Symmetries in Quantum Mechanics: from Klein-Gordon Equation to Higgs Mechanism

Symmetry has guided the development of advances in physics, from early quantum theory to modern field theories. The challenge is to understand the relationship of quantum mechanics to relativistic invariance and ultimately to apply it to particle mass generation. By tracing the historical developments from the Klein-Gordon equation to spontaneous symmetry breaking, the unifying role of symmetry among the different formulations is demonstrated. First, the 0-spin relativistic Klein-Gordon wave equation is derived. Second, the Dirac equation is derived from it. Third, by introducing gauge principles, the Higgs mechanism is naturally arrived at, explaining massive gauge bosons in the Standard Model. The paper shows the central role of symmetry both from a historical perspective and the implication in modern frameworks.