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I wrote a paper about this at an REU I attended two years ago. At the time, I had the goal of making the transition from classical mechanics to quantum mechanics as natural as possible, motivating the axioms of quantum mechanics from those of classical mechanics, which are a lot more intuitive. Unfortunately, given the time constraints of the REU (just a couple of months) and being relatively new to the subject, I found myself mostly following other sources (in particular, Strocchi), which made use of the $C^*$-algebraic formalism. I have to admit, at the time, I did feel a bit uncomfortable throwing away all the unbounded operators, because, having studied "physicist's" quantum mechanics before, these are some things I naturally wanted to include.

Come two years later at the same REU, I tried to tackle the same problem, but this time I wanted to develop the axioms of quantum mechanics so to as allow unbounded operators in the theory. This led me to develop the notion of what I call an $F^*$-algebra ($F$ for Fréchet). I am actually still working on the paper at the moment, but, after reading your post, I decided to upload a preliminary copy to my academia.edu account. You should read it, check it out, and see what you think. I'd be very happy to hear any comments you have.

Be warned though, I am still in the process of editing and revising it, so dare I say, there may be some errors. Read it with a skeptical eye and let me know if you catch anything.

Cheers! Jonny Gleason

Here's the link: http://chicago.academia.edu/JonathanGleason/Papers/857632/From_Classical_to_Quantum_The_F_-algebraic_Approach

I wrote a paper about this at an REU I attended two years ago. At the time, I had the goal of making the transition from classical mechanics to quantum mechanics as natural as possible, motivating the axioms of quantum mechanics from those of classical mechanics, which are a lot more intuitive. Unfortunately, given the time constraints of the REU (just a couple of months) and being relatively new to the subject, I found myself mostly following other sources (in particular, Strocchi), which made use of the $C^*$-algebraic formalism. I have to admit, at the time, I did feel a bit uncomfortable throwing away all the unbounded operators, because, having studied "physicist's" quantum mechanics before, these are some things I naturally wanted to include.

Come two years later at the same REU, I tried to tackle the same problem, but this time I wanted to develop the axioms of quantum mechanics so to as allow unbounded operators in the theory. This led me to develop the notion of what I call an $F^*$-algebra ($F$ for Fréchet). I am actually still working on the paper at the moment, but, after reading your post, I decided to upload a preliminary copy to my academia.edu account. You should read it, check it out, and see what you think. I'd be very happy to hear any comments you have.

Be warned though, I am still in the process of editing and revising it, so dare I say, there may be some errors. Read it with a skeptical eye and let me know if you catch anything.

Cheers! Jonny Gleason

Here's the link: https://drive.google.com/open?id=0B6xfgYpCM4U3UXRNVkhPTGc5RDA

EDIT: I actually did find a couple of errors in the paper at some point, but I had since lost the .tex file and never got around to just rewriting the entire thing with corrections in place. I don't remember exactly where they are at this point (it's been a couple of years), but if I recall they're not subtle, so if you just make sure to check the proofs you should be okay.

I wrote a paper about this at an REU I attended two years ago. At the time, I had the goal of making the transition from classical mechanics to quantum mechanics as natural as possible, motivating the axioms of quantum mechanics from those of classical mechanics, which are a lot more intuitive. Unfortunately, given the time constraints of the REU (just a couple of months) and being relatively new to the subject, I found myself mostly following other sources (in particular, Strocchi), which made use of the $C^*$-algebraic formalism. I have to admit, at the time, I did feel a bit uncomfortable throwing away all the unbounded operators, because, having studied "physicist's" quantum mechanics before, these are some things I naturally wanted to include.

Come two years later at the same REU, I tried to tackle the same problem, but this time I wanted to develop the axioms of quantum mechanics so to as allow unbounded operators in the theory. This led me to develop the notion of what I call an $F^*$-algebra ($F$ for Frechet). I am actually still working on the paper at the moment, but, after reading your post, I decided to upload a preliminary copy to my academia.edu account. You should read it, check it out, and see what you think. I'd be very happy to hear any comments you have.

Be warned though, I am still in the process of editing and revising it, so dare I say, there may be some errors. Read it with a skeptical eye and let me know if you catch anything.

Cheers! Jonny Gleason

Here's the link: http://chicago.academia.edu/JonathanGleason/Papers/857632/From_Classical_to_Quantum_The_F_-algebraic_Approach

P.S.: How do you make the accent in Frechet on mathoverflow?

I wrote a paper about this at an REU I attended two years ago. At the time, I had the goal of making the transition from classical mechanics to quantum mechanics as natural as possible, motivating the axioms of quantum mechanics from those of classical mechanics, which are a lot more intuitive. Unfortunately, given the time constraints of the REU (just a couple of months) and being relatively new to the subject, I found myself mostly following other sources (in particular, Strocchi), which made use of the $C^*$-algebraic formalism. I have to admit, at the time, I did feel a bit uncomfortable throwing away all the unbounded operators, because, having studied "physicist's" quantum mechanics before, these are some things I naturally wanted to include.

Come two years later at the same REU, I tried to tackle the same problem, but this time I wanted to develop the axioms of quantum mechanics so to as allow unbounded operators in the theory. This led me to develop the notion of what I call an $F^*$-algebra ($F$ for Fréchet). I am actually still working on the paper at the moment, but, after reading your post, I decided to upload a preliminary copy to my academia.edu account. You should read it, check it out, and see what you think. I'd be very happy to hear any comments you have.

Be warned though, I am still in the process of editing and revising it, so dare I say, there may be some errors. Read it with a skeptical eye and let me know if you catch anything.

Cheers! Jonny Gleason

Here's the link: http://chicago.academia.edu/JonathanGleason/Papers/857632/From_Classical_to_Quantum_The_F_-algebraic_Approach

I believe I might be the Gleason you are referring to.

I think the paper you might be referring to is one I wrote for an REU I attended two years ago. At the time, I had the goal of making the transition from classical mechanics to quantum mechanics as natural as possible, motivating the axioms of quantum mechanics from those of classical mechanics, which are a lot more intuitive. Unfortunately, given the time constraints of the REU (just a couple of months) and being relatively new to the subject, I found myself mostly following other sources (in particular, Strocchi), which made use of the $C^*$-algebraic formalism. I have to admit, at the time, I did feel a bit uncomfortable throwing away all the unbounded operators, because, having studied "physicist's" quantum mechanics before, these are some things I naturally wanted to include.

Come two years later at the same REU, I tried to tackle the same problem, but this time I wanted to develop the axioms of quantum mechanics so to as allow unbounded operators in the theory. This led me to develop the notion of what I call an $F^*$-algebra ($F$ for Frechet). I am actually still working on the paper at the moment, but, after reading your post, I decided to upload a preliminary copy to my academia.edu account. You should read it, check it out, and see what you think. I'd be very happy to hear any comments you have.

Be warned though, I am still in the process of editing and revising it, so dare I say, there may be some errors. Read it with a skeptical eye and let me know if you catch anything.

Cheers! Jonny Gleason

Here's the link: http://chicago.academia.edu/JonathanGleason/Papers/857632/From_Classical_to_Quantum_The_F_-algebraic_Approach

P.S.: How do you make the accent in Frechet on mathoverflow?

I wrote a paper about this at an REU I attended two years ago. At the time, I had the goal of making the transition from classical mechanics to quantum mechanics as natural as possible, motivating the axioms of quantum mechanics from those of classical mechanics, which are a lot more intuitive. Unfortunately, given the time constraints of the REU (just a couple of months) and being relatively new to the subject, I found myself mostly following other sources (in particular, Strocchi), which made use of the $C^*$-algebraic formalism. I have to admit, at the time, I did feel a bit uncomfortable throwing away all the unbounded operators, because, having studied "physicist's" quantum mechanics before, these are some things I naturally wanted to include.

Come two years later at the same REU, I tried to tackle the same problem, but this time I wanted to develop the axioms of quantum mechanics so to as allow unbounded operators in the theory. This led me to develop the notion of what I call an $F^*$-algebra ($F$ for Frechet). I am actually still working on the paper at the moment, but, after reading your post, I decided to upload a preliminary copy to my academia.edu account. You should read it, check it out, and see what you think. I'd be very happy to hear any comments you have.

Be warned though, I am still in the process of editing and revising it, so dare I say, there may be some errors. Read it with a skeptical eye and let me know if you catch anything.

Cheers! Jonny Gleason

Here's the link: http://chicago.academia.edu/JonathanGleason/Papers/857632/From_Classical_to_Quantum_The_F_-algebraic_Approach

P.S.: How do you make the accent in Frechet on mathoverflow?