Mathematical Foundations of Quantum Field Theory and Perturbative String Theory

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Email, fax, or send via postal mail to:. Conceptual progress in fundamental theoretical physics is linked with the search for the suitable mathematical structures that model the physical systems. Quantum field theory QFT has proven to be a rich source of ideas for mathematics for a long time. This has been obtained at the price of introducing heavy mathematical tools, often unfamiliar to the average physicist, perhaps widening the language gap between the quantum gravity and the high energy physics community.

The reason for searching a mathematical-physics level of precision is that in quantum gravity one moves on a very unfamiliar terrain —quantum field theory on manifolds— where the experience accumulated in conventional quantum field theory is often useless and sometimes even misleading. Given the unlikelihood of finding direct experimental corroboration, the research can only aim, at least for the moment, at the goal of finding a consistent theory, with the correct limits in the regimes that we control experimentally.

In these conditions, high mathematical rigor is the only assurance of the consistency of the theory. One may object that a rigorous definition of quantum gravity is a vain hope, given that we do not even have a rigorous definition of QED, presumably a much simpler theory. But the serious difficulties of QED and of the other conventional field theories are ultraviolet. The physical hope supporting the quantum gravity research program is that the ultraviolet structure of a diffeomorphism invariant quantum field theory is profoundly different from the one of conventional theories.

Indeed, recall that in a very precise sense there is no short distance limit in the theory; the theory naturally cuts itself off at the Planck scale, due to the very quantum discreteness of spacetime. Thus the hope that quantum gravity could be defined rigorously may be optimistic, but it is not ill founded. After these comments, let me briefly mention some of the structures that have been explored in. This fact is often misunderstood: recoupling theory lives in 2d and is associated by Kauffman to knot theory by means of the usual projection of knots from 3d to 2d.

Here, the Kauffman axioms are not satisfied at the intersections created by the 2d projection of the spin network, but only at the nodes in 3d. See [ 77 ] for a detailed discussion. For instance, consider a 4-valent node of four links colored a, b, c, d. The color of the node is determined by expanding the 4-valent node into a trivalent tree; in this case, we have a single internal links. The expansion can be done in different ways by pairing links differently. These are related to each other by the recoupling theorem of pg. Equation 29 follows just from the definitions given above.

Recoupling theory provides a powerful computational tool in this context. Since spin network states satisfy recoupling theory, they form a Temperley-Lieb algebra [ ]. The scalar product 14 in is given also by the Temperley-Lieb trace of the spin networks, or, equivalently by the Kauffman brackets, or, equivalently, by the chromatic evaluation of the spin network. See Ref.

Next, admits a rigorous representation as an L 2 space, namely a space of square integrable functions. The space of the distributional connections is the closure of the space of smooth connection in a certain topology. Thus, distributional connections can be seen as limits of sequences of connections, in the same manner in which distributions can be seen as limits of sequences of functions.

Usual distributions are defined as elements of the topological dual of certain spaces of functions. The group elements satisfy certain properties. The space of such generalized connections is denoted. In fact, one may show that 31 defines by linearity and continuity a well-defined absolutely continuous measure on. Then, one can prove that , under the natural isomorphism given by identifying cylindrical functions. This is the loop transform formula that was derived heuristically in [ ]; here it becomes rigorously defined.

One can show that this is a suitable norm closure of the space of smooth SU 2 connections over physical space, modulo gauge transformations. Thus, a number of powerful mathematical tools are at hand for dealing with nonperturbative quantum gravity. The next step in the construction of the theory is to factor away diffeomorphism invariance. This is a key step for two reasons. However, most of this redundancy is gauge, and disappears when one solves the diffeomorphism constraint, defining the diff-invariant Hilbert space. This is the reason for which the loop representation, as defined here, is of great value in diffeomorphism invariant theories only.

The second reason is that turns out to have a natural basis labeled by knots. An s-knot s is an equivalence class of spin networks S under diffeomorphisms. Thus, the physical quantum states of the gravitational field turn out to be essentially classified by knot theory. There are various equivalent ways of obtaining from.

One can use regularization techniques for defining the quantum operator corresponding to the classical diffeomorphism constraint in terms of elementary loop operators, and then find the kernel of such operator. Equivalently, one can factor by the natural action of the diffeomorphism group that it carries. There are several rigorous ways for defining the quotient of a Hilbert space by the unitary action of a group. See in particular the construction in [ 19 ], which follows the ideas of Marolf and Higuchi [ , , , ].

This, however, is a false problem. This can be seen in a variety of equivalent ways. For instance, it can be seen from the following theorem. This is discussed, for instance, in [ ]. There are two distinct possibilities for factoring away the diffeomorphisms in the quantum theory, yielding two distinct version of the theory. The first possibility is to factor away smooth transformations of the manifold. In doing so, finite dimensional moduli spaces associated with high valence nodes appear [ 98 ], so that the resulting Hilbert space is still non-separable.

The physical relevance of these moduli parameters is unclear at this stage, since they do not seem to play any role in the quantum theory. Alternatively, one can consistently factor away continuous transformations of the manifold. This possibility has been explored by Zapata in [ , ], and seems to lead to a consistent theory free of the residual non separability. Finally, the definition of the theory is completed by giving the hamiltonian constraint. A number of approaches to the definition of a hamiltonian constraint have been attempted in the past, with various degrees of success.

Recently, however, Thiemann has succeeded in providing a onstraint that yields a well defined, finite operator. I will not describe it here. Here i labels the nodes of the s-knot s; I J labels couples of distinct links emerging from i. This is illustrated in Figure 2. Some of these coefficients have been explicitly computed [ 52 ].

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The operator defined above is obtained by introducing a regularized expression for the classical hamiltonian constraint, written in terms of elementary loop observables, turning these observables into the corresponding operators and taking the limit. Thus, here diff invariance plays again a crucial role in the theory.

For a discussion of the problems raised by the Thiemann operator and of the variant proposed, see section 8.

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A recent development in the formalism is the translation of loop quantum gravity into spacetime covariant form. More precisely, it was proven in [ ] that the matrix elements of the operator U T. In conventional QFT each term of a Feynman sum corresponds naturally to a certain Feynman diagram, namely a set of lines in spacetime meeting at vertices branching points.

A similar natural structure of the terms appears in quantum gravity, but surprisingly the diagrams are now given by surfaces is spacetime that branch at vertices. Thus, one has a formulation of quantum gravity as a sum over surfaces in spacetime. Intuitively, the time evolution of a spin network in spacetime is given by a colored surface. The surfaces capture the gravitational degrees of freedom. They turn out to depend only on the colors of the surface components immediately adjacent the vertex v.

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The sum turns out to be finite and explicitly computable order by order. As in the usual Feynman diagrams, the vertices describe the elementary interactions of the theory. In particular, here one sees that the complicated structure of the Thiemann hamiltonian, which makes a node split into three nodes, corresponds to a geometrically very simple vertex. Figure 3 is a picture of the elementary vertex. Notice that it represents nothing but the spacetime evolution of the elementary action of the hamiltonian constraint, given in Figure 2.

An example of a surface in the sum is given in Figure 4. The sum over surfaces version of loop quantum gravity provides a link with certain topological quantum field theories and in particular with the the Crane-Yetter model [ 71 , 72 , 73 , 74 , 75 ], which admit an extremely similar representation. For a discussion on the precise relation between topological quantum field theory and diffeomorphism invariant quantum field theory, see [ ] and [ , , 83 ].

The idea of expressing the theory as a sum over surfaces has been developed by Baez [ 26 ], who has studied the general form of generally covariant quantum field theories formulated in this manner, and by Smolin and Markopoulou [ ], who have studied how to directly capture the Lorentzian causal structure of general relativity modifying the elementary vertices. They have also explored the idea that the long range correlations of the low energy regime of the theory are related to the existence of a phase transition in the microscopic dynamics, and have found intriguing connections with the theoretical description of percolation.

In section 6, I have sketched the basic structure of loop quantum gravity. This structure has been developed in a number of directions, and has been used to derive a number of results. Without any ambition of completeness, I list below some of these developments. Solutions of the hamiltonian constraints. One of the most surprising results of the theory is that it has been possible to find exact solutions of the hamiltonian constraint.

This follows from the key result that the action of the hamiltonian constraints is non vanishing only over nodes of the s-knots [ , ]. Therefore s-knots without nodes are physical states that solve the quantum Einstein dynamics. There is an infinite number of independent states of this sort, classified by conventional knot theory.

The physical interpretation of these solutions is still rather obscure. Various other solutions have been found.

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See the recent review [ 82 ] and references therein. In particular, Pullin has studied in detail solutions related to the Chern-Simon term in the connection representation and to the Jones polynomial in the loop representation. According to a celebrated result by Witten [ ], the two are the loop transform of each other. Time evolution. Strong field perturbation expansion. Trying to describe the temporal evolution of the quantum gravitational field by solving the hamiltonian constraint yields the conceptually well-defined [ ], but notoriously non-transparent, frozen-time formalism. This approach has lead to the tentative definition of a physical hamiltonian [ , 49 ], and to a preliminary investigation of the possibility of transition amplitudes between s-knot states, order by order in a strong coupling perturbative expansion [ ].

Fermions have been added to the theory [ , , 36 , ]. Maxwell and Yang-Mills. The extension of the theory to the Maxwell field has been studied in [ , 91 ]. The extension to Yang-Mills theory has been explored recently in [ ]. In [ ], Thiemann shows that the Yang-Mills term in the quantum hamiltonian constraint can be defined in a rigorous manner, extending the methods of [ , , ]. A remarkable result in this context is that ultraviolet divergences do not seem to appear, strongly supporting the expectation that the natural cut off introduced by quantum gravity might cure the ultraviolet difficulties of conventional quantum field theory.

Application to other theories. Lattice and simplicial models. A number of interesting discretized versions of the theory are being studied. See in particular [ , , 94 , 79 ]. Planck scale discreteness of space. The most remarkable physical result obtained from loop quantum gravity is, in my opinion, evidence for a physical quantum discreteness of space at the Planck scale.

This is manifested in the fact that certain operators corresponding to the measurement of geometrical quantities, in particular area and volume, have discrete spectrum. According to the standard interpretation of quantum mechanics which we adopt , this means that the theory predicts that a physical measurement of an area or a volume will necessarily yield quantized results. Since the smallest eigenvalues are of Planck scale, this implies that there is no way of observing areas or volumes smaller than Planck scale. The spectra of the area and volume operators have been computed with much detail in loop quantum gravity.

These spectra have a complicated structure, and they constitute detailed quantitative physical predictions of loop quantum gravity on Planck scale physics. If we had experimental access to Planck scale physics, they would allow the theory to be empirically tested in great detail. A few comments are in order. Later, the result has been recovered by alternative techniques and extended by a number of authors. In particular, Ashtekar and Lewandowski [ 18 ] have repeated the derivation, using the connection representation, and have completed the computation of the spectrum adding the sector which was not computed in [ ].

The Ashtekar-Lewandowski component of the spectrum has then been rederived in the loop representation by Frittelli Lehner and Rovelli in [ 84 ]. Loll has employed lattice techniques to point out a numerical error in [ ] corrected in the Erratum in the eigenvalues of the volume.

The analysis of the volume eigenvalues has been performed in [ 77 ], where general techniques for performing these calculations are described in detail. The spectrum of the volume has then been analyzed also in [ ]. There are also a few papers that have anticipated the main result presented in [ ]. In particular, Ashtekar Rovelli and Smolin have argued for a physical discreteness of space emerging from the loop representation in [ 24 ], where some of the eigenvalues of the area already appear, although in implicit form.

The first explicit claim that area eigenvalues might in principle be observable in the presence of matter is by Rovelli in [ ]. The reason is that the length operator is difficult to define and of more difficult physical interpretations. For attempts in this direction, see [ ].

Whether this is simply a technical difficulty, or it reflects some deep fact, is not clear to me. Therefore, we cannot directly interpret them as representing physical measurements. Realistic physical measurements of areas and volumes always refer to physical surfaces and spatial regions, namely surfaces and spatial regions determined by some physical object. For instance, I can measure the area of the surface of a certain table.

In the dynamical theory that describes the gravitational field as well as the table, the area of the surface of the table is a diffeomorphism invariant quantity A, which depends on gravitational as well as matter variables. In the quantum theory, A will be represented by a diffeomorphism invariant operator. Now, as first realized in [ ], it is plausible that the operator A is, mathematically, the same operator as the pure gravity area operator.

The discreteness of area and volume is derived as follows. The result can easily be worked out by writing the standard formula for the area of a surface, and replacing the metric with the appropriate function of the loop variables. The actual construction of this operator requires regularizing the classical expression and then taking the limit of a sequence of operators, in a suitable operator topology. An alternative regularization technique is discussed in [ 18 ]. This result shows that the spin network states with a finite number of intersection points with the surface and no nodes on the surface are eigen-states of the area operator.

For the full spectrum, see [ 18 ] connection representation and [ 84 ] loop representation.

Mathematical Foundations Of Quantum Field And Perturbative String Theory Pdf

A similar result can be obtained for the volume [ , , , 77 , ]. Let us restrict ourselves here, for simplicity, to spin networks S with non-degenerate four-valent nodes, labeled by an index i. Let a i , b i , c i , d i be the colors of the links adjacent to the i - th node and let J i label the basis in the intertwiner space. See [ 52 ]. The volume eigenvalues v i are obtained by diagonalizing these matrices.

For more details, and the full derivation of these formulas, see [ 52 , ]. Classical limit.

Quantum states representing flat spacetime. Discrete small scale structure of space. A natural problem is then how flat space or any other smooth geometry might emerge from the theory. Notice that in a general rela-tivistic context the Minkowski solution does not have all the properties of the conventional field theoretical vacuum.

In gravitational physics there is no real equivalent of the conventional vacuum, particularly in the spatially compact case. One then expects that flat space is represented by some highly excited state in the theory. States in that describe flat space when probed at low energy large distance have been studied in [ 24 , , 50 , 99 ]. These have a discrete structure at the Planck scale. The Bekenstein-Mukhanov effect. Bekenstein and Mukhanov observe that in most approaches to quantum gravity the area can take only quantized values [ ].

Since the area of the black hole surface is connected to the black hole mass, black hole mass is likely to be quantized as well. The mass of the black hole decreases when radiation is emitted. Therefore emission happens when the black hole makes a quantum leap from one quantized value of the mass energy to a lower quantized value, very much as atoms do.

A consequence of this picture is that radiation is emitted at quantized frequencies, corresponding to the differences between energy levels. Thus, quantum gravity implies a discretized emission spectrum for the black hole radiation. I will denote this effect as the kinematical Bekenstein-Mukhanov effect. Unfortunately, however, the kinematical Bekenstein-Mukhanov effect disappears if we replace the naive ansatz with the spectrum 41 computed from loop quantum gravity. This is due to the details of the spectrum 41 of the area.

A detailed discussion of this result is in [44], but the result was already contained implicitly, in the first version in [18]. It is important to notice that the density of the eigenvalues shows only that the simple kinematical argument of Bekenstein and Mukhanov is not valid in this theory, and not that their conclusions is necessarily wrong.

As emphasized by Mukhanov, a discretization of the emitted spectrum could still be originated dynamically. A physical understanding and a first principles derivation of this relation require quantum gravity, and therefore represent a challenge for every candidate theory of quantum theory. A derivation of the Bekenstein-Hawking expression 46 for the entropy of a Schwarzschild black hole of surface area A via a statistical mechanical computation, using loop quantum gravity, was obtained in [ , , ].

This derivation is based on the ideas that the entropy of the hole originates from the microstates of the horizon that correspond to a given macroscopic configuration [ , 65 , 64 , 38 , 39 ]. Physical arguments indicate that the entropy of such a system is determined by an ensemble of configurations of the horizon with fixed area [ ].

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In the quantum theory these states are finite in number, and can be counted [ , ]. Counting these microstates using loop quantum gravity yields. An alternative derivation of this result has been announced from Ashtekar, Baez, Corichi and Krasnov [ 12 ]. Thus, the theory is compatible with the numerical constant in the Bekenstein-Hawking formula, but does not lead to it univocally.

The precise significance of this fact is under discussion. On the issue of entropy in loop gravity, see also [ ]. Hamiltonian constraint. The kinematics of the theory is well understood, both physically quanta of area and volume, polymer-like geometry and from the mathematical point of view , s-knot states, area and volume operators.

The part of the theory which is not yet fully under control is the dynamics, which is determined by the hamiltonian constraint. A plausible candidate for the quantum hamiltonian constraint is the operator introduced by Thiemann [ , , ]. The commutators of the Thiemann operator with itself and with the diffeomorphism constraints close, and therefore the operator defines a complete and consistent quantum theory. However, doubts have been raised on the physical correctness of this theory, and some variants of the operator have been considered.

The doubts originate from various considerations. First, Lewandowski, Marolf and others have stressed the fact the quantum constraint algebra closes, but it is not isomorphic to the classical constraint algebra of GR [ ]. Recently, a detailed analysis of this problem has been completed by Marolf, Lewandowski, Gambini and Pullin [ 90 ]. The failure to reproduce the classical constraint algebra has been disputed, and is not necessarily a problem, since the only strict requirement on the quantum theory, besides consistency, is that its gauge invariant physical predictions match the ones of classical general relativity in the appropriate limit.

Still, the difference in the algebras may be seen as circumstantial evidence not a proof for the failure of the classical limit. The issue is technically delicate and still controversial. I hope I will be able to say something more definitive in the next update of this review. Finally, by translating the Thiemann operator into a spacetime covariant four-dimensional formalism, Reisenberger and Rovelli have noticed a suspicious lack of manifest 4-d covariance in the action of the operator [ ], a fact pointing again to the possibility of anomalies in the quantum constraint algebra.

There are two versions of the volume operator in the literature: V RL , introduced in [ ] and V AL , introduced in [ 15 , 17 , 16 ]. See [ ] for a detailed comparison. Originally, Thiemann thought that using V RS in the hamiltonian constraint would yield difficulties, but it later became clear that this is not the case [ ]. Both versions of the volume can be used in the definition, yielding two alternative versions of the hamiltonian [ ]. Next, in its simplest version the operator is non-symmetric.

Since the classical hamiltonian constraint is real on SU 2 gauge invariant states , one might expect a corresponding self-adjoint quantum operator. Next, Smolin has considered some ad hoc modifications of the constraint in [ ]. This covariantisation amounts to adding to the vertex described in Figure 3 the vertices, described in Figure 5 , which are simply obtained by re-orienting Figure 3 in spacetime. Ultimately, the final tests of any proposal for the hamiltonian constraint operator must be consistency and a correct classical limit.

Thus, the solution of the hamiltonian constraint puzzle is likely to be subordinate to the solution of the problem of extracting the classical limit of the dynamics from the theory. The basics of the description of matter in the loop formalism have been established in [ , , , 31 , , ]. Work needs to be done in order to develop a full description of the basic matter couplings. In particular, there are strong recurring indications that the Planck scale discreteness naturally cuts the traditional quantum field infinities off. In particular, in [ ], Thiemann argues that the Hamiltonian constraint governing the coordinate time evolution of the Yang-Mills field is a well defined operator I recall that, due to the ultraviolet divergences, no rigorously well defined hamiltonian operator for conventional Yang-Mills theory is known in 4 dimensions.

If these indications are confirmed, the result would be very remarkable. What is still missing are calculational techniques that could allow us to connect the well-defined constraint with finite observables quantities such as scattering amplitudes. Spacetime formalism. In my view, the development of continuous spacetime formalisms, [ , , 26 , ], is one of the most promising areas of development of the theory, because it might be the key for addressing most of the open problems. First, a spacetime formalism frees us from the obscurities of the frozen time formalism, and allows an intuitive, Feynman-style, description of the dynamics of quantum spacetime.

I think that the classical limit, the quantum description of black holes, or graviton-graviton scattering, just to mention a few examples, could be addressed much more easily in the covariant picture. Second, it allows the general ideas of Hartle [ ] and Isham [ , , , ] on the interpretation of generally covariant quantum theories to be applied in loop quantum gravity.

This could drastically simplify the complications of the canonical way of dealing with general covariant observables [ , ]. Third, the spacetime formalism should suggest solutions to the problem of selecting the correct hamiltonian constraint: it is usually easier to deal with invariances in the Lagrangian rather than in the hamiltonian formalisms. The spacetime formalism is just born, and much has to be done. See the original papers for suggestions and open problems. Black holes. The derivation of the Bekenstein-Hawking entropy formula is a major success of loop quantum gravity, but much remains to be understood.

A clean derivation from the full quantum theory is not yet available. Such a derivation would require us to understand what, precisely, the event horizon in the quantum theory is. To do so, we should understand how to effectively deal with the quantum dynamics, how to describe the classical limit in order to find the quantum states corresponding to classical black hole solutions , as well as how to describe asymptotically flat quantum states.

Besides these formal issues, at the roots of the black hole entropy puzzle there is a basic physical problem, which, to my understanding, is still open. The problem is to understand how we can use basic thermodynamical and statistical ideas and techniques in a general covariant context. Energy is an extremely slippery notion in gravity.

Thus, how do we define the statistical ensemble? Put in other words: To compute the entropy say in the microcanonical of a normal system, we count the states with a given energy. In GR we should count the states with a given what? One may say: black hole states with a given area. But why so? We do understand why the number of states with given energy governs the thermodynamical behavior of normal systems. But why should the number of states with given area govern the thermodynamical behavior of the system, namely govern its heat exchanges with the exterior?

A tentative physical discussion of this last point can be found in [ ].

Mathematical Foundations Of Quantum Field And Perturbative String Theory Pdf

How to extract physics from the theory. Assume we pick a specific hamiltonian constraint. Then we have, in principle, a well defined quantum theory. How do we extract physical information from it? Some physical consequences of the theory, such as the area and volume eigenvalues, or the entropy formula, have been extracted from the theory by various ad hoc methods.

But is there a general technique, say corresponding to the traditional QFT perturbation expansion of the S matrix, for describing the full dynamics of the gravitational field? Presumably, such general techniques should involve some kind of expansion, since we could not hope to solve the theory exactly.

Attempts to define physical expansions have been initiated in [ ] and, in different form, in [ ]. Ideally, one would want a general scheme for computing transition amplitudes in some expansion parameter around some state. Computing scattering amplitudes would be of particular interest, in order to make connection with particle physics language and to compare the theory with string predictions. Finally, to prove that loop quantum gravity is a valuable candidate for describing quantum spacetime, we need to prove that its classical limit is GR or at least overlaps GR in the regime where GR is well tested.

However, the weaves studied so far [ 24 , 99 ] are 3d weaves, in the sense that they are eigenstates of the three dimensional metric. Such a state corresponds to an eigenstate of the position for a particle. Classical behavior is recovered not by these states but rather by wave packets which have small spread in position as well as in momentum. Similarly, the quantum Minkowski spacetime should have small spread in the three metric as well as in its momentum — as the quantum electromagnetic vacuum has small quantum spread in the electric and magnetic field.

To recover classical GR from loop quantum gravity, we must understand such states.

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Preliminary investigation in this direction can be found in [ , ], but these papers are now several years old, and they were written before the more recent solidification of the basics of the theory. Another direction consists in the direct study of coherent states in the state space of the theory. As these brief notes indicate, the various open problems in loop quantum gravity are interconnected. In a sense, loop quantum gravity grew aiming at the nonperturbative regime, and the physical results obtained so far are in this regime.

The main issue is then to recover the long distance behavior of the theory. That is, to study its classical limit and the dynamics of the low energy excitations over a semiclassical background. These are especially relevant in applications where usually the data and the associated scientific questions, and not a single method class is in the focus of interest current project and collaboration domains include the medical sciences, sports and prevention, geoscience, physics and finance.

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