March  2017, 9(1): 47-82. doi: 10.3934/jgm.2017002

Covariant Hamiltonian field theories on manifolds with boundary: Yang-Mills theories

1. 

ICMAT and Depto. de Matemáticas, Univ. Carlos Ⅲ de Madrid, Avda. de la Universidad 30,28911 Leganés, Madrid, Spain

2. 

Dept. of Mathematics, Univ. of California at Berkeley, 903 Evans Hall, 94720 Berkeley CA, USA

* Corresponding author: A. Ibort

Received  June 2015 Revised  January 2017 Published  March 2017

The multisymplectic formalism of field theories developed over the last fifty years is extended to deal with manifolds that have boundaries. In particular, a multisymplectic framework for first-order covariant Hamiltonian field theories on manifolds with boundaries is developed. This work is a geometric fulfillment of Fock's formulation of field theories as it appears in recent work by Cattaneo, Mnev and Reshetikhin [11]. This framework leads to a geometric understanding of conventional choices for boundary conditions and relates them to the moment map of the gauge group of the theory.

It is also shown that the natural interpretation of the Euler-Lagrange equations as an evolution system near the boundary leads to a presymplectic Hamiltonian system in an extended phase space containing the natural configuration and momenta fields at the boundary together with extra degrees of freedom corresponding to the transversal components at the boundary of the momenta fields of the theory. The consistency conditions for evolution at the boundary are analyzed and the reduced phase space of the system is shown to be a symplectic manifold with a distinguished isotropic submanifold corresponding to the boundary data of the solutions of Euler-Lagrange equations. This setting makes it possible to define well-posed boundary conditions, and provides the adequate setting for the canonical quantization of the system.

The notions of the theory are tested against three significant examples: scalar fields, Poisson σ-model and Yang-Mills theories.

Citation: Alberto Ibort, Amelia Spivak. Covariant Hamiltonian field theories on manifolds with boundary: Yang-Mills theories. Journal of Geometric Mechanics, 2017, 9 (1) : 47-82. doi: 10.3934/jgm.2017002
References:
[1]

M. AsoreyA. Ibort and G. Marmo, Global theory of quantum boundary conditions and topology change, International Journal of Modern Physics A, 20 (2005), 1001-1025. doi: 10.1142/S0217751X05019798. Google Scholar

[2]

M. AsoreyD. Garcıa-Alvarez and J. M. Muñoz-Castañeda, Boundary effects in bosonic and fermionic field theories, International Journal of Geometric Methods in Modern Physics, 12 (2015), 1560004, 12pp. doi: 10.1142/S021988781560004X. Google Scholar

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[4]

E. Binz and J. Sniatycki, Geometry of Classical Fields, Courier Dover Publications, 2011.Google Scholar

[5]

F. CantrijnA. Ibort and M. de Léon, On the geometry of multisymplectic manifolds, Journal of the Australian Mathematical Society (Series A), 66 (1999), 303-330. doi: 10.1017/S1446788700036636. Google Scholar

[6]

J. F. CariñenaM. Crampin and A. Ibort, On the multisymplectic formalism for first-order field theories, Diff. Geom., Appl., 1 (1991), 345-374. doi: 10.1016/0926-2245(91)90013-Y. Google Scholar

[7]

A. S. Cattaneo and G. Felder, A Path Integral approach to the Kontsevich quantization formula, Comm. in Math. Phys., 212 (2000), 591-611. doi: 10.1007/s002200000229. Google Scholar

[8]

A. S. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, In Quantization of singular symplectic quotients, Birkhäuser Basel, (2001), 61-93; ibid. Poisson sigma models and deformation quantization, Modern Physics Letters A, 16 (2001), 179-189. doi: 10.1142/S0217732301003255. Google Scholar

[9]

A. S. Cattaneo, P. Mnev and N. Reshetikhin, Classical and Quantum Lagrangian Field Theories with Boundary, PoS (CORFU2011) 044, arXiv preprint, arXiv: 1207.0239.Google Scholar

[10]

A. S. Cattaneo and I. Contreras, Groupoids and Poisson sigma models with boundary, Geometric, Algebraic and Topological Methods for Quantum Field Theory, (2014), 315-330. doi: 10.1142/9789814460057_0009. Google Scholar

[11]

A. CattaneoP. Mnev and N. Reshetikhin, Classical BV theories on manifolds with boundaries, Commun. Math. Phys., 332 (2014), 535-603. doi: 10.1007/s00220-014-2145-3. Google Scholar

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I. Contreras, Relational Symplectic Groupoids and Poisson Sigma Models with Boundary, Ph. D. Thesis (2013). See also: A. Cattaneo, I. Contreras, Relational symplectic groupoids and Poisson sigma models with boundary, arXiv: 1401.7319v1 [math. SG] (2014).Google Scholar

[13]

A. Echeverrıa-EnrıquezM. C. Muñoz-Lecanda and N. Román-Roy, Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys., 41 (2000), 7402-7444. doi: 10.1063/1.1308075. Google Scholar

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A. Echeverrıa-EnrıquezM. De LéonM. C. Muñoz-Lecanda and N. Román-Roy, Extended Hamiltonian systems in multisymplectic field theories, J. Math. Phys., 48 (2007), 112901, 30pp. doi: 10.1063/1.2801875. Google Scholar

[15]

J. GasetP. D. Prieto-Martınez and N. Román-Roy, Variational principles and symmetries on fibered multisymplectic manifolds, Communications in Mathematics, 24 (2016), 137-152. doi: 10.1515/cm-2016-0010. Google Scholar

[16]

K. Gawedzki, On the geometrization of the canonical formalism in the classical field theory, Rep. Math. Phys., 3 (1972), 307-326. doi: 10.1016/0034-4877(72)90014-6. Google Scholar

[17]

G. Giachetta, L. Mangiarotti and G. A. Sardanashvili, Advanced Classical Field Theory, World Scientific, 2009. doi: 10.1142/9789812838964. Google Scholar

[18]

H. Goldschmidt and S. Sternberg, The Hamilton-Jacobi formalism in the calculus of variations, Ann. Inst. Fourier, 23 (1973), 203-267. doi: 10.5802/aif.451. Google Scholar

[19]

M. J. GotayJ. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, Journal of Mathematical Physics, 19 (1978), 2388-2399. doi: 10.1063/1.523597. Google Scholar

[20]

M. J. Gotay, J. Isenberg, J. E. Marsden and R. Montgomery, Momentum maps and classical relativistic fields. Part Ⅰ: Covariant field theory, arXiv preprint physics/9801019 (1998);Google Scholar

[21]

M. J. Gotay, J. Isenberg, J. E. Marsden and R. Montgomery, Momentum maps and classical relativistic fields. Part Ⅱ: Canonical analysis of field theories, arXiv preprint mathph/0411032 (2004).Google Scholar

[22]

K. Grabowska, A Tulczyjew triple for classical fields, J. Phys. A: Math. Theor., 45 (2012), 145207 (35pp). doi: 10.1088/1751-8113/45/14/145207. Google Scholar

[23]

K. Grabowska and L. Vitagliano, Tulczyjew triples in higher derivative field theory, J. Geom. Mech., 7 (2015), 1-33. doi: 10.3934/jgm.2015.7.1. Google Scholar

[24]

A. Ibort and A. Spivak, Covariant Hamiltonian first-order field theories with constraints, on manifolds with boundary: The case of Hamiltonian dynamics, Banach Center Publications, 110 (2016), 87-104. doi: 10.4064/bc110-0-6. Google Scholar

[25]

J. Kijowski and W. Szczyrba, A canonical structure for classical field theories, Commun. Math. Phys., 46 (1976), 183-206. doi: 10.1007/BF01608496. Google Scholar

[26]

J. Kijowski and W. Tulczyjew, A Symplectic Framework for Field Theories, Lecture Notes in Physics 107, Springer-Verlag, Berlin, 1979. Google Scholar

[27]

M. de LéonJ. Marın-SolanoJ. C. MarreroM. C. Muñoz-Lecanda and N. Román-Roy, Premultisymplectic constraint algorithm for field theories, Int. J. Geom. Methods Mod. Phys., 2 (2005), 839-871. doi: 10.1142/S0219887805000880. Google Scholar

[28]

P. D. Prieto-Martınez and N. Román-Roy, A multisymplectic unified formalism for secondorder classical field theories, (2014). J. Geom. Mech., 7 (2015), 203-253, arXiv: 1402.4087; ibid., Variational Principles for multisymplectic second-order classical field theories (2014). arXiv preprint arXiv: 1412.1451. doi: 10.3934/jgm.2015.7.203. Google Scholar

[29]

N. Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories, SIGMA, 5 (2009), 100 (25pp). doi: 10.3842/SIGMA.2009.100. Google Scholar

[30]

C. Rovelli, Dynamics without time for quantum gravity: Covariant Hamiltonian formalism and Hamilton-Jacobi equation on the space $\mathcal{G}$, Decoherence and Entropy in Complex Systems. Springer Berlin Heidelberg, 633 (2004), 36-62. doi: 10.1007/978-3-540-40968-7_4. Google Scholar

[31]

C. Rovelli, A note on the foundation of relativistic mechanics: Covariant Hamiltonian general relativity, Topics in Mathematical Physics, General Relativity and Cosmology in Honor of Jerzy Pleban'ski: Proceedings of 2002 International Conference, Cinvestav, Mexico City, 17-20 September 2002. World Scientific, (2006), 397-407. doi: 10.1142/9789812772732_0033. Google Scholar

[32] D. J. Saunders, The Geometry of Jet Bundles, Cambridge University Press, 1989. doi: 10.1017/CBO9780511526411. Google Scholar
[33]

L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher order field theories, J. Geom. Phys., 60 (2010), 857-873. doi: 10.1016/j.geomphys.2010.02.003. Google Scholar

show all references

References:
[1]

M. AsoreyA. Ibort and G. Marmo, Global theory of quantum boundary conditions and topology change, International Journal of Modern Physics A, 20 (2005), 1001-1025. doi: 10.1142/S0217751X05019798. Google Scholar

[2]

M. AsoreyD. Garcıa-Alvarez and J. M. Muñoz-Castañeda, Boundary effects in bosonic and fermionic field theories, International Journal of Geometric Methods in Modern Physics, 12 (2015), 1560004, 12pp. doi: 10.1142/S021988781560004X. Google Scholar

[3]

M. Atiyah, The Geometry and Physics of Knots, Lezioni Lincei. CUP, Cambridge, 1990. doi: 10.1017/CBO9780511623868. Google Scholar

[4]

E. Binz and J. Sniatycki, Geometry of Classical Fields, Courier Dover Publications, 2011.Google Scholar

[5]

F. CantrijnA. Ibort and M. de Léon, On the geometry of multisymplectic manifolds, Journal of the Australian Mathematical Society (Series A), 66 (1999), 303-330. doi: 10.1017/S1446788700036636. Google Scholar

[6]

J. F. CariñenaM. Crampin and A. Ibort, On the multisymplectic formalism for first-order field theories, Diff. Geom., Appl., 1 (1991), 345-374. doi: 10.1016/0926-2245(91)90013-Y. Google Scholar

[7]

A. S. Cattaneo and G. Felder, A Path Integral approach to the Kontsevich quantization formula, Comm. in Math. Phys., 212 (2000), 591-611. doi: 10.1007/s002200000229. Google Scholar

[8]

A. S. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, In Quantization of singular symplectic quotients, Birkhäuser Basel, (2001), 61-93; ibid. Poisson sigma models and deformation quantization, Modern Physics Letters A, 16 (2001), 179-189. doi: 10.1142/S0217732301003255. Google Scholar

[9]

A. S. Cattaneo, P. Mnev and N. Reshetikhin, Classical and Quantum Lagrangian Field Theories with Boundary, PoS (CORFU2011) 044, arXiv preprint, arXiv: 1207.0239.Google Scholar

[10]

A. S. Cattaneo and I. Contreras, Groupoids and Poisson sigma models with boundary, Geometric, Algebraic and Topological Methods for Quantum Field Theory, (2014), 315-330. doi: 10.1142/9789814460057_0009. Google Scholar

[11]

A. CattaneoP. Mnev and N. Reshetikhin, Classical BV theories on manifolds with boundaries, Commun. Math. Phys., 332 (2014), 535-603. doi: 10.1007/s00220-014-2145-3. Google Scholar

[12]

I. Contreras, Relational Symplectic Groupoids and Poisson Sigma Models with Boundary, Ph. D. Thesis (2013). See also: A. Cattaneo, I. Contreras, Relational symplectic groupoids and Poisson sigma models with boundary, arXiv: 1401.7319v1 [math. SG] (2014).Google Scholar

[13]

A. Echeverrıa-EnrıquezM. C. Muñoz-Lecanda and N. Román-Roy, Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys., 41 (2000), 7402-7444. doi: 10.1063/1.1308075. Google Scholar

[14]

A. Echeverrıa-EnrıquezM. De LéonM. C. Muñoz-Lecanda and N. Román-Roy, Extended Hamiltonian systems in multisymplectic field theories, J. Math. Phys., 48 (2007), 112901, 30pp. doi: 10.1063/1.2801875. Google Scholar

[15]

J. GasetP. D. Prieto-Martınez and N. Román-Roy, Variational principles and symmetries on fibered multisymplectic manifolds, Communications in Mathematics, 24 (2016), 137-152. doi: 10.1515/cm-2016-0010. Google Scholar

[16]

K. Gawedzki, On the geometrization of the canonical formalism in the classical field theory, Rep. Math. Phys., 3 (1972), 307-326. doi: 10.1016/0034-4877(72)90014-6. Google Scholar

[17]

G. Giachetta, L. Mangiarotti and G. A. Sardanashvili, Advanced Classical Field Theory, World Scientific, 2009. doi: 10.1142/9789812838964. Google Scholar

[18]

H. Goldschmidt and S. Sternberg, The Hamilton-Jacobi formalism in the calculus of variations, Ann. Inst. Fourier, 23 (1973), 203-267. doi: 10.5802/aif.451. Google Scholar

[19]

M. J. GotayJ. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, Journal of Mathematical Physics, 19 (1978), 2388-2399. doi: 10.1063/1.523597. Google Scholar

[20]

M. J. Gotay, J. Isenberg, J. E. Marsden and R. Montgomery, Momentum maps and classical relativistic fields. Part Ⅰ: Covariant field theory, arXiv preprint physics/9801019 (1998);Google Scholar

[21]

M. J. Gotay, J. Isenberg, J. E. Marsden and R. Montgomery, Momentum maps and classical relativistic fields. Part Ⅱ: Canonical analysis of field theories, arXiv preprint mathph/0411032 (2004).Google Scholar

[22]

K. Grabowska, A Tulczyjew triple for classical fields, J. Phys. A: Math. Theor., 45 (2012), 145207 (35pp). doi: 10.1088/1751-8113/45/14/145207. Google Scholar

[23]

K. Grabowska and L. Vitagliano, Tulczyjew triples in higher derivative field theory, J. Geom. Mech., 7 (2015), 1-33. doi: 10.3934/jgm.2015.7.1. Google Scholar

[24]

A. Ibort and A. Spivak, Covariant Hamiltonian first-order field theories with constraints, on manifolds with boundary: The case of Hamiltonian dynamics, Banach Center Publications, 110 (2016), 87-104. doi: 10.4064/bc110-0-6. Google Scholar

[25]

J. Kijowski and W. Szczyrba, A canonical structure for classical field theories, Commun. Math. Phys., 46 (1976), 183-206. doi: 10.1007/BF01608496. Google Scholar

[26]

J. Kijowski and W. Tulczyjew, A Symplectic Framework for Field Theories, Lecture Notes in Physics 107, Springer-Verlag, Berlin, 1979. Google Scholar

[27]

M. de LéonJ. Marın-SolanoJ. C. MarreroM. C. Muñoz-Lecanda and N. Román-Roy, Premultisymplectic constraint algorithm for field theories, Int. J. Geom. Methods Mod. Phys., 2 (2005), 839-871. doi: 10.1142/S0219887805000880. Google Scholar

[28]

P. D. Prieto-Martınez and N. Román-Roy, A multisymplectic unified formalism for secondorder classical field theories, (2014). J. Geom. Mech., 7 (2015), 203-253, arXiv: 1402.4087; ibid., Variational Principles for multisymplectic second-order classical field theories (2014). arXiv preprint arXiv: 1412.1451. doi: 10.3934/jgm.2015.7.203. Google Scholar

[29]

N. Román-Roy, Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories, SIGMA, 5 (2009), 100 (25pp). doi: 10.3842/SIGMA.2009.100. Google Scholar

[30]

C. Rovelli, Dynamics without time for quantum gravity: Covariant Hamiltonian formalism and Hamilton-Jacobi equation on the space $\mathcal{G}$, Decoherence and Entropy in Complex Systems. Springer Berlin Heidelberg, 633 (2004), 36-62. doi: 10.1007/978-3-540-40968-7_4. Google Scholar

[31]

C. Rovelli, A note on the foundation of relativistic mechanics: Covariant Hamiltonian general relativity, Topics in Mathematical Physics, General Relativity and Cosmology in Honor of Jerzy Pleban'ski: Proceedings of 2002 International Conference, Cinvestav, Mexico City, 17-20 September 2002. World Scientific, (2006), 397-407. doi: 10.1142/9789812772732_0033. Google Scholar

[32] D. J. Saunders, The Geometry of Jet Bundles, Cambridge University Press, 1989. doi: 10.1017/CBO9780511526411. Google Scholar
[33]

L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher order field theories, J. Geom. Phys., 60 (2010), 857-873. doi: 10.1016/j.geomphys.2010.02.003. Google Scholar

Figure 1.  Bundles, sections and fields: configurations and momenta
Figure 2.  The space of fields at the boundary $\mathcal{M}$ and its relevant structures.
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