A Generalized Forces and QED/QCD

[JohnH]

In Lagrangian mechanics we learn about generalized forces. However, I haven't seen these explicitly mentioned in books on QFT. Can the Lagrangians of QED or QCD be expressed in terms of generalized forces or is there some connection there, in particular to the Nielsen form.

 

[Vanadium 50]

You can certainly write down Lagrangians, if that's your question.

 

[JohnH]

No, I mean, the Nielsen form of the Lagrange's equations is

d/dt (∂T/∂˙q_j− ∂T/∂q_j= Q_j

Where Q_j are generalized forces, and the Lagrangian in QED is

L = −1/4 F_μνF^μν + ψ(bar) (iγ^μ∂_μ − m) ψ − q ψ(bar)γ^μA_μψ

I understand that the former equation is a form of the Euler-Lagrange equation while the latter equation is an equation for the Lagrangian density, but I'm still curious about the connection between the generalized forces of Lagrangian mechanics and forces as described by QFT.

 

[vanhees71]

In relativistic physics, is much more natural to work with field equations than with point-particle equations as in Newtonian mechanics. There's even a no-go theorem telling us that there is no fully consistent dynamics of interacting point particles in relativistic physics. All we can do is the approximation that you have a single particle moving in an external field (like charged particles in an external electromagnetic field). A fully self-consistent set of equations for a closed point-particle system, however, does not exist:

H. Leutwyler, A no-interaction theorem in classical
relativistic Hamiltonian particle mechanics, Nuovo Cim. 37, 556 (1965),
https://doi.org/10.1007/BF02749856

 

[andresB]

In relativistic physics, is much more natural to work with field equations than with point-particle equations as in Newtonian mechanics. There's even a no-go theorem telling us that there is no fully consistent dynamics of interacting point particles in relativistic physics. All we can do is the approximation that you have a single particle moving in an external field (like charged particles in an external electromagnetic field). A fully self-consistent set of equations for a closed point-particle system, however, does not exist:

H. Leutwyler, A no-interaction theorem in classical
relativistic Hamiltonian particle mechanics, Nuovo Cim. 37, 556 (1965),
https://doi.org/10.1007/BF02749856

It is outside of the original question from the OP, but lately, I feel quite uncomfortable with the Currie-Jordan-Sudarshan no-go theorem. In this Hamiltonian setting, with the given Poisson brackets, I don't get the correct Lorentz transformation of the coordinates under the exponential action of the generator of Lorentz boost (instead, coordinates have the same transformation law as the Newton-Wigner operator), so everything after that feels suspicious to me.

 

[vanhees71]

Do you mean this paper?

https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.35.350

It's a long time ago, I've read it, and I don't remember the details.

What do you mean by "exponential action"? Here a Lorentz boost in 1-direction should be something like $\exp (\mathcal{L}_{K_1} \eta)$ where $\mathcal{L}_{K_1} A=\{K_1,A \}$ is the usual Poisson bracket as the "Lie bracket". Shouldn't this get you the correct Lorentz transformation for all four-vector quantities, particularly the space-time coordinates and four-momentum?

 

[andresB]

Do you mean this paper?

https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.35.350

It's a long time ago, I've read it, and I don't remember the details.

What do you mean by "exponential action"? Here a Lorentz boost in 1-direction should be something like $\exp (\mathcal{L}_{K_1} \eta)$ where $\mathcal{L}_{K_1} A=\{K_1,A \}$ is the usual Poisson bracket as the "Lie bracket". Shouldn't this get you the correct Lorentz transformation for all four-vector quantities, particularly the space-time coordinates and four-momentum?

Yes, that is the paper.

Indeed the Lorentz boost should be as you write it. It gives the correct transformation of the momentum and the Hamiltonian, but not for the spatial coordinates, not to mention that it doesn't even begin to work on the time (since time is not taken as a canonical variable).

And it is for the same reasons that the Newton-Wigner operator doesn't transform as the spatial components of a 4-vector, because time is not taken symmetrically in the formalism.

The only way to implement a Lorentz transformation as a canonical transformation (that I'm aware of) use the extended phase-space formalism https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.135.9755&rep=rep1&type=pdf

 

[JohnH]

Alright, thanks for the replies, but I am a bit confused. If the Euler-Lagrange equation is used in QFT, why can't the Nielsen form be used, the two equating. You seem to be saying that the Nielsen form requires a model that uses point particles, but why doesn't the same logic apply to the Euler-Lagrange equation?

 

[PeroK]

You seem to be saying that the Nielsen form requires a model that uses point particles, but why doesn't the same logic apply to the Euler-Lagrange equation?

The Lagrangian formalism and the Euler-Lagrange equations can apply to fields. E.g. in the case of classical fields:

https://en.wikipedia.org/wiki/Lagrangian_(field_theory)

 

[JohnH]

I understand second quantization is applied to the Lagrangian formalism. What I don't understand is why it can't be applied to the Nielsen form if it is interchangeable with the Euler-Lagrange equation.

 

[andresB]

First and foremost, you seem to be confusing point dynamics with field dynamics. The Lagrangian formulation of both is different. The Euler-Lagrange equations for field are different from the one you wrote.

Now, In Lagrangian mechanics, a distinction is made between the forces that can be derived from potential and the ones that can't.

The forces that can be derived from a potential can then be incorporated into the Lagrangian. In QED and QCD all interactions can be written in the Lagrangian. I'm confident you could write a "generalized force" for those interactions, but I don't think you gain anything from doing so.

 

[JohnH]

Thank you. That's clarifying.

 

[vanhees71]

I understand second quantization is applied to the Lagrangian formalism. What I don't understand is why it can't be applied to the Nielsen form if it is interchangeable with the Euler-Lagrange equation.

The problem with a point-particle picture is that there is no Hamiltonian formalism for relativistic interacting point particles and thus it's also impossible to use some heuristic argument like "canonical quantization" to formulate a relativistic quantum mechanics, and indeed it turns out that except for free particles there is no way to formulate a first-quantization formalism.

The physical reason is also obvious today: Whenever you deal with particles interacting at "relativistic energies", i.e., if the collision energy gets at or above the mass threshold for the lightest particles which participate in the interactions the scattering particles are involved in, with some probability it's possible that these new particles get created and/or the original particles get destroyed. That means you need a formalism, where creation and annihilation processes of particles can be described.

Historically the first formalism was Dirac's hole-theoretical formulation of QED. That's a very cumbersome way, but at least formally it's equivalent to the modern formalism: Dirac ad hoc introduced the idea that all states referring to negative-frequency modes of the free Dirac equation are occupied, and he defined that to be the vacuum, and thus these filled states only manifistate themselves if at some relativistic reaction an electron out of this "Dirac sea" is excited to the positive-frequency domain, and the corresponding hole acts like a particle with the opposite charge of the electron and positive energy. All this works, afaik, at least for QED, but it's very cumbersome to work with.

Much more adequate and elegant is the use of quantum field theory, i.e., "second quantization". This was already clear in the very beginning of quantum mechanics, when Jordan introduced the idea of quantizing the electromagnetic field in 1926 (in the famous "Dreimännerarbeit"), but at this time most of the other physicists didn't like this idea too much, and that's why a few years later Dirac had to reinvent the idea to describe spontaneous emission of photons in addition to induced emission and absorption, which can also be treated in the semiclassical approximation, where the particles ("electrons") are quantized but the em. field kept classical.

Today relativistic QT is exclusively treated in terms of relativistic, local QFTs, where annihilation and creation processes are very naturally described.

 

[JohnH]

I appreciate the detailed answer. I guess I was confused because visually Lagrangian mechanics resembles a field, but I see now that one must have a classical field in order to canonically quantize it. Seems like a silly mistake now, but again, thank you for clarifying things.

 

[JohnH]

I'm confident you could write a "generalized force" for those interactions, but I don't think you gain anything from doing so.

Would it look something like this?
∂_μ(∂L/∂(∂_μψ)− ∂L/∂ψ=ψ_Q