Lagrangian vs. Hamiltonian in QFT

[copernicus1]

I'm a little confused about why the Lagrangian is Lorentz invariant and the Hamiltonian is not. I keep reading that the Lagrangian is "obviously" Lorentz invariant because it's a scalar, but isn't the Hamiltonian a scalar also?

I've been thinking this issue must be somewhat more complex, because mass is an invariant between frames, and it's a scalar, but energy obviously isn't invariant, and it's a scalar too, so I guess I'm missing something. Is it related to the fact that energy is a component of the 4-momentum?

Any help is appreciated!

 

[Avodyne]

The hamiltonian is the time component of the energy-momentum four-vector; hence it is not Lorentz invariant.

In QFT, the action is Lorentz invariant, and the lagrangian density is Lorentz invariant, but the lagrangian itself is not. (In QFT books, "lagrangian" often really means "lagrangian density".)

 

[The_Duck]

I'm a little confused about why the Lagrangian is Lorentz invariant and the Hamiltonian is not. I keep reading that the Lagrangian is "obviously" Lorentz invariant because it's a scalar, but isn't the Hamiltonian a scalar also?

Not in the sense that is meant here. I believe you're taking "scalar" to mean "single-component object." But almost always in QFT, "scalar" means "Lorentz scalar," i.e., "object that is invariant under Lorentz transformations." So isn't saying that "the Lagrangian density is Lorentz invariant because it is a scalar" circular? What people mean here is: "look at our form for the Lagrangian density. All the Lorentz indices are contracted. Therefore this thing is invariant under Lorentz transformations."

Similarly, "vector" in QFT almost always means "4-vector," an object with specific transformation rules under Lorentz transformations.

I've been thinking this issue must be somewhat more complex, because mass is an invariant between frames, and it's a scalar, but energy obviously isn't invariant, and it's a scalar too, so I guess I'm missing something. Is it related to the fact that energy is a component of the 4-momentum?

Right; the Hamiltonian is the total energy, which is a component of the 4-momentum, which changes under Lorentz transformations, i.e., is not a scalar in the QFT sense. In relativistic field theory, we say, "the Hamiltonian is not a scalar: it is the zeroth component of a vector."

 

[tom.stoer]

To show that a theory in canonical i.e. Hamiltonmian formulation is Lorentz-invariant requires some work: in addition to the Hamiltonian H = P^° the other components Pⁱ as well as the generators of the Lorentz group Lⁱ (angular momentum - rotations) and Kⁱ (boosts) have to be constructed. In addition it has to be shown that these objects fulfil the required Poincare operator algebra.