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My question is about the generators of the Lorentz group: signature $(-,+,+,+)$. I have found the well known Lorentz generators (intended as elements of its algebra evaluated in the identity element of the group)

Boosts: \begin{equation*} K_{1} =\begin{pmatrix} 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} \ \ \ K_{2} =\begin{pmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} \ \ K_{3} =\begin{pmatrix} 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 \end{pmatrix} \end{equation*}

Rotations: \begin{equation*} J_{1} =\begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0 \end{pmatrix} \ \ \ J_{2} =\begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{pmatrix} \ \ J_{3} =\begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} \end{equation*} with the following commutation relations \begin{equation*} [ J_{i} ,J_{j}] =\epsilon _{ijk} J_{k} \ \ \ \ [ K_{i} ,K_{j}] =-\epsilon _{ijk} J_{k} \ \ \ \ [ J_{i} ,K_{j}] =\epsilon _{ijk} K_{k} \end{equation*}

Then I want to find the generators of the action of the Lorentz group on spacetime, the induced vector fields, defined as \begin{equation} V^\sharp |_x=\frac{\operatorname{d}}{\operatorname{d}t}\exp (tV)x \Bigl|_{t=0} \end{equation} where $V$ is any of the previus generators, $x$ is a point of the spacetime and $t$ is a generic parameter.

For the Lorentz group I found the following generators of the action \begin{gather*} J^{\sharp }_{i} =\epsilon _{ijk} x^{j} \partial _{k} \ \Rightarrow \ J^{\sharp }_{1} =x^{2} \partial _{3} -x^{3} \partial _{2} \ \ \ \ J^{\sharp }_{2} =x^{3} \partial _{1} -x^{1} \partial _{3} \ \ \ \ J^{\sharp }_{3} =x^{1} \partial _{2} -x^{2} \partial _{1}\\ K^{\sharp }_{i} =x^{i} \partial _{0} +x^{0} \partial _{i} \ \Rightarrow \ K^{\sharp }_{1} =x^{1} \partial _{0} +x^{0} \partial _{1} \ \ \ \ K^{\sharp }_{2} =x^{2} \partial _{0} +x^{0} \partial _{2} \ \ \ \ K^{\sharp }_{3} =x^{3} \partial _{0} +x^{0} \partial _{3} \end{gather*}

My problem is that these generators don't have the same commutation relations of the generators of the group. For example $[J_1^\sharp,J_2^\sharp]=-J_3^\sharp$.

What am I wrong?

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1 Answer 1

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Suppose that $G$ is a Lie group, $M$ is a manifold and $\lambda:G\times M\rightarrow M$ is a smooth left action. There is then an induced mapping $\lambda_\ast:\mathfrak g\rightarrow\mathfrak X(M)$ of the Lie algebra $\mathfrak g$ into the Lie algebra of vector fields $\mathfrak X(M)$ that is an antihomomorphism, i. e. $$ [\lambda_\ast X,\lambda_\ast Y]=-\lambda_\ast[X,Y] $$ for any $ X,Y\in\mathfrak g$.

By contrast, for a right action, this relation is a genuine Lie algebra homomorphism. It is easy to turn a left action into a right action - use every element's inverse, i.e. $\rho_gx=\lambda_{g^{-1}}x$, then $\rho$ is a right action.

So using OP's notation, if $V^\sharp$ is defined as $$ V^\sharp_x=\frac{d}{dt}\exp(-tV)x|_{t=0}, $$ then the commutation relations will come out as intended.

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  • $\begingroup$ Thanks! I was wondering if it is always necessary to work with induced vector fields whose commutation relations are the same of the group generators or I can just leave everything how it is. I thought that something was wrong in my method, but now I see that my result is correct. Being aware of the reasons and of the (legitimate) minus sign in the commutation relations, can I use my definition of induced vector fields? $\endgroup$
    – Nabla
    Commented Jun 18, 2020 at 10:00

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