Editing Second quantization (section)

=== Insertion and deletion operation ===

The creation and annihilation of a particle is implemented by the insertion and deletion of the single-particle state from the first quantized wave function in an either symmetric or anti-symmetric manner. Let <math>\psi_\alpha</math> be a single-particle state, let 1 be the tensor identity (it is the generator of the zero-particle space '''C''' and satisfies <math>\psi_\alpha\equiv1\otimes\psi_\alpha\equiv\psi_\alpha\otimes1</math> in the [[tensor algebra]] over the fundamental Hilbert space), and let <math>\Psi =\psi_{\alpha_1}\otimes\psi_{\alpha_2}\otimes\cdots</math> be a generic tensor product state. The insertion <math>\otimes_\pm</math>  and the deletion <math>\oslash_\pm</math> operators are linear operators defined by the following recursive equations
:<math>\psi_\alpha\otimes_\pm 1=\psi_\alpha,\quad\psi_\alpha\otimes_\pm(\psi_\beta\otimes\Psi)= \psi_\alpha\otimes\psi_\beta\otimes\Psi\pm\psi_\beta\otimes(\psi_\alpha\otimes_\pm\Psi);</math>
:<math>\psi_\alpha\oslash_\pm 1=0,\quad\psi_\alpha\oslash_\pm(\psi_\beta\otimes\Psi)= \delta_{\alpha\beta}\Psi\pm\psi_\beta\otimes(\psi_\alpha\oslash_\pm\Psi).</math>
Here <math>\delta_{\alpha\beta}</math> is the [[Kronecker delta]] symbol, which gives 1 if <math>\alpha=\beta</math>, and 0 otherwise. The subscript <math>\pm</math> of the insertion or deletion operators indicates whether symmetrization (for bosons) or anti-symmetrization (for fermions) is implemented.