WHY NOETHER'S THEOREM
Noether’s Theorem claims to find a justification for conservation laws in symmetries. It has been said that the conservation laws follow from the symmetry properties of nature. It has also been argued that symmetries cause conservation laws. I will show that none of these interpretations stands up to any rigorous analysis.
IS A TAUTOLOGY
by Miles Mathis
Noether’s Theorem historically evolved out of the action principle, through the Lagrangian and Hamiltonian mathematics. I don’t think anyone will dispute this. What is not so clear to most people is how Lagrangian and Hamiltonian mechanics is related, at the foundational level, to Newtonian mechanics. The conservation laws are all laws derived from Newtonian mechanics. The symmetries are all laws that are derived from the action principle and from Lagrange and Hamilton. In this way they are often parallel explanations. Hamiltonian mathematics and mechanics has been preferred for the past century or more since it simplifies many calculations. Theoretically its justifications are slimmer. Conceptually, the action principle is more abstract than the concepts of Newton like force and energy and momentum. In the simplest situation (no potential or acceleration) action is just KΔt, where K is the kinetic energy. This is not terribly abstract. But still, it is a compound variable of more complexity than K or F. And the Lagrangian quickly becomes a dense variable. It is the integral of a differential of two energies—kinetic and potential. In this way, action is a concept that is quite abstract. And, although the Lagrangian is a useful mathematical distillation, it is easy to argue that we have paid for that with clarity (it is not easy to convince, but it is easy to argue).
What I will show is that the symmetries restate the conservation laws in a new mathematical language. This is the language of action rather than force. Noether’s Theorem then rediscovers this basic equivalence, proving it with more complex mathematics. The problem is that this equivalence is not proof that the symmetries cause the conservation laws. It is only proof that they are necessarily linked. The symmetries and the conservation laws are the same concepts in different mathematical language. Noether’s Theorem provides the bridge from one language to another. But it does not prove what physicists now claim it proves. The conservation laws are not explained by the symmetries anymore than the symmetries are explained by the conservation laws. In the end, Noether’s Theorem is a tautology. It proves nothing more than that A = A.
I am not so sure that Noether herself would have disagreed. She wanted to mathematically prove a connection between the conservation laws and the symmetries, which she did. She showed that where you have a symmetry, you also have a conservation law. They are necessarily linked. However, the 20th century interpretation of this to mean that the symmetries are the theoretical basis for the conservation laws is not justified by her math. Contemporary physics has overstated the importance of Noether’s Theorem, calling it the key to a new unification theory, the most important foundational theorem ever, etc. It is an elegant theorem, but it might have been proved with much simpler math. Noether might have remembered that Lagrangian mechanics grew directly out of Newtonian mechanics. It is therefore no surprise to find that their basic laws are equivalent at the foundational level. Newton, I suspect, would have proved the equivalence of the symmetries and the conservation laws by returning to the genesis of the action concept out of his own concepts. He would have pulled apart the Lagrangian into its building blocks, showing the direct relationship between the two maths at the lowest level. Noether preferred to use what might be called the historical end-math of the evolution from Newton to Hamilton. This gave her a very impressive proof indeed. But dense and unwieldy. Prone to be judged on its bulk and its novelty.
As just a sample of how Newton might cut through to the heart of the matter, consider the link found by Noether’s Theorem between time symmetry and the conservation of energy. All we need to do is look at the simple equation I mentioned above.
S = KΔt
Where S is the action in a zero potential field. In this stripped down equation it is quite clear that time symmetry implies a conservation of energy. If Δt is a constant then K must be “conserved” since it is the only variable you have left. If the integral of this action S is required to be stationary—since that is how objects “must” move, according to the theory—then of course the kinetic energy is going to be conserved. This is equally true if we have a potential energy as well, since the energies are summed before being integrated. They are integrated with respect to time. The connection between energy and time cannot be a mystery, since together they define the action, as above.
The connection between spatial symmetry and momentum is equally non-mysterious. Let us rewrite the above equation to express momentum.
K = pv [I ignore the ˝]
Where p is momentum
S = pvΔt
vΔt = Δx
S = pΔx
By the same argument I just used, the variables are necessarily linked. If space is symmetrical and the integral of S is defined as being stationary, then momentum must be conserved. It is axiomatic. The very way that the action was defined gives us these relations. The relationships between symmetries and conservation laws don’t need to be proved by Noether’s Theorem since they are definitional. They are tautologies. Noether proved the link between symmetries and conservation laws in a paper filled with higher math and difficult equations and dense concepts. I have deduced the fundamental equivalence of the symmetries and conservation laws in half a page with six algebraic variables. You may extrapolate from my method and do the angular momentum/rotation equivalence if it appeals to you. I think I have already proved my point, and since I have just made a claim to efficiency, I will stop here.
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