# Lévy-Leblond equation

In quantum mechanics, the Lévy-Leblond equation describes the dynamics of a spin-1/2 particle. It is a linearized version of the Schrödinger equation and of the Pauli equation. It was derived by French physicist Jean-Marc Lévy-Leblond in 1967.
Lévy-Leblond equation was obtained under similar heuristic derivations as the Dirac equation, but contrary to the latter, Lévy-Leblond equation is not relativistic. As both equations recover the electron gyromagnetic ratio, it is suggested that spin is not necessarily a relativistic phenomenon.

## Equation

For a nonrelativistic spin-1/2 particle of mass m, a representation of the time-independent Lévy-Leblond equation reads:
    $\left\{{\begin{matrix}E\psi +({\boldsymbol {\sigma }}\cdot \mathbf {p} c)\chi =0\\({\boldsymbol {\sigma }}\cdot \mathbf {p} c)\psi +2mc^{2}\chi =0\end{matrix}}\right.$
where c is the speed of light, E is the nonrelativistic particle energy, $\mathbf {p} =-i\hbar \nabla$ is the momentum operator, and ${\boldsymbol {\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})$ is the vector of Pauli matrices, which is proportional to the spin operator $\mathbf {S} ={\tfrac {1}{2}}\hbar {\boldsymbol {\sigma }}$. Here $\psi ,\chi$ are two components functions (spinors) describing the wave function of the particle.
By minimal coupling, the equation can be modified to account for the presence of an electromagnetic field,
    $\left\{{\begin{matrix}(E-qV)\psi +[{\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} )c]\chi =0\\{[{\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} )c]}\psi +2mc^{2}\chi =0\end{matrix}}\right.$
where q is the electric charge of the particle. V is the electric potential, and A is the magnetic vector potential. This equation is linear in its spatial derivatives.

## Relation to spin

In 1928, Paul Dirac linearized the relativistic dispersion relation and obtained Dirac equation, described by a bispinor. This equation can be decoupled into two spinors in the non-relativistic limit, leading to predict the electron magnetic moment with a gyromagnetic ratio g=2{\textstyle g=2}. The success of Dirac theory has led to some textbooks to erroneously claim that spin is necessarily a relativistic phenomena.
Jean-Marc Lévy-Leblond applied the same technique to the non-relativistic energy relation showing that the same prediction of g=2{\textstyle g=2} can be obtained. Actually to derive the Pauli equation from Dirac equation one has to pass by Lévy-Leblond equation. Spin is then a result of quantum mechanics and linearization of the equations but not necessarily a relativistic effect.
Lévy-Leblond equation is Galilean invariant. This equation demonstrates that one does not need the full Poincaré group to explain the spin 1/2. In the classical limit where c→∞{\textstyle c\to \infty }, quantum mechanics under the Galilean transformation group are enough. Similarly, one can construct a non-relativistic linear equation for any arbitrary spin. Under the same idea one can construct equations for Galilean electromagnetism.

## Relation to other equations
### Schrödinger's and Pauli's equation

Taking the second line of Lévy-Leblond equation and inserting it back into the first line, one obtains through the algebra of the Pauli matrices, that
    ${\frac {1}{2m}}({\boldsymbol {\sigma }}\cdot \mathbf {p} )^{2}\psi -E\psi =\left[{\frac {1}{2m}}\mathbf {p} ^{2}-E\right]\psi =0$,
which is the Schrödinger equation for a two-valued spinor. Note that solving for $\chi$ also returns another Schrödinger's equation. Pauli's expression for spin-1⁄2 particle in an electromagnetic field can be recovered by minimal coupling:
    $\left\{{\frac {1}{2m}}[{\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} )]^{2}+qV\right\}\psi =E\psi$.
While Lévy-Leblond is linear in its derivatives, Pauli's and Schrödinger's equations are quadratic in the spatial derivatives.

### Dirac equation

Dirac equation can be written as:
    $\left\{{\begin{matrix}({\mathcal {E}}-mc^{2})\psi +({\boldsymbol {\sigma }}\cdot \mathbf {p} c)\chi =0\\({\boldsymbol {\sigma }}\cdot \mathbf {p} c)\psi +({\mathcal {E}}+mc^{2})\chi =0\end{matrix}}\right.$
where E{\textstyle {\mathcal {E}}} is the total relativistic energy. In the non-relativistic limit, E≪mc$^2${\textstyle E\ll mc^{2}} and E≈mc$^2$+E+⋯{\textstyle {\mathcal {E}}\approx mc^{2}+E+\cdots } one recovers, Lévy-Leblond equations.

## Heuristic derivation

Similar to the historical derivation of Dirac equation by Paul Dirac, one can try to linearize the non-relativistic dispersion relation E=p$^2$/2m {\textstyle E={\frac {\mathbf {p} ^{2}}{2m}}}. We want two operators Θ and Θ' linear in p{\textstyle \mathbf {p} } (spatial derivatives) and E, like
    $\left\{{\begin{matrix}\Theta \Psi =[AE+\mathbf {B} \cdot \mathbf {p} c+2mc^{2}C]\Psi =0\\\Theta '\Psi =[A'E+\mathbf {B} '\cdot \mathbf {p} c+2mc^{2}C']\Psi =0\end{matrix}}\right.$
for some A, A$^′$, B=(B$_x$, B$_y$, B$_z$), B$^′$=(B$_x$$^′$, B$_y$$^′$, B$_z$$^′$), C, C$^′${\textstyle A, A', \mathbf {B} =(B_{x}, B_{y}, B_{z}), \mathbf {B} '=(B_{x}', B_{y}', B_{z}'), C, C'}, such that their product recovers the classical dispersion relation, that is
    ${\frac {1}{2mc^{2}}}\Theta '\Theta =E-{\frac {\mathbf {p} ^{2}}{2m}}$,
where the factor 2mc$^2$ is arbitrary an it is just there for normalization. By carrying out the product, one find that there is no solution if A, A$^′$, B$_i$, B$_i$$^′$, C, C$^′${\textstyle A, A', B_{i}, B_{i}', C, C'} are one dimensional constants. The lowest dimension where there is a solution is 4. Then A, A$^′$, B, B$^′$, C, C$^′${\textstyle A, A', \mathbf {B} , \mathbf {B} ', C, C'} are matrices that must satisfy the following relations:
    $\left\{{\begin{matrix}A'A=0\\C'C=0\\A'B_{i}+B_{i}'A=0\\C'B_{i}+B_{i}'C=0\\A'C+C'A=I_{4}\\B_{i}'B_{j}+B_{j}'B_{i}=-2\delta _{ij}\end{matrix}}\right.$
these relations can be rearranged to involve the gamma matrices from Clifford algebra. I$_N${\textstyle I_{N}} is the Identity matrix of dimension N. One possible representation is
    $A=A'={\begin{pmatrix}0&0\\I_{2}&0\end{pmatrix}},B_{i}=-B_{i}'={\begin{pmatrix}\sigma _{i}&0\\0&\sigma _{i}\end{pmatrix}},C=C'={\begin{pmatrix}0&I_{2}\\0&0\end{pmatrix}}$,
such that ΘΨ=0{\textstyle \Theta \Psi =0}, with Ψ=(ψ, χ){\textstyle \Psi =(\psi , \chi )} , returns Lévy-Leblond equation. Other representations can be chosen leading to equivalent equations with different signs or phases.