In the early days of quantum mechanics, Erwin Shroedinger was developing his famous wave equation. I may need to double check this, but I believe he initially tried to develop a relativistic wave equation but essentially came up with a prototype of the Klein-Gordon equation and abandoned it. Klein-Gordon actually developed the equation and apparently it doesn't work for the electron but it works for spin-0 bosons, such as the Higgs. Shroedinger went on to develop his famous wave equation, which ostensibly contains all the information you would ever want to know about a particle/system, and if you're really interested, you can just apply a measurable operator to that equation and find your position, momentum, or energy. My question here regards the structure of these equations. Klein-Gordon has two derivatives of time and two derivatives of space. This is the logical first stab at formulating a relativistic wave equation just looking at the Einstein relation, E^2=(pc)^2+(mc^2)^2. The cononical Shroedinger equation has one derivative of time and two derivatives of space. And the Dirac equation has one derivative of time and one derivative of space. So again, my question is basically this...How can we be so cavalier about differentially differentiating these wave equations? Is there any larger model of quantum mechanics that can justify arbitrarily taking different time and space derivatives for these three equations?