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elsafo
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Anyone can help me how to argue that interaction lagrangian is invariant under gauge transformation?
elsafo said:Anyone can help me how to argue that interaction lagrangian is invariant under gauge transformation?
Vanishing of ##J_\mu## at spatial infinity is uncontroversial. But... why should it vanish as ##|t|\to\infty## ?samalkhaiat said:[tex]\delta \int_{ D } d^{ 4 } x \ A_{ \mu } J^{ \mu } = \int_{ D } d^{ 4 } x \ \partial_{ \mu } ( \Lambda J^{ \mu } ) = \int_{ \partial D } d S_{ \mu } \ \Lambda J^{ \mu } = 0[/tex]
The first equality follows from the fact that gauge fields couple to conserved (matter) current, the second equality is just the divergence theorem, and the last one follow because [itex]J_{ \mu }[/itex] vanishes at the boundary [itex]\partial D[/itex] at infinity.
Hi,strangerep said:Hi Sam,
(Since the OP seems happy with the answers, I'll venture a clarification question...)
Vanishing of ##J_\mu## at spatial infinity is uncontroversial. But... why should it vanish as ##|t|\to\infty## ?
I had presumed that gauge transformations are required to approach the identity in that limit, but is there perhaps another reason?
OK, thanks.samalkhaiat said:Hi,
[...] If [itex]\Lambda = \lambda[/itex] is constant at [itex]\partial D[/itex], then the statement reduces to that of charge conservation.
[...]
Thus
[tex]\delta \int_{ D } d^{ 4 } x \ A^{ \mu } J_{ \mu } = \lambda \left( \int_{ \partial D_{ 1 } } d^{ 3 } x \ J_{ 0 } ( t_{ 1 } , x ) - \int_{ \partial D_{ 2 } } d^{ 3 } x \ J_{ 0 } ( t_{ 2 } , x ) \right) = 0 .[/tex]
Gauge invariance is a fundamental principle in physics that states that the laws of physics should remain unchanged regardless of the choice of a specific mathematical representation or gauge. In other words, physical quantities should not depend on the specific coordinate system or units used to measure them.
The interaction Lagrangian is a mathematical expression used to describe the interactions between particles in a physical system. Gauge invariance ensures that the interaction Lagrangian remains unchanged under a gauge transformation, meaning that the physical laws and predictions derived from it are still valid.
Gauge invariance is essential because it allows us to formulate and understand physical theories in a way that is independent of the chosen mathematical representation. Without it, we would not be able to make consistent and accurate predictions about the behavior of particles and systems.
In certain cases, gauge invariance can be violated. This can happen, for example, in theories that involve massless particles or when dealing with extreme conditions such as high energies or strong gravitational fields. However, these violations are typically resolved by introducing more complex mathematical frameworks.
Gauge invariance is a fundamental principle and is not directly tested experimentally. However, the predictions and calculations derived from theories that incorporate gauge invariance can be tested and verified through experiments and observations. If these predictions are consistent with the results of experiments, it provides evidence for the validity of gauge invariance in the theory.