Unbounded operators in non-relativistic QM of one spin-0 particle

[Fredrik]

What exactly are the axioms of non-relativistic QM of one spin-0 particle? The mathematical model we're working with is the Hilbert space L^2(\mathbb R^3) (at least in one formulation of the theory). But then what? Do we postulate that observables are represented by self-adjoint operators? Do we say that a measurement of an operator A on a system prepared in state |\psi\rangle yields result a_n and leaves the system in the eigenstate |n\rangle with probability |\langle n|\psi\rangle|^2? Then how do we handle e.g. the position and momentum operators, which don't have eigenvectors?

Can the problem of unbounded operators be solved without the concept of a "rigged Hilbert space"? Is it easy to solve when we do use a rigged Hilbert space? What is a rigged Hilbert space anyway?

I think I brought this up a few years ago, but apparently I wasn't able to understand it even after discussing it. I think I will this time, because of what I've learned since then. Don't hold back on technical details. I want a complete answer, or the pieces that will help me figure it out for myself.

 

[jensa]

I too would like a clarification on the subject of "rigged Hilbert space". Sometimes it seems like it is just a word people throw into justify introducing non-normalizable eigenstates and treating them in a similar way as other eigenstates with the substitution \langle n|m\rangle =\delta_{nm}\rightarrow\langle x|x'\rangle=\delta(x-x'). Is this just some trick or is there more to it.

Sometimes people introduce boxes with periodic boundary conditions and let the size of those boxes go to infinity at the end... is this more rigorous? Probably not ..?

 

[strangerep]

What exactly are the axioms of non-relativistic QM of one spin-0
particle? The mathematical model we're working with is the Hilbert
space L^2(\mathbb R^3) (at least in one formulation of
the theory). But then what? Do we postulate that observables are
represented by self-adjoint operators?

More axiomatically, one can start with a complete normed algebra
of operators (a Banach algebra), satisfying some extra axioms
that make it into a C* algebra. Then construct a Hilbert space
on which the elements of the algebra act as operators. This is
called the "GNS" construction. The algebraic approach avoids
some of the operator ambiguities that can arise with the
"Hilbert space first" approach.

Do we say that a measurement of
an operator A on a system prepared in state |\psi\rangle
yields result a_n and leaves the system in the eigenstate
|n\rangle with probability
|\langle n|\psi\rangle|^2 ? Then how do we handle
e.g. the position and momentum operators, which don't have eigenvectors?

Using Rigged Hilbert Space (RHS), aka "Gelfand Triple".
(Personally I dislike both names, and prefer the more
explicit "Gelfand Triple Space", though I think I'm alone in
that usage.)

Can the problem of unbounded operators be solved without the concept of a
"rigged Hilbert space"? Is it easy to solve when we do use a rigged Hilbert
space? What is a rigged Hilbert space anyway?

Without an RHS, you've got to pay careful attention to the domains of
operators. The general spectral theorem for s.a. operators on inf-dim
Hilbert space is littered with domain stuff. But the whole point of
the RHS idea is to avoid that stuff and provide a rigorous mathematical
underpinning of Dirac's original bra-ket stuff that uses such improper
eigenvectors.

Do you have a copy of Ballentine's QM textbook? It's one of
the few that explain and emphasize how all the Dirac-style QM
we know and love is really all being done in an RHS.
(Ballentine also shows how some of the operators in non-rel
QM arise by considering unitary representations of the
Galilei group, which was another part of your question.)

I just looked at the RHS Wiki page but it's very brief and doesn't
tell you much. Although Ballentine describes RHS, it's only at an
introductory level. There's an old book by Bohm & Gadella,
"Dirac Kets, Gamow Vectors, and Gel'fand Triplets" which explains
a bit more, but they too don't get into the mathematical guts.

I think I brought this up a few years ago, but apparently I
wasn't able to understand it even after discussing it. I
think I will this time, because of what I've learned since
then. Don't hold back on technical details. I want a
complete answer, or the pieces that will help me figure it
out for myself.

There's no way I can fit a complete technical answer in a Physics
Forums post, but maybe I can get you started...

The basic idea is to start with a Hilbert space "H" and then construct
a family of subspaces. To do this, take the formula for your
Hilbert space norm, and then modify it to make it harder for all
states to have a finite norm. E.g., change the usual norm from
<br /> \int dx \psi^*(x) \psi(x)<br />
to something like
<br /> \int dx |x|^n \psi^*(x) \psi(x)<br />
Clearly, for n>0, only a subset of the original \psi
functions still have finite norm. It is therefore a "seminorm" (meaning
that it's defined only a subset of H). This family of seminorms,
indexed by n, define a family of progressing smaller and smaller
subsets of the original Hilbert space H. It turns out that each such
subspace is a linear space, and is dense in the next larger one.

More generally, this construction comes under the heading of
"Nuclear Space", with a corresponding family of "seminorms".
The Wiki page for Nuclear Space has some more info.

Now, to proceed further, you need to know a couple of things about
inf-dim vector spaces and their duals. First a Hilbert space H is
isomorphic to its dual (i.e., isomorphic to the set of linear
mappings from H to C). Then, if you restrict to a linear subspace
of H, (let's call it \Omega_1, corresponding the
case n=1 above), the dual of \Omega_1, which I'll
denote as \Omega^*_1, is generally larger than H.
I.e., we have \Omega_1 \subset H \subset \Omega^*_1.

Note that the usual norm and inner product are ill-defined between
vectors belonging to the dual space \Omega^*_n, but
we still have well defined dual-pairing between a vector from
\Omega^*_n and a vector from \Omega_n.
This is enough for Dirac-style quantum theory.

Actually, I'm getting a bit ahead of myself. First, we should take an
inductive limit n\to\infty of the \Omega_n
spaces, which I'll denote simply as plain \Omega without
the subscript. This is the subspace of functions from H which vanish
faster than any power of x.

The "Rigged Hilbert Space", or "Gel'fand Triple", is the name given
to the triplet of densely nested spaces:
<br /> \Omega \subset H \subset \Omega^*<br />
The word "rigged" should be understood to mean "equipped and ready for
action". (Even with this explanation I personally still think it's a
poor name.)

[Continued in next post because of "Database error"...]

 

[strangerep]

[Continuation of previous post...]

The master stroke now comes in that the so-called improper states of
position, momentum, etc, in Dirac's bra-ket formalism correspond to
vectors in \Omega^*. It is possible to take a s.a.
operator on \Omega, and extend to an operator on
\Omega^*, where the extension is defined in terms of its
action on elements of \Omega via the dual-pairing.

Taking this further, there is a generalization of the usual spectral
theorem, called the Gelfand-Maurin Nuclear Spectral Theorem which
shows that eigenvectors of A in the dual space \Omega^*
are "complete" in a generalized sense, even though they're not
normalizable.

So although people "throw around" the phrase "rigged Hilbert space"
it's actually very important to the mathematical underpinnings of QM,
though perhaps less so if you just want to do Dirac-style basic
calculations. The RHS *is* the arena for modern QM, rather than the
simpler Hilbert space as widely believed. The RHS, with the G-M Nuclear
Spectral Theorem, is a far more general mathematical foundation than
the trick of "finite boxes", etc, that jensa asked about.

There's also an old textbook by Maurin "General Eigenfunction Expansions..."
which gives the rigorous proof (though not very clearly, imho). But I
think I'll stop here and see what followup questions arise.

 

[jostpuur]

This connection between unboundedness of operators and the non-normalizable eigenvectors is non-existing, and indicates some confusion.

Example 1:

If you consider the system defined by a Hamiltonian (this is the infinitely deep potential well)

<br /> H = -\frac{\hbar^2}{2m}\partial_x^2 + \infty\;\chi_{]-\infty,0[\cup ]L,\infty[}(x)<br />

and solve it's eigenstates, you get a sequence of normalizable eigenvectors |\psi_1\rangle,|\psi_2\rangle,\ldots, and in this basis the Hamiltonian is

<br /> H = \frac{\hbar^2\pi^2}{2mL^2}\left(\begin{array}{ccccc}<br /> 1 &amp; 0 &amp; 0 &amp; 0 &amp; \cdots \\<br /> 0 &amp; 4 &amp; 0 &amp; 0 &amp; \cdots \\<br /> 0 &amp; 0 &amp; 9 &amp; 0 &amp; \cdots \\<br /> 0 &amp; 0 &amp; 0 &amp; 16 &amp; \cdots \\<br /> \vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots \\<br /> \end{array}\right)<br />

This is an unbounded operator, but still it can be diagonalized in the Hilbert space in the standard sense.

Example 2:

If you regularize the differential operator \partial^2_x by making some cut-off in the Fourier-space, you obtain some pseudo-differential operator which will be approximately the same as the \partial_x^2 for wave packets containing only large wave lengths. So fix some large R and set

<br /> (H_R \hat{\psi})(p) = \frac{p^2}{2m} \chi_{[-R,R]}(p)\hat{\psi}(p),<br />

which is the same thing as

<br /> (H_R \psi)(x) = \frac{1}{2m} \frac{1}{2\pi\hbar} \int\limits_{-R}^R\Big(<br /> \int\limits_{-\infty}^{\infty} p^2 \psi(x&#039;) e^{i(x-x&#039;)p/\hbar} dx&#039;\Big) dp.<br />

This operator is bounded and \|H_R\| = \frac{R^2}{2m} &lt; \infty. However, it's eigenvectors are outside the Hilbert space L^2(\mathbb{R}).

Conclusion:

So it is possible to have an unbounded operator so that its eigenvectors are inside the Hilbert space, and it is possible to have a bounded operator so that its eigenvectors are outside the Hilbert space.

 

[jostpuur]

If we ask a question that what is the probability for a momentum to be in an interval [p_0-\Delta, p_0+\Delta], we get the answer from the expression

<br /> \frac{1}{2\pi\hbar} \int\limits_{p_0-\Delta}^{p_0+\Delta} |\hat{\psi}(p)|^2 dp.<br />

I'm not convinced that it is useful to insist on being able to deal with probabilities of precise eigenstates. Experimentalists cannot measure such probabilities either.

 

[Fredrik]

Good answers, both of you. I appreciate that you are taking the time to explain these things to me. I need some time to think about the technically advanced parts of Strangerep's posts, so for now I'll just reply to Jostpuur. I'll reply to Strangerep later today, or tomorrow.

Jostpuurs post #6 brings up one of the things I was thinking about when wrote the OP and when I was reading Strangerep's reply. The RHS stuff is interesting, and I definitely want to learn it, but I feel that as long as we're just talking about the non-relativistic quantum theory of one spinless particle, it should be possible to avoid the complications by stating the axioms of the theory of physics carefully, instead of changing the mathematical model (by replacing the Hilbert space by a Gelfand triple). What I mean by the "axioms of the theory of physics" are the statements that tell us how to interpret the mathematics as predictions of probabilities of possible results of experiments. Am I right about this, or do we absolutely need something like a RHS just to state the simplest possible quantum theory in a logically consistent way?

Joostpur, I agree that your example 1 proves that it's possible for an unbounded operator on a Hilbert space to have eigenvectors. I didn't expect that. The Hilbert space in your example is L^2([0,L]), not L^2(\mathbb R^3), but those two spaces are isomorphic (unless I have misunderstood that too), so this should mean that there's an unbounded operator on L^2(\mathbb R^3) that has an eigenvector.

I don't understand 100% of example 2, but I accept it as a convincing argument that it's possible for a bounded operator to fail to have eigenvectors. The part that's confusing me is that I don't see what the "eigenvectors" of H_R are. I assume that they are some sort of distributions.

 

[strangerep]

This connection between unboundedness of operators and
the non-normalizable eigenvectors is non-existing, and indicates some
confusion.

Confusion in terminology perhaps, but there is certainly a connection.

Expressed more precisely, let me quote the Hellinger-Toeplitz
theorem (from Lax, p377):

"An operator M that is defined everywhere on a Hilbert space H and
is its own adjoint, (Mx,y) = (x,My), is necessarily bounded."

The proof is only a few lines.
There follows a corollary (further down on p377):

"It follows from this [...] that unbounded operators that are their
own adjoints can be defined only on a subspace of the Hilbert space".

Example 1:

[...]
you get a sequence of normalizable eigenvectors
|\psi_1\rangle,|\psi_2\rangle,\ldots, and in this basis the
Hamiltonian is

<br /> H = \frac{\hbar^2\pi^2}{2mL^2}\left(\begin{array}{ccccc}<br /> 1 &amp; 0 &amp; 0 &amp; 0 &amp; \cdots \\<br /> 0 &amp; 4 &amp; 0 &amp; 0 &amp; \cdots \\<br /> 0 &amp; 0 &amp; 9 &amp; 0 &amp; \cdots \\<br /> 0 &amp; 0 &amp; 0 &amp; 16 &amp; \cdots \\<br /> \vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots \\<br /> \end{array}\right)<br />

This is an unbounded operator, but still it can be
diagonalized in the Hilbert space in the standard sense.

The last bit about "in the Hilbert space" is incorrect. Let me
simplify your example...

Your eigenvectors can be written as infinite-length vectors:
<br /> e_1 := (1,0,0,0,...)~~<br /> <br /> e_2 := (0,1,0,0,...)~~<br /> <br /> \dots~<br /> <br /> e_k := (0,0,...,0,1,0,0,...)~~<br />
Forgetting the constants, your Hamiltonian can be written as
<br /> H_{nm} ~=~ n^2 ~ \delta_{nm}<br />

Now, I can construct a particular linear combination of the e_k:
<br /> \psi := \sum_{k=1}^\infty \frac{1}{k} ~ e_k<br />
whose squared norm is
<br /> (\psi,\psi) ~=~ \sum_{k=1}^\infty \frac{1}{k^2} ~&lt;~ \infty<br />
So \psi is in the Hilbert space. Now consider
<br /> \phi := H \psi ~=~ \sum_{k=1}^\infty k^2 ~ \frac{1}{k} ~ e_k <br /> ~=~ \sum_{k=1}^\infty k ~ e_k<br />
whose squared norm is
<br /> (\phi,\phi) ~=~ \sum_{k=1}^\infty k^2 ~\to~ \infty<br />
and therefore \phi is not in the Hilbert space.

Hence, it's incorrect to say that this Hamiltonian is an
operator on the entire Hilbert space. It's only a well-defined
operator on a subspace of the Hilbert space.

The rigged Hilbert space formalism was developed to make sense of
this. The Hamiltonian *can* be diagonalized in a sense, but must
be done in terms of generalized eigenvectors in a larger space
of tempered distributions (the \Omega^* from my
earlier post).

Example 2:
[...]
This operator is bounded and \|H_R\| = \frac{R^2}{2m} &lt; \infty.
However, it's eigenvectors are outside the Hilbert space
L^2(\mathbb{R}).

Your Hamiltonian in example 2 (Edit: in the limit R\to\infty )
is not well-defined on all of L^2(\mathbb{R}). I.e., it's not
an operator on all of L^2(\mathbb{R}).

You seem to be defining
the eigenvectors on a subset of L^2(\mathbb{R}) (with
finite "R") and then assuming they remain well-defined when you
take R\to\infty. But the limit
<br /> \lim_{R\to\infty} \|H_R\| ~=~ \lim_{R\to\infty} \frac{R^2}{2m}<br />
does not exist.

I'm not convinced that it is useful to insist on being able to deal with
probabilities of precise eigenstates. Experimentalists cannot measure such
probabilities either.

Sure, plenty of people get along fine without understanding
these subtleties. Dirac was one of them. He just knew intuitively
that it was ok, somehow. Later, some mathematicians came along
and made it more rigorous and respectable, using rigged Hilbert
space and related concepts. And Fredrick's question was clearly
asking about the mathematically precise stuff.

 

[jostpuur]

I was already aware of the fact that unbounded operators are often{}^* defined on some subsets of the original Hilbert space, although I thought it would not be necessary to get into that matter in my post. For example the domain of H=-\frac{\hbar^2}{2m}\partial_x^2 is

<br /> D(\partial_x^2) = \big\{\psi\in L^2(\mathbb{R})\;|\; \int\limits_{-\infty}^{\infty} p^4 |\hat{\psi}(p)|^2 dp &lt; \infty \big\},<br />

but this doesn't usually get mentioned in every post that is concerned with this Hamiltonian.

Expressed more precisely, let me quote the Hellinger-Toeplitz
theorem (from Lax, p377):

"An operator M that is defined everywhere on a Hilbert space H and
is its own adjoint, (Mx,y) = (x,My), is necessarily bounded."

The proof is only a few lines.
There follows a corollary (further down on p377):

"It follows from this [...] that unbounded operators that are their
own adjoints can be defined only on a subspace of the Hilbert space".

I can admit that I didn't know this theorem.

({}^*: Or let's say that I was under belief that this "often" the case, while not being aware of the fact that it is always the case, with self-adjoint operators.)

The last bit about "in the Hilbert space" is incorrect. Let me
simplify your example...

Your eigenvectors can be written as infinite-length vectors:
<br /> e_1 := (1,0,0,0,...)~~<br /> <br /> e_2 := (0,1,0,0,...)~~<br /> <br /> \dots~<br /> <br /> e_k := (0,0,...,0,1,0,0,...)~~<br />
Forgetting the constants, your Hamiltonian can be written as
<br /> H_{nm} ~=~ n^2 ~ \delta_{nm}<br />

Now, I can construct a particular linear combination of the e_k:
<br /> \psi := \sum_{k=1}^\infty \frac{1}{k} ~ e_k<br />
whose squared norm is
<br /> (\psi,\psi) ~=~ \sum_{k=1}^\infty \frac{1}{k^2} ~&lt;~ \infty<br />
So \psi is in the Hilbert space. Now consider
<br /> \phi := H \psi ~=~ \sum_{k=1}^\infty k^2 ~ \frac{1}{k} <br /> ~=~ \sum_{k=1}^\infty k<br />
whose squared norm is
<br /> (\phi,\phi) ~=~ \sum_{k=1}^\infty k^2 ~\to~ \infty<br />
and therefore \phi is not in the Hilbert space.

Hence, it's incorrect to say that this Hamiltonian is an
operator on the entire Hilbert space. It's only a well-defined
operator on a subspace of the Hilbert space.

I had not thought about this example carefully, and was not aware of the fact that the domain is not the full space, but now when I look my post, I don't think that I would have very explicitly claimed the domain to be the full space either.

Your Hamiltonian in example 2 is not well-defined
on all of L^2(\mathbb{R}). I.e., it's not an operator
on all of L^2(\mathbb{R}).

I don't agree on this. Let (X,\mu) be some measure space, and f\in L^{\infty}(X) some measurable function. Then the formula

<br /> \psi\mapsto M_f\psi,\quad (M_f\psi)(x) = f(x)\psi(x)<br />

defines a bounded operator M_f:L^2(X)\to L^2(X), and \|M_f\|\leq \|f\|_{\infty}. My example belongs to this class of operators.

 

[jostpuur]

Joostpur, I agree that your example 1 proves that it's possible for an unbounded operator on a Hilbert space to have eigenvectors. I didn't expect that. The Hilbert space in your example is L^2([0,L]), not L^2(\mathbb R^3), but those two spaces are isomorphic (unless I have misunderstood that too), so this should mean that there's an unbounded operator on L^2(\mathbb R^3) that has an eigenvector.

The harmonic oscillator is an example of operator which is defined on some subspace of L^2(\mathbb{R}^n), and which has a sequence of eigenvectors whose span is dense in L^2(\mathbb{R}^n).

I don't understand 100% of example 2, but I accept it as a convincing argument that it's possible for a bounded operator to fail to have eigenvectors. The part that's confusing me is that I don't see what the "eigenvectors" of H_R are. I assume that they are some sort of distributions.

I'll continue with the example from my previous post, where M_f was defined. Suppose for simplicity that the f was also injective, so that it doesn't get same values at different locations. Now we ask whether \lambda is an eigenvalue. If it is, then an equation

<br /> f(x)\psi(x) =\lambda \psi(x)<br />

must be true for a.e. x\in X. This cannot happen unless f(\overline{x})=\lambda with some \overline{x}\in X, and unless \psi(x)=0 for a.e. x\neq \overline{x}. So the eigenvectors would have to be \psi(x) = \chi_{\{\overline{x}\}}(x). If the measure \mu is such measure that \mu(\{\overline{x}\})=0, then the eigenvector doesn't exist because it is zero. What happens in the non-rigorous formalism is that this kind of eigenvector is multiplied with an infinite constant so that it becomes non-zero.

If the Hamiltonian is defined in the Fourier-space by a multiplication

<br /> (H\hat{\psi})(p) = \frac{p^2}{2m}\hat{\psi}(p)<br />

then in the non-rigorous formalism the eigenvectors are delta-functions \delta(p-p&#039;), and in the spatial representation they are the plane waves e^{ixp/\hbar}. If the Hamiltonian is made bounded by force by multiplying the operated function with \chi_{[-R,R]}(p), then the same eigenvectors still work, but the eigenvalues are different. The eigenvalues are the same for -R\leq p\leq R, but go to zero for other p.

 

[Fredrik]

I can construct a particular linear combination of the e_k:
...
...and therefore \phi is not in the Hilbert space.
...
Hence, it's incorrect to say that this Hamiltonian is an
operator on the entire Hilbert space.

I don't see how this observation implies anything more than that H is unbounded, and we already knew that.

Edit: I do now, thanks to Jostpuur. See my next post.

Your Hamiltonian in example 2 is not well-defined
on all of L^2(\mathbb{R}). I.e., it's not an operator
on all of L^2(\mathbb{R}).

It looks well-defined to me. It's just not injective on all of L^2(\mathbb R) and I guess that means it's not self-adjoint on L^2(\mathbb R).

H_R\psi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dp\ e^{-ipx}\frac{p^2}{2m}\tilde\psi(x)\chi_{[-R,R]}(p)=\frac{1}{2\pi}\int_{-R}^{R}dp\ e^{-ipx}\frac{p^2}{2m}\int_{-\infty}^{\infty}dx&#039;\ e^{ipx&#039;}\psi(x&#039;)

We can take the expression on the right-hand side as the definition of H_R, and if we do I think it's clear that this operator is well-defined. The middle expression implies that it's not injective. The integral doesn't depend on what values the Fourier transform \tilde\psi has outside of the interval [-R,R]. (LOL, the tilde is invisible in itex mode).

then in the non-rigorous formalism the eigenvectors are delta-functions \delta(p-p&#039;), and in the spatial representation they are the plane waves e^{ixp/\hbar}. If the Hamiltonian is made bounded by force by multiplying the operated function with \chi_{[-R,R]}(p), then the same eigenvectors still work, but the eigenvalues are different. The eigenvalues are the same for -R\leq p\leq R, but go to zero for other p.

If I insert \psi(x&#039;)=e^{ikx&#039;} into the right-hand side of my equation above, I don't get a constant times e^{ikx}. I get zero. But I might be doing something wrong.

 

[jostpuur]

I don't see how this observation implies anything more than that H is unbounded, and we already knew that.

In my opinion strangerep wrote a relevant comment concerning my example 1, but made a mistake with the example 2.

Recall that if an operator T:V\to V between two norm spaces is unbounded, then it does not mean that \|T\psi\|=\infty for some \psi\in V. This would be contradictory with the implicit assumption T(V)\subset V. Instead it means that there is a sequence of vectors \psi_1,\psi_2,\psi_3,\ldots\in V such that \|\psi_n\|\leq 1 for all n, and

<br /> \sup_{n\in\mathbb{N}} \|T\psi_n\| = \infty.<br />

When V is a subset of some larger vector space in which the norms of vectors are infinite, and the image point T\psi is defined with some formula in this larger vector space, it can happen that \|T\psi\|=\infty for some \psi\in V. In this case we don't obtain an operator T:V\to V, but instead an operator T:D(T)\to V where

<br /> D(T) = \{\psi\in V\;|\; \|T\psi\|&lt;\infty\}<br />

is the domain of the operator. This is what happens in the example of the infinite potential well. The Hamiltonian is not defined on the entire Hilbert space L^2([0,L]), but only on some subspace. However, this subspace is dense in the Hilbert space.

I don't think that I said anything wrong in my example 1 though. Despite the fact that the Hamiltonian is not defined in the entire Hilbert space, the Hamiltonian has a sequence of orthogonal eigenvectors, whose span is dense in the Hilbert space, so the Hamiltonian is pretty diagonalizable there.

 

[Fredrik]

...\|T\psi\|=\infty for some \psi\in V. This would be contradictory with the implicit assumption T(V)\subset V.

D'oh. This is what I missed. Thanks. \|H\psi\| must be finite when the codomain of H is a Hilbert space, so Strangrep's calculation does prove that H can't be defined on all of L^2([0,L]). It's a proof by contradiction:

Assume that H=-(\hbar/2m)d^2/dx^2 is a linear operator from L^2([0,L]) into L^2([0,L]). Then it's defined on the specific \psi that Strangrep defined, but that \psi satisfies \|H\psi\|=\infty, and that contradicts the assumption that the range of H is a subset of L^2([0,L]).

 

[strangerep]

Your Hamiltonian in example 2 is not well-defined
on all of L^2(\mathbb{R}) .

It looks well-defined to me. [...]

I intended (but neglected to say) "in the limit R\to\infty".
I've now edited my earlier post to fix this. Sorry for my lack of care.

 

[Fredrik]

...a C* algebra. Then construct a Hilbert space
on which the elements of the algebra act as operators. This is
called the "GNS" construction.

I have heard about it (e.g. in a reply you wrote to me months ago), but I haven't studied it yet. It's on page 250 of the functional analysis book I bought a while ago (Conway), and I have read very little of that book so far. It's going to take a while before I get there, so maybe we can skip the details (especially proofs), and just talk about what the point is. What is the point? I thought it had something to do with relativity and causality.

Edit: I think I see the point you were going for. I was asking about how to introduce observables into the theory, and you just meant that this is one way to define them. Is it a better way than to define the observables as self-adjoint operators on a separable Hilbert space?

Does the C*-algebra/GNS approach have anything to do with the RHS concept, or are the two unrelated? (They seem unrelated to me).

Without an RHS, you've got to pay careful attention to the domains of
operators. The general spectral theorem for s.a. operators on inf-dim
Hilbert space is littered with domain stuff. But the whole point of
the RHS idea is to avoid that stuff and provide a rigorous mathematical
underpinning of Dirac's original bra-ket stuff that uses such improper
eigenvectors.

Spectral theorem...page 262...and that's probably not the most general one. It's about normal operators (A*A=AA*). I think it would be easier for me to learn the RHS stuff than to get through a whole book of functional analysis. (I think I know what I need to know about measures, integration and distributions).

Do you have a copy of Ballentine's QM textbook?

Unfortunately no. It's one of several books that I'm thinking about buying, but I haven't done it yet. I checked it out after reading Demystifier's thread about it (I noticed you did too), and I think it looks great. Fortunately, the relevant pages are available at Google books.

(Ballentine also shows how some of the operators in non-rel
QM arise by considering unitary representations of the
Galilei group, which was another part of your question.)

I'm trying to find a specific set of statements that can be said to define the theory. I'm sure there are many different sets of statements that do the job (in the sense that each set is logically consistent and makes the same predictions about the results of experiments as some other set). I'd like to see the simplest set of statements that can define the theory, and also the set of statements that's the easiest to generalize to the relativistic case.

I chose to ask specifically about non-relativistic QM of one spin-0 particle because it's the simplest of all relevant quantum theories, and I felt that it should be possible to define it in a pretty simple way. The traditional way (which is kind of sloppy) is to postulate among other things that states are represented by the rays of a (separable) Hilbert space (or specifically L^2(\mathbb R^3)) and that the time evolution of a state is given by the Schrödinger equation. I think I would prefer to drop the explicit stuff about the Schrödinger equation, and instead postulate something about inertial observers and unitary representations of the Galilei group. This would give us both the Schrödinger equation and a definition of the Hamiltonian, the momentum operators and the spin operators (and probably the position operator too, but I haven't fully understood that part...something about central charges of the Lie algebra).

I'm also interested in how the axioms must be changed when we go from non-relativistic to special relativistic quantum mechanics, and finally to general relativistic quantum mechanics. (But we can ignore that last one in this thread ).

I just looked at the RHS Wiki page but it's very brief and doesn't
tell you much. Although Ballentine describes RHS, it's only at an
introductory level.

There are some parts of of Ballentine's explanation where I feel that he dumbs it down a bit too much, but I think I understand what he should have said instead, so it's not a problem. His explanation, combined with yours, is very helpful actually.

Actually, I'm getting a bit ahead of myself. First, we should take an
inductive limit n\to\infty of the \Omega_n
spaces, which I'll denote simply as plain \Omega without
the subscript. This is the subspace of functions from H which vanish
faster than any power of x.

Hm, this part sounds familiar. I read the part about tempered distributions in Streater and Wightman recently, but I didn't try to understand every word. They defined a space of test functions that vanish faster than any power of x, and defined a tempered distribution to be a member of its dual space. The part I didn't understand was the exact definition of "vanish faster than any power of x". I'm going to read that part again, and see if I can understand it.

Is the bottom line here that the members of H* are distributions with H (square integrable functions) as the test function space, and that the members of \Omega^* are tempered distributions? Hm, what you said to Jostpuur in #8 looks like a "yes" to that question.

I just realized that there's one small difference. The members of L^2(\mathbb R^3) are not all infinitely differentiable, and test functions are usually assumed to be.

It seems a bit strange and complicated to define a sequence \Omega_n instead of defining \Omega right away, but then I didn't understand S & W on a first read, and they seem to go straight for \Omega (if I remember correctly). Maybe that's why I didn't understand them.

Taking this further, there is a generalization of the usual spectral
theorem, called the Gelfand-Maurin Nuclear Spectral Theorem which
shows that eigenvectors of A in the dual space \Omega^*
are "complete" in a generalized sense, even though they're not
normalizable.

That sounds interesting, but it's not even in my book.

So although people "throw around" the phrase "rigged Hilbert space"
it's actually very important to the mathematical underpinnings of QM,
though perhaps less so if you just want to do Dirac-style basic
calculations. The RHS *is* the arena for modern QM, rather than the
simpler Hilbert space as widely believed.

I'm definitely going to have to learn the details then. I really appreciate your effort in this thread. I still don't get it completely, but I'm getting closer.

 

[strangerep]

What is the point [re C*-algebras]? I thought it had
something to do with relativity and causality.

Does the C*-algebra/GNS approach have anything to do with the
RHS concept, or are the two unrelated?

Hmmm. That's a rather large question. It's not specifically about
relativity and causality, but more about constructing a quantum theory
starting from an algebra of observables, instead of starting from a
Hilbert space.

You can read the axioms for C*-algebras for yourself, but briefly,
they're a subclass of Banach *-algebras, which in turn are *-algebras
with a norm defined on every element satisfying certain extra axioms
(e.g., the norm is submultiplicative -- See the top of Wiki Banach
algebra page for what that means).

One then considers linear functionals in the dual space of this normed
algebra to arrive at a way of mapping observables to numbers, thus
getting a quantum theory.

When starting from a Heisenberg algebra, one usually employs the
regularized (Weyl) form of the canonical commutation relations to
banish the pathological behaviour that caused the need for RHS in the
other formalism. The GNS construction basically means being given a
vacuum vector (and its linear multiples, i.e., a 1D linear space), and
multiplying it by all the elements of the algebra to generate a full
Hilbert space (of course, I'm skipping lots of technicalities here).

Spectral theorem...page 262...

Of which book? (I don't have Conway.)

and that's probably not the most general one.
It's about normal operators (A*A=AA*).

Lax does the same, and stops short of talking about RHS and the
Gelfand-Maurin generalization. (Looking at the index, I couldn't
even find "nuclear spaces" mentioned.) Reed & Simon Vol-1 talk
about nuclear stuff, but in the context of tempered distributions.
That's why I asked a question here a while ago about proofs of the
G-M theorem. Eventually I got hold of Maurin's book, difficult
though it is.

I think it would be easier for me to learn the RHS stuff than to get
through a whole book of functional analysis. (I think I know what I
need to know about measures, integration and distributions).

Specific examples of the \Omega, \Omega^* are indeed
(respectively) the Schwartz space of test functions, and its dual space
of tempered distributions. The important thing about functional
analysis is that abstracts away from specific spaces to general
properties of inf-dim linear spaces, abstracting a lot of messy
integration stuff by expressing things as linear operators instead.
Although I too found functional analysis quite challenging and
bewildering initially, I've now come to prefer it immensely and I only
drop back to explicit integral stuff when considering specific examples.
Functional analysis is an essential tool for the mathematical
physicist, imho.

I chose to ask specifically about non-relativistic QM of one spin-0 particle
because it's the simplest of all relevant quantum theories, and I felt that it
should be possible to define it in a pretty simple way. The traditional way
(which is kind of sloppy) is to postulate among other things that states are
represented by the rays of a (separable) Hilbert space (or specifically
L^2(\mathbb R^3)) and that the time evolution of a state
is given by the Schrdinger equation. I think I would prefer to drop the
explicit stuff about the Schrdinger equation, and instead postulate
something about inertial observers and unitary representations of the
Galilei group. This would give us both the Schrdinger equation and a
definition of the Hamiltonian, the momentum operators and the spin
operators

Yes, so far. It boils down to: (1) pick a algebra of observables (actually
their universal enveloping algebra), and (2) find all unitary irreducible
representations of this algebra. (3) Construct tensor-product spaces thereof.
The details fill many books of course. Weinberg takes this approach
(more-or-less) in his volumes.

(and probably the position operator too, but I haven't fully
understood that part...something about central charges of the Lie
algebra).

Ah, the position operator (and localization) can get tricky. It's not
too bad for the Galilei case (Ballentine covers it), but constructing a
relativistic position operator is still controversial and problematic.

Is the bottom line here that the members of H* are distributions with H
(square integrable functions) as the test function space,

No, H is self-dual. The test function space (Schwartz space) is an
example of my \Omega space (i.e., a dense subspace of H).

and that the members
of \Omega^* are tempered distributions?

Yes to that part.

It seems a bit strange and complicated to define a sequence
\Omega_n instead of defining \Omega right away,

This (and the sequence of progressively stricter norms I mentioned
originally) are just a rigorous way to define and generalize the notion
of "...functions vanishing faster than any power of x...".

The Wiki page on "nuclear space" has a bit more, though it doesn't
mention the G-M theorem. Try to find Maurin's book if you can.
(or maybe vol-4 in the series by Gelfand & Vilenkin -- I couldn't
obtain the latter, but many authors reference it).

Edit: I just remembered... there's some old ICTP lecture
notes by Maurin on this stuff, available as:

streaming.ictp.trieste.it/preprints/P/66/012.pdf

It covers a lot of the theorems, but skips the lengthy proofs.

 

[Avodyne]

I don't see why it's necessary to go beyond Hilbert space. Rather than defining a position operator, we could define projection operators with eigenvalue 1 if the particle is in some particular volume V, and 0 otherwise; heuristically, these would be

P_{x\in V} \equiv \int_V d^3\!x\,|x\rangle\langle x|

Similarly for a volume in momentum space. Then, instead of defining a hamiltonian whose action could take a state out of the Hilbert space, we could define a unitary time evolution operator.

 

[George Jones]

This is a very interesting thread in which I would like to participate actively, and from which I would like to learn, but I'm too busy with work for the next week or two to do the necessary reading and thinking.

Sometimes people introduce boxes with periodic boundary conditions and let the size of those boxes go to infinity at the end... is this more rigorous? Probably not ..?

Even for a non-relativistic particle in a box, there is a lot a technical "grit". The position operator doesn't have any eigenstates that live in the Hilbert space of states, it only has distributional eigenstates. Also, the momentum operator is unbounded, and thus, by the Hellinger-Toeplitz theorem (as already posted by strangerep), the momentum operator cannot act on all the states in the Hilbert space of states.

This is an example of something much more general. If two self-adjoint operators satisfy a canonical commutation relation, then it is easy to show that at least one of the operators must be unbounded.

This would give us both the Schrödinger equation and a definition of the Hamiltonian, the momentum operators and the spin operators (and probably the position operator too, but I haven't fully understood that part...something about central charges of the Lie algebra).

I think you're referring to the fact that (unlike the case for the Poincare group) non-relativistic quantum mechanics deals with representations of a central extension of the Galilean group, not with representations of the Galilean group. This is related to mass in non-relativistic quantum mechanics. Ballentine never uses the term "central extension," but, unlike most (all?) standard quantum mechanics texts, he does give a non-rigorous version. See: page 73, Multiples of identity (c); page 76; pages 80-81.

Hm, this part sounds familiar. I read the part about tempered distributions in Streater and Wightman recently, but I didn't try to understand every word. They defined a space of test functions that vanish faster than any power of x, and defined a tempered distribution to be a member of its dual space. The part I didn't understand was the exact definition of "vanish faster than any power of x". I'm going to read that part again, and see if I can understand it.

I think that you would like chapter 9, Generalized Functions, form the book Fourier Analysis and Its Applications by Gerald B. Folland.

I don't see why it's necessary to go beyond Hilbert space.

I think that it's just a matter of taste whether one uses Hilbert spaces or rigged Hilbert spaces as a rigourous basis for quantum mechanics. For example, Reed and Simon write (in v1 of their infamous work):

"We must emphasize that we regard the spectral theorem as sufficient for any argument where a nonrigorous approach might rely on Dirac notation; thus, we only recommend the abstract rigged space approach to readers with a strong emotional attachment to the Dirac formalism."

I also think that the reason for the popularity of the Hilbert space approach is historical.

In the early 1930s, before the work of Schwartz and Gelfand on distributions and Gelfand triples, von Neumann came up a rigorous Hilbert space formalism for quantum theory.

I think if a rigorous rigged Hilbert space version of quantum theory had come along before the rigorous Hilbert space version of quantum theory, then the Hilbert space version might today be even less well-known than the rigged Hilbert space version actually is. Students would now be hearing vague mutterings about "making things rigourous with Gelfand triples," instead of hearing vague mutterings about "making things rigourous with Hilbert spaces."

 

[meopemuk]

In my understanding, the reason why standard Hilbert space formalism is not suitable for QM is rather simple. Let's say I want to define an eigenfunction of the momentum operator. In the position space such an eigenfunction has the form (I work in 1D for simplicity)

\psi(x) = N \exp(ipx)

where N is a normalization factor. This wavefunction must be normalized to unity, which gives


1 =\int \limits_V |\psi(x)|^2 dx = N^2V

where V is the "volume of space", which is, of course, infinite. This means that the normalization factor is virtually zero

N = 1/\sqrt{V}

So, the value of the wavefunction at each space point is virtually zero too, and it can't belong to the Hilbert space. But the wavefunction is not EXACTLY zero, because its normalization integral is equal to 1. So, here we are dealing with resolving uncertain expressions like "zero x infinity".

As far as I know, there is a branch of mathematics called "non-standard analysis", which tries to assign a definite meaning to such "virtually zero" or "virtually infinite" quantities and to define mathematical operations with them. I guess that using methods of non-standard analysis in quantum mechanics could be an alternative solution for the "improper states" in QM (instead of the rigged Hilbert space formalism). Did anyone hear about applying non-standard analysis to QM?

 

[Fredrik]

I'm still a bit confused by distributions and tempered distributions. Let's see if we can sort this out.

We define D to be the set of all C^\infty functions from \mathbb R^n to \mathbb C with compact support. (This D isn't used in the construction of a rigged Hilbert space. I'm defining it just for completeness). We say that \phi_n\rightarrow\phi if there's a compact set K that contains the supports of all the \phi_n, and every D^\alpha\phi_n converges uniformly on \mathbb R^n to D^\alpha\phi. Here we're using the notation

|\alpha|=\alpha_1+\cdots+\alpha_n

D^\alpha f(x)=\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}f(x)

The members of D are called test functions. Now we define a distribution as a linear function T:D\rightarrow\mathbb C, which is continuous in the following sense:

\lim_{n\rightarrow\infty}T(\phi_n)=T(\lim_{n\rightarrow\infty}\phi_n)

I think I would prefer to do it a bit differently (if the following is in fact equivalent to the above, but it might not be). We define an inner product and the associated norm by

\langle f,g\rangle=\int_D f g\ d\mu

where \mu is the Lebesgue measure on \mathbb R^n. Now we define the space of distributions to be the dual space of D. Is this definition equivalent to the first?

 

[Fredrik]

Now let's consider tempered distributions. We define S as the set of all C^\infty functions from \mathbb R^n to \mathbb C that satisfy

\sup_x\{|x^\alpha D^\beta\phi(x)|}&lt;\infty

for all \alpha and \beta. I'm using the notation

x^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n}

The set S is called the Schwarz class (at least by Folland). Now, I would like to say that a member of its dual space is called a tempered distribution, but that doesn't seem to be how it's done. Instead of using the standard norm, we define an infinite number of new norms:

\|\phi\|_{r,s}=\sum_{\alpha\leq r}\sum_{\beta\leq s}\sup_x\{x^\alpha D^\beta \phi(x)\}

and define a tempered distribution as a linear functional that's bounded with respect to at least one of these norms. I'm pretty confused by this. Is the set of tempered distributions the dual space of the Schwarz class or not?

 

[strangerep]

So, the value of the wavefunction at each space point is virtually zero too, and it can't belong to the Hilbert space. But the wavefunction is not EXACTLY zero, because its normalization integral is equal to 1. So, here we are dealing with resolving uncertain expressions like "zero x infinity".

Yes, but that's why the more general versions of the spectral theorem are formulated
in terms of "projection-valued measures" to which Lebesgue integration theory is applicable.

Did anyone hear about applying non-standard analysis to QM?

I remember a fairly recent paper:

math-ph/0612082

Title: An approach to nonstandard quantum mechanics
Authors: Andreas Raab

Abstract: We use nonstandard analysis to formulate quantum mechanics in
hyperfinite-dimensional spaces. Self-adjoint operators on
hyperfinite-dimensional spaces have complete eigensets, and bound
states and continuum states of a Hamiltonian can thus be treated on an
equal footing. We show that the formalism extends the standard
formulation of quantum mechanics. To this end we develop the
Loeb-function calculus in nonstandard hulls. The idea is to perform
calculations in a hyperfinite-dimensional space, but to interpret
expectation values in the corresponding nonstandard hull. We further
apply the framework to non-relativistic quantum scattering theory. For
time-dependent scattering theory, we identify the starting time and the
finishing time of a scattering experiment, and we obtain a natural
separation of time scales on which the preparation process, the
interaction process, and the detection process take place. For
time-independent scattering theory, we derive rigorously explicit
formulas for the M{\o}ller wave operators and the S-Matrix.

I only skimmed it at the time. I should take a closer look.

 

[Fredrik]

This is an example of something much more general. If two self-adjoint operators satisfy a canonical commutation relation, then it is easy to show that at least one of the operators must be unbounded.

That rings a bell. Link.

I think you're referring to the fact that (unlike the case for the Poincare group) non-relativistic quantum mechanics deals with representations of a central extension of the Galilean group, not with representations of the Galilean group. This is related to mass in non-relativistic quantum mechanics. Ballentine never uses the term "central extension," but, unlike most (all?) standard quantum mechanics texts, he does give a non-rigorous version. See: page 73, Multiples of identity (c); page 76; pages 80-81.

I think that settles it...I'm going to have buy Ballentine's book.

I think that you would like chapter 9, Generalized Functions, form the book Fourier Analysis and Its Applications by Gerald B. Folland.

Thanks for the tip. It looks good. Link.

 

[Fredrik]

I don't see why it's necessary to go beyond Hilbert space. Rather than defining a position operator, we could define projection operators with eigenvalue 1 if the particle is in some particular volume V, and 0 otherwise; heuristically, these would be

P_{x\in V} \equiv \int_V d^3\!x\,|x\rangle\langle x|

Similarly for a volume in momentum space. Then, instead of defining a hamiltonian whose action could take a state out of the Hilbert space, we could define a unitary time evolution operator.

Yes, this makes a lot of sense. Ballentine seems to agree. Quote from page 28:

We now have two mathemathcally sound solutions to the problem that a self-adjoint operator need not possesses a complete set of eigenvectors in the Hilbert space of vectors with finite norms. The first, based on the spectral theorem (Theorem 4 of Sec. 1.3), is to restate our equations in terms of projection operators which are well defined in Hilbert space, even if they cannot be expressed as sums of outer products of eigenvectors in Hilberbert space. The second, based on the generalized spectral theorem, is to enlarge our mathematical framework from Hilbert space to rigged Hilbert space, in which a complete set of eigenvectors (of possibly infinite norm) is guaranteed to exist.

 

[Fredrik]

...more about constructing a quantum theory
starting from an algebra of observables, instead of starting from a
Hilbert space.

OK, but why would we want to? If QM can be made rigorous using either a Hilbert space and the spectral theorem, or a rigged Hilbert space and the generalized spectral theorem (see the Ballentine quote in my previous post), then we don't need C*-algebras for mathematical rigor, and in the end, it gives us a Hilbert space anyway.

When starting from a Heisenberg algebra, one usually employs the
regularized (Weyl) form of the canonical commutation relations to
banish the pathological behaviour that caused the need for RHS in the
other formalism. The GNS construction basically means being given a
vacuum vector (and its linear multiples, i.e., a 1D linear space), and
multiplying it by all the elements of the algebra to generate a full
Hilbert space (of course, I'm skipping lots of technicalities here).

I don't really understand any of this, but you don't have to break your back trying to explain it all to me. I'm probably going to have to postpone a serious attempt to understand some of the things we're discussing in this thread until I have studied some more functional analysis. I will of course try to understand as much as possible now, but I might not always be successful.

Of which book? (I don't have Conway.)

Yes, Conway. I only mentioned the page number to indicate how much I would have to read to really understand it.

Functional analysis is an essential tool for the mathematical
physicist, imho.

I don't doubt that. It kind of bugs me though when I think about how things were handled at my university. We weren't really encouraged to take more math classes. Most of us didn't even study complex analysis, and probably only about one or two physics students per year bothered to study the analysis class based on "Principles of mathematical analysis" by Rudin. I'm glad I was one of them.

Edit: I just remembered... there's some old ICTP lecture
notes by Maurin on this stuff, available as:

streaming.ictp.trieste.it/preprints/P/66/012.pdf

Cool, I'll check it out tomorrow.

 

[strangerep]

Is this definition equivalent to the first?

There's a distinction to be made be the "algebraic dual", which seems
to be what your 2nd defn is (by default), and the "topological dual"
which corresponds to your 1st defn.

The linear functionals in the topological dual space are required to be
continuous. Therefore, the topological dual is (in general) a subspace
of the algebraic dual (since there's an extra restriction is imposed
upon the former).

FAPP, you might as well just concentrate on the topological dual.

(I'll respond to the tempered distribution question in a later post.)

 

[strangerep]

Is the set of tempered distributions the dual space of the Schwarz class or not?

I don't have Folland unfortunately, but my understanding is that the
space of tempered distributions is indeed the (topological) dual of the
Schwartz space.

This coincides with Lax (p559) and also the Wiki page
http://en.wikipedia.org/wiki/Distribution_(mathematics ) .
See the section "Tempered distributions and Fourier transform".
Warning: read the paragraph starting "The space of tempered
distributions is defined ..." very carefully since the wording
is a little confusing if read too quickly.

 

[strangerep]

Yes, this makes a lot of sense. Ballentine seems to agree. Quote from page 28:

We now have two mathemathcally sound solutions to the problem that a self-adjoint operator need not possesses a complete set of eigenvectors in the Hilbert space of vectors with finite norms. The first, based on the spectral theorem (Theorem 4 of Sec. 1.3), is to restate our equations in terms of projection operators which are well defined in Hilbert space, even if they cannot be expressed as sums of outer products of eigenvectors in Hilberbert space. The second, based on the generalized spectral theorem, is to enlarge our mathematical framework from Hilbert space to rigged Hilbert space, in which a complete set of eigenvectors (of possibly infinite norm) is guaranteed to exist.

I think Ballentine's spectral theorem (Theorem 4 of Sec. 1.3) is distinctly different
from what Avodyne was saying (which involves a restriction to finite domains in both
position and momentum space, and is thus only a subspace of the unrestricted Hilbert space
that Ballentine refers to.

 

[strangerep]

...more about constructing a quantum theory starting from an algebra of
observables, instead of starting from a Hilbert space.

OK, but why would we want to? If QM can be made rigorous using either a Hilbert space and the spectral theorem, or a rigged Hilbert space and the generalized spectral theorem (see the Ballentine quote in my previous post), then we don't need C*-algebras for mathematical rigor, and in the end, it gives us a Hilbert space anyway.

For QFT in infinite-dimensions, even the rigged Hilbert space is not big enough.
The singular behaviours arising from the interaction terms in the Hamiltonian
become far more pathological than mere delta distributions.

Secondly, there are general theorems about how not all operators are sensible
observables. There's a little more on this at the end of a previous thread:
https://www.physicsforums.com/showthread.php?t=262821&page=2

Thirdly, there are so-called Bogoliubov transformations of the a/c ops in QFT
which take you from one Hilbert space into another unitarily inequivalent one.
This stuff is important in condensed matter physics, and other stuff like
Hawking-Unruh.

Hence, plenty of people believe that it's better to start with the algebra of
relevant physical observables, and then construct states as linear mappings
from the algebra to C. In that formalism, issues like those above are more
out in the open.

 

[Fredrik]

There's a distinction to be made be the "algebraic dual", which seems
to be what your 2nd defn is (by default), and the "topological dual"
which corresponds to your 1st defn.

I wasn't aware of the distinction. (I checked the Wikipedia article, so I am now). The type of "dual" I had in mind was the one that consists of bounded linear functionals, with the word "bounded" meaning that there exists an M>0 such that |Tx|\leq M\|x\| for all x. It's fairly easy to show that a functional is bounded in this sense if and only if it's continuous with respect to the metric induced by the norm.

So what I had in mind is the topological dual of a vector space with the topology induced by the inner product. What's confusing me is the definition of continuity. It seems natural to me to define the standard inner product, and use the concept of continuity induced by it. But instead, we choose to define convergence of a sequence in D, and use that to define continuity. These definitions may be equivalent, but it's not obvious that they are.

It gets even more weird in the case of tempered distributions. We have defined an infinite number of norms, which should give us infinitely many kinds of continuity. Maybe I'm just overlooking something simple, but it seems weird that none of the texts (http://en.wikipedia.org/wiki/Distribution_(mathematics)#Test_function_space") say anything about it.

 

[Fredrik]

For QFT in infinite-dimensions, even the rigged Hilbert space is not big enough.
The singular behaviours arising from the interaction terms in the Hamiltonian
become far more pathological than mere delta distributions.

Interesting...and strange. The strange part is that we can actually make predictions in spite of all of this.

Secondly, there are general theorems about how not all operators are sensible
observables. There's a little more on this at the end of a previous thread:
https://www.physicsforums.com/showthread.php?t=262821&page=2

I don't see anything about it in that thread. but I recognize the claim from http://en.wikipedia.org/wiki/Density_matrix#C.2A-algebraic_formulation_of_states", but I don't really understand it. I think I understood one thing though. A self-adjoint operator is only an observable if it preserves each superselection sector. For example, it can't take a bosonic state to a fermionic state.

 

[strangerep]

I don't see anything about it in that thread.

OK. There's some papers by Gotay, e.g., paper math-ph/9809011,
that talk about this in excruciating detail. It's peripheral to the
main subject of the current thread, so you probably won't want
to wade through 50-pages of nontrivial math. Maybe just look
at p18: the paragraph starting with

In particular, one can see
at the outset that it is impossible for a prequantization to satisfy
the “product → anti-commutator” rule.

and the manipulations that follow. And maybe look through the
early sections to get the context.

EDIT: A better reference might be math-ph/9809015.

But like I said, this goes off on an tangent from the current thread.

 

[strangerep]

So what I had in mind is the topological dual of a vector space with the topology induced by the inner product. What's confusing me is the definition of continuity. It seems natural to me to define the standard inner product, and use the concept of continuity induced by it. But instead, we choose to define convergence of a sequence in D, and use that to define continuity. These definitions may be equivalent, but it's not obvious that they are.

I'm starting to lose track of what you're referring to, hence unable to offer a pinpoint
answer. Maybe you should re-summarize?

It gets even more weird in the case of tempered distributions. We have defined an infinite
number of norms, which should give us infinitely many kinds of continuity.

Note that each of these norms defines a subspace. I.e., the sequence of norms defines
a sequence of spaces, each densely nested in the previous (larger) one. So "continuity"
applies in the context of the norm topology on each subspace. The tempered distribution
case applies to the (dual of) the "inductive limit" of this sequence of spaces (afaiu).

 

[Fredrik]

I'm starting to lose track of what you're referring to, hence unable to offer a pinpoint
answer. Maybe you should re-summarize?

OK, let's focus on "distributions" first, not "tempered distributions". We define the test function space (I'll call it \mathcal D) as the set of C^\infty functions from \mathbb R^n into \mathbb C that have compact support. I would like to define a "distribution" as a member of the dual space \mathcal D^*, defined as the set of continuous linear functions T:\mathcal D\rightarrow\mathbb C. But to do that, we need to define what "continuous" means. There are at least two ways to do that.

Option 1: Define the usual inner product. The inner product gives us a norm, and the norm gives us a metric. Now we can use the definition of continuity that applies to all metric spaces.

T:\mathcal D\rightarrow\mathbb C is continuous at g\in\mathcal D if for each \epsilon&gt;0 there's a \delta&gt;0 such that \|f-g\|&lt;\delta\implies|T(f)-T(g)|&lt;\epsilon. T is continuous on a set U\subset\mathcal D if it's continuous at each point in U. Alternatively, and equivalently, T is continuous on U\subset\mathcal D if T^{-1}(E) is open for every open E\subset\mathbb C.

Weird, it takes a lot less to cause a database error now than a couple of weeks ago. I'll continue in the next post.

 

[Fredrik]

Option 2: This is the option that every text on the subject seems to prefer, and I don't see why. We define what it means for a sequence in \mathcal D to be convergent by saying that \phi_n\rightarrow\phi if there's a compact set K in \mathbb R^n that contains the supports of all the \phi_n, and D^\alpha \phi_n converges uniformly on K to D^\alpha\phi, for all \alpha. Recall that D^\alpha is defined by

|\alpha|=\alpha_1+\cdots+\alpha_n

D^\alpha f(x)=\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}f(x)

Now we define "continuous" by saying that T:\mathcal D\rightarrow\mathbb C is continuous at \phi\in\mathcal D if

\lim_{n\rightarrow\infty}T(\phi_n)=T(\phi)

for every sequence \{\phi_n\} that converges to \phi. Maybe these two options are actually equivalent, but I would expect that they're not, mostly because I think option 1 is more natural and intuitive. I don't see why these authors would all go for option 2 if they are equivalent.

 

[Fredrik]

Note that each of these norms defines a subspace. I.e., the sequence of norms defines
a sequence of spaces, each densely nested in the previous (larger) one. So "continuity"
applies in the context of the norm topology on each subspace. The tempered distribution
case applies to the (dual of) the "inductive limit" of this sequence of spaces (afaiu).

Ah, that's actually very helpful. I just re-read a few statements from the part of Streater & Wightman where they define tempered distributions, keeping in mind what you just said, and suddenly what they're saying makes a lot more sense. Unfortunately I have to go to bed now, but I'll continue tomorrow.

 

[Fredrik]

The answer to my concerns in #34-35 (or at least a partial answer) is probably that there's nothing weird or unexpected about the fact that the concept of continuity depends on what topology you define on the set you're working with. The space of distributions isn't the dual space of the test functions. It's a dual space of the test functions. Or to be more accurate, the topological dual space of a set isn't defined. Only the topological dual space of a topological vector space is defined, so we have to choose a topology first, and then we get a dual space.

It still makes me wonder why their version of continuity is more desirable. It probably has something to do with the idea that derivatives of distributions should always exist, but I haven't thought that idea through yet.

Edit: I'm satisfied that I understand the definition of distributions and tempered distributions now, so we don't have to discuss them unless someone else wants to. I'm not saying that I understand every detail perfectly, but my understanding is good enough.

By they way, a thought occurred to me when I was replying to another thread. Isn't it weird to start the formulation of the simplest meaningful quantum theory (the non-relativistic QM of one spin-0 particle) by postulating that the states are rays in L^2(\mathbb R^3) when the next postulate says that the time evolution is given by the Schrödinger equation? The Schrödinger equation only makes sense if the partial derivatives exist, and that implies continuity, but L^2(\mathbb R) contains ridiculous functions like the one Hurkyl thought of in another thread to disprove a claim I made there:

Consider the function:

<br /> \psi(x) = <br /> \begin{cases}<br /> 0 &amp; x &lt; 1 \\<br /> 1 &amp; x \in [n, n + n^{-2}) \\<br /> 0 &amp; x \in [n + n^{-2}, n+1)<br /> \end{cases}<br />

where n ranges over all positive integers. \psi(x) does not converge to zero at +\infty. However,

<br /> \int_{-\infty}^{+\infty} |\psi(x)|^2 \, dx<br /> = \sum_{n = 1}^{+\infty} \int_{n}^{n + n^{-2}} 1 \, dx<br /> = \sum_{n = 1}^{+\infty} n^{-2} = \pi^2 / 6

 

[strangerep]

Maybe these two options are actually equivalent, but I would expect that they're not, mostly because I think option 1 is more natural and intuitive. I don't see why these authors would all go for option 2 if they are equivalent.

First, you can probably separate out the differentiability stuff from the topological
continuity stuff.

I think (in the absence of precise definitions of those two cases) that both notions
of convergence are equivalent. The first is just saying that you can construct a
Cauchy sequence and the second is saying that a sequence converges.
(Topologically, a sequence "converges to a point z" if, given an arbitrary open
set O containing z, all points of the sequence are eventually "in" O -- after some
integer n in the sequence.

The thing about restricting to infinitely differentiable functions is so that you
can do integral calculus, i.e., solve a DE given an initial condition, in such
a way that as to be compatible with the (rigged) Hilbert space structure.

the concept of continuity depends on what topology you define on
the set you're working with

Yes. This is a crucial point to understand in order to work with other
topologies (weak, weak*, and others). Generally speaking, convergence
is easier in weaker (coarser) topologies, but more theorems can be
proven with stronger (finer) topologies. Bit of a tradeoff depending on
exactly what one is trying to achieve.

 

[Fredrik]

I think I'm going to have to study some more functional analysis before I post a bunch of new questions. I'll read some more of Conway first, and maybe I'll read Maurin's lecture notes. Strangerep and Jostpuur, thanks for the answers so far. George, feel free to bump the thread when you're less busy. I'm interested in what you have to say about all of this.

 

[dextercioby]

Brilliant discussion on a very interesting topic, I must say. I was wondering, how would I build the RHS for the hydrogen atom ? The hamiltonian for the H atom has a mixed spectrum and I've never seen an application of the RHS formalism for an operator with mixed spectrum.

 

[jostpuur]

OK, let's focus on "distributions" first, not "tempered distributions". We define the test function space (I'll call it \mathcal D) as the set of C^\infty functions from \mathbb R^n into \mathbb C that have compact support. I would like to define a "distribution" as a member of the dual space \mathcal D^*, defined as the set of continuous linear functions T:\mathcal D\rightarrow\mathbb C. But to do that, we need to define what "continuous" means. There are at least two ways to do that.

Option 1: Define the usual inner product. The inner product gives us a norm, and the norm gives us a metric. Now we can use the definition of continuity that applies to all metric spaces.

Equipping the \mathcal{D} with a norm

<br /> \|\psi\|^2 = \int\limits_{\mathbb{R}^n} dx\; |\psi(x)|^2<br />

and defining \mathcal{D}^* as a topological dual

<br /> \mathcal{D}^* = \big\{ \phi^*\in \mathbb{C}^{\mathcal{D}} \;|\; \phi^*\;\textrm{is linear},\; \sup_{\|\psi\|\leq 1} |\phi^*(\psi)| &lt; \infty\big\}<br />

is probably not what anyone wants, because for example this collection of linear forms doesn't contain delta-functions \delta_x:\mathcal{D}\to\mathbb{C}, \delta_x(\psi)=\psi(x).

Was that what you meant by your option 1?

Option 2: This is the option that every text on the subject seems to prefer, and I don't see why. We define what it means for a sequence in \mathcal D to be convergent by saying that \phi_n\rightarrow\phi if there's a compact set K in \mathbb R^n that contains the supports of all the \phi_n, and D^\alpha \phi_n converges uniformly on K to D^\alpha\phi, for all \alpha.

Now we define "continuous" by saying that T:\mathcal D\rightarrow\mathbb C is continuous at \phi\in\mathcal D if

\lim_{n\rightarrow\infty}T(\phi_n)=T(\phi)

for every sequence \{\phi_n\} that converges to \phi. Maybe these two options are actually equivalent, but I would expect that they're not, mostly because I think option 1 is more natural and intuitive. I don't see why these authors would all go for option 2 if they are equivalent.

I'm not 100% sure of this, but I've heard that it is possible prove the existence of such topology in \mathcal{D}, that the convergence \psi_n\to\psi in that topology is equivalent with this definition that you described here. When \mathcal{D} is equipped with such topology, then the standard collection of distributions \mathcal{D}^* can be defined according to the definition of topological dual.

 

[jostpuur]

In my understanding, the reason why standard Hilbert space formalism is not suitable for QM is rather simple. Let's say I want to define an eigenfunction of the momentum operator. In the position space such an eigenfunction has the form (I work in 1D for simplicity)

\psi(x) = N \exp(ipx)

where N is a normalization factor. This wavefunction must be normalized to unity, which gives


1 =\int \limits_V |\psi(x)|^2 dx = N^2V

where V is the "volume of space", which is, of course, infinite. This means that the normalization factor is virtually zero

N = 1/\sqrt{V}

While I have understood this problem, it has not yet become clear to me how this problem is supposed to be solved in a more sophisticated formalism. The only solution that I have understood so far is this:

If we ask a question that what is the probability for a momentum to be in an interval [p_0-\Delta, p_0+\Delta], we get the answer from the expression

<br /> \frac{1}{2\pi\hbar} \int\limits_{p_0-\Delta}^{p_0+\Delta} |\hat{\psi}(p)|^2 dp.<br />

I'm not convinced that it is useful to insist on being able to deal with probabilities of precise eigenstates. Experimentalists cannot measure such probabilities either.

I don't see why it's necessary to go beyond Hilbert space. Rather than defining a position operator, we could define projection operators with eigenvalue 1 if the particle is in some particular volume V, and 0 otherwise; heuristically, these would be

P_{x\in V} \equiv \int_V d^3\!x\,|x\rangle\langle x|

Similarly for a volume in momentum space. Then, instead of defining a hamiltonian whose action could take a state out of the Hilbert space, we could define a unitary time evolution operator.

I'm not convinced that the distributions are very useful. The distributions are usually defined by using rather strange topologies or norms. The Swartz norm seems to be totally different from the original idea of a Hilbert space norm. However, the Hilbert space norm is highly relevant for the probability interpretation. How are we supposed to deal with quantum mechanical probabilities if the Hilbert space norm has been replaced with some of these distribution norms?

 

[Fredrik]

is probably not what anyone wants, because for example this collection of linear forms doesn't contain delta-functions \delta_x:\mathcal{D}\to\mathbb{C}, \delta_x(\psi)=\psi(x).

Was that what you meant by your option 1?

Ahh, excellent point. Yes, that's what I meant. I get it now. (I didn't until I read your post). Delta functions wouldn't be bounded (and therefore not continous) linear functionals if we go for option 1. For delta to be bounded, there must exist M>0 such that

|\delta f|\leq M\|f\|

for all test functions f, but the inequality is equivalent to

\frac{|f(0)|}{\|f\|}\leq M

and we don't have to look hard to find an f that violates this. Even some constant functions will do. For example, define f_r(x)=1 when |x|<r, and f_r(x)=0 outside that region. Now shrink that region (i.e. choose a smaller r) until the norm of f_r gets small enough to violate the inequality.

 

[Fredrik]

I'm not 100% sure of this, but I've heard that it is possible prove the existence of such topology in \mathcal{D}, that the convergence \psi_n\to\psi in that topology is equivalent with this definition that you described here. When \mathcal{D} is equipped with such topology, then the standard collection of distributions \mathcal{D}^* can be defined according to the definition of topological dual.

I don't think the construction of such a topology is very difficult. It would be harder to prove that the result is equivalent to what we already have, but even that looks doable. (I'm not sure I care enough to give it a try though ).

We can e.g. define a subset of \mathcal D to be open if it can be expressed as T^{-1}(U) where U is an open subset of the complex numbers, and T is continuous in the sense defined above.

Alternatively, I think we can define a subset E of \mathcal D to be open if for every sequence in \mathcal D that converges to a a point in E (in the sense defined above), E contains all but a finite number of members of the sequence.

I'm guessing that these definitions are adequate (in the sense that they both define a topology on \mathcal D), and equivalent (in the sense that those topologies are the same), but I haven't made any attempt to prove it or disprove it.

 

[strangerep]

We can e.g. define a subset of \mathcal D to be open if it can be expressed as T^{-1}(U) where U is an open subset of the complex numbers, and T is continuous in the sense defined above.

Yes, that defines the "T-weak" topology on \mathcal D. (Weak topologies are
defined implicitly/indirectly by a demand that a certain set of functions be continuous
under it.)

Alternatively, I think we can define a subset E of \mathcal D to be open if for every sequence in \mathcal D that converges to a a point in E (in the sense defined above), E contains all but a finite number of members of the sequence.

I don't follow this. If you haven't yet specifed a topology on \mathcal D
then there's no meaning in the statement that a sequence of elements of
\mathcal D "converges". But maybe I misunderstood and you
meant something else?

 

[strangerep]

I was wondering, how would I build the RHS for the hydrogen atom ? The hamiltonian for the H atom has a mixed spectrum and I've never seen an application of the RHS formalism for an operator with mixed spectrum.

The discrete part of the spectrum doesn't pose the same difficulties as encountered
with the continuous part. One has a resolution of unity as usual, consisting of an
integral over the continuous part, plus a sum over the discrete part. Rigged Hilbert
space and the nuclear spectral theorem give meaning to the continuous part.
(Not sure whether that's what you were asking, though.)

BTW, for anyone who's still a bit perplexed about the role of all this
rigged Hilbert space stuff, I noticed this pedagogical introductory paper:

Rafael de la Madrid,
"The role of the rigged Hilbert space in Quantum Mechanics",
Available as: quant-ph/0502053

It's written in a very physicist-friendly way, using the example of a rectangular
potential barrier to make everything concrete. It also explains the connections
between the bras and kets of rigged Hilbert space theory and distributions
quite clearly (while minimzing the heavy pure math that probably turns some
people off).

 

[Fredrik]

I don't follow this. If you haven't yet specifed a topology on \mathcal D
then there's no meaning in the statement that a sequence of elements of
\mathcal D "converges". But maybe I misunderstood and you
meant something else?

You understood me right. We define convergence of sequences first, and use that to define the topology. Convergence is defined as in post #35. I got that definition and the idea that it defines a topology on the test function space from the Wikipedia article on distributions. This is a direct link to the relevant section. Note in particular the sentence "It can be given a topology by defining the limit of a sequence of elements of D(U)." Unfortunately, Wikipedia doesn't say how the definition of convergence defines a topology. That's why I came up with those two guesses about how it can be done.

 

[Fredrik]

BTW, for anyone who's still a bit perplexed about the role of all this
rigged Hilbert space stuff, I noticed this pedagogical introductory paper:

Rafael de la Madrid,
"The role of the rigged Hilbert space in Quantum Mechanics",
Available as: quant-ph/0502053

Thanks. I have read the first 9 pages now, and I'm very pleased with it so far. I'm definitely going to read the rest later.

 

[Fredrik]

I found a review of Maurin's books, and the reviewer is complaining a lot about how careless Maurin's presentations are. Link. I'm guessing it would be better to try to find the original articles than to read his books.

 

[DarMM]

For QFT in infinite-dimensions, even the rigged Hilbert space is not big enough.
The singular behaviours arising from the interaction terms in the Hamiltonian
become far more pathological than mere delta distributions.

Interesting...and strange. The strange part is that we can actually make predictions in spite of all of this.

I thought I'd add a bit to this since it's very interesting.

As you know the Hilbert Space for 1 particle spin-0 QM is L^2(\mathbb R^3, dx). Where dx is the Lesbesgue measure.

\mathbb R^3[/tex] coming from the fact that a particle can occupy any point in three dimensional space. So the set of all points \mathbb R^3[/tex] is the classical configuration space.&lt;br /&gt; &lt;br /&gt; For Quantum Field Theory the classical configurations are the set of all possible field configurations. This set turns out to be \mathcal{D}^*(\mathbb R^3), the space of distributions. So the Hilbert space of QFT is:&lt;br /&gt; L^2(\mathcal{D}^*(\mathbb R^3), d\nu), the space of square integrable functions over the space of distributions with respect to some measure d\nu. A free QFT and an interacting QFT differ by their choice of d\nu. &lt;br /&gt; Quantum Fields \phi(x) are then objects that when integrated against a function f(x) give an object \int{\phi(x)f(x)}dx = \phi(f), which is an unbounded operator on L^2(\mathcal{D}^*(\mathbb R^3), d\nu).&lt;br /&gt; &lt;br /&gt; Who said rigorous QFT was hard?