

In [30]:

    
from IPython.display import Math, display
def displayWithTitle(multivector, title):
    display(Math(title + '=' + multivector._repr_latex_()))
def CheckProperties(i,j,k, ilabel, jlabel, klabel):
    displayWithTitle(i, title=ilabel)
    displayWithTitle(j, title=jlabel)
    displayWithTitle(k, title=klabel)
    displayWithTitle((i*i), title=ilabel+'^2')
    displayWithTitle((j*j), title=jlabel+'^2')
    displayWithTitle((k*k), title=klabel+'^2')
    displayWithTitle((i*j), title=ilabel + jlabel)
    displayWithTitle((j*i), title=jlabel + ilabel)
    displayWithTitle((j*k), title=jlabel + klabel)
    displayWithTitle((k*j), title=klabel + jlabel)
    displayWithTitle((k*i), title=klabel + ilabel)
    displayWithTitle((i*k), title=ilabel + klabel)
    displayWithTitle((i*j*k), title=ilabel + jlabel + klabel)



In [31]:

    
from sympy.matrices import *
from galgebra.ga import Ga
from galgebra.printer import *

ALGEBRA & DEFINITIONS

Clifford algebra is $$Cl_{1,4}(\mathbb{R})$$

Flat space, no metric, just signature

All constants are equal to 1



In [32]:

    
from sympy import *
variables = (t, x, y, z, w) = symbols('t x y z w', real=True)
print(variables)









    



(t, x, y, z, w)



In [33]:

    
metric=[1
        ,-1
        ,-1
        ,-1
        ,-1]

myBasis='gamma_t gamma_x gamma_y gamma_z gamma_w'

sp5d = Ga(myBasis, g=metric, coords=variables,norm=True)
(gamma_t, gamma_x, gamma_y, gamma_z, gamma_w) = sp5d.mv()
(grad, rgrad) = sp5d.grads()

Quaternions

http://en.wikipedia.org/wiki/Quaternion



In [34]:

    
iquat=gamma_y*gamma_z
jquat=gamma_z*gamma_x
kquat=gamma_x*gamma_y
iquat.texLabel='\\mathit{\\boldsymbol{i}}'
jquat.texLabel='\\mathit{\\boldsymbol{j}}'
kquat.texLabel='\\mathit{\\boldsymbol{k}}'

display(Math('(1,'+iquat.texLabel+','+jquat.texLabel+','+kquat.texLabel+')'))
CheckProperties(iquat,jquat,kquat,iquat.texLabel,jquat.texLabel,kquat.texLabel)









    





$$(1,\mathit{\boldsymbol{i}},\mathit{\boldsymbol{j}},\mathit{\boldsymbol{k}})$$






    





$$\mathit{\boldsymbol{i}}=\begin{equation*}  \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \end{equation*}$$






    





$$\mathit{\boldsymbol{j}}=\begin{equation*} - \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} \end{equation*}$$






    





$$\mathit{\boldsymbol{k}}=\begin{equation*}  \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} \end{equation*}$$






    





$$\mathit{\boldsymbol{i}}^2=\begin{equation*} -1 \end{equation*}$$






    





$$\mathit{\boldsymbol{j}}^2=\begin{equation*} -1 \end{equation*}$$






    





$$\mathit{\boldsymbol{k}}^2=\begin{equation*} -1 \end{equation*}$$






    





$$\mathit{\boldsymbol{i}}\mathit{\boldsymbol{j}}=\begin{equation*}  \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} \end{equation*}$$






    





$$\mathit{\boldsymbol{j}}\mathit{\boldsymbol{i}}=\begin{equation*} - \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} \end{equation*}$$






    





$$\mathit{\boldsymbol{j}}\mathit{\boldsymbol{k}}=\begin{equation*}  \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \end{equation*}$$






    





$$\mathit{\boldsymbol{k}}\mathit{\boldsymbol{j}}=\begin{equation*} - \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \end{equation*}$$






    





$$\mathit{\boldsymbol{k}}\mathit{\boldsymbol{i}}=\begin{equation*} - \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} \end{equation*}$$






    





$$\mathit{\boldsymbol{i}}\mathit{\boldsymbol{k}}=\begin{equation*}  \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} \end{equation*}$$






    





$$\mathit{\boldsymbol{i}}\mathit{\boldsymbol{j}}\mathit{\boldsymbol{k}}=\begin{equation*} -1 \end{equation*}$$

Imaginary unit



In [35]:

    
imag=gamma_w
imag.texLabel='i'

displayWithTitle(imag, title=imag.texLabel)
displayWithTitle((imag*imag), title=imag.texLabel+'^2')









    





$$i=\begin{equation*}  \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$i^2=\begin{equation*} -1 \end{equation*}$$

Associative Hyperbolic Quaternions



In [36]:

    
ihquat=gamma_t
jhquat=gamma_t*gamma_x*gamma_y*gamma_z*gamma_w
khquat=gamma_x*gamma_y*gamma_z*gamma_w
ihquat.texLabel='\\mathbf{i}'
jhquat.texLabel='\\mathbf{j}'
khquat.texLabel='\\mathbf{k}'

display(Math('(1,'+ihquat.texLabel+','+jhquat.texLabel+','+khquat.texLabel+')'))
CheckProperties(ihquat,jhquat,khquat,ihquat.texLabel,jhquat.texLabel,khquat.texLabel)









    





$$(1,\mathbf{i},\mathbf{j},\mathbf{k})$$






    





$$\mathbf{i}=\begin{equation*}  \boldsymbol{\gamma }_{t} \end{equation*}$$






    





$$\mathbf{j}=\begin{equation*}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$\mathbf{k}=\begin{equation*}  \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$\mathbf{i}^2=\begin{equation*} 1 \end{equation*}$$






    





$$\mathbf{j}^2=\begin{equation*} 1 \end{equation*}$$






    





$$\mathbf{k}^2=\begin{equation*} 1 \end{equation*}$$






    





$$\mathbf{i}\mathbf{j}=\begin{equation*}  \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$\mathbf{j}\mathbf{i}=\begin{equation*}  \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$\mathbf{j}\mathbf{k}=\begin{equation*}  \boldsymbol{\gamma }_{t} \end{equation*}$$






    





$$\mathbf{k}\mathbf{j}=\begin{equation*}  \boldsymbol{\gamma }_{t} \end{equation*}$$






    





$$\mathbf{k}\mathbf{i}=\begin{equation*}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$\mathbf{i}\mathbf{k}=\begin{equation*}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$\mathbf{i}\mathbf{j}\mathbf{k}=\begin{equation*} 1 \end{equation*}$$

DIRAC

http://en.wikipedia.org/wiki/Dirac_equation

Regular $$0=({\gamma}_0 \frac{\partial}{\partial t}+{\gamma}_1 \frac{\partial}{\partial x}+{\gamma}_2 \frac{\partial}{\partial y}+{\gamma}_3 \frac{\partial}{\partial z}+im) {\psi}$$

Adjoint $$0={\overline{\psi}}(\frac{\partial}{\partial t}{\gamma}_0+\frac{\partial}{\partial x}{\gamma}_1+\frac{\partial}{\partial y}{\gamma}_2+\frac{\partial}{\partial z}{\gamma}_3+im)$$

Gradient definition



In [37]:

    
displayWithTitle(grad,title='\overrightarrow{D} = \Sigma e_i \partial x_i')
displayWithTitle(rgrad,title='\overleftarrow{D} = \Sigma \partial x_i e_i')









    





$$\overrightarrow{D} = \Sigma e_i \partial x_i=\begin{equation*} gamma_t D{t} -gamma_x D{x} -gamma_y D{y} -gamma_z D{z} -gamma_w D{w} \end{equation*}$$






    





$$\overleftarrow{D} = \Sigma \partial x_i e_i=\begin{equation*} D{t} gamma_t -D{x}gamma_x -D{y}gamma_y -D{z}gamma_z -D{w}gamma_w \end{equation*}$$

PHYSICS

The following symbols are defined :

Energy $$E \in \mathbb{R}$$

Mass $$m \in \mathbb{R}$$



In [38]:

    
m, E, p_x, p_y, p_z = symbols('m E p_x p_y p_z', real=True)

rquat = [iquat, jquat, kquat]
pv =[p_x, p_y, p_z]
p = S(0)
for (dim, var) in zip(pv, rquat):
    p += var * dim
p.texLabel='\\mathbf{p}'

display(Latex('Momentum $'+p.texLabel+'$ is defined with $p_x, p_y, p_z \\in \\mathbb{R}$'))
display(Math(p.texLabel+'=p_x'+iquat.texLabel+'+p_y'+jquat.texLabel+'+p_z'+kquat.texLabel))
displayWithTitle(p, title=p.texLabel)









    





Momentum $\mathbf{p}$ is defined with $p_x, p_y, p_z \in \mathbb{R}$






    





$$\mathbf{p}=p_x\mathit{\boldsymbol{i}}+p_y\mathit{\boldsymbol{j}}+p_z\mathit{\boldsymbol{k}}$$






    





$$\mathbf{p}=\begin{equation*} p_{z} \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} - p_{y} \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} + p_{x} \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \end{equation*}$$



In [39]:

    
f=m*w-imag*(E*t+p_x*x+p_y*y+p_z*z)

displayWithTitle(f, title='f')
displayWithTitle(grad*f, title='\overrightarrow{D}f')
displayWithTitle(f*rgrad, title='f\overleftarrow{D}')
displayWithTitle(grad*f*grad*f, title='\overrightarrow{D}f\overrightarrow{D}f')
displayWithTitle(f*rgrad*f*rgrad, title='f\overleftarrow{D}f\overleftarrow{D}')
displayWithTitle((grad*f - f*rgrad)/2, title='1/2 (\overrightarrow{D}f-f\overleftarrow{D})')
displayWithTitle((grad*f + f*rgrad)/2, title='1/2 (\overrightarrow{D}f+f\overleftarrow{D})')









    





$$f=\begin{equation*} m w  + \left ( - E t - p_{x} x - p_{y} y - p_{z} z\right ) \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$\overrightarrow{D}f=\begin{equation*} - m \boldsymbol{\gamma }_{w} - E \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{w} + p_{x} \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{w} + p_{y} \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} + p_{z} \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$f\overleftarrow{D}=\begin{equation*} - m \boldsymbol{\gamma }_{w} + E \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{w} - p_{x} \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{w} - p_{y} \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} - p_{z} \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$\overrightarrow{D}f\overrightarrow{D}f=\begin{equation*} E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2} \end{equation*}$$






    





$$f\overleftarrow{D}f\overleftarrow{D}=\begin{equation*} E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2} \end{equation*}$$






    





$$1/2 (\overrightarrow{D}f-f\overleftarrow{D})=\begin{equation*} - E \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{w} + p_{x} \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{w} + p_{y} \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} + p_{z} \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$1/2 (\overrightarrow{D}f+f\overleftarrow{D})=\begin{equation*} - m \boldsymbol{\gamma }_{w} \end{equation*}$$

Density element combined derivatives

See here http://www.bbk.ac.uk/tpru/BasilHiley/Bohm-Vienna.pdf



In [40]:

    
K={}

K[1]=(ihquat*imag*E-imag*m+khquat*p)*(1+jhquat)
K[1].texLabel='('+ihquat.texLabel+imag.texLabel+'E-'+imag.texLabel+'m+'+khquat.texLabel+p.texLabel+')(1+'+jhquat.texLabel+')'

texLabel='('+ihquat.texLabel+imag.texLabel+'E+'+imag.texLabel+'m+'+khquat.texLabel+p.texLabel+')'+'(1-'+jhquat.texLabel+')'+imag.texLabel
K[2]=(ihquat*imag*E+imag*m+khquat*p)*(1-jhquat)*imag
K[2].texLabel=texLabel
texLabel='-'+'('+ihquat.texLabel+imag.texLabel+'E+'+imag.texLabel+'m+'+khquat.texLabel+p.texLabel+')'+imag.texLabel


def showDerivatives(f, i, k):
    displayWithTitle(k, 'k_' + str(i) + '=' + k.texLabel)
    displayWithTitle(k*k, 'k_' + str(i) + '^2')
    rho=(f*k)*(k*f)
    displayWithTitle(rho, title='\\rho')
    left=grad*rho
    right=rho*rgrad
    display(Latex('Combine regular and adjoint Dirac equations with difference'))
    displayWithTitle((left-right)/2, '1/2 (\overrightarrow{D}\\rho-\\rho\overleftarrow{D})')
    display(Latex('Combine regular and adjoint Dirac equations with addition'))
    displayWithTitle((left+right)/2, '1/2 (\overrightarrow{D}\\rho+\\rho\overleftarrow{D})')

$$k_1$$

but density element seems null



In [41]:

    
showDerivatives(f, 1, K[1])









    





$$k_1=(\mathbf{i}iE-im+\mathbf{k}\mathbf{p})(1+\mathbf{j})=\begin{equation*} - m \boldsymbol{\gamma }_{w} + E \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{w} - p_{x} \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{w} - p_{y} \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} - p_{z} \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + p_{z} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} - p_{y} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} + p_{x} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} - E \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + m \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \end{equation*}$$






    





$$k_1^2=\begin{equation*} \left ( 2 E^{2} - 2 m^{2} - 2 {\left ( p_{x} \right )}^{2} - 2 {\left ( p_{y} \right )}^{2} - 2 {\left ( p_{z} \right )}^{2}\right )  + \left ( 2 E^{2} - 2 m^{2} - 2 {\left ( p_{x} \right )}^{2} - 2 {\left ( p_{y} \right )}^{2} - 2 {\left ( p_{z} \right )}^{2}\right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$\rho=\begin{equation*} \left ( - 2 E^{4} t^{2} - 4 E^{3} p_{x} t x - 4 E^{3} p_{y} t y - 4 E^{3} p_{z} t z + 2 E^{2} m^{2} t^{2} + 2 E^{2} m^{2} w^{2} + 2 E^{2} {\left ( p_{x} \right )}^{2} t^{2} - 2 E^{2} {\left ( p_{x} \right )}^{2} x^{2} - 4 E^{2} p_{x} p_{y} x y - 4 E^{2} p_{x} p_{z} x z + 2 E^{2} {\left ( p_{y} \right )}^{2} t^{2} - 2 E^{2} {\left ( p_{y} \right )}^{2} y^{2} - 4 E^{2} p_{y} p_{z} y z + 2 E^{2} {\left ( p_{z} \right )}^{2} t^{2} - 2 E^{2} {\left ( p_{z} \right )}^{2} z^{2} + 4 E m^{2} p_{x} t x + 4 E m^{2} p_{y} t y + 4 E m^{2} p_{z} t z + 4 E {\left ( p_{x} \right )}^{3} t x + 4 E {\left ( p_{x} \right )}^{2} p_{y} t y + 4 E {\left ( p_{x} \right )}^{2} p_{z} t z + 4 E p_{x} {\left ( p_{y} \right )}^{2} t x + 4 E p_{x} {\left ( p_{z} \right )}^{2} t x + 4 E {\left ( p_{y} \right )}^{3} t y + 4 E {\left ( p_{y} \right )}^{2} p_{z} t z + 4 E p_{y} {\left ( p_{z} \right )}^{2} t y + 4 E {\left ( p_{z} \right )}^{3} t z - 2 m^{4} w^{2} - 2 m^{2} {\left ( p_{x} \right )}^{2} w^{2} + 2 m^{2} {\left ( p_{x} \right )}^{2} x^{2} + 4 m^{2} p_{x} p_{y} x y + 4 m^{2} p_{x} p_{z} x z - 2 m^{2} {\left ( p_{y} \right )}^{2} w^{2} + 2 m^{2} {\left ( p_{y} \right )}^{2} y^{2} + 4 m^{2} p_{y} p_{z} y z - 2 m^{2} {\left ( p_{z} \right )}^{2} w^{2} + 2 m^{2} {\left ( p_{z} \right )}^{2} z^{2} + 2 {\left ( p_{x} \right )}^{4} x^{2} + 4 {\left ( p_{x} \right )}^{3} p_{y} x y + 4 {\left ( p_{x} \right )}^{3} p_{z} x z + 2 {\left ( p_{x} \right )}^{2} {\left ( p_{y} \right )}^{2} x^{2} + 2 {\left ( p_{x} \right )}^{2} {\left ( p_{y} \right )}^{2} y^{2} + 4 {\left ( p_{x} \right )}^{2} p_{y} p_{z} y z + 2 {\left ( p_{x} \right )}^{2} {\left ( p_{z} \right )}^{2} x^{2} + 2 {\left ( p_{x} \right )}^{2} {\left ( p_{z} \right )}^{2} z^{2} + 4 p_{x} {\left ( p_{y} \right )}^{3} x y + 4 p_{x} {\left ( p_{y} \right )}^{2} p_{z} x z + 4 p_{x} p_{y} {\left ( p_{z} \right )}^{2} x y + 4 p_{x} {\left ( p_{z} \right )}^{3} x z + 2 {\left ( p_{y} \right )}^{4} y^{2} + 4 {\left ( p_{y} \right )}^{3} p_{z} y z + 2 {\left ( p_{y} \right )}^{2} {\left ( p_{z} \right )}^{2} y^{2} + 2 {\left ( p_{y} \right )}^{2} {\left ( p_{z} \right )}^{2} z^{2} + 4 p_{y} {\left ( p_{z} \right )}^{3} y z + 2 {\left ( p_{z} \right )}^{4} z^{2}\right )  + 4 m w \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z + E m^{2} t + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t + m^{2} p_{x} x + m^{2} p_{y} y + m^{2} p_{z} z + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{w} + 4 m w \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z - E m^{2} t - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t - m^{2} p_{x} x - m^{2} p_{y} y - m^{2} p_{z} z - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + \left ( - 2 E^{4} t^{2} - 4 E^{3} p_{x} t x - 4 E^{3} p_{y} t y - 4 E^{3} p_{z} t z + 2 E^{2} m^{2} t^{2} + 2 E^{2} m^{2} w^{2} + 2 E^{2} {\left ( p_{x} \right )}^{2} t^{2} - 2 E^{2} {\left ( p_{x} \right )}^{2} x^{2} - 4 E^{2} p_{x} p_{y} x y - 4 E^{2} p_{x} p_{z} x z + 2 E^{2} {\left ( p_{y} \right )}^{2} t^{2} - 2 E^{2} {\left ( p_{y} \right )}^{2} y^{2} - 4 E^{2} p_{y} p_{z} y z + 2 E^{2} {\left ( p_{z} \right )}^{2} t^{2} - 2 E^{2} {\left ( p_{z} \right )}^{2} z^{2} + 4 E m^{2} p_{x} t x + 4 E m^{2} p_{y} t y + 4 E m^{2} p_{z} t z + 4 E {\left ( p_{x} \right )}^{3} t x + 4 E {\left ( p_{x} \right )}^{2} p_{y} t y + 4 E {\left ( p_{x} \right )}^{2} p_{z} t z + 4 E p_{x} {\left ( p_{y} \right )}^{2} t x + 4 E p_{x} {\left ( p_{z} \right )}^{2} t x + 4 E {\left ( p_{y} \right )}^{3} t y + 4 E {\left ( p_{y} \right )}^{2} p_{z} t z + 4 E p_{y} {\left ( p_{z} \right )}^{2} t y + 4 E {\left ( p_{z} \right )}^{3} t z - 2 m^{4} w^{2} - 2 m^{2} {\left ( p_{x} \right )}^{2} w^{2} + 2 m^{2} {\left ( p_{x} \right )}^{2} x^{2} + 4 m^{2} p_{x} p_{y} x y + 4 m^{2} p_{x} p_{z} x z - 2 m^{2} {\left ( p_{y} \right )}^{2} w^{2} + 2 m^{2} {\left ( p_{y} \right )}^{2} y^{2} + 4 m^{2} p_{y} p_{z} y z - 2 m^{2} {\left ( p_{z} \right )}^{2} w^{2} + 2 m^{2} {\left ( p_{z} \right )}^{2} z^{2} + 2 {\left ( p_{x} \right )}^{4} x^{2} + 4 {\left ( p_{x} \right )}^{3} p_{y} x y + 4 {\left ( p_{x} \right )}^{3} p_{z} x z + 2 {\left ( p_{x} \right )}^{2} {\left ( p_{y} \right )}^{2} x^{2} + 2 {\left ( p_{x} \right )}^{2} {\left ( p_{y} \right )}^{2} y^{2} + 4 {\left ( p_{x} \right )}^{2} p_{y} p_{z} y z + 2 {\left ( p_{x} \right )}^{2} {\left ( p_{z} \right )}^{2} x^{2} + 2 {\left ( p_{x} \right )}^{2} {\left ( p_{z} \right )}^{2} z^{2} + 4 p_{x} {\left ( p_{y} \right )}^{3} x y + 4 p_{x} {\left ( p_{y} \right )}^{2} p_{z} x z + 4 p_{x} p_{y} {\left ( p_{z} \right )}^{2} x y + 4 p_{x} {\left ( p_{z} \right )}^{3} x z + 2 {\left ( p_{y} \right )}^{4} y^{2} + 4 {\left ( p_{y} \right )}^{3} p_{z} y z + 2 {\left ( p_{y} \right )}^{2} {\left ( p_{z} \right )}^{2} y^{2} + 2 {\left ( p_{y} \right )}^{2} {\left ( p_{z} \right )}^{2} z^{2} + 4 p_{y} {\left ( p_{z} \right )}^{3} y z + 2 {\left ( p_{z} \right )}^{4} z^{2}\right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





Combine regular and adjoint Dirac equations with difference






    





$$1/2 (\overrightarrow{D}\rho-\rho\overleftarrow{D})=\begin{equation*} 4 E m w \left(- E^{2} + m^{2} + {\left ( p_{x} \right )}^{2} + {\left ( p_{y} \right )}^{2} + {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{w} + 4 m p_{x} w \left(E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{w} + 4 m p_{y} w \left(E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} + 4 m p_{z} w \left(E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 4 m p_{z} w \left(- E^{2} + m^{2} + {\left ( p_{x} \right )}^{2} + {\left ( p_{y} \right )}^{2} + {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} + 4 m p_{y} w \left(E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} + 4 m p_{x} w \left(- E^{2} + m^{2} + {\left ( p_{x} \right )}^{2} + {\left ( p_{y} \right )}^{2} + {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + 4 E m w \left(E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \end{equation*}$$






    





Combine regular and adjoint Dirac equations with addition






    





$$1/2 (\overrightarrow{D}\rho+\rho\overleftarrow{D})=\begin{equation*} 4 m \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z + E m^{2} t + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t + m^{2} p_{x} x + m^{2} p_{y} y + m^{2} p_{z} z + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right)  + 4 E \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z + E m^{2} t + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t + m^{2} p_{x} x + m^{2} p_{y} y + m^{2} p_{z} z + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{t} + 4 p_{x} \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z - E m^{2} t - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t - m^{2} p_{x} x - m^{2} p_{y} y - m^{2} p_{z} z - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{x} + 4 p_{y} \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z - E m^{2} t - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t - m^{2} p_{x} x - m^{2} p_{y} y - m^{2} p_{z} z - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{y} + 4 p_{z} \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z - E m^{2} t - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t - m^{2} p_{x} x - m^{2} p_{y} y - m^{2} p_{z} z - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{z} + 4 m^{2} w \left(- E^{2} + m^{2} + {\left ( p_{x} \right )}^{2} + {\left ( p_{y} \right )}^{2} + {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{w} + 4 m^{2} w \left(E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + 4 p_{z} \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z - E m^{2} t - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t - m^{2} p_{x} x - m^{2} p_{y} y - m^{2} p_{z} z - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} + 4 p_{y} \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z + E m^{2} t + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t + m^{2} p_{x} x + m^{2} p_{y} y + m^{2} p_{z} z + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 4 p_{x} \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z - E m^{2} t - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t - m^{2} p_{x} x - m^{2} p_{y} y - m^{2} p_{z} z - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 4 E \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z + E m^{2} t + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t + m^{2} p_{x} x + m^{2} p_{y} y + m^{2} p_{z} z + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 4 m \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z + E m^{2} t + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t + m^{2} p_{x} x + m^{2} p_{y} y + m^{2} p_{z} z + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$

$$k_2$$

but density element seems null



In [42]:

    
showDerivatives(f, 2, K[2])









    





$$k_2=(\mathbf{i}iE+im+\mathbf{k}\mathbf{p})(1-\mathbf{j})i=\begin{equation*} - m  - E \boldsymbol{\gamma }_{t} + p_{x} \boldsymbol{\gamma }_{x} + p_{y} \boldsymbol{\gamma }_{y} + p_{z} \boldsymbol{\gamma }_{z} - p_{z} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} + p_{y} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} - p_{x} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + E \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + m \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$k_2^2=\begin{equation*} \left ( 2 E^{2} + 2 m^{2} - 2 {\left ( p_{x} \right )}^{2} - 2 {\left ( p_{y} \right )}^{2} - 2 {\left ( p_{z} \right )}^{2}\right )  + 4 E m \boldsymbol{\gamma }_{t} - 4 m p_{x} \boldsymbol{\gamma }_{x} - 4 m p_{y} \boldsymbol{\gamma }_{y} - 4 m p_{z} \boldsymbol{\gamma }_{z} + 4 m p_{z} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} - 4 m p_{y} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 4 m p_{x} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} - 4 E m \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + \left ( - 2 E^{2} - 2 m^{2} + 2 {\left ( p_{x} \right )}^{2} + 2 {\left ( p_{y} \right )}^{2} + 2 {\left ( p_{z} \right )}^{2}\right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$\rho=\begin{equation*} \left ( - 2 E^{4} t^{2} - 4 E^{3} p_{x} t x - 4 E^{3} p_{y} t y - 4 E^{3} p_{z} t z - 2 E^{2} m^{2} t^{2} + 2 E^{2} m^{2} w^{2} + 2 E^{2} {\left ( p_{x} \right )}^{2} t^{2} - 2 E^{2} {\left ( p_{x} \right )}^{2} x^{2} - 4 E^{2} p_{x} p_{y} x y - 4 E^{2} p_{x} p_{z} x z + 2 E^{2} {\left ( p_{y} \right )}^{2} t^{2} - 2 E^{2} {\left ( p_{y} \right )}^{2} y^{2} - 4 E^{2} p_{y} p_{z} y z + 2 E^{2} {\left ( p_{z} \right )}^{2} t^{2} - 2 E^{2} {\left ( p_{z} \right )}^{2} z^{2} - 4 E m^{2} p_{x} t x - 4 E m^{2} p_{y} t y - 4 E m^{2} p_{z} t z + 4 E {\left ( p_{x} \right )}^{3} t x + 4 E {\left ( p_{x} \right )}^{2} p_{y} t y + 4 E {\left ( p_{x} \right )}^{2} p_{z} t z + 4 E p_{x} {\left ( p_{y} \right )}^{2} t x + 4 E p_{x} {\left ( p_{z} \right )}^{2} t x + 4 E {\left ( p_{y} \right )}^{3} t y + 4 E {\left ( p_{y} \right )}^{2} p_{z} t z + 4 E p_{y} {\left ( p_{z} \right )}^{2} t y + 4 E {\left ( p_{z} \right )}^{3} t z + 2 m^{4} w^{2} - 2 m^{2} {\left ( p_{x} \right )}^{2} w^{2} - 2 m^{2} {\left ( p_{x} \right )}^{2} x^{2} - 4 m^{2} p_{x} p_{y} x y - 4 m^{2} p_{x} p_{z} x z - 2 m^{2} {\left ( p_{y} \right )}^{2} w^{2} - 2 m^{2} {\left ( p_{y} \right )}^{2} y^{2} - 4 m^{2} p_{y} p_{z} y z - 2 m^{2} {\left ( p_{z} \right )}^{2} w^{2} - 2 m^{2} {\left ( p_{z} \right )}^{2} z^{2} + 2 {\left ( p_{x} \right )}^{4} x^{2} + 4 {\left ( p_{x} \right )}^{3} p_{y} x y + 4 {\left ( p_{x} \right )}^{3} p_{z} x z + 2 {\left ( p_{x} \right )}^{2} {\left ( p_{y} \right )}^{2} x^{2} + 2 {\left ( p_{x} \right )}^{2} {\left ( p_{y} \right )}^{2} y^{2} + 4 {\left ( p_{x} \right )}^{2} p_{y} p_{z} y z + 2 {\left ( p_{x} \right )}^{2} {\left ( p_{z} \right )}^{2} x^{2} + 2 {\left ( p_{x} \right )}^{2} {\left ( p_{z} \right )}^{2} z^{2} + 4 p_{x} {\left ( p_{y} \right )}^{3} x y + 4 p_{x} {\left ( p_{y} \right )}^{2} p_{z} x z + 4 p_{x} p_{y} {\left ( p_{z} \right )}^{2} x y + 4 p_{x} {\left ( p_{z} \right )}^{3} x z + 2 {\left ( p_{y} \right )}^{4} y^{2} + 4 {\left ( p_{y} \right )}^{3} p_{z} y z + 2 {\left ( p_{y} \right )}^{2} {\left ( p_{z} \right )}^{2} y^{2} + 2 {\left ( p_{y} \right )}^{2} {\left ( p_{z} \right )}^{2} z^{2} + 4 p_{y} {\left ( p_{z} \right )}^{3} y z + 2 {\left ( p_{z} \right )}^{4} z^{2}\right )  + 4 E m \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{t} - 4 m p_{x} \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{x} - 4 m p_{y} \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{y} - 4 m p_{z} \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{z} + 4 m w \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z - E m^{2} t + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t - m^{2} p_{x} x - m^{2} p_{y} y - m^{2} p_{z} z + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{w} + 4 m w \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z - E m^{2} t + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t - m^{2} p_{x} x - m^{2} p_{y} y - m^{2} p_{z} z + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + 4 m p_{z} \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} - 4 m p_{y} \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 4 m p_{x} \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} - 4 E m \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + \left ( 2 E^{4} t^{2} + 4 E^{3} p_{x} t x + 4 E^{3} p_{y} t y + 4 E^{3} p_{z} t z + 2 E^{2} m^{2} t^{2} - 2 E^{2} m^{2} w^{2} - 2 E^{2} {\left ( p_{x} \right )}^{2} t^{2} + 2 E^{2} {\left ( p_{x} \right )}^{2} x^{2} + 4 E^{2} p_{x} p_{y} x y + 4 E^{2} p_{x} p_{z} x z - 2 E^{2} {\left ( p_{y} \right )}^{2} t^{2} + 2 E^{2} {\left ( p_{y} \right )}^{2} y^{2} + 4 E^{2} p_{y} p_{z} y z - 2 E^{2} {\left ( p_{z} \right )}^{2} t^{2} + 2 E^{2} {\left ( p_{z} \right )}^{2} z^{2} + 4 E m^{2} p_{x} t x + 4 E m^{2} p_{y} t y + 4 E m^{2} p_{z} t z - 4 E {\left ( p_{x} \right )}^{3} t x - 4 E {\left ( p_{x} \right )}^{2} p_{y} t y - 4 E {\left ( p_{x} \right )}^{2} p_{z} t z - 4 E p_{x} {\left ( p_{y} \right )}^{2} t x - 4 E p_{x} {\left ( p_{z} \right )}^{2} t x - 4 E {\left ( p_{y} \right )}^{3} t y - 4 E {\left ( p_{y} \right )}^{2} p_{z} t z - 4 E p_{y} {\left ( p_{z} \right )}^{2} t y - 4 E {\left ( p_{z} \right )}^{3} t z - 2 m^{4} w^{2} + 2 m^{2} {\left ( p_{x} \right )}^{2} w^{2} + 2 m^{2} {\left ( p_{x} \right )}^{2} x^{2} + 4 m^{2} p_{x} p_{y} x y + 4 m^{2} p_{x} p_{z} x z + 2 m^{2} {\left ( p_{y} \right )}^{2} w^{2} + 2 m^{2} {\left ( p_{y} \right )}^{2} y^{2} + 4 m^{2} p_{y} p_{z} y z + 2 m^{2} {\left ( p_{z} \right )}^{2} w^{2} + 2 m^{2} {\left ( p_{z} \right )}^{2} z^{2} - 2 {\left ( p_{x} \right )}^{4} x^{2} - 4 {\left ( p_{x} \right )}^{3} p_{y} x y - 4 {\left ( p_{x} \right )}^{3} p_{z} x z - 2 {\left ( p_{x} \right )}^{2} {\left ( p_{y} \right )}^{2} x^{2} - 2 {\left ( p_{x} \right )}^{2} {\left ( p_{y} \right )}^{2} y^{2} - 4 {\left ( p_{x} \right )}^{2} p_{y} p_{z} y z - 2 {\left ( p_{x} \right )}^{2} {\left ( p_{z} \right )}^{2} x^{2} - 2 {\left ( p_{x} \right )}^{2} {\left ( p_{z} \right )}^{2} z^{2} - 4 p_{x} {\left ( p_{y} \right )}^{3} x y - 4 p_{x} {\left ( p_{y} \right )}^{2} p_{z} x z - 4 p_{x} p_{y} {\left ( p_{z} \right )}^{2} x y - 4 p_{x} {\left ( p_{z} \right )}^{3} x z - 2 {\left ( p_{y} \right )}^{4} y^{2} - 4 {\left ( p_{y} \right )}^{3} p_{z} y z - 2 {\left ( p_{y} \right )}^{2} {\left ( p_{z} \right )}^{2} y^{2} - 2 {\left ( p_{y} \right )}^{2} {\left ( p_{z} \right )}^{2} z^{2} - 4 p_{y} {\left ( p_{z} \right )}^{3} y z - 2 {\left ( p_{z} \right )}^{4} z^{2}\right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





Combine regular and adjoint Dirac equations with difference






    





$$1/2 (\overrightarrow{D}\rho-\rho\overleftarrow{D})=\begin{equation*} 4 E m w \left(- E^{2} + m^{2} + {\left ( p_{x} \right )}^{2} + {\left ( p_{y} \right )}^{2} + {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{w} + 4 m p_{x} w \left(E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{w} + 4 m p_{y} w \left(E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} + 4 m p_{z} w \left(E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 4 m p_{z} w \left(E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} + 4 m p_{y} w \left(- E^{2} + m^{2} + {\left ( p_{x} \right )}^{2} + {\left ( p_{y} \right )}^{2} + {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} + 4 m p_{x} w \left(E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + 4 E m w \left(- E^{2} + m^{2} + {\left ( p_{x} \right )}^{2} + {\left ( p_{y} \right )}^{2} + {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \end{equation*}$$






    





Combine regular and adjoint Dirac equations with addition






    





$$1/2 (\overrightarrow{D}\rho+\rho\overleftarrow{D})=\begin{equation*} 4 m \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z - E m^{2} t - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t - m^{2} p_{x} x - m^{2} p_{y} y - m^{2} p_{z} z - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right)  + 4 E \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z - E m^{2} t + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t - m^{2} p_{x} x - m^{2} p_{y} y - m^{2} p_{z} z + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{t} + 4 p_{x} \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z + E m^{2} t - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t + m^{2} p_{x} x + m^{2} p_{y} y + m^{2} p_{z} z - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{x} + 4 p_{y} \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z + E m^{2} t - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t + m^{2} p_{x} x + m^{2} p_{y} y + m^{2} p_{z} z - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{y} + 4 p_{z} \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z + E m^{2} t - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t + m^{2} p_{x} x + m^{2} p_{y} y + m^{2} p_{z} z - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{z} + 4 m^{2} w \left(- E^{2} - m^{2} + {\left ( p_{x} \right )}^{2} + {\left ( p_{y} \right )}^{2} + {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{w} + 4 m^{2} w \left(- E^{2} - m^{2} + {\left ( p_{x} \right )}^{2} + {\left ( p_{y} \right )}^{2} + {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + 4 p_{z} \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z - E m^{2} t + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t - m^{2} p_{x} x - m^{2} p_{y} y - m^{2} p_{z} z + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} + 4 p_{y} \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z + E m^{2} t - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t + m^{2} p_{x} x + m^{2} p_{y} y + m^{2} p_{z} z - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 4 p_{x} \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z - E m^{2} t + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t - m^{2} p_{x} x - m^{2} p_{y} y - m^{2} p_{z} z + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 4 E \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z + E m^{2} t - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t + m^{2} p_{x} x + m^{2} p_{y} y + m^{2} p_{z} z - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 4 m \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z + E m^{2} t + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t + m^{2} p_{x} x + m^{2} p_{y} y + m^{2} p_{z} z + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$

$$k_1 + k_2$$

density element is not null



In [43]:

    
sum=K[1]+K[2]
sum.texLabel='k_1+k_2'
showDerivatives(f, 3, sum)









    





$$k_3=k_1+k_2=\begin{equation*} - m  - E \boldsymbol{\gamma }_{t} + p_{x} \boldsymbol{\gamma }_{x} + p_{y} \boldsymbol{\gamma }_{y} + p_{z} \boldsymbol{\gamma }_{z} - m \boldsymbol{\gamma }_{w} + E \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{w} - p_{x} \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{w} - p_{y} \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} - p_{z} \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + p_{z} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} - p_{y} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} + p_{x} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} - E \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + m \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} - p_{z} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} + p_{y} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} - p_{x} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + E \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + m \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$k_3^2=\begin{equation*} \left ( 4 E^{2} - 4 {\left ( p_{x} \right )}^{2} - 4 {\left ( p_{y} \right )}^{2} - 4 {\left ( p_{z} \right )}^{2}\right )  + 4 E m \boldsymbol{\gamma }_{t} - 4 m p_{x} \boldsymbol{\gamma }_{x} - 4 m p_{y} \boldsymbol{\gamma }_{y} - 4 m p_{z} \boldsymbol{\gamma }_{z} + 4 m p_{z} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} - 4 m p_{y} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 4 m p_{x} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} - 4 E m \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} - 4 m^{2} \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





$$\rho=\begin{equation*} \left ( - 4 E^{4} t^{2} - 8 E^{3} p_{x} t x - 8 E^{3} p_{y} t y - 8 E^{3} p_{z} t z + 4 E^{2} m^{2} w^{2} + 4 E^{2} {\left ( p_{x} \right )}^{2} t^{2} - 4 E^{2} {\left ( p_{x} \right )}^{2} x^{2} - 8 E^{2} p_{x} p_{y} x y - 8 E^{2} p_{x} p_{z} x z + 4 E^{2} {\left ( p_{y} \right )}^{2} t^{2} - 4 E^{2} {\left ( p_{y} \right )}^{2} y^{2} - 8 E^{2} p_{y} p_{z} y z + 4 E^{2} {\left ( p_{z} \right )}^{2} t^{2} - 4 E^{2} {\left ( p_{z} \right )}^{2} z^{2} + 8 E {\left ( p_{x} \right )}^{3} t x + 8 E {\left ( p_{x} \right )}^{2} p_{y} t y + 8 E {\left ( p_{x} \right )}^{2} p_{z} t z + 8 E p_{x} {\left ( p_{y} \right )}^{2} t x + 8 E p_{x} {\left ( p_{z} \right )}^{2} t x + 8 E {\left ( p_{y} \right )}^{3} t y + 8 E {\left ( p_{y} \right )}^{2} p_{z} t z + 8 E p_{y} {\left ( p_{z} \right )}^{2} t y + 8 E {\left ( p_{z} \right )}^{3} t z - 4 m^{2} {\left ( p_{x} \right )}^{2} w^{2} - 4 m^{2} {\left ( p_{y} \right )}^{2} w^{2} - 4 m^{2} {\left ( p_{z} \right )}^{2} w^{2} + 4 {\left ( p_{x} \right )}^{4} x^{2} + 8 {\left ( p_{x} \right )}^{3} p_{y} x y + 8 {\left ( p_{x} \right )}^{3} p_{z} x z + 4 {\left ( p_{x} \right )}^{2} {\left ( p_{y} \right )}^{2} x^{2} + 4 {\left ( p_{x} \right )}^{2} {\left ( p_{y} \right )}^{2} y^{2} + 8 {\left ( p_{x} \right )}^{2} p_{y} p_{z} y z + 4 {\left ( p_{x} \right )}^{2} {\left ( p_{z} \right )}^{2} x^{2} + 4 {\left ( p_{x} \right )}^{2} {\left ( p_{z} \right )}^{2} z^{2} + 8 p_{x} {\left ( p_{y} \right )}^{3} x y + 8 p_{x} {\left ( p_{y} \right )}^{2} p_{z} x z + 8 p_{x} p_{y} {\left ( p_{z} \right )}^{2} x y + 8 p_{x} {\left ( p_{z} \right )}^{3} x z + 4 {\left ( p_{y} \right )}^{4} y^{2} + 8 {\left ( p_{y} \right )}^{3} p_{z} y z + 4 {\left ( p_{y} \right )}^{2} {\left ( p_{z} \right )}^{2} y^{2} + 4 {\left ( p_{y} \right )}^{2} {\left ( p_{z} \right )}^{2} z^{2} + 8 p_{y} {\left ( p_{z} \right )}^{3} y z + 4 {\left ( p_{z} \right )}^{4} z^{2}\right )  + 4 E m \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{t} - 4 m p_{x} \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{x} - 4 m p_{y} \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{y} - 4 m p_{z} \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{z} + 8 m w \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{w} - 8 m^{3} w \left(E t + p_{x} x + p_{y} y + p_{z} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + 4 m p_{z} \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} - 4 m p_{y} \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 4 m p_{x} \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} - 4 E m \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z + m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 4 m^{2} \left(E^{2} t^{2} + 2 E p_{x} t x + 2 E p_{y} t y + 2 E p_{z} t z - m^{2} w^{2} + {\left ( p_{x} \right )}^{2} x^{2} + 2 p_{x} p_{y} x y + 2 p_{x} p_{z} x z + {\left ( p_{y} \right )}^{2} y^{2} + 2 p_{y} p_{z} y z + {\left ( p_{z} \right )}^{2} z^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





Combine regular and adjoint Dirac equations with difference






    





$$1/2 (\overrightarrow{D}\rho-\rho\overleftarrow{D})=\begin{equation*} 8 E m w \left(- E^{2} + m^{2} + {\left ( p_{x} \right )}^{2} + {\left ( p_{y} \right )}^{2} + {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{w} + 8 m p_{x} w \left(E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{w} + 8 m p_{y} w \left(E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} + 8 m p_{z} w \left(E^{2} - m^{2} - {\left ( p_{x} \right )}^{2} - {\left ( p_{y} \right )}^{2} - {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$






    





Combine regular and adjoint Dirac equations with addition






    





$$1/2 (\overrightarrow{D}\rho+\rho\overleftarrow{D})=\begin{equation*} 8 E \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{t} + 8 p_{x} \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{x} + 8 p_{y} \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{y} + 8 p_{z} \left(E^{3} t + E^{2} p_{x} x + E^{2} p_{y} y + E^{2} p_{z} z - E {\left ( p_{x} \right )}^{2} t - E {\left ( p_{y} \right )}^{2} t - E {\left ( p_{z} \right )}^{2} t - {\left ( p_{x} \right )}^{3} x - {\left ( p_{x} \right )}^{2} p_{y} y - {\left ( p_{x} \right )}^{2} p_{z} z - p_{x} {\left ( p_{y} \right )}^{2} x - p_{x} {\left ( p_{z} \right )}^{2} x - {\left ( p_{y} \right )}^{3} y - {\left ( p_{y} \right )}^{2} p_{z} z - p_{y} {\left ( p_{z} \right )}^{2} y - {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{z} + 8 m^{2} w \left(- E^{2} + {\left ( p_{x} \right )}^{2} + {\left ( p_{y} \right )}^{2} + {\left ( p_{z} \right )}^{2}\right) \boldsymbol{\gamma }_{w} - 8 m^{4} w \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} - 8 m^{2} p_{z} \left(E t + p_{x} x + p_{y} y + p_{z} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{w} + 8 m^{2} p_{y} \left(E t + p_{x} x + p_{y} y + p_{z} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} - 8 m^{2} p_{x} \left(E t + p_{x} x + p_{y} y + p_{z} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 8 E m^{2} \left(E t + p_{x} x + p_{y} y + p_{z} z\right) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} + 8 m \left(- E^{3} t - E^{2} p_{x} x - E^{2} p_{y} y - E^{2} p_{z} z + E m^{2} t + E {\left ( p_{x} \right )}^{2} t + E {\left ( p_{y} \right )}^{2} t + E {\left ( p_{z} \right )}^{2} t + m^{2} p_{x} x + m^{2} p_{y} y + m^{2} p_{z} z + {\left ( p_{x} \right )}^{3} x + {\left ( p_{x} \right )}^{2} p_{y} y + {\left ( p_{x} \right )}^{2} p_{z} z + p_{x} {\left ( p_{y} \right )}^{2} x + p_{x} {\left ( p_{z} \right )}^{2} x + {\left ( p_{y} \right )}^{3} y + {\left ( p_{y} \right )}^{2} p_{z} z + p_{y} {\left ( p_{z} \right )}^{2} y + {\left ( p_{z} \right )}^{3} z\right) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}\wedge \boldsymbol{\gamma }_{w} \end{equation*}$$



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