| Summary: | In this paper, we argue in favor of first-order homogeneous Lagrangians in the velocities. The relevant form of such Lagrangians is discussed and justified physically and geometrically. Such Lagrangian systems possess Reparametrization Invariance (RI) and explain the observed common Arrow of Time as related to the non-negative mass for physical particles. The extended Hamiltonian formulation, which is generally covariant and applicable to reparametrization-invariant systems, is emphasized. The connection between the explicit form of the extended Hamiltonian <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">H</mi></semantics></math></inline-formula> and the meaning of the process parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> is illustrated. The corresponding extended Hamiltonian <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">H</mi></semantics></math></inline-formula> defines the classical phase space-time of the system via the Hamiltonian constraint <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">H</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and guarantees that the Classical Hamiltonian <i>H</i> corresponds to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mn>0</mn></msub></semantics></math></inline-formula>—the energy of the particle when the coordinate time parametrization is chosen. The Schrödinger’s equation and the principle of superposition of quantum states emerge naturally. A connection is demonstrated between the positivity of the energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>=</mo><mi>c</mi><msub><mi>p</mi><mn>0</mn></msub><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and the normalizability of the wave function by using the extended Hamiltonian that is relevant for the proper-time parametrization.
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