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The positivity of the energy in relativistic quantum mechanics implies thatwave functions can be continued analytically to the forward tube T in complexspacetime. For Klein-Gordon particles, we interpret T as an extended (8D)classical phase space containing all 6D classical phase spaces as symplecticsubmanifolds. The evaluation maps $e_z: f\to f(z)$ of wave functions on T arerelativistic coherent states reducing to the Gaussian coherent states in thenonrelativistic limit. It is known that no covariant probability interpretationexists for Klein-Gordon particles in real spacetime because the time componentof the conserved "probability current" can attain negative values even forpositive-energy solutions. We show that this problem is solved very naturallyin complex spacetime, where $|f(x-iy)|^2$ is interpreted as a probabilitydensity on all 6D phase spaces in T which, when integrated over the "momentum"variables y, gives a conserved spacetime probability current whose timecomponent is a positive regularization of the usual one. Similar results areobtained for Dirac particles, where the evaluation maps $e_z$ are spinor-valuedrelativistic coherent states. For free quantized Klein-Gordon and Dirac fields,the above formalism extends to n-particle/antiparticle coherent states whosescalar products are Wightman functions. The 2-point function plays the role ofa reproducing kernel for the one-particle and antiparticle subspaces.

Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis