- The Relationship Between Tensor Product Representation of Identity Matrix and Dirac Notation

1. Tensor Operations #

$$(e^i \otimes e_j)(v,f) = e^i(v) e_j(f)$$

2. Tensors as Linear Operators #

\begin{equation*} \begin{aligned} (e^i \otimes e_j)(\boldsymbol{v},f) &= f((e^i \otimes e_j)(\boldsymbol{v})) \\ \text{,where }\ (e^i \otimes e_j)(\boldsymbol{v}) = e^i(\boldsymbol{v}) e_j \\ &= f(e^i(\boldsymbol{v}) e_j) \\ &= e^i(\boldsymbol{v}) f(e_j) \\ \text{,where } \ f(e_j) = e_j(f) \\ &= e^i(\boldsymbol{v}) e_j(f) \end{aligned} \end{equation*}

3. Using Metric Duality #

$$(e^i(v) e_j(f) = (e_i | v) (e^j | f)$$

4. Definition of Tensor Product Inner Product #

$$(v_1 \otimes w_1|v_2 \otimes w_2) = (v_1|v_2) (w_1|w_2)$$

5. Deriving the Relationship #

$$(e^i \otimes e_j)(v,f) = e^i(v) e_j(f)$$

$$= (e_i|v) (e^j|f) = (e_i \otimes e^j|v \otimes f)$$

6. Tensor Product Representation of Identity Operator Using Linear Operator Properties #

$$(e^i \otimes e_i)(v) = e^i(v) e_i = v^i e_i,$$

therefore,

$$\sum_i e^i \otimes e_i = I_{(1,1)} = I$$

7. Identity Operator Representation in Dirac Notation #

$$\bra{a}\ket{b} = \sum_i a_i^* b_i = \sum_i \bra{a}\ket{i} \bra{i}\ket{b}$$

$$= \bra{a} \left( \sum_i \ket{i}\bra{i}\right) \ket{b},$$

therefore,

$$\sum_i \ket{i}\bra{i} = I$$

8. Comparison of Tensor Product Representation and Dirac Notation for Identity Matrix #

$$I : \sum_i \ket{i}\bra{i}\quad \leftrightarrow \quad \sum_i e^i \otimes e_i$$

9. Additional Comparison of Tensor Product Representation and Dirac Notation #

$$(e^i \otimes e_j)(v)=e^i(v) e_j = v^i e_j ,$$

meanwhile,

$$\ket{j}\bra{i}\ket{a} = a_i \ket{j} ,$$

therefore,

$$\ket{j}\bra{i} \quad \leftrightarrow \quad e^i \otimes e_j$$