Perfect scalars on the lattice

We perform renormalization group transformations to construct optimally loc al perfect lattice actions for free scalar fields of any mass. Their coupli ngs decay exponentially. The spectrum is identical to the continuum spectru m, while thermodynamic quantities have tiny lattice artifacts. To make such actions applicable in simulations, we truncate the couplings to a unit hyp ercube and observe that spectrum and thermodynamics are still drastically i mproved compared to the standard lattice action. We show how preconditionin g techniques can be applied successfully to this type of action. We also co nsider a number of variants of the perfect lattice action, such as the use of an anisotropic or triangular lattice, and modifications of the renormali zation group transformations motivated by wavelets. Along the way we illumi nate the consistent treatment of gauge fields, and we find a new fermionic fixed point action with attractive properties.