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I am confused in the notation on page 67 and page 70 a text (http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures-2018.pdf), whether it's talking about a direct product or an outer product:

  1. On page 67, it mentioned that

    "you can take a direct product of two $j = 1/2$ representations"

    and build representations of higher j.

  2. On page 70, it mentioned

    "we can think of [the Lorentz Group] as the direct product $SU(2) \times SU(2)$."

In each of the above, does the author mean Direct Product or Tensor Product?

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  • $\begingroup$ Minor comment to the post (v3): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$
    – Qmechanic
    Dec 14, 2018 at 19:36
  • $\begingroup$ “The universe is an enormous direct product of representations of symmetry groups.” – Steven Weinberg . $\endgroup$ Jul 1, 2020 at 13:59

2 Answers 2

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  1. On p. 67 Tseng means a tensor product of representations.

  2. On p. 70 Tseng means a direct product of groups.

    Note however that the actual statement about the Lorentz group is wrong/imprecise as explained in e.g. this Phys.SE post.

    Concerning direct product vs. tensor product of groups, see also my related Phys.SE answer here.

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In both the cases the author is talking about direct product. Addition of two angular momentums $j_1$ and $j_2$ is represented using the direct product of the two angular momentum spaces. The wave functions or kets in the product space are represented using tensor product notation.

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