In David Simmons-Duffin's TASI lectures on conformal bootstrap, in section 2.4, the author derives the Killing equation (eq. 20) corresponding to spacetime translation symmetry (spacetime dimensions $D>2$), relying on the fact that there is a conserved symmetric stress tensor. I reproduce below the relevant part of the discussion from section 2.4 of the notes.

Given a vector field $\epsilon = \epsilon^\mu(x)\partial_\mu$, the charge $$ Q_\epsilon(\Sigma) = -\int_\Sigma dS_\mu \epsilon_\nu(x)T^{\mu\nu}(x)$$ will be conserved whenever \begin{align} 0&=\partial_\mu(\epsilon_\nu T^{\mu\nu})\\ &=\partial_\mu\epsilon_\nu T^{\mu\nu}+\epsilon_\nu\partial_\mu T^{\mu\nu}\\ &=\frac{1}{2}(\partial_\mu\epsilon_\nu + \partial_\nu\epsilon_\mu)T^{\mu\nu} \end{align} i.e. $$\partial_\mu\epsilon_\nu + \partial_\nu\epsilon_\mu=0$$

In flat space, the author states that the solutions to this Killing equation are \begin{align} p_\mu &= \partial_\mu\\ m_{\mu\nu}&= x_\nu\partial_\mu-x_\mu\partial_nu \end{align}

I don't see how these solutions satisfy the Killing equation if I substitute $\epsilon_\mu=p_\mu$, for example. How do I correctly interpret these solutions? By contrast, Ginsparg's notes describe the solutions to the Killing equations in terms of polynomials in the coordinate $x^\mu$, e.g. $\epsilon_\mu(x)=a_\mu$ for translations. How do I reconcile these two ways of writing solutions to the Killing equations?

Also, what does it mean to say that $\epsilon=\epsilon^{\mu}(x)\partial_\mu$ is a "vector field"? (mentioned above eq. 18 in Simmons-Duffin's notes)


1 Answer 1


1) $\partial_\mu$ provide a coordinate basis for any vector field, and we identify the tangent space at a point on the manifold as the span of $\partial_\mu$ at that point, exploiting the correspondence between directional derivatives and tangent vectors. For details, look at for example Carroll's book on GR. This allows us to write any vector in the tangent space as $\vec{v}=v^\mu\partial_\mu$.

2) These are solutions in the sense that these are the generators of the transformations you are considering. For example, the transformation corresponding to $\epsilon=const$, i.e. translations, is generated by $p_\mu$. Similarly if $\epsilon$ was an infinitesimal lorentz transformation on the coordinates, the transformation would be generated by $m_{\mu\nu}$. These generators act on fields, out of which the energy momentum tensor is constructed. The generators capture the consequence of the transformations on $x$ via $x\to x+\epsilon$ on the fields.

3) As an aside, the more general equation is the conformal killing equation, where the RHS of your killing equation is not zero but proportional to $\partial\cdot\epsilon$. In that case, there will be additional solutions-namely dilatations and special conformal transformations.


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