×

zbMATH — the first resource for mathematics

Quantum extended crystal super PDEs. (English) Zbl 1258.81064
Summary: We generalize our geometric theory on extended crystal PDEs and their stability, to the category \(\mathfrak Q_{S}\) of quantum supermanifolds. By using the algebraic topologic techniques, obstructions to the existence of global quantum smooth solutions for such equations are obtained. Applications are given to encode quantum dynamics of nuclear nuclides, identified with graviton-quark-gluon plasmas, and to study their stability. We prove that such quantum dynamical systems are encoded by suitable quantum extended crystal Yang-Mills super PDEs. In this way stable nuclear-charged plasmas and nuclides are characterized as suitable stable quantum solutions of such quantum Yang-Mills super PDEs. An existence theorem of local and global solutions with mass-gap, is given for quantum super Yang-Mills PDEs, \(\widehat {(\text{YM})}\), by identifying a suitable constraint, \(\widehat {(\text{Higgs})}\subset \widehat {(\text{YM})}\), Higgs quantum super PDE, bounded by a quantum super partial differential relation \(\widehat {(\text{Goldstone})}\subset \widehat {(\text{YM})}\), quantum Goldstone-boundary. A global solution \(V\subset \widehat {(\text{YM})}\), crossing the quantum Goldstone-boundary acquires (or loses) mass. Stability properties of such solutions are characterized.

MSC:
81T13 Yang-Mills and other gauge theories in quantum field theory
46S60 Functional analysis on superspaces (supermanifolds) or graded spaces
81T60 Supersymmetric field theories in quantum mechanics
35Q40 PDEs in connection with quantum mechanics
81V22 Unified quantum theories
83E50 Supergravity
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Einstein, A., Die grundlage der allgemeinen relativitätstheorie, Ann. phys., 49, 4, 769-822, (1916) · JFM 46.1293.01
[2] Maxwell, J.C., A treatise on electricity and magnetisme, vols. 1 and 2, (1873), Oxford, Third ed. 1904
[3] Maxwell, J.C., Matter and motion, (1895), Society for Promoting Christian Knowledge London
[4] Ricci-Curbastro, G., Opere. vol. I. un. mat. ital & cons. naz. ricerche, (1956), Ed. Cremonese Roma
[5] Ricci-Curbastro, G., Opere. vol. II. un. mat. ital & cons. naz. ricerche, (1957), Ed. Cremonese Roma
[6] Schouten, J.A., Ricci-calculus, (1954), Springer-Verlag Berlin, Gottingen, Heidelberg · Zbl 0057.37803
[7] Wheeler, J.A., On the nature of quantum geometrodynamics, Ann. phys., 2, 604-614, (1957) · Zbl 0078.19201
[8] Wheeler, J.A., Geometrodynamics and the problem of motion, Rev. modern phys., 44, 1, (1961) · Zbl 0097.42601
[9] Wheeler, J.A., Geometrodynamics, (1963), Academic Press New York · Zbl 0124.22207
[10] Misner, C.W.; Thorne, K.S.; Wheeler, J.A., Gravitation, (1973), W.H. Freeman San Francisco
[11] Prástaro, A., (un)stability and bordism groups in PDE’s, Banach J. math. anal., 1, 1, 139-147, (2007) · Zbl 1130.58014
[12] Prástaro, A., Extended crystal PDE’s, [math.AT] · Zbl 1253.35135
[13] Prástaro, A., Extended crystal PDE’s stability. I: the general theory, Math. comput. modelling, 49, 9-10, 1759-1780, (2009) · Zbl 1171.35322
[14] Prástaro, A., Extended crystal PDE’s stability. II: the extended crystal MHD-PDE’s, Math. comput. modelling, 49, 9-10, 1781-1801, (2009) · Zbl 1171.35323
[15] Prástaro, A., On the extended crystal PDE’s stability. I: the \(n\)-d’alembert extended crystal PDE’s, Appl. math. comput., 204, 1, 63-69, (2008) · Zbl 1161.35054
[16] Prástaro, A., On the extended crystal PDE’s stability. II: entropy-regular-solutions in MHD-PDE’s, Appl. math. comput., 204, 1, 82-89, (2008) · Zbl 1161.35462
[17] Prástaro, A., Geometry of PDEs and mechanics, (1996), World Scientific Publishing Co Singapore · Zbl 0866.35007
[18] Prástaro, A., (co)bordisms in PDE’s and quantum PDE’s, Rep. math. phys., 38, 3, 443-455, (1996) · Zbl 0885.58094
[19] Prástaro, A., (co)bordism groups in quantum PDE’s, Acta appl. math., 64, 2/3, 111-217, (2000) · Zbl 0978.58016
[20] Prástaro, A., Quantum manifolds and integral (co)bordism groups in quantum partial differential equations, Nonlinear anal. TMA, 47/4, 2609-2620, (2001) · Zbl 1042.35610
[21] Prástaro, A., Quantized partial differential equations, (2004), World Scientific Publishing Co · Zbl 1067.58022
[22] A. Prástaro, Quantum super Yang-Mills equations: global existence and mass-gap, (eds. G.S. Ladde, N.G. Madhin & M. Sambandham), Proceedings Dynamic Systems Appl. 4 (2004) 227-234.
[23] A. Prástaro, Conservation laws in quantum super PDE’s, Proceedings of the Conference on Differential & Difference Equations and Applications, (eds. R. P. Agarwal & K. Perera), Hindawi Publishing Corporation, New York (2006), 943-952.
[24] Prástaro, A., (co)bordism groups in quantum super PDE’s. I: quantum supermanifolds, Nonlinear anal. RWA, 8, 2, 505-533, (2007) · Zbl 1152.58313
[25] Prástaro, A., (co)bordism groups in quantum super PDE’s. II: quantum super PDE’s, Nonlinear anal. RWA, 8, 2, 480-504, (2007) · Zbl 1152.58312
[26] Prástaro, A., (co)bordism groups in quantum super PDE’s. III: quantum super yang – mills equations, Nonlinear anal. RWA, 8, 2, 447-479, (2007) · Zbl 1152.58311
[27] A. Prástaro, On quantum black-hole solutions of quantum super Yang-Mills equations, Proceedings Dynamic Systems Appl. 5 (2008) 407-414.
[28] Prástaro, A., Surgery and bordism groups in quantum partial differential equations. I: the quantum Poincaré conjecture, Nonlinear anal. TMA, 71, 12, 502-525, (2009)
[29] Prástaro, A., Surgery and bordism groups in quantum partial differential equations. II: variational quantum PDE’s, Nonlinear anal. TMA, 71, 12, 526-549, (2009)
[30] Prástaro, A., Quantum extended crystal PDE’s, Nonlinear stud., 18, 3, 447-485, (2011), [math.AT] · Zbl 1253.35135
[31] Prástaro, A., Quantum exotic PDE’s, [math.AT] · Zbl 1270.81115
[32] de Witt, B., Supermanifolds, (1986), Cambridge Univers. Press Cambridge
[33] Green, M.B.; Schwarz, J.H.; Witten, E., ()
[34] van Nieuvwenhuizen, P., Supergravity, Phys. rep., 68, 4, 189-339, (1981)
[35] Wess, J.; Bagger, J., ()
[36] West, P., Introduction to supersymmetry and supergravity, (1986), World Scientific Publishing Co Singapore
[37] Witten, E., Supersymmetric yang – mills theory on a four-manifold, J. math. phys., 35, 5101-5135, (1994) · Zbl 0822.58067
[38] Sharpe, R.W., Differential geometry: cartan’s generalization of klein’s erlangen program, (1997), Springer New York · Zbl 0876.53001
[39] C. Ehresmann, Les connexions infinitésimales dans un espace fibfré différentiable (French), Centre Belge Rech. Math., Colloque de Topologie, Bruxelles, du 5 au 8 juin 1950, 29-55, 1951.
[40] Abe, E., Hopf algebras, (1980), Cambridge Univ. Press Cambridge · Zbl 0476.16008
[41] Atiyah, M., Topological quantum field theory, Pub. math. inst. hautes études sci., The geometry and physics of knots, 68, 175-186, (1990), Cambridge University Press Gambridge · Zbl 0692.53053
[42] Boardman, J.M., Singularities of differentiable maps, Pub. math. inst. hautes études sci., 33, 21-57, (1967) · Zbl 0165.56803
[43] Bourbaki, N., Theorie spectrales, (1969), Hermann Paris
[44] (), Papers dedicated to C.T.C. Wall
[45] Golubitsky, M.; Guillemin, V., Stable mappings and their singularities, (1973), Springer-Verlag New York · Zbl 0294.58004
[46] Hirsh, M.W., Differential topology, (1976), Springer-Verlag Berlin
[47] Levine, H.I., The singularities of \(S_1^q\), Illinois. J. math., 8, 152-168, (1964) · Zbl 0124.38801
[48] Levine, H.I., Elimination of cusps, Topology, 3, 263, (1965) · Zbl 0146.20001
[49] Madsen, I.B.; Milgram, R.J., ()
[50] Mather, J.N., Stability of \(C^\infty\) mappings. I: the division theorem, Ann. of math., 87, 1, 89-104, (1968) · Zbl 0159.24902
[51] Mather, J.N., Stability of \(C^\infty\) mappings. II: infinitesimal stability implies stability, Ann. of math., 87, 2, 254-291, (1969) · Zbl 0177.26002
[52] Mather, J.N., Stability of \(C^\infty\) mappings. III: finitely determined maps germs, Publ. math. inst. hautes études sci., 35, 127-156, (1968) · Zbl 0159.25001
[53] Mather, J.N., Stability of \(C^\infty\) mappings. IV: classification of stable germs by \(R\)-algebras, Publ. math. inst. hautes études sci., 37, 223-248, (1969) · Zbl 0202.55102
[54] Mather, J.N., Stability of \(C^\infty\) mappings. V: transversality, Adv. math., 4, 3, 301-336, (1970) · Zbl 0207.54303
[55] McCleary, J., User’s guide to spectral sequences, (1985), Publish or Perish in USA · Zbl 0577.55001
[56] McConnel, J.C.; Robson, J.C., Noncommutative Noetherian rings, (), 636 pp · Zbl 0644.16008
[57] Milnor, J.; Stasheff, J., ()
[58] Quillen, D., Elementary proofs of some results of cobordism theory using Steenrod operators, Adv. math., 7, 29-56, (1971) · Zbl 0214.50502
[59] Rotman, J.J., An introduction to homological algebra, (1979), Academic Press San Diego · Zbl 0441.18018
[60] Rudyak, Y.B., On thom spectra, orientability and cobordism, (1998), Springer-Verlag Berlin · Zbl 0906.55001
[61] Stong, R.E., ()
[62] Sullivan, D., René thom’s work on geometric homology and bordism, Bull. amer. math. soc., 41, 3, 341-350, (2004) · Zbl 1045.57001
[63] Sweedler, M.E., Hopf algebras, (1969), Benjamin New York · Zbl 0194.32901
[64] Switzer, R., Algebraic topology-homotopy and homology, (1975), Springer-Verlag Berlin · Zbl 0305.55001
[65] Thom, R., Quelques propriété globales des variétés différentieles, Comm. math. helv., 28, 17-86, (1954) · Zbl 0057.15502
[66] Thom, R., Remarques sur LES problemes comportant des inéqualities différentielles globales, Bull. soc. math. France, 87, 455-468, (1954) · Zbl 0213.25302
[67] Thom, R., LES singularités des applications differentiables, Ann. inst. Fourier (Grenoble), 6, 43-87, (1955-56)
[68] Wall, C.T.C., Determination of the cobordism ring, Ann. of math., 72, 292-311, (1960) · Zbl 0097.38801
[69] Wall, C.T.C., (), Amer. math. soc. surveys and monographs, vol. 69, (1999), Amer. Math. Soc.
[70] Manin, Yu., Topics in noncommutative geometry, (1991), Princeton University Press Princeton, NJ · Zbl 0724.17007
[71] E.S. Fedorov, The Symmetry and Structure of Crystals. Fundamental Works, Moscow, 1949, pp. 111-255 (in Russian: “Simmetria i Struktura Kristallov”, Osnovnye roboty. [Moscow] (1949)). Translated from the 1949 Russian edition. American Crystallographic Association, New York, 1971.
[72] Hahn, Th., International tables for crystallography. vol. A: space-groups symmetry, (2006), Springer
[73] Plesken, W.; Pest, M., On maximal finite irreducible subgroup of \(G L(n, \mathbb{Z})\), I, II, Math. comp., 31, 536-551, (1977), 552-573
[74] Raghunathan, M.S., Discrete subgroups of Lie groups, (1972), Springer-Verlag New York · Zbl 0254.22005
[75] Schönflies, A., Krystallsysteme und kristallstruktur, (1891), Teubner · JFM 23.0554.02
[76] Schwartzenberger, R.L.E., \(N\)-dimensional crystallography, (1980), Pitman
[77] Sunada, T., Crystals that nature might miss creating, Notices amer. math. soc., 55, 2, 209-215, (2008) · Zbl 1144.82071
[78] Keller, H.H., ()
[79] Bryant, R.L.; Chern, S.S.; Gardner, R.B.; Goldschmidt, H.L.; Griffiths, P.A., Exterior differential systems, (1991), Springer-Verlag New York · Zbl 0726.58002
[80] Goldshmidt, H., Integrability criteria for systems of non-linear partial differential equations, J. differential geom., 1, 269-307, (1967)
[81] Goldshmidt, H.; Spencer, D., Submanifolds and over-determined differential operators, (), 319-356
[82] Gromov, M., Partial differential relations, (1986), Springer-Verlag Berlin · Zbl 0651.53001
[83] Krasilśhchik, I.S.; Lychagin, V.; Vinogradov, A.M., Geometry of jet spaces and nonlinear partial differential equations, (1986), Gordon and Breach Science Publishers S.A Amsterdam
[84] Kuranishi, M., On E. cartan’s prolongation theorem of exterior differential systems, Amer. J. math., 79, 1-47, (1957) · Zbl 0077.29701
[85] Kuranishi, M., Lectures on exterior differantial systems, Tata inst. fund. res. stud. math., (1962)
[86] Kuranishi, M., Sheaves defined by deformation theory of pseudogroup structures, Amer. J. math., 86, 379-391, (1964) · Zbl 0119.07704
[87] Kuranishi, M., Lectures on involutive systems, (1969), Publications Institute Mathematics Sao Paolo · Zbl 0163.12001
[88] Libermann, P., Introduction à l’étude de certaines systèmes différentiels, (), Cah. topol. Géom. différ. catég., 23, 55-72, (1982)
[89] Lychagin, V.; Prástaro, A., Singularities of Cauchy data, characteristics, cocharacteristics and integral cobordism, Differential geom. appl., 4, 287-300, (1994) · Zbl 0808.58039
[90] Palais, R.S., Manifolds of sections of the fiber bundles and the calculus of variations, (), Foundations of global non-linear analysis, 195-205, (1968), W.A. Benjamin Inc New York-Amsterdam, 131 pp
[91] Prástaro, A., Gauge geometrodynamics, Riv. nuovo cimento, 5, 4, 1-122, (1982) · Zbl 0695.58028
[92] Prástaro, A., Dynamic conservation laws, () · Zbl 1131.35381
[93] Prástaro, A., Cobordism of PDE’s, Boll. unione mat. ital., 30, 5-B, 977-1001, (1991) · Zbl 0746.57015
[94] Prástaro, A., Quantum geometry of PDE’s, Rep. math. phys., 30, 3, 273-354, (1991) · Zbl 0771.58024
[95] Prástaro, A.; Regge, T., The group structure of supergravity, Ann. inst. H. Poincaré phys. théor., 44, 1, 39-89, (1986) · Zbl 0588.53066
[96] Spencer, D.C., Determination of structures on manifolds defined by transitive continuous pseudogroups, I. II. III., Ann. of math., 7, 51-114, (1965)
[97] Spencer, D.C., Overdetermined systems of linear partial differential equations, Bull. amer. math. soc., 75, 79-239, (1969) · Zbl 0185.33801
[98] Takens, F., A global version of the inverse problem of the calculus of variations, J. differential geom., 14, 4, 543-562, (1979) · Zbl 0463.58015
[99] Prástaro, A., Geometry of super PDE’s., (), 259-315 · Zbl 0879.58080
[100] Prástaro, A., Geometry of quantized super PDE’s, Amer. math. soc. transl., 167, 2, 165-192, (1995) · Zbl 0844.58012
[101] Prástaro, A., Quantum geometry of super PDE’s, Rep. math. phys, 37, 1, 23-140, (1996) · Zbl 0887.58064
[102] Prástaro, A., Quantum and integral (co)bordism groups in partial differential equations, Acta appl. math., 51, 3, 243-302, (1998) · Zbl 0924.58103
[103] Prástaro, A., (co)bordism groups in PDE’s, Acta appl. math., 59, 2, 111-202, (1999) · Zbl 0949.35011
[104] Prástaro, A., Geometry of PDE’s. I: integral bordism groups in PDE’s, J. math. anal. appl., 319, 547-566, (2006) · Zbl 1100.35007
[105] Prástaro, A., Geometry of PDE’s. II: variational PDE’s and integral bordism groups, J. math. anal. appl., 321, 930-948, (2006) · Zbl 1160.58301
[106] Prástaro, A., Geometry of PDE’s. IV: navier – stokes equation and integral bordism groups, Math. anal. appl., 338, 2, 1140-1151, (2008) · Zbl 1135.35064
[107] Prástaro, A., Exotic heat PDE’s, Comput. math. appl., 10, 1, 64-81, (2011), [math.GT] · Zbl 1276.55008
[108] A. Prástaro, Exotic heat PDE’s. II, In: P.M. Pardalos and Th.M. Rassias (Eds.), Essays in Mathematics and its Applications. (Dedicated to Stephen Smale.) Springer, New York. arXiv: 1009.1176 [math.AT] (in press).
[109] A. Prástaro, Exotic n-d’Alembert PDE’s and stability, In: G. Georgiev (USA), P. Pardalos (USA) and H.M. Srivastava (Canada) (Eds.), Stability, Approximation and Inequalities. (Dedicated to Themistocles M. Rassias for his 60th), Springer, New York. arXiv:1011.0081 [math.AT] (in press).
[110] Prástaro, A., Exotic PDE’s, [math.AT] · Zbl 1326.55004
[111] Agarwal, R.P.; Prástaro, A., Geometry of PDE’s. III(I): webs on PDE’s and integral bordism groups. the general theory, Adv. math. sci. appl., 17, 1, 239-266, (2007) · Zbl 1143.53017
[112] Agarwal, R.P.; Prástaro, A., Geometry of PDE’s. III(II): webs on PDE’s and integral bordism groups. applications to Riemannian geometry PDE’s, Adv. math. sci. appl., 17, 1, 267-281, (2007) · Zbl 1140.53005
[113] Agarwal, R.P.; Prástaro, A., Singular PDE’s geometry and boundary value problems, J. nonlinear convex anal., 9, 3, 417-460, (2008) · Zbl 1171.35006
[114] Agarwal, R.P.; Prástaro, A., On singular PDE’s geometry and boundary value problems, Appl. anal., 88, 8, 1115-1131, (2009) · Zbl 1180.35012
[115] Prástaro, A.; Rassias, Th.M., A geometric approach to an equation of J.d’alembert, Proc. amer. math. soc., A geometric approach of the generalized d’alembert equation, J. comput. appl. math., 113, 1-2, 93-122, (2000)
[116] Prástaro, A.; Rassias, Th.M., A geometric approach to a noncommutative generalized d’alembert equation, C. R. acad. sci. Paris, 330, I-7, 545-550, (2000) · Zbl 0966.35105
[117] Prástaro, A.; Rassias, Th.M., Ulam stability in geometry of PDE’s, Nonlinear funct. anal. appl., 8, 2, 259-278, (2003) · Zbl 1096.39028
[118] Prástaro, A., On the general structure of continuum physics. I: derivative spaces, Boll. unione mat. ital., On the general structure of continuum physics. II: differential operators, Boll. unione mat. ital., On the general structure of continuum physics. III: the physical picture, Boll. unione mat. ital., 5, S-FM, 107-129, (1981) · Zbl 0478.58005
[119] Hyers, D.H., On the stability of the linear functional equation, Proc. natl. acad. sci. USA, 27, 222-224, (1941) · JFM 67.0424.01
[120] Ulam, S.M., A collection of mathematical problems, (1960), Interscience Publ. New York · Zbl 0086.24101
[121] Ljapunov, A.M., Stability of motion, with an contribution by V.A. pliss and an introduction by V.P. basov, ()
[122] Grönwall, T.H., Note on the derivative with respect to a parameter of the solutions of a system of differential equations, Ann. of math., 20, 292-296, (1919) · JFM 47.0399.02
[123] Dunford, N.; Shwartz, J.T., Linear operators, part II, (1958), Interscience
[124] M.N. Chernodub, A. Nakamura, V.I. Zakharov, Manifestations of magnetic vortices in equation of state of Yang-Mills plasma. arXiv:0807.5012v1 [hep-ph] 31 Jul. 2008.
[125] M.N. Chernodub, A. Nakamura, V.I. Zakharov, Abelian monopoles and center vortices in Yang-Mills plasmas. arXiv:0812.4633v1 [hep-ph] 25 Dec. 2008.
[126] E. Rondio, Gluons’ polarization in the nucleon, In: SPIN 2008, The 18th International Spin Physics Symposium, University of Virginia, October 6-11, 2008, USA.
[127] von Neumann, J., Mathematical foundations of quantum mechanics, (1955), Princeton Univ. Press Princeton
[128] Demuth, M.; Krishna, M., ()
[129] Dirac, P.M.A., The principles of quantum mechanics, (1958), Oxford University Press London
[130] Kato, T., Perturbation theory for linear operators, (1976), Springer-Verlag Berlin
[131] von Neumann, J., ()
[132] Weyl, H., The theory of groups and quantum mechanics, (1931), University Press Princeton, N.Y · JFM 58.1374.01
[133] Hazewinkel, M., Encyclopaedia of mathematics. supplement volume I, (1988), Springer · Zbl 0676.00027
[134] Michael, E.A., Locally multiplactivelly-convex topological algebras, Mem. amer. math. soc., 11, (1952)
[135] Goldstone, J., Field theories with “superconductor” solutions, Nuovo cimento, X, 19, 154-164, (1961) · Zbl 0099.23006
[136] Goldstone, J.; Salam, A.; Weinberg, S., Broken symmetries, Phys. rev., 127, 3, 965-970, (1962) · Zbl 0106.20601
[137] Higgs, P.W., Broken symmetries, massless particles and gauge fields, Phys. lett., 12, 2, 132-133, (1964)
[138] Higgs, P.W., Broken symmetries and the masses of gauge bosons, Phys. rev. lett., 13, 16, 508-509, (1964)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.