Representations on Hilbert space

[Jip]

Hello,
I have some troubles understanding Hilbert representations for the standard free quantum particle

On the one hand, we can represent Heisenberg algebra [Xi,Pj]= i delta ij on the space of square integrable functions on, say, R^3, with the X operator represented as multiplication and P operator as i times the gradient.
On the other hand, we also represent the Galilean Lie group on the same Hilbert space

So my question is : is that obvious that these two representations are "compatible"? Are they any constraints that one may derive by asking that the Hilbert must be the representation of both Lie Heisenberg Lie algebra + Galilean Lie algebra (in particular because P is both the momentum in [X,P] = i, and the generator of translations of the galilean group)?

For instance, I could choose to represent QM on a finite interval; then it would break translation invariance. So which come first? Do we build the Hilbert as a representation of Heisenberg, and then impose some symmetry, or construct the Hilbert space as a representation of symmetry group and then define X via [X,P]= i ?
Maybe this is not very clear, for I am indeed puzzled by the fact that P appears in both algebras.

Thanks for any help!

 

[dextercioby]

This is by far the most interesting question I've seen here since 2008, perhaps. :) (joke!).

Let's get down to serious business.
Your question is really about the connection between the standard Dirac-von Neumann formulation in terms of Hilbert spaces, linear operators and state vectors and the Weyl-Wigner symmetry-groups approach (this was later extended by A.O. Barut in the 1960s through the introduction of the so-called "dynamical algebras") . Are these 2 completely equivalent? (from a rigorous mathematical standpoint)
Yes, they are and the proof in the case of free n-D particles is an interesting array of theorems and proofs in functional (harmonic) analysis. If you decide to consider systems in interaction, then indeed you break one of the symmetries of the Galilean space-time (rotation or translation), but the equivalence is still there. The momentum operator is a derived concept in the W-W formulation, which exists as a conserved obserable only for translation-invariant systems, while the Hamiltonian operator is fundamental, for there's no break to the time invariance.

If you study for example the famous Stone-von Neumann uniqueness theorem, then you need to start with the basic assumptions behind it, one of them being: is the system allowed to have a finite classical configurations space or not? Physically, motion should be unbounded, right? This piece of info tells you then what Hilbert space to look for. So the S-vN thm. is one of the fundamental results of the Dirac-vN formulation. You can easily connect it with the group representation theory, once you consider the abstract Heisenberg group and its unitary irreds on a certain separable Hilbert space.

So nothing really comes first of the 2, you simply have to make a choice: you either start off with a classical theory based on the symmetries of the Hamiltonian, or start with the fundamental Poisson brackets on the phase space + a certain quantization scheme (the so-called Dirac or canonical quantization prescription) to lead you to the quantum Hamiltonian, hence dynamics via the Schrödinger equation etc.

Imagine you want to "solve" the Hydrogen atom in its simplest formulation (no special relativity, no spin, infinite mass nucleus). If I'm telling you the 2 formulations are completely equivalent, how would you prove it?

 

[aleazk]

You can find the answer to this interesting question in the book "Geometry of Quantum Theory", by V.S.Varadarajan.

I will summarize some of the points.

Suppose we have some configuration manifold $M$(i.e., "space") and we want to study the localization properties of a quantum system on this space. This means that the system will "show its position in this space", i.e., there's a quantum observable for the position of the system. If this observable exists, we say that the quantum system is localizable. In more precise terms, in the Hilbert space of the system there exists a PVM $P$ (proyector valued measure) over $M$ such that $P(E)$ represents the proposition "the position of the system lies in the Borel set $E\subseteq M$".

Now, suppose we have some Lie group $G$ acting on this configuration space. In classical phase space (the cotangent bundle of $M$), we know that this action $\alpha$ gives us observables called momentum observables associated to the this action (for the traslations, we get linear momentum; for the rotations, SO(3), we get angular momentum). Also in this classical phase space, this action induces an action on the classical propositions in an obvious way.

We want to implement something similar but in the quantum case. Without entering into the sophistications (like the definition of quantum symmetries, etc.), the result is that the action in the space of quantum propositions is implemented by taking a proyective unitary representation $U$ of the group $G$ in the Hilbert space that satisfies the Imprimitivity condition (for the sake of simplicity, I will assume it's actually a true representation; a lot of mathematical machinery is devoted to this aspect, though, and it's a pretty important and non-trivial aspect):

$$U_{g}P(E)U_{g}^{-1}=P(\alpha E)$$

The infinitesimal generators of the representation are self-adjoint operators, i.e., quantum observables and we will give to them the same physical interpretation as their classical counterparts. Notice that, as in the classical case, this induces the group's Lie algebra in the space of observables.

So, our definition of quantum localization in a manifold $M$ led us to consider a mathematical object called System of Imprimitivity (which is the name given to all the structure I just mentioned). Note that we are looking for some pretty specific thing: a Hilbert space, a representation of the group, and a PVM such that the Imprimitivity condition holds. This latter condition makes restrictions on the possible concrete Hilbert spaces and concrete representations of the group that we are allowed to take if we want to build a concrete system of imprimitivity, i.e., a concrete Hilbert space eqquiped with a concrete representation of the group may or may not admit a PVM so that the imprimitivity condition holds.

Let us consider a simple example: $M$ is the real line, $G$ the group of traslations and $\alpha$ the usual action of this group in the real line. The infinitesimal generator of the representation will be the quantum linear momentum. So, we have:

$$\hat{\mathrm{U}}_{x}\hat{\mathrm{P}}(E)\hat{\mathrm{U}}_{x}^{-1}=\hat{\mathrm{P}}(E-x)$$

$$\hat{\mathrm{U}}_{x}=e^{2\pi ix\hat{p}}$$

$$\hat{\mathrm{V}}_{y}=e^{2\pi iy\hat{q}}\doteq\intop_{\mathbb{R}}e^{2\pi iyz}d\hat{\mathrm{P}}(z)$$

$$\hat{\mathrm{U}}_{x}\hat{\mathrm{V}}_{y}\hat{\mathrm{U}}_{x}^{-1}=\intop_{\mathbb{R}}e^{2\pi iyz}d\left(\hat{\mathrm{U}}_{x}\hat{\mathrm{P}}(z)\hat{\mathrm{U}}_{x}^{-1}\right)=\intop_{\mathbb{R}}e^{2\pi iyz}d\hat{\mathrm{P}}(z-x)$$

$$\hat{\mathrm{U}}_{x}\hat{\mathrm{V}}_{y}\hat{\mathrm{U}}_{x}^{-1}=\intop_{\mathbb{R}}e^{2\pi iy(z'+x)}d\hat{\mathrm{P}}(z')=e^{2\pi iyx}\intop_{\mathbb{R}}e^{2\pi iyz'}d\hat{\mathrm{P}}(z')=e^{2\pi iyx}\hat{\mathrm{V}}_{y}$$

$$\hat{\mathrm{U}}_{x}\hat{\mathrm{V}}_{y}=e^{2\pi iyx}\hat{\mathrm{V}}_{y}\hat{\mathrm{U}}_{x}$$

Which are the Weyl relations. So, with our definition of quantum localization, we recover the usual definitions and the CCR (the Heisenberg algebra) arising as the generators of the Weyl relations.

This is pretty nice, but how many concrete systems that satisfy the Weyl relations exist? The answer is given by a very powerful theorem called the Imprimitivity Theorem. This theorem allows us to determine and classify all of the possible different systems of imprimitivity. And in this way, it allows us to solve the problem of quantum localization: we defined our notion of quantum localization and we classified all of the possible concrete systems that satisfy this definition.

In the example above, we, of course, get the usual Stone-von Neumann Theorem about the uniqueness of the CCR as stated in terms of the Weyl relations. If we now consider $G$ equal to the rotations plus translations, considerations related to the proyective representation thingy lead us to consider the covering of this group. The Imprimitivity Theorem now says that the family of non-equivalent systems is parametrized by the spin of the system and we actually get the usual results of non-relativistic QM (i.e., the CCR plus orbital angular momentum plus spin).

With this, the study of quantum localization ends.

Notice that we have been dealing only with space and its symmetries, what about spacetime? Let's forget about all of the above (localization, etc.) for a moment and adopt the following point of view for QM: fix a spacetime and take its special relativity group (for Galilean spacetime, it's the Galilei group; for Minkowski spacetime, it's the Poincaré group). We will call a quantum system to be specially covariant if the special relativity group can be implemented as quantum symmetries in the Hilbert space (notice that I take the groups as the fundamental things, representations of the algebra only arise through the generators of the group representations). Usually, we get some proyective unitary representation of the group. For the case of the Poincaré group, one passes to the universal covering because true representations of this group characterize what we need (there are theorems for this, etc., I don't want to enter into these details now). For the Galilei group one also takes a central extention, the charge is the mass, etc.

The infinitesimal generators are interpreted as momentum too, but now we have a new one: energy, associated to the time traslations. But it's the hamiltonian for the free particle only.

The problem is now to classify all of the possible concrete covariant systems. The relativity groups are usually semidirect products of an abelian normal subgroup (the spacetime traslations) and a closed subgroup (e.g., the proper Lorentz group for the case of the Poincaré group). Incredibly enough, one can obtain from the Imprimitivity Theorem a corollary called the Mackey Machine. The Mackey Machine deals precisely with the representation theory of these type of semidirect product groups. So, we kill two unrelated birds (quantum localization in space and spacetime covariant systems) with one mathematical shot!

The Mackey Machine can be used to successfully determine and classify all of the representations of both the Galilei group and the Poincaré group.

Let's restrict to Galilei. So, we have two different lists in our hands: 1) the list of all of the possible concrete localizable quantum systems (as per our definition of localization via systems of imprimitivity); 2) the list of all of the possible concrete specially or spacetime covariant quantum systems (as per our definition via representations of the special relativity group as quantum symmetries).

And, in this way, we finally arrived to your question!: can we compare both lists? the answer is yes!

Take the relativity group. Now, this group has, as a subgroup, the group of symmetries of configuration space (understood as a hypersurface at constant time in spacetime). We can take this configuration space with its group of symmetries and perform the systems of imprimitivity construction, classification of allowed systems and final list. The list consists of concrete triples of the form: Hilbert space; representation of the group; PVM such that the imprimitivity condition holds. These systems define quantum localization on this particular configuration space.

Now take some concrete spacetime covariant system from the other list. It's a concrete Hilbert space equipped with a concrete representation of the relativity group. And now it comes the thing: restrict the representation of the relativity group to the subgroup of the configuration space. Suppose that we get in this way a representation of this subgroup. We get a pair: Hilbert space; representation of the subgroup. But the list in the previous paragraph is also of elements of this form! If we manage to find a pair in this list that is unitarily equivalent to the pair we just constructed from the spacetime covariant system, we could take the corresponding PVM (which integrated gives us the position operator), define it in the spacetime covariant system and get a system of imprimitivity, i.e., the spacetime covariant system would be localizable! If we can't find such a pair in the list, then this particular spacetime covariant system is not localizable (since the list exhausts the possible localizable systems).

For the case of the Galilei group, one easily sees that all of the spacetime covariant systems are localizable.

For the Poincaré group, in general only the massive systems are localizable. So, in a sense, you were right: there are cases in which the two things are not "compatible". And, even more, those cases have physical meaning: those systems are spacetime covariant but not localizable. Such systems actually exist in nature, e.g., the photons. The advantage of all of this framework I mentioned in this post (due to mathematician George Mackey, but which is based on the previous work of many people, like von Neumann, Wigner, etc.) is that it allows us to study localization, spacetime covariance, and when covariant systems are localizable and when they are not.

As an interesting mathematical note, the Weyl relations are equivalent to a representation of the Heisenberg group (the unique connected, simply connected Lie group with the Heisenberg algebra as Lie algebra). The Heisenberg group is a semidirect product of an abelian normal subgroup and a closed subgroup, so we can use the Mackey Machine to prove the Stone-von Neumann theorem too.

The actual details of all this can be tremendously technical, I have omitted almost everything in order to give you a bird eye view. I myself have only a partial grasp of some of the topics involved.

Some references (I learned these topics from these references):

-Geometry of Quantum Theory, by V.S.Varadarajan (it develops the full theory; both the mathematical theory and its physical application; to read and grasp this book in all of its profundity is an authentic tour de force, though; I only succeeded in partial readings of the most accessible sections, but found everything I understood to be very enlightening to my understanding of QM)

-A Course in Abstract Harmonic Analysis, by G.B.Folland (it's a math book for mathematicians; it develops the mathematical theory of systems of imprimitivity and the Mackey Machine; also covers usual topics like the Peter-Weyl theorem for compact groups, etc., a very nice book if you are interested in representation theory and affine topics)

-Foundations of Quantum Mechanics, by J.Jauch (a very gentle introduction to the advanced mathematical formulation of QM; the quantum lattice of propositions; a pedagogical introduction to the formulation of localization via systems of imprimitivity)

-Quantum field theory, a tourist guide for mathematicians, by G.B.Folland (it contains a very quick but effective introduction to the use of the Mackey Machine in the problem about obtaining the possible representations of the Poincaré group)

 

[strangerep]

On the one hand, we can represent Heisenberg algebra [Xi,Pj]= i delta ij on the space of square integrable functions on, say, R^3, with the X operator represented as multiplication and P operator as i times the gradient.
On the other hand, we also represent the Galilean Lie group on the same Hilbert space

So my question is : is that obvious that these two representations are "compatible"?

Since your question was about the "standard free quantum particle", I'll offer a simpler answer. In Ballentine, ch3, he shows how the position operator can be expressed in terms of the mass and the Galilean boost operator. In that sense, if one has a (unitary, irreducible) Galilean representation, one also has a rep of the usual CCRs. The fine print is that one is thereby working in the centrally extended Galilei group, and must stay in an irreducible rep so that the mass is a scalar constant. (A "standard free quantum particle" is essentially exactly that.)

In the relativistic case, if one is similarly in an irreducible rep, one can perform a contraction to the (centrally extended) Galilei group, and arrive at the CCRs the same way. (Can dig out references if anyone wants them.)

 

[DrDu]

If you decide to consider systems in interaction, then indeed you break one of the symmetries of the Galilean space-time (rotation or translation), but the equivalence is still there.

I don't see this. An interaction of the particles doesn't break Galilean symmetry (this is an important difference to Poincare, where only point interactions are allowed).

 

[Jip]

Thanks for your replies! Aleazk, this is a great, great answer. First time I hear about this stuff.
I see there is quite a lot to learn, indeed. And now I can come to my real business: deformation of Heisenberg algebra vs deformation of symmetries.
The physics litterature has come up with a generalized uncertainty principle Delta x Delta p > 1 + Delta p^2, which can come from generalized comutation relations :
[x_i,p_j] = f_ij (P_i)

Many papers available on arxiv. AFAIK however, the question of symmetry is barely adressed, and I think this is a mistake. So I'm trying to clarify the links between localization (indeed), the fact that X shall not be self adjoint in general, but only symmetric, possible space-time like symmetries via deformed Galilean group. As I see it, this imprimitivity stuff would be the only way to actually build the possible theories.

 

[strangerep]

Many papers available on arxiv. AFAIK however, the question of symmetry is barely adressed, [...]

I truly have no idea how you could reach that conclusion. Symmetries are ubiquitous in modern physics, from the advanced treatments of classical mechanics (cf. Marsden & Ratiu), to the foundations of QFT (cf. the early sections of Weinberg vol 1). E.g., Wigner realized many decades ago that the types of elementary particles could be classified according to the unitary irreducible representations of the Poincare group. (This is also explained in Weinberg vol 1.)

 

[bhobba]

Symmetries are ubiquitous in modern physics,

Indeed eg the importance and usefulness of Noethers Theorem - but that barely scratches the surface.

Thanks
Bill

 

[Jip]

I truly have no idea how you could reach that conclusion. Symmetries are ubiquitous in modern physics, from the advanced treatments of classical mechanics (cf. Marsden & Ratiu), to the foundations of QFT (cf. the early sections of Weinberg vol 1). E.g., Wigner realized many decades ago that the types of elementary particles could be classified according to the unitary irreducible representations of the Poincare group. (This is also explained in Weinberg vol 1.)

Sure! Don't misunderstand me, I was talking about the symmetries of deformed Quantum Mechanics according to the Generalized uncertainty principle, not standard QM :)