Classical field theory is Lagrangian theory where field equations are Euler – Lagrange equations derived from a Lagrangian (see my post Classical field theory is complete: the strict geometric formulation). This is the case of all observable classical fields, which are electromagnetic, gravitational and Dirac fermion fields. One also considers classical non-Abelian gauge fields described by the Yang – Mills Lagrangian. A problem is Hamiltonian field theory.

Applied to field theory, the familiar symplectic Hamiltonian technique takes the form of instantaneous Hamiltonian formalism on an infinite-dimensional phase space, where canonical variables are field functions at some instant of time. The corresponding Hamilton equations are ill defined, and they are not differential equations. Therefore, this Hamiltonian formalism fails to be a counterpart of the Lagrangian one, and it is applied only to quantization of fields, e.g., in quantum gauge theory.

In mechanics, Hamilton equations are first order dynamic equations, and Hamilton formalism is associated to the first order Lagrangian one. Therefore, let us restrict our consideration to first order Lagrangian field theory when a Lagrangian depends on derivatives of fields of not more than first order. Note that Lagrangians of all realistic field models are of this type. These are Lagrangians of gauge and Dirac fermion fields. The second order Hilbert – Einstein Lagrangian of General Relativity is equivalent to the first order one, and Lagrangians of metric-affine gravitation theory, where dynamic variables are both a pseudo-Riemannian metric and a linear connection, also are of first order.

Let us follow the conventional geometric formulation of classical field theory where fields are represented by sections of some fibre bundle Hamilton equations are equivalent to the Euler – Lagrange equations in the case of hyperregular Lagrangians

*Y->X*. Their configuration space of first order Lagrangian theory on*Y->X*is the first order jet manifold*JY*of*Y->X*. A first order Lagrangian*L*is defined as a density on*JY*, and it provides the Legendre map*of the configuration space***L***JY*to the phase space*P*which is a finite-dimensional manifold endowed a polysymplectic form. The true Hamiltonian counterpart of classical first order Lagrangian field theory is covariant Hamiltonian formalism on*P*where canonical momenta correspond to derivatives of fields with respect to all world coordinates, not only the time. Covariant*L*when the Legendre map*is a diffeomorphism. If a Lagrangian***L***L*is degenerate, we come to a multi-Hamiltonian theory on the constraint subspace*of the phase space***L**(JY)*P*in general. This is the case of all the above mentioned field theories.**References:**

Giachetta, G., Mangiarotti, L., Sardanashvily, G., Advanced Classical Field Theory (World Scientific, 2009)

Giachetta, G., Mangiarotti, L., Sardanashvily, G., Covariant Hamilton equations for field theory, Journal of Physics A: Theoretical and Mathematical,

**32**(1999) 6629-6642WikipediA: Covariant Hamiltonian field theory

Dear Gennadi,

ReplyDeletein the contemporary approaches that use the (differencial) geometric formulation for classical field theory, which is presented in numerous papers, we always worried about the influence of defining all those mathematicall objects, which in some case introduce new assumptions, constraints, structures. Often enough, they then cause redundancies, and non-uniqueness and have to be curtailed afterwards again, to reduce the results to those which are really physical.

Therefore we choosed an other approach than the differencial geometrical one. Instead we treated the DeDonder Weyl covariant Hamilton Field Theory in the realm of local coordinates. This local coordinate representation enabled us to keep the whole mathmatics on the basis of pure tensor calculus. Nevertheless, this description is chart-independent and thus applies to all local coordinate systems.

It turned out, that previous approaches in this way failed because of additional assumptions that stoped further progress on this path. First of all, we find e.g. at Lanczos the statement: "Indeed, the Hamiltonian function H_1, of the extended problem vanishes identically". But not all information is erased by that. Only the value of the Hamiltonian is zero, not the Hamiltonian itself (the partitial derivatives still carry information, and need not to be 0). So it was possible to derive a complete consistent formalism only on the basis of simple tensor calculus without this or other artificial assumptions. We saw that this path was quite straight forward, since no new degrees of freedom or ununiqueness have shown up.

The results are an extended Hamilton Formalism in point mechanics, which treats space and time as dynamical variables on equal footing, a covariant Hamilton Field Formalism which lead us to a full-fledged canonical gauge theory (with everything in it: Poisson-brackets, Liouville, Noether, etc.). First we did this for a fixed spacetime. Many concepts that are normally introduced as assumptions in the convenient weak gauge theory showed up naturally (e.g. minimal coupling) by itself. Then we also extended our gauge theory for local U(N) gauge variance by the embedding of of a variation of the space time metric. The resulting canonical field equations then contain completely naturally, without additional assumptions, terms that are able to describe massive gauge fields. In our description it turned out, that the gauge ﬁelds constitute the source of a space time curvature which on its part creates a mass term. Potentially this explains, why the in other respects so successful conventional gauge theory, fails by explaining massive gauge bosons. The a priori determination of a ﬁxed space-time metric is obviously not compatible with massive gauge bosons, since massive particles according to the Einstein-equations are always connected with a dynamical space-time. In our extended canonical transformation formalism, general relativity is in a natural way integrated in gauge theory. Since also transformations of space-time now are considered, the existence of massive gauge ﬁleds does not longer break the claim for local gauge invariance.

You can find the covariant Hamilton Field theory and canonical gauge theory at

arXiv:1205.5754

and the extended theory, which shows up field equations related to massive gauge fields at

arXiv:1206.4452

best wishes

Hermine

This comment has been removed by the author.

DeleteDear Professor Reichau,

DeleteThank you for your comment. I looked through your works.

There is the intensive literature on this subject since 90th, including different variants: polysymplectic, multisymplectic, k-cosymplectic and others. In particular, a key point is that the jet manifold technique conventionally provides a strict mathematical formulation of theory of nonlinear differential operators and equations, including Lagrangian theory.

Sincerely yours

Gennadi Sardanashvily

gennadi.sardanashvily@unicam.it

http://www.g-sardanashvily.ru/

Greetings! I can notice the fact that you really understand what you are speaking about here. Do you have a special education that is somehow related with the topic of the blog entry? Waiting forward to hear your answer.

ReplyDelete