Lagrangian for a relativistic free particle 472 16.6 Example: Relativistic particle in an … fundamental to Hamiltonian mechanics which is based on the Hamiltonian H (q, p ,t). For a conservative … total energy 187 7.10 Hamiltonian invariance 188 7.11 Hamiltonian for cyclic coordinates

A simple mathematical procedure is introduced which allows redefining in an exact way divergent integrals and limits that appear in the basic equations of classical electrodynamics with point charges. In this way all divergences are at once removed without affecting the locality and the relativistic covariance of the theory, and with no need for mass renormalization. The procedure is first used to obtain a finite expression for the electromagnetic energy-momentum of the system. We show that the relativistic Lorentz-Dirac equation can be deduced from the conservation of this electromagnetic energy-momentum plus the usual mechanical term. Then we derive a finite lagrangian, which depends on the particle variables and on the actual electromagnetic potentials at a given time. From this lagrangian the equations of motion of both particles and fields can be derived via Hamilton's variational principle. The hamiltonian formulation of the theory can be obtained in a straightforward way. This leads to an interesting comparison between the resulting divergence-free expression of the hamiltonian functional and the standard renormalization rules for perturbative quantum electrodynamics.

This paper demonstrates that the standard spacetime-first formulation of relativistic physics contains a fundamental structural circularity. Using Dirac’s reparametrization-invariant first-order Hamiltonian formalism, we show that spacetime coordinates arise as accumulated functionals of the energy–momentum trajectory and therefore cannot serve as an independent geometric background. Proper time and worldline geometry emerge only after gauge choice and are not primitive geometric inputs. This dependency inversion clarifies long-standing tensions involving background dependence, the problem of time, and generally covariant observables. It establishes energy and momentum as the logically prior variables and positions spacetime as a derived relational construct. The paper concludes by indicating how this viewpoint naturally extends to a background-free geometric framework defined on the Spectral Manifold, where Lorentzian structure appears directly in (E, p). This work serves as a conceptual bridge between classical relativistic mechanics and emerging energy-first approaches to spacetime and geometry. Functional Accumulation of Energy–Momentum Trajectories Precludes Background Spacetime | Zenodo Skip to main You are using an outdated browser. Please upgrade your browser to improve your experience. Published November 16, 2025 | Version v1 Journal article Open Functional Accumulation of Energy–Momentum Trajectories Precludes Background Spacetime Authors/Creators Lee, Michael (Researcher) Description This paper demonstrates that the standard spacetime-first formulation of relativistic physics contains a fundamental structural circularity. Using Dirac’s reparametrization-invariant first-order Hamiltonian formalism, we show that spacetime coordinates arise as accumulated functionals of the energy–momentum trajectory and therefore cannot serve as an independent geometric background. Proper time and worldline geometry emerge only after gauge choice and are not primitive geometric inputs. This dependency inversion clarifies long-standing tensions involving background dependence, the problem of time, and generally covariant observables. It establishes energy and momentum as the logically prior variables and positions spacetime as a derived relational construct. The paper concludes by indicating how this viewpoint naturally extends to a background-free geometric framework defined on the Spectral Manifold, where Lorentzian structure appears directly in (E, p). This work serves as a conceptual bridge between classical relativistic mechanics and emerging energy-first approaches to spacetime and geometry.

Dirac formulation of open relativistic strings as systems with constraints is made explicitly. Classical theory is given in the standard light-cone and covariant center-of-mass gauges. It is mentioned that the well-known result D = 26 is affected by using the standard quantization of the mutually independent nonphysical boson creation and annihilation operators. It is shown that in the Dirac formulation these operators are not independent in both the gauges. Concepts of Physics, Vol. IV, No. 4 (2007) DOI: 10.2478/v10005-007-0023-x 487 We give some new conditions on these operators and show that the theory is consistent with Poincare algebra in any dimension D. 488 Concepts of Physics, Vol. IV, No. 4 (2007) Dirac Formulation of Free Open String 1 Hamilton description of classical open string We will study the Nambu–Goto [1] free open string in dimension D. We assume the sign convention gμν = diag(−1, 1, . . . , 1), where μ, ν = 0, 1, . . . , D − 1. The string is described by the functions X(τ, σ), where τ ∈ R and σ ∈ 〈0, π〉. The classical string is described by Lagrangian L(X) = −ω ∫ π 0 Ldσ, where ω > 0 is a constant and the Lagrangian density L(τ, σ) is L = √( ẊX ′ )2 − (Ẋ)2(X ′)2 . A dot means partial derivation with respect to τ , a dash with respect to σ, and XY = gμνXY ν = XμY ν . The boundary conditions are X ′ μ(τ, 0) = X ′ μ(τ, π) = 0. In the Hamiltonian formulation we define momenta Pμ(τ, σ) = δL δẊμ(σ) = ω Ẋμ(X ′X ′)−X ′ μ(ẊX ′) √( ẊX ′ )2 − (Ẋ)2(X ′)2 . (1) From (1