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Fredrik said:
<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>.
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