Stochastic quantisation of Yang-Mills-Higgs in 3D. (English) Zbl 07887295
The purpose of this paper is to construct and study the Langevin dynamic associated to the Euclidean Yang-Mills-Higgs (YMH) measure on the torus \(\mathbf{T}^d\), for \(d=3\). The authors establish both a state space and a Markov process associated to the stochastic quantisation equation of Yang-Mills-Higgs (YMH) theories. The state space \(S\) is a nonlinear metric space of distributions that can serve as initial conditions for the (deterministic and stochastic) YMH flow with good continuity properties. By taking into account the gauge covariance of the deterministic YMH flow, they extend the gauge equivalence relation \(\sim\) to \(S\) in a canonic way, creating a quotient space of “gauge orbits” \(\mathcal{O}\). Using the theory of regularity structures, they demonstrate local in time solutions to the renormalised stochastic YMH flow. Additionally, symmetry arguments in the small noise limit result in a unique selection of renormalization counterterms, ensuring that these solutions are gauge covariant in law. This enables the definition of a canonical Markov process on the space of gauge orbits \(\mathcal{O}\) up to a potential finite time blow-up, associated with the stochastic YMH flow. The work aloso includes four appendices. Appendix A discusses modelled distributions with singular behaviour at \(t=0\) (allowing for the construction of solutions starting from suitable singular initial conditions). Appendix B contains results on the deterministic YMH flow (helpful for defining the gauge equivalence). Appendix C extends the well-posedness result from \(\S\) 5 to the coupled system discussed in \(\S\) 6 and \(\S\) 7. Appendix D proves that the solution maps are injective in law as functions of renormalisation constants (which is useful for showing gauge covariance in \(\S\) 6).
Reviewer: Vladimir Balan (Bucureşti)
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60L30 | Regularity structures |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
Keywords:
Markov process; state space; stochastic quantisation; Yang-Mills-Higgs theories; gauge covariance; gauge orbits; theory of regularity structures; renormalised stochastic flowReferences:
| [1] | Albeverio, S.; Kusuoka, S., The invariant measure and the flow associated to the \(\Phi^4_3\)-quantum field model, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), 20, 4, 1359-1427, 2020 · Zbl 1480.81079 · doi:10.2422/2036-2145.201809_008 |
| [2] | Balaban, T., Ultraviolet stability of three-dimensional lattice pure gauge field theories, Commun. Math. Phys., 102, 2, 255-275, 1985 · Zbl 0595.58047 · doi:10.1007/BF01229380 |
| [3] | Balaban, T., Renormalization group approach to lattice gauge field theories. I. Generation of effective actions in a small field approximation and a coupling constant renormalization in four dimensions, Commun. Math. Phys., 109, 2, 249-301, 1987 · Zbl 0611.53080 · doi:10.1007/BF01215223 |
| [4] | Balaban, T., Large field renormalization. II. Localization, exponentiation, and bounds for the \(\mathbf{R}\) operation, Commun. Math. Phys., 122, 3, 355-392, 1989 · Zbl 0704.58060 · doi:10.1007/BF01238433 |
| [5] | Barashkov, N.; Gubinelli, M., A variational method for \(\Phi^4_3\), Duke Math. J., 169, 17, 3339-3415, 2020 · Zbl 1508.81928 · doi:10.1215/00127094-2020-0029 |
| [6] | Barashkov, N.; Gubinelli, M., The \({\Phi_3^4}\) measure via Girsanov’s theorem, Electron. J. Probab., 26, 1-29, 2021 · Zbl 1469.81040 · doi:10.1214/21-EJP635 |
| [7] | Bern, Z.; Halpern, M. B.; Sadun, L.; Taubes, C., Continuum regularization of quantum field theory. II. Gauge theory, Nucl. Phys. B, 284, 1, 35-91, 1987 · doi:10.1016/0550-3213(87)90026-5 |
| [8] | Bogachev, V. I., Measure Theory. Vol. I, II, 2007, Berlin: Springer, Berlin · Zbl 1120.28001 · doi:10.1007/978-3-540-34514-5 |
| [9] | Bringmann, B., Cao, S.: Global well-posedness of the stochastic Abelian-Higgs equations in two dimensions. ArXiv e-prints (2024). arXiv:2403.16878 |
| [10] | Bruned, Y.; Hairer, M.; Zambotti, L., Algebraic renormalisation of regularity structures, Invent. Math., 215, 3, 1039-1156, 2019 · Zbl 1481.16038 · doi:10.1007/s00222-018-0841-x |
| [11] | Bruned, Y.; Chandra, A.; Chevyrev, I.; Hairer, M., Renormalising SPDEs in regularity structures, J. Eur. Math. Soc., 23, 3, 869-947, 2021 · Zbl 1465.60057 · doi:10.4171/jems/1025 |
| [12] | Bruned, Y.; Gabriel, F.; Hairer, M.; Zambotti, L., Geometric stochastic heat equations, J. Am. Math. Soc., 35, 1, 1-80, 2021 · Zbl 1476.60101 · doi:10.1090/jams/977 |
| [13] | Brydges, D.; Fröhlich, J.; Seiler, E., On the construction of quantized gauge fields. I. General results, Ann. Phys., 121, 1-2, 227-284, 1979 · doi:10.1016/0003-4916(79)90098-8 |
| [14] | Brydges, D. C.; Fröhlich, J.; Seiler, E., Construction of quantised gauge fields. II. Convergence of the lattice approximation, Commun. Math. Phys., 71, 2, 159-205, 1980 · doi:10.1007/BF01197918 |
| [15] | Brydges, D. C.; Fröhlich, J.; Seiler, E., On the construction of quantized gauge fields. III. The two-dimensional Abelian Higgs model without cutoffs, Commun. Math. Phys., 79, 3, 353-399, 1981 · doi:10.1007/BF01208500 |
| [16] | Cao, S., Wilson loop expectations in lattice gauge theories with finite gauge groups, Commun. Math. Phys., 380, 3, 1439-1505, 2020 · Zbl 1483.81114 · doi:10.1007/s00220-020-03912-z |
| [17] | Cao, S.; Chatterjee, S., The Yang-Mills heat flow with random distributional initial data, Commun. Partial Differ. Equ., 48, 2, 209-251, 2023 · Zbl 1512.35011 · doi:10.1080/03605302.2023.2169937 |
| [18] | Cao, S.; Chatterjee, S., A state space for 3D Euclidean Yang-Mills theories, Commun. Math. Phys., 405, 1, 2024 · Zbl 1537.81146 · doi:10.1007/s00220-023-04870-y |
| [19] | Chandra, A., Chevyrev, I.: Gauge field marginal of an Abelian Higgs model. ArXiv e-prints (2022). arXiv:2207.05443 · Zbl 07868802 |
| [20] | Chandra, A., Ferdinand, L.: A flow approach to the generalized KPZ equation. ArXiv e-prints (2024). arXiv:2402.03101 |
| [21] | Chandra, A., Hairer, M.: An analytic BPHZ theorem for regularity structures. ArXiv e-prints (2016). arXiv:1612.08138 |
| [22] | Chandra, A.; Chevyrev, I.; Hairer, M.; Shen, H., Langevin dynamic for the 2D Yang-Mills measure, Publ. Math. Inst. Hautes Études Sci., 136, 1-147, 2022 · Zbl 1518.70029 · doi:10.1007/s10240-022-00132-0 |
| [23] | Charalambous, N.; Gross, L., The Yang-Mills heat semigroup on three-manifolds with boundary, Commun. Math. Phys., 317, 3, 727-785, 2013 · Zbl 1279.58005 · doi:10.1007/s00220-012-1558-0 |
| [24] | Charalambous, N.; Gross, L., Neumann domination for the Yang-Mills heat equation, J. Math. Phys., 56, 7, 2015 · Zbl 1321.81041 · doi:10.1063/1.4927250 |
| [25] | Chatterjee, S., The leading term of the Yang-Mills free energy, J. Funct. Anal., 271, 10, 2944-3005, 2016 · Zbl 1364.70042 · doi:10.1016/j.jfa.2016.04.032 |
| [26] | Chatterjee, S., Yang-Mills for probabilists, Probability and Analysis in Interacting Physical Systems, 1-16, 2019, Cham: Springer, Cham · Zbl 1430.81054 · doi:10.1007/978-3-030-15338-0_1 |
| [27] | Chatterjee, S., Wilson loops in Ising lattice gauge theory, Commun. Math. Phys., 377, 1, 307-340, 2020 · Zbl 1441.81122 · doi:10.1007/s00220-020-03738-9 |
| [28] | Chatterjee, S., A probabilistic mechanism for quark confinement, Commun. Math. Phys., 385, 2, 1007-1039, 2021 · Zbl 1467.82010 · doi:10.1007/s00220-021-04086-y |
| [29] | Chen, X.-Y., A strong unique continuation theorem for parabolic equations, Math. Ann., 311, 4, 603-630, 1998 · Zbl 0990.35028 · doi:10.1007/s002080050202 |
| [30] | Chevyrev, I., Yang-Mills measure on the two-dimensional torus as a random distribution, Commun. Math. Phys., 372, 3, 1027-1058, 2019 · Zbl 1434.35134 · doi:10.1007/s00220-019-03567-5 |
| [31] | Chevyrev, I., Norm inflation for a non-linear heat equation with Gaussian initial conditions, Stoch. Partial Differ. Equ., Anal. Computat., 2023 · Zbl 07914048 · doi:10.1007/s40072-023-00317-6 |
| [32] | Chevyrev, I., Shen, H.: Invariant measure and universality of the 2D Yang-Mills Langevin dynamic. ArXiv e-prints (2023). arXiv:2302.12160 |
| [33] | Del Debbio, L.; Patella, A.; Rago, A., Space-time symmetries and the Yang-Mills gradient flow, J. High Energy Phys., 212, 2013 · Zbl 1342.81271 · doi:10.1007/JHEP11(2013)212 |
| [34] | DeTurck, D. M., Deforming metrics in the direction of their Ricci tensors, J. Differ. Geom., 18, 1, 157-162, 1983 · Zbl 0517.53044 · doi:10.4310/JDG/1214509286 |
| [35] | Donaldson, S. K., Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. (3), 50, 1, 1-26, 1985 · doi:10.1112/plms/s3-50.1.1 |
| [36] | Donaldson, S. K.; Kronheimer, P. B., The Geometry of Four-Manifolds, 1990, New York: The Clarendon Press, New York · Zbl 0820.57002 · doi:10.1093/oso/9780198535539.001.0001 |
| [37] | Driver, B. K., Convergence of the \({\text{U}}(1)_4\) lattice gauge theory to its continuum limit, Commun. Math. Phys., 110, 3, 479-501, 1987 · Zbl 0638.58031 · doi:10.1007/BF01212424 |
| [38] | Driver, B. K., YM_2: continuum expectations, lattice convergence, and lassos, Commun. Math. Phys., 123, 4, 575-616, 1989 · Zbl 0819.58043 · doi:10.1007/BF01218586 |
| [39] | Duch, P.: Flow equation approach to singular stochastic PDEs. ArXiv e-prints (2021). arXiv:2109.11380 |
| [40] | Durhuus, B., On the structure of gauge invariant classical observables in lattice gauge theories, Lett. Math. Phys., 4, 6, 515-522, 1980 · Zbl 0463.22019 · doi:10.1007/BF00943439 |
| [41] | Fine, D. S., Quantum Yang-Mills on a Riemann surface, Commun. Math. Phys., 140, 2, 321-338, 1991 · Zbl 0734.53069 · doi:10.1007/BF02099502 |
| [42] | Fodor, Z.; Holland, K.; Kuti, J.; Nogradi, D.; Wong, C. H., The Yang-Mills gradient flow in finite volume, J. High Energy Phys., 2012, 11, 2012 · Zbl 1397.81196 · doi:10.1007/JHEP11(2012)007 |
| [43] | Forsström, M. P.; Lenells, J.; Viklund, F., Wilson loops in finite Abelian lattice gauge theories, Ann. Inst. Henri Poincaré Probab. Stat., 58, 4, 2129-2164, 2022 · Zbl 1513.81102 · doi:10.1214/21-aihp1227 |
| [44] | Forsström, M. P.; Lenells, J.; Viklund, F., Wilson loops in the Abelian lattice Higgs model, Probab. Math. Phys., 4, 2, 257-329, 2023 · Zbl 1538.81044 · doi:10.2140/pmp.2023.4.257 |
| [45] | Friz, P. K.; Hairer, M., A Course on Rough Paths. With an Introduction to Regularity Structures, 2014, Cham: Springer, Cham · Zbl 1327.60013 · doi:10.1007/978-3-319-08332-2 |
| [46] | Garban, C.; Sepúlveda, A., Improved spin-wave estimate for Wilson loops in \(U(1)\) lattice gauge theory, Int. Math. Res. Not., 2023, 21, 18142-18198, 2023 · Zbl 1537.81120 · doi:10.1093/imrn/rnac356 |
| [47] | Gerencsér, M.; Hairer, M., Singular SPDEs in domains with boundaries, Probab. Theory Relat. Fields, 173, 3-4, 697-758, 2019 · Zbl 1411.60097 · doi:10.1007/s00440-018-0841-1 |
| [48] | Gross, L., Convergence of \({\text{U}}(1)_3\) lattice gauge theory to its continuum limit, Commun. Math. Phys., 92, 2, 137-162, 1983 · Zbl 0571.60011 · doi:10.1007/BF01210842 |
| [49] | Gross, L., Lattice gauge theory; heuristics and convergence, Stochastic Processes—Mathematics and Physics (Bielefeld, 1984), 130-140, 1986, Berlin: Springer, Berlin · Zbl 0579.60002 · doi:10.1007/BFb0080213 |
| [50] | Gross, L.: Stability of the Yang-Mills heat equation for finite action. ArXiv e-prints (2017). arXiv:1711.00114 |
| [51] | Gross, L., The Yang-Mills heat equation with finite action in three dimensions, Mem. Am. Math. Soc., 275, 2022 · Zbl 1482.35001 · doi:10.1090/memo/1349 |
| [52] | Gross, L.; King, C.; Sengupta, A., Two-dimensional Yang-Mills theory via stochastic differential equations, Ann. Phys., 194, 1, 65-112, 1989 · Zbl 0698.60047 · doi:10.1016/0003-4916(89)90032-8 |
| [53] | Gubinelli, M.; Hofmanová, M., A PDE construction of the Euclidean \(\phi_3^4\) quantum field theory, Commun. Math. Phys., 384, 1, 1-75, 2021 · Zbl 1514.81190 · doi:10.1007/s00220-021-04022-0 |
| [54] | Gubinelli, M., Meyer, S.-J.: The FBSDE approach to sine-Gordon up to \(6\pi \). ArXiv e-prints (2024). arXiv:2401.13648 |
| [55] | Gubinelli, M.; Imkeller, P.; Perkowski, N., Paracontrolled distributions and singular PDEs, Forum Math. Pi, 3, 2015 · Zbl 1333.60149 · doi:10.1017/fmp.2015.2 |
| [56] | Hairer, M., A theory of regularity structures, Invent. Math., 198, 2, 269-504, 2014 · Zbl 1332.60093 · doi:10.1007/s00222-014-0505-4 |
| [57] | Hairer, M.; Mattingly, J., The strong Feller property for singular stochastic PDEs, Ann. Inst. Henri Poincaré Probab. Stat., 54, 3, 1314-1340, 2018 · Zbl 1426.60088 · doi:10.1214/17-AIHP840 |
| [58] | Hairer, M.; Schönbauer, P., The support of singular stochastic partial differential equations, Forum Math. Pi, 10, 2022 · Zbl 1490.60186 · doi:10.1017/fmp.2021.18 |
| [59] | Hairer, M.; Steele, R., The \(\Phi_3^4\) measure has sub-Gaussian tails, J. Stat. Phys., 186, 3, 2022 · Zbl 1490.81104 · doi:10.1007/s10955-021-02866-3 |
| [60] | Hall, B., Lie groups, Lie Algebras, and Representations: An Elementary Introduction, 2015, Cham: Springer, Cham · Zbl 1316.22001 · doi:10.1007/978-3-319-13467-3 |
| [61] | Hong, M.-C.; Tian, G., Global existence of the m-equivariant Yang-Mills flow in four dimensional spaces, Commun. Anal. Geom., 12, 1, 183-211, 2004 · Zbl 1068.58006 · doi:10.4310/CAG.2004.V12.N1.A10 |
| [62] | Jaffe, A.; Witten, E., Quantum Yang-Mills theory, The Millennium Prize Problems, 129-152, 2006, Cambridge: Clay Math. Inst, Cambridge · Zbl 1194.81002 |
| [63] | Kallenberg, O., Foundations of Modern Probability, 2021, Cham: Springer, Cham · Zbl 1478.60001 · doi:10.1007/978-3-030-61871-1 |
| [64] | King, C., The \({\text{U}}(1)\) Higgs model. I. The continuum limit, Commun. Math. Phys., 102, 4, 649-677, 1986 · Zbl 0593.46063 · doi:10.1007/BF01221651 |
| [65] | King, C., The \({\text{U}}(1)\) Higgs model. II. The infinite volume limit, Commun. Math. Phys., 103, 2, 323-349, 1986 · Zbl 0593.46064 · doi:10.1007/BF01206942 |
| [66] | Kupiainen, A., Renormalization group and stochastic PDEs, Ann. Henri Poincaré, 17, 3, 497-535, 2016 · Zbl 1347.81063 · doi:10.1007/s00023-015-0408-y |
| [67] | Lévy, T., Yang-Mills measure on compact surfaces, Mem. Am. Math. Soc., 166, 2003 · Zbl 1036.58009 · doi:10.1090/memo/0790 |
| [68] | Lévy, T., Wilson loops in the light of spin networks, J. Geom. Phys., 52, 4, 382-397, 2004 · Zbl 1078.81059 · doi:10.1016/j.geomphys.2004.04.003 |
| [69] | Lévy, T.; Sengupta, A., Four chapters on low-dimensional gauge theories, Stochastic Geometric Mechanics, 115-167, 2017, Cham: Springer, Cham · Zbl 1454.81002 · doi:10.1007/978-3-319-63453-1_7 |
| [70] | Lüscher, M., Properties and uses of the Wilson flow in lattice QCD, J. High Energy Phys., 2010, 8, 2010 · Zbl 1291.81393 · doi:10.1007/JHEP08(2010)071 |
| [71] | Magnen, J.; Rivasseau, V.; Sénéor, R., Construction of \({\text{YM}}_4\) with an infrared cutoff, Commun. Math. Phys., 155, 2, 325-383, 1993 · Zbl 0790.53056 · doi:10.1007/BF02097397 |
| [72] | Migdal, A. A., Recursion equations in gauge theories, Sov. Phys. JETP, 42, 1975 |
| [73] | Moinat, A.; Weber, H., Space-time localisation for the dynamic \(\Phi^4_3\) model, Commun. Pure Appl. Math., 73, 12, 2519-2555, 2020 · Zbl 1455.35309 · doi:10.1002/cpa.21925 |
| [74] | Mourrat, J.-C.; Weber, H., The dynamic \(\Phi^4_3\) model comes down from infinity, Commun. Math. Phys., 356, 3, 673-753, 2017 · Zbl 1384.81068 · doi:10.1007/s00220-017-2997-4 |
| [75] | Mujica, J., Holomorphic functions on Banach spaces, Note Mat., 25, 2, 113-138, 2005 · Zbl 1195.46046 |
| [76] | Narayanan, R.; Neuberger, H., Infinite \(N\) phase transitions in continuum Wilson loop operators, J. High Energy Phys., 2006, 3, 2006 · Zbl 1226.81149 · doi:10.1088/1126 |
| [77] | Nelson, E., Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev., 150, 1079-1085, 1966 · doi:10.1103/PhysRev.150.1079 |
| [78] | Parisi, G.; Wu, Y. S., Perturbation theory without gauge fixing, Sci. Sin., 24, 4, 483-496, 1981 · Zbl 1480.81051 · doi:10.1360/ya1981-24-4-483 |
| [79] | Sengupta, A., The Yang-Mills measure for \(S^2\), J. Funct. Anal., 108, 2, 231-273, 1992 · Zbl 0769.60009 · doi:10.1016/0022-1236(92)90025-E |
| [80] | Sengupta, A., Gauge invariant functions of connections, Proc. Am. Math. Soc., 121, 3, 897-905, 1994 · Zbl 0823.58007 · doi:10.2307/2160291 |
| [81] | Sengupta, A., Gauge theory on compact surfaces, Mem. Am. Math. Soc., 126, 1997 · Zbl 0873.58076 · doi:10.1090/memo/0600 |
| [82] | Sepanski, M. R., Compact Lie groups, 2007, New York: Springer, New York · Zbl 1246.22001 · doi:10.1007/978-0-387-49158-5 |
| [83] | Shen, H., Stochastic quantization of an Abelian gauge theory, Commun. Math. Phys., 384, 3, 1445-1512, 2021 · Zbl 1509.81567 · doi:10.1007/s00220-021-04114-x |
| [84] | Shen, H.; Zhu, R.; Zhu, X., A stochastic analysis approach to lattice Yang-Mills at strong coupling, Commun. Math. Phys., 400, 2, 805-851, 2023 · Zbl 1522.81127 · doi:10.1007/s00220-022-04609-1 |
| [85] | Shen, H., Smith, S.A., Zhu, R.: A new derivation of the finite N master loop equation for lattice Yang-Mills. Electron. J. Probab. 29 (2024). doi:10.1214/24-ejp1090 · Zbl 1535.60107 |
| [86] | Shen, H., Zhu, R., Zhu, X.: Langevin dynamics of lattice Yang-Mills-Higgs and applications. ArXiv e-prints (2024). arXiv:2401.13299 |
| [87] | Waldron, A., Long-time existence for Yang-Mills flow, Invent. Math., 217, 3, 1069-1147, 2019 · Zbl 1507.53096 · doi:10.1007/s00222-019-00877-2 |
| [88] | Zwanziger, D., Covariant quantization of gauge fields without Gribov ambiguity, Nucl. Phys. B, 192, 1, 259-269, 1981 · doi:10.1016/0550-3213(81)90202-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
