6 Summary

Loop quantum cosmology as an induced representation of loop quantum gravity describes quantum gravitational dynamics by difference equations, which allows a discussion of the singularity problem and provides a dynamical solution in many models. To understand properties of solutions, techniques such as generating functions or continued fractions are available for a direct analysis of difference equations. But especially in view of the more complicated extension to inhomogeneous models with many coupled difference equations, more tools are needed. Moreover, lattice effects of inhomogeneous states [14] can lead to difference equations with non-equidistant step size [25].

An effective treatment allows the computation of interesting properties of a solution, such as the peak position and spread of wave packets, without knowing the full solution. In this way one can extract physically interesting properties, analogously to “low energy” aspects in particle physics, without having to know explicit solutions for states.

These techniques are especially powerful in a solvable bounce model. This provides an intuitive bounce picture in special models, sourced by a free scalar, as exact effective systems. As a solvable system it provides the basis for a perturbation scheme to include matter interactions and inhomogeneities. Even physical inner product issues to properly normalize wave functions are addressable at the effective level through reality conditions. This gives hope that the notorious physical inner product problem can be dealt with in full quantum gravity, too.

Acknowledgements This work was supported in part by NSF grant PHY0554771.

References

1.A. Ashtekar, New Hamiltonian formulation of general relativity. Phys. Rev. D 36, 1587–1602 (1987)

2.A. Ashtekar and M. Bojowald, Quantum Geometry and the Schwarzschild Singularity. Class. Quantum Gravity 23, 391–411 (2006). gr-qc/0509075

3.A. Ashtekar and J. Lewandowski, Background independent quantum gravity: A status report. Class. Quantum Gravity 21, R53–R152 (2004). gr-qc/0404018

4.A. Ashtekar and T.A. Schilling, Geometrical Formulation of Quantum Mechanics. pp. 23–65. Springer, New York (1999). gr-qc/9706069

5.A. Ashtekar, M. Bojowald, and J. Lewandowski, Mathematical structure of loop quantum cosmology. Adv. Theor. Math. Phys. 7, 233–268 (2003). gr-qc/0304074

6.A. Ashtekar, T. Pawlowski, and P. Singh, Quantum nature of the big bang: Improved dynamics. Phys. Rev. D 74, 084003 (2006). gr-qc/0607039

7.J.F.G. Barbero, Real Ashtekar variables for Lorentzian signature space-times. Phys. Rev. D 51, 5507–5510 (1995). gr-qc/9410014

8.M. Bojowald, Absence of a singularity in loop quantum cosmology. Phys. Rev. Lett. 86, 5227– 5230 (2001). gr-qc/0102069

9.M. Bojowald, Isotropic loop quantum cosmology. Class. Quantum Gravity 19, 2717–2741 (2002). gr-qc/0202077

10.M. Bojowald, Homogeneous loop quantum cosmology. Class. Quantum Gravity 20, 2595– 2615 (2003). gr-qc/0303073

11.M. Bojowald, Spherically symmetric quantum geometry: States and basic operators. Class. Quantum Gravity 21, 3733–3753 (2004). gr-qc/0407017

12.M. Bojowald, Non-singular black holes and degrees of freedom in quantum gravity. Phys. Rev. Lett. 95, 061301 (2005). gr-qc/0506128

13.M. Bojowald, Loop quantum cosmology. Living Rev. Relativ. 11, 4 (2008). http://www. livingreviews.org/lrr-2008-4

14.M. Bojowald, Loop quantum cosmology and inhomogeneities. Gen. Relativ. Gravit. 38, 1771– 1795 (2006). gr-qc/0609034

15.M. Bojowald, Quantum cosmology. In: Encyclopedia of Mathematical Physics, vol. 4, p. 153. Elsevier, London (2006). gr-qc/0603110

16.M. Bojowald, Large scale effective theory for cosmological bounces. Phys. Rev. D 75, 081301(R) (2007)

17.M. Bojowald, Dynamical coherent states and physical solutions of quantum cosmological bounces. Phys. Rev. D 75, 123512 (2007)

18.M. Bojowald, Singularities and quantum gravity. In: Proceedings of the XIIth Brazilian School on Cosmology and Gravitation. AIP Conf. Proc. 910, 294–333 (2007)

19.M. Bojowald and H.A. Kastrup, Symmetry reduction for quantized diffeomorphism invariant theories of connections. Class. Quantum Gravity 17, 3009–3043 (2000). hep-th/9907042

20.M. Bojowald and A. Rej, Asymptotic properties of difference equations for isotropic loop quantum cosmology. Class. Quantum Gravity 22, 3399–3420 (2005). gr-qc/0504100

21.M. Bojowald and A. Skirzewski, Effective equations of motion for quantum systems. Rev. Math. Phys. 18, 713–745 (2006). math-ph/0511043

22.M. Bojowald and R. Swiderski, Spherically symmetric quantum geometry: Hamiltonian constraint. Class. Quantum Gravity 23, 2129–2154 (2006). gr-qc/0511108

23.M. Bojowald, G. Date, and K. Vandersloot, Homogeneous loop quantum cosmology: The role of the spin connection. Class. Quantum Gravity 21, 1253–1278 (2004). gr-qc/0311004

24.M. Bojowald, H.H. Hernández, and H.A. Morales-Técotl, Perturbative degrees of freedom in loop quantum gravity: anisotropies. Class. Quantum Gravity 23, 3491–3516 (2006). gr-qc/0511058

25.M. Bojowald, D. Cartin, and G. Khanna, Lattice refining loop quantum cosmology, anisotropic models and stability. Phys. Rev. D 76, 064018 (2007). arXiv:0704.1137

26.O. Brodbeck, On symmetric gauge fields for arbitrary gauge and symmetry groups. Helv. Phys. Acta 69, 321–324 (1996). gr-qc/9610024

27.D. Cartin and G. Khanna, Absence of pre-classical solutions in Bianchi I loop quantum cosmology. Phys. Rev. Lett. 94, 111302 (2005). gr-qc/0501016

28.D. Cartin and G. Khanna, Wave functions for the Schwarschild black hole interior. Phys. Rev. D 73, 104009 (2006). gr-qc/0602025

29.D. Cartin, G. Khanna, and M. Bojowald, Generating function techniques for loop quantum cosmology. Class. Quantum Gravity 21, 4495–4509 (2004). gr-qc/0405126

30.G. Date, Pre-classical solutions of the vacuum Bianchi I loop quantum cosmology. Phys. Rev. D 72, 067301 (2005). gr-qc/0505030

31.B.S. DeWitt, Quantum theory of gravity. I. The canonical theory. Phys. Rev. 160, 1113–1148 (1967)

32.J. Engle, Quantum field theory and its symmetry reduction. Class. Quantum Gravity 23, 2861– 2893 (2006). gr-qc/0511107

33.J. Engle, On the physical interpretation of states in loop quantum cosmology. Class. Quantum Gravity 24, 5777–5802 (2007)

34.S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)

35.A. Heslot, Quantum mechanics as a classical theory. Phys. Rev. D 31, 1341–1348 (1985)

36.V. Husain and O. Winkler, On singularity resolution in quantum gravity. Phys. Rev. D 69, 084016 (2004). gr-qc/0312094

37.T.W.B. Kibble, Geometrization of quantum mechanics. Commun. Math. Phys. 65, 189–201 (1979)

38.S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol. 1. Wiley, New York (1963)

39.T. Koslowski, A cosmological sector in loop quantum gravity. arXiv:0711.1098

40.T. Koslowski, Reduction of a quantum theory. gr-qc/0612138

41.L. Modesto, Loop quantum black hole. Class. Quantum Gravity 23, 5587–5601 (2006). gr-qc/0509078

42.K. Noui, A. Perez, and K. Vandersloot, On the physical Hilbert space of loop quantum cosmology. Phys. Rev. D 71, 044025 (2005). gr-qc/0411039

43.C. Rovelli, Quantum Gravity. Cambridge University Press, Cambridge (2004)

44.A. Skirzewski, Effective equations of motion for quantum systems. PhD thesis, HumboldtUniversität Berlin (2006)

45.T. Thiemann, Introduction to Modern Canonical Quantum General Relativity. Cambridge University Press, Cambridge (2007)

46.D.L. Wiltshire, An introduction to quantum cosmology. In: Robson, B., Visvanathan, N., Woolcock, W.S. (eds.) Cosmology: The Physics of the Universe, pp. 473–531. World Scientific, Singapore (1996). gr-qc/0101003

Boundary Effects on the Interface Dynamics for the Stochastic Allen–Cahn Equation

Lorenzo Bertini, Stella Brassesco and Paolo Buttà

Abstract We consider a stochastic perturbation of the Allen–Cahn equation in a bounded interval [−a, b] with boundary conditions fixing the different phases at a

and b. We investigate the asymptotic behavior of the front separating the two stable

√

phases in the limit ε → 0, when the intensity of the noise is ε and a, b → ∞ with ε. In particular, we prove that it is possible to choose a = a(ε) such that in a suitable time scaling limit, the front evolves according to a one-dimensional diffusion process with a nonlinear drift accounting for a “soft” repulsion from a. We finally show that a “hard” repulsion can be obtained by an extra diffusive scaling.

1 Introduction

for V a two well non-degenerate symmetric potential that attains its minimum at ±m is a well studied equation that appears in different contexts to study the formation and evolution of interfaces for systems where two stable phases coexist. (See [1] for

Lorenzo Bertini

Dipartimento di Matematica, Università di Roma ‘La Sapienza’, P.le Aldo Moro 2, 00185 Rome, Italy, email: bertini@mat.uniroma1.it

Stella Brassesco

Departamento de Matemáticas, Instituto Venezolano de Investigaciones Científicas, Apartado Postal 20632, Caracas 1020-A, Venezuela, email: sbrasses@ivic.ve

Paolo Buttà

Dipartimento di Matematica, Università di Roma ‘La Sapienza’, P.le Aldo Moro 2, 00185 Rome, Italy, email: butta@mat.uniroma1.it

an instance.) The two phases are identified with +m and −m, and they are known to be stationary stable solutions of (1). There is also a family M of stationary solutions

mζ of (1) that interpolate between −m and +m, that is, they satisfy 12 mζ = V (mζ ) and mζ (±∞) = ±m. We have that

where mζ (x) = m(x − ζ ), m(x) being the centered solution, that is, m(0) = 0. The convergence of mζ (x) to the value of the pure phases as x → ±∞ is known to be exponentially fast (see [13]) so mζ is thought to represent a diffuse interface located at ζ .

A natural problem is then to study the evolution of an initial datum belonging (or close) to M under the dynamics given by a stochastic perturbation of (1). It is the

subject of a series of papers (see [6, 7, 9, 8, 15] for precise statements), where it is

√

shown that, if a space-time white noise W (dx, dt ) of intensity is added to (1),

with initial datum m(x, 0) = mx0 (x), satisfies m( ) as → 0, where B(t ) is a Browian motion.

The previous result is proved in [9] in fact for the spatial variable x [−a, b], with a = −α and b = −β for α, β > 0, and Neumann boundary conditions at the endpoints. In the time scaling −1, the fluctuations introduced by the noise govern the evolution of the interface, and the scaling of the endpoints make them so distant that the effects of the boundary conditions are not present in the limiting behavior. Moreover, it follows from the analysis in [9] that this picture remains true if for instance a = C log −1 for sufficiently large C.

On the other hand, from the results in [10] and [17] for the deterministic evolution given by (1) with Neumann boundary conditions on a = c log −1 for sufficiently small c and b = −γ , a deterministic drift to the left becomes dominant in the time scaling −1, and the +m phase takes over finally. This is a finite size effect, contrasting to the slow motion for the interface when far enough from the endpoints, reminiscent of the stability of M in the infinite volume situation. Recall that in a bounded interval, the pure phases are the unique global attractors.

We consider here the effect of non-homogeneous Dirichlet boundary conditions that fix the two different phases at the endpoints of [−a, b] on the evolution of a front located initially at zero, under the dynamics given by (3). In this case, there is

a unique stationary stable solution of (1) m , which is close to m b−a as a, b → ∞.

2

In particular, initial data close to mζ are attracted to this profile, and the evolution is

known to occur along a set Ma,b , which is close to M as a, b → ∞.

√

When a space-time white noise of intensity is added, we look at the evolution of the interface at a time scaling of order −1, in the case a = C0 log( −1) and