Lagrangian vs. Hamiltonian in QFT

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Discussion Overview

The discussion centers on the differences in Lorentz invariance between the Lagrangian and Hamiltonian formulations in quantum field theory (QFT). Participants explore the implications of these differences and the definitions of scalars and vectors in this context.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the Lagrangian being Lorentz invariant while the Hamiltonian is not, questioning whether the Hamiltonian can also be considered a scalar.
  • Another participant clarifies that the Hamiltonian is the time component of the energy-momentum four-vector, which is not Lorentz invariant.
  • A different participant notes that while the action and Lagrangian density are Lorentz invariant, the Lagrangian itself is not, suggesting that "Lagrangian" often refers to "Lagrangian density" in QFT literature.
  • There is a discussion about the meaning of "scalar" in QFT, with one participant arguing that it refers to Lorentz invariance rather than simply being a single-component object.
  • Another participant agrees that the Hamiltonian is the total energy and emphasizes that it is not a scalar in the QFT sense, as it is the zeroth component of a vector that changes under Lorentz transformations.
  • One participant mentions that demonstrating Lorentz invariance in the Hamiltonian formulation requires constructing additional components and showing they fulfill the Poincare operator algebra.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the definitions and implications of Lorentz invariance in the context of the Lagrangian and Hamiltonian. There is no consensus on the clarity of these concepts, and multiple viewpoints are presented.

Contextual Notes

Participants highlight the complexity of the relationship between energy, mass, and their invariance under Lorentz transformations, indicating that assumptions about these properties may not be straightforward.

copernicus1
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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!
 
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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".)
 
copernicus1 said:
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.

copernicus1 said:
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."
 
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.
 

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