Fundamentals of Time and Relativity

Noether's Theorem

  • Emmy Noether's first theorem states that a symmetry of the action necessarily leads to a corresponding conservation law.
  • The general rule is that a symmetry in physics indicates the existence of a conserved quantity.

The action (integral) of a physical system is a local functional of a dynamical variable x from which the system's behavior can be determined by the principle of least action. A symmetry of the system is then a transformation of the dynamical variable x→x+δx that leaves the action unchanged:

10.11

δS:=S[x+δx]−S[x]=0

For example, because of the homogeneity of space and time, the action should be invariant under spatial and temporal translations x→x+δa , where δa=δ a μ e μ is an infinitesimal constant spacetime vector. Emmy Noether's first theorem states that such a symmetry of the action necessarily leads to a corresponding conservation law, in this case, conservation of energy and momentum.

Essentially, the proof of this statement follows already from the variational equation (10.5) for a free particle. The equation of motion in the second term at the right-hand side is obviously satisfied by the actual path x ¯ (τ) . So the translational variation of the action integral around this path takes the form

10.12

δ S free [ x ¯ (τ)]=−m d x ¯ (τ) dτ ⋅ δa| 1 2 =− p ¯ (τ)⋅ δa| 1 2

where p ¯ (τ)= p ¯ μ (τ) e μ is the actual four-momentum of the system. Due to the translational invariance of the laws of physics, the variation (10.12) must vanish. Hence, the four-momentum p ¯ μ (τ) along the worldline of the particle is conserved:

10.13

p ¯ μ ( τ 1 )− p ¯ μ ( τ 2 )=0
  • Since (10.13) is a relationship between four-vectors, the conservation law of four-momentum is valid in any inertial frame.
  • The general connection between symmetry and conservation law was established in 1915 by Emmy Noether in her famous (first) theorem published in 1918.