The Bootstrap
There is a way to solve a conformal field theory without writing down a Lagrangian. Without specifying a path integral. Without choosing a gauge or choosing a renormalization scheme. You simply demand that the theory be consistent. You demand that the four-point function satisfy crossing symmetry. You demand that the operator spectrum be positive and that the OPE coefficients be real. And from these demands alone, you constrain — sometimes completely determine — the theory. This is the conformal bootstrap.
The idea is simple enough to state on a napkin. Consider the four-point function of identical scalar primaries $\mathcal{O}$ with dimension $\Delta$:
$$\langle \mathcal{O}(x_1) \mathcal{O}(x_2) \mathcal{O}(x_3) \mathcal{O}(x_4) \rangle = \frac{1}{(x_{12}^2 x_{34}^2)^\Delta} G(z, \bar{z})$$
where $z, \bar{z}$ are the conformally invariant cross-ratios. The function $G(z, \bar{z})$ encodes all the dynamics. It contains the information about the spectrum of operators in the OPE of $\mathcal{O} \times \mathcal{O}$ and their OPE coefficients.
Now, the OPE can be performed in two channels. In the $s$-channel, you expand $\mathcal{O}(x_1)\mathcal{O}(x_2)$ first, producing a sum over conformal families. In the $t$-channel, you expand $\mathcal{O}(x_1)\mathcal{O}(x_4)$ first. Crossing symmetry demands that these two expansions give the same result:
$$G(z, \bar{z}) = \left(\frac{z}{\bar{z}}\right)^\Delta G\left(\frac{1-z}{1-\bar{z}}, \frac{\bar{z}}{1-\bar{z}}\right)$$
This is a single equation, but it is an equation about an infinite tower of operators. Each operator in the OPE contributes a conformal block — a universal function determined entirely by the symmetry — and the coefficient of that block is the square of the OPE coefficient (for identical external operators). The crossing equation is:
$$\sum_{\mathcal{O}k} \lambda_k^2 F{\Delta, \ell}(z, \bar{z}) = \left(\frac{z}{\bar{z}}\right)^\Delta \sum_{\mathcal{O}k} \lambda_k^2 F{\Delta, \ell}\left(\frac{1-z}{1-\bar{z}}, \frac{\bar{z}}{1-\bar{z}}\right)$$
where $F_{\Delta, \ell}$ is the conformal block for an operator of dimension $\Delta$ and spin $\ell$. The sum runs over all operators appearing in the $\mathcal{O} \times \mathcal{O}$ OPE. The coefficients $\lambda_k^2$ must be positive. The dimensions $\Delta$ must satisfy the unitarity bound.
The bootstrap does not solve for the conformal blocks. They are known, at least in principle, from representation theory. It does not solve for the OPE coefficients or the spectrum. Instead, it uses the crossing equation to ask: what sets of ${\Delta_k, \lambda_k}$ are consistent with crossing symmetry and unitarity?
The answer depends on the external dimension $\Delta$ and on the central charge $c$ (in $d = 2$). For a given $\Delta$, you can draw exclusion plots: regions of the $(\Delta_{\text{gap}}, \Delta_*)$ plane that are ruled out, where $\Delta_{\text{gap}}$ is the dimension of the lightest operator other than the stress tensor, and $\Delta_*$ is the dimension of the lightest operator with spin greater than two. The exclusion is achieved by finding a functional $\alpha$ that acts on the conformal blocks such that $\alpha[F_{\Delta,\ell}] \geq 0$ for all operators consistent with the bound, but $\alpha[F_\Delta] < 0$ for the operator you are testing. If such a functional exists, the operator cannot appear in the OPE.
This is the bootstrap: turn a consistency condition into an exclusion problem. And the exclusion is powerful. In the 3D Ising model, the bootstrap has determined the dimensions of the leading scalar operators to 6-8 significant digits, in agreement with the best Monte Carlo and series expansion results. In the 3D XY model, similar results hold. The bootstrap has also placed tight bounds on the dimensions of operators in various theories, ruling out large classes of putative CFTs.
In two dimensions, the bootstrap is even more powerful. For rational CFTs, the space of consistent solutions is discrete, and the bootstrap can completely classify them. The minimal models are the solutions. But the bootstrap goes beyond the minimal models: it has been used to study Liouville theory, which is not rational but is still completely solvable, and more recently to study non-Lagrangian theories in $d = 4$ and $d = 6$ that arise in the context of the AdS/CFT correspondence.
The bootstrap was born in the work of Ferrara, Roulet, and Zumino (1972) and of Polyakov (1974), who recognized that the conformal bootstrap could be used as a non-perturbative tool. It was dormant for decades, then revived by Rychkov and Tanner in 2009, who showed that numerical methods could be used to solve the bootstrap equations in $d > 2$. Since then, the bootstrap has become one of the most powerful non-perturbative tools in theoretical physics, with a community of researchers who use semidefinite programming, analytical functionals, and numerical optimization to explore the space of consistent CFTs.
The bootstrap is a form of reasoning that does not appeal to dynamics. It does not ask "what are the equations of motion?" It asks "what are the constraints?" And in a conformal theory, the constraints are so strong that they sometimes determine the answer. This is the bootstrap's claim to fame: that consistency alone, applied to the simplest possible assumptions (locality, unitarity, crossing symmetry), can constrain a theory to the point of near-uniqueness.
It is also a form of humility. The bootstrap does not assume that you can solve the theory from a Lagrangian. It does not assume that perturbation theory will work. It does not assume that the theory has a weak coupling limit. It assumes only that the theory is a conformal field theory, and it asks what follows. The answer is often: the theory is more constrained than you thought.
The bootstrap is not a method for all QFTs. It is a method for CFTs. But CFTs are everywhere: they are the fixed points of RG flows, the universal descriptions of critical phenomena, the boundary theories of AdS/CFT, the worldsheet theories of string theory. And the bootstrap is the most powerful tool we have for understanding them non-perturbatively. It is the art of constraining from consistency, and it is the purest expression of what theoretical physics can achieve when it relies not on models but on principles.