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- ICRA Seminars on Quantum Gravity (Spring 2006) - supervision of Dr. G. Montani
Aim of this seminars was to fix some relevant links between different approach to the cut-off physics. Particular attention has been devoted to the mathematical framework underlying Loop Quantum Gravity in view of clarifying how the notion of a minimal lenght lives within the local Lorentz gauge symmetry. In this respect, some implications of Loop Quantum Cosmology are discussed and the features of a Big-Bounce, replacing a Big-Bang, are outlined in some detail. As alternative approach to the cut-off physics, we dealt with the non-commutative structure of the space-time, especially on the base of a Generalized Uncertainty Principle (GUP) formalism. The issues of a GUP quantum dynamics were described in the case of a non-relativistic particles, while the proposal for a real quantum field theory was critically revised. Finally interesting aspects of a path integral theory for the gravitational field were analyzed within a metric formulation and attention to the gauge fixing procedure was devoted.
Speaker: M.V. Battisti
Generalized uncertainty principle and noncommutative spacetime
The existence of a minimal observable length has long been suggested when we try to unify Enistein’s theory of classical gravity with quantum mechanics principles [1]. The first attempt to describe a Lorentz invariant spacetime with a minimal lenght was made by Snyder in 1947 [2]. This approach is compared with that one [3] in which the authors study in full detail the quantum mechanical structure which underlies a generalized uncertainty relation, which implement the appearance of a nonzero minimal uncertainty in position. At the end, introducing the Moyal star-product, we discuss about the criteria for preserving Poncare‘ invariance in noncommutative gauge theories [4].
1- S.Doplicher, K.Fredenhagen, J.E.Roberts, 1994 “Spacetime quantization induced by classical gravity” Phys. Lett. B331 39.
2- H.S.Snyder, “Quantized space-time” 1947 Phys. Rev. 71 38.
3- A.Kempf, G.Mangano, R.B.Mann, 1995 “Hilbert space representation of the minimal uncertainty relation” Phys. Rev. 52 1108.
4- R.Banerjee, B. Chakraborty, K. Kumar, 2004 “Noncommutative gauge theories and Lorentz symmetry” Phys. Rev. D70 125004.Framework of Loop Quantum Cosmology
Is showed how the classical singularity, of k=0 FRW cosmological model, is removed by quantum geometry. We begin by singling out ‘elementary functions’ on the classical phase space which are to have unambiguous quantum analogs: the almost periodic function and the momenta p. No operator corresponding to connections is defined. Then,we will express the physically interesting operators in terms of these variables. We will show that the WDW equation is trasformed in a difference equation, because the area operators has a lowest eigenvalue and, thus, is physically inappropriate to try to localize the curvature on arbitrary small surfaces. References:
1-M.Bojowald Phys.Rev.Lett f86 (2001) 5227.
2-A.Ashtekar, M.Bojowald and J.Levandowski Adv.Theor.Math.Phys. 7 (2003) 233
3-A.Ashtekar, T.Pawlowski and P.Singh Phys.Rev.Lett. 96 (2006) 141301
Speaker: R. Benini
Wheeler-DeWitt equation
We briefly discuss some features of the Wheeler-DeWitt [WDW] equation: starting from the Arnowitt-Deser-Misner approach to the Einstein action, we discuss the meaning, the problems and some solutions of the WDW. We derive in particular the solutions in the mini super-space of the flat FRW model, with particular attention to the role that the initial conditions have in a cosmological model. Then, the problem of classical constraints while quantizing a theory is discussed from the Multi-time point of view; in this framework some features of the Bianchi type I and IX homogeneous models are presented. The last remark is about the semiclassical states in the quantum Mixmaster.
References
1-B. S. DeWitt Phys. Rev. 160, 1113-1148 (1967)
2- R. Arnowitt, S. Deser, C.W.Misner Gravitation :An Introduction to current Research. L. Witten (ed.), Wiley, New York,p.227 (1962)
3-K.Kuchar,Quantum Gravity II, a second Oxford symposium. eds C.Isham et al. Clarendom press (1981)
4- E.Kolb, M.Turner The Early Universe ,Ch.11. Addison-Wesley (1990)
5- T.Thiemann Introduction to Modern Canonical Quantum General Relativity. Available on arXiv:grqc/0110034 v1 (2001)
6- C.Isham Canonical Quantum Gravity and the Problem of Time. Available arXiv:gr-qc/9201011(1992)
7- C. Misner,Phys. Rev 186 5 (1969).
Speaker: F. Cianfrani
Phenomenology of Lorentz violations
The fate of Lorentz invariance in Quantum Gravity is an open issue; however, if it will induce Lorentz violating terms in the Lagrangian density, because of radiative corrections, they will be uppressed by the size of Standard Model couplings, unless some sort of fine-tuning on the parameters of such terms. Therefore, the quantum gravity phenomenology of possible Lorentz-violations has already been ruled out by experimental data.
Reference:
‘Lorentz Invariance Violation and its Role in Quantum Gravity Phenomenology’ by John Collins ,
Alejandro Perez , Daniel Sudarsky (Draft chapter contributed to the book Towards quantum gravity,
being prepared by Daniele Oriti for Cambridge University Press ); available: arXiv:hep-th/0603002Covariant formulation of Loop Quantum Gravity
It is possible to derive the formulation of General Relativity in terms of Barbero-Immirzi connections starting from a variational principle. In particular, the ADM splitting, in the time gauge, of the Holst’s action gives the set of constraints predicted by Barbero. In particular, the Super-Hamiltonian and the Super-Momentum ones reproduce the algebra of time-diffeomorphisms and three-diffeomorphisms, respectively. Then, the machinery of the quantization by virtue of holonomies can start. However, holonomies themselves, despite the case of Ashtekar variables, depend on the splitting, therefore connections have no well-defined behavior under time reparametrization. In order to solve this aesthetical issue, Alexandrov introduce a covariant theory, where the splitting is not performed in the time gauge. At the end, the ambiguity due to the Immirzi parameter disappears, but arise second class constraints, which in general cannot be solved and predict modified commutation relations beetwen fields and their conjugates momenta.
References:
1-‘Barbero’s Hamiltonian derived from a generalized Hilbert-Palatini action’, Soren Holst, Phys. Rev. D53, 10, (1996), 5965.
2-’Comment on Holst’s Lagrangian formulation’, Joseph Samuel, Phys. Rev. D63, (2001), 068501.
3-’SU(2) Loop Quantum Gravity seen from Covariant Theory’, Sergei Alexandrov, Etera Livine, Phys.Rev. D67, (2003), 044009, arXiv: gr-qc/0209105Resolution of the cosmological singularity in Loop Quantum Gravity
The quantization of the k=0 Freedman-Robertson-Walker space-time, in a minisuperspace approach, can be performed in the Wheeler DeWitt or in the Loop Quantum Gravity framework. The main distinction between the two procedures deals with the choice of fundamental variables (in the former case they are holonomies and smeard densitized triads) and the definition of the kinematical Hilbert space; in the dynamical sector, we end up with differential and difference equations, respectively. The origin of difference equations can be traced to the impossibility to take the limit of vanishing area for loops, when the hamiltonian constraint is rewritten in terms of holonomies, since no operator corresponding to connections is defined. Hence, by the group averaging tecniques a scalar product can be introduced, so amplitudes, Dirac observables and semiclassical states can be determined. By introducing a massless scalar field as the internal time, the evolution of semiclassical states is studied: while for the Wheeler DeWitt equation, once evolved backward in time, they remain semi-classical till they reach the classical singularity, this is replaced by a bounce in Loop Quantum Cosmology. Moreover, before the bounce a semi-classical contracting phase is predicted. This way, Loop Quantum Cosmology solves the Big Bang singularity.
References:
1- A.Ashtekar ,T.Pawlowski, P. Singh, Quantum Nature of the Big Bang: An Analytical and Numerical Investigation I , available ArXix: gr-qc/0604013
2- A.Ashtekar ,T.Pawlowski, P. Singh, Quantum Nature of the Big Bang’, Phys.Rev.Lett. 96 (2006) 141301.
Speaker: V. Lacquaniti
A derivation of ADM splitting via the Space-Ambient embedding of a manifold
The embedding of a 4D manifold in a 5D external Minkowskyan space ( our so-called Space-Ambient) allows us to recover the whole features of General Relativity in a simple vectorial picture. Within this picture it’s easy to stress the geometrical meaning of the metrics and the covariant derivative. A similar technique allows us to derive in a fast way the rules of the ADM splitting. It is showed hot to get the rules for the sinchronous splitting, for the generic ADM splitting of the metrics and for the proijection of a generic tensor; also, it is examined the splitting of the covariant derivative, the definition of the extrinsic curvature and its geometrical meaning. Finally, the Gauss-Codacci formula is provided, with a discussion of the dynamical properties of the Einstein-Hilbert action that arises from this picture.
References:
1- R. Arnowitt, S. Deser, C.W.Misner Gravitation : An Introduction to current Research. L. Witten (ed.), Wiley, New York,p.227 (1962)
2- Misner, Thorne, Wheeler Gravitation,Ch.21. Twenty four Printing (2002)
3- K.Kuchar, in Quantum Gravity II, a second Oxford symposium. eds C.Isham et al. Clarendom press (1981)
4- E.Kolb, M.Turner The Early Universe,Ch.11. Addison-Wesley (1990)
5- T.Thiemann Introduction to Modern Canonical Quantum General Relativity. Available on arXiv:grqc/0110034 v1 (2001)
6- R. M. Wald General Relativity,Appendix E., The University of Chicago Press (1989)
7- C.Isham Canonical Quantum Gravity and the Problem of Time. Available arXiv:gr-qc/9201011 (1992)
Speaker: O.M. Lecian
Generalized uncertainty principle, noncommutative spacetime and the scalar field
The gravity-induced breakdown of canonical quantum mechanics in the description of the spacetime (1) at the Plank scale and the emergence of a cut off is described by different mathematical structures. In the framework of a generalized uncertainty principle (2), the quantization of fields is discussed, and two approaches are presented. The standard QFT established by A.Kempf (3) ,based on nonzero minimal lengths and momenta and achieved in the framework of a generalized Bargman Fock representation, is compared to the proposal by T.Matsuo et al. (4) , where nonzero minimal momenta only are taken into account, and canonical and path integral quantization are extended to higher dimensions by means of the introduction of new parameters. k-Minkowski and canonical noncommutative spacetimes are presented, and the Moyal product is introduced. In particular, in a canonical noncommutative spacetime scenario, the quantization of the scalar field is studied (5), and the problem of microcausality ( and its possible violation) is investigated between vacuum states and between non-vacuum states (6): for particular choices of the commutation relations only the microcausality of the scalar field is satisfied.
References:
1- S.Doplicher, K.Fredenhagen. J.E.Roberts, ‘Spacetime quantization induced by classical gravity’, Phys.Lett. B331 (1994) 39-44.
2- A. Kempf, G. Mangano, R. B. Mann ‘Hilbert Space Representation of the Minimal Length Uncertainty Relation’, Phys.Rev. D52 (1995) 1108-1118, hep-th/9412167
3- A.Kempf ‘On Quantum Field Theory with Nonzero Minimal Uncertainties in Positions and Momenta’, Math.Phys. 38 (1997) 1347-1372, hep-th/9602085
4- T.Matsuo, Y. Shibusa ‘Quantization of fields based on Generalized Uncertainty Principle’, hepth/0511031.
5- G.Amelino-Camelia, M.Arzano, L.Doplicher ‘ Field Theories on Canonical and Lie-Algebra Noncommutative Spacetimes’, hep-th/0205047.
6- Z.Z.Ma ‘Microcausality of scalar field on noncommutative spacetime’, hep-th/0603054Functional approach to quantum gravity
Different formalisms in quantum gravity are aimed to describe the dynamics of physical processes by means of integration over all the possible states (geometries), once the initial and the final states are given. In this context, Minkowski vacuum has been studied (1). Minkowski vacuum, the zero-particle state, is expressed in terms of a functional that depends on the field boundary value and on the geometry of the surface that encloses the region where experiments are performed. This functional is preliminary expressed in terms of a functional Schroedinger representation; then, the concept of Hilbert space is extended to include the initial and final states of the measure operation: here a covariant vacuum is defined, which maps the initial state into the final one, and a relation connecting the two vacuum statesis found. Neither spacial nor time infinities are needed: since macroscopic scales are much larger than the mass gap that ensures convergence, local particles can be treated like global particles for these purposes. Analogous formulas are worked out for background independent quantum gravity, where the Minkowski vacuum can be expressed by a Euclidean gravitational functional integral. Faddeev Popovmethod in the temporal gauge is applied to the propagation Kernel (2,3) : after fixing the notation in the YM case, General relativity is taken into account. Changing all the fields into a synchronous gauge and transforming the action accordingly solve the problem of fixing such a gauge without removing all four-metrics. In the compact case, the functional integral is independent of the proper time, which can be determined from the 00 component of the Einstein equations. In the asymptotically flat case, on the other hand, the functional integral depends also on the asymptotic proper time, because of the boundary conditions.
1- F.Conrady, L.Doplicher, R.Oeckl, C.Rovelli, M.Testa, ‘Minkowski vacuum in background independent quantum gravity’, gr-qc/0307118
2- M.Pawlowski, V.N.Pervushin, V.I.Smirichinski, ‘Invariant Hamiltonian Quantization of General Relativity’, hep-th/9909009
3 F.Mattei, C.Rovelli, S.Speziale, M.Testa, ‘From 3-geometry transition amplitudes to graviton states’, gr-qc/0508007
Speaker: E. Magliaro
Loop Quantum space
In this talk I show how to quantize (in LQG) the canonical formulation of General Relativity. Starting from the Hamiltonian system defined by three constraints equations, a quantization of the theory can be obtained in terms of complex valued Schrödinger-like wave functionals [A]; the Gaussian constraint and the vectorial constraint simply force [A] to be invariant under SU(2) gauge transformations and 3d diffeomorphisms, the Hamiltonian constraint gives the Wheeler-De Witt equation. In order to construct the kinematical Hilbert space is necessary to find suitable functionals of the connection (cylindrical functions) and then to require internal gauge invariance (spin network states) and diffeomorphism invariance (linear functionals of spin network states) of the states and of their scalar product. The next step is to find well defined operators in our Hilbert space that are invariant and self-adjoint. We obtain two operators of this kind which are diagonal on the spin networks with discrete spectrum and have a precise physical interpretation: they are the physical area of the surface intersected by spin networks’ links, and the physical volume that get contribution only from the nodes of the spin network states; we notice that in the context of Loop Quantum Gravity this discreteness is a direct consequence of a (conceptually) straightforward quantization of General Relativity. Finally I present relational interpretation of Quantum Mechanics and I observe that there is a connection between relationalism of QM and of GR due to the connection between contiguity and interaction.
Reference:
C.Rovelli, ‘Quantum Gravity’, Cambridge University Press
Speaker: S. Mercuri
Lorentz invariance and space(-time) discretization
One of the most interesting aspects of non-perturbative quantum gravity theories is the discretization of the space(-time). In particular in Loop Quantum Gravity the area and volume operators are not only hermitian and regularizable, but have also discrete eigenvalues in the base of spin-network. The question is: Can the Lorentz symmetry be reconciliable with such a discretization? Or, as suggested by the non-commutative geometry theories, we might expect a modification in the Lorentz-Fitzgerald trasformation at the Planck scale? We present arguments in favour of the former and latter hypothesis accordingly to the up to now results existing in literature.The Ashtekar-Barbero-Immirzi connections
The canonical formulation of General Relativity leads to a consistent formulation of a Quantum Gravity theory. Even though many aspects of Quantum Gravity can be studied in the framework of ADM phase space, it is in general useful to operate a canonical transformation, passing to the Ashtekar formulation. This trasformation reduces the phase space of General Relativity to that of a Yang-Mills gauge theory of the SU(2,C) group, allowing the implementation of many well known technics developed in gauge theories to the gravitational theory. We give a detailed description of the construction of Ashtekar phase space, dwelling upon the problem of Immirzi parameter and to the definition of Barbero connections, which actually complicate the constraints, but being real do not need any reality condition.