I'm a little bit confused about the difference between the spinor and vector representations of SU(N)--I guess I could start with asking how a spinor and a vector differ: is this only a matter of how they transform under Lorentz transformations?(adsbygoogle = window.adsbygoogle || []).push({});

Following up, the covariant derivative for a spinor of SU(2) is (i.e. for a scalar field [tex]\phi[/tex] that transforms as a spinor of SU(2)):

[tex] D_\mu \phi = (\partial_\mu - i g A^a_\mu \tau ^a )\phi [/tex]

While the covariant derivative for the vector representation of a scalar [tex]\phi[/tex] is:

[tex] D_\mu \phi = \partial_\mu_a + g \epsilon_{abc} A^a_\mu \phi_c [/tex]

(these are from Peskin and Schroeder p. 694-5, eq. (20.22) and (20.27) resp.)

My understanding is that this means we have a scalar field [tex]\phi[/tex] that has a nonabelian gauge symmetry in some abstract (internal) SU(2) space.

The spinor covariant derivative seems to make sense from the general definition of the covariant derivative:

[tex]D_\mu = \partial_\mu - igA^a_\mu t^a[/tex]

where [tex]t^a[/tex] is a generator of the gauge group. does this mean that the generator of the vector representation is something like [tex]\epsilon_{abc}[/tex]? Where does this [tex]\epsilon_{abc}[/tex] come from?

Thanks,

Flip

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# SU(N) Vector vs. Spinor Representations

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