Approximate symmetries

Next we come to the special case of symmetries, i.e., invariance under under coordinate transformations. Symmetry transformations ${\bf S}$ change the arguments of a function $\phi$. For example for the translation of a function $\phi(x)\rightarrow\tilde \phi(x)={\bf S} \phi(x) = \phi(x-c)$. Therefore it is useful to see how ${\bf S}$ acts on the arguments of a function. Denoting the (possibly improper) eigenvectors of the coordinate operator ${\bf x}$ with eigenvalue $x$ by $(\cdot,\,x)$ = $\vert x)$, i.e., ${\bf x} \vert x) = x \vert x)$, function values can be expressed as scalar products, e.g. $\phi(x)$ = $(x,\, \phi)$ for a function in $x$, or, in two variables, $\phi(x,y)$ = $(x\otimes y,\, \phi)$. (Note that in this `eigenvalue' notation, frequently used by physicists, for example $2\vert x)\ne\vert 2x)$.) Thus, we see that the action of ${\bf S}$ on some function $h(x)$ is equivalent to the action of ${\bf S}^T$ ( = ${\bf S}^{-1}$ if orthogonal) on $\vert x)$

$${\bf S} \phi(x)=(x,{\bf S} \phi)=({\bf S}^T x, \phi) ,$$ (214)

or for $\phi(x,y)$

$${\bf S} \phi(x,y) =\left( {\bf S}^T (x \otimes y), \, \phi \right) .$$ (215)

Assuming ${\bf S}$ = ${\bf S}_x{\bf S}_y$ we may also split the action of ${\bf S}$,

$${\bf S} \phi(x,y) =\left( ({\bf S}_x^T x) \otimes y, \, {\bf S}_y \phi \right) .$$ (216)

This is convenient for example for vector fields in physics where $x$ and $\phi(\cdot,y)$ form three dimensional vectors with $y$ representing a linear combination of component labels of $\phi$.

Notice that, for a general operator ${\bf S}$, the transformed argument ${\bf S}\vert x)$ does not have to be an eigenvector of the coordinate operator ${\bf x}$ again. In the general case ${\bf S}$ can map a specific $\vert x)$ to arbitrary vectors being linear combinations of all $\vert x^\prime)$, i.e., ${\bf S}\vert x)$ = $\int \!dx^\prime\, S(x,x^\prime) \vert x^\prime)$. A general orthogonal ${\bf S}$ maps an orthonormal basis to another orthonormal basis. Coordinate transformations, however, are represented by operators ${\bf S}$, which map coordinate eigenvectors $\vert x)$ to other coordinate eigenvectors $\vert\sigma (x))$. Hence, such coordinate transformations ${\bf S}$ just changes the argument $x$ of a function $\phi$ into $\sigma(x)$, i.e.,

$${\bf S} \phi(x) = \phi(\sigma (x)) ,$$ (217)

with $\sigma(x)$ a permutation or a one-to-one coordinate transformation. Thus, even for an arbitrary nonlinear coordinate transformation $\sigma$ the corresponding operator ${\bf S}$ in the space of $\phi$ is linear. (This is one of the reasons why linear functional analysis is so useful.)

A special case are linear coordinate transformations for which we can write $\phi(x)\rightarrow\tilde \phi(x) = {\bf S} \phi (x) = \phi(Sx)$, with $S$ (in contrast to ${\bf S}$) acting in the space of $x$. An example of such $S$ are coordinate rotations which preserve the norm in $x$-space, analogously to Eq. (211) for $\phi$, and form a Lie group $S(\theta)=e^{\sum_i\theta_i A_i}$ acting on coordinates, analogously to Eq. (212).