical approach because one can define observables as quantities that commute with all of the constraints.
The same condition will hold for the four-dimensional versions, only they will have to be spacetime diffeomorphism invariant in that case. This will still rule out local observables since any quantities defined at points or regions of the spacetime manifold will clearly fail to be diffeomorphism invariant. Hence, the problems of observables and the result that they must be either global or relational in general relativity is not a special feature of the canonical formulation, but a generic feature of theories possessing diffeomorphism invariance.
Quantum theory in general resists any straightforward ontological reading, and this goes double for quantum gravity. In quantum 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 the face 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 in individuating objects and their properties—in 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.
Quantum Structure of Spacetime
One area that philosophers might profit from is in the investigation of the relational observables that appear to be necessitated by diffeomorphism invariance. For example, since symmetries such as the gauge symmetries associated with the constraints come with quite a lot of metaphysical baggage attached as philosophers of physics know from the hole argument , such a move involves philosophically weighty assumptions.
For example, the presence of symmetries in a theory would appear to allow for more possibilities than one without, so eradicating the symmetries by solving the constraints and going to the reduced, physical phase space means eradicating a chunk of possibility space: in particular, one is eradicating states that are deemed to be physically equivalent, despite having some formal differences in terms of representaton. Hence, imposing the constraints involves some serious modal assumptions.
Belot and Earman have argued that since the traditional positions on the ontology of spacetime relationalism and substantivalism involve a commitment to a certain way of counting possibilities, the decision to eliminate symmetries can have serious implications for the ontology one can then adopt. Further, if some particular method out of retaining or eliminating symmetries were shown to be successful in the quest for quantizing gravity, then, they argue, one could have good scientific reasons for favouring one of substantivalism or relationalism.
See Belot, a, for more on this argument; Rickles, c, explicitly argues against the idea that possibility spaces have any relevance for spacetime ontology. In the loop quantum gravity program, the area and volume operators have discrete spectra. Thus, like electron 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 the structure of a differential manifold, but rather that it only approximates such a manifold on large scales, or at low energies. A similar idea, that classical spacetime is an emergent entity, can be found in several approaches to quantum gravity see Butterfield and Isham, and , for a discussion of emergence in quantum gravity.
M-Theory and Quantum Geometry
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 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 vicious circle. 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 the many-worlds or relative-state interpretations.