Standard Model: Lagrangian vs. Hamiltonian

[suyver]

I was wondering: why is the SM always written with a Lagrangian? Couldn't you just as well write it with a Hamiltonian? The way I understand, the Lagrangian gives me the kinetic energy minus the potential energy (basically a measure for the "free energy", though not in the thermodynamical sense), while the Hamiltonian gives me the total energy of the system. Are these quantities interchangable, or is it really neccecary to write the SM in terms of a Lagrangian?

 

[beacon]

I think they are interchangealbe.
Probably Lagrangian is more fundamental when considering path integral and gauge symmetry. But I think it's just a kind of notation. You can also start from Lagrangian and correspondingly modify the forms of path integral and gauge symmetry.

 

[Yustas]

Doesn't matter, they are related by a canonical transformation. Depending on the situation you use one or another: for canonical quantization one uses Hamiltonian formalism, for path integrals -- Lagrangian.

 

[suyver]

Cool, thanks for the clear answers!

 

[Janitor]

I realize the question here has to do with the Standard Model, so the information I am giving is not relevant. But in case some first-year physics student wanders through this thread, I will quote something that I stumbled upon today in a Schaum's outline book that might be of some use in clarifying similar questions outside of the realm of particle physics. This has to do with plain old ordinary Newtonian physics, and was written in 1967--a time when there wasn't yet a Standard Model, if memory serves.

The basic laws of dynamics can be formulated in several ways other than that given by Newton. The most important of these are: (a) D'Alembert's principle; (b) Lagrange's equations, (c) Hamilton's equations, (d) Hamilton's principle. All are basically equivalent.

 

[humanino]

It appears to me slightly more subtle than just "OK, trivial transformation between the two formalism". The problem is : how do you treat time !? In the Hamiltonian formalism, time is clearly separated from other coordinate (space). This is very sick to make lorentz symmetry obvious. On the contrary, Lagragian formalism from the beginning respects lorentz symmetry.

 

[Haelfix]

Except, the actual process of quantizing a classical system is completely ambigous in the lagrangian formalism. Moreover, its not clear mathematically that they the hamiltonian and lagrangian of a system are related by canonical transformations, in fact it is not true in general.

In classical mechanics, Hamiltons principle is more general than the lagrangian. So too is it in quantum field theory. The lagrangian manifold is typically a subspace of the more general symplectic space

But like you said, it is more convenient to work with the lagrangian since it admits a path integral formulation, and is manifestly covariant.