How do vector functions behave under transformations for symmetry?

[Nikratio]

Hi,

How does one define symmetry of a system?

I believe that a scalar function $$g(\vec x)$$ is called symmetric under a transformation $$\vec F(\vec x)$$ if and only if $$g(\vec x) = g(\vec F(\vec x))$$

If there is an equivalent criteria for vector functions, I would be inclined to define a system as symmetric under a transformation if all its observables are symmetric under this transformation. Is that correct?




However, how does a vector function need to behave in order to be called symmetric? If the transformation is just a translation in space, we we want all the cartesian components to be invariant, just like individual scalar functions. On the other hand, if the transformation is a rotation, we want the cartesian components to rotate accordingly. But what is the general pattern here? How do the components of a vector valued function have to transform under a general transformation, in order for the function to be called symmetric under this transformation?

 

[zush]

Symmetry is defined by invariance of the integral of a function over a transformation. (i.e. linear, rotation, gauge, etc.) If the integral does not change over these transformations, then it is invariant, and the function is said to have that kind of symmetry. (i.e. translational symmetry, rotational symmetry, gauge symmetry.) I suggest that you read about calculus of variations, and then about invariance and Noether's theorem.

 

[Nikratio]

Symmetry is defined by invariance of the integral of a function over a transformation. (i.e. linear, rotation, gauge, etc.) If the integral does not change over these transformations, then it is invariant, and the function is said to have that kind of symmetry. (i.e. translational symmetry, rotational symmetry, gauge symmetry.) I suggest that you read about calculus of variations, and then about invariance and Noether's theorem.

Could you be more specific? What is "the integral of a function over a transformation"?

I think I am reasonably familiar with calculus of variations and Noether's theorem, but maybe I should refresh my memory because I don't see how either would help me with my question. Could you give more concrete references?