Hi, I'm interested in constructive QFT and I'd like to pose a question about construction of the one-particle Hilbert space of states for bosons. For fermions satisfying the Dirac equation, the inner product is [tex] \langle \psi, \phi \rangle = \int d^3 x \; \psi^\dagger(x) \phi(x). [/tex] The inner product is a constant of motion (and Lorentz invariant) because the integrand is the 0th component of a conserved current. The inner product is positive definite, and defines the Hilbert space L^2. Is there a similar inner product for bosons satisfying the Klein-Gordan equation? I previously thought that boson state spaces required an indefinite inner product (i.e., existence of negative-norm states), which is a problem because the triangle inequality is broken. This basically ruins any chance of applying Hilbert space theory. Recently I found a paper  which refers to an answer, but doesn't give it explicitly. To quote from the introduction: Can somebody help me out here? Thanks very much, Dave  Shale, Trans. Am. Math. Soc. 103 (1962) 149, Linear Symmetries of Free Boson Fields.