# Angular momentum and rotation symmetry

In my book, it is written that for any vector $$\mathbf{v}$$, we have

$$\{\mathbf{v},\mathbf{L}\cdot \mathbf{n}\}=\mathbf{n}\times\mathbf{v}.\tag{1}$$

For me it is absurd... For example, if we take $$\mathbf{v}=(\cos(x),0,0)$$

the Poisson bracket, say we consider $$L_z=xp_y-yp_x$$:

$$(\{\mathbf{v},\mathbf{L}\cdot \mathbf{n}\})_x=\frac{\partial v_1}{\partial x}\frac{\partial L_z}{\partial p_x}-\frac{\partial L_z}{\partial x}\frac{\partial v_1}{\partial p_x}+\frac{\partial v_1}{\partial y}...=-\sin(x)x.$$ I can see that it already fails sincer the left-hand side is $$\mathbf{n}\times\mathbf{v}$$ can not contain $$\sin(x)$$...

Or maybe I misunderstood the proposition?

Screenshot of the book's content:

• could you cite the book? – Fabio Di Nocera Jul 16 '19 at 16:37
• Poisson brackets are normally written with curly braces, not square ones. – G. Smith Jul 17 '19 at 0:43
• @G.Smith isn't curly brackets anticommutator? $\{a,b\}=ab+ba$. – Upc Jul 18 '19 at 0:17
• @Upc Can you still cite the book? A screenshot is not a replacement for a citation. It is more like a replacement for a quotation. – Dvij Mankad Jul 18 '19 at 0:31
• – G. Smith Jul 18 '19 at 2:09

2. The notion of vectors $${\bf v}$$ that seems to be relevant here are maps $$\{\text{phase space}\}\times\{\text{time axis}\}\quad\stackrel{\bf v}{\longrightarrow}\quad \mathbb{R}^3$$ that transform in the vector representation of the 3D rotation group $$SO(3)$$. Here the angular momentum $$L_x$$, $$L_y$$, and $$L_z$$ belong to the corresponding $$so(3)$$ Lie algebra. OP's eq. (1) is the characteristic property of such vectors $${\bf v}$$. [Here $${\bf n}$$ is a so-called rotation vector, although it is not a vector in above sense].
3. Similarly, scalars $$s$$ in this context are maps $$\{\text{phase space}\}\times\{\text{time axis}\}\quad\stackrel{s}{\longrightarrow}\quad \mathbb{R}$$ that transform in the trivial representation of $$SO(3)$$.
5. Similarly, a constant 3-tuple $${\bf v}=(1,-5,2)$$ is not a vector in the sense of point 2 because it stays constant rather than rotate under 3D rotations.