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

In summary: Omega \subset H \subset \Omega^*_1Note that a vector v\in H is also a linear functional, so youcan think of H as being a subspace of its dual, and thenthe RHS relation becomes\Omega \subset H \subset H^*,where H^* is the topological dual, and is generated by the"bra" vectors, \langle \psi |. Also note thatthe RHS relation is the same as saying that \Omega is the"topological dual" of H.From this general starting point, you can then move on toformalisms that use all sorts of unbounded operators asobservables, with domain issues and all
  • #71
Fredrik said:
Now Wigner's theorem about symmetries says that given a simply connected symmetry group G (i.e. a group of functions that map the set of rays of the Hilbert space H bijectively onto itself), we can find a map U:G→GL(H) such that U(g) is either linear and unitary for all g in G or antilinear and antiunitary for all g in G, and U(g)U(h)=U(gh) for all g,h in G.

<Note for moderator> I revive this thread because I currently have in interest in group theory and its applications to QM.

Wigner's theorem does not refer to group theory and has nothing to do with <simple connectedness>. For the original form, please, see Wigner's book in the English translation of 1959, Section 20 and especially its appendix (pp. 220 until 236 - English 1959 translation from original German).

The modern form appears in several sources. I pick the version from Thaller's book which is in the attachment to this post.

Now, how can we formulate quantum mechanics from a symmetry-based approach ?
I may call this the Wigner-Weyl formulation of quantum mechanics. The basic postulate is this:

If the system under quantization is described at a classical level by an irreducible set of observables and whose classical equations of motion are invariant under a set of continuous transformations called symmetries which form a Lie group G, then the set of all possible representatives of quantum states is given by an infinite-dimensional, complex , separable Hilbert space on which the operators of a linear unitary* irreducbile representation of a group G' act.

Notes:
1. According to the laws of classical mechanics in the Newtonian formulation, a possible group of symmetries for them is given by the full Galilei group (the specially-relativistic correspondent of it is the full Poincare group). Apparently, this is not the largest symmetry group of Newton's equations.
2. For G a connected Lie-group, G' is the universal covering group of a non-trivial central extension of G. (example: the set of all proper classical rotations form a group isomorphic to SO(3) which is connected and path-connected).
3. If G is a non-connected Lie-group, then G' is the universal covering group of a non-trivial central extension of the connected component of G (example: the set of all Lorentz transformations forms a group isomorphic to O(1,3). The group O(1,3) has 4 distinct components, thus being disconnected).
4. G can admit no non-trivial central extensions. For G simply-connected, the classes of inequivalent non-trivial central extensions of G are bijectively related to the classes of inequivalent non-trivial central extensions of Lie(G). (an example of group with non-trivial central extensions is the full Galilei group of classical mechanics. An example of group with only trivial central extensions: the restricted Poincare group (= the component of the full Poincare group connected to identity).
5. If G is simply-connected, its universal covering group is isomorphic to G. (example, SU(2) and SL(2,C)).
6. If Psi is a unit ray of the Hilbert space, any modulus one vector in Psi is called a representative of a ray Psi (representative of a quantum state Psi).
7.*: Discrete symmetry transformations such as temporal inversion are handled based on Wigner's theorem using antiunitary operators.
8. The fundamental mathematical theorem underlying the postulate is the main result of Bargmann's famous 1954 article (transparent or not in the article): <Let G be a connected Lie group. Then its (projective) representations on a projective Hilbert space are in bijective correspondence with the vector unitary representations of G' (described above in notes 2-->5 and mentioned in the postulate)>.
9. For an exact symmetry, the Hamilton operator must be a central element in Lie(G') (actually a representation of Lie(G') through essentially self-adjoint operators on the Hilbert space of the quantum system).

Another comment to which I don't have a reference right now (but hope to have in the future) is this:
10. The postulate can be extended for rigged Hilbert spaces.

Without the sources for 10, I can only speculate on the identification of the G@rding domain of Lie(G') and the <small space> in the Gelfand triple.

I'm in search of mathematical proofs of the statements in notes 1--> 10 and any other theorem of group theory (harmonic analysis) which can be used in a symmetry-approach to QM. For this I've opened a thread on March 19th in the <Algebra> subforum on a particular theorem which I quote here as well (No answer, no group theorist there, I guess).

<Let G be a connected and simply-connected Lie group. Then all the linear irreducible representations of G on a TVS are faithful (= single-valued = true representations = representation morphism is injective)>.

In the meantime, I found a proof, but for it to make sense I first need to find out a rigorous definition of <multi-valued> representation of a Lie group. I've searched the literature and came up with this one: "A multi-valued linear representation of a multiple-connected and connected Lie group G on a TVS is a linear representation of the universal covering group of G". But I'm dissatisfied, for it seems to assume the theorem already: it doesn't define a multi-valued representation for a simply-connected group (which doesn't exist as specified by the theorem). So I can't use it to prove the theorem.

Another definition would be then: A multi-valued linear representation of a connected Lie group is defined through the following property: there exist at least two distinct elements of G, a and b, such as U(a)=U(b).

The proof I was seeking is given below for SU(2) (adaptation from a text by Elie Cartan) but applies to all G's in the hypothesis (if G is non-compact, then the matrices mentioned in the text are infinite dimensional, if one insists on unitarity).

<SU(2) is compact, then all its irreducible linear representations are finite-dimensional and equivalent to unitary ones. Let's assume SU(2) had multi-valued representations {U} (whatever that means, see the issue with the definition above) and for g(t) a continuous curve in SU(2), U(g(t)) is a continuous curve in Aut(V) (V is the TVS of the multi-valued representation). Even in the absence of unitarity of U, the set of all U's in Aut(V) is imposed to have a structure of a topological space. (If U are unitary operators, then the set of all U's in Aut(V) is a Banach space in the strong topology). Following the continuous variation of the representing matrix U_ij as the point in the group space describes a suitable closed contour starting and finishing at some origin, the matrix would start as the unit matrix and end as a different matrix. Continuously deforming the contour in Aut(V), the final matrix will stay the same, i.e. different than the unit matrix. But SU(2) is simply-connected, so any closed contour in it can be continuously deformed to a point. Using the continuity of the representation morphism, the representation matrices would coincide, thus a contradiction>.

This proof is crystal clear to me and leads me to think that multi-valued representations of connected Lie-groups indeed exist iff the manifold of a connected Lie-group is multiple-connected when seen as a topological space.

So one can go further and issue a corollary:
<Let G be a connected, m-connected Lie group. Then all its irreducible linear representations in a TVS are at most m-valued>.
 

Attachments

  • Scan-Thaller.jpg
    Scan-Thaller.jpg
    30 KB · Views: 361
Last edited:
Physics news on Phys.org
  • #72
I can't edit on my post, so I'll have to quote it.

dextercioby said:
1. According to the laws of classical mechanics in the Newtonian formulation, a possible group of symmetries for them is given by the full Galilei group (the specially-relativistic correspondent of it is the full Poincare group). Apparently, this is not the largest symmetry group of Newton's equations.

Indeed, the symmetry group of Newton's equations is larger than the full Galilei group, it is a 13-parameter Lie group (connected or not, I will investigate, definitely the connected component of the identity is non simply-connected as it would necessarily contain a subgroup isomorphic to SO(3)).

dextercioby said:
Another comment to which I don't have a reference right now (but hope to have in the future) is this:
10. The postulate can be extended for rigged Hilbert spaces.

Without the sources for 10, I can only speculate on the identification of the G@rding domain of Lie(G') and the <small space> in the Gelfand triple.

I promise to come back on this issue, as I consider it to be of upmost importance.

For the interested reader, if any, I will also make some comments on previous posts in this thread, hopefully to enchance its content, so that it would turn out in some sort of learning material.
 
  • #73
dextercioby said:
Indeed, the symmetry group of Newton's equations is larger than the full Galilei group, it is a 13-parameter Lie group (connected or not, I will investigate, definitely the connected component of the identity is non simply-connected as it would necessarily contain a subgroup isomorphic to SO(3)).

Is there any particular reason you start with the Newtonian formulation rather than
a more modern Hamiltonian formulation? For the latter, there's a large body of work
on symmetries, starting from the basic inhomogeneous symplectic group, and
progressing to more sophisticated stuff. Have you tried the following book?

Marsden & Ratiu,
"Introduction to Mechanics & Symmetry,
Springer 1991, (2nd Ed), ISBN 0-387-98643-X)

Armed with such insights, the relation between classical and quantum
becomes much more transparent in this setting. (Quantum commutators
as deformations of classical brackets.)

It's more powerful to try and quantize in terms of a full dynamical group
rather than just a symmetry group.

Arnold Neumaier treats some of this in his book, but I'm sure he'll have
more to say on this when he has time... :-)

Another comment to which I don't have a reference right now (but hope to
have in the future) is this:
10. The postulate can be extended for rigged Hilbert spaces.

Try this:

S. Wickramasekara & A. Bohm,
"Symmetry Representations in the Rigged
Hilbert Space Formulation of Quantum Mechanics",
(Available as math-ph/0302018)

Wickramasekara & Bohm said:
Abstract:

We discuss some basic properties of Lie group representations in
rigged Hilbert spaces. In particular, we show that a differentiable
representation in a rigged Hilbert space may be obtained as the
projective limit of a family of continuous representations in a nested scale
of Hilbert spaces. We also construct a couple of examples illustrative
of the key features of group representations in rigged Hilbert spaces.
Finally, we establish a simple criterion for the integrability of an
operator Lie algebra in a rigged Hilbert space.


dextercioby said:
Without the sources for 10, I can only speculate on the identification of the
G@rding domain of Lie(G') and the <small space> in the Gelfand triple.

That's the way RHS usually works; -- find the largest space for which all quantities
of interest are defined everywhere (i.e., the <small space> in the Gelfand triple),
then take its closure and dual to get the other spaces in the triple. More sophisticated
treatments involve scales of Hilbert spaces, or partial-inner product (PIP) spaces,
but I'm not yet convinced that the boil down to much more than good old RHS (except
of course for the greater rigor obtained thereby).
 
  • #74
strangerep said:
Marsden & Ratiu,
"Introduction to Mechanics & Symmetry,
Springer 1991, (2nd Ed), ISBN 0-387-98643-X)

Armed with such insights, the relation between classical and quantum
becomes much more transparent in this setting. (Quantum commutators
as deformations of classical brackets.)

It's more powerful to try and quantize in terms of a full dynamical group
rather than just a symmetry group.

Thanks for the input. I will try to find the book and see if one indeed obtains more things from such a perspective than the symmetry-based approach to QM offers.

strangerep said:
S. Wickramasekara & A. Bohm,
"Symmetry Representations in the Rigged
Hilbert Space Formulation of Quantum Mechanics",
(Available as math-ph/0302018)

I knew about this article, but I had some older articles in mind.
 
  • #75
dextercioby said:
[...] the symmetry group of Newton's equations is larger than the full Galilei group, it is a 13-parameter Lie group (connected or not, I will investigate, definitely the connected component of the identity is non simply-connected as it would necessarily contain a subgroup isomorphic to SO(3)).

Since you recently mentioned these posts elsewhere, I'll re-revive this unfinished thread...

Did you finish investigating the larger group? If so, what did you find?
 
  • #76
strangerep said:
Since you recently mentioned these posts elsewhere, I'll re-revive this unfinished thread...

Did you finish investigating the larger group? If so, what did you find?

Yes for the first question, for the second: arxiv.org article under <math-ph/0102011v2> says it better than me. (I don't have access, however, to Niederer's article in HPA. It would be nice to see the original argument, too.)
 
  • #77
dextercioby said:
[...] arxiv.org article under <math-ph/0102011v2>[...]

Thanks. I remembered the group vaguely in the context of the
Schrodinger equation, but I'd forgotten the name "Niederer".
 
<h2>1. What are unbounded operators in non-relativistic QM of one spin-0 particle?</h2><p>Unbounded operators are mathematical objects used in quantum mechanics to represent physical observables, such as position, momentum, and energy. In non-relativistic QM of one spin-0 particle, these operators are not limited to a specific range of values and can take on any real number.</p><h2>2. How are unbounded operators different from bounded operators?</h2><p>Unlike bounded operators, which have a finite range of values, unbounded operators have an infinite range. This means that the eigenvalues of unbounded operators are not limited to a specific set of values and can take on any real number.</p><h2>3. Can unbounded operators have discrete and continuous spectra?</h2><p>Yes, unbounded operators can have both discrete and continuous spectra. The discrete spectrum consists of a countable set of eigenvalues, while the continuous spectrum consists of an uncountable set of values.</p><h2>4. How are unbounded operators related to the uncertainty principle?</h2><p>The uncertainty principle states that certain pairs of physical observables, such as position and momentum, cannot be known simultaneously with arbitrary precision. Unbounded operators play a crucial role in the formulation of the uncertainty principle, as they represent these observables and their corresponding uncertainties.</p><h2>5. Can unbounded operators be used to describe systems with multiple particles?</h2><p>Yes, unbounded operators can be used to describe systems with multiple particles. In this case, the operators act on the combined Hilbert space of all the particles, and their eigenvalues correspond to the collective properties of the system, such as total energy and momentum.</p>

1. What are unbounded operators in non-relativistic QM of one spin-0 particle?

Unbounded operators are mathematical objects used in quantum mechanics to represent physical observables, such as position, momentum, and energy. In non-relativistic QM of one spin-0 particle, these operators are not limited to a specific range of values and can take on any real number.

2. How are unbounded operators different from bounded operators?

Unlike bounded operators, which have a finite range of values, unbounded operators have an infinite range. This means that the eigenvalues of unbounded operators are not limited to a specific set of values and can take on any real number.

3. Can unbounded operators have discrete and continuous spectra?

Yes, unbounded operators can have both discrete and continuous spectra. The discrete spectrum consists of a countable set of eigenvalues, while the continuous spectrum consists of an uncountable set of values.

4. How are unbounded operators related to the uncertainty principle?

The uncertainty principle states that certain pairs of physical observables, such as position and momentum, cannot be known simultaneously with arbitrary precision. Unbounded operators play a crucial role in the formulation of the uncertainty principle, as they represent these observables and their corresponding uncertainties.

5. Can unbounded operators be used to describe systems with multiple particles?

Yes, unbounded operators can be used to describe systems with multiple particles. In this case, the operators act on the combined Hilbert space of all the particles, and their eigenvalues correspond to the collective properties of the system, such as total energy and momentum.

Similar threads

  • Quantum Physics
Replies
7
Views
548
  • Quantum Physics
Replies
24
Views
594
  • Quantum Physics
Replies
7
Views
1K
Replies
0
Views
446
  • Quantum Physics
Replies
1
Views
876
  • Quantum Physics
Replies
2
Views
923
  • Quantum Physics
Replies
4
Views
961
Replies
9
Views
918
  • Quantum Physics
Replies
11
Views
877
Back
Top