I encountered a statement that "while Lorentz invariance is apparent in the Lagrangian formulation, it is not so in the Hamiltonian formulation of a classical field." I do not completely understand this statement, though I thought this statement was essentially pointing to the two questions I asked.
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Since this question is about appearances, it would be clearer if you showed the Lagrangian and the Hamiltonian for a relativistic scalar field. – G. Smith Sep 02 '20 at 16:55
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Here is a MathJax tutorial so that you can write them. – G. Smith Sep 02 '20 at 16:58
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More generally (not just for scalar field theory), the main points are:
The Lagrangian formulation of a relativistic$^1$ field theory is manifestly Lorentz-covariant as it is constructed from manifestly Lorentz-covariant objects.
Therefore the corresponding Hamiltonian formalism, defined via a Legendre transform, must in principle also be Lorentz-covariant, as the two formalisms describe the same underlying theory. However, such Hamiltonian formalism is not manifestly Lorentz-covariant as it singles out the time-coordinate.
Concerning a manifestly covariant Hamiltonian formalism, see e.g. Ref. 1 and this Phys.SE post.
References:
- C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories. Published in Three hundred years of gravitation (Eds. S. W. Hawking and W. Israel), (1987) 676.
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$^1$ The Lagrangian and Hamiltonian formulations of non-relativistic theories are not Lorentz-covariant (and we shall not discuss them further).
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