- #1

marcus

Science Advisor

Gold Member

Dearly Missed

- 24,770

- 790

Several papers have appeared in the last few months suggesting a development in the quantum theory of general relativity which is analogous to the Stone-von Neumann theorem of 1931.

The clearest and most representative of the bunch is probably the Okolow-Lewandowski paper dated February 2003:

http://arxiv.org/gr-qc/0302059 [Broken]

this thread is intended to describe the S-vN theorem and its importance in connecting Heisenberg and Schroedinger pictures and to draw the analogy with current work in quantum GR.

[Lubos Motl kindly pointed out an error in the above link where I had written 0202... rather than 0302..., and I have corrected the typo.]

I'll begin with a brief sketch of the S-vN theorem.

Then, about 10 posts further along the thread, there will be some discussion of the Okolow-Lewandowski paper, the work of Sahlmann, and mention of a forthcoming paper by the four principal researchers involved in this development: Okolow-Lewandowski-Sahlmann-Thiemann. this last paper is referred to by O-L but is not yet in the preprint archive.

Eric Weissteins MathWorld says this about the S-vN theorem:

<<Stone-von Neumann theorem---A theorem which specifies the structure of the generic unitary representation of the Weyl relations and thus establishes the equivalence of Heisenberg's matrix mechanics and Schrödinger's wave mechanics formulations of quantum mechanics in Euclidean space.

Neumann, J. von. "Die Eindeutigkeit der Schrödingerschen Operationen." Math. Ann. 104, 570-578, 1931.>>

Weisstein omits to reference M.H.Stone, "Linear Transformations in Hilbert Space and Their Applications to Analysis," 1932

Here is something Baez said about it:

<<... the Stone-von Neumann theorem I know and love. The version I know says: all strongly continuous irreducible unitary representations of the Weyl relations for a finite-dimensional symplectic vector space are unitarily equivalent.

This is the theorem that guarantees the equivalence between Schrodinger's and Heisenberg's approaches to quantum mechanics.

Recall that the Weyl relations are the exponentiated version of Heisenberg's "canonical commutation relations". We exponentiate them to turn the nasty unbounded self-adjoint P's and Q's into nice one-parameter groups of unitary operators exp(isP) and exp(isQ), which are technically easier to deal with. The Stone-von Neumann theorem doesn't hold for infinite-dimensional

symplectic vector spaces, which is why quantum field theory is trickier than quantum mechanics....>>

http://www.lns.cornell.edu/spr/1999-04/msg0016106.html

Von Neumann was born in Budapest Hungary in 1903. His name was Janos. This is maybe the most famous theorem in Representation Theory---there is essentially only one irred representation of the Weyl algebra.

Heisenberg formulated QM in terms that amounted to a matrix algebra. The important objects were operators called "observables". The Heisenberg picture did not have "wave functions". There was some technical trouble with unbounded operators, which were tamed by exponentiating them--the Weyl algebra resulted.

The Schroedinger picture had wave functions from which you could build a Hilbert space of quantum states. You could translate the (Heisenberg) Weyl algebra into operators on that Hilbert space. But maybe there was some ambiguity about this! Maybe there were several essentially different ways of doing that?

No, von Neumann showed that there is essentially only one way to set the Weyl algebra up in business operating on the Schroedinger wave functions or quantum state space. There was only one way to "represent" the algebra as operators on the Hilbert space. So even though matrices dont LOOK like wave functions the two were giving us the same picture of the world.

Something interesting happened this year that was reminiscent of von Neumann. Jerzy Lewandowski drew the analogy. Hanno Sahlmann used the same exponentiation trick to tame some unbounded LQG observables to get an algebra Lewandowski called the "Sahlmann algebra" and proved a uniqueness theorem about its representations. the Sahlmann algebra is just big enough to contain the holonomies on a manifold and the (tamed) electric fluxes and all its reasonable representations are equivalent to one called the Ashtekar-Lewandowski representation. the theorem is on page 16 of

http://arxiv.org/gr-qc/0302090 [Broken]

(this is one that sA mentioned he came across back in February or so when it came out, maybe I should try to read it?)

The clearest and most representative of the bunch is probably the Okolow-Lewandowski paper dated February 2003:

http://arxiv.org/gr-qc/0302059 [Broken]

this thread is intended to describe the S-vN theorem and its importance in connecting Heisenberg and Schroedinger pictures and to draw the analogy with current work in quantum GR.

[Lubos Motl kindly pointed out an error in the above link where I had written 0202... rather than 0302..., and I have corrected the typo.]

I'll begin with a brief sketch of the S-vN theorem.

Then, about 10 posts further along the thread, there will be some discussion of the Okolow-Lewandowski paper, the work of Sahlmann, and mention of a forthcoming paper by the four principal researchers involved in this development: Okolow-Lewandowski-Sahlmann-Thiemann. this last paper is referred to by O-L but is not yet in the preprint archive.

Eric Weissteins MathWorld says this about the S-vN theorem:

<<Stone-von Neumann theorem---A theorem which specifies the structure of the generic unitary representation of the Weyl relations and thus establishes the equivalence of Heisenberg's matrix mechanics and Schrödinger's wave mechanics formulations of quantum mechanics in Euclidean space.

Neumann, J. von. "Die Eindeutigkeit der Schrödingerschen Operationen." Math. Ann. 104, 570-578, 1931.>>

Weisstein omits to reference M.H.Stone, "Linear Transformations in Hilbert Space and Their Applications to Analysis," 1932

Here is something Baez said about it:

<<... the Stone-von Neumann theorem I know and love. The version I know says: all strongly continuous irreducible unitary representations of the Weyl relations for a finite-dimensional symplectic vector space are unitarily equivalent.

This is the theorem that guarantees the equivalence between Schrodinger's and Heisenberg's approaches to quantum mechanics.

Recall that the Weyl relations are the exponentiated version of Heisenberg's "canonical commutation relations". We exponentiate them to turn the nasty unbounded self-adjoint P's and Q's into nice one-parameter groups of unitary operators exp(isP) and exp(isQ), which are technically easier to deal with. The Stone-von Neumann theorem doesn't hold for infinite-dimensional

symplectic vector spaces, which is why quantum field theory is trickier than quantum mechanics....>>

http://www.lns.cornell.edu/spr/1999-04/msg0016106.html

Von Neumann was born in Budapest Hungary in 1903. His name was Janos. This is maybe the most famous theorem in Representation Theory---there is essentially only one irred representation of the Weyl algebra.

Heisenberg formulated QM in terms that amounted to a matrix algebra. The important objects were operators called "observables". The Heisenberg picture did not have "wave functions". There was some technical trouble with unbounded operators, which were tamed by exponentiating them--the Weyl algebra resulted.

The Schroedinger picture had wave functions from which you could build a Hilbert space of quantum states. You could translate the (Heisenberg) Weyl algebra into operators on that Hilbert space. But maybe there was some ambiguity about this! Maybe there were several essentially different ways of doing that?

No, von Neumann showed that there is essentially only one way to set the Weyl algebra up in business operating on the Schroedinger wave functions or quantum state space. There was only one way to "represent" the algebra as operators on the Hilbert space. So even though matrices dont LOOK like wave functions the two were giving us the same picture of the world.

Something interesting happened this year that was reminiscent of von Neumann. Jerzy Lewandowski drew the analogy. Hanno Sahlmann used the same exponentiation trick to tame some unbounded LQG observables to get an algebra Lewandowski called the "Sahlmann algebra" and proved a uniqueness theorem about its representations. the Sahlmann algebra is just big enough to contain the holonomies on a manifold and the (tamed) electric fluxes and all its reasonable representations are equivalent to one called the Ashtekar-Lewandowski representation. the theorem is on page 16 of

http://arxiv.org/gr-qc/0302090 [Broken]

(this is one that sA mentioned he came across back in February or so when it came out, maybe I should try to read it?)

Last edited by a moderator: