Conformal Algebra from Angle Preservation

1. What is a Conformal Transformation?

In physics, symmetries are transformations that leave the laws of physics unchanged. You know from your study of relativity that the Poincaré group, consisting of translations and Lorentz transformations (rotations and boosts), represents the isometries of Minkowski spacetime. Isometries are transformations that preserve distances.

A conformal transformation is a less restrictive, and therefore more general, type of symmetry. It is a coordinate transformation that preserves angles, but not necessarily distances. Visually, this means that if you draw a grid and apply a conformal transformation to it, the grid lines might curve and stretch, but at every point where they intersect, they will still be perpendicular.

Mathematically, this means that a coordinate transformation $x^{\mu }\to x^{\prime \mu }$ is conformal if it rescales the metric tensor by a position-dependent factor, $\Omega (x)$. For a flat spacetime with metric $g_{\mu \nu }$, the new metric in the primed coordinates must satisfy:

The fact that the scale factor $\Omega (x)$ is the same in all directions at a given point is what ensures angles are preserved.

To understand this definition more deeply, please watch the following segment from a lecture by Tobias Osborne.

2. Infinitesimal Conformal Transformations

To understand the structure of the conformal group, we will employ a standard technique from your work with Lie groups: we'll analyze its infinitesimal transformations. This allows us to find the group's generators and their commutation relations, which define the Lie algebra.

We consider a transformation that is infinitesimally close to the identity:

where ${ϵ}^{\mu }(x)$ is a small, position-dependent vector field.

The requirement that this transformation is conformal leads to a powerful constraint on the vector field ${ϵ}^{\mu }(x)$. This constraint is known as the conformal Killing equation. The following reading derives this equation and classifies its solutions for spacetimes of dimension $d>2$.

As you've seen in the reading, solving the conformal Killing equation for $d>2$ dimensions shows that ${ϵ}^{\mu }(x)$ can at most be a quadratic function of the coordinates $x^{\mu }$. This gives rise to four distinct types of infinitesimal transformations:

1. Translations: ${ϵ}^{\mu }=a^{\mu }$ (constant vector)
   $x^{\mu }\to x^{\mu }+a^{\mu }$
2. Rotations / Lorentz Transformations: ${ϵ}^{\mu }={\omega }^{\mu }{}_{\nu }{x}^{\nu }$ (with ${\omega }_{\mu \nu }$ antisymmetric)
   $x^{\mu }\to ({\delta }_{\nu }^{\mu }+{\omega }^{\mu }{}_{\nu })x^{\nu }$
3. Dilatations (Scale Transformations): ${ϵ}^{\mu }=\lambda x^{\mu }$ (constant scalar $\lambda$)
   $x^{\mu }\to (1+\lambda )x^{\mu }$
4. Special Conformal Transformations (SCTs): ${ϵ}^{\mu }=2(b\cdot x)x^{\mu }-x^2{b}^{\mu }$ (constant vector $b^{\mu }$)
   $x^{\mu }\to x^{\mu }+2(b\cdot x)x^{\mu }-x^2{b}^{\mu }$

The first two generate the Poincaré group, which is a subgroup of the conformal group. Dilatations and SCTs are the new additions.

3. The Conformal Algebra

Each of these infinitesimal transformations is generated by a corresponding operator. The set of these generators and their commutation relations forms the conformal algebra.

In quantum field theory, these generators are operators acting on the space of fields or, equivalently, as differential operators on functions of the coordinates. The following reading provides their explicit form and the algebra they satisfy.

To summarize, the generators are:

• Translations: $P_{\mu }=i{\partial }_{\mu }$
• Rotations/Boosts: $M_{\mu \nu }=i(x_{\mu }{\partial }_{\nu }-x_{\nu }{\partial }_{\mu })$
• Dilatations: $D=i{x}^{\mu }{\partial }_{\mu }$
• Special Conformal Transformations: $K_{\mu }=i(2x_{\mu }{x}^{\nu }{\partial }_{\nu }-x^2{\partial }_{\mu })$

Their commutation relations define the conformal algebra, denoted $\mathfrak{so}(d+1,1)$ for Euclidean space or $\mathfrak{so}(d,2)$ for Minkowski space. Here are the key relations:

• The Poincaré subalgebra is preserved:

  $$[M_{\mu \nu },P_{\rho }]=-i(g_{\mu \rho }{P}_{\nu }-g_{\nu \rho }{P}_{\mu })$$ $$[M_{\mu \nu },M_{\rho \sigma }]=-i(g_{\mu \rho }{M}_{\nu \sigma }-\dots )$$

• $D$ is a scalar, while $P_{\mu }$ and $K_{\mu }$ are vectors under rotations:

  $$[M_{\mu \nu },D]=0,[M_{\mu \nu },K_{\rho }]=-i(g_{\mu \rho }{K}_{\nu }-g_{\nu \rho }{K}_{\mu })$$

• $D$ measures the "scaling dimension":

  $$[D,P_{\mu }]=-i{P}_{\mu },[D,K_{\mu }]=i{K}_{\mu }$$

  This implies $P_{\mu }$ acts like a lowering operator for the eigenvalue of $D$, while $K_{\mu }$ acts as a raising operator.

• The final, crucial commutator that closes the algebra:

  $$[P_{\mu },K_{\nu }]=2i(g_{\mu \nu }D-M_{\mu \nu })$$

All other commutators (e.g., $[P_{\mu },P_{\nu }]$, $[K_{\mu },K_{\nu }]$) are zero. The total number of generators is $\frac{(d+1)(d+2)}{2}$. For $d=4$, this gives 15 generators.

4. The Special Case: Two Dimensions

Our entire discussion so far has been for $d>2$. You may have noticed hints in the resources that $d=2$ is special. Let's briefly touch upon why.

In two dimensions, the conformal Killing equation becomes much less restrictive. In fact, it is equivalent to the Cauchy-Riemann equations of complex analysis.

This connection to holomorphic functions means that in two dimensions, the conformal symmetry group is infinite-dimensional. Any transformation $z\to f(z)$ where $z=x_1+i{x}_2$ and $f$ is a holomorphic function is a local conformal transformation. This vast symmetry makes 2D CFTs particularly tractable and powerful, which is a major reason for their importance in string theory and statistical mechanics. We will explore this in great detail in Module 4.

Conclusion

In this lesson, we have laid the groundwork for our study of conformal field theory.

Key Takeaways:

• Conformal transformations are coordinate transformations that preserve angles by rescaling the metric with a local factor $\Omega (x)$.
• For dimensions $d>2$, analyzing infinitesimal transformations reveals a finite-dimensional Lie algebra with $\frac{(d+1)(d+2)}{2}$ generators.
• This conformal algebra contains the Poincaré algebra and is extended by a generator for dilatations ($D$) and a vector of generators for special conformal transformations ($K_{\mu }$).
• The structure of the algebra is defined by the commutation relations between these generators, notably $[P_{\mu },K_{\nu }]=2i(g_{\mu \nu }D-M_{\mu \nu })$.
• In two dimensions, the symmetry algebra becomes infinite-dimensional due to its connection with holomorphic functions.

In our next lesson, we will look at the finite versions of these transformations and show how the conformal group can be identified with the group $SO(d+1,1)$, providing a more geometric and unified picture of these symmetries.