Lie Group Representations
Overview
A Lie group representation is a
Generators
This discussion follows the discussion and notation in
Robinson.
The parameterization of the Lie group is such that the indentity of the group is
specified with the parameters set to zero. (if this is not the case, a transformation of
the parameters will yield the desired effect)
{% g(\alpha_i) |_{\alpha_i = 0} = e %}
Because a representation is a
group homomorphism,
we have
{% D(g(\alpha_i)) |_{\alpha_i = 0} = \mathbb{I} %}
{% D(g(\delta \alpha_i)) = \mathbb{I} + \delta \alpha_i \frac{\partial D(g(\alpha_i))}{\partial \alpha_i} + ... %}
Next we define the generator {% X_i %}
{% X_i =\frac{\partial D}{\partial \alpha_i} |_{\alpha_i = 0} %}
or alternatively, {% X_i %} can be defined as
{% X_i = -i \frac{\partial D}{\partial \alpha_i} |_{\alpha_i = 0} %}
the extra {% -i %} makes {% X_i %}
Hermitian.
{% D(\delta \alpha_i) = \mathbb{I} + i\delta \alpha_i X_i + ... %}
{% lim _{n \to \infty} (1 + i \delta \alpha_i X_i)^n = lim _{n \to \infty} (1+i\frac{\alpha_i}{n} X_i )^n = e^{i\alpha_i X_i} %}
