Noether's Theorem: the Origin of the Conservation Laws and the Meaning of Spin


The conservation of energy, the conservation of momentum and angular momentum, and the conservation of electric charge are not assumptions but can be proved. They follow from the symmetries of a dynamical system expressed in Lagrangian terms. That any continuous symmetry results in a corresponding conservation law was first enunciated clearly, and proved, by Emmy Noether in 1915. Noether's Theorem is probably the most important and profound theorem in the whole of mathematical physics.

For example, the absence of an explicit time dependence in the Lagrangian implies that the dynamical behaviour of the system will be the same tomorrow as it is today and was yesterday. This means that the Lagrangian is invariant under the action of translation in time. Hence there must be a conserved quantity. It turns out that this conserved quantity is energy.

Similarly, we expect invariance of our theories under spatial translations. The physical laws controlling the behaviour of a system are expected to be the same in the UK as in Australia - or as on Mars, or our space programmes will be in big trouble. What is the corresponding conserved quantity? Since there are three spatial dimensions to generate symmetries, there are three conserved quantities the three components of momentum.

The third example is invariance under spatial rotations. The isotropy of space implies there should be a corresponding conserved quantity. There is: it is angular momentum.

All the above examples are for symmetries of the Poincare group. So what about invariance under boosts - the symmetry due to the equivalence of inertial observers? What quantity is conserved as a result? There is such a conserved quantity. There must be by Noether's Theorem. However the status of this quantity is rather different which is why it has no familiar name. Many texts avoid discussion of this issue. Those that do discuss it tend to leave the reader still rather perplexed as to why its interpretation differs from the other Poincare generated conservation laws. This is discussed in the pdf below.

The Poincare group does not provide the only possible symmetries. There may also be "internal" symmetries, independent of spacetime. For example, a Lagrangian for a complex field may depend only upon absolute magnitudes and hence be invariant if the field is multiplied by a phase factor. This leads to a conserved quantity which can be interpreted as electric charge. In a similar manner the various internal symmetry groups of particle physics [SU(2), SU(3)] lead to a multitude of conserved quantum numbers.

In the pdf below, the quantities conserved as a result of the Poincare symmetries, as they apply to fields of arbitrary spin, are derived in detail. It also indicates how the conserved quantities corresponding to internal symmetries can be derived. The derivation can be interpreted as either classical or quantum, it makes no difference.

One of the important by-products of this analysis is clarification of the origin of spin. The starting point is that the field at a point generally has several components (e.g., it might be a vector field). We already know that these components relate to the transformation properties of the field under rotations and boosts (Lorentz transforms). But why do these different field components relate to spin in the sense of a localised angular momentum? This identification of the field components with spin in this sense follows from the Noether analysis.

Moreover it is worth noting that this interpretation of spin can be made purely classically. Spin itself is not intrinsically quantum mechanical as is sometimes thought. It is only the confining of spin measurements to discrete values which is quantum mechanical.


Read the pdf Noether's Theorem: the Origin of the Conservation Laws and the Meaning of Spin

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At last something interesting comes out of strings? An "overzealous" powering up of a tiny motor caused a cotton string to twist in an unusual pattern, creating the above image. The string whirled some 10 to 20 times during the two seconds of exposure used for the image. [Image courtesy of Andrew Davidhazy, Rochester Institute of Technology/Science]