[Paper Review] A Note on the scale symmetry and Noether current

Naohisa Ogawa|ArXiv.org|Jul 13, 1998

Theoretical and Computational Physics1 references3 citations

TL;DR

This paper investigates scale symmetry in classical mechanics and field theory where the action is not invariant under scale transformations, yet the equations of motion remain invariant. It proves that the action can be expressed as the difference of formal (non-conserved) Noether charges at time boundaries, and in field theory, as a boundary integral of the formal Noether current, offering a new formulation of scale symmetry beyond standard Noether's theorem.

ABSTRACT

Usually we consider the symmetry of action as the symmetry of the theory, however, in the Keplar problem the scaling symmetry existing in equa tion of motion is not the ones for action. It changes the multiplicative c onstant of action and the time boundary. In such a case that the scale tran sformation does not leave the action invariant but keeping the equation inva riant, the following statement is proved. The time integration of Lagrangian is explicitly performed and the action ca n be expressed by the difference of formal (non-conserved) Noether charges a t time boundaries. In field theory the action can be expressed by the bound ary integration of the formal Noether current.

Motivation & Objective

• To analyze scale symmetry in systems where the action is not invariant under scale transformations, yet the equations of motion remain symmetric.
• To address the breakdown of standard Noether's theorem in scale-symmetric systems with non-invariant actions.
• To reformulate the action in terms of boundary values of formal Noether charges, even when the current is not conserved.
• To extend the formalism to field theory, expressing the action as a boundary integral of the Noether current.
• To clarify the role of time boundary conditions and multiplicative constants in scale symmetry when the action is not invariant.

Proposed method

• Derives the time integral of the Lagrangian to express the action as the difference of formal Noether charges at initial and final times.
• Introduces the concept of 'formal' Noether charges that do not satisfy conservation laws due to scale non-invariance of the action.
• Applies the formalism to the Kepler problem as a key example where scale symmetry exists in equations of motion but not in the action.
• Extends the construction to field theory by expressing the action as a boundary integral of the Noether current.
• Uses variational calculus to show that the action remains invariant under scale transformations despite the non-invariance of the action functional.
• Analyzes the transformation behavior of the action under scale changes, including multiplicative constants and time boundary shifts.

Experimental results

Research questions

• RQ1 How can scale symmetry be consistently defined in systems where the action is not invariant under scale transformations?
• RQ2 What is the role of boundary terms in the action when the Noether current is not conserved?
• RQ3 Can the action be reconstructed from formal Noether charges even when the symmetry does not leave the action invariant?
• RQ4 How does the formalism extend from mechanics to field theory in the context of scale symmetry?
• RQ5 What is the physical significance of the time boundary dependence in scale-symmetric systems with non-invariant actions?

Key findings

• The action in scale-symmetric systems with non-invariant actions can be expressed as the difference of formal Noether charges evaluated at time boundaries.
• Even when the Noether current is not conserved, the time integral of the Lagrangian yields a well-defined boundary expression.
• The formalism successfully accounts for scale symmetry in the Kepler problem, where the action is not invariant but the equations of motion are.
• In field theory, the action is shown to be expressible as a boundary integral of the formal Noether current, generalizing the mechanical result.
• The transformation of the action under scale changes includes a multiplicative constant and a shift in time boundaries, which are essential for symmetry preservation.
• The results demonstrate that Noether's theorem can be extended beyond conserved currents by considering formal charges in non-invariant settings.