Physics:Polar form of the Dirac equation

The polar form of the Dirac equation is the obtained when the Dirac equation is written in terms of a spinor in polar form, that is the form in which each of its four components are expressed as the product of a module times a unitary complex phase.

The Dirac field is mathematically described by a spinor, an object that consists of four complex scalar components that transform in terms of complex representations of the Lorentz group. Because each component is a complex scalar function, it can be written as the product of a module times a unitary complex phase. When this is done for all components of the spinor, while still respecting the manifest co-variance, the result is the polar form of the Dirac spinor field. Its mathematical expression can be given by first assigning some preliminary definitions.

The gamma matrices [math]\displaystyle{ \boldsymbol{\gamma}^{a} }[/math] are introduced as the element of the Clifford algebra [math]\displaystyle{ \{\boldsymbol{\gamma}_{a},\boldsymbol{\gamma}_{b}\}=2\mathbb{I}\eta_{ab} }[/math] and from that we can define

[math]\displaystyle{ \frac{1}{4}\left[\boldsymbol{\gamma}_{a}\!,\!\boldsymbol{\gamma}_{b}\right]\!=\!\boldsymbol{\sigma}_{ab} }[/math]

as well as the [math]\displaystyle{ \boldsymbol{\gamma^5} }[/math] matrix: we also have [math]\displaystyle{ \overline{\psi}\!=\!\psi^{\dagger}\boldsymbol{\gamma}_{0} }[/math] as the operation of spinor conjugation. Spinor bi-linear quantities are defined according to

[math]\displaystyle{ S^{a}\!=\!\overline{\psi}\boldsymbol{\gamma}^{a}\boldsymbol{\gamma^5}\psi }[/math]

[math]\displaystyle{ U^{a}\!=\!\overline{\psi}\boldsymbol{\gamma}^{a}\psi }[/math]

[math]\displaystyle{ \Theta\!=\!i\overline{\psi}\boldsymbol{\gamma^5}\psi }[/math]

[math]\displaystyle{ \Phi\!=\!\overline{\psi}\psi }[/math]

all of which are real tensors.

According to the values of these bi-linear quantities, spinor fields can be classified in various ways: if both [math]\displaystyle{ \Theta }[/math] and [math]\displaystyle{ \Phi }[/math] vanish identically, the spinor field is called singular, and the class of singular spinor fields contains the notable cases of Majorana spinor and Weyl spinor. However, if in general, [math]\displaystyle{ \Theta }[/math] and [math]\displaystyle{ \Phi }[/math] do not vanish simultaneously, the spinor field is called regular, and in this case it is always possible to find a local frame in which the regular spinor field acquires the form:

[math]\displaystyle{ \psi\!=\!\phi\ e^{-\frac{i}{2}\beta\boldsymbol{\gamma^5}}\left(\begin{array}{c}1\\0\\1\\0\end{array}\right) }[/math]

up to a transformation of the type [math]\displaystyle{ \psi\rightarrow\boldsymbol{\gamma^5}\psi }[/math] and up to the reversal of the third axis.

Therefore, a regular spinor can, in general, be written according to

[math]\displaystyle{ \psi\!=\!\phi\ e^{-\frac{i}{2}\beta\boldsymbol{\gamma^5}}\boldsymbol{S}\left(\begin{array}{c}1\\0\\1\\0\end{array}\right) }[/math]

where [math]\displaystyle{ \phi }[/math] is called module and [math]\displaystyle{ \beta }[/math] is called Yvon-Takabayashi angle, and where [math]\displaystyle{ \boldsymbol{S} }[/math] is the generic complex representation of the Lorentz transformation that is needed to obtain the generic spinor in its most reduced form as written above.

When the spinor is written in polar form, the bi-linear quantities can be written according to

[math]\displaystyle{ S^{a}\!=\!2\phi^{2}s^{a} }[/math]

[math]\displaystyle{ U^{a}\!=\!2\phi^{2}u^{a} }[/math]

where [math]\displaystyle{ u^{a} }[/math] is the velocity vector and [math]\displaystyle{ s^{a} }[/math] is the spin axial velocity, and

[math]\displaystyle{ \Theta\!=\!2\phi^{2}\sin{\beta} }[/math]

[math]\displaystyle{ \Phi\!=\!2\phi^{2}\cos{\beta} }[/math]

in terms of module and Yvon-Takabayashi angle.

The polar form given in the case in which [math]\displaystyle{ \boldsymbol{S}=\mathbb{I} }[/math] corresponds to the case in which the velocity vector has only its temporal component and the spin axial-vector has only its third component, so that the full polar form is the one associated to the spinor rest frame and spin eigenstate. This shows that the module and Yvon-Takabayashi angle are the only two real degrees of freedom.

The frame in which the polar form is obtained is locally defined, meaning that the transformation [math]\displaystyle{ \boldsymbol{S} }[/math] is generally point-dependent: so it is always possible to write

[math]\displaystyle{ \boldsymbol{S}\partial_{\mu}\boldsymbol{S}^{-1}\!=\!i\partial_{\mu}\theta\mathbb{I}\!+\!\frac{1}{2}\partial_{\mu}\theta_{ij}\boldsymbol{\sigma}^{ij} }[/math]

where [math]\displaystyle{ \theta }[/math] is a generic complex phase and [math]\displaystyle{ \theta_{ij}\!=\!-\theta_{ji} }[/math] are the six parameters of Lorentz transformations (namely, the three Euler angles and the three rapidities). By introducing a spin connection [math]\displaystyle{ \Omega_{ij\mu} }[/math] as well as an electrodynamic gauge potential [math]\displaystyle{ A_{\mu} }[/math] it is possible to define

[math]\displaystyle{ \partial_{\mu}\theta_{ij}\!-\!\Omega_{ij\mu}\!\equiv\!R_{ij\mu} }[/math]

[math]\displaystyle{ \partial_{\mu}\theta\!-\!qA_{\mu}\!\equiv\!P_{\mu} }[/math]

which can be proven to be gauge covariant real tensors.

In terms of these elements, it is possible to prove that the Dirac equation for a spinor field in polar form implies the two equations

[math]\displaystyle{ \frac{1}{2}\varepsilon_{\mu\alpha\nu\iota}R^{\alpha\nu\iota}\!-\!2P^{\iota}u_{[\iota}s_{\mu]}\!+\!2(\nabla\beta/2\!-\!XW)_{\mu}\!+\!2s_{\mu}m\cos{\beta}\!=\!0 }[/math]

[math]\displaystyle{ R_{\mu a}^{\ \ \ a}\!-\!2P^{\rho}u^{\nu}s^{\alpha}\varepsilon_{\mu\rho\nu\alpha}\!+\!2s_{\mu}m\sin{\beta}\!+\!\nabla_{\mu}\ln{\phi^{2}}\!=\!0 }[/math]

where for generality [math]\displaystyle{ W_{\mu} }[/math] serves as the axial-vector dual of the completely anti-symmetric part of the torsion tensor; conversely, it is possible to prove that these two equations imply the Dirac equation with the spinor written in the polar form, proving their equivalence.

These two equations specify all first-order derivatives of the two physical components given by the module and the Yvon-Takabayashi angle.

Therefore, the Dirac spinor field equation, which is four complex equations and therefore eight equations in total, is equivalently written in terms of two real vectorial equations, which are two four-dimensional equations and therefore eight equations in total.

This pair of coupled first-order differential equations is the polar form of the Dirac equation.