Quantum structure emerging from the transformation design of the Dirac equation

• We perform the transformation design of space and time for spin-1/2 matter waves.
• A quantum rule could naturally emerge as constraints imposed on the Dirac equation itself.
• New fermion states share the quantum spectrum of a 2D Coulomb system.
• Transformation design uncovers a new exact solvable model.
• A quantum spectrum can be created by a geometric structure.

It is shown that a quantum structure can be created by a set of chosen constraint conditions that emerge from the transformation design of the Dirac equation in general relativity. As an explanation, the constraints that cause novel bound states with the quantization rule of a 2D Coulomb system are presented. The discussion in this paper provides a systematic way to look for constraints that generate a required quantization rule.