On a geometric framework for Lagrangian supermechanics

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December 2017, 9(4): 411-437. doi: 10.3934/jgm.2017016

On a geometric framework for Lagrangian supermechanics

Andrew James Bruce ^1,, , Katarzyna Grabowska ^2, and Giovanni Moreno ^3,

1.	Mathematics Research Unit, University of Luxembourg, Maison du Nombre 6, avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg
2.	Faculty of Physics, University of Warsaw, Pasteura 5, 02-093, Warszawa, Poland
3.	Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656, Warszawa, Poland

* Corresponding author: Andrew James Bruce

Received September 2016 Revised March 2017 Published October 2017

Fund Project: KG was supported by the Polish National Science Centre grant DEC-2012/06/A/ST1/00256. GM supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska–Curie grant agreement No 654721 'GEOGRAL'.

We re-examine classical mechanics with both commuting and anticommuting degrees of freedom. We do this by defining the phase dynamics of a general Lagrangian system as an implicit differential equation in the spirit of Tulczyjew. Rather than parametrising our basic degrees of freedom by a specified Grassmann algebra, we use arbitrary supermanifolds by following the categorical approach to supermanifolds.

Keywords: Supermechanics, supermanifolds, Lagrangian systems, Tulczyjew triples, curves.

Mathematics Subject Classification: Primary: 58A50; Secondary: 70H03.

Citation: Andrew James Bruce, Katarzyna Grabowska, Giovanni Moreno. On a geometric framework for Lagrangian supermechanics. Journal of Geometric Mechanics, 2017, 9 (4) : 411-437. doi: 10.3934/jgm.2017016

References:

[1]	V. P. Akulov and S. Duplij, Nilpotent marsh and SUSY QM, In Supersymmetries and quantum symmetries (Dubna, 1997), volume 524 of Lecture Notes in Phys., pages 235–242. Springer, Berlin, 1999. doi: 10.1007/BFb0104604. Google Scholar
[2]	M. Batchelor, The structure of supermanifolds, Trans. Amer. Math. Soc., 253 (1979), 329-338. doi: 10.1090/S0002-9947-1979-0536951-0. Google Scholar
[3]	F. A. Berezin and M. S. Marinov, Particle spin dynamics as the grassmann variant of classical mechanics, Annals of Physics, 104 (1977), 336-362. doi: 10.1016/0003-4916(77)90335-9. Google Scholar
[4]	A. J. Bruce, On curves and jets of curves on supermanifolds, Arch. Math. (Brno), 50 (2014), 115-130. doi: 10.5817/AM2014-2-115. Google Scholar
[5]	J. F. Cariñena and H. Figueroa, A geometrical version of Noether's theorem in supermechanics, Rep. Math. Phys., 34 (1994), 277-303. doi: 10.1016/0034-4877(94)90002-7. Google Scholar
[6]	J. F. Cariñena and H. Figueroa, Hamiltonian versus Lagrangian formulations of supermechanics, J. Phys. A, 30 (1997), 2705-2724. doi: 10.1088/0305-4470/30/8/017. Google Scholar
[7]	J. F. Cariñena and H. Figueroa, Singular Lagrangians in supermechanics, Differential Geom. Appl., 18 (2003), 33-46. doi: 10.1016/S0926-2245(02)00096-7. Google Scholar
[8]	C. Carmeli, L. Caston and R. Fioresi, Mathematical Foundations of Supersymmetry， EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich, 2011. doi: 10.4171/097. Google Scholar
[9]	R. Casalbuoni, The classical mechanics for bose-fermi systems, Il Nuovo Cimento A (1965-1970), 33 (1976), 389-431. Google Scholar
[10]	P. Deligne and J. W. Morgan, Notes on supersymmetry (following Joseph Bernstein), In Quantum fields and strings: A course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), Amer. Math. Soc., Providence, RI, (1999), 41–97. Google Scholar
[11]	F. Dumitrescu, Superconnections and parallel transport, Pacific J. Math., 236 (2008), 307-332. doi: 10.2140/pjm.2008.236.307. Google Scholar
[12]	D. S. Freed, Five Lectures on Supersymmetry, AMS, Providence, USA, 1999 Google Scholar
[13]	S. Garnier and T. Wurzbacher, The geodesic flow on a Riemannian supermanifold, J. Geom. Phys., 62 (2012), 1489-1508. doi: 10.1016/j.geomphys.2012.02.002. Google Scholar
[14]	K. Gawędzki, Supersymmetries-mathematics of supergeometry, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 335-366. Google Scholar
[15]	O. Goertsches, Riemannian supergeometry, Math. Z., 260 (2008), 557-593. doi: 10.1007/s00209-007-0288-z. Google Scholar
[16]	J. Grabowski and P. Urbański, Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids, Ann. Global Anal. Geom., 15 (1997), 447-486. doi: 10.1023/A:1006519730920. Google Scholar
[17]	J. Grabowski, M. de León, J. C. Marrero and D. Martín de Diego, Nonholonomic constraints: A new viewpoint, J. Math. Phys., 50 (2009), 013520, 17pp. doi: 10.1063/1.3049752. Google Scholar
[18]	R. Heumann and N. S. Manton, Classical supersymmetric mechanics, Ann. Physics, 284 (2000), 52-88. doi: 10.1006/aphy.2000.6057. Google Scholar
[19]	L. A. Ibort and J. Marín-Solano, Geometrical foundations of Lagrangian supermechanics and supersymmetry, Rep. Math. Phys., 32 (1993), 385-409. doi: 10.1016/0034-4877(93)90031-9. Google Scholar
[20]	G. Junker, Supersymmetric Methods in Quantum and Statistical Physics, Texts and Monographs in Physics. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61194-0. Google Scholar
[21]	G. Junker and S. Matthiesen, Supersymmetric classical mechanics, J. Phys. A, 27 (1994), L751-L755. doi: 10.1088/0305-4470/27/19/006. Google Scholar
[22]	G. Junker and S. Matthiesen, Pseudoclassical mechanics and its solution. Addendum to: "Supersymmetric classical mechanics" [J. Phys. A 27 (1994), no. 19, L751–L755]. J. Phys. A, 28 (1995), 1467–1468.Google Scholar
[23]	S. Mac Lane, Categories for the Working Mathematician volume 5 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1998. Google Scholar
[24]	D. A. Leǐtes, Introduction to the theory of supermanifolds, Uspekhi Mat. Nauk, 35 (1980), 3-57,255. Google Scholar
[25]	M. A. Lledo, Superfields, nilpotent superfields and superschemes, preprint, arXiv: 1702.00755 [hep-th].Google Scholar
[26]	Y. I. Manin, Gauge Field Theory and Complex Geometry volume 289 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second edition, 1997. Translated from the 1984 Russian original by N. Koblitz and J. R. King, With an appendix by Sergei Merkulov. doi: 10.1007/978-3-662-07386-5. Google Scholar
[27]	N. S. Manton, Deconstructing supersymmetry, J. Math. Phys., 40 (1999), 736-750. doi: 10.1063/1.532682. Google Scholar
[28]	G. Marmo, G. Mendella and W. M. Tulczyjew, Symmetries and constants of the motion for dynamics in implicit form, Annales de l'I.H.P. Physique théorique, 57 (1992), 147-166. Google Scholar
[29]	F. Ongay-Larios and O. A. Sánchez-Valenzuela, R^1\|1-supergroup actions and superdifferential equations, Proc. Amer. Math. Soc., 116 (1992), 843-850. doi: 10.2307/2159456. Google Scholar
[30]	C. Sachse and C. Wockel, The diffeomorphism supergroup of a finite-dimensional supermanifold, Adv. Theor. Math. Phys., 15 (2011), 285-323. doi: 10.4310/ATMP.2011.v15.n2.a2. Google Scholar
[31]	G. Salgado and J. A. Vallejo-Rodríguez, The meaning of time and covariant superderivatives in supermechanics, Adv. Math. Phys., (2009), Art. ID 987524, 21pp. Google Scholar
[32]	A. S. Schwarz, On the definition of superspace, Teoret. Mat. Fiz., 60 (1984), 37-42. Google Scholar
[33]	W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114. Google Scholar
[34]	V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, volume 11 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2004. doi: 10.1090/cln/011. Google Scholar
[35]	A. A. Voronov, Maps of supermanifolds, Teoret. Mat. Fiz., 60 (1984), 43-48. Google Scholar
[36]	Th. Voronov, Geometric Integration Theory on Supermanifolds, Soviet Scientific Reviews, Section C: Mathematical Physics Reviews, 9, Part 1. Harwood Academic Publishers, Chur, 1991. iv+138 pp. Google Scholar
[37]	Th. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids, In Quantization, Poisson brackets and beyond (Manchester, 2001), volume 315 of Contemp. Math., pages 131–168. Amer. Math. Soc., Providence, RI, 2002.Google Scholar
[38]	E. Witten, Dynamical Breaking of Supersymmetry, Nucl. Phys. B, 188 (1981), 513-554. doi: 10.1016/0550-3213(81)90006-7. Google Scholar
[39]	K. Yano and S. Ishihara, Tangent and Cotangent Bundles: Differential Geometry, Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, No. 16. Google Scholar

show all references

References:

[1]	V. P. Akulov and S. Duplij, Nilpotent marsh and SUSY QM, In Supersymmetries and quantum symmetries (Dubna, 1997), volume 524 of Lecture Notes in Phys., pages 235–242. Springer, Berlin, 1999. doi: 10.1007/BFb0104604. Google Scholar
[2]	M. Batchelor, The structure of supermanifolds, Trans. Amer. Math. Soc., 253 (1979), 329-338. doi: 10.1090/S0002-9947-1979-0536951-0. Google Scholar
[3]	F. A. Berezin and M. S. Marinov, Particle spin dynamics as the grassmann variant of classical mechanics, Annals of Physics, 104 (1977), 336-362. doi: 10.1016/0003-4916(77)90335-9. Google Scholar
[4]	A. J. Bruce, On curves and jets of curves on supermanifolds, Arch. Math. (Brno), 50 (2014), 115-130. doi: 10.5817/AM2014-2-115. Google Scholar
[5]	J. F. Cariñena and H. Figueroa, A geometrical version of Noether's theorem in supermechanics, Rep. Math. Phys., 34 (1994), 277-303. doi: 10.1016/0034-4877(94)90002-7. Google Scholar
[6]	J. F. Cariñena and H. Figueroa, Hamiltonian versus Lagrangian formulations of supermechanics, J. Phys. A, 30 (1997), 2705-2724. doi: 10.1088/0305-4470/30/8/017. Google Scholar
[7]	J. F. Cariñena and H. Figueroa, Singular Lagrangians in supermechanics, Differential Geom. Appl., 18 (2003), 33-46. doi: 10.1016/S0926-2245(02)00096-7. Google Scholar
[8]	C. Carmeli, L. Caston and R. Fioresi, Mathematical Foundations of Supersymmetry， EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich, 2011. doi: 10.4171/097. Google Scholar
[9]	R. Casalbuoni, The classical mechanics for bose-fermi systems, Il Nuovo Cimento A (1965-1970), 33 (1976), 389-431. Google Scholar
[10]	P. Deligne and J. W. Morgan, Notes on supersymmetry (following Joseph Bernstein), In Quantum fields and strings: A course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), Amer. Math. Soc., Providence, RI, (1999), 41–97. Google Scholar
[11]	F. Dumitrescu, Superconnections and parallel transport, Pacific J. Math., 236 (2008), 307-332. doi: 10.2140/pjm.2008.236.307. Google Scholar
[12]	D. S. Freed, Five Lectures on Supersymmetry, AMS, Providence, USA, 1999 Google Scholar
[13]	S. Garnier and T. Wurzbacher, The geodesic flow on a Riemannian supermanifold, J. Geom. Phys., 62 (2012), 1489-1508. doi: 10.1016/j.geomphys.2012.02.002. Google Scholar
[14]	K. Gawędzki, Supersymmetries-mathematics of supergeometry, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 335-366. Google Scholar
[15]	O. Goertsches, Riemannian supergeometry, Math. Z., 260 (2008), 557-593. doi: 10.1007/s00209-007-0288-z. Google Scholar
[16]	J. Grabowski and P. Urbański, Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids, Ann. Global Anal. Geom., 15 (1997), 447-486. doi: 10.1023/A:1006519730920. Google Scholar
[17]	J. Grabowski, M. de León, J. C. Marrero and D. Martín de Diego, Nonholonomic constraints: A new viewpoint, J. Math. Phys., 50 (2009), 013520, 17pp. doi: 10.1063/1.3049752. Google Scholar
[18]	R. Heumann and N. S. Manton, Classical supersymmetric mechanics, Ann. Physics, 284 (2000), 52-88. doi: 10.1006/aphy.2000.6057. Google Scholar
[19]	L. A. Ibort and J. Marín-Solano, Geometrical foundations of Lagrangian supermechanics and supersymmetry, Rep. Math. Phys., 32 (1993), 385-409. doi: 10.1016/0034-4877(93)90031-9. Google Scholar
[20]	G. Junker, Supersymmetric Methods in Quantum and Statistical Physics, Texts and Monographs in Physics. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61194-0. Google Scholar
[21]	G. Junker and S. Matthiesen, Supersymmetric classical mechanics, J. Phys. A, 27 (1994), L751-L755. doi: 10.1088/0305-4470/27/19/006. Google Scholar
[22]	G. Junker and S. Matthiesen, Pseudoclassical mechanics and its solution. Addendum to: "Supersymmetric classical mechanics" [J. Phys. A 27 (1994), no. 19, L751–L755]. J. Phys. A, 28 (1995), 1467–1468.Google Scholar
[23]	S. Mac Lane, Categories for the Working Mathematician volume 5 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1998. Google Scholar
[24]	D. A. Leǐtes, Introduction to the theory of supermanifolds, Uspekhi Mat. Nauk, 35 (1980), 3-57,255. Google Scholar
[25]	M. A. Lledo, Superfields, nilpotent superfields and superschemes, preprint, arXiv: 1702.00755 [hep-th].Google Scholar
[26]	Y. I. Manin, Gauge Field Theory and Complex Geometry volume 289 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second edition, 1997. Translated from the 1984 Russian original by N. Koblitz and J. R. King, With an appendix by Sergei Merkulov. doi: 10.1007/978-3-662-07386-5. Google Scholar
[27]	N. S. Manton, Deconstructing supersymmetry, J. Math. Phys., 40 (1999), 736-750. doi: 10.1063/1.532682. Google Scholar
[28]	G. Marmo, G. Mendella and W. M. Tulczyjew, Symmetries and constants of the motion for dynamics in implicit form, Annales de l'I.H.P. Physique théorique, 57 (1992), 147-166. Google Scholar
[29]	F. Ongay-Larios and O. A. Sánchez-Valenzuela, R^1\|1-supergroup actions and superdifferential equations, Proc. Amer. Math. Soc., 116 (1992), 843-850. doi: 10.2307/2159456. Google Scholar
[30]	C. Sachse and C. Wockel, The diffeomorphism supergroup of a finite-dimensional supermanifold, Adv. Theor. Math. Phys., 15 (2011), 285-323. doi: 10.4310/ATMP.2011.v15.n2.a2. Google Scholar
[31]	G. Salgado and J. A. Vallejo-Rodríguez, The meaning of time and covariant superderivatives in supermechanics, Adv. Math. Phys., (2009), Art. ID 987524, 21pp. Google Scholar
[32]	A. S. Schwarz, On the definition of superspace, Teoret. Mat. Fiz., 60 (1984), 37-42. Google Scholar
[33]	W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114. Google Scholar
[34]	V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, volume 11 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2004. doi: 10.1090/cln/011. Google Scholar
[35]	A. A. Voronov, Maps of supermanifolds, Teoret. Mat. Fiz., 60 (1984), 43-48. Google Scholar
[36]	Th. Voronov, Geometric Integration Theory on Supermanifolds, Soviet Scientific Reviews, Section C: Mathematical Physics Reviews, 9, Part 1. Harwood Academic Publishers, Chur, 1991. iv+138 pp. Google Scholar
[37]	Th. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids, In Quantization, Poisson brackets and beyond (Manchester, 2001), volume 315 of Contemp. Math., pages 131–168. Amer. Math. Soc., Providence, RI, 2002.Google Scholar
[38]	E. Witten, Dynamical Breaking of Supersymmetry, Nucl. Phys. B, 188 (1981), 513-554. doi: 10.1016/0550-3213(81)90006-7. Google Scholar
[39]	K. Yano and S. Ishihara, Tangent and Cotangent Bundles: Differential Geometry, Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, No. 16. Google Scholar

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