# Hamiltonian formulation of *classical* field theory

Hi,

I was looking for a book that would explain classical field theory in a Hamiltonian setting. What I mean by this is that there be no *actions* around, no *Lagrangians* and *Legendre transforms* to define the Hamiltonian and so on. What I'm looking fo is an exposition of (classical) field theory that starts with the Hamiltonian and the Poisson bracket, from which one can obviously derive the equations of motion and everything else. In this setting I can then *quantize* the theory in the usual way. And if the exposition is mathematically (semi-)rigorous, even better!

Can anyone suggest a reference? Review paper, book or similar?

(Mods, sorry if the post is not in the appropriate forum. It seemed the most appropriate to me as I'm interested in the quantization of the theory.)

I know Goldstein has a section on classical field theory, but I don't know whether or not it develops it in the way you suggest; I'd guess not however.

Can I ask why you're so averse to obtaining it via the Legendre transformation of the Lagrangian?

muppet -> Because of "intellectual elegance". I need to write up something about canonical (Hamiltonian) quantization and it's conceptually much more elegant if I could avoid talking about Lagrangians and actions. Much like in ordinary QM where the Lagrangian doesn't enter till you want to talk about path integrals. Not to mention that the Legendre transform is not really well defined if the Lagrangian contains interaction terms with derivatives. I could put what I know together and a physicists would be happy about it, but mathematical rigour is a bit of an issue, so I wanted some reference on the subject. Thanks for the suggetions tho', I'll check it out.

I see. The trouble is that it's defined as the Legendre transformation of the Lagrangian. In the setting of classical mechanics, the Legendre transformation only works out to be
$$T-V \rightarrow T+V$$
when the kinetic energy is a quadratic form in the velocity. If the hamiltonian were defined as the energy operator, then the Schroedinger equation $$H\Psi=E\Psi$$ would be trivial; it's because the energy operator can be defined as the generator of infinitesmal time translations, and the hamiltonian as the legendre transformation of the lagrangian, that it isn't.

Also, if you want to be careful about how you do things, I'd suggest having a look at chapter 7 of vol.1 of Weinberg's Quantum theory of fields; he's careful to emphasise some subtleties a lot of authors suppress for expedience.

samalkhaiat
 I.V. Kanatchikov: On Field Theoretic Generalizations of a Poisson Algebra,
Rep. Math. Phys. 40 (1997) 225-234, hep-th/9710069.
 I.V. Kanatchikov: Canonical Structure of Classical Field Theory in the Polymomentum
Phase Space, Rep. Math. Phys. 41 (1998) 49-90, hep-th/9709229.
 J.F. Cari˜nena, M. Crampin & L.A. Ibort: On the Multisymplectic Formalism
for First Order Field Theories, Diff. Geom. Appl. 1 (1991) 345-374.
 M.J. Gotay, J. Isenberg & J.E. Marsden: Momentum Maps and Classical
Relativistic Fields I: Covariant Field Theory, physics/9801019.
 M. Forger & H. R¨omer: A Poisson Bracket on Multisymplectic Phase Space,
Rep. Math. Phys. 48 (2001) 211-218; math-ph/0009037.
 H. Goldschmidt & S. Sternberg: The Hamilton-Cartan Formalism in the
Calculus of Variations, Ann. Inst. Four. 23 (1973) 203-267.
 V. Guillemin & S. Sternberg: Geometric Asymptotics, Mathematical Surveys,
Vol. 14, American Mathematical Society, Providence 1977.
 J. Kijowski: A Finite-dimensional Canonical Formalism in the Classical Field
Theory, Commun. Math. Phys. 30 (1973) 99-128; Multiphase Spaces and Gauge
in Calculus of Variations, Bull. Acad. Pol. Sci. SMAP 22 (1974) 1219-1225.
 J. Kijowski & W. Szczyrba: Multisymplectic Manifolds and the Geometrical
Construction of the Poisson Brackets in the Classical Field Theory, in: “G´eometrie
Symplectique et Physique Math´ematique”, pp. 347-379, ed.: J.-M. Souriau,
C.N.R.S., Paris 1975.
 J. Kijowski & W. Szczyrba: Canonical Structure for Classical Field Theories,
Commun. Math. Phys. 46 (1976) 183-206.
 J. Kijowski & W. Tulczyjew: A Symplectic Framework for Field Theories,
Lecture Notes in Physics, Vol. 107, Springer-Verlag, Berlin 1979.
 M. Forger, C. Paufler & H. R¨omer: More about Poisson Brackets and
Poisson Forms in Multisymplectic Field Theory.
 C. Crnkovi´c & E. Witten: Covariant Description of Canonical Formalism
in Geometrical Theories, in: “Three Hundred Years of Gravitation”, pp. 676-684,
eds: W. Israel & S. Hawking, Cambridge University Press, Cambridge 1987.
 C. Crnkovi´c: Symplectic Geometry of Covariant Phase Space, Class. Quant.
Grav. 5 (1988) 1557-1575.
 G. Zuckerman: Action Principles and Global Geometry, in: “Mathematical Aspects
of String Theory”, pp. 259-288, ed.: S.-T. Yau, World Scientific, Singapore
1987.
 R.E. Peierls: The Commutation Laws of Relativistic Field Theory, Proc. Roy.
Soc. Lond. A 214 (1952) 143-157.
 B. de Witt: Dynamical Theory of Groups and Fields, in: “Relativity, Groups
and Topology, 1963 Les Houches Lectures”, pp. 585-820, eds: B. de Witt & C. de
Witt, Gordon and Breach, New York 1964.
 B. de Witt: The Spacetime Approach to Quantum Field Theory, in: “Relativity,
Groups and Topology II, 1983 Les Houches Lectures”, pp. 382-738, eds.: B. de
Witt & R. Stora, Elsevier, Amsterdam 1984.
 S.V. Romero: Colchete de Poisson Covariante na Teoria Geom´etrica dos Campos,
PhD thesis, Institute for Mathematics and Statistics, University of S˜ao Paulo,
June 2001.
 M. Forger & S.V. Romero: Covariant Poisson Brackets in Geometric Field
Theory.
 R. Abraham & J.E. Marsden: Foundations of Mechanics, 2nd edition, Benjamin/
 V. Arnold: Mathematical Methods of Classical Mechanics, 2nd edition, Springer,
Berlin 1989.
 H.A. Kastrup: Canonical Theories of Lagrangian Dynamical Systems in Physics
Phys. Rep. 101 (1983) 3-167.
 G. Martin: A Darboux Theorem for Multisymplectic Manifolds, Lett. Math.
Phys. 16 (1988) 133-138.
 G. Martin: Dynamical Structures for k-Vector Fields, Int. J. Theor. Phys. 41
(1988) 571-585.
 F. Cantrijn, A. Ibort & M. de Le´on: On the Geometry of Multisymplectic
Manifolds, J. Austral. Math. Soc. (Series A) 66 (1999) 303-330.
 C. Paufler & H. R¨omer: Geometry of Hamiltonian n-Vector Fields in Multisymplectic
Field Theory, math-ph/0102008, to appear in J. Geom. Phys.
 C. Paufler: A Vertical Exterior Derivative in Multisymplectic Geometry and a
Graded Poisson Bracket for Nontrivial Geometries, Rep. Math. Phys. 47 (2001)
101-119; math-ph/0002032.
 L. Faddeev, in Quantum Fields and Strings: A Course for Mathematicians, Vol 1, P513-550. American Mathematical Society.

regards

sam

Last edited:
 I.V. Kanatchikov: On Field Theoretic Generalizations of a Poisson Algebra,
Rep. Math. Phys. 40 (1997) 225-234, hep-th/9710069.
 I.V. Kanatchikov: Canonical Structure of Classical Field Theory in the Polymomentum
Phase Space, Rep. Math. Phys. 41 (1998) 49-90, hep-th/9709229.
 J.F. Cari˜nena, M. Crampin & L.A. Ibort: On the Multisymplectic Formalism
for First Order Field Theories, Diff. Geom. Appl. 1 (1991) 345-374.
 M.J. Gotay, J. Isenberg & J.E. Marsden: Momentum Maps and Classical
Relativistic Fields I: Covariant Field Theory, physics/9801019.
 M. Forger & H. R¨omer: A Poisson Bracket on Multisymplectic Phase Space,
Rep. Math. Phys. 48 (2001) 211-218; math-ph/0009037.
 H. Goldschmidt & S. Sternberg: The Hamilton-Cartan Formalism in the
Calculus of Variations, Ann. Inst. Four. 23 (1973) 203-267.
 V. Guillemin & S. Sternberg: Geometric Asymptotics, Mathematical Surveys,
Vol. 14, American Mathematical Society, Providence 1977.
 J. Kijowski: A Finite-dimensional Canonical Formalism in the Classical Field
Theory, Commun. Math. Phys. 30 (1973) 99-128; Multiphase Spaces and Gauge
in Calculus of Variations, Bull. Acad. Pol. Sci. SMAP 22 (1974) 1219-1225.
 J. Kijowski & W. Szczyrba: Multisymplectic Manifolds and the Geometrical
Construction of the Poisson Brackets in the Classical Field Theory, in: “G´eometrie
Symplectique et Physique Math´ematique”, pp. 347-379, ed.: J.-M. Souriau,
C.N.R.S., Paris 1975.
 J. Kijowski & W. Szczyrba: Canonical Structure for Classical Field Theories,
Commun. Math. Phys. 46 (1976) 183-206.
 J. Kijowski & W. Tulczyjew: A Symplectic Framework for Field Theories,
Lecture Notes in Physics, Vol. 107, Springer-Verlag, Berlin 1979.
 M. Forger, C. Paufler & H. R¨omer: More about Poisson Brackets and
Poisson Forms in Multisymplectic Field Theory.
 C. Crnkovi´c & E. Witten: Covariant Description of Canonical Formalism
in Geometrical Theories, in: “Three Hundred Years of Gravitation”, pp. 676-684,
eds: W. Israel & S. Hawking, Cambridge University Press, Cambridge 1987.
 C. Crnkovi´c: Symplectic Geometry of Covariant Phase Space, Class. Quant.
Grav. 5 (1988) 1557-1575.
 G. Zuckerman: Action Principles and Global Geometry, in: “Mathematical Aspects
of String Theory”, pp. 259-288, ed.: S.-T. Yau, World Scientific, Singapore
1987.
 R.E. Peierls: The Commutation Laws of Relativistic Field Theory, Proc. Roy.
Soc. Lond. A 214 (1952) 143-157.
 B. de Witt: Dynamical Theory of Groups and Fields, in: “Relativity, Groups
and Topology, 1963 Les Houches Lectures”, pp. 585-820, eds: B. de Witt & C. de
Witt, Gordon and Breach, New York 1964.
 B. de Witt: The Spacetime Approach to Quantum Field Theory, in: “Relativity,
Groups and Topology II, 1983 Les Houches Lectures”, pp. 382-738, eds.: B. de
Witt & R. Stora, Elsevier, Amsterdam 1984.
 S.V. Romero: Colchete de Poisson Covariante na Teoria Geom´etrica dos Campos,
PhD thesis, Institute for Mathematics and Statistics, University of S˜ao Paulo,
June 2001.
 M. Forger & S.V. Romero: Covariant Poisson Brackets in Geometric Field
Theory.
 R. Abraham & J.E. Marsden: Foundations of Mechanics, 2nd edition, Benjamin/
 V. Arnold: Mathematical Methods of Classical Mechanics, 2nd edition, Springer,
Berlin 1989.
 H.A. Kastrup: Canonical Theories of Lagrangian Dynamical Systems in Physics
Phys. Rep. 101 (1983) 3-167.
 G. Martin: A Darboux Theorem for Multisymplectic Manifolds, Lett. Math.
Phys. 16 (1988) 133-138.
 G. Martin: Dynamical Structures for k-Vector Fields, Int. J. Theor. Phys. 41
(1988) 571-585.
 F. Cantrijn, A. Ibort & M. de Le´on: On the Geometry of Multisymplectic
Manifolds, J. Austral. Math. Soc. (Series A) 66 (1999) 303-330.
 C. Paufler & H. R¨omer: Geometry of Hamiltonian n-Vector Fields in Multisymplectic
Field Theory, math-ph/0102008, to appear in J. Geom. Phys.
 C. Paufler: A Vertical Exterior Derivative in Multisymplectic Geometry and a
Graded Poisson Bracket for Nontrivial Geometries, Rep. Math. Phys. 47 (2001)
101-119; math-ph/0002032.
 L. Faddeev, in Quantum Fields and Strings: A Course for Mathematicians, Vol 1, P513-550. American Mathematical Society.

regards

sam

Well, now I feel inadequate :tongue:

samalkhaiat -> Wow, I'm aboslutely impressed. You've given me more references than I could hope for! Thanks A LOT!!!

Fredrik
Staff Emeritus