Mode d’emploi de la théorie constructive des champs bosoniques : avec une application aux chemins rugueux
Confluentes Mathematici, Tome 4 (2012) no. 1.

Nous développons dans cet article les principaux arguments constructifs utilisés en théorie quantique des champs, en nous cantonnant aux théories bosoniques, pour lesquelles il n'existe pas de présentation générale récente. L'article s'adresse d'abord et avant tout à des mathématiciens ou physiciens mathématiciens connaissant les arguments de base de la théorie perturbative des champs, et souhaitant connaître un cadre général dans lequel ils peuvent être rendus rigoureux. Il fournit également un aperçu d'une série d'articles récents [50, 51] visant à donner une définition constructive des chemins rugueux et du calcul stochastique fractionnaire.

We develop in this article the principal constructive arguments used in quantum field theory, limiting us to bosonic theories, for which there does not exist any recent general presentation. The article is primarily written for mathematicians or mathematical physicists knowing the basic arguments of quantum field theory, and desiring to discover a general framework in which they can be made rigorous. It also provides a glimpse of a recent series of articles [50, 51] whose aim is to give a constructive definition of rough paths and fractionary stochastic calculus.

Publié le :
DOI : 10.1142/S179374421240004X

Jérémie Unterberger 1

1
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Jérémie Unterberger. Mode d’emploi de la théorie constructive des champs bosoniques : avec une application aux chemins rugueux. Confluentes Mathematici, Tome 4 (2012) no. 1. doi : 10.1142/S179374421240004X. https://cml.centre-mersenne.org/articles/10.1142/S179374421240004X/

[1] A. Abdesselam, Explicit constructive renormalization, Ph.D. Thesis (1997).

[2] A. Abdesselam and V. Rivasseau, Trees, Forests and Jungles : A Botanical Garden for Cluster Expansions, Lecture Notes in Physics, Vol. 446 (Springer, 1995).

[3] A. Abdesselam and V. Rivasseau, An explicit large versus small field multiscale cluster expansion, Rev. Math. Phys. 9 (1997) 123–199.

[4] A. Abdesselam and V. Rivasseau, Explicit fermionic tree expansions, Lett. Math. Phys. 44 (1998) 77–88.

[5] J. Baez and J. Muniain, Gauge Fields, Knots and Gravity, Series on Knots and Every- thing, Vol. 4 (World Scientific, 1994).

[6] G. Benfatto, M. Cassandro, G. Gallavotti, F. Nicol‘o, E. Olivieri, E. Pressutti and E. Scacciatelli, On the ultraviolet stability in the Euclidean scalar field theories, Com- mun. Math. Phys. 71 (1980) 95–130.

[7] G. Benfatto, G. Gallavotti and V. Mastropietro, Renormalization group and the Fermi surface in the Luttinger model, Phys. Rev. B 45 (1992) 5468–5480.

[8] D. Bernard, K. Gawedzki and A. Kupiainen, Anomalous scaling in the N-point func- tion of passive scalar.

[9] J. Bricmont, K. Gawedzki and A. Kupiainen, KAM theorem and quantum field theory, Commun. Math. Phys. 201 (1999) 699–727.

[10] C. Brouder, Runge–Kutta methods and renormalization, Euro. Phys. J. C 12 (2000) 521–534.

[11] D. C. Brydges and T. Kennedy, Mayer expansion of the Hamilton–Jacobi equation, J. Stat. Phys. 49 (1987) 19–49.

[12] E. Bacry and J. F. Muzy, Log-infinitely divisible multifractal processes, Commun. Math. Phys. 236 (2003) 449–475.

[13] P. Cartier and C. DeWitt-Morette, Brydges’ operator in renormalization theory, in Mathematical Physics and Stochastic Analysis, eds. S. Albeverio et al. (World Scien- tific, 2000), pp. 165–168.

[14] A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199 (1998) 203–242.

[15] L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and fractional Brow- nian motions, Probab. Th. Relat. Fields 122 (2002) 108–140.

[16] B. Duplantier and S. Sheffield, Liouville quantum gravity and KPZ, arXiv :0808.1560.

[17] Constructive Quantum Field Theory, Proc. of the 1973 Erice Summer School, eds. G. Velo and A. Wightman, Lecture Notes in Physics, Vol. 25 (Springer, 1973).

[18] G. Falkovich, K. Gawedzki and M. Vergassola, Particles and fields in fluid turbulence.

[19] L. Foissy and J. Unterberger, Ordered forests, permutations and iterated integrals, arXiv :1004.5208.

[20] J. Feldman, J. Magnen, V. Rivasseau and R. Sénéor, Bounds on completely convergent Euclidean Feynman graphs, Commun. Math. Phys. 98 (1985) 273–288.

[21] J. Feldman, J. Magnen, V. Rivasseau and R. Sénéor, Bounds on renormalized Feyn- man graphs, Commun. Math. Phys. 100 (1985) 23–55.

[22] J. Feldman, J. Magnen, V. Rivasseau and R. Sénéor, Construction and Borel summa- bility of infrared Φ4 4 by a phase space expansion, Commun. Math. Phys. 109 (1987) 437–480.

[23] J. Feldman, V. Rivasseau, J. Magnen and E. Trubowitz, An infinite volume expansion for many Fermions Green’s functions, Helv. Phys. Acta 65 (1992) 679.

[24] U. Frisch, Turbulence : The Legacy of A. N. Kolmogorov (Cambridge Univ. Press, 1995).

[25] P. Friz and N. Victoir, Multidimensional Dimensional Processes Seen as Rough Paths (Cambridge Univ. Press, 2010).

[26] G. Gallavotti, Invariant tori : A field theoretic point of view on Eliasson’s work, in Advances in Dynamical Systems and Quantum Physics, éd. R. Figari (World Scientific, 1995), pp. 117–132.

[27] G. Gallavotti and K. Nicol‘o, Renormalization theory in 4-dimensional scalar fields, Commun. Math. Phys. 100 (1985) 545–590 ; 101 (1985) 247–282.

[28] K. Gawedzki and A. Kupiainen, Massless lattice ϕ4 4 theory : Rigorous control of a renormalizable asymptotically free model, Commun. Math. Phys. 99 (1985) 197–252.

[29] J. Glimm and A. Jaffe, Quantum Physics, A Functionnal Point of View (Springer, 1987).

[30] J. Glimm and A. Jaffe, Positivity of the ϕ4 3 Hamiltonian, Fortschr. Phys. 21 (1973) 327–376.

[31] J. Glimm, A. Jaffe and T. Spencer, The particle structure of the weakly coupled P(ϕ)2 model and other applications of high temperature expansions : Part II. The cluster expansion, in Constructive Quantum Field Theory (Erice 1973), op. cit. [17].

[32] N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992).

[33] M. Gubinelli, Controlling rough paths, J. Funct. Anal. 216 (2004) 86–140.

[34] R. Gurau, J. Magnen and V. Rivasseau, Tree quantum field theory, arXiv :0807.4122.

[35] B. Hambly and T. Lyons, Stochastic area for Brownian motion on the Sierpinski gasket, Ann. Probab. 26 (1998) 132–148.

[36] K. Hepp, Proof of the Bogoliubov–Parasiuk theorem on renormalization, Commun. Math. Phys. 2 (1966) 301–326.

[37] C. Itzykson and J.-M. Drouffe, Statistical Field Theory (Cambridge Univ. Press, 1989).

[38] I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus (Springer, 1991).

[39] M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003) 157–216.

[40] A. Kupiainen and P. Muratore-Ginanneschi, Scaling, renormalization and statistical conservation laws in the Kraichnan model of turbulent advection.

[41] S. Lando and A. Zvonkin, Graphs on Surfaces and Their Applications (Springer, 2003).

[42] M. Laguës and A. Lesne, Invariance d’échelle. Des changements d’état à la turbulence (Belin, 2003).

[43] M. Le Bellac, Quantum and Statistical Field Theory (Oxford Univ. Press, 1991).

[44] A. Lejay, An Introduction to Rough Paths, Séminaire de Probabilités XXXVII, 1–59, Lecture Notes in Math., Vol. 1832 (Springer, 2003).

[45] A. Lejay, Yet another introduction to rough paths, Sém. Probab. 1979 (2009) 1–101.

[46] T. Lyons, Differential equations driven by rough signals, Rev. Mat. Ibro. 14 (1998) 215–310.

[47] T. Lyons and Z. Qian, System Control and Rough Paths (Oxford Univ. Press, 2002).

[48] T. Lyons and N. Victoir, An extension theorem to rough paths, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) 835–847.

[49] J. Magnen and V. Rivasseau, Constructive φ4 -theory without tears, arXiv :0706.2457.

[50] J. Magnen and J. Unterberger, From constructive theory to fractional stochastic cal- culus. (I) An introduction : Rough path theory and perturbative heuristics, Ann. Henri Poincaré 12 (2011) 1199–1226.

[51] J. Magnen and J. Unterberger, From constructive theory to fractional stochastic cal- culus. (II) The rough path for 1 6 < α < 1 4 : Constructive proof of convergence, à paraˆıtre à, Ann. Henri Poincaré, arXiv :1103.1750.

[52] J. Magnen and J. Unterberger, Renormalized rough paths : A stochastic differential equation approach, in preparation.

[53] V. Mastropietro, Non-Perturbative Renormalization (World Scientific, 2008).

[54] J. Magnen and D. Iagolnitzer, Weakly self avoiding polymers in four dimensions, Commun. Math. Phys. 162 (1994) 85–121.

[55] J. Moore, Lectures on Seiberg–Witten Invariants, Lecture Notes in Mathematics, Vol. 1629 (Springer, 1996).

[56] E. Nelson, A quartic interaction in two dimensions, in Mathematical Theory of Ele- mentary Particles, eds. R. Goodman and I. Segal (MIT Press, 1966).

[57] D. Nualart, Stochastic calculus with respect to the fractional Brownian motion and applications, Contemp. Math. 336 (2003) 3–39.

[58] R. Peltier and J. Lévy-Véhel, Multifractional Brownian motion : Definition and pre- liminary results, INRIA research report, RR-2645 (1995).

[59] M. Peskine and D. Schröder, An Introduction to Quantum Field Theory (Addison- Wesley, 1995).

[60] V. Rivasseau, From Perturbative to Constructive Renormalization (Princeton Univ. Press, 1991).

[61] V. Rivasseau, F. Vignes-Tourneret and R. Wulkenhaar, Renormalization of noncom- mutative φ4-theory by multi-scale analysis, Commun. Math. Phys. 262 (2006) 565–594.

[62] D. Ruelle, Statistical Mechanics, Rigorous Results (Benjamin, 1969).

[63] M. Salmhofer, Renormalization : An Introduction (Springer, 1999).

[64] H. Triebel, Spaces of Besov–Hardy–Sobolev Type (Teubner, 1978).

[65] J. Unterberger, Stochastic calculus for fractional Brownian motion with Hurst param- eter H > 1/4 : A rough path method by analytic extension, Ann. Probab. 37 (2009) 565–614.

[66] J. Unterberger, A central limit theorem for the rescaled Lévy area of two-dimensional fractional Brownian motion with Hurst index H < 1/4, arXiv :0808.3458.

[67] J. Unterberger, A renormalized rough path over fractional Brownian motion, arXiv :1006.5604.

[68] J. Unterberger, A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering, Stoch. Proc. Appl. 120 (2010) 1444–1472.

[69] J. Unterberger, Hölder-continuous paths by Fourier normal ordering, Commun. Math. Phys. 298 (2010) 1–36.

[70] J. Unterberger, A Lévy area by Fourier normal ordering for multidimensional frac- tional Brownian motion with small Hurst index, arXiv :0906.1416.

[71] F. Vignes-Tourneret, Renormalisation des théories de champs non commutatives, Thèse de doctorat de l’Université Paris 11, arXiv :math-ph/0612014.

[72] A. S. Wightman, Remarks on the present state of affairs in the quantum theory of ele- mentary particles, in Mathematical Theory of Elementary Particles, eds. R. Goodman and I. Segal (MIT Press, 1966).

[73] K. G. Wilson, Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture, Phys. Rev. B 4 (1971) 3174–3184.

[74] K. G. Wilson and J. Kogut, The renormalization group and the ε expansion, Phys. Rep. 12 (1974) 75–200.

[75] E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic inte- grals, Ann. Math. Statist. 36 (1965) 1560–1564.

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