Bootstrap crossing and positivity

Take four identical Hermitian scalar primary operators with scaling dimension $\Delta_\phi$ in a unitary Euclidean conformal field theory. Factor the two-point-distance dependence so that the remaining function is $G(u,v)$, where u and v are conformal cross ratios. Expanding the operator product of the first pair gives a sum of conformal blocks. Exchanging an appropriate pair of insertion points yields another expansion of the same correlator. Equality of these descriptions is crossing symmetry, rather than a new dynamical force.

With the conventional prefactor $1/(x_{12}^{2\Delta_\phi}x_{34}^{2\Delta_\phi})$, crossing reads $v^{\Delta_\phi}G(u,v)=u^{\Delta_\phi}G(v,u)$. The block sum separates known kinematic functions from unknown dimensions, spins, and OPE coefficients. The identity contribution has fixed normalization once the external two-point function is normalized. Changing the prefactor convention changes the displayed crossing equation, so a numerical implementation must use mutually consistent blocks and prefactors.

Reflection positivity and an appropriate real operator basis make the identical-scalar OPE coefficients enter as nonnegative squares. Move crossing into a vector equation containing the identity vector plus a nonnegative sum of exchanged-operator vectors. A linear functional can rule out a proposed spectrum if it evaluates positively on the identity and nonnegatively on every exchanged vector allowed by the assumptions. Applying it would make zero equal a strictly positive quantity, which is impossible.

The word every is doing essential work. A sampled grid of a few dimensions and spins is not automatically a certificate over a continuous unbounded spectrum. Practical numerical methods use controlled block approximations, derivative bases, spin treatment, and positive-polynomial or semidefinite constraints. Numerical tolerances and approximation errors must be assessed before calling a result rigorous. In mixed-correlator systems, positivity becomes a matrix condition rather than the simple scalar-square form used here.

Bootstrap bounds depend on dimension, external dimensions, global symmetry representations, unitarity, and any assumed gaps or operator multiplicities. A gap assumption is additional physical input, even when motivated by a known model. An isolated allowed island can strongly identify candidate data, but survival at finite derivative order is still necessary consistency evidence rather than a complete constructive definition of the theory. Convergence under stronger tests makes that evidence more persuasive.

The worked example compresses the functional logic to one scalar constraint. It teaches why nonnegative coefficients can make a proposed cancellation impossible, without pretending that two arbitrary numbers are conformal blocks. Actual research calculations combine infinitely many potential operators and exploit their analytic structure. Extending sharp bounds to less symmetric theories, real-time observables, and situations without ordinary positivity remains challenging; those directions require new arguments rather than silently reusing the unitary identical-scalar setup.