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Scalar product and isospace

  1. Mar 2, 2009 #1
    In some textbooks you can find that a term
    [tex] \vec{\tau} \cdot \vec{A}_{\mu} = \sum_{a=1}^{3} \tau_{a} \, A_{\mu}^{a} [/tex]
    is called scalar product in isospace (where the tau's denotes the Pauli matrices and [tex]A_{\mu}^{a}[/tex] is a four-vector). But how can one call this "scalar" product. The product is a matrix and not a scalar. And the usual definition of a scalar product requires that the product has to be a scalar.

    Or take another example: [tex] \gamma^{\mu} \, A_{\mu} [/tex] is called a scalar product of four-vectors in space-time. It is confusing. Could anyone explain that to me? Do we really have a scalar product (in a strict mathematical sense) or is it just a convention done by physicists?
  2. jcsd
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