A Quantum Algebra Supplement - Sample Applications - Continuation

Sample Applications - Continuation

Dirac - Equation

The free fields that solve the field equations are again specified:

1. The Hamilton operator is defined and evaluated:

   $$\begin{array}{rl}& \mathfrak{H}:=-i(\overline{\mathrm{\Psi }}//\text{QASimplify[]})\cdot \sum _{i=1}^3({\gamma }^i\cdot ({\mathfrak{D}}_{x_i}(\mathrm{\Psi }//\text{Evaluate})/.p\to p1))+m(\overline{\mathrm{\Psi }}//\text{QASimplify[]})\cdot (\mathrm{\Psi }/.p\to p1)\\ & H:={\int }_V\mathfrak{H}{d}^{3}x\\ & H//\text{QAEvaluate2}//\text{Timing}\\ & \{154.141,\\ & -2\hat{1}\sum _p{p}_0+\sum _p{\hat{c}}_{p_1,p_2,p_3}^{1,†}\cdot {\hat{c}}_{p_1,p_2,p_3}^1{p}_0+\sum _p{\hat{c}}_{p_1,p_2,p_3}^{2,†}\cdot {\hat{c}}_{p_1,p_2,p_3}^2{p}_0+\sum _p{\hat{d}}_{p_1,p_2,p_3}^{1,†}\cdot {\hat{d}}_{p_1,p_2,p_3}^1{p}_0+\sum _p{\hat{d}}_{p_1,p_2,p_3}^{2,†}\cdot {\hat{d}}_{p_1,p_2,p_3}^2{p}_0\}\end{array}$$

2. The 3 - momentum is defined and evaluated:
   

   $\begin{array}{rll}& {\mathfrak{P}}_{{\mathrm{i\_}}_{}}:=i(\overline{\mathrm{\Psi }}\text{// QASimplify}[])\cdot ({\gamma }^0\cdot ({\mathfrak{D}}_{{\mathrm{x}}_i}(\mathrm{\Psi }\text{// Evaluate})\mathrm{/.\; p}\to \mathrm{p1}))& \\ & P_{{\mathrm{i\_}}_{}}:={\int }_V{\mathfrak{P}}_i{d}^{3}x& \\ & \text{Map [}\left(P_i\text{// QAEvaluate2}\right)\mathrm{\&},\{1,2,3\}]\text{// Timing}& \\ & \{116.688,\{& \\ & -\sum _p{\hat{c}}_{p_1,p_2,p_3}^{1,†}\cdot {\hat{c}}_{p_1,p_2,p_3}^1{p}_1-\sum _p{\hat{c}}_{p_1,p_2,p_3}^{2,†}\cdot {\hat{c}}_{p_1,p_2,p_3}^2{p}_1-\sum _p{\hat{d}}_{p_1,p_2,p_3}^{1,†}\cdot {\hat{d}}_{p_1,p_2,p_3}^1{p}_1-\sum _p{\hat{d}}_{p_1,p_2,p_3}^{2,†}\cdot {\hat{d}}_{p_1,p_2,p_3}^2{p}_1& ,\\ & -\sum _p{\hat{c}}_{p_1,p_2,p_3}^{1,†}\cdot {\hat{c}}_{p_1,p_2,p_3}^1{p}_2-\sum _p{\hat{c}}_{p_1,p_2,p_3}^{2,†}\cdot {\hat{c}}_{p_1,p_2,p_3}^2{p}_2-\sum _p{\hat{d}}_{p_1,p_2,p_3}^{1,†}\cdot {\hat{d}}_{p_1,p_2,p_3}^1{p}_2-\sum _p{\hat{d}}_{p_1,p_2,p_3}^{2,†}\cdot {\hat{d}}_{p_1,p_2,p_3}^2{p}_2& ,\\ & -\sum _p{\hat{c}}_{p_1,p_2,p_3}^{1,†}\cdot {\hat{c}}_{p_1,p_2,p_3}^1{p}_3-\sum _p{\hat{c}}_{p_1,p_2,p_3}^{2,†}\cdot {\hat{c}}_{p_1,p_2,p_3}^2{p}_3-\sum _p{\hat{d}}_{p_1,p_2,p_3}^{1,†}\cdot {\hat{d}}_{p_1,p_2,p_3}^1{p}_3-\sum _p{\hat{d}}_{p_1,p_2,p_3}^{2,†}\cdot {\hat{d}}_{p_1,p_2,p_3}^2{p}_3& \\ & \}\}& \end{array}$

3. The 4 - current is defined and the charge is evaluated (without multiplicative charge constant):
   

   $\begin{array}{l}{\mathfrak{J}}_{i_{-}}:=(\bar{\mathrm{\Psi }}⁄⁄\text{QASimplify}[])\cdot ({\gamma }^i\cdot (\mathrm{\Psi }/.\mathrm{p}\to \mathrm{p}1))\\ J_{i_{-}}:={\int }_{\mathrm{v}}{\mathfrak{J}}_i{d}^3\mathrm{x}\\ \text{QAEvaluate2}\left[J_0\right]//\text{Timing}\\ \{41.5313,2\hat{1}\sum _{p}1+\sum _p{\hat{c}}_{p_1,p_2,p_3}^{1,†}\cdot {\hat{c}}_{p_1,p_2,p_3}^1+\sum _p{\hat{c}}_{p_1,p_2,p_3}^{2,†}\cdot {\hat{c}}_{p_1,p_2,p_3}^2-\sum _p{\hat{d}}_{p_1,p_2,p_3}^{1,†}\cdot {\hat{d}}_{p_1,p_2,p_3}^1-\sum _p{\hat{d}}_{p_1,p_2,p_3}^{2,†}\cdot {\hat{d}}_{p_1,p_2,p_3}^2\}\end{array}$