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After a comment of John Baez to a question I asked on MathOverflow, I would like to ask what the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) \otimes U(1)$ is. The $\times$ is the Cartesian product while the $\otimes$ is the tensor product. I gave the example of the Standard Model gauge group but it can be any product of groups. My question is when talking about global or gauge groups, do we mean Cartesian products or tensor products? And what is the real difference between them anyway?

Qmechanic
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Marion
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    Physicists always mean the Cartesian product of groups when talking about groups, but for some arcane reason they sometimes use $\otimes$ for it. As for the difference, what about e.g. the Wikipedia articles on Cartesian products of groups and tensor products is unclear to you? (The use of the tensor product for groups is rare) – ACuriousMind Sep 13 '15 at 18:16
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    @ACuriousMind And then those same physicists will call $\otimes$ a direct product. –  Sep 13 '15 at 19:58
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    Related : Total spin of two spin- 1/2 particles, SECOND_ANSWER \begin{equation} \mathbb{C}^{\boldsymbol{r}}\times \mathbb{C}^{\boldsymbol{s}}\equiv \mathbb{C}^{\boldsymbol{r+s}}\neq \mathbb{C}^{rs}\equiv \mathbb{C}^{\boldsymbol{r}}\boldsymbol{\otimes} \mathbb{C}^{\boldsymbol{s}} \tag{31}
    \end{equation}     – Frobenius Oct 03 '21 at 13:13

1 Answers1

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  1. The main point is that we usually only consider tensor products $V \otimes W$ of vector spaces $V$, $W$ (as opposed to general sets $V$, $W$). But groups (say $G$, $H$) are often not vector spaces. If we only consider tensor products of vector spaces, then the object $G \otimes H$ is nonsense, mathematically speaking.

    With further assumptions on the groups $G$ and $H$, it is sometimes possible to define a tensor product $G \otimes H$ of groups, cf. my Phys.SE answer here and links therein.

  2. If $V$ and $W$ are two vector spaces, then the tensor product $V \otimes W$ is again a vector space. Also the direct or Cartesian product $V\times W$ of vector spaces is isomorphic to the direct sum $V \oplus W$ of vector spaces, which is again a vector space.

    In fact, if $V$ is a representation space for the group $G$, and $W$ is a representation space for the group $H$, then both the tensor product $V\otimes W$ and the direct sum $V\oplus W$ are representation spaces for the Cartesian product group $G\times H$.

    (The direct sum representation space $V\oplus W\cong (V\otimes \mathbb{F}) \oplus(\mathbb{F}\otimes W)$ for the Cartesian product group $G\times H$ can be viewed as a direct sum of two $G\times H$ representation spaces, and is hence a composite concept. Here $\mathbb{F}$ denotes the field. Recall that any group has a trivial representation.)

    This interplay between the tensor product $V\otimes W$ and the Cartesian product $G\times H$ may persuade some authors into using the misleading notation $G\otimes H$ for the Cartesian product $G\times H$. Unfortunately, this often happens in physics and in category theory.

  3. In contrast to groups, note that Lie algebras (say $\mathfrak{g}$, $\mathfrak{h}$) are always vector spaces, so tensor products $\mathfrak{g}\otimes\mathfrak{h}$ of Lie algebras do make sense. However due to exponentiation, it is typically the direct sum $\mathfrak{g}\oplus\mathfrak{h}$ of Lie algebras that is relevant. If $\exp:\mathfrak{g}\to G$ and $\exp:\mathfrak{h}\to H$ denote exponential maps, then $\exp:\mathfrak{g}\oplus\mathfrak{h}\to G\times H$.

Qmechanic
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  • Therefore it is basically wrong to write $\otimes$ when talking about gauge group products in physics. What about the algebras though, which are elements of vector spaces? Also, I checked your nice answer, would you be able to give me some good reference for this? – Marion Sep 13 '15 at 19:14
  • therefore the Lie algebra of the SM can be written as $\mathfrak{su}(3)\otimes \mathfrak{su}(2) \otimes \mathfrak{u}(1)$ or $\mathfrak{su}(3)\times \mathfrak{su}(2) \times \mathfrak{u}(1)$? – Marion Sep 13 '15 at 20:58
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    Neither. The Lie algebra of the SM is $\mathfrak{su}(3)\oplus \mathfrak{su}(2) \oplus \mathfrak{u}(1)$. – Qmechanic Sep 13 '15 at 21:01
  • Is this related to the fact that the algebra is given by the exponential? And in which case and for what reason do you form a product (direct or tensorial) of elements of the algebra? Could you give an example maybe? – Marion Sep 13 '15 at 22:43
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    $V\times W $ is a vector space always if you define addition and scalar multiplication appropriately, which actually just happens to be $V\oplus W$. – snulty May 09 '16 at 15:53
  • What does $\mathbb{F}$ above stand for? 2. And do the representation spaces constructed from direct product $V\times W$ and tensor product $V\otimes W$ both have physical implications? 3. If so, would you please give some concrete example?
    • – Jason Jul 24 '23 at 02:37