Conformal Group: Construction and Relation to Poincaré Group

From Non-Linear Transformations to Linear Rotations

Let's begin by recalling the problem. The transformations we found, especially the SCTs, are non-linear functions of the coordinates $x^{\mu }$. As you know from your study of group theory in physics, representations are far easier to handle when the group acts linearly. The central idea we'll explore is that these complicated non-linear transformations in $d$-dimensional space can be represented as simple linear rotations in a higher, $(d+2)$-dimensional space.

The video below provides a good conceptual introduction to why we need to think beyond standard ${\mathbb{R}}^d$ and consider a compactified space where infinity is just another point.

The Embedding Space Formalism

The idea of "adding the point at infinity" can be made precise by embedding our $d$-dimensional space into a $(d+2)$-dimensional one. In this larger space, the conformal transformations will act as simple linear transformations (rotations) that preserve a particular metric.

For clarity in the following discussion, let's work in $d$-dimensional Euclidean space, ${\mathbb{R}}^d$. The corresponding conformal group is $SO(d+1,1)$. The logic for Minkowski space ${\mathbb{R}}^{1,d-1}$ is identical, leading to the group $SO(d,2)$.

The following reading provides a detailed construction of this embedding space.

Let's summarize the essential construction. We introduce a $(d+2)$-dimensional space with coordinates $X^A=(X^{\mu },X^{d+1},X^{d+2})$. We define a metric ${\eta }_{AB}$ with signature $(d+1,1)$ by the line element:
$d{s}^2={\delta }_{\mu \nu }d{X}^{\mu }d{X}^{\nu }+(d{X}^{d+1}{)}^2-(d{X}^{d+2}{)}^2$
It's often convenient to use light-cone coordinates $X^{\pm }=X^{d+1}\pm X^{d+2}$, so the line element becomes $d{s}^2={\delta }_{\mu \nu }d{X}^{\mu }d{X}^{\nu }+d{X}^{+}d{X}^{-}$.

Our physical $d$-dimensional space is identified with the projective null cone in this embedding space, defined by two conditions:

1. Null Cone: $X^A{X}_A={\eta }_{AB}{X}^A{X}^B=X^{\mu }{X}_{\mu }+X^{+}{X}^{-}=0$
2. Projective Identification: $X^A\sim \lambda X^A$ for any $\lambda \in \mathbb{R}\setminus \{0\}$. This means all points along a given ray from the origin are identified.

A point $x^{\mu }$ in ${\mathbb{R}}^d$ is mapped to a ray on this cone by the section:
$X^A(x)=(X^{\mu },X^{+},X^{-})=(x^{\mu },1,-x^2)$
You can verify this choice satisfies the null cone condition: $x^{\mu }{x}_{\mu }+(1)(-x^2)=x^2-x^2=0$. The choice $X^{+}=1$ fixes the scaling freedom.

The crucial insight is that a linear transformation $X^{\prime A}=G^A{}_B{X}^B$ that preserves the embedding space metric (i.e., a Lorentz transformation in ${\mathbb{R}}^{d+1,1}$, so $G\in O(d+1,1)$) induces a transformation $x^{\mu }\to x^{\prime \mu }$ that is a conformal transformation in ${\mathbb{R}}^d$. As the reading showed, this leads to $d{x}^{\prime 2}=(\frac{X^{+}}{X^{\prime +}}{)}^{2}d{x}^2$, which is precisely the definition of a conformal transformation with scale factor $\Omega (x{)}^2=(X^{+}/X^{\prime +}{)}^2$.

Since the dimension of the group $O(d+1,1)$ is $\frac{1}{2}(d+2)(d+1)$, this matches the number of generators we found for the conformal algebra. We have successfully unified all conformal transformations into a single group of higher-dimensional rotations.

The Conformal Algebra and its Poincaré Subalgebra

Now that we have identified the conformal group as $SO(d+1,1)$ (the connected component of $O(d+1,1)$), we can examine its Lie algebra, $so(d+1,1)$. This will allow us to make the connection to the Poincaré group explicit.

The generators of $so(d+1,1)$ are the antisymmetric tensors $J_{AB}$. Our task is to relate these $d(d+1)/2$ generators to the familiar generators $P_{\mu },M_{\mu \nu },D,K_{\mu }$.

As the text explains, the generators are identified as follows:

• Rotations: $M_{\mu \nu }=J_{\mu \nu }$
• Dilatation: $D=J_{d+1,d+2}$
• Translations: $P_{\mu }=J_{\mu ,d+1}+J_{\mu ,d+2}$
• Special Conformal Transformations: $K_{\mu }=J_{\mu ,d+1}-J_{\mu ,d+2}$

The standard commutation relations for the $so(d+1,1)$ algebra,
$[J_{MN},J_{RS}]=-i({\eta }_{MR}{J}_{NS}-{\eta }_{MS}{J}_{NR}-{\eta }_{NR}{J}_{MS}+{\eta }_{NS}{J}_{MR})$
when written in terms of $M,P,K,D$, reproduce exactly the conformal algebra we derived in the first lesson.

Now, let's address the second part of today's learning outcome: the relation to the Poincaré group. The Poincaré group consists of rotations/boosts ($M_{\mu \nu }$) and translations ($P_{\mu }$). Let's look at the subalgebra generated by these operators within the full conformal algebra:

• $[M_{\mu \nu },M_{\rho \sigma }]=-i(g_{\mu \rho }{M}_{\nu \sigma }-\dots )$
• $[M_{\mu \nu },P_{\rho }]=-i(g_{\mu \rho }{P}_{\nu }-g_{\nu \rho }{P}_{\mu })$
• $[P_{\mu },P_{\nu }]=0$ (This follows from the $J_{AB}$ definition)

These are precisely the commutation relations of the Poincaré algebra. This demonstrates that the Poincaré algebra is a subalgebra of the conformal algebra. Correspondingly, the Poincaré group is a subgroup of the conformal group. The conformal group extends the Poincaré group by adding dilatations and special conformal transformations.

Conclusion

In this lesson, we have achieved a significant milestone in our understanding of conformal symmetry. By moving to a higher-dimensional embedding space, we have found an elegant and unified description of the conformal group.

Key Takeaways:

• The non-linear action of the conformal group in $d$-dimensional Euclidean space can be linearized by representing it as the group of linear rotations in a $(d+2)$-dimensional space, $SO(d+1,1)$.
• The physical $d$-space is identified with the projective null cone ($X^2=0,X\sim \lambda X$) within this embedding space.
• The generators of $SO(d+1,1)$, $J_{AB}$, decompose neatly into the generators of translations ($P_{\mu }$), rotations ($M_{\mu \nu }$), dilatations ($D$), and SCTs ($K_{\mu }$).
• The Poincaré algebra, generated by $M_{\mu \nu }$ and $P_{\mu }$, is a subalgebra of the conformal algebra. The conformal group is therefore a beautiful extension of the spacetime symmetries of special relativity.

Now that we have a solid grasp of the symmetry group and its algebraic structure, our next step is to study its representations. In the next lesson, we will see how fields and operators in a quantum theory transform under this symmetry. This will lead us to the central concept of primary operators, which are the fundamental objects upon which Conformal Field Theories are built.