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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, R1|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, R1|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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