Lagrangian invariant but Action is gauge invariant

In summary, the conversation discusses the concept of having a massive photon in 3D and how it relates to gauge invariance. The Lagrangian (6) is shown to be gauge invariant, but there are leftover terms that need to be integrated. The speaker is unsure of how these terms will vanish and asks for clarification on the mathematical process.
  • #1
creepypasta13
375
0

Homework Statement



So I'm having some difficulty with my QFT assignment. I have to solve the following problem.

In three spacetime dimensions (two space plus one time) an antisymmetric Lorentz tensor
F[itex]^{\mu\nu}[/itex] = -F[itex]^{\nu\mu}[/itex] is equivalent to an axial Lorentz vector, F[itex]^{\mu\nu}[/itex] = e[itex]^{\mu\nu\lambda}[/itex]F[itex]_{\lambda}[/itex]. Consequently, in 3D
one can have a massive photon despite unbroken gauge invariance of the electromagnetic
field A[itex]_{\mu}[/itex]. Indeed, consider the following Lagrangian:

L = -(1/2)*F[itex]_{\lambda}[/itex]F[itex]^{\lambda}[/itex] + (m/2)*F[itex]_{\lambda}[/itex]A[itex]^{\lambda}[/itex] (6)

where

F[itex]_{\lambda}[/itex](x) = (1/2)*[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex]F[itex]^{\mu\nu}[/itex] = [itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex],

or in components, F[itex]_{0}[/itex] = -B, F1 = +E[itex]^{2}[/itex], F[itex]_{2}[/itex] = -E[itex]^{1}[/itex].

(a) Show that the action S = [itex]\int[/itex]d[itex]^{3}[/itex]x*L is gauge invariant (although the Lagrangian (6) is not invariant).
So I tried substituting A[itex]^{\lambda}[/itex] -> A[itex]^{\lambda'}[/itex] = A[itex]^{\lambda}[/itex] + [itex]\partial[/itex][itex]^{\lambda}[/itex][itex]\Lambda[/itex]
and F[itex]^{\lambda}[/itex] -> F[itex]^{\lambda'}[/itex] = [itex]\epsilon[/itex][itex]^{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]_{\mu}[/itex]A[itex]_{\nu}[/itex]'

then I obtained L' = L + (1/2)*[ F[itex]_{\lambda}[/itex] [itex]\epsilon^{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]_{\mu}[/itex] [itex]\partial[/itex][itex]_{\nu}[/itex] [itex]\Lambda[/itex] + some other terms]

What I don't understand is how these leftover terms would vanish after being integrated (to obtain S'), but they don't all vanish if they are not integrated (since L is not invariant). Is there some kind of special mathematical trick I have to use? I just don't see how I can integrate terms like [itex]\int[/itex]d[itex]^{3}[/itex]x F[itex]_{\lambda}[/itex][itex]\epsilon[/itex][itex]^{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]_{\mu}[/itex][itex]\partial[/itex][itex]_{\nu}[/itex][itex]\Lambda[/itex]
 
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  • #2
Well, I can see how Fλ ϵλμν∂μ ∂ν Λ would vanish. The \mu and \nu are symmetric wrt to exchange for the partial derivatives, but the indices of the \epsilon symbol are totally antisymmetric. Multiplying a symmetric tensor and an antisymmetry one gives zero always.
 

What is a Lagrangian invariant but Action is gauge invariant?

A Lagrangian invariant refers to a physical quantity or property that remains unchanged under transformations within a particular framework or system, specifically within the Lagrangian formalism. On the other hand, action refers to the fundamental quantity that describes the dynamics of a system. A gauge invariant action means that the action remains unchanged under gauge transformations, which are mathematical transformations that do not affect the physical predictions of a theory.

What is the significance of Lagrangian invariance and gauge invariance?

Lagrangian invariance and gauge invariance play crucial roles in modern physics, particularly in fields such as quantum field theory and particle physics. These concepts allow for the consistent description of physical systems and help to ensure that physical laws remain unchanged under different mathematical frameworks or transformations.

How do Lagrangian invariance and gauge invariance relate to each other?

Lagrangian invariance and gauge invariance are closely related concepts. In fact, gauge invariance is a type of Lagrangian invariance. This means that if a physical system is described by a Lagrangian that is invariant under certain transformations, it will also be invariant under gauge transformations.

What are some examples of Lagrangian invariant but Action is gauge invariant theories?

One example is the electromagnetic field theory, where the Lagrangian describing the dynamics of the electromagnetic field is invariant under gauge transformations. Another example is the Standard Model of particle physics, which is based on gauge theories and describes the interactions between elementary particles.

What are the implications of Lagrangian invariance and gauge invariance on physical theories?

Lagrangian invariance and gauge invariance have significant implications on the predictions and interpretations of physical theories. They allow for the consistency and accuracy of predictions, and also provide a mathematical framework for understanding the fundamental forces and particles in the universe.

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