Tree quantum field theory. (English) Zbl 1206.81073
Summary: We propose a new formalism for quantum field theory (QFT) which is neither based on functional integrals, nor on Feynman graphs, but on marked trees. This formalism is constructive, i.e., it computes correlation functions through convergent rather than divergent expansions. It applies both to fermionic and bosonic theories. It is compatible with the renormalization group, and it allows to define non-perturbatively differential renormalization group equations. It accommodates any general stable polynomial Lagrangian. It can equally well treat noncommutative models or matrix models such as the Grosse-Wulkenhaar model. Perhaps most importantly it removes the space-time background from its central place in QFT, paving the way for a non-perturbative definition of field theory in non-integer dimension.
MSC:
| 81T08 | Constructive quantum field theory |
| 81T05 | Axiomatic quantum field theory; operator algebras |
| 05C05 | Trees |
| 81T17 | Renormalization group methods applied to problems in quantum field theory |
References:
| [1] | Bergeron, F.; Labelle, G.; Leroux, P., Combinatorial species and tree-like structures. (Encyclopedia of mathematics and its applications) (1997), London: Cambridge University Press, London · Zbl 0888.05001 |
| [2] | Brydges, D.; Kennedy, T., Mayer expansions and the Hamilton-Jacobi equation, J. Stat. Phys., 48, 19 (1987) · doi:10.1007/BF01010398 |
| [3] | Abdesselam, A.; Rivasseau, V.; Rivasseau, V., Trees, forests and jungles: a botanical garden for cluster expansions, Constructive physics, Lecture Notes in Physics, vol. 446 (1995), Berlin: Springer, Berlin · Zbl 0822.60095 |
| [4] | ‘T Hooft, G.; Veltman, M., Regularization and renormalization of gauge fields, Nucl. Phys., B44, 1, 189-213 (1972) · doi:10.1016/0550-3213(72)90279-9 |
| [5] | Wilson, K., Quantum field—theory models in less than 4 dimensions, Phys. Rev. D, 7, 2911-2926 (1973) · doi:10.1103/PhysRevD.7.2911 |
| [6] | Abdesselam, A., The Grassmann-Berezin calculus and theorems of the matrix-tree type, Adv. Appl. Math., 33, 1, 51-70 (2004) · Zbl 1052.05044 · doi:10.1016/j.aam.2003.07.002 |
| [7] | Gurau, R.; Rivasseau, V., Parametric representation of noncommutative field theory. math-ph/0606030, Commun. Math. Phys., 272, 811-835 (2007) · Zbl 1156.81465 · doi:10.1007/s00220-007-0215-5 |
| [8] | Rivasseau, V.; Tanasa, A., Parametric representation of critical noncommutative QFT models, Commun. Math. Phys., 279, 355-379 (2008) · Zbl 1158.81026 · doi:10.1007/s00220-008-0437-1 |
| [9] | Lesniewski, A., Effective action for the Yukawa_2 quantum field theory, Commun. Math. Phys., 108, 437 (1987) · doi:10.1007/BF01212319 |
| [10] | Feldman, J.; Magnen, J.; Rivasseau, V.; Trubowitz, E., An infinite volume expansion for many Fermion Green’s functions, Helv. Phys. Acta, 65, 679 (1992) · Zbl 0921.35140 |
| [11] | Abdesselam, A.; Rivasseau, V., Explicit fermionic cluster expansion, Lett. Math. Phys., 44, 77-88 (1998) · Zbl 0907.60087 · doi:10.1023/A:1007413417112 |
| [12] | Disertori, M.; Rivasseau, V., Continuous constructive fermionic renormalization, Annales Henri Poincaré, 1, 1 (2000) · Zbl 1083.81546 · doi:10.1007/PL00000998 |
| [13] | Disertori, M.; Rivasseau, V., Interacting Fermi liquid in two dimensions at finite temperature, Part I: Convergent attributions, Commun. Math. Phys., 215, 251 (2000) · Zbl 1043.82014 · doi:10.1007/s002200000300 |
| [14] | Disertori, M.; Rivasseau, V., Interacting Fermi liquid in two dimensions at finite temperature, Part II: Renormalization, in two dimensions at finite temperature, Commun. Math. Phys., 215, 291 (2000) · Zbl 1043.82013 · doi:10.1007/s002200000301 |
| [15] | Feldman, J.; Knörrer, H.; Trubowitz, E., A two dimensional Fermi liquid. Part 1: Overview, Commun. Math. Phys., 247, 1-47 (2004) · Zbl 1068.82004 · doi:10.1007/s00220-003-0996-0 |
| [16] | Feldman, J.; Knörrer, H.; Trubowitz, E., A two dimensional Fermi Liquid. Part 2: Convergence, Commun. Math. Phys., 247, 49-111 (2004) · Zbl 1070.82005 · doi:10.1007/s00220-003-0997-z |
| [17] | Feldman, J.; Knörrer, H.; Trubowitz, E., A two dimensional Fermi Liquid. Part 3: The Fermi Surface, Commun. Math. Phys., 247, 113-177 (2004) · Zbl 1071.82005 · doi:10.1007/s00220-003-0998-y |
| [18] | Feldman, J.; Knörrer, H.; Trubowitz, E., Particle hole ladders, Commun. Math. Phys., 247, 179-194 (2004) · Zbl 1070.82006 · doi:10.1007/s00220-004-1038-2 |
| [19] | Feldman, J.; Knörrer, H.; Trubowitz, E., Convergence of perturbation expansions in Fermionic models. Part 1: Nonperturbative bounds, Commun. Math. Phys., 247, 195-242 (2004) · Zbl 1069.82001 · doi:10.1007/s00220-004-1039-1 |
| [20] | Feldman, J.; Knörrer, H.; Trubowitz, E., Convergence of perturbation expansions in Fermionic models. Part 2: Overlapping loops, Commun. Math. Phys., 247, 243-319 (2004) · Zbl 1068.82005 · doi:10.1007/s00220-004-1040-8 |
| [21] | Rivasseau, V., The two dimensional Hubbard Model at half-filling: I. Convergent contributions, J. Stat. Phys., 106, 693-722 (2002) · Zbl 1001.82046 · doi:10.1023/A:1013770608643 |
| [22] | Afchain, S.; Magnen, J.; Rivasseau, V., Renormalization of the 2-point function of the Hubbard Model at half-filling, Ann. Henri Poincaré, 6, 399 (2005) · Zbl 1096.82020 · doi:10.1007/s00023-005-0213-0 |
| [23] | Afchain, S.; Magnen, J.; Rivasseau, V., The Hubbard model at half-filling, part III: the lower bound on the self-energy, Ann. Henri Poincaré, 6, 449 (2005) · Zbl 1255.82021 · doi:10.1007/s00023-005-0214-z |
| [24] | Benfatto, G.; Giuliani, A.; Mastropietro, V., Low temperature analysis of two-dimensional Fermi systems with symmetric Fermi surface, Ann. Henri Poincaré, 4, 137-193 (2003) · Zbl 1021.82001 · doi:10.1007/s00023-003-0125-9 |
| [25] | Benfatto, G.; Giuliani, A.; Mastropietro, V., Fermi liquid behavior in the 2D Hubbard model at low temperatures, Ann. Henri Poincaré, 7, 809-898 (2006) · Zbl 1137.82004 · doi:10.1007/s00023-006-0270-z |
| [26] | Grosse, H.; Wulkenhaar, R., Renormalization of \({\phi^4}\) -theory on noncommutative \({{\mathbb R}^4}\) in the matrix base, Commun. Math. Phys., 256, 305-374 (2005) · Zbl 1075.82005 · doi:10.1007/s00220-004-1285-2 |
| [27] | Grosse, H.; Wulkenhaar, R., Power-counting theorem for non-local matrix models and renormalization, Commun. Math. Phys., 254, 91-127 (2005) · Zbl 1079.81049 · doi:10.1007/s00220-004-1238-9 |
| [28] | Rivasseau, V.; Vignes-Tourneret, F.; Wulkenhaar, R., Renormalization of noncommutative \({\phi^4}\) -theory by multi-scale analysis, Commun. Math. Phys., 262, 565-594 (2006) · Zbl 1109.81056 · doi:10.1007/s00220-005-1440-4 |
| [29] | Gurau, R.; Magnen, J.; Rivasseau, V.; Vignes-Tourneret, F., Renormalization of non-commutative \({\phi^4_4}\) field theory in x space, Commun. Math. Phys., 267, 515-542 (2006) · Zbl 1113.81101 · doi:10.1007/s00220-006-0055-8 |
| [30] | Grosse, H.; Wulkenhaar, R., The beta-function in duality-covariant noncommutative \({\phi^4 }\) -theory, Eur. Phys. J. C, 35, 277-282 (2004) · Zbl 1191.81207 · doi:10.1140/epjc/s2004-01853-x |
| [31] | Disertori, M.; Rivasseau, V., Two and Three Loops Beta Function of Non Commutative \({\phi^4_4}\) Theory Eur, Phys. J. C, 50, 661 (2007) · Zbl 1248.81254 |
| [32] | Disertori, M.; Gurau, R.; Magnen, J.; Rivasseau, V., Vanishing of beta function of non commutative \({\Phi_4^4}\) to all orders, Phys. Lett. B, 649, 1, 95-102 (2007) · Zbl 1248.81253 · doi:10.1016/j.physletb.2007.04.007 |
| [33] | Glimm, J.; Jaffe, A., Quantum physics. A functional integral point of view (1987), Berlin: Springer, Berlin |
| [34] | Rivasseau, V., From perturbative to constructive renormalization (1991), New Jersey: Princeton University Press, New Jersey |
| [35] | Brydges, D.: Weak perturbations of massless Gaussian measures. In: Constructive physics, LNP 446, Springer, Berlin (1995) · Zbl 0829.60035 |
| [36] | Abdesselam, A.; Rivasseau, V., An explicit large versus small field multiscale cluster expansion, Rev. Math. Phys., 9, 123 (1997) · Zbl 0870.60100 · doi:10.1142/S0129055X97000063 |
| [37] | Rivasseau, V., Constructive matrix theory, JHEP, 09, 008 (2007) · doi:10.1088/1126-6708/2007/09/008 |
| [38] | Magnen, J.; Rivasseau, V., Constructive \({\phi^4}\) field theory without tears, Ann. Henri Poincaré, 9, 403-424 (2008) · Zbl 1141.81022 · doi:10.1007/s00023-008-0360-1 |
| [39] | Speer, E., Generalized Feynman amplitudes (1969), New Jersey: Princeton University Press, New Jersey · Zbl 0172.27301 |
| [40] | Letac, G., Massam, H.: A tutorial on non central Wishart distributions. http://www.lsp.ups-tlse.fr/Fp/Letac/Wishartnoncentrales.pdf · Zbl 1386.60068 |
| [41] | Ishi, H., Positive Riesz distributions on homogeneous cones, J. Math. Soc. Jpn., 52, 1, 161186 (2000) · Zbl 0954.43003 · doi:10.2969/jmsj/05210161 |
| [42] | Speer, E.; Rivasseau, V., The Borel transform in Euclidean \({\phi^4_4 }\) , local existence for Re ν < 4, Commun. Math. Phys., 72, 293 (1980) · doi:10.1007/BF01197553 |
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