The standard approach to quantum field theory begins with a Lagrangian—a mathematical object encoding the dynamics of fields and their interactions. From this starting point, we derive equations of motion, compute correlation functions, and extract physical predictions. Yet this Lagrangian-first philosophy, powerful as it has proven, harbors a profound limitation: it assumes we know the microscopic definition of our theory before we can say anything about it.

The conformal bootstrap represents a radical departure from this paradigm. Rather than beginning with a specific Lagrangian and computing its consequences, the bootstrap approach asks: what constraints do consistency and symmetry alone impose on quantum field theories? The answer, it turns out, is far more restrictive than anyone initially imagined. Conformal field theories—those enjoying both Poincaré symmetry and scale invariance—can be cornered, constrained, and in some cases completely solved by demanding only that they make internal sense.

This philosophical shift has yielded spectacular practical results. Critical exponents in the three-dimensional Ising model have been computed to unprecedented precision. New conformal field theories have been discovered without ever writing down their Lagrangians. And perhaps most remarkably, the bootstrap has opened windows into quantum gravity through the AdS/CFT correspondence, constraining holographic theories in ways that illuminate the quantum structure of spacetime itself.

The conformal bootstrap rests on a deceptively simple observation: correlation functions in conformal field theories are not arbitrary objects but must satisfy powerful consistency conditions. Chief among these is crossing symmetry—the requirement that different ways of computing the same correlation function yield identical results. This constraint, combined with unitarity (the positivity of quantum mechanical probabilities), turns out to dramatically restrict what conformal field theories can exist.

Consider the four-point function of identical scalar operators. Using the operator product expansion, we can decompose this correlator into contributions from operators appearing in the OPE of two scalars. The conformal symmetry fixes the form of each contribution up to overall coefficients—the OPE coefficients—and the scaling dimensions of exchanged operators. These constitute the CFT data: the fundamental numbers defining any conformal field theory.

Crossing symmetry demands that the OPE expansion in the s-channel (grouping operators 1,2 and 3,4) equals the expansion in the t-channel (grouping operators 1,4 and 2,3). This equality must hold for all values of the conformal cross-ratios, yielding infinitely many constraints on the finite amount of CFT data. The system is wildly overdetermined, which is precisely the source of the bootstrap's power.

Unitarity adds further restrictions by requiring that OPE coefficients squared be positive and that scaling dimensions satisfy lower bounds determined by spin. These positivity constraints transform the bootstrap equations from abstract functional relations into a concrete mathematical problem: which combinations of CFT data are consistent with all crossing and unitarity conditions?

The remarkable fact is that these consistency conditions alone—without any reference to a Lagrangian, path integral, or microscopic definition—can determine CFT data uniquely or constrain it to narrow allowed regions. The bootstrap does not assume the existence of a particular theory; it derives what theories can exist from the requirement of internal consistency.

Takeaway

Consistency is not merely a check on known theories—it is a generative principle that can determine what theories exist without ever specifying their microscopic definition.

The abstract power of bootstrap constraints becomes concrete through a remarkable connection to convex optimization. The crossing equations, combined with unitarity bounds, define a search problem: find CFT data satisfying infinitely many polynomial inequalities in infinitely many variables. This infinite-dimensional feasibility problem would seem hopelessly intractable, yet a systematic approach exists.

The key insight is that the crossing equations can be reorganized as conditions on a finite-dimensional space of functionals. If we can find a linear functional that is positive on all contributions from operators above the unitarity bound yet negative when applied to the crossing equation, we have derived a contradiction—proving that no consistent CFT with the assumed properties can exist. This transforms the bootstrap into a problem of finding such ruling-out functionals.

Semidefinite programming provides the computational framework. By discretizing the functional space and truncating to finite operator spin, the bootstrap becomes a semidefinite program: optimize a linear objective subject to the constraint that certain matrices remain positive semidefinite. Modern interior-point algorithms solve such programs efficiently, enabling systematic exploration of the space of allowed CFT data.

The results have been extraordinary. For the three-dimensional Ising model—the paradigmatic example of a critical phase transition—bootstrap methods have determined the scaling dimension of the spin operator to eight significant figures: Δ_σ = 0.5181489(10). This exceeds the precision of any other theoretical method and rivals the best experimental measurements. The conformal bootstrap has become the most accurate technique for computing universal properties of critical phenomena.

Beyond precision, the numerical bootstrap has revealed unexpected structure. Allowed regions in CFT parameter space often exhibit sharp kinks or cusps where known theories sit. The three-dimensional Ising model appears at precisely such a kink, suggesting that nature selects for theories at special points in the landscape of consistent CFTs. Whether this reflects some deeper principle remains an open question.

Takeaway

The marriage of ancient consistency conditions with modern optimization algorithms has transformed philosophical constraints into the most precise computational tool for non-perturbative physics.

The AdS/CFT correspondence asserts that certain conformal field theories are exactly equivalent to theories of quantum gravity in anti-de Sitter space. This duality maps CFT operators to gravitational fields, correlation functions to scattering amplitudes, and CFT data to gravitational couplings. Constraining holographic CFTs through the bootstrap therefore constrains quantum gravity itself.

Large-N conformal field theories with sparse spectra of low-dimension operators are believed to have weakly-coupled gravitational duals. The bootstrap imposes severe restrictions on such theories. Causality in the bulk spacetime translates to bounds on OPE coefficients; the absence of negative-norm states requires specific relations between CFT data. These constraints have derived results about quantum gravity that would be difficult or impossible to obtain by direct gravitational calculation.

One striking application involves higher-spin theories. Conformal field theories with weakly broken higher-spin symmetry are constrained to have specific correlators fixed by the symmetry. This determines properties of Vasiliev theory—an exotic theory of higher-spin gauge fields in AdS—from pure CFT reasoning. The bootstrap perspective suggests that the space of consistent quantum gravitational theories is far more constrained than Lagrangian-level analysis would indicate.

The bootstrap has also illuminated the black hole information paradox and related puzzles. Thermal correlators in holographic CFTs encode black hole physics; bootstrap constraints on these correlators constrain the quantum mechanics of black holes. Recent work has connected bootstrap bounds to chaos, scrambling, and the emergence of semiclassical geometry.

Perhaps most profoundly, the bootstrap suggests that quantum gravity might be uniquely determined by consistency. If the conformal field theory dual to string theory on AdS sits at a special point in the bootstrap landscape—a kink or isolated point—then string theory would emerge not from postulates about fundamental strings but from the requirement that a quantum gravitational theory make mathematical sense.

Takeaway

The conformal bootstrap transforms the study of quantum gravity from the invention of new theories to the exploration of what consistency permits, suggesting that the structure of spacetime may be dictated by logical necessity.

The conformal bootstrap represents a profound shift in how we think about quantum field theory. Rather than viewing theories as defined by their microscopic constituents—fields and Lagrangians—the bootstrap perspective sees theories as constrained by their macroscopic consistency. What survives the gauntlet of crossing symmetry and unitarity earns the right to exist.

This shift carries philosophical weight beyond its technical successes. If physical theories are selected not by the particular atoms of their construction but by the requirement that they make sense, then the structure of nature may be far more inevitable than contingent. The bootstrap hints that the laws of physics are not arbitrary choices but mathematical necessities.

For the search for unification, this perspective offers both hope and challenge. Hope, because it suggests that the ultimate theory may be uniquely determined by consistency. Challenge, because finding that theory requires not intuition about microscopic physics but mathematical virtuosity in exploring the constraints. The conformal bootstrap shows that such exploration is possible—and that the results can exceed our expectations.