Physical dimensions in math












6












$begingroup$


I was interested in the idea of of formalising the idea of physical dimensions with an algebraic structure containing "all physical quantities of any type". You'd need:




  • Scalar multiplication over the reals (so you can get "2 kg" from "2 * kg")

  • Addition within the same dimension (so you can have "kg + kg = 2 kg")

  • Multiplication of any two elements (so you can have "J = N m = N * m")

  • Inverses (so you can have "m/s = m * s^(-1)")


A tensor algebra could formalise this system -- but then you'd get all sorts of objects like "1 kg + 1 m", which make no sense.



A group would make sense -- with sub-groups like "mass measurements", "time measurements", "real numbers", "units" -- but then you can't have zero. Plus, I'd like to have some notion of units or "unit vectors"/"unit tensors".



What's a good way to formalise this?






abstract-algebra group-theory physics tensor-products dimensional-analysis







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Also consider that some physical measurements are unitless.
    $endgroup$
    – David K
    Nov 29 '18 at 18:20






  • 2




    $begingroup$
    You might be interested in the "mathematical properties" section of the Wikipedia article on dimensional analysis: en.wikipedia.org/wiki/…
    $endgroup$
    – awkward
    Nov 29 '18 at 19:55










  • $begingroup$
    I would suggest a bundle of Abelian groups over the rational vector space of dimensions mentioned in the above wikipedia article: there's an assigned Abelian group to each dimension.
    $endgroup$
    – Berci
    Nov 29 '18 at 23:25
















6












$begingroup$


I was interested in the idea of of formalising the idea of physical dimensions with an algebraic structure containing "all physical quantities of any type". You'd need:




  • Scalar multiplication over the reals (so you can get "2 kg" from "2 * kg")

  • Addition within the same dimension (so you can have "kg + kg = 2 kg")

  • Multiplication of any two elements (so you can have "J = N m = N * m")

  • Inverses (so you can have "m/s = m * s^(-1)")


A tensor algebra could formalise this system -- but then you'd get all sorts of objects like "1 kg + 1 m", which make no sense.



A group would make sense -- with sub-groups like "mass measurements", "time measurements", "real numbers", "units" -- but then you can't have zero. Plus, I'd like to have some notion of units or "unit vectors"/"unit tensors".



What's a good way to formalise this?






abstract-algebra group-theory physics tensor-products dimensional-analysis







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Also consider that some physical measurements are unitless.
    $endgroup$
    – David K
    Nov 29 '18 at 18:20






  • 2




    $begingroup$
    You might be interested in the "mathematical properties" section of the Wikipedia article on dimensional analysis: en.wikipedia.org/wiki/…
    $endgroup$
    – awkward
    Nov 29 '18 at 19:55










  • $begingroup$
    I would suggest a bundle of Abelian groups over the rational vector space of dimensions mentioned in the above wikipedia article: there's an assigned Abelian group to each dimension.
    $endgroup$
    – Berci
    Nov 29 '18 at 23:25














6












6








6


3



$begingroup$


I was interested in the idea of of formalising the idea of physical dimensions with an algebraic structure containing "all physical quantities of any type". You'd need:




  • Scalar multiplication over the reals (so you can get "2 kg" from "2 * kg")

  • Addition within the same dimension (so you can have "kg + kg = 2 kg")

  • Multiplication of any two elements (so you can have "J = N m = N * m")

  • Inverses (so you can have "m/s = m * s^(-1)")


A tensor algebra could formalise this system -- but then you'd get all sorts of objects like "1 kg + 1 m", which make no sense.



A group would make sense -- with sub-groups like "mass measurements", "time measurements", "real numbers", "units" -- but then you can't have zero. Plus, I'd like to have some notion of units or "unit vectors"/"unit tensors".



What's a good way to formalise this?






abstract-algebra group-theory physics tensor-products dimensional-analysis







share|cite|improve this question









$endgroup$




I was interested in the idea of of formalising the idea of physical dimensions with an algebraic structure containing "all physical quantities of any type". You'd need:




  • Scalar multiplication over the reals (so you can get "2 kg" from "2 * kg")

  • Addition within the same dimension (so you can have "kg + kg = 2 kg")

  • Multiplication of any two elements (so you can have "J = N m = N * m")

  • Inverses (so you can have "m/s = m * s^(-1)")


A tensor algebra could formalise this system -- but then you'd get all sorts of objects like "1 kg + 1 m", which make no sense.



A group would make sense -- with sub-groups like "mass measurements", "time measurements", "real numbers", "units" -- but then you can't have zero. Plus, I'd like to have some notion of units or "unit vectors"/"unit tensors".



What's a good way to formalise this?






abstract-algebra group-theory physics tensor-products dimensional-analysis




abstract-algebra group-theory physics tensor-products dimensional-analysis






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 29 '18 at 17:44









Abhimanyu Pallavi SudhirAbhimanyu Pallavi Sudhir

894619




894619








  • 1




    $begingroup$
    Also consider that some physical measurements are unitless.
    $endgroup$
    – David K
    Nov 29 '18 at 18:20






  • 2




    $begingroup$
    You might be interested in the "mathematical properties" section of the Wikipedia article on dimensional analysis: en.wikipedia.org/wiki/…
    $endgroup$
    – awkward
    Nov 29 '18 at 19:55










  • $begingroup$
    I would suggest a bundle of Abelian groups over the rational vector space of dimensions mentioned in the above wikipedia article: there's an assigned Abelian group to each dimension.
    $endgroup$
    – Berci
    Nov 29 '18 at 23:25














  • 1




    $begingroup$
    Also consider that some physical measurements are unitless.
    $endgroup$
    – David K
    Nov 29 '18 at 18:20






  • 2




    $begingroup$
    You might be interested in the "mathematical properties" section of the Wikipedia article on dimensional analysis: en.wikipedia.org/wiki/…
    $endgroup$
    – awkward
    Nov 29 '18 at 19:55










  • $begingroup$
    I would suggest a bundle of Abelian groups over the rational vector space of dimensions mentioned in the above wikipedia article: there's an assigned Abelian group to each dimension.
    $endgroup$
    – Berci
    Nov 29 '18 at 23:25








1




1




$begingroup$
Also consider that some physical measurements are unitless.
$endgroup$
– David K
Nov 29 '18 at 18:20




$begingroup$
Also consider that some physical measurements are unitless.
$endgroup$
– David K
Nov 29 '18 at 18:20




2




2




$begingroup$
You might be interested in the "mathematical properties" section of the Wikipedia article on dimensional analysis: en.wikipedia.org/wiki/…
$endgroup$
– awkward
Nov 29 '18 at 19:55




$begingroup$
You might be interested in the "mathematical properties" section of the Wikipedia article on dimensional analysis: en.wikipedia.org/wiki/…
$endgroup$
– awkward
Nov 29 '18 at 19:55












$begingroup$
I would suggest a bundle of Abelian groups over the rational vector space of dimensions mentioned in the above wikipedia article: there's an assigned Abelian group to each dimension.
$endgroup$
– Berci
Nov 29 '18 at 23:25




$begingroup$
I would suggest a bundle of Abelian groups over the rational vector space of dimensions mentioned in the above wikipedia article: there's an assigned Abelian group to each dimension.
$endgroup$
– Berci
Nov 29 '18 at 23:25










1 Answer
1






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oldest

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2












$begingroup$

You could just use a multivariate polynomial ring $mathbb R[x_1, x_2,x_3,x_4,ldots]$ and map units of measure to a specific $x_i$. To obtain inverses you can go one step further to the field of fractions of this ring $mathbb R(x_1,x_2,x_3,x_4ldots)$.



That allows you to do addition and $mathbb R$ scaling inside the subspace $langle x_irangle$. Inversions are obviously possible in the field of fractions.



This allows you to freely construct different products of different units, and you can tell what combination is going on by the makeup of its monomial coefficient. If $x_1$ is kg and $x_2$ is s, then $frac{x_1}{x_2^2}$ is kg/s^2.



Until you're a little more specific about what you hope to achieve, and why just using $mathbb R$ with conversions between units doesn't satisfy your problem, I'm not sure what else can be said.



I'm also don't know why you are worried about addition between objects where addition isn't appropriate. Why do you want to eliminate it? It happens all the time in physical interpretations of Clifford algebra, and Clifford algebra is still useful. Extra nonsense that we don't use isn't a justification to throw out the stuff that works.






share|cite|improve this answer









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    $begingroup$

    You could just use a multivariate polynomial ring $mathbb R[x_1, x_2,x_3,x_4,ldots]$ and map units of measure to a specific $x_i$. To obtain inverses you can go one step further to the field of fractions of this ring $mathbb R(x_1,x_2,x_3,x_4ldots)$.



    That allows you to do addition and $mathbb R$ scaling inside the subspace $langle x_irangle$. Inversions are obviously possible in the field of fractions.



    This allows you to freely construct different products of different units, and you can tell what combination is going on by the makeup of its monomial coefficient. If $x_1$ is kg and $x_2$ is s, then $frac{x_1}{x_2^2}$ is kg/s^2.



    Until you're a little more specific about what you hope to achieve, and why just using $mathbb R$ with conversions between units doesn't satisfy your problem, I'm not sure what else can be said.



    I'm also don't know why you are worried about addition between objects where addition isn't appropriate. Why do you want to eliminate it? It happens all the time in physical interpretations of Clifford algebra, and Clifford algebra is still useful. Extra nonsense that we don't use isn't a justification to throw out the stuff that works.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      You could just use a multivariate polynomial ring $mathbb R[x_1, x_2,x_3,x_4,ldots]$ and map units of measure to a specific $x_i$. To obtain inverses you can go one step further to the field of fractions of this ring $mathbb R(x_1,x_2,x_3,x_4ldots)$.



      That allows you to do addition and $mathbb R$ scaling inside the subspace $langle x_irangle$. Inversions are obviously possible in the field of fractions.



      This allows you to freely construct different products of different units, and you can tell what combination is going on by the makeup of its monomial coefficient. If $x_1$ is kg and $x_2$ is s, then $frac{x_1}{x_2^2}$ is kg/s^2.



      Until you're a little more specific about what you hope to achieve, and why just using $mathbb R$ with conversions between units doesn't satisfy your problem, I'm not sure what else can be said.



      I'm also don't know why you are worried about addition between objects where addition isn't appropriate. Why do you want to eliminate it? It happens all the time in physical interpretations of Clifford algebra, and Clifford algebra is still useful. Extra nonsense that we don't use isn't a justification to throw out the stuff that works.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        You could just use a multivariate polynomial ring $mathbb R[x_1, x_2,x_3,x_4,ldots]$ and map units of measure to a specific $x_i$. To obtain inverses you can go one step further to the field of fractions of this ring $mathbb R(x_1,x_2,x_3,x_4ldots)$.



        That allows you to do addition and $mathbb R$ scaling inside the subspace $langle x_irangle$. Inversions are obviously possible in the field of fractions.



        This allows you to freely construct different products of different units, and you can tell what combination is going on by the makeup of its monomial coefficient. If $x_1$ is kg and $x_2$ is s, then $frac{x_1}{x_2^2}$ is kg/s^2.



        Until you're a little more specific about what you hope to achieve, and why just using $mathbb R$ with conversions between units doesn't satisfy your problem, I'm not sure what else can be said.



        I'm also don't know why you are worried about addition between objects where addition isn't appropriate. Why do you want to eliminate it? It happens all the time in physical interpretations of Clifford algebra, and Clifford algebra is still useful. Extra nonsense that we don't use isn't a justification to throw out the stuff that works.






        share|cite|improve this answer









        $endgroup$



        You could just use a multivariate polynomial ring $mathbb R[x_1, x_2,x_3,x_4,ldots]$ and map units of measure to a specific $x_i$. To obtain inverses you can go one step further to the field of fractions of this ring $mathbb R(x_1,x_2,x_3,x_4ldots)$.



        That allows you to do addition and $mathbb R$ scaling inside the subspace $langle x_irangle$. Inversions are obviously possible in the field of fractions.



        This allows you to freely construct different products of different units, and you can tell what combination is going on by the makeup of its monomial coefficient. If $x_1$ is kg and $x_2$ is s, then $frac{x_1}{x_2^2}$ is kg/s^2.



        Until you're a little more specific about what you hope to achieve, and why just using $mathbb R$ with conversions between units doesn't satisfy your problem, I'm not sure what else can be said.



        I'm also don't know why you are worried about addition between objects where addition isn't appropriate. Why do you want to eliminate it? It happens all the time in physical interpretations of Clifford algebra, and Clifford algebra is still useful. Extra nonsense that we don't use isn't a justification to throw out the stuff that works.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 29 '18 at 18:25









        rschwiebrschwieb

        106k12102249




        106k12102249






























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