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On a geometric framework for Lagrangian supermechanics
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 |
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.
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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