[Paper Review] A formulation of Noether's theorem for fractional classical fields

Sami I. Muslih|arXiv (Cornell University)|Mar 2, 2010

Fractional Differential Equations Solutions4 references3 citations

TL;DR

This paper formulates Noether's theorem for fractional classical fields by extending variational principles to fractional-order field systems, deriving fractional Euler-Lagrange equations and corresponding conservation laws. The key contribution is the derivation of conserved currents for fractional Dirac fields, generalizing Noether’s theorem to non-integer order field theories.

ABSTRACT

This paper presents a formulation of Noether's theorem for fractional classical fields. We extend the variational formulations for fractional discrete systems to fractional field systems. By applying the variational principle to a fractional action $S$, we obtain the fractional Euler-Lagrange equations of motion. Considerations of the Noether's variational problem for discrete systems whose action is invariant under gauge transformations will be extended to fractional variational problems for classical fields. The conservation laws associated with fractional classical fields are derived. As an example we present the conservation laws for the fractional Dirac fields.

Motivation & Objective

• To extend Noether's theorem from integer-order to fractional-order classical field theories.
• To formulate a variational principle for fractional field actions and derive the corresponding fractional Euler-Lagrange equations.
• To identify symmetries in fractional field systems and derive associated conservation laws.
• To apply the framework to the specific case of fractional Dirac fields and derive their conserved currents.

Proposed method

• Adapt the variational principle to fractional-order actions using Riemann-Liouville or Caputo-type fractional derivatives.
• Derive the fractional Euler-Lagrange equations of motion from the stationarity condition of the fractional action functional.
• Extend the Noether variational problem from discrete fractional systems to continuous fractional field systems.
• Identify continuous symmetries in the fractional action and derive the corresponding conserved currents via the Noether current construction.
• Apply the formalism to the fractional Dirac field Lagrangian to compute explicit conservation laws.
• Use gauge invariance and symmetry transformations to determine conserved quantities in fractional field theories.

Experimental results

Research questions

• RQ1 How can Noether's theorem be generalized to fractional-order classical field theories?
• RQ2 What are the fractional Euler-Lagrange equations derived from a fractional action principle in field theory?
• RQ3 What conservation laws emerge from continuous symmetries in fractional field systems?
• RQ4 How do the conserved currents for fractional Dirac fields differ from their integer-order counterparts?
• RQ5 What role do fractional derivatives play in preserving symmetry and conservation in field theories?

Key findings

• The paper derives the fractional Euler-Lagrange equations for classical fields using a variational principle applied to a fractional action functional.
• It establishes a general framework for deriving conservation laws in fractional field theories from continuous symmetries.
• The conserved currents for fractional Dirac fields are explicitly computed, demonstrating the applicability of the formalism.
• The method preserves the structure of Noether's theorem in the fractional setting, linking symmetries to conservation laws.
• The results show that fractional field theories admit conserved quantities under gauge transformations, analogous to integer-order theories.
• The framework extends the scope of Noether’s theorem to non-integer order field systems, enabling the study of fractional symmetries and conservation.