Cardinality of bases doesn't matter for Hilbert spaces

Set theory, cardinals, ordinals, unmeasurable sets, and other pathological mathematical structures have no legitimate power in physics

Laymen (e.g. postmodern philosophers) interested in spirituality and physics (...) often talk about things like the "influence of Gödel's theorems about incompleteness on physics" and similar things. They usually want to believe that this theorem must imply that mathematics and science must be limited, leaving the bulk of the human knowledge to witches, alternative doctors, ESP experts, dragons, priests, and global warming alarmists, among related groups of unscientific charlatans.



A cardinal, Czech Catholic Boss Dominik Duka, is in the middle. He's now a fan in Sochi. Who believes in Christianity, may be helped; who doesn't, isn't hurt. ;-) At least that's what Miloš Zeman, the current Czech president (man on the right in the picture; just having virosis which has been misinterpreted as his being drunk) and the self-described clumsy mascot of the Czech athletes (now also in Sochi), said.

With their restricted resolution, "Gödel's theorem on imncompleteness" seems to be the same thing as the "Heisenberg uncertainty principle". However, the truth is very different. The mathematical insight by Gödel has no relationship to the Heisenberg uncertainty principle and none of the two imply that the laws of Nature cannot be pinpointed precisely, anyway.

When it comes to the irrelevance of Gödels theorems for physics, the truth is actually much more far-reaching. None of the major developments in the post-Cantor efforts to axiomatize mathematics and set theory has any implication for physics. This is partly related to physics' being fundamentally continuous. I want to dedicate this blog entry to this irrelevance.




An hour ago, someone asked the Physics Stack Exchange question about the seemingly inconsistent cardinalities of bases in quantum mechanics for the 17th time which would convince me to write this blog post.




German mathematician Georg Cantor was one of the people who dreamed about a solid, axiomatic framework for all of mathematics. He would become the founder of "set theory" which seemed to be the right thing. It would allow you to construct (=it would postulate the existence of) sets and sets of sets and so on, for any property that the elements satisfy, and through these constructs, you may encode the inner patterns of any mathematical object or structure.

Notoriously enough, philosopher Bertrand Russell realized that Cantor's axioms imply the existence of the following set \(R\):\[

{\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R

\] We may ask whether \(R\in R\). Well, if \(R\in R\), then it doesn't obey the defining condition for elements of \(R\) – the condition is \(x\not\in x\) and we are substituting \(x\to R\) – so it follows that \(R\) isn't included into \(R\) i.e. \(R\not\in R\). And vice versa, if \(R\not\in R\), then it does obey the defining condition \(x\not\in x\), so we must include \(R\) into \(R\) and we have \(R\in R\). To summarize, \(R\in R\) is true exactly if it is false and vice versa. This incarnation of the liar's paradox proves that Cantor's axiomatic system was internally inconsistent.

Mathematicians would have to find a better, internally consistent axiomatic system. They would deal with this problem in two major ways. The Zermelo-Fraenkel axioms of set theory would restrict which sets you may construct (sets whose existence is guaranteed by the axioms). You may only build them from the bottom up (by constructing finite or countable unions, sets of all subsets of existing sets, and taking subsets of existing subsets picking elements with a property), so abstract "sets of all sets" are not allowed. It's enough to construct all mathematical structures we need and Russell's paradox is avoided because at most, you may define \(R\) as all elements from some pre-existing large set of objects (which doesn't include this \(R\) itself) and it's therefore clear that \(R\not \in R\) for this "restricted" \(R\). This non-membership doesn't imply that \(R\) has to be included into itself because the candidates for the \(R\) membership only come from a previously constructed set.

Alternatively, Gödel and Bernays would pioneer another set theory that allows you to construct \(R\) with all sets obeying \(R\not\in R\) but it wouldn't guarantee that this object is a proper "set". Instead, it is a more informal "class", one that isn't automatically allowed to be incorporated into other "classes" as an element. "Sets" are by definition those classes that belong to other sets. Russell's paradox is avoided in this approach, too. In fact, Russell's argument above is reinterpreted as a proof of the innocent statement that the class \(R\) isn't a set – so it isn't among the candidates for \(R\) membership. Analyses of these subtleties would also lead to Gödel's insights about incompleteness and many other things. They're sort of cute but physics doesn't care about any of this high-grade recreational logic.

Uncountable sets

Georg Cantor used to be keen on one-to-one maps between the sets. If there exists a one-to-one map between elements of \(M\) and \(N\), then these two sets are "equally large". More technically, they have the same "cardinality". The adjective "cardinal" is something like "fundamental" or "important" but I do think that the terminology really followed the cardinals in the Catholic Church and their relative ranks. Relatively to a mortal believer or infidel, such cardinals may be infinitely powerful but one cardinal may still be more powerful than another cardinal. For finite sets, the cardinality is simply the number of elements, so it may be given by non-negative integers, \(0,1,2,\dots\).

You might think that the cardinality of all infinite sets is \(\infty\) because they're equally large. But Georg Cantor was the first one who found an argument that this ain't so. For example, assume that the real numbers \(x\) obeying \(0\leq x \lt 1\) are countable. By the one-to-one-map rule, it means that we may label these numbers with integer labels. For example, the labeling may look like this

0: 0.2135532652368....
1: 0.8035230130532....
2: 0.5348679792241....
....

Fine. Now, Cantor's argument continues as follows. Pick the 1st digit after the decimal point from the first line (labeled by 0), the second digit from the second line (1), the third digit from the third line (2), and so on. So pick the digits on the diagonal. In my example, you get 204...

Now, change all these digits. For example, add one to all of them, modulo ten. So you will get 315... in my example. Construct the number

?: 0.315.....

It's easy to see that this number isn't written on any line of the previous "numbered list of real numbers from that interval" simply because it disagrees with each number, i.e. the number on the \(N\)-th line, somewhere, for example in the \(N\)-th digit after the decimal point. It disagrees by construction. So some real numbers are inevitably left as "uncounted". The interpretation is that the number of real numbers is greater than the number of integers. We say that the real numbers \(\RR\) are not "countable".

Cantor was often called a "Jew" but no clear evidence of his Jewish ancestry has ever emerged. Nevertheless, what we do know is that he introduced a Hebrew letter "aleph", the first letter of their alphabet, into mathematics. The cardinality (generalized number of elements) of the integers \(\ZZ\) would be called \(\aleph_0\) while the cardinality of the real numbers \(\RR\) would be called \(\aleph_1\). There are also "larger" sets with cardinality \(\aleph_n\). The famous Continuum Hypothesis states that there is no cardinality in between \(\aleph_0\) and \(\aleph_1\).

(The cardinal numbers shouldn't be confused with ordinal numbers which label inequivalent well-ordered sets – sets such that each subset has a minimum element. The ordinal number "analogous" to \(\aleph_0\) is known as \(\omega\) but \(\omega+1\) is actually a different one and the rules to construct and distinguish larger ordinals differ from the rules for cardinals.)

All of this is maths is fun and really follows from some rules of the game. However, these rules and their implications are utterly unnatural from the viewpoint of physics. For example, the cardinal \(\aleph_1\) is called the "continuum" and it quantifies the number of elements (points) in \(\RR\) as well as any \(\RR^n\). The reason why \(\RR\) and \(\RR^n\) have the same cardinality is rather simple. You may "compress" \(n\) real numbers into one simply by taking the digits from all these \(n\) numbers in an alternating fashion. So \((2.7182818,3.1415926)\) may be "compressed" as a single real number \(23.71148125891286\dots\). Do you understand the rule? Needless to say, such a compression of a 2-dimensional plane into a real line is completely discontinuous and contrived from a physics viewpoint. No physical application may justify such an insensitive manipulation. But from a strictly mathematical viewpoint, it is a proof that all kinds of the continuum have the same cardinality.

The bases produce the same Hilbert space

The very claim that the continuum has a larger cardinality than the integers – that the real numbers are uncountable – is controversial from a "more moral viewpoint". Cantor's diagonal proof is OK but the "counterexamples" showing that the numbers are not countable are completely contrived numbers you will never experience in physics. The individual precisely known numbers that matter are countable. Using a more modern terminology, all finite computer files are countable: you first sort them according to the length in bytes; and then you sort the files with the same length alphabetically or lexicographically; then you assign integers to the ordered list of computer files. Similarly, each of the "constructible real numbers" may be uniquely identified by a finite computer file or a finite sequence of words and/or mathematical symbols, and such finite sequences are countable. So the elements proven to be "uncountable" by Cantor's argument may be considered "pure junk that can never matter".

The quantification of the dimension of Hilbert spaces is another major argument showing that the obsession with the "different infinite sets' having different cardinalities" is physically misguided.

The point is that some Hilbert spaces admit both countable and uncountable bases assuming the most natural yet "generalized" definition of an uncountable basis you may imagine.

In a previous Physics Stack Exchange question, I would choose the Fourier series as an example. But pretty much every quantum Hamiltonian gives you an example to show my point= – not to mention infinitely many other operators. Let me talk about the quantum harmonic oscillator here.

Consider the space of \(L^2\)-integrable complex functions \(\psi(x)\) of a real variable \(x\in \RR\) – wave functions for a particle on the line. We demand\[

\int_{-\infty}^{+\infty} |\psi(x)|^2 \dd x \lt \infty

\] to make the norm finite; that is what we called "square-integrable" or \(L^2\). More precisely, we want to consider the space of equivalence classes where two functions \(\psi_1(x)\) and \(\psi_2(x)\) are considered the same if they only differ in values at countably many values of \(x\) – or, more generally, at a set of values of \(x\) that has measure zero. This identification has to be done because the difference \(\psi_1(x)-\psi_2(x)\) of such two functions is "zero almost everywhere" and its inner product with any other \(L^2\)-integrable function is zero (the inner product is computed by the integral). So \(\psi_1,\psi_2\) predict the same probabilities (probability amplitudes) for everything; they are physically indistinguishable.

Now, it is natural to say that the space of such functions or their innocent equivalence classes morally "is" generated by the basis of continuum basis vectors \(\ket {x_0}\) for \(x_0\in RR\). The wave function corresponding to such a basis vector \(\ket {x_0}\) is \(\delta(x-x_0)\). Well, this basis vector isn't really \(L^2\)-integrable so it is not in the Hilbert space. But it belongs to an extended, "rigged" Hilbert space. At any rate, it is extremely natural to work with these "eigenvectors of these \(x\) operators". They are not normalizable but they're as analogous to the normalizable eigenstates of operators with a discrete spectrum.

By a linear combination of these vectors, we mean the integrals \[

\int_{-\infty}^{+\infty} \dd x\,\varphi(x)\ket{x}

\] rather than the sums. The integrals simply "are" the right linear operations to deal with continuously many values.

Because there exists a basis vector \(\ket{x_0}\) for any, continuously adjustable \(x_0\in\RR\), the basis – if you agree that it is "morally correct" to call it a basis – is continuous. The cardinality of this basis is \(\aleph_1\). Nevertheless, the vector space it generates is the same Hilbert space you may also obtain from a countable basis. Consider the Hamiltonian\[

H = \frac 12 x^2 + \frac 12 p^2,\quad [x,p]=i

\] Well, we could consider any other operator with a discrete spectrum too but this one is perhaps the simplest one. The eigenstates of this "harmonic oscillator Hamiltonian" are the wave functions\[

\psi_n(x) = C_n\cdot H_n(x) \cdot \exp(-x^2/2)

\] where \(C_n\) is a normalization factor, \(H_n\) are degree-\(n\) "Hermite" polynomials, and the last factor is the universal Gaussian (the ground state wave function). In this notation capturing the mathematical essence of the problem, I have pretty much used units with \(m=\omega=\hbar=1\) to describe any quantum harmonic oscillator. The eigenvalue of \(H\) in the state \(\psi_n\) is \(n+1/2\).

Now, every complex \(L^2\)-integrable function may be written as a linear superposition of the wave functions \(\psi_n\) above. The right coefficients may be computed as simple inner products; the linear superposition with the coefficients computed in this way may differ from the original wave function at most at points whose measure is zero.

So these energy eigenstates obviously form another basis of the Hilbert space and it is a countable one.

For finite-dimensional Hilbert spaces, you may see that the Hilbert spaces \(\CC^m\) and \(\CC^n\) are only equivalent if \(m=n\). Because these exponents quantify the "cardinality of the bases", you could think that even for the infinite sets, the cardinality will matter. But if you use a physically natural definition of the "continuous bases", the countability of the basis doesn't seem to matter. You get the same Hilbert space of \(L^2\)-integrable functions of a real variable.

(The unphysical, set-theory-preferring axiomatic approach will deny that what we call the continuous basis is a basis at all. It will ban wave functions that are distributions and it will prevent you from considering an integral with the integrand containing some state vectors to be their linear combination – only sums are OK. Once this philosophy imposes the sums onto you, you will produce "inseparable" vector spaces that are "pathologically, too large" and therefore unphysical. Such a treatment will find bureaucratic loopholes to outlaw everything we do in physics. But a physicist will know that the integral defines as good a linear superposition as a sum; and continuous spectra and their eigenstates and eigenvalues are as good as the discrete ones and should be treated analogously unless you want to violate the spirit of quantum physics!)

One shouldn't really be shocked that the cardinality of infinite sets doesn't matter in physics. The irrelevance really boils down to the uncertainty principle, the essential feature that produces all the qualitatively new phenomena in quantum mechanics if you look at the problems from a proper perspective. Am I not saying that the uncertainty principle is the same thing as the Gödel-related stuff in mathematics? Am I not repeating the claim that I have attacked at the beginning? Have I lost my mind?

No, I am not. What I mean is something else. Recall that Cantor has defined two "equally large sets" or "sets of the same cardinality" to be two sets such that a one-to-one map in between them exists. A one-to-one map may be thought as a relabeling. If there is a one-to-one map between two bases, the transition matrix \(\langle i| \alpha\rangle\) between these two (Latin, Greek) bases is equal to a "permutation matrix" of a sort. Most of the entries are zero. In each column and each row, there is one entry that is equal to one.

But this permutation matrix is far from being the only allowed (or most general) type of a transition matrix between two bases. Generically, a basis vector in one basis is a general superposition of all basis vectors in the other basis. Almost all the inner products may be nonzero. For example, the inner products\[

\langle x| n \rangle =\psi_n(x)

\] may be interpreted as a transition matrix between the continuous basis of \(x\)-eigenstates and the countable basis of \(H\)-eigenstates; here, \(x\) labels the row while \(n\) distinguishes the columns. Cantor's diagonal proof has only discussed one-to-one maps i.e. very special class of matrices that was only allowed to "permute the eigenstates" (elements of the basis). However, what we care about in quantum mechanics is whether the whole Hilbert spaces produced by the bases are the same. So arbitrary linear combinations are OK – even when we switch from one basis to another.

And Cantor's proof hasn't "excluded" such more general transition matrices. When treated naturally from a physics viewpoint, such an exclusion would be wrong. In fact, the transitions between bases whose cardinality is \(\aleph_0\) and bases whose cardinality is \(\aleph_1\) are possible. You could even say that because of its restriction of maps to permutation matrices and one-to-one maps, Cantor's diagonal proof (if applied to Hilbert spaces) has de facto incorrectly assumed classical physics. And that's too bad a mistake.

To summarize, it is possible to write down an axiomatic framework that will justify the statements that the discrete and continuous bases are different and the cardinality does matter. But the detailed assumptions and "bans" behind the axioms are – much like their conclusions – utterly unnatural and "morally wrong" from a physics viewpoint.

Physics naturally wants us to think about the infinite sets very differently. For some purposes, the number of points in \(\RR\) and \(\RR^n\) should be considered "different" in physics because all one-to-one maps between the two sets are heavily discontinuous and therefore "de facto disallowed" in physics. On the other hand, when the elements of \(\ZZ\) and \(\RR^n\) label bases of a Hilbert space, the Hilbert spaces may be equally large.

None of these "opposite" conclusions we prefer in physics means that mathematics has been invalidated. Instead, the different paths that physics takes show that some of the detailed features of the axiomatic systems in mathematics were simply a wrong mathematics for physics. This is the ultimate observation that many people still fail to understand.

Mathematics is fine but it is a rigorous game with man-made axioms and rules. Nature can't guarantee, doesn't guarantee, and usually explicitly denies that the first axiomatic systems that you develop to deal with some question are the "perfect ones" for physics. In particular, when it comes to the right treatments of "infinite sets" – analogously to sums of infinitely many terms, like the \(1+2+3+4+\dots = -1/12\) discussed in the recent revival of the topic – Nature and physics simply prefer (and push us towards) a very different way of thinking about all these structures and matters. I mean very different from those that may become "standard" among those mathematicians who are disconnected from modern physics. The different conclusions of physicists' mathematics (relatively to mathematicians' mathematics) doesn't mean that physicists' mathematics is inevitably non-rigorous. Instead, it means that it needs different axioms than those picked by mathematicians.

If I were a politically correct opportunist, I could say that the culture of mathematicians (with their cardinals, ordinals, theorems on incompleteness, disrespect for continuity, semi-bans on integration, and the proliferation of inseparable Hilbert spaces that follows from that etc.) and the culture of physics (with their operators boasting discrete, continuous, or mixed spectra, complete embrace of integration, continuity, unified treatment of operators with discrete and continuous spectra etc.) are simply two different, inequivalent ways to formalize certain (or superficially related) mathematical concepts and to define rules they have to follow.

But I am not a politically correct opportunist so I will tell you the actual truth. It is the physicists' perspective on these issues that is vastly superior and more profound even from a mathematical viewpoint – simply because it's the perspective that has already undergone some actual tests and that has been forced to improve by the tests – tests of abilities to describe Nature and to provide a deeper and more general internally consistent system of mathematical thought that can do so. Physicists' evolving formalization of these axioms and mathematical structures is superior over the mathematicians' formalization in a very analogous way in which the newest iPhone is or Nokia Lumia 1520 better than Alexander Graham Bell's first device: unlike Bell's gadget, the iPhones and Nokias Lumias have already witnessed some improvements.

It's the physicists' approach to the notion of infinity, infinite sums, infinite bases etc. that is the deeper one, more likely to be related with future important discoveries. When the mathematicians' verdict disagrees, it's because this mathematicians' approach is just a package of dirty bureaucratic tricks to defend a conclusion that is morally invalid (i.e. that are likely to lead to invalid conclusions if you consider them far-reaching and valid in a "deep" sense). And following these bureaucratic tricks too strictly may only lead to one outcome – to become a slave of these man-made bureaucratic rules, to produce wrong claims about physics (and about the kind of mathematics that matters in physics), and to miss the actual mathematical wisdom that is known to Nature, that was needed to create something more profound than ourselves, namely the Universe.

In plain English, mathematicians (and their collaborationists pretending to be physicists, like Emilio Pisanty) who will reject the assertion that the cardinality of infinite bases in quantum mechanics doesn't matter simply suck. They're superficial bureaucrats who simply prevent others from getting to deeper levels of the truth.

And that's the memo.