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    I'm having a bit of difficulty trying to combine these four equations into one "covariant" type equation:

     \sigma^0 \sigma^0 = \sigma^0

     \sigma^0 \sigma^i = \sigma^i  \;\;;\;\; i = 1,2,3

     \sigma^i \sigma^0 = \sigma^i  \;\;;\;\; i = 1,2,3

     \sigma^i \sigma^j = \delta^{ij} \sigma^0 + i \epsilon^{ijk} \sigma^k \;\;;\;\; i = 1,2,3\;\; j =1,2,3\;\; k = 1,2,3

    I want an equation in the form:

     \sigma^{\mu} \sigma^{\nu} = ?
    where  \mu,\nu = 0,1,2,3

    The difficultly comes when you spot that  \delta^{\mu \nu} \sigma^0 fits a good part of the problem but then when you add in expressions like  \delta^{0\mu} \sigma^{\nu} it is spoilt when  \nu = 0 as you start to over count.

    Any ideas or answers?

    For those interested, the sigmas are the pauli matrices inc. 0 which is the 2x2 identity matrix.
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    (Original post by div curl F = 0)
    I'm having a bit of difficulty trying to combine these four equations into one "covariant" type equation:

     \sigma^0 \sigma^0 = \sigma^0

     \sigma^0 \sigma^i = \sigma^i  \;\;;\;\; i = 1,2,3

     \sigma^i \sigma^0 = \sigma^i  \;\;;\;\; i = 1,2,3

     \sigma^i \sigma^j = \delta^{ij} \sigma^0 + i \epsilon^{ijk} \sigma^k \;\;;\;\; i = 1,2,3\;\; j =1,2,3\;\; k = 1,2,3

    I want an equation in the form:

     \sigma^{\mu} \sigma^{\nu} = ?
    where  \mu,\nu = 0,1,2,3

    The difficultly comes when you spot that  \delta^{\mu \nu} \sigma^0 fits a good part of the problem but then when you add in expressions like  \delta^{0\mu} \sigma^{\nu} it is spoilt when  \nu = 0 as you start to over count.

    Any ideas or answers?

    For those interested, the sigmas are the pauli matrices inc. 0 which is the 2x2 identity matrix.
    That issue can be dealt with by multiplying by (1  - \delta^{0 \nu})

    To be honest the expression is going to be so messy that I'm not sure what use it will be, but for what it's worth here's what I get:

    \sigma^{\mu} \sigma^{\nu} = \delta^{\mu \nu}\sigma^{0} + (1 - \delta^{0 \mu})(1 - \delta^{0 \nu}) i \epsilon^{\mu \nu \rho} \sigma^{\rho} + \delta^{0 \mu}(1 - \delta^{0 \nu})\sigma^{\nu} + \delta^{0 \nu}(1 - \delta^{0 \mu})\sigma^{\mu}
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    (Original post by Dystopia)
    That issue can be dealt with by multiplying by (1  - \delta^{0 \nu})

    To be honest the expression is going to be so messy that I'm not sure what use it will be, but for what it's worth here's what I get:

    \sigma^{\mu} \sigma^{\nu} = \delta^{\mu \nu}\sigma^{0} + (1 - \delta^{0 \mu})(1 - \delta^{0 \nu}) i \epsilon^{\mu \nu \rho} \sigma^{\rho} + \delta^{0 \mu}(1 - \delta^{0 \nu})\sigma^{\nu} + \delta^{0 \nu}(1 - \delta^{0 \mu})\sigma^{\mu}

    The only trouble with this type of equation is the doubling up of indices in each term implies summation and its not strictly covariant. I did get something similar when I went down this path but run into trouble later on in my work because it doesn't use the same conventions as the rest of my work.

    I am toying with the idea of inventing a tensor, say M, so it gives the above relations and lets the relation be written as:
     \sigma^{\alpha} \sigma^{\beta} = \delta^{\alpha \beta} \sigma^0 + i M^{\alpha \beta \gamma} \sigma_{\gamma}

    Thanks for the reply though, much appreciated.
 
 
 
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