Analogy of Exponential Map for Jordan Algebras
$begingroup$
Today I gave a talk about this paper that constructs a Jordan algebra (more precisely, a JB algebra) to model (bounded) physical observables.
It cites this paper, that proves that every JB algebra $A$ has a uniquely determined closed Jordan ideal $J$ such that $A/J$ (again a JB algebra) can be realized as a subalgebra of the bounded self-adjoint operators on a Hilbert space.
Afterwards I got this awesome question:
Can one model the Schrödinger time-evolution using only JB algebra concepts?
The intuition behind the question is that: Suppose we could represent a JB algebra on a Hilbert space as above (i.e $J$ is trivial). Then, by Stone's theorem, every element of the algebra corresponds to a strongly continuous one-parameter unitary group which in turn corresponds to a continuous automorphism on the JB algebra.
The first correspondence is given by the exponential
begin{equation}
e^{ i H t }
end{equation}
for some self-adjoint operator $H$ on the Hilbert space corresponding to an element of the JB algebra and $t in mathbb{R}$.
This suggests that (maybe) one could expect to find a canonical mapping between elements of a JB algebra and its continuous one-parameter groups of automorphisms.
Is this true? Note that if $J$ is trivial, it has to be true. Is the nontriviality of $J$ a necessary condition for this to fail? Otherwise what is the proper obstruction?
I tried to google for answers, but I only found remotely related findings.
quantum-mechanics jordan-algebras
asked Nov 29 '18 at 18:20
ioloiolo
6011416
$endgroup$
add a comment |
$begingroup$
Today I gave a talk about this paper that constructs a Jordan algebra (more precisely, a JB algebra) to model (bounded) physical observables.
It cites this paper, that proves that every JB algebra $A$ has a uniquely determined closed Jordan ideal $J$ such that $A/J$ (again a JB algebra) can be realized as a subalgebra of the bounded self-adjoint operators on a Hilbert space.
Afterwards I got this awesome question:
Can one model the Schrödinger time-evolution using only JB algebra concepts?
The intuition behind the question is that: Suppose we could represent a JB algebra on a Hilbert space as above (i.e $J$ is trivial). Then, by Stone's theorem, every element of the algebra corresponds to a strongly continuous one-parameter unitary group which in turn corresponds to a continuous automorphism on the JB algebra.
The first correspondence is given by the exponential
begin{equation}
e^{ i H t }
end{equation}
for some self-adjoint operator $H$ on the Hilbert space corresponding to an element of the JB algebra and $t in mathbb{R}$.
This suggests that (maybe) one could expect to find a canonical mapping between elements of a JB algebra and its continuous one-parameter groups of automorphisms.
Is this true? Note that if $J$ is trivial, it has to be true. Is the nontriviality of $J$ a necessary condition for this to fail? Otherwise what is the proper obstruction?
I tried to google for answers, but I only found remotely related findings.
quantum-mechanics jordan-algebras
asked Nov 29 '18 at 18:20
ioloiolo
6011416
$endgroup$
add a comment |
$begingroup$
Today I gave a talk about this paper that constructs a Jordan algebra (more precisely, a JB algebra) to model (bounded) physical observables.
It cites this paper, that proves that every JB algebra $A$ has a uniquely determined closed Jordan ideal $J$ such that $A/J$ (again a JB algebra) can be realized as a subalgebra of the bounded self-adjoint operators on a Hilbert space.
Afterwards I got this awesome question:
Can one model the Schrödinger time-evolution using only JB algebra concepts?
The intuition behind the question is that: Suppose we could represent a JB algebra on a Hilbert space as above (i.e $J$ is trivial). Then, by Stone's theorem, every element of the algebra corresponds to a strongly continuous one-parameter unitary group which in turn corresponds to a continuous automorphism on the JB algebra.
The first correspondence is given by the exponential
begin{equation}
e^{ i H t }
end{equation}
for some self-adjoint operator $H$ on the Hilbert space corresponding to an element of the JB algebra and $t in mathbb{R}$.
This suggests that (maybe) one could expect to find a canonical mapping between elements of a JB algebra and its continuous one-parameter groups of automorphisms.
Is this true? Note that if $J$ is trivial, it has to be true. Is the nontriviality of $J$ a necessary condition for this to fail? Otherwise what is the proper obstruction?
I tried to google for answers, but I only found remotely related findings.
quantum-mechanics jordan-algebras
asked Nov 29 '18 at 18:20
ioloiolo
6011416
$endgroup$
Today I gave a talk about this paper that constructs a Jordan algebra (more precisely, a JB algebra) to model (bounded) physical observables.
It cites this paper, that proves that every JB algebra $A$ has a uniquely determined closed Jordan ideal $J$ such that $A/J$ (again a JB algebra) can be realized as a subalgebra of the bounded self-adjoint operators on a Hilbert space.
Afterwards I got this awesome question:
Can one model the Schrödinger time-evolution using only JB algebra concepts?
The intuition behind the question is that: Suppose we could represent a JB algebra on a Hilbert space as above (i.e $J$ is trivial). Then, by Stone's theorem, every element of the algebra corresponds to a strongly continuous one-parameter unitary group which in turn corresponds to a continuous automorphism on the JB algebra.
The first correspondence is given by the exponential
begin{equation}
e^{ i H t }
end{equation}
for some self-adjoint operator $H$ on the Hilbert space corresponding to an element of the JB algebra and $t in mathbb{R}$.
This suggests that (maybe) one could expect to find a canonical mapping between elements of a JB algebra and its continuous one-parameter groups of automorphisms.
Is this true? Note that if $J$ is trivial, it has to be true. Is the nontriviality of $J$ a necessary condition for this to fail? Otherwise what is the proper obstruction?
I tried to google for answers, but I only found remotely related findings.
quantum-mechanics jordan-algebras
quantum-mechanics jordan-algebras
asked Nov 29 '18 at 18:20
ioloiolo
6011416
asked Nov 29 '18 at 18:20
ioloiolo
6011416
asked Nov 29 '18 at 18:20
ioloiolo
6011416
asked Nov 29 '18 at 18:20
ioloiolo
6011416
asked Nov 29 '18 at 18:20
ioloiolo
6011416
6011416
add a comment |
add a comment |
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