# Really, what is the minimum number of postulates of quantum mechanics?

I have a question that has been annoying me for a while. Going across many textbooks on quantum mechanics, looking at the postulates they list, we find that the number of postulates vary from one text to another. That means either some the postulates listed in some books are either not sufficient or are redundant.

That makes me wonder, what is really the minimum sufficient number of postulates of QM and what are they? isn't there supposed to be a consensus on that?

• Let's flip the problem around for a minute. Can you enumerate the basic postulates from which classical mechanics spring? For that many how many make up E&M? Are Maxwell's equations and the Lorentz force law sufficient? Too many? Why or why not? The trick with establishing consensus is of course one of categorization. When should two statements in natural language be counted as one postulate and when as two. And you can't avoid the natural language problem because equations need interpretation. Aug 24, 2012 at 2:29
• People of a highly mathematical bent will want to count the boundary conditions, and what of those who feel that Lagrangian or Hamiltonian mechanics are in some sense better for approaching axiomatic mechanics. Aug 24, 2012 at 2:42
• Do you include Galileo transform/space metrics into postulates of classical mechanics? Aug 24, 2012 at 5:25
• Physics is not only mathematics but "mathematics + its interpretations to physical systems"; and i think dmckee is right in that interpretation part of a physical theory makes it difficult to analyse this question of independence of axioms on a purely logical basis. For example in postulates of QM one makes use of a term "measurement" (e.g. in "after a measurement state of system will collapse to an eigenstate of observable being measured") which (in usual QM) has a meaning only after we interpret it to a physical system. Aug 24, 2012 at 8:31
• Euclid used five very elegant postulates to express planar geometry. But I could also define the plane by saying "$\mathbb{R}^2$ with the metric $d(p,q) = \sqrt{(p_1-q_1)^2+(p_2-q_2)^2}$". It is worthwhile to try and find the minimal assumptions behind things, but the actual number of postulates mostly depends on the language used to express them, so it's not a very helpful measure of how minimal the assumptions really are. Aug 24, 2012 at 12:19

People sometimes care about authorities, unlike Richard Feynman, so let me borrow an authority, Richard Feynman, who said the following about this very question (minimum set of postulates behind quantum mechanics) in his Feynman Lectures on Physics, page 5-11 in this edition:

Equation (5.25), i.e. $\langle j|i\rangle =\delta_{ji}$, is not independent of the other laws we have mentioned. It happens that we are not particularly interested in the mathematical problem of finding the minimum set of independent axioms that will give all the laws as consequences. (Footnote: Redundant truth doesn't bother us!) We are satisfied if we have a set that is complete and not apparently inconsistent. We can, however, show that equations (5.24) i.e. $\langle \chi| \phi\rangle = \sum_{{\rm all}\,i} \langle \chi|i\rangle \langle i|\phi\rangle$ and (5.25) are not independent. Suppose...

Feynman only says that he's not terribly interested in the problem and he shows that he may find one relevant fact for such a problem. But if he really tried to solve it fully, he would of course find out that it depends on the choice which parts of the framework are considered intrinsic parts of the theory and which are just examples or "external mathematical tricks" that allow us to explain it or solve it; which objects are fundamentally the same and can be unified; whether we should include some "philosophical" issues that many people misinterpret just because they misinterpret them and even though the other axioms give us no direct reason to make mistakes or misinterpretations, and so on.

It's a messy problem. The counting of rules and principles always depends on "social conventions". One can surely reformulate the ten commandments in an equivalent way that has 9 or 12 commandments and some other religious traditions actually do exactly that. The same holds for postulates of the Euclidean geometry.

In quantum mechanics, one may list e.g. these 6 principles of the Copenhagen interpretation. But one may still find people who will claim they're incomplete and people who will say things that contradict these basic pillars of modern physics. So extra explanations and more specific comments are needed for students who are slower, and so on. Also note that the list of 6 principles of the Copenhagen interpretation doesn't contain Feynman's "completeness identity" and the "orthogonality of a basis" equations at all, probably because the "Copenhagen interpretation explanation" would treat them as technical details or auxiliary mathematical tools to solve things that one should have learned elsewhere.

It's an extremely subtle task to "minimize the list of axioms", especially in topics such as the foundations of quantum mechanics where some people still fail to agree what is actually right and what is not.

• In this beautiful lecture by Nima, streamer.perimeterinstitute.ca/Flash/… at the min 3:30 he is saying that states evolve in time according to Schrodinger equation is not a postulate because it follows from the basic definitions. Could you please explain how that dynamical equation follows from the basic definitions. Then he moves on talking about that there is only one postulate..etc. Beautiful lecture.
– Revo
Aug 24, 2012 at 13:30
• He probably meant that the Schr. equation follows from the definition of the Hamiltonian which is the generator of translations in time (via Noether-like links) much like the angular momentum is by definition the generator of rotations etc. Aug 25, 2012 at 8:06

The point of a list is to give a simple memorizable starting point in terms of which one can organizing the innumerable additional details that come later. Clearly, this can be done in different ways, and authors take their liberty to do it the way they think is most useful for their readers.

As there is no consensus about the foundations, you can't expect a single answer. (This doesn't even exist for much more well-defined subjects such as group theory, where you can formulate the basic assumptions in a number of different but equivalent ways.)

Your answer also depends on how much is packed into a single postualte, and how much of QM foundations the postulates should cover.

The six postulates in the section ''Postulates for the formal core of quantum mechanics'' of Chapter A1: Fundamental concepts in quantum mechanics of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html are more numerous and more detailed than those in most lists I have seen, but they also contain much more information that is usually left implicit but without which one can do hardly any useful QM.

There is an article by Vojciech H Zurek in Physics Today page 44 October 2014 which provides an answer. He enumerates postulates and shows that some of the standard postulates (say as stated by Cohen-Tanoudji for example) are redundant. He derives these postulates from 3 postulates by separating the parts of Hilbert space which describe the quantum system, quantum apparatus and their quantum environment (postulate 0). The view point Zurek advocates is gaining acceptance, but not by all.

Zurek is a well-known Physicist, but more importantly Zurek's article seems to be compelling (apart from a few glossed over points).

• How many postulates does Zurek require? Could you provide a link to the Physics Today article and maybe include the postulates he claims? Dec 5, 2014 at 17:16