Problem of Quantum Gravity

Transcription

1 New Mexico Tech (January 11, 2012) DRAFT Problem of Quantum Gravity Ivan G. Avramidi Department of Mathematics New Mexico Institute of Mining and Technology Socorro, NM 87801, USA These are some notes on the foundational problems of quantum gravity.

2 1 1 Problems of the Fundamental Physics Four of the most important problems of the fundamental physics, which are probably related to each other, are [1]: 1. Quantization of gravity: find a way to combine the basic principles of general relativity with the basic principles of quantum mechanics. 2. Foundations of quantum mechanics: develop a consistent interpretation of quantum mechanics. 3. Unified field theory: combine all known interactions and particles in a single theory. 4. Cosmology: explain the origin of the dark matter and the dark energy. 2 Classical Mechanics Any physical theory consists of two parts: kinematics and dynamics. Kinematics describes the states of a system and dynamics predicts the evolution of the state in time, so given an initial state at an initial time we should be able to predict the state of the system at any time in the future. This is exactly what happens in classical mechanics. The configuration of a mechanical system can be described by a finite number of independent parameters called the coordinates. Each such coordinate is called a degree of freedom and the total number of them is called the number of degrees of freedom. That is why, a classical mechanical system is a system with finitely many degrees of freedom. However, the knowledge of all coordinates at an initial time is not enough to predict the dynamical evolution of the system since the prescribing the coordinates only does not describe the state of the system. Two systems could have the same coordinates but be in completely different states since one system could be at rest and the other could be moving. That is why, a complete description of the state of a classical mechanical system requires prescribing not only the coordinates but also the velocities (or momenta). Then, given the coordinates and momenta at an initial time uniquely determines the evolution of the system in the future. To develop the mathematical formalism one introduces the set of all states of a system called the phase space, which is a finite dimensional symplectic manifold. Then the state if a point in the phase space and the dynamics is a curve in the qg.tex; November 20, 2012; 16:49; p. 1

3 2 phase space, which is governed by the Hamiltonian equations (a system of first order ordinary differential equations) dx i dt = H p i, dp i dt = H x i, (2.1) where H(x, p) is the total energy of the system called the Hamiltonian. The equations of classical mechanics are invariant under the Galilean transformations, which include translation of time, translation of space, rotation of space and the uniform motion of space (the Galilean boost). Let us stress one very important point here. Suppose that our system consists of two subsystems. If these subsystems interact then, strictly speaking, the whole system should be treated as a whole. In other words the equations of motion above do not decouple. However, if the interaction of these subsystems becomes negligible, let say, they move apart at large distance, then the equations of motion decouple and we have two subsystems whose dynamics is completely independent. Then the knowledge of the evolution of each subsystem determines the dynamics of the whole system. 3 Classical Field Theory Besides classical mechanical systems the classical physics also deals with the classical electromagnetic fields. A field can be viewed as a mechanical system with infinitely many degrees of freedom, one for each point in space. That is why, the state of a classical field is described by specifying the field and its time derivative at each point in space. Thus, the phase space is now the set of all, let say, smooth functions on space and the dynamical evolution is a curve in this space governed by the Maxwell equations µ F µν = J ν, (3.2) where F µν = µ A ν ν A µ and J ν is the electric current. Here the indices are the space-time indices that range over 0, 1, 2, 3 with x 0 denoting the time coordinate, more precisely, ct where c is the speed of light. These equations are experimentally verified with a very good precision and there is no doubt that they are valid in all macroscopic phenomena including electromagnetism, in particular, light propagation. The problem is that they are not invariant under the Galilean transformations of classical mechanics but rather under the Lorentz transformations, which include translation of time, translation of space, rotation of space and the pseudo-rotation of space-time (Lorentz boost). qg.tex; November 20, 2012; 16:49; p. 2

4 3 4 Special Relativity Einstein analyzed the physics of the classical mechanics and showed that classical mechanics is inconsistent with the properties of the light propagation. He modified the laws of the classical mechanics and formulated a new theory called Special Relativity. In this theory all physical laws are the same in all inertial reference frames, that is, all equations of physics are invariant under the Lorentz transformation of the space-time coordinates (ct, x, y, z). It is an experimental fact that light propagates with the same speed c in all inertial reference frames. A reference frame is a collection of clocks in space located sufficiently dense in the whole space, say on a lattice. One of the major problems is the problem of synchronization of clocks in a reference frame. To do this one needs to start the clock at the origin and somehow to submit this information to all other clocks. The fastest way to submit such information is by sending a light signal. Of course, it will take some time for the light to reach distant clocks. That is why the whole notion of simultaneity becomes relative. That is why the theory is called relativity theory. Thus in order to construct reference frames in space-time one needs the light to synchronize the clocks in space. The propagation of light in free space is described by the wave equation ( ϕ = 1 ) c 2 2 t + ϕ = 0, (4.3) where ϕ is any of the components of the electromagnetic field and is called the D Alambert operator. This operator can be written in more compact form as = η µν µ ν, (4.4) where (x µ ) = (ct, x i ) are the coordinates of space-time and (η µν ) = diag ( 1, 1, 1, 1) is the so-called Minkowski metric of space-time. Therefore, the pseudo-euclidean metric of special relativity is naturally built-in in the wave equation for the light propagation. By studying the properties of wave propagation one can reconstruct all concepts of special relativity, including causality, light cones, The parameter that measures the tendency of an object to stay at rest is called the mass. According to special relativity when the speed of an object increases then the inertia increases and becomes infinite when the speed reaches the speed of light. That is why, nothing can travel faster than light and the new relativistic effects become important for objects moving close to the speed of light. The only such objects are elementary particles. If the mass of an elementary particle qg.tex; November 20, 2012; 16:49; p. 3

5 4 is m then it is relativistic if its speed is large enough so that its kinetic energy is comparable to mc 2. Of course, for massless particles this means that they are always relativistic, so they always travel with the speed of light. Beside the photon, there are some other massless particles like neutrino (according to recent data it may have a small mass), graviton (not discovered yet) and others. There was a controversy recently about new experimental data that neutrino could propagate faster than light, but now almost everybody agrees that that was ax experimental error. If that were true, then the whole body of modern physics should have been reevaluated and modified somehow. Mathematically, one could say that special relativity is a deformation of classical mechanics, which is recovered in the limit c. Note that classical Maxwell field theory is already relativistic. 5 Quantum Mechanics It is an experimental fact that microscopic objects exhibit wave phenomena (like diffraction and interference) that cannot be explained by the classical mechanics. One of the most important features of a quantum system is that it cannot be reduced to its parts but should be treated as a whole; one says, that different parts of a quantum system are entangled. This is a very new feature that is absent in classical mechanics. If a quantum system consists of two subsystems then even if their interaction vanishes (let say, they move apart at large distance) they are still not independent. The dynamics of the whole system is not determined by the dynamics of its (even not interacting) subsystems. Of course, one could say that this means that these subsystems do interact, but the interaction is not the usual local one but rather some kind of a non-local interaction. This is what is meant when one says that the subsystems are entangled. This leads to a number of surprising apparent paradoxes, like the Schrödinger cat paradox, which lead some authors to conclude that quantum mechanics is an incomplete theory with some deep foundational problems which need to be resolved before it can be united with gravity. The state of a quantum system is described by specifying a complex-valued function on the physical space called the wave function. The wave function is determined only up to a constant normalizing factor which can be chosen so that it has a unit norm, and, additionally, up to a constant phase factor. Given an initial state, that is, an initial wave function, at an initial time quantum mechanics predicts the wave function at all future times. qg.tex; November 20, 2012; 16:49; p. 4

6 5 The standard approach to quantum mechanics (called the Copenhagen interpretation) is somewhat minimalistic. It is assumed that the only way to know the state of a quantum system is by providing a set of measurements. A measurement is an interaction of a quantum system with a classical system governed by the classical mechanics. That is why, quantum mechanics requires classical mechanics for its logical formulation. The main problem of quantum mechanics is therefore reduced to predicting the outcomes of measurements. Another very important feature of quantum mechanics is that it cannot predict the outcome of a measurement with complete certainty; the most it can do is to predict the probabilities of various outcomes of a measurement. There two main physical principles in quantum mechanics: the superposition principle and the uncertainty principle. The uncertainty principle says that in general two physical observables cannot have specific values in a given state, there is a limit at which they can be determined. The product of uncertainties in these two physical variables is bounded from above, that is, if one approaches zero then the other must go to infinity. Suppose that we have two initial states ψ 1 (0) and ψ 2 (0) of a system. Then quantum mechanics determines the evolution of each of these states with time, say ψ 1 (t) and ψ 2 (t). The superposition principle says that the time evolution of the initial state ψ(0) = a 1 ψ 1 (0) + a 2 ψ(0) (where a 1 and a 2 are complex numbers such that a a 2 2 = 1) is determined by the same linear combination ψ(t) = a 1 ψ 1 (t) + a 2 ψ(t). Moreover, if the measurement of a physical observable A in the state ψ 1 always returns the value A 1 and the measurement of A in the state ψ 2 always returns the value A 2, then the measurement of A in the state ψ(t) = a 1 ψ 1 (t) + a 2 ψ(t) can only give either A 1 and A 2 with probabilities a 1 2 and a 2 2. The mathematical formalism of quantum mechanics is provided by the theory of self-adjoint operators in a Hilbert space. The states of a quantum system (the wave function) are described by unit vectors in a Hilbert space. The physical observables are described by self-adjoint operators in the Hilbert space. The possible values of a physical observable A are given by the eigenvalues A n of the operator A. The probability that a measurement of a physical quantity in a state ψ returns A n is determined by P n = (ψ n, ψ) 2, where ψ n are the eigenfunctions of the operator A corresponding to the eigenvalue A n. The average value of the observable A in a state ψ is given by the expectation value A = (ψ, Aψ) = n P n A n. The dynamical evolution of a state of a quantum system is described by the Schrödinger equation (i t H) Ψ = 0, (5.5) qg.tex; November 20, 2012; 16:49; p. 5

7 6 or Ψ(t) = exp ( i ) th Ψ(0), (5.6) where H is a self-adjoint operator in the Hilbert space called the Hamiltonian; in the coordinate representation it is a second-order partial differential operator, and is the Planck constant. One could say that quantum mechanics is a deformation of classical mechanics, which is recovered in the limit 0. The wave function of a classical system is determined by its classical action S (evaluated along the classical trajectory) ( i ) Ψ classical = a exp S. (5.7) 6 Quantum Field Theory Of course, quantum mechanics is not a relativistic theory. It does not apply to physical phenomena involving relativistic quantum objects like elementary particles. The most important property of elementary particles is that the number particles is not conserved; the particles are being created and annihilated all the time. This is what one usually sees in the experiments. There are certain ways to register an elementary particle and to measure its mass, its spin, its charge, its momentum and other characteristics. One collides say a beam of electrons with a beam of positrons (or something else) coming from large distances at certain angles. The states of the particles in both beams are measured in advance and known, like their momentum, the polarization, etc. The interaction occurs in a compact space region in a very small amount of time. Then one registers the products of this collision going in all possible directions with all possible momenta. What one sees is not just the two particles that collided (such a collision is called an elastic scattering; it happens usually at low energies) but a lot of new particles that were created during the collision. The original particles may have disappeared totally. One could get a pretty good visual picture of what is going on by drawing so called Feynman diagrams. Quite often the results of the collision further decay in some other particles which are detected. So, if one knows in what it could decay one can judge of the existence of some new particles. This is how usually the discoveries of new elementary particles are made. The mathematical theory of relativistic quantum mechanics is called Quantum Field Theory. The most common problems in quantum theory are scattering problems. One prepares particles in a certain specified state and collides them either qg.tex; November 20, 2012; 16:49; p. 6

8 7 with each other or with a target and then observes the outcome, that is, registers the states of the outgoing particles. If ψ in is the initial state and ψ out is the final state then (ψ in, ψ out ) is called the amplitude of this process, its square (ψ in, ψ out ) 2 gives the probability of the initial state ψ in becoming the final state ψ out. The set of all such amplitudes is called the scattering matrix. The primary goal of quantum field theory is the calculation of the scattering matrix, that is, such amplitudes. The quantum field theories that describe the interaction of elementary particles are non-linear, and, therefore, cannot be solved exactly. The only reasonable way to carry out calculations is the perturbation theory. In this approach one splits the field in a free non-interacting part and small quantum perturbations that interact with each other. Then one expands in powers of the perturbation and gets an infinite series in powers of a parameter called a coupling constant that describes the strength of the interaction. Such a series can be represented graphically by the famous Feynmann diagrams. It turns out that roughly speaking the order of the perturbation theory is related to the number of loops in Feynman diagrams. The tree diagrams are purely classical, they do not take into account the quantum nature of the elementary particles. All loop diagrams represent quantum corrections to the classical processes. And then one realizes that there are two main problems with such an expansion. First, the whole series is only an asymptotic series and it diverges. Second each term in this series in all orders of the perturbation theory, if computed formally, also diverges. The reason for this divergences, called the ultra-violet divergences, is the local nature of the interaction. For example, let us consider a self-interacting neutral scalar field of mass m in flat space with the action S = dx { 12 } gµν µ ϕ(x) ν ϕ(x) m2 R 2 ϕ2 (x) λϕ 4 (x) (6.8) 4 Here x = (t, x) are the points in space-time, dx = dtdx, g µν = diag ( 1, 1, 1, 1) is the Minkowski metric aand λ is a coupling constant that describes the strength of the self-interaction. One should mention that in quantum field theory one usually chooses natural units in which the speed of light and the Planck constant are both equal to one, that is, = c = 1. The propagation of the scalar particles is described by so called Feynmann propagator G(x, y) which is the Green function of the Klein-Gordon operator + m 2 = 2 t + m 2 (6.9) qg.tex; November 20, 2012; 16:49; p. 7

9 8 with the radiation conditions at t and t +. It has the form G(x, x ) = D(x x dq ) = eiq (x x ) 1 (6.10) (2π) 4 q 2 + m 2 where m is a constant parameter called the mass, dq = dq 0 dq, the dot product is defined as usual q (x x ) = q 0 (t t ) + q (x x ) and q 2 = q q = q q2. This integral should be computed as follows. One rotates the contour of integration over q 0 from the real axis to the imaginary axis so that q 2 becomes the Euclidean square. Then one can use to get G(x, x ) = 1 q 2 + m 2 = 0 R 4 0 dse sm2 ds e s(q2 +m 2 ) R 4 dq (2π) 4 e sq2 +iq (x x ) (6.11) (6.12) This integral is Gaussian and gives a very nice integral representation of the Feynmann propagator G(x, x ) = D(x x ) = ds(4πs) 2 exp { sm 2 (x } x ) 2. (6.13) 4s 0 It can be computed in terms of Bessel functions but it is much better to live it in this form. It is important to recall that the Feynmann propagator is a specific Green function of the wave equation and is, therefore, a distribution. All the problems of quantum field theory come from the attempt to multiply such distributions which is not well defined and leads to all kind of divergences at small distances. One of the simplest examples of one-loop Feynman diagrams is the so-called self-energy integral ( ) ( ) x dxe ipx + x G, x x x + x G, x x = dx e ipx D(x)D(x) (6.14) R R 4 or, which is the same thing, dq (2π) 4 1 [ q2 + m 2] [ (p q) 2 + m2], (6.15) R 4 It is not difficult to see that these integrals diverge as q or x 0. The reason is that the integrand decreases as q as q 4 and the volume of integration qg.tex; November 20, 2012; 16:49; p. 8

10 9 grows like q 4. Therefore these integrals diverge logarithmically. This divergence is called the ultraviolet divergence since it corresponds to large momenta. It is local because it diverges as the distance x 0. A reasonable thing to do is to regularize the integrals representing the Feynmann diagrams to make them finite and then to take off the regularization at the end of the calculations. For example one could just cut off the integrals at some large momenta (or small distances) or by introducing a smooth cutoff function. It turns out that in some important cases of quantum field theories (quantum electrodynamics, Yang-Mills theory, Dirac theory, Higgs theory, etc) only finitely many types of singularities occur, and, therefore, it is possible to isolate all singularities and then absorb them in the redefinition of the classical parameters of the fields (like mass, coupling constant, the field itself etc.). This means that one classifies all Feynmann diagrams according to their potential for the divergences. The high-order diagrams will usually converge but all of them have some diverging subdiagrams. The problem is complicated even further by the overlapping of the divergences from one subdiagram with another. All this can be done and has been done. It is not easy. People like Feynmann and Schwinger got Nobel prizes for doing that in quantum electrodynamics and t Hooft got another Nobel prize for doing the same thing in Yang-Mills theory. This is done as follows. In the example above one assume that the mass m, the coupling constant λ and the field redefinition constant Z (after doing the rescaling ϕ Zϕ) depend on the regularization parameters in a singular way such they cancel precisely all divergences in all Feynmann diagrams. The parameters m, λ, Z are not physically measurable, that is why one can do that. What are measurable parameters are the renormalized values m ren and λ ren. If this is possible, then the theory is called renormalizable. In such theories one can compute all quantities of interest and compare them with the experiments, which makes them consistent quantum field theories. The meaning of the renormalizability is that the physics at low energies does not depend on the unknown physics at high energies. All other theories are called non-renormalizable. In such theories the number of different types of divergences is infinite and it is impossible to get rid of them all by redefining only finitely many parameters of the classical fields. Unfortunately, this is a generic case, and General Relativity is a perfect example of a non-renormalizable theory. In such theories the details of the physics at high energies impact the physical phenomena at low energies. Now, one could think of quantum field theory as a deformation of the quantum mechanics by relativizing it or the deformation of special relativity by quantizing qg.tex; November 20, 2012; 16:49; p. 9

11 10 it. In either way, in the limits 0 and c one should recover the classical mechanics. 7 Classical Gravity Classical Newtonian Gravity is a theory of gravitational interaction of massive objects. In the classical theory the gravitational phenomena are described by a scalar gravitational field. Every massive body creates a gravitational field whose gradient determines the force exerted by the gravitational field on another massive body. This interaction is instantaneous, in other words, it propagates with infinite speed. A distribution of matter with the mass density ρ creates the gravitational field ϕ which is determined from the Poission equation ϕ = 4πGρ, (7.16) where G is the Newton s gravitational constant. Then the gravitational force exerted on a body of mass m is equal to F = m ϕ. (7.17) 8 General Relativity Einstein showed that the classical theory of gravitation contradicts special relativity and found a way to modify the gravity theory so that it becomes relativistic. One of the most important physical principles of general relativity is the Equivalence Principle that asserts that locally the effects of the gravitational field is equivalent to the acceleration of the reference frame. That is why, he required that the equation of physics should be invariant not only under linear Lorentz transformations but also under general coordinate transformations. This lead him to introduce the pseudo-riemannian metric and then the connection and the curvature of the space-time. Einstein applied the analysis of simultaneity carried out in special relativity to the case when light propagates not in a free space but in a space with a gravitational field. The wave equation for light propagation in presence of a gravitational field should be described by a more general wave equation = g µν (x) µ ν + b µ (x) µ + h(x), (8.18) qg.tex; November 20, 2012; 16:49; p. 10

12 11 where the coefficients with the highest derivative form a space and time dependent symmetric matrix with one negative eigenvalue and three positive eigenvalues. The eigenvector corresponding to the negative eigenvalue determines the time direction. Such a matrix is nothing but the pseudo-riemannian metric of space-time. Thus, by studying the propagation of light in space one can reconstruct all concepts of general relativity, including the light cone, the geodesics, the connection the curvature etc. Therefore, according to Einstein gravity is described not by a single scalar field but by a metric, which is a symmetric 2-tensor and all gravitational phenomena are the manifestations of the curvature of the space-time. In particular, there is no instantaneous gravitational interaction; instead, all massive bodies (as well as light) move along the geodesics of the curved space-time. The Einstein field equations are R µν 1 2 g µνr = 8πGT µν, (8.19) where T µν is the energy-momentum tensor. The equations of motion (the geodesics) are d 2 x µ dτ + dx α dx β 2 Γµ αβ dτ dτ = 0, (8.20) where τ is an affine parameter. One could think of general relativity as a deformation of the special relativity such that the special relativity is recovered as G 0 (no gravity). In the limit as c general relativity turns to a non-relativistic classical gravity theory. 9 Quantum Gravity General relativity is constructed by using the following fundamental objects and concepts: 1. events, 2. epacetime, 3. topology of spacetime, 4. manifold structure of spacetime, 5. smooth differentiable structure of spacetime, qg.tex; November 20, 2012; 16:49; p. 11

13 12 6. diffeomorphism group invariance, 7. causal structure (global hyperbolicity), 8. dimension of spacetime, 9. (pseudo)-riemannian metric with the signature ( + +), 10. canonical connections on spin-tensor bundles over the spacetime. Every attempt to quantize general relativity immediately encounters many problems. Here are some of them: 1. what is the space-time? 2. how relevant are the space-time concepts of general relativity? 3. what exactly should be quantized (metric, connection, topology)? 4. should one consider connection an independent quantum object? 5. is it possible to use the standard interpretation of quantum mechanics? 6. is it possible to quantize gravity separately from all other interactions? 7. how much of the space-time structure should remain fixed: topology, smooth structure, causal structure, etc? 8. are the continuum concepts of standard theory valid at ultra-small scales? 9. is space-time discrete? 10. does the continuum structure appear only in coarse-grained sense? 11. does the topology change? 12. what exactly is quantum (does fluctuate)? 13. what is the dimension of the space-time? 14. can the signature of the metric change? qg.tex; November 20, 2012; 16:49; p. 12

14 13 There are many approaches to quantum gravity based on how they answer the above questions. The very first one is the attempt to quantize general relativity in the same way as an ordinary quantum field theory. We fix the topology and decompose the metric in the Minkowski metric and a fluctuation, which describes the propagation of gravitons, massless particles of spin 2. This is the minimalistic approach since we do not change anything else, including the interpretation of quantum mechanics. The main problem with this approach is that general relativity is non-renormalizable and, therefore, this approach in its simplest form just does not work. One can modify the theory by including some other fields and some extra dimensions and require very specific interactions, eventually, this approach led to the string theory. This is the approach of high-energy physicists to the gravitational problem. One of the reasons for non-renormalizability of general relativity is its nonpolynomial behavior. There are infinitely many types of interaction of gravitons with each other. However, this is not the main one. The main reason is the dimension of its coupling constant (which is G, the Newton constant), [G] = 1 M 2 (9.21) where M is the unit of energy (or mass). This means that the expansion in powers of G produces more and more divergent diagrams. One of the attempts to cure this problem was to add terms quadratic in curvature to the Einstein action functional. The action of such theory has the form S = M { 1 dxg 1/2 1 (R 2Λ) 16πG 2a 2C µναβc µναβ + 1 } 6b 2 R2, (9.22) where Λ is the cosmological constant, C µναβ is the Weyl tensor, a is the tensor coupling constant and b is the conformal coupling constant. At the linearized level this theory describes the graviton (massless tensor field of spin 2), as well as two new particles: massive tensor field of spin 2 with mass am p and a massive scalar field of spin 0 and mass bm p, where M p = (16πG) 1/2 is the Planck mass. Note that here we use the natural units in which = c = 1. With these constants inserted the Planck mass is M p = (16πG/( c)) 1/2. This theory is called the R + R 2 theory. Its field equations are differential equations of fourth order. That is why, it is also called the quantum gravity with higher derivatives. Exactly because of this reason the propagators in momentum representation decrease at infinity not as p 2 but as p 4, fast enough to make the qg.tex; November 20, 2012; 16:49; p. 13

15 14 theory renormalizable. This is so because the coupling constants a and b are dimensionless Unfortunately, this theory suffers from other problems, like non-unitarity or ghost states, these are the states with negative norm. All these problems come roughly speaking from the wrong sign of the R 2 term. As shown in [3] even if this theory is renormalizable it is not asymptotically free like the Yang-Mills theory. At high energies the coupling a goes to zero and the coupling b goes to infinity at a specific energy. Other approaches to quantum gravity include: loop quantum gravity, dynamical triangulations, non-commutative geometry and others. References [1] L. Smolin, The Trouble with Physics, Mariner Books, 2007 [2] I. G. Avramidi, Matrix general relativity: a new look at old problems, Class. Quant. Grav. 21 (2004) [3] I. G. Avramidi and A. O. Barvinsky, Asymptotic freedom in higher-derivative quantum gravity, Phys. Lett. B 159 (1985) qg.tex; November 20, 2012; 16:49; p. 14