- 5

- 0

## Main Question or Discussion Point

Let's have a theory involving Dirac field [tex]\psi[/tex]. This theory is decribed by some Lagrangian density [tex]\mathcal{L}(\psi,\partial_\mu\psi)[/tex]. Taking [tex]\psi[/tex] as the canonical dynamical variable, its conjugate momentum is defined as

[tex]\pi=\frac{\partial\mathcal{L}}{\partial(\partial_0\psi)}[/tex]

Than the quantization simply means to impose the canonical anticommatation relation of the type [tex]\{\psi,\pi \}_{e.t.}=i\delta^3(x-y)[/tex].

OK. But I wonder, whether this procedure is unique. It is well known that free (massless) Dirac field can be described by Lagrangian [tex]\mathcal{L}=i\bar\psi\gamma^\mu\partial_\mu\psi[/tex], as well as by Lagrangian

[tex]\mathcal{L}=\frac{i}{2}\bar\psi\gamma^\mu(\partial_\mu\psi)-\frac{i}{2}(\partial_\mu\bar\psi)\gamma^\mu\psi[/tex]

That's because both Lagranians differ only by a total divergence and hence give the same equations of motion. The problem is that they give different conjugate momenta: the former gives [tex]\pi=i\psi^\dag[/tex], the latter

[tex]\pi=\frac{i}{2}\psi^\dag[/tex]

Where's the problem? Does it mean that the creation/anihilation operators within each quantization have different anticommatation relations (differing by factor of 1/2)? Or something else? Please help, thanks..

[tex]\pi=\frac{\partial\mathcal{L}}{\partial(\partial_0\psi)}[/tex]

Than the quantization simply means to impose the canonical anticommatation relation of the type [tex]\{\psi,\pi \}_{e.t.}=i\delta^3(x-y)[/tex].

OK. But I wonder, whether this procedure is unique. It is well known that free (massless) Dirac field can be described by Lagrangian [tex]\mathcal{L}=i\bar\psi\gamma^\mu\partial_\mu\psi[/tex], as well as by Lagrangian

[tex]\mathcal{L}=\frac{i}{2}\bar\psi\gamma^\mu(\partial_\mu\psi)-\frac{i}{2}(\partial_\mu\bar\psi)\gamma^\mu\psi[/tex]

That's because both Lagranians differ only by a total divergence and hence give the same equations of motion. The problem is that they give different conjugate momenta: the former gives [tex]\pi=i\psi^\dag[/tex], the latter

[tex]\pi=\frac{i}{2}\psi^\dag[/tex]

Where's the problem? Does it mean that the creation/anihilation operators within each quantization have different anticommatation relations (differing by factor of 1/2)? Or something else? Please help, thanks..