# Joint Spectral Measure theorem

I want to gain an intuition to understand the joint spectral measure theorem. In the case that operators involved in this theorem have purely discrete spectrum, the theorem should be reduced to the fact that if operators pairwise commute then they share eigenspaces and therefore they can be simultaneously measured (being their spectrum their eigenvalues). However if the spectrum is continuous, then it has to be their spectral measures that commute in order that they can be simultaneously measured. This operators have a common spectral measure. But, does the reason they can be simultaneously measured is due to the fact that they share subspaces (some sort of "eigenspaces")? Also the joint spectrum in this case should be subsets of the real line of very small length. Each of these subsets should have a one to one correspondence with each possible subspace where the wave function can collapse (the shared subspaces). (some sort of "eigenvalues"). Please let me know if what I am thinking is correct.

## 1 Answer

Please let me know if what I am thinking is correct.

Yes, the thinking is essentially correct. To help validate and maybe refine what's written in the question, I'll offer some thoughts about how the spectral measure theorem relates to the following facts:

• Fact 1: Real measurements have finite resolution.

• Fact 2: Real measurements apply only to one part of a larger system.

To be specific, consider two mutually commuting observables, say $$A$$ and $$B$$, both with continuous spectra given by the whole real line. To be more explicit, consider a Hilbert space of $$N$$-variable functions $$\psi(x,y,z,...)$$, and define $$A$$ and $$B$$ by $$A\,\psi(x,y,z,...)=x\,\psi(x,y,z,...) \hskip2cm B\,\psi(x,y,z,...)=y\,\psi(x,y,z,...). \tag{1}$$ Real measurements have finite resolution, so we shouldn't think of the outcome of an $$A$$-measurement as yielding an eigenvector/eigenvalue of $$A$$, which don't exist mathematically anyway. A better way to think about it (but still artificial; see the footnote) is to think of a measurement of $$A$$ as a collection of a finite number of mutually compatible yes/no questions. Each yes/no question asks "Is the value of $$x$$ greater that $$x_0$$ or less than $$x_0$$?" for some given value of $$x_0$$. This question corresponds to a dichotomic observable, represented by a pair of mutually complementary projection operators $$\{P,\,1-P\}$$ from the spectral decomposition of $$A$$. The projection operators $$P$$ and $$1-P$$ both have infinite-dimensional eigenspaces. After measuring the dichotomic observable $$\{P,\,1-P\}$$, the original state-vector $$|\psi\rangle$$ should be replaced by either $$P|\psi\rangle$$ or $$(1-P)|\psi\rangle$$ for the purpose of making predictions about future measurements. Since all of the projection operators in the spectral decomposition of $$A$$ commute with each other, we can model a measurement of $$A$$ as a large (but finite) number of these dichotomic measurements whose sequence doesn't matter. And since all of the projection operators in the spectral decomposition of $$B$$ commute with those of $$A$$, we can handle a joint finite-resolution measurement of $$A$$ and $$B$$ in the same way we handle a finite-resolution measurement of $$A$$ itself — namely as a large but finite collection of mutually compatible dichotomic measurements. In this way, the treatment of observables with continuous spectra is effectively reduced to the case of observables with discrete spectra.

That addresses Fact 1, and Fact 2 can be addressed similarly. According to equation (1), every non-zero projection operator in the spectral decomposition of $$A$$ (or $$B$$) has an infinite-dimensional eigenspace. None of them have finite-dimensional eigenspaces, much less one-dimensional eigenspaces. Physically, this corresponds to the fact that the observable $$A$$ (or $$B$$) is associated with just one part of a larger system (represented in this contrived example by the long list of variables $$x,y,z,...$$), and a measurement of $$A$$ cannot completely determine the state of the rest of the system, no matter how fine the resolution of the $$A$$-measurement might be. (A more careful version of this statement would lead in interesting directions involving entanglement and things like the Reeh-Schlieder theorem, but I'll resist the temptation to be that careful.)

Altogether, I think this agrees with the understanding that was spelled out in the question. The purpose of this answer was simply to validate what was written in the question.

Footnote

As mentioned above, the collection-of-dichotomic-observables view is still artificial, because real measurements don't have perfectly sharp boundaries between the different possible outcomes. For example, although a real measurement of a particle's position has finite resolution, that finite resolution isn't really realized by a subdivision of space into finite-size cells with perfectly sharp boundaries. An even better way to think about measurement is to use a model that includes the measurement equipment and other environmental entities as part of the overall quantum system, so that the physical act of measurement is encompassed by the Schrödinger equation for the whole system. This does not solve the infamous measurement problem, but it does allow us to defer application of the projection $$|\psi\rangle\rightarrow P|\psi\rangle$$ until after the real, messy physical process of measurement is practically complete. The conceptual advantage of doing this is that now the model itself "knows" that the boundaries between different possible measurement outcomes are fuzzy, just as we know they must be in the real world. We may still ultimately apply a projection $$|\psi\rangle\rightarrow P|\psi\rangle$$, which still imposes artificially sharp boundaries, but this way at least the artificial-ness of it can be quantified using the model itself. In practice, nobody actually does things this way, because the math is prohibitively difficult: solving the Schrödinger equation for all of the molecules in a whole laboratory is, well, pretty hard. The point here is simply that it could be done in principle. This can help put things into a slightly more satisfying perspective, even though it does not solve the infamous measurement problem.