# On the correspondence rules between the physical world and mathematical frameworks (quantum mechanics)

At the start of Chapter 2 of his Quantum Mechanics: A modern development, Ballentine gives the framework which sets the stage for the questions I have:

Every physical theory involves some basic physical concepts, a mathemati- cal formalism, and set of correspondence rules which map the physical concepts onto the mathematical objects that represent them. The correspondence rules are first used to express a physical problem in mathematical terms. Once the mathematical version of the problem is formulated, it may be solved by purely mathematical techniques that need not have any physical interpretation. The formal solution is then translated back into the physical world by means of the correspondence rules.

My question is about the correspondence between self-adjoint operators and dynamical variables, and between state (density) operators and physical states.

On page 45, Ballentine says

An observable is a dynamical variable that can, in principle, be measured.

where, at some earlier time, he has also given the following postulate of quantum mechanics:

Postulate 1. To each dynamical variable (physical concept) there corresponds a linear operator (mathematical object), and the possible values of the dynamical variable are the eigenvalues of the operator.

I give this latter postulate just to demonstrate Ballentine's definition/usage of the word "dynamical variable", which is of course not in general the same as its traditional usage as a function on phase space. At any rate, the point is that there is some word for the physical quantities of interest in a given system. My questions here are twofold:

(1) Do we customarily assume (or can one prove?) that every self-adjoint can be measured in the physical world, in principle? I'm not even sure what such a proof would look like, but I do think it's conceivable that there is not a bijective correspondence between self-adjoint operators on Hilbert space and dynamical variables. Indeed, I think there mustn't be, seeing as density operators for example are self-adjoint and yet don't correspond to any dynamical variable?

(2) The more important question is one in the physical world entirely. Ballentine seems to make a distinction between observables and dynamical variables. But is there any reason to suspect that there are physical quantities which cannot, in principle, be measured?

State (density) operators and physical states

Ballentine shows quite clearly in Chapter 2 how each state can be represented by a unit trace, positive, self-adjoint operator. My question is therefore:

(3) Do we assume that, in principle, every unit trace, positive, self-adjoint operator corresponds to some physical state? Is there a proof that a preparation procedure can exist for each such state?

Perhaps the answers to these questions are in a quantum information text somewhere, but I am unfortunately not at present an owner of such a text.

• Tell me if these responses are trivial: for (2), one cannot in principle simultaneously measure the value of position and momentum of a particle with arbitrary precision; for (3) , assuming one can perform any operation then one can prepare any state, anything can be prepared, per dx.doi.org/10.1088/1751-8113/40/21/015 (this can also be asked in classical mechanics: if I can write down a state of $N$ particles in exactly $N$ places with $N$ momenta, does this mean one can prepare the state?) Jan 23 at 16:47
• @QuantumMechanic Your response for (3) is immensely helpful, thank you. For (2), that is unfortunately not what I'm getting at. Quite apart from incompatible observables, I am wondering if any measurement is (assumed?) possible to be done in isolation.
– EE18
Jan 23 at 16:50
• Related: This, this and this and the links therein. Jan 23 at 20:10
• Thank you @TobiasFünke -- I think my main question would be whether density operators are observables, as it allows for this answer (math.stackexchange.com/questions/4624350/…) to be correct.
– EE18
Jan 23 at 20:19
• I think your linked question is a duplicate of this. And whether or not density operators are observables is discussed in the linked questions from my previous comments... Jan 23 at 20:31

The significance of these rules is in what they prohibit, not what they allow. In a classical world, you can't read the contents of the books of the library of Alexandria in the state of the world today, but at the most fundamental level, before any particular dynamical laws have been defined, it isn't prohibited. Whether it's allowed in principle depends on what your principles are. It's disallowed if your principles include the principles of thermodynamics. Quantum mechanics just adds some additional limitations to the limitations that already exist classically.

(1) Do we customarily assume (or can one prove?) that every self-adjoint can be measured in the physical world, in principle?

Quantum mechanics definitely asserts that any quantity not representable by a self-adjoint operator can't be measured. That's a significant limitation because it implies that there are individually measurable quantities that can't be measured jointly, which never happens classically. To prohibit any more than that, at this level, would make the theory something other than QM.

[...] density operators for example are self-adjoint and yet don't correspond to any dynamical variable

Density matrices always have an interpretation as in-principle-valid-at-the-lowest-level measurements. E.g., if a flip of a biased coin results in a mixed state of diag(2/3, 1/3) in the (head,tail) basis, the corresponding measurement returns 2/3 if the coin came up heads and 1/3 if tails. I don't think that interpretation has any deep significance, but it exists.

(2) [...] Ballentine seems to make a distinction between observables and dynamical variables. But is there any reason to suspect that there are physical quantities which cannot, in principle, be measured?

I don't understand what distinction Ballentine is making between observables and dynamical variables, if any. He seems to imply in one of your quotes that some dynamical variables aren't observables, and in the other that all dynamical variables are observables.

(3) Do we assume that, in principle, every unit trace, positive, self-adjoint operator corresponds to some physical state? Is there a proof that a preparation procedure can exist for each such state?

The answer to this is essentially the same: any state not so representable can't be prepared, and which states so representable can in principle be prepared depends on how many principles you have.