Conformable Dirac Equation for Graphene: Hamiltonian Formulation and Exact Solutions

We formulate the low-energy Dirac equation for graphene using conformable derivatives, introducing reference length and time scales to ensure dimensional consistency. The conformable derivative is shown to be equivalent to a coordinate scaling, which maps the conformable Dirac equation exactly onto the ordinary Dirac equation in scaled coordinates. A Hamiltonian formulation is developed via Dirac’s constraint analysis in the scaled coordinates. The consistency conditions of the constrained system are examined, revealing a second-class constraint structure that requires the introduction of Dirac brackets to properly define the reduced phase space. This Hamiltonian derivation is physically meaningful because it provides the framework for canonical quantization, ensures the correct counting of physical degrees of freedom, and rigorously establishes the unitary equivalence between the conformable and ordinary Dirac pictures. Exact analytical solutions are obtained for free massless fermions, massive Dirac fermions, a nanoribbon with infinite-mass boundary conditions, and Landau levels for a field uniform in the scaled coordinates. The energy spectrum in the nanoribbon scales as [Formula: see text], while the Landau levels are independent of [Formula: see text] when the field is uniform in the scaled coordinates. The work clarifies the conformable deformation as a local coordinate scaling and links it to a power-law strain field in graphene, providing a physical interpretation for the conformable parameter [Formula: see text].