Abstract
We study quartic matrix models with partition function \({\mathcal{Z}[E, J] = \int dM}\) exp(trace\({(JM - EM^{2} - \frac{\lambda}{4} M^4)}\)). The integral is over the space of Hermitean \({\mathcal{N} \times \mathcal{N}}\) -matrices, the external matrix E encodes the dynamics, \({\lambda > 0}\) is a scalar coupling constant and the matrix J is used to generate correlation functions. For E not a multiple of the identity matrix, we prove a universal algebraic recursion formula which gives all higher correlation functions in terms of the 2-point function and the distinct eigenvalues of E. The 2-point function itself satisfies a closed non-linear equation which must be solved case by case for given E. These results imply that if the 2-point function of a quartic matrix model is renormalisable by mass and wavefunction renormalisation, then the entire model is renormalisable and has vanishing β-function.
As the main application we prove that Euclidean \({\phi^4}\) -quantum field theory on four-dimensional Moyal space with harmonic propagation, taken at its self-duality point and in the infinite volume limit, is exactly solvable and non-trivial. This model is a quartic matrix model, where E has for \({\mathcal{N} \to \infty}\) the same spectrum as the Laplace operator in four dimensions. Using the theory of singular integral equations of Carleman type we compute (for \({\mathcal{N} \to \infty}\) and after renormalisation of \({E, \lambda}\)) the free energy density (1/volume) log\({(\mathcal{Z}[E, J]/\mathcal{Z}[E, 0])}\) exactly in terms of the solution of a non-linear integral equation. Existence of a solution is proved via the Schauder fixed point theorem.
The derivation of the non-linear integral equation relies on an assumption which in subsequent work is verified for coupling constants \({\lambda \leq 0}\) .
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References
Abdesselam, A., Rivasseau, V.: Trees, forests and jungles: a botanical garden for cluster expansions. In: Constructive physics. Lect. Notes Phys., Vol. 446, Berlin: Springer, 1994, pp. 7–36. [hep-th/9409094]
Aizenman M.: Proof of the triviality of \({{\phi^4_d}}\) field theory and some mean field features of Ising models for \({d > 4}\) . Phys. Rev. Lett. 47, 1–4 (1981)
Avramescu C.: Sur l’existence des solutions convergentes des systèmes d’équations différentielles non linéaires. Ann. Mat. Pura Appl. 81, 147–168 (1969)
Banks, T., Fischler, W., Shenker, S.H., Susskind, L.: M theory as a matrix model: a conjecture. Phys. Rev. D 55, 5112–5128 (1997). [hep-th/9610043]
Baxter R.J.: Eight-vertex model in lattice statistics. Phys. Rev. Lett. 26, 832–833 (1971)
Baxter R.J.: Hard hexagons: exact solution. J. Phys. A Math. Gen. 13, L61–L70 (1980)
Becchi, C., Giusto, S., Imbimbo, C.: The Wilson–Polchinski renormalization group equation in the planar limit. Nucl. Phys. B 633, 250–270 (2002). [hep-th/0202155]
Becchi, C., Giusto, S., Imbimbo, C.: The renormalization of noncommutative field theories in the limit of large noncommutativity. Nucl. Phys. B 664, 371–399 (2003). [hep-th/0304159]
Belavin A.A., Polyakov A.M., Zamolodchikov A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333–380 (1984)
Bethe H.: Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Phys. 71, 205–226 (1931)
Brezin E., Kazakov V.A.: Exactly solvable field theories of closed strings. Phys. Lett. B 236, 144–150 (1990)
Brunetti, R., Fredenhagen, K.: Quantum field theory on curved backgrounds. Lect. Notes Phys. 786, 129–155 (2009). [arXiv:0901.2063 [gr-qc]]
Brydges D.C., Kennedy T.: Mayer expansions and the Hamilton–Jacobi equation. J. Stat. Phys. 48, 19–49 (1987)
Carleman T.: Sur la résolution de certaines équations intégrales. Arkiv Mat. Astron. och Fysik 16, 19 (1922)
Chamseddine, A.H., Connes, A.: The spectral action principle. Commun. Math. Phys. 186, 731–750 (1997). [hep-th/9606001]
Cianciaruso, F., Colao, V., Marino, G., Xu, H.-K.: A compactness result for differentiable functions with an application to boundary value problems. Ann. Mat. Pura Appl. (2011). doi:10.1007/s10231-011-0230-1
Connes A.: Noncommutative geometry. Academic Press, San Diego (1994)
Connes, A.: Gravity coupled with matter and foundation of noncommutative geometry. Commun. Math. Phys. 182, 155–176 (1996). [hep-th/9603053]
Connes, A.: On the spectral characterization of manifolds. J. Noncommut. Geom. 7, 1–82 (2013). [arXiv:0810.2088 [math.OA]]
Di Francesco, P., Ginsparg, P.H., Zinn-Justin, J.: 2D gravity and random matrices. Phys. Rept. 254, 1–133 (1995). [hep-th/9306153]
Disertori, M., Rivasseau, V.: Two and three loops beta function of non commutative \({{\phi^4_4}}\) theory. Eur. Phys. J. C 50, 661–671 (2007). [hep-th/0610224]
Disertori, M., Gurau, R., Magnen, J., Rivasseau, V.: Vanishing of beta function of non commutative \({{\phi^4_4}}\) theory to all orders, Phys. Lett. B 649, 95–102 (2007). [hep-th/0612251]
Doplicher, S., Fredenhagen, K., Roberts, J.E.: The Quantum structure of space–time at the Planck scale and quantum fields. Commun. Math. Phys. 172, 187–220 (1995). [hep-th/0303037]
Douglas M.R., Shenker S.H.: Strings in less than one dimension. Nucl. Phys. B 335, 635–654 (1990)
Faddeev L.D.: Instructive history of the quantum inverse scattering method. Acta Appl. Math. 39, 69–84 (1995)
Feldman J., Magnen J., Rivasseau V., Seneor R.: Massive Gross–Neveu model: a rigorous perturbative construction. Phys. Rev. Lett. 54, 1479–1481 (1985)
Feldman J.S., Osterwalder K.: The Wightman axioms and the mass gap for weakly coupled \({{\phi^4_3}}\) quantum field theories. Ann. Phys. 97, 80–135 (1976)
Filk T.: Divergencies in a field theory on quantum space. Phys. Lett. B 376, 53–58 (1996)
Friedan D., Qiu Z.-a., Shenker S.H.: Conformal invariance, unitarity and two-dimensional critical exponents. Phys. Rev. Lett. 52, 1575–1578 (1984)
Fröhlich J.: On the triviality of \({{\lambda \phi^4_d}}\) theories and the approach to the critical point in \({{d \geq 4}}\) dimensions. Nucl. Phys. B 200, 281–296 (1982)
Gawedzki K., Kupiainen A.: Gross–Neveu model through convergent perturbation expansions. Commun. Math. Phys. 102, 1–30 (1985)
Gayral, V., Gracia-Bondía, J.M., Iochum, B., Schücker, T., Várilly, J.C.: Moyal planes are spectral triples. Commun. Math. Phys. 246, 569–623 (2004). [hep-th/0307241]
Gayral, V., Wulkenhaar, R.: Spectral geometry of the Moyal plane with harmonic propagation. J. Noncommut. Geom. 7, 939–979 (2013). [arXiv:1108.2184 [math.OA
Ginibre J.: General formulation of Griffiths’ inequalities. Commun. Math. Phys. 16, 310–328 (1970)
Glimm, J., Jaffe, A.M.: The \({{\lambda \phi^4_2}}\) quantum field theory without cut-offs, I. Phys. Rev. 176, 1945–1951 (1968)
Glimm, J., Jaffe, A.M.: The \({{\lambda \phi^4_2}}\) quantum field theory without cut-offs. II. The field operators and the approximate vacuum. Ann. Math. 91, 362–401 (1970)
Glimm, J., Jaffe, A.M.: The \({{\lambda \phi^4_2}}\) quantum field theory without cut-offs, III. The physical vacuum. Acta Math. 125, 203–267 (1970)
Glimm, J., Jaffe, A.M.: The \({{\lambda \phi^4_2}}\) quantum field theory without cut-offs, IV. Perturbations of the Hamiltonian. J. Math. Phys. 13, 1568–1584 (1972)
Glimm J., Jaffe A.M.: Positivity of the \({{\phi^4_3}}\) Hamiltonian. Fortsch. Phys. 21, 327–376 (1973)
Glimm J., Jaffe A.M., Spencer T.: The Wightman axioms and particle structure in the \({{{P(\phi)_2}}}\) quantum field model. Ann. Math. 100, 585–632 (1974)
Glimm J., Jaffe A.M.: Quantum physics. A functional integral point of view. Springer, New York (1987)
Gracia-Bondía, J.M., Várilly, J.C.: Algebras of distributions suitable for phase space quantum mechanics. I.. J. Math. Phys. 29, 869–879 (1988)
Gracia-Bondía, J.M., Várilly, J.C.: Algebras of distributions suitable for phase space quantum mechanics. II. Topologies on the Moyal algebra. J. Math. Phys. 29, 880–887 (1988)
Gradshteyn I.S., Ryzhik I.M.: Table of integrals, series, and products. Academic Press, Boston, MA (1994)
Gross D.J., Migdal A.A.: Nonperturbative two-dimensional quantum gravity. Phys. Rev. Lett. 64, 127–130 (1990)
Gross D.J., Neveu A.: Dynamical symmetry breaking in asymptotically free field theories. Phys. Rev. D 10, 3235–3253 (1974)
Gross D.J., Wilczek F.: Ultraviolet behavior of nonabelian gauge theories. Phys. Rev. Lett. 30, 1343–1346 (1973)
Grosse, H., Steinacker, H.: Renormalization of the noncommutative \({{\phi^3}}\) -model through the Kontsevich model. Nucl. Phys. B 746, 202–226 (2006). [hep-th/0512203]
Grosse, H., Steinacker, H.: Exact renormalization of a noncommutative \({{\phi^3}}\) model in 6 dimensions. Adv. Theor. Math. Phys. 12, 605–639 (2008). [hep-th/0607235]
Grosse, H., Wulkenhaar, R.: The β-function in duality-covariant noncommutative \({{\phi^4}}\) -theory. Eur. Phys. J. C 35, 277–282 (2004). [hep-th/0402093]
Grosse, H., Wulkenhaar, R.: Power-counting theorem for non-local matrix models and renormalisation. Commun. Math. Phys. 254, 91–127 (2005). [hep-th/0305066]
Grosse, H., Wulkenhaar, R.: Renormalisation of \({{\phi^4}}\) -theory on noncommutative \({{\mathbb{R}^4}}\) in the matrix base. Commun. Math. Phys. 256, 305–374 (2005). [hep-th/0401128]
Grosse, H., Wulkenhaar, R.: Renormalization of \({{\phi^4}}\) -theory on noncommutative \({{\mathbb{R}^4}}\) to all orders. Lett. Math. Phys. 71, 13–26 (2005). [hep-th/0403232]
Grosse, H., Wulkenhaar, R.: Progress in solving a noncommutative quantum field theory in four dimensions. arXiv:0909.1389. [hep-th]
Grosse, H., Wulkenhaar, R.: 8D-spectral triple on 4D-Moyal space and the vacuum of noncommutative gauge theory. J. Geom. Phys. 62, 1583–1599 (2012). [arXiv:0709.0095 [hep-th]]
Grosse, H., Wulkenhaar, R.: Solvable limits of a 4D noncommutative QFT. arXiv:1306.2816 [math-ph]
Grosse, H., Wulkenhaar, R.: Construction of the \({{\phi^4_4}}\) -quantum field theory on noncommutative Moyal space (2014). arXiv:1402.1041 [math-ph]
Guerra, F., Rosen, L., Simon, B.: The \({{P(\phi)_2}}\) Euclidean quantum field theory as classical statistical mechanics. Ann. Math. 101, 111–189 (1975) and Ann. of Math. 101, 191–259 (1975)
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). [hep-th/0512271]
Haag R.: Local quantum physics: fields, particles, algebras. Springer, Berlin (1992)
Haag R., Kastler D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848–861 (1964)
Hagen C.R.: The Thirring model. Nuovo Cim. B 51, 169–186 (1967)
Ishibashi, N., Kawai, H., Kitazawa, Y., Tsuchiya, A.: A large-N reduced model as superstring. Nucl. Phys. B 498, 467–491 (1997). [hep-th/9612115]
Ising E.: Beitrag zur Theorie des Ferromagnetismus. Z. Phys. 31, 253–258 (1925)
Jaffe, A.M.: Constructive quantum field theory. In: Fokas, A. et al. (eds.) Mathematical Physics 2000. London: Imperial College Press, 2000, pp. 111–127
Jaffe, A.M., Witten, E.: Quantum Yang–Mills theory. In: Carlson, J., et al. (eds.) The millenium prize problems. Providence, Amer. Math. Soc., 2006, pp. 129–152
Jimbo, M. (ed.): Yang–Baxter equation in integrable systems, Singapore: World Scientific, 1990
Johnson K.: Solution of the equations for the Green’s functions of a two-dimensional relativistic field theory. Nuovo Cim. 20, 773–790 (1961)
Kac M.: On distributions of certain Wiener functionals. Trans. Am. Math. Soc. 65, 1–13 (1949)
Klaiber, B.: The Thirring model. In: Boulder 1967 lectures in theoretical physics, Vol. Xa. Quantum theory and statistical physics, New York, 1968, pp. 141–176
Kontsevich M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1–23 (1992)
Korepin, V.E.: Direct calculation of the S-matrix in the massive Thirring model. Theor. Math. Phys. 41, 953–967 (1979) [Teor. Mat. Fiz. 41 (1979) 169–189]
Kramers, H.A., Wannier, G.H.: Statistics of the two-dimensional ferromagnet. I+II, Phys. Rev. 60, 252–262, 263–276 (1941)
Landau, L.D., Abrikosov, A.A., Khalatnikov, I.M.: On the removal of infinities in quantum electrodynamics. (in russ.) Dokl. Akad. Nauk SSSR 95, 497–500 (1954)
Landau, L.D., Abrikosov, A.A., Khalatnikov, I.M.: Asymptotic expression of the electron Green function in quantum electrodynamics. (in russ.) Dokl. Akad. Nauk SSSR 95, 773–776 (1954)
Landau, L.D., Abrikosov, A.A., Khalatnikov, I.M.: An asymptotic expression for the photon Green function in quantum electrodynamics. (in russ.) Dokl. Akad. Nauk SSSR 95, 1117–1120 (1954)
Langmann, E., Szabo, R.J.: Duality in scalar field theory on noncommutative phase spaces. Phys. Lett. B 533, 168–177 (2002). [hep-th/0202039]
Langmann, E., Szabo, R.J., Zarembo, K.: Exact solution of quantum field theory on noncommutative phase spaces. JHEP 0401, 017 (2004). [hep-th/0308043]
Lieb E.H.: Residual entropy of square ice. Phys. Rev. 162, 162–172 (1967)
Magnen, J., Rivasseau, V.: Constructive \({{\phi^4}}\) field theory without tears. Ann. Henri Poincaré 9, 403–424 (2008) [arXiv:0706.2457 [math-ph]]
Minwalla, S., Van Raamsdonk, M., Seiberg, N.: Noncommutative perturbative dynamics. JHEP 0002, 020 (2000). [hep-th/9912072]
Muskhelishvili N.I.: Singul��re Integralgleichungen. Akademie-Verlag, Berlin (1965)
Nelson, E.: Quantum fields and Markoff fields. In: Spencer, D.C. (ed.) Partial differential equations. Providence: Amer. Math. Soc., 1973, pp. 413–420
Osterwalder K., Schrader R.: Axioms for Euclidean Green’s functions. Commun. Math. Phys. 31, 83–112 (1973)
Osterwalder K., Schrader R.: Axioms for Euclidean Green’s functions II. Commun. Math. Phys. 42, 281–305 (1975)
Onsager L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117–149 (1944)
Politzer, H.D.: Reliable perturbative results for strong interactions? Phys. Rev. Lett. 30, 1346–1349 (1973)
Rieffel, M.A.: Deformation quantization for actions of \({{\mathbb{R}^d}}\) . Memoirs AMS 506, 1–96 (1993)
Rivasseau V.: Construction and Borel summability of planar four-dimensional Euclidean field theory. Commun. Math. Phys. 95, 445–486 (1984)
Rivasseau V.: From perturbative to constructive renormalization. Princeton University Press, Princeton (1991)
Rivasseau, V., Vignes-Tourneret, F., Wulkenhaar, R.: Renormalisation of noncommutative \({{\phi^4}}\) -theory by multi-scale analysis. Commun. Math. Phys. 262, 565–594 (2006). [hep-th/0501036]
Rivasseau, V.: Non-commutative renormalization. In: Duplantier, B., Rivasseau, V. (eds.) Quantum spaces (Séminaire Poincaré X). Basel: Birkhäuser Verlag, 2007, pp 19–109. [arXiv:0705.0705 [hep-th]]
Rivasseau, V.: Constructive matrix theory. JHEP 0709, 008 (2007). [arXiv:0706.1224 [hep-th]]
Rivasseau, V., Wang, Z.: Constructive renormalization for \({{\Phi^{4}_2}}\) theory with loop vertex expansion. J. Math. Phys. 53, 042302 (2012) [arXiv:1104.3443 [math-ph]]
Schwinger J.: Euclidean quantum electrodynamics. Phys. Rev. 115, 721–731 (1959)
Schwinger J.S.: Gauge invariance and mass. II. Phys. Rev. 128, 2425–2429 (1962)
Streater R.F., Wightman A.S.: PCT, spin and statistics, and all that. Benjamin, New York (1964)
Symanzik, K.: A modified model of Euclidean quantum field theory. Courant Institute of Mathematical Sciences, New York University, Report IMM-NYU 327 (1964)
Thirring W.E.: A soluble relativistic field theory?. Ann. Phys. 3, 91–112 (1958)
Hooft G.’t.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461–473 (1974)
Hooft G.’t.: Rigorous construction of planar diagram field theories in four-dimensional Euclidean space. Commun. Math. Phys. 88, 1–25 (1983)
Tricomi F.G.: Integral equations. Interscience, New York (1957)
Wang, Z.: Constructive renormalization of 2-dimensional Grosse–Wulkenhaar model. [arXiv:1205.0196 [hep-th]]
Wess J., Zumino B.: Consequences of anomalous Ward identities. Phys. Lett. B 37, 95–97 (1971)
Wightman A.S.: Quantum field theory in terms of vacuum expectation values. Phys. Rev. 101, 860–866 (1956)
Wightman A.S., Gårding L.: Fields as operator-valued distributions in quantum field theory. Ark. Fys. 28, 129–184 (1964)
Wilson K.G., Kogut J.B.: The renormalization group and the \({{\epsilon}}\) -expansion. Phys. Rept. 12, 75–200 (1974)
Witten E.: Nonabelian bosonization in two dimensions. Commun. Math. Phys. 92, 455–472 (1984)
Witten E.: Two-dimensional gravity and intersection theory on moduli space. Surveys Diff. Geom. 1, 243–310 (1991)
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Grosse, H., Wulkenhaar, R. Self-Dual Noncommutative \({\phi^4}\) -Theory in Four Dimensions is a Non-Perturbatively Solvable and Non-Trivial Quantum Field Theory. Commun. Math. Phys. 329, 1069–1130 (2014). https://doi.org/10.1007/s00220-014-1906-3
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DOI: https://doi.org/10.1007/s00220-014-1906-3