Solve (igamma_{mu}partial^mu-m)psi=0

$$( i \gamma _ { \mu } \partial ^ { \mu } - m ) \psi = 0$$

Solve for m

$$\left\{\begin{matrix}\\m=i\gamma _{μ}∂^{\mu }\text{, }&\text{unconditionally}\\m\in \mathrm{C}\text{, }&\psi =0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply i\gamma _{μ}∂^{\mu }-m by \psi .

$$i\gamma _{μ}∂^{\mu }\psi -m\psi =0$$

Subtract i\gamma _{μ}∂^{\mu }\psi from both sides. Anything subtracted from zero gives its negation.

$$-m\psi =-i\gamma _{μ}∂^{\mu }\psi$$

The equation is in standard form.

$$\left(-\psi \right)m=-i\gamma _{μ}\psi ∂^{\mu }$$

Divide both sides by -\psi .

$$\frac{\left(-\psi \right)m}{-\psi }=-\frac{i\gamma _{μ}\psi ∂^{\mu }}{-\psi }$$

Dividing by -\psi undoes the multiplication by -\psi .

$$m=-\frac{i\gamma _{μ}\psi ∂^{\mu }}{-\psi }$$

Divide -i\gamma _{μ}∂^{\mu }\psi by -\psi .

$$m=i\gamma _{μ}∂^{\mu }$$

Solve for γ_μ

$$\left\{\begin{matrix}\gamma _{μ}=-\frac{im}{∂^{\mu }}\text{, }&\mu =0\text{ or }∂\neq 0\\\gamma _{μ}\in \mathrm{C}\text{, }&\psi =0\text{ or }\left(m=0\text{ and }∂=0\text{ and }\mu \neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply i\gamma _{μ}∂^{\mu }-m by \psi .

$$i\gamma _{μ}∂^{\mu }\psi -m\psi =0$$

Add m\psi to both sides. Anything plus zero gives itself.

$$i\gamma _{μ}∂^{\mu }\psi =m\psi$$

The equation is in standard form.

$$i\psi ∂^{\mu }\gamma _{μ}=m\psi$$

Divide both sides by i∂^{\mu }\psi .

$$\frac{i\psi ∂^{\mu }\gamma _{μ}}{i\psi ∂^{\mu }}=\frac{m\psi }{i\psi ∂^{\mu }}$$

Dividing by i∂^{\mu }\psi undoes the multiplication by i∂^{\mu }\psi .

$$\gamma _{μ}=\frac{m\psi }{i\psi ∂^{\mu }}$$

Divide m\psi by i∂^{\mu }\psi .

$$\gamma _{μ}=-\frac{im}{∂^{\mu }}$$