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1. • CommentRowNumber1.
   • CommentAuthorgmlewis
   • CommentTimeOct 7th 2016
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   Author: gmlewis
   Format: TextSuppose it's the early 1900s and you're a theoretical physicist trying to explain the research results from studies of black-body radiation and/or of specific heats. So you're someone like Planck, Einstein, Lorentz or others. Further suppose that you have current (2016) category theory in your mathematical toolkit. Might a different form of quantum mechanics have evolved from that mix compared to the QM that actually began to appear in the early 1900s? If so, what might it look like? Do those questions even make any sense? I suppose it's a kind of thought experiment. Would anyone care to volunteer an opinion on whether any value might occur in such a study? Please Note: I am not trying to lay claim to this or any other "experiment". I don't know nearly enough about category theory to even attempt it. But I'm curiouus as to whether it's a totally stupid idea, or one that might be useful, or one that might even have some historical precedence in other areas of research. Thanks very much for your consideration of this question, and for your tolerance to an off-beat question. Best regards, Gary Lewis

   Suppose it's the early 1900s and you're a theoretical physicist trying to explain the research results from studies of black-body radiation and/or of specific heats. So you're someone like Planck, Einstein, Lorentz or others. Further suppose that you have current (2016) category theory in your mathematical toolkit. Might a different form of quantum mechanics have evolved from that mix compared to the QM that actually began to appear in the early 1900s? If so, what might it look like?
   
   Do those questions even make any sense? I suppose it's a kind of thought experiment. Would anyone care to volunteer an opinion on whether any value might occur in such a study?
   
   Please Note: I am not trying to lay claim to this or any other "experiment". I don't know nearly enough about category theory to even attempt it. But I'm curiouus as to whether it's a totally stupid idea, or one that might be useful, or one that might even have some historical precedence in other areas of research.
   
   Thanks very much for your consideration of this question, and for your tolerance to an off-beat question.
   
   Best regards,
   Gary Lewis
2. • CommentRowNumber2.
   • CommentAuthorNikolajK
   • CommentTimeOct 10th 2016
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   Author: NikolajK
   Format: MarkdownItexAfaik is was only around that time or later that physicist would get a more mathematical perspective on group representation theory and isomorphisms, so in the sceanrio where such topics algebra were common knowledge, you can assume that things would at least have evolved quicker. Might be worth to look into topics such as the history of the Dirac equation to get more perspective.

   Afaik is was only around that time or later that physicist would get a more mathematical perspective on group representation theory and isomorphisms, so in the sceanrio where such topics algebra were common knowledge, you can assume that things would at least have evolved quicker. Might be worth to look into topics such as the history of the Dirac equation to get more perspective.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 10th 2016
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   Author: David_Corfield
   Format: MarkdownItexYou'd have your classical theory [captured](https://ncatlab.org/schreiber/show/Classical+field+theory+via+Cohesive+homotopy+types) in cohesive HoTT. So then what sparks the idea to [quantize](https://ncatlab.org/schreiber/show/Quantization+via+Linear+homotopy+types)?

   You’d have your classical theory captured in cohesive HoTT.

   So then what sparks the idea to quantize?

4. • CommentRowNumber4.
   • CommentAuthorfastlane69
   • CommentTimeOct 10th 2016
   • (edited Oct 10th 2016)
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   Author: fastlane69
   Format: Text"Further suppose that you have current (2016) category theory in your mathematical toolkit. Might a different form of quantum mechanics have evolved from that mix compared to the QM that actually began to appear in the early 1900s? " HoTT as above and papers such as these answer your question in the affirmative IMO. [Kindergarten Quantum Mechanics](https://arxiv.org/abs/quant-ph/0510032) [Categories for practicing physcists](http://www.cs.ox.ac.uk/bob.coecke/Cats.pdf) [Topos Quantum Mechanics](https://arxiv.org/abs/1207.1744)

   "Further suppose that you have current (2016) category theory in your mathematical toolkit. Might a different form of quantum mechanics have evolved from that mix compared to the QM that actually began to appear in the early 1900s? "
   
   HoTT as above and papers such as these answer your question in the affirmative IMO.
   
   [Kindergarten Quantum Mechanics](https://arxiv.org/abs/quant-ph/0510032)
   
   [Categories for practicing physcists](http://www.cs.ox.ac.uk/bob.coecke/Cats.pdf)
   
   [Topos Quantum Mechanics](https://arxiv.org/abs/1207.1744)
5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 11th 2016
   • (edited Oct 11th 2016)
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   Author: Urs
   Format: MarkdownItexDavid Corfield wrote: > You'd have your classical theory [captured](https://ncatlab.org/schreiber/show/Classical+field+theory+via+Cohesive+homotopy+types) in cohesive HoTT. Or phrased more independently of my note: generally, in order see how concepts want to proceed, it is helpful to organize what you already have in the most abstract way. Incidentally, a maybe more comprehensive account of classical field theory as something that naturally comes out differential cohesion is in the chapter _[[schreiber:Higher Prequantum Geometry]]_ that I wrote for Anel-Catren's book. I am busy finalizing the first actual article in the corresponding series, which is why I am a bit quiet here lately. > So then what sparks the idea to [quantize](https://ncatlab.org/schreiber/show/Quantization+via+Linear+homotopy+types)? At least in broad outline it is very suggestive: in the formulation of classical (or rather pre-quantum) field theory then time evolution is naturally expressed in terms of correspondences carrying prequantum data- Such as discussed also at _[[prequantized Lagrangian correspondence]]_. Any correspondence allows to do [[integral transforms]] and secondary integral transforms through it. Now secondary integral transforms are very rich, but only when applied to linear homotopy types, otherwise they degenerate to something trivial. This naturally suggests to read the correspondences appearing in classical field theory as prescriptions for secondary integral transforms in linear (stable) homotopy types. This leads to quantum theory.

   David Corfield wrote:

   You’d have your classical theory captured in cohesive HoTT.

   Or phrased more independently of my note: generally, in order see how concepts want to proceed, it is helpful to organize what you already have in the most abstract way.

   Incidentally, a maybe more comprehensive account of classical field theory as something that naturally comes out differential cohesion is in the chapter Higher Prequantum Geometry (schreiber) that I wrote for Anel-Catren’s book. I am busy finalizing the first actual article in the corresponding series, which is why I am a bit quiet here lately.

   So then what sparks the idea to quantize?

   At least in broad outline it is very suggestive: in the formulation of classical (or rather pre-quantum) field theory then time evolution is naturally expressed in terms of correspondences carrying prequantum data- Such as discussed also at prequantized Lagrangian correspondence. Any correspondence allows to do integral transforms and secondary integral transforms through it. Now secondary integral transforms are very rich, but only when applied to linear homotopy types, otherwise they degenerate to something trivial. This naturally suggests to read the correspondences appearing in classical field theory as prescriptions for secondary integral transforms in linear (stable) homotopy types. This leads to quantum theory.