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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.

[2]

M. Batchelor, The structure of supermanifolds, Trans. Amer. Math. Soc., 253 (1979), 329-338.  doi: 10.1090/S0002-9947-1979-0536951-0.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

R. Casalbuoni, The classical mechanics for bose-fermi systems, Il Nuovo Cimento A (1965-1970), 33 (1976), 389-431. 

[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.

[11]

F. Dumitrescu, Superconnections and parallel transport, Pacific J. Math., 236 (2008), 307-332.  doi: 10.2140/pjm.2008.236.307.

[12]

D. S. Freed, Five Lectures on Supersymmetry, AMS, Providence, USA, 1999

[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.

[14]

K. Gawędzki, Supersymmetries-mathematics of supergeometry, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 335-366. 

[15]

O. Goertsches, Riemannian supergeometry, Math. Z., 260 (2008), 557-593.  doi: 10.1007/s00209-007-0288-z.

[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.

[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.

[18]

R. Heumann and N. S. Manton, Classical supersymmetric mechanics, Ann. Physics, 284 (2000), 52-88.  doi: 10.1006/aphy.2000.6057.

[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.

[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.

[21]

G. Junker and S. Matthiesen, Supersymmetric classical mechanics, J. Phys. A, 27 (1994), L751-L755.  doi: 10.1088/0305-4470/27/19/006.

[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.

[23]

S. Mac Lane, Categories for the Working Mathematician volume 5 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1998.

[24]

D. A. Leǐtes, Introduction to the theory of supermanifolds, Uspekhi Mat. Nauk, 35 (1980), 3-57,255. 

[25]

M. A. Lledo, Superfields, nilpotent superfields and superschemes, preprint, arXiv: 1702.00755 [hep-th].

[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.

[27]

N. S. Manton, Deconstructing supersymmetry, J. Math. Phys., 40 (1999), 736-750.  doi: 10.1063/1.532682.

[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. 

[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.

[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.

[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.

[32]

A. S. Schwarz, On the definition of superspace, Teoret. Mat. Fiz., 60 (1984), 37-42. 

[33]

W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114. 

[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.

[35]

A. A. Voronov, Maps of supermanifolds, Teoret. Mat. Fiz., 60 (1984), 43-48. 

[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.

[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.

[38]

E. Witten, Dynamical Breaking of Supersymmetry, Nucl. Phys. B, 188 (1981), 513-554.  doi: 10.1016/0550-3213(81)90006-7.

[39]

K. Yano and S. Ishihara, Tangent and Cotangent Bundles: Differential Geometry, Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, No. 16.

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.

[2]

M. Batchelor, The structure of supermanifolds, Trans. Amer. Math. Soc., 253 (1979), 329-338.  doi: 10.1090/S0002-9947-1979-0536951-0.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

R. Casalbuoni, The classical mechanics for bose-fermi systems, Il Nuovo Cimento A (1965-1970), 33 (1976), 389-431. 

[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.

[11]

F. Dumitrescu, Superconnections and parallel transport, Pacific J. Math., 236 (2008), 307-332.  doi: 10.2140/pjm.2008.236.307.

[12]

D. S. Freed, Five Lectures on Supersymmetry, AMS, Providence, USA, 1999

[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.

[14]

K. Gawędzki, Supersymmetries-mathematics of supergeometry, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 335-366. 

[15]

O. Goertsches, Riemannian supergeometry, Math. Z., 260 (2008), 557-593.  doi: 10.1007/s00209-007-0288-z.

[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.

[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.

[18]

R. Heumann and N. S. Manton, Classical supersymmetric mechanics, Ann. Physics, 284 (2000), 52-88.  doi: 10.1006/aphy.2000.6057.

[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.

[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.

[21]

G. Junker and S. Matthiesen, Supersymmetric classical mechanics, J. Phys. A, 27 (1994), L751-L755.  doi: 10.1088/0305-4470/27/19/006.

[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.

[23]

S. Mac Lane, Categories for the Working Mathematician volume 5 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1998.

[24]

D. A. Leǐtes, Introduction to the theory of supermanifolds, Uspekhi Mat. Nauk, 35 (1980), 3-57,255. 

[25]

M. A. Lledo, Superfields, nilpotent superfields and superschemes, preprint, arXiv: 1702.00755 [hep-th].

[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.

[27]

N. S. Manton, Deconstructing supersymmetry, J. Math. Phys., 40 (1999), 736-750.  doi: 10.1063/1.532682.

[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. 

[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.

[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.

[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.

[32]

A. S. Schwarz, On the definition of superspace, Teoret. Mat. Fiz., 60 (1984), 37-42. 

[33]

W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114. 

[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.

[35]

A. A. Voronov, Maps of supermanifolds, Teoret. Mat. Fiz., 60 (1984), 43-48. 

[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.

[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.

[38]

E. Witten, Dynamical Breaking of Supersymmetry, Nucl. Phys. B, 188 (1981), 513-554.  doi: 10.1016/0550-3213(81)90006-7.

[39]

K. Yano and S. Ishihara, Tangent and Cotangent Bundles: Differential Geometry, Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, No. 16.

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