- #1
victorvmotti
- 152
- 5
I'm following the lecture notes by https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf.
On page 169, section 6.2 he is briefly touching on the non-abelian gauge symmetry in the SM.
The fundamental representation makes sense to me. For example, for ##SU(3)##, we define the object or column vector with three component, suppressing spinor indices, ##\psi(x)=(\psi_1, \psi_2, \psi_3)^T##. The fundamental representation, a ##3*3## matrix ##V(h)##, acts on this column vector, with each component itself a Dirac spinor that has 4 components of complex numbers. Lagrangian density then is showed to remain invariant when the ##\psi## is multiplied by the ##3*3## matrix ##V(h)##.
But the next step puzzles me a little, when we use the adjoint representation. Here instead of a 3 component column vector, we use a ##3*3## matrix ##\psi(x)##, whose elements ##\psi_{ij}(x)## are complex numbers based on the definition given in (6.32). It is a ##3*3## matrix or representation of the ##SU(3)## Lie algebra.
But when in (6.34) the Lagranian density is defined as ##tr(\bar{\psi}(i\gamma^\mu\partial_\mu-m)\psi)= \bar{ \psi_{ij}}(i\gamma^\mu\partial_\mu-m)\psi_{ji}##. Here ##\psi## is a ##3*3## matrix whose elements, ##\psi_{ji}##, are not complext numbers but Dirac spinors.
In another word, is this what we mean by the spinor indices are suppressed in the adjoint representation? How can we think of ##\psi(x)## both as a ##3*3## matrix with components to be both complex numbers as defined in (6.32) and Dirac spinors as used in (6.34) that have themselves four components, each a complex number.
On page 169, section 6.2 he is briefly touching on the non-abelian gauge symmetry in the SM.
The fundamental representation makes sense to me. For example, for ##SU(3)##, we define the object or column vector with three component, suppressing spinor indices, ##\psi(x)=(\psi_1, \psi_2, \psi_3)^T##. The fundamental representation, a ##3*3## matrix ##V(h)##, acts on this column vector, with each component itself a Dirac spinor that has 4 components of complex numbers. Lagrangian density then is showed to remain invariant when the ##\psi## is multiplied by the ##3*3## matrix ##V(h)##.
But the next step puzzles me a little, when we use the adjoint representation. Here instead of a 3 component column vector, we use a ##3*3## matrix ##\psi(x)##, whose elements ##\psi_{ij}(x)## are complex numbers based on the definition given in (6.32). It is a ##3*3## matrix or representation of the ##SU(3)## Lie algebra.
But when in (6.34) the Lagranian density is defined as ##tr(\bar{\psi}(i\gamma^\mu\partial_\mu-m)\psi)= \bar{ \psi_{ij}}(i\gamma^\mu\partial_\mu-m)\psi_{ji}##. Here ##\psi## is a ##3*3## matrix whose elements, ##\psi_{ji}##, are not complext numbers but Dirac spinors.
In another word, is this what we mean by the spinor indices are suppressed in the adjoint representation? How can we think of ##\psi(x)## both as a ##3*3## matrix with components to be both complex numbers as defined in (6.32) and Dirac spinors as used in (6.34) that have themselves four components, each a complex number.