Conformal field theory and the bootstrap

Take a nonidentity scalar primary $O$ of dimension $\Delta$. Translation invariance makes its two-point function depend only on separation, and rotation invariance reduces that dependence to $r=|x|$. Under a dilation, each insertion contributes a factor set by its dimension. The correlator must therefore satisfy $G(\lambda r)=\lambda^{-2\Delta}G(r)$. Differentiating with respect to the positive scale factor at one gives an ordinary differential equation, $rG'(r)=-2\Delta G(r)$.

Solving gives $G(r)=C_O/r^{2\Delta}$ away from coincident points. The coefficient depends on the chosen operator normalization and can be set to one for a suitable nonzero operator. Contact terms at coincident points require separate distributional care. Special conformal symmetry further restricts two-point functions of different primaries: in an orthogonal basis their correlator vanishes unless the dimensions and other quantum numbers match appropriately. These are symmetry statements, not perturbative approximations.

Radial quantization treats distance from an insertion as a time-like ordering coordinate. A local primary creates a state, and derivatives create descendant states. Reflection positivity translates into nonnegative norms of the physical states in this Euclidean construction. Computing norms using the conformal algebra then bounds allowed dimensions. For a nonidentity scalar primary in a unitary theory with $d>2$, the bound is $\Delta\ge(d-2)/2$.

The identity is an important exception: its dimension is zero and it is not constrained by that nonidentity bound. Saturation by a scalar produces a null descendant and the corresponding free-field equation under the usual unitary CFT assumptions. In nonunitary theories or in two dimensions the representation analysis changes. A number called an anomalous dimension cannot be assigned arbitrarily while maintaining positivity and crossing; it is part of the consistent operator spectrum of a particular theory.

The operator product expansion expresses nearby insertions as a sum over local operators and their descendants. Applying it in different pairings of a four-point function must give the same result. Conformal blocks package the descendant contributions of each primary, leaving scaling dimensions and three-point coefficients as the unknown data. For identical scalars the squared coefficients are nonnegative in a suitable unitary normalization, giving numerical bootstrap methods something stronger than an arbitrary fitting problem.

A bootstrap bound excludes spectra inconsistent with crossing and the imposed assumptions. A small allowed island can become very informative, but it depends on assumptions such as symmetry, gaps, and which correlators were included. Finite derivative or spin truncations and numerical tolerances must be reported. Some special CFTs are exactly solvable; generic interacting higher-dimensional CFTs are not. The research achievement is often a rigorous or systematically strengthened constraint, not a full closed-form answer for every correlation function.