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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. 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. |
[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. 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. |
[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]. 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. |
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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]. 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. |
[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. Google Scholar |
[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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