One-Dimensional Conformal Symmetry and SL(2,R)

A one-dimensional conformal theory has only three connected infinitesimal spacetime symmetries—translation, dilation, and special conformal transformation—but their global action is subtle. They generate $\mathfrak{sl}(2,\mathbb R)$, act projectively on the compactified line, and organize every local primary into a lowest-weight tower. This page separates that kinematics from the stronger assumptions of unitarity, locality, or a particular quantum-mechanical realization.

Required background. The Conformal Algebra and Its Generators fixes the generator convention. State–Operator Correspondence supplies the primary-state interpretation. Helpful background. Representations and Intertwiners supplies the language of modules and covers.

The compactified line and its global action

Adjoin one point at infinity to $\mathbb R$. The result $\mathbb R\cup\{\infty\}\simeq\mathbb{RP}^1$ is a circle, conveniently described by homogeneous coordinates $[X^1:X^2]$ with $x=X^1/X^2$. A matrix

$$g=\begin{pmatrix}a&b\\ c&d\end{pmatrix}, \qquad ad-bc=1,$$

acts by the Möbius transformation

$$x\longmapsto g\cdot x=\frac{ax+b}{cx+d}.$$

The matrices $g$ and $-g$ give the same map, so the faithful connected action on points is $PSL(2,\mathbb R)=SL(2,\mathbb R)/\{\pm1\}$. The universal cover can nevertheless matter for states: a projective representation of $PSL(2,\mathbb R)$ may lift to an ordinary representation only on a cover. One must therefore distinguish the group acting on the coordinate from the group represented on the Hilbert space.

The derivative is $(cx+d)^{-2}$ away from the pole. Thus a connected Möbius map preserves cyclic order on the compactified line, even though its affine expression can send a point through infinity. Orientation reversal, for example $x\mapsto-x$, belongs to a disconnected extension and requires separate parity or antiunitary data. It does not follow from the connected algebra.

For a primary $\mathcal O$ of scaling dimension $\Delta$, covariance under an orientation-preserving map is

$$U_g\,\mathcal O(x)\,U_g^{-1} =\left|\frac{d(g\cdot x)}{dx}\right|^\Delta \mathcal O(g\cdot x).$$

The absolute value makes the Euclidean scaling factor real and positive. A field with an orientation-reversal sign needs an additional label $\varepsilon_{\mathcal O}=\pm1$; it is not encoded by $\Delta$.

Infinitesimal generators

On a primary insertion, the Ward operators may be represented as

$$\mathscr P=-\partial_x, \qquad \mathscr D=-(x\partial_x+\Delta), \qquad \mathscr K=-(x^2\partial_x+2\Delta x).$$

Their signs reflect that these differential operators annihilate correlators. The abstract state-space generators use the volume convention

$$[D,P]=P, \qquad [D,K]=-K, \qquad [K,P]=2D.$$

At the origin, a primary state obeys

$$D|\mathcal O\rangle=\Delta|\mathcal O\rangle, \qquad K|\mathcal O\rangle=0.$$

The descendants are $P^n|\mathcal O\rangle$ with dimensions $\Delta+n$. For a reflection-positive theory, $P^\dagger=K$, and induction gives

$$\langle\mathcal O|K^nP^n|\mathcal O\rangle =n!\,(2\Delta)_n\,\langle\mathcal O|\mathcal O\rangle.$$

Consequently a nonidentity scalar lowest-weight module has $\Delta>0$ in a positive Hilbert space; $\Delta=0$ is the identity module after null descendants are removed. Without reflection positivity the same algebraic tower exists, but this norm argument and the sign restriction do not.

The conformal-algebra, primary, and descendant construction used here is the one developed in Simmons-Duffin 2017, §§3–4, pp. 12–20, Open PDF, specialized to the one-dimensional subalgebra.

From the line to the circle

Choose a circle coordinate $\theta$ by

$$x=\tan\frac{\theta}{2}, \qquad -\pi<\theta<\pi.$$

The point $x=\infty$ closes the circle at $\theta=\pi$. A primary transported to the circle is

$$\mathcal O_{S^1}(\theta) =\left(\frac{dx}{d\theta}\right)^\Delta\mathcal O_{\mathbb R}(x) =\left(\frac{1+x^2}{2}\right)^\Delta\mathcal O_{\mathbb R}(x).$$

This is a Weyl-covariant change of conformal frame, not yet a choice of real-time Hamiltonian. In conformal quantum mechanics, the physical Hamiltonian may be the parabolic generator $P$, while a compact evolution operator proportional to $P+K$ has a discrete lowest-weight spectrum. Confusing those generators can turn a representation statement into a false dynamical claim; the original de Alfaro–Fubini–Furlan construction is a concrete warning de Alfaro, Fubini, and Furlan 1976, §§2–4.

A representation checklist

Before interpreting a proposed one-dimensional conformal system, record:

Question	Required datum	What it controls
Which global group acts?	$PSL(2,\mathbb R)$, $SL(2,\mathbb R)$, or a cover	Allowed monodromy of states around the circle
Is orientation reversal a symmetry?	Unitary or antiunitary action and $\varepsilon_{\mathcal O}$	Reflection relations and statistics signs
Which generator is the Hamiltonian?	$P$, $D$, or a compact combination	Whether energy is continuous, dilatational, or discrete
Which module occurs?	Lowest weight, continuous series, reducible, or indecomposable	Spectrum and descendant structure
Is the inner product positive?	Reflection-positive adjoint and domain	Positivity of norms and OPE squares

Local operators prepared at the origin naturally furnish lowest-weight modules. Other unitary irreducible representations of a cover of $SL(2,\mathbb R)$ are mathematically legitimate, but they are not automatically local primary modules. The distinction is essential when comparing an intrinsic 1D CFT with a quantum-mechanical model Andrzejewski 2016, §§2–4.

Common pitfalls

Treating the affine line as globally invariant. A special conformal transformation can pass through infinity. Global statements belong on $\mathbb{RP}^1$, with cyclic order tracked through the added point.

Equating the coordinate group with the state-space group. Points see $PSL(2,\mathbb R)$, while states can require a nontrivial cover. State monodromy and orientation reversal must be specified independently.

Calling every $\mathfrak{sl}(2,\mathbb R)$ model a 1D CFT. The algebra alone does not provide a local operator expansion, a reflection-positive vacuum, or crossing-symmetric correlators. Those are additional structures tested in the rest of the chapter.

Exercises

Show that $x\mapsto(ax+b)/(cx+d)$ preserves cyclic order on $\mathbb{RP}^1$ when $ad-bc=1$.

Solution

Away from its pole the derivative is $(cx+d)^{-2}>0$, so the map preserves local orientation. Crossing the pole moves continuously through the single point at infinity on $\mathbb{RP}^1$; it does not reverse the orientation of the circle. Hence cyclic order is preserved globally.

Derive the descendant norm $\langle\mathcal O|K^nP^n|\mathcal O\rangle=n!(2\Delta)_n\langle\mathcal O|\mathcal O\rangle$.

Solution

Using $K|\mathcal O\rangle=0$ and the algebra gives $KP^n|\mathcal O\rangle=n(2\Delta+n-1)P^{n-1}|\mathcal O\rangle$. Iterate this identity $n$ times. The product is $n!\prod_{j=0}^{n-1}(2\Delta+j)=n!(2\Delta)_n$.

References

• Andrzejewski, K. “Quantum Conformal Mechanics.” Annals of Physics 367 (2016): 227–250. arXiv. DOI.
• de Alfaro, V., Fubini, S., and Furlan, G. “Conformal Invariance in Quantum Mechanics.” Il Nuovo Cimento A 34 (1976): 569–612. DOI.
• Simmons-Duffin, D. “TASI Lectures on the Conformal Bootstrap.” In Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, 1–74. World Scientific, 2017. arXiv. DOI.