Standard Model: Lagrangian vs. Hamiltonian

In summary: So, in summary, the Lagrangian and Hamiltonian formulations of classical mechanics and quantum field theory are equivalent and can be related by canonical transformations. The Lagrangian is more fundamental when considering path integrals and gauge symmetry, but it is just a matter of notation and convenience. Both formulations are valid and can be used depending on the situation.
  • #1
suyver
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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?
 
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  • #2
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.
 
  • #3
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.
 
  • #4
Cool, thanks for the clear answers!
 
  • #5
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.
 
  • #6
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.
 
  • #7
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.
 

1. What is the Standard Model?

The Standard Model is a theoretical framework in physics that describes the fundamental particles and their interactions, as well as the forces that govern the behavior of these particles.

2. What is the difference between the Lagrangian and Hamiltonian formulations of the Standard Model?

The Lagrangian and Hamiltonian formulations are two different mathematical approaches used to describe the behavior of the particles in the Standard Model. The Lagrangian approach is based on the principle of least action, while the Hamiltonian approach is based on the principle of least energy.

3. Which formulation is more commonly used in the Standard Model?

The Lagrangian formulation is more commonly used in the Standard Model as it allows for easier calculations and is more flexible in incorporating new theories or particles.

4. What is the significance of the Lagrangian and Hamiltonian formulations in the Standard Model?

The Lagrangian and Hamiltonian formulations play a crucial role in the Standard Model as they provide the mathematical framework for understanding the behavior of particles and the fundamental forces that govern their interactions. They also help in predicting and testing new phenomena and theories.

5. Are there any limitations to the Standard Model's Lagrangian and Hamiltonian formulations?

While the Lagrangian and Hamiltonian formulations have been successful in describing many fundamental particles and their interactions, they have some limitations. For example, they do not include gravity, which is described by a separate theory called general relativity. Additionally, the Standard Model is unable to explain certain phenomena, such as dark matter and dark energy, which remain as open questions in physics.

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