Rebooting the S-matrix bootstrap

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I will discuss constraints on the S-matrix of gapped, Lorentz invariant quantum field theories due to crossing symmetry, analyticity and unitarity.  In particular I will bound cubic couplings, quartic couplings and scattering lengths relevant for the elastic scattering amplitude of two identical scalar particles. After a warm-up in 1+1 dimensions I will move to 3+1 dimensions.  In the cases where the results can be compared with results in the older S-matrix literature they are in excellent agreement.