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import sympy


    

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from __future__ import absolute_import, division
from __future__ import print_function
from galgebra.printer import Format, xpdf
from galgebra.ga import Ga


    

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Format()


    

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g4d = Ga('a b c d')


    

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(a, b, c, d) = g4d.mv()


    

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g4d.g


    

    
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$\displaystyle \left[\begin{matrix}(a.a) & (a.b) & (a.c) & (a.d)\\(a.b) & (b.b) & (b.c) & (b.d)\\(a.c) & (b.c) & (c.c) & (c.d)\\(a.d) & (b.d) & (c.d) & (d.d)\end{matrix}\right]$

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a | (b * c)


    

    
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\begin{equation*} - \left ( a\cdot c\right )  \boldsymbol{b} + \left ( a\cdot b\right )  \boldsymbol{c} \end{equation*}

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a | (b ^ c)


    

    
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\begin{equation*} - \left ( a\cdot c\right )  \boldsymbol{b} + \left ( a\cdot b\right )  \boldsymbol{c} \end{equation*}

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a | (b ^ c ^ d)


    

    
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\begin{equation*} \left ( a\cdot d\right )  \boldsymbol{b}\wedge \boldsymbol{c} - \left ( a\cdot c\right )  \boldsymbol{b}\wedge \boldsymbol{d} + \left ( a\cdot b\right )  \boldsymbol{c}\wedge \boldsymbol{d} \end{equation*}

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(a | (b ^ c)) + (c | (a ^ b)) + (b | (c ^ a))


    

    
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\begin{equation*}  0  \end{equation*}

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a * (b ^ c) - b * (a ^ c) + c * (a ^ b)


    

    
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\begin{equation*} 3 \boldsymbol{a}\wedge \boldsymbol{b}\wedge \boldsymbol{c} \end{equation*}

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a * (b ^ c ^ d) - b * (a ^ c ^ d) + c * (a ^ b ^ d) - d * (a ^ b ^ c)


    

    
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\begin{equation*} 4 \boldsymbol{a}\wedge \boldsymbol{b}\wedge \boldsymbol{c}\wedge \boldsymbol{d} \end{equation*}

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(a ^ b) | (c ^ d)


    

    
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\begin{equation*} - \left ( a\cdot c\right )  \left ( b\cdot d\right )  + \left ( a\cdot d\right )  \left ( b\cdot c\right )  \end{equation*}

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((a ^ b) | c) | d


    

    
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\begin{equation*} - \left ( a\cdot c\right )  \left ( b\cdot d\right )  + \left ( a\cdot d\right )  \left ( b\cdot c\right )  \end{equation*}

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Ga.com(a ^ b, c ^ d)


    

    
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\begin{equation*} - \left ( b\cdot d\right )  \boldsymbol{a}\wedge \boldsymbol{c} + \left ( b\cdot c\right )  \boldsymbol{a}\wedge \boldsymbol{d} + \left ( a\cdot d\right )  \boldsymbol{b}\wedge \boldsymbol{c} - \left ( a\cdot c\right )  \boldsymbol{b}\wedge \boldsymbol{d} \end{equation*}

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from __future__ import absolute_import, division
from __future__ import print_function
import sys
from sympy import symbols, sin, cos
from galgebra.printer import Format, xpdf, Get_Program, Print_Function
from galgebra.ga import Ga


    

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Format()
coords = symbols('t x y z', real=True)
coords


    

    
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$\displaystyle \left( t, \  x, \  y, \  z\right)$

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(st4d, g0, g1, g2, g3) = Ga.build(
    'gamma*t|x|y|z', g=[1, -1, -1, -1], coords=coords)


    

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g0


    

    
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\begin{equation*}  \boldsymbol{\gamma }_{t} \end{equation*}

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g1


    

    
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\begin{equation*}  \boldsymbol{\gamma }_{x} \end{equation*}

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g2


    

    
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\begin{equation*}  \boldsymbol{\gamma }_{y} \end{equation*}

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g3


    

    
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\begin{equation*}  \boldsymbol{\gamma }_{z} \end{equation*}

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I = st4d.i
I


    

    
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\begin{equation*}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \end{equation*}

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(m, e) = symbols('m e')


    

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m


    

    
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$\displaystyle m$

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e


    

    
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$\displaystyle e$

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# 4-Vector Potential
A = st4d.mv('A', 'vector', f=True)
A


    

    
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\begin{equation*} A = A^{t}  \boldsymbol{\gamma }_{t} + A^{x}  \boldsymbol{\gamma }_{x} + A^{y}  \boldsymbol{\gamma }_{y} + A^{z}  \boldsymbol{\gamma }_{z} \end{equation*}

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# 8-componentrealspinor
psi = st4d.mv('psi', 'spinor', f=True)
psi


    

    
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\begin{equation*} psi = \psi    + \psi ^{tx}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x} + \psi ^{ty}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y} + \psi ^{tz}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{z} + \psi ^{xy}  \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} + \psi ^{xz}  \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} + \psi ^{yz}  \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + \psi ^{txyz}  \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \end{equation*}

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sig_z = g3 * g0
sig_z


    

    
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\begin{equation*} - \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{z} \end{equation*}

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dirac_eq = (st4d.grad * psi) * I * sig_z - e * A * psi - m * psi * g0
dirac_eq


    

    
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\begin{equation*} \left ( - e A^{t}  \psi   - e A^{x}  \psi ^{tx}  - e A^{y}  \psi ^{ty}  - e A^{z}  \psi ^{tz}  - m \psi   - \partial_{y} \psi ^{tx}  - \partial_{z} \psi ^{txyz}  + \partial_{x} \psi ^{ty}  + \partial_{t} \psi ^{xy} \right ) \boldsymbol{\gamma }_{t} + \left ( - e A^{t}  \psi ^{tx}  - e A^{x}  \psi   - e A^{y}  \psi ^{xy}  - e A^{z}  \psi ^{xz}  + m \psi ^{tx}  + \partial_{y} \psi   - \partial_{t} \psi ^{ty}  - \partial_{x} \psi ^{xy}  + \partial_{z} \psi ^{yz} \right ) \boldsymbol{\gamma }_{x} + \left ( - e A^{t}  \psi ^{ty}  + e A^{x}  \psi ^{xy}  - e A^{y}  \psi   - e A^{z}  \psi ^{yz}  + m \psi ^{ty}  - \partial_{x} \psi   + \partial_{t} \psi ^{tx}  - \partial_{y} \psi ^{xy}  - \partial_{z} \psi ^{xz} \right ) \boldsymbol{\gamma }_{y} + \left ( - e A^{t}  \psi ^{tz}  + e A^{x}  \psi ^{xz}  + e A^{y}  \psi ^{yz}  - e A^{z}  \psi   + m \psi ^{tz}  + \partial_{t} \psi ^{txyz}  - \partial_{z} \psi ^{xy}  + \partial_{y} \psi ^{xz}  - \partial_{x} \psi ^{yz} \right ) \boldsymbol{\gamma }_{z} + \left ( - e A^{t}  \psi ^{xy}  + e A^{x}  \psi ^{ty}  - e A^{y}  \psi ^{tx}  - e A^{z}  \psi ^{txyz}  - m \psi ^{xy}  - \partial_{t} \psi   + \partial_{x} \psi ^{tx}  + \partial_{y} \psi ^{ty}  + \partial_{z} \psi ^{tz} \right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} + \left ( - e A^{t}  \psi ^{xz}  + e A^{x}  \psi ^{tz}  + e A^{y}  \psi ^{txyz}  - e A^{z}  \psi ^{tx}  - m \psi ^{xz}  + \partial_{x} \psi ^{txyz}  + \partial_{z} \psi ^{ty}  - \partial_{y} \psi ^{tz}  - \partial_{t} \psi ^{yz} \right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} + \left ( - e A^{t}  \psi ^{yz}  - e A^{x}  \psi ^{txyz}  + e A^{y}  \psi ^{tz}  - e A^{z}  \psi ^{ty}  - m \psi ^{yz}  - \partial_{z} \psi ^{tx}  + \partial_{y} \psi ^{txyz}  + \partial_{x} \psi ^{tz}  + \partial_{t} \psi ^{xz} \right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + \left ( - e A^{t}  \psi ^{txyz}  - e A^{x}  \psi ^{yz}  + e A^{y}  \psi ^{xz}  - e A^{z}  \psi ^{xy}  + m \psi ^{txyz}  + \partial_{z} \psi   - \partial_{t} \psi ^{tz}  - \partial_{x} \psi ^{xz}  - \partial_{y} \psi ^{yz} \right ) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} \end{equation*}

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dirac_eq.Fmt(2)


    

    
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 \begin{align*}  & \left ( - e A^{t}  \psi   - e A^{x}  \psi ^{tx}  - e A^{y}  \psi ^{ty}  - e A^{z}  \psi ^{tz}  - m \psi   - \partial_{y} \psi ^{tx}  - \partial_{z} \psi ^{txyz}  + \partial_{x} \psi ^{ty}  + \partial_{t} \psi ^{xy} \right ) \boldsymbol{\gamma }_{t} + \left ( - e A^{t}  \psi ^{tx}  - e A^{x}  \psi   - e A^{y}  \psi ^{xy}  - e A^{z}  \psi ^{xz}  + m \psi ^{tx}  + \partial_{y} \psi   - \partial_{t} \psi ^{ty}  - \partial_{x} \psi ^{xy}  + \partial_{z} \psi ^{yz} \right ) \boldsymbol{\gamma }_{x} + \left ( - e A^{t}  \psi ^{ty}  + e A^{x}  \psi ^{xy}  - e A^{y}  \psi   - e A^{z}  \psi ^{yz}  + m \psi ^{ty}  - \partial_{x} \psi   + \partial_{t} \psi ^{tx}  - \partial_{y} \psi ^{xy}  - \partial_{z} \psi ^{xz} \right ) \boldsymbol{\gamma }_{y} + \left ( - e A^{t}  \psi ^{tz}  + e A^{x}  \psi ^{xz}  + e A^{y}  \psi ^{yz}  - e A^{z}  \psi   + m \psi ^{tz}  + \partial_{t} \psi ^{txyz}  - \partial_{z} \psi ^{xy}  + \partial_{y} \psi ^{xz}  - \partial_{x} \psi ^{yz} \right ) \boldsymbol{\gamma }_{z} \\  &  + \left ( - e A^{t}  \psi ^{xy}  + e A^{x}  \psi ^{ty}  - e A^{y}  \psi ^{tx}  - e A^{z}  \psi ^{txyz}  - m \psi ^{xy}  - \partial_{t} \psi   + \partial_{x} \psi ^{tx}  + \partial_{y} \psi ^{ty}  + \partial_{z} \psi ^{tz} \right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} + \left ( - e A^{t}  \psi ^{xz}  + e A^{x}  \psi ^{tz}  + e A^{y}  \psi ^{txyz}  - e A^{z}  \psi ^{tx}  - m \psi ^{xz}  + \partial_{x} \psi ^{txyz}  + \partial_{z} \psi ^{ty}  - \partial_{y} \psi ^{tz}  - \partial_{t} \psi ^{yz} \right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} + \left ( - e A^{t}  \psi ^{yz}  - e A^{x}  \psi ^{txyz}  + e A^{y}  \psi ^{tz}  - e A^{z}  \psi ^{ty}  - m \psi ^{yz}  - \partial_{z} \psi ^{tx}  + \partial_{y} \psi ^{txyz}  + \partial_{x} \psi ^{tz}  + \partial_{t} \psi ^{xz} \right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + \left ( - e A^{t}  \psi ^{txyz}  - e A^{x}  \psi ^{yz}  + e A^{y}  \psi ^{xz}  - e A^{z}  \psi ^{xy}  + m \psi ^{txyz}  + \partial_{z} \psi   - \partial_{t} \psi ^{tz}  - \partial_{x} \psi ^{xz}  - \partial_{y} \psi ^{yz} \right ) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}  \end{align*} 

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dirac_eq = dirac_eq.simplify()
dirac_eq


    

    
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 \begin{align*}  & \left ( - e A^{t}  \psi   - e A^{x}  \psi ^{tx}  - e A^{y}  \psi ^{ty}  - e A^{z}  \psi ^{tz}  - m \psi   - \partial_{y} \psi ^{tx}  - \partial_{z} \psi ^{txyz}  + \partial_{x} \psi ^{ty}  + \partial_{t} \psi ^{xy} \right ) \boldsymbol{\gamma }_{t} + \left ( - e A^{t}  \psi ^{tx}  - e A^{x}  \psi   - e A^{y}  \psi ^{xy}  - e A^{z}  \psi ^{xz}  + m \psi ^{tx}  + \partial_{y} \psi   - \partial_{t} \psi ^{ty}  - \partial_{x} \psi ^{xy}  + \partial_{z} \psi ^{yz} \right ) \boldsymbol{\gamma }_{x} + \left ( - e A^{t}  \psi ^{ty}  + e A^{x}  \psi ^{xy}  - e A^{y}  \psi   - e A^{z}  \psi ^{yz}  + m \psi ^{ty}  - \partial_{x} \psi   + \partial_{t} \psi ^{tx}  - \partial_{y} \psi ^{xy}  - \partial_{z} \psi ^{xz} \right ) \boldsymbol{\gamma }_{y} + \left ( - e A^{t}  \psi ^{tz}  + e A^{x}  \psi ^{xz}  + e A^{y}  \psi ^{yz}  - e A^{z}  \psi   + m \psi ^{tz}  + \partial_{t} \psi ^{txyz}  - \partial_{z} \psi ^{xy}  + \partial_{y} \psi ^{xz}  - \partial_{x} \psi ^{yz} \right ) \boldsymbol{\gamma }_{z} \\  &  + \left ( - e A^{t}  \psi ^{xy}  + e A^{x}  \psi ^{ty}  - e A^{y}  \psi ^{tx}  - e A^{z}  \psi ^{txyz}  - m \psi ^{xy}  - \partial_{t} \psi   + \partial_{x} \psi ^{tx}  + \partial_{y} \psi ^{ty}  + \partial_{z} \psi ^{tz} \right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} + \left ( - e A^{t}  \psi ^{xz}  + e A^{x}  \psi ^{tz}  + e A^{y}  \psi ^{txyz}  - e A^{z}  \psi ^{tx}  - m \psi ^{xz}  + \partial_{x} \psi ^{txyz}  + \partial_{z} \psi ^{ty}  - \partial_{y} \psi ^{tz}  - \partial_{t} \psi ^{yz} \right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} + \left ( - e A^{t}  \psi ^{yz}  - e A^{x}  \psi ^{txyz}  + e A^{y}  \psi ^{tz}  - e A^{z}  \psi ^{ty}  - m \psi ^{yz}  - \partial_{z} \psi ^{tx}  + \partial_{y} \psi ^{txyz}  + \partial_{x} \psi ^{tz}  + \partial_{t} \psi ^{xz} \right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + \left ( - e A^{t}  \psi ^{txyz}  - e A^{x}  \psi ^{yz}  + e A^{y}  \psi ^{xz}  - e A^{z}  \psi ^{xy}  + m \psi ^{txyz}  + \partial_{z} \psi   - \partial_{t} \psi ^{tz}  - \partial_{x} \psi ^{xz}  - \partial_{y} \psi ^{yz} \right ) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}  \end{align*} 

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dirac_eq.Fmt(2)


    

    
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 \begin{align*}  & \left ( - e A^{t}  \psi   - e A^{x}  \psi ^{tx}  - e A^{y}  \psi ^{ty}  - e A^{z}  \psi ^{tz}  - m \psi   - \partial_{y} \psi ^{tx}  - \partial_{z} \psi ^{txyz}  + \partial_{x} \psi ^{ty}  + \partial_{t} \psi ^{xy} \right ) \boldsymbol{\gamma }_{t} + \left ( - e A^{t}  \psi ^{tx}  - e A^{x}  \psi   - e A^{y}  \psi ^{xy}  - e A^{z}  \psi ^{xz}  + m \psi ^{tx}  + \partial_{y} \psi   - \partial_{t} \psi ^{ty}  - \partial_{x} \psi ^{xy}  + \partial_{z} \psi ^{yz} \right ) \boldsymbol{\gamma }_{x} + \left ( - e A^{t}  \psi ^{ty}  + e A^{x}  \psi ^{xy}  - e A^{y}  \psi   - e A^{z}  \psi ^{yz}  + m \psi ^{ty}  - \partial_{x} \psi   + \partial_{t} \psi ^{tx}  - \partial_{y} \psi ^{xy}  - \partial_{z} \psi ^{xz} \right ) \boldsymbol{\gamma }_{y} + \left ( - e A^{t}  \psi ^{tz}  + e A^{x}  \psi ^{xz}  + e A^{y}  \psi ^{yz}  - e A^{z}  \psi   + m \psi ^{tz}  + \partial_{t} \psi ^{txyz}  - \partial_{z} \psi ^{xy}  + \partial_{y} \psi ^{xz}  - \partial_{x} \psi ^{yz} \right ) \boldsymbol{\gamma }_{z} \\  &  + \left ( - e A^{t}  \psi ^{xy}  + e A^{x}  \psi ^{ty}  - e A^{y}  \psi ^{tx}  - e A^{z}  \psi ^{txyz}  - m \psi ^{xy}  - \partial_{t} \psi   + \partial_{x} \psi ^{tx}  + \partial_{y} \psi ^{ty}  + \partial_{z} \psi ^{tz} \right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y} + \left ( - e A^{t}  \psi ^{xz}  + e A^{x}  \psi ^{tz}  + e A^{y}  \psi ^{txyz}  - e A^{z}  \psi ^{tx}  - m \psi ^{xz}  + \partial_{x} \psi ^{txyz}  + \partial_{z} \psi ^{ty}  - \partial_{y} \psi ^{tz}  - \partial_{t} \psi ^{yz} \right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{z} + \left ( - e A^{t}  \psi ^{yz}  - e A^{x}  \psi ^{txyz}  + e A^{y}  \psi ^{tz}  - e A^{z}  \psi ^{ty}  - m \psi ^{yz}  - \partial_{z} \psi ^{tx}  + \partial_{y} \psi ^{txyz}  + \partial_{x} \psi ^{tz}  + \partial_{t} \psi ^{xz} \right ) \boldsymbol{\gamma }_{t}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z} + \left ( - e A^{t}  \psi ^{txyz}  - e A^{x}  \psi ^{yz}  + e A^{y}  \psi ^{xz}  - e A^{z}  \psi ^{xy}  + m \psi ^{txyz}  + \partial_{z} \psi   - \partial_{t} \psi ^{tz}  - \partial_{x} \psi ^{xz}  - \partial_{y} \psi ^{yz} \right ) \boldsymbol{\gamma }_{x}\wedge \boldsymbol{\gamma }_{y}\wedge \boldsymbol{\gamma }_{z}  \end{align*} 

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