When [itex]\alpha[/itex] is infinitesimal, i.e. [itex]\alpha^{2} = 0[/itex], you can write [itex]e^{i\alpha} = 1 + i\alpha[/itex]. So, [itex]\Phi \to \Phi^{'} = \Phi + i\alpha \Phi[/itex], leads to [itex]\delta\Phi = \Phi^{'} - \Phi = i\alpha \Phi[/itex].marcom said:Hi,
Could you please explain me why, under the transformation of a complex valued field Φ→eiαΦ, for an infinitesimal transformation we have the following relation?
δΦ=iαΦ
Thanks a lot
An infinitesimal transformation of a field is a small, continuous change in the values or properties of a field. It is often used in physics to describe the behavior of a system over time.
An infinitesimal transformation involves a very small change in the field, while a finite transformation involves a larger, more significant change. Infinitesimal transformations are often used in physics because they allow for a more precise and accurate description of the behavior of a system.
Some examples of fields that undergo infinitesimal transformations include electric and magnetic fields, gravitational fields, and quantum fields.
Infinitesimal transformations are essential in physics because they help us understand how a system changes over time. They allow us to study the behavior of a system in small increments, which can then be used to make predictions about its behavior in larger increments.
While infinitesimal transformations are a powerful tool in physics, there are some limitations to their use. For example, they may not be applicable in systems with discontinuous or abrupt changes. Additionally, the assumptions made in using infinitesimal transformations may not always hold true in complex systems.