For example, sometimes one sees $SU(3)\otimes SU(2)\otimes U(1)$ instead of $SU(3)\times SU(2)\times U(1)$.

My understanding is the the product here is just the usual direct product (aka Cartesian product), which is completely different from a tensor product or Kronecker product which the $\otimes$ symbol is usually used for. So why is $\otimes$ often (especially in a high energy physics context) for the direct product--is it just historical convention or is there any deep reason?

  • $\begingroup$ This is a common annoyance, cf. physics.stackexchange.com/q/447342/50583 and its linked questions. $\endgroup$
    – ACuriousMind
    May 11, 2019 at 22:37
  • 1
    $\begingroup$ I'm voting to close this question as off-topic because asking for the reasons for notation is off-topic here. If you are interested in the actual historical development, consider asking at History of Science and Mathematics instead. $\endgroup$
    – ACuriousMind
    May 11, 2019 at 22:38
  • 1
    $\begingroup$ @ACuriousMind I think this should be closed as a duplicate of the question you linked. It is essentially the same question, the OP is interested in terminology (not history per se), and there is no way for the OP to know whether it is a physics point or a pure terminology point a priori (thus, the question). $\endgroup$
    – user87745
    May 11, 2019 at 22:55
  • 3
    $\begingroup$ What would one even mean by the tensor product of noncommutative groups? $\endgroup$
    – WillO
    May 12, 2019 at 0:23
  • 2
    $\begingroup$ It’s because the tensor product looks more “mathy”, so it presumably makes the writer look cooler. $\endgroup$
    – knzhou
    May 12, 2019 at 1:12


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