# Is $[\hat x, \hat p_x] = i\hbar\, \mathbb{I}$ contradicting a fact about commutators?

My background in quantum mechanics is minimal, and I had seen the canonical commutation relation $$[\hat x, \hat p_x] = i\hbar\, \mathbb{I}$$ in a course, I took about two years ago. I'm doing pure mathematics, and in Operator Theory, I learned that the commutator of two bounded linear operators on a Hilbert space cannot be a non-zero scalar multiple of the identity. For references, see Halmos' exposition here. The result was first proved by Wintner, though.

Question: What's going wrong in the case of the canonical commutation relation? Is at least one of the two operators not bounded or linear? Also, what's the Hilbert space on which the operators $$\hat x$$ and $$\hat p_x$$ act?

• Have you worked through the proof in Halmos, applying it to these particular operators, to see what goes wrong? Jun 3 at 19:09
• @WillO I don't know these operators well enough to apply the proof to them, I believe. That's why I'm asking the question! Jun 3 at 19:13
• these operators are manifestly unbounded, their spectrum being the whole real line... Jun 3 at 19:17
• They're linear but unbounded. Jun 3 at 19:18
• Related/possible duplicates: physics.stackexchange.com/q/149786/50583, physics.stackexchange.com/q/19447/50583 Jun 3 at 20:47

Just to develop what people already said in the comment: in fact, you put your finger on the proof that position and momentum operators can't have bounded spectra.

Since those operators have an infinite number of eigenvalues, their trace is undefined (the sum doesn't converge), so the commutation relation can hold without violating any algebric property.

Such operators actually act inside a space that is larger than a simple Hilbert space, a structure called a rigged Hilbert space.

Fun fact: a pair of operators governed by a relation like $$[A,B]=i$$, then they also verify a Heisenberg-type property, $$\Delta a\,\Delta b\geqslant\frac{\lvert[A,B]\rvert}{2}$$.

• Is there a name for the "rigged" Hilbert space on which these operators act? Jun 3 at 19:32
• In English, I believe that the proper name is rigged Hilbert space. In my language (French), it can be translated as a "Gelfand triplet". Jun 3 at 19:34
• @connected-subgroup It should really be '"rigged Hilbert" space,' not '"rigged" Hilbert space.' The rigged space is not a Hilbert space, since it contains non-normalizable states: physics.stackexchange.com/a/614466/70245
– Buzz
Jun 3 at 21:29
• Minor remark, there are plenty of bounded operators with an infinite number of eigenvalues (even continuum infinite if you pass me the jargon, since in this case they are not eigenvalues).
– lcv
Jun 3 at 23:16
• No problem. But to be precise there are also plenty of bounded operators with purely continuous spectrum.
– lcv
Jun 3 at 23:27