Comments on an essay by Wigner

[PSA: Happy 4th of July. Juno arrives at Jupiter tonight!]

This is short and worth reading:

This essay has no formal abstract; the above is the second paragraph, which I find to be profound. Here is the PDF. The essay shares the same name and much of the material with Wigner’s 1963 Nobel lecture [PDF].The Nobel lecture has a nice bit contrasting invariance principles with covariance principles, and dynamical invariance principles with geometrical invariance principles.^a

Some comments:

• It is very satisfying to see Wigner — the titan — highlight the deep importance of the seminal work by the grandfather of my field, Dieter Zeh. Likewise for his comments on Bell:

  As to the J.S. Bell inequalities, I consider them truly important, inasmuch as they prove that in the case considered by him, one cannot define a non-negative probability function which describes the state of his system in the classical sense, i.e., gives nonnegative probabilities for all possible events….

  This is a very interesting and very important observation and it is truly surprising that it has not been made before. Perhaps some of those truly interested in the epistemology of quantum mechanics took it for granted but they did not demonstrate it.

• I like the hierarchy of regularity that Wigner draws: data ➢ laws ➢ symmetries. Symmetries are strong restrictions on, but do not determine, laws in the same way that laws are strong restrictions on, but do not determine, data.
• It is interesting that Wigner tried to embed relativistic restrictions into the description of initial states:

  Let me mention, finally, one effect which the theory of relativity should have introduced into the description of the initial conditions and perhaps also into the description of all states. The state vectors, as we use them, describe the properties of the states which they assume at a definite time. But these are unobservable – we can not get signals instantaneously from a distance. It would be, therefore, more reasonable for the state vector, or the wave function, to describe the state on the negative light cone from the points of which signals may reach the observer. I have proposed this and tried to do it also but with very little success, at least so far.

• I found something things to disagree with. Wigner comments on the subtleties of specifying the classical state of a gauge field like E&M, which must either be overcomplete, undercomplete, or violate a symmetry of the theory. But then he suggests this is better in quantum mechanics

  The states of quantum mechanics are described by complex so called state vectors in the infinite dimensional Hilbert space which does not give a greater complexity than the electromagnetic field specifications of Maxwell’s theory….In this regard, the description of the states, hence also of the initial conditions, of quantum mechanical theory comes closer to the ideal described originally than that of Maxwell’s theory.

  But the quantum state is strictly more complicated than a corresponding classical state. All of the classical gauge issues persist, right?

• Finally, Wigner surprisingly downplays Noether’s contribution

  But the invariance principles do have other, less obvious applications. The oldest one of these is the establishment of conservation laws. This is usually credited to E. Noether but was proposed even earlier by Hamel.

  I don’t know anything about this history except the conventional wisdom, in which Noether is canonized. A critical discussion of Wigner’s views about who contributed most to the symmetry-conservation connection (Noether’s theorem) can be found here.