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Here is the equation

$$\{x_i,L_j\}=\sum\limits_{k}ϵ_{ijk} x_k.$$

Is this equation generalised for any number of dimensions? In which case, would the following example be correct assuming 4 dimensions?

$$\{x_2, L_1\}=ϵ_{213} x_3 + ϵ_{214} x_4.$$

This, however contradicts Susskind stating that the indices $i$, $j$ and $k$ represent the three directions of space.

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    $\begingroup$ The notation makes it clear that it is only applicable to 3D. There is definitely a generalisation to higher dimensions, but it is not obvious how to do it $\endgroup$
    – naturallyInconsistent
    Commented Jul 21 at 7:51
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    $\begingroup$ In four dimensions you need a four-index Levi-Civita symbol $\epsilon_{ijkl}$. $\endgroup$
    – mike stone
    Commented Jul 21 at 7:55

2 Answers 2

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  1. In any dimension (not necessarily 3), the angular momentum is a bi-vector $$L^{ij}~=~x^ip^j-x^jp^i,\tag{1}$$ cf. e.g. this Phys.SE post.

  2. Together with the canonical Poisson bracket $$\{x^i,x^j\}~=~0,\quad\{x^i,p_j\}~=~\delta^i_j, \quad \{p_i,p_j\}~=~0, \tag{2}$$ def. (1) implies the following form of OP's sought-for equation $$\{L^{ij},x^k\}~\stackrel{(1)+(2)}{=}~x^j\eta^{ik}-x^i\eta^{jk},\tag{3}$$ where $\eta_{ij}$ are components of a metric tensor.

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    $\begingroup$ Let me add that specifically in 3 dimensions, because of the existence of a 3-tensor $\epsilon_{ijk}$, we can equivalently describe a bi-vector as a vector $L_i = \frac{1}{2} \sum_{j,k} \epsilon_{ijk} L^{jk}$. Using this definition, we can derive the bracket in the question from the bracket in this answer. $\endgroup$
    – Prahar
    Commented Jul 21 at 9:55
  • $\begingroup$ $\uparrow$ Right. $\endgroup$
    – Qmechanic ♦
    Commented Jul 21 at 10:03
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All the answers and comments above are brilliant and professional, however I probably get your point here, as I am also very new to these topics and notations.

For the summation, it is not about any physical meaning or adding any physical quantity along 3 axes.

The summation is here only to work with the Levi-Civita Symbol ϵ, which is defined to have non-zero value only when i , j , k are all different.

Moreover, as others said, the notation and equation only works in 3D, because the design of ϵ is specificly for the right-hand rule in 3D space.

The example in the same page of the book:

{x2, L1} = ϵ213 * x3

which is from the summation:

{x2, L1} = ϵ211 * x1 + ϵ212 * x2 + ϵ213 * x3 = 0 * x1 + 0 * x2 + (-1) * x3 = -x3

This one powerful equation can describe all the different situations of directions with the aid of summation and the Levi-Civita Symbol ϵ.

This is my first day and first answer in stachexchange and I don't understand how to use the style or latex very well so please forgive me for the bad arrangement.

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