In the previous section we have established the convention we will follow about spinors and Clifford matrices \(\varvec{\gamma }_{a}\) defined to be such that \(\{\varvec{\gamma }_{a},\varvec{\gamma }_{b}\}\!=\!2\mathbb {I}\eta _{ab}\) and from which \(\varvec{\sigma }^{ik}\!=\![\varvec{\gamma }^{i},\varvec{\gamma }^{k}]/4\) are the generators of the Lorentz group. With the identity \(2i\varvec{\sigma }_{ab}\!=\!\varepsilon _{abcd}\varvec{\pi }\varvec{\sigma }^{cd}\) one can implicitly define the fifth gamma matrix, which here we will indicate with \(\varvec{\pi }\) (the reasons for this convention are that there is no justification for the index five since we are not here in pentadimensional spaces and that denoting it as a gamma with no index would make it impossible to distinguish it from a normal gamma in which the index has been suppressed for compactness. The choice of the letter pi for \(\varvec{\pi }\) stands for parity or projector much in the same way the choice of sigma for \(\varvec{\sigma }_{ab}\) stands for spin). By exponentiation of the generators of the Lorentz group we get an element of the Lorentz group \(\varvec{\Lambda }\) and so we can define the element of the spin group as \(\varvec{S}\!=\!\varvec{\Lambda }e^{iq\alpha }\) where \(\alpha \) is the phase accounting for the gauge transformation of q charge. The general property of such transformations is that they verify \(\varvec{S}\varvec{\gamma }^{a}\varvec{S}^{-1}\!=\! \varvec{\Lambda }\varvec{\gamma }^{a}\varvec{\Lambda }^{-1}\!=\! \varvec{\gamma }^{b}(\Lambda ^{-1})^{a}_{b}\) where \((\Lambda ^{-1})^{i}_{a}(\Lambda ^{-1})^{j}_{b}\eta _{ij}\!=\!\eta _{ab}\) so that, according to the usual denominations, \((\Lambda )^{i}_{j}\) is the real Lorentz transformation, \(\varvec{\Lambda }\) the complex Lorentz transformation and \(\varvec{S}\) the spinorial transformation. The spinor field is an object that transforms under a spinorial transformation \(\varvec{S}\) according to \(\psi \!\rightarrow \!\varvec{S}\psi \) and \(\overline{\psi }\!\rightarrow \!\overline{\psi }\varvec{S}^{-1}\) where \(\overline{\psi }\!=\!\psi ^{\dagger }\varvec{\gamma }^{0}\) is the adjoint procedure.
With this pair of adjoint spinors, we can construct the spinor bi-linears
$$\begin{aligned}{} & {} \Sigma ^{ab}\!=\!2\overline{\psi }\varvec{\sigma }^{ab}\varvec{\pi }\psi \ \ \ \ \ \ \ \ \ \ \ \ M^{ab}\!=\!2i\overline{\psi }\varvec{\sigma }^{ab}\psi \end{aligned}$$
(10)
$$\begin{aligned}{} & {} S^{a}\!=\!\overline{\psi }\varvec{\gamma }^{a}\varvec{\pi }\psi \ \ \ \ \ \ \ \ \ \ \ \ U^{a}\!=\!\overline{\psi }\varvec{\gamma }^{a}\psi \end{aligned}$$
(11)
$$\begin{aligned}{} & {} \Theta \!=\!i\overline{\psi }\varvec{\pi }\psi \ \ \ \ \ \ \ \ \ \ \ \ \Phi \!=\!\overline{\psi }\psi \end{aligned}$$
(12)
which are all real tensors. As clear they are not all linearly independent and in fact we have the relation
$$\begin{aligned} \Sigma ^{ij}\!=\!-\frac{1}{2}\varepsilon ^{abij}M_{ab} \end{aligned}$$
(13)
showing that the two antisymmetric tensors \(\Sigma ^{ij}\) and \(M_{ab}\) are the Hodge duals of one another. Additionally, one has
$$\begin{aligned} M_{ab}(\Phi ^{2}\!+\!\Theta ^{2})\!=\!\Phi U^{j}S^{k}\varepsilon _{jkab}\!+\!\Theta U_{[a}S_{b]} \end{aligned}$$
(14)
showing that if \(\Phi ^{2}\!+\!\Theta ^{2}\!\ne \!0\) then also \(M_{ab}\) can be dropped in favor of the two vectors and the two scalars. Axial-vector and vector with pseudo-scalar and scalar are also not independent since
$$\begin{aligned} 2U_{\mu }S_{\nu }\varvec{\sigma }^{\mu \nu }\varvec{\pi }\psi \!+\!U^{2}\psi =0 \end{aligned}$$
(15)
as well as
$$\begin{aligned} U_{a}U^{a}= & {} -S_{a}S^{a}\!=\!\Theta ^{2}\!+\!\Phi ^{2} \end{aligned}$$
(16)
$$\begin{aligned} U_{a}S^{a}= & {} 0 \end{aligned}$$
(17)
and in the case in which \(\Phi ^{2}\!+\!\Theta ^{2}\!\ne \!0\) we can see that the axial-vector is space-like, while the vector is time-like.
Throughout this paper we will systematically deal with spinor fields for which \(\Phi ^{2}+\Theta ^{2}\!\ne \!0\) called regular spinor fields (spinors for which \(\Phi \!\equiv \!\Theta \!\equiv \!0\) are called singular spinor fields, or flag-dipole spinors, and they are of considerable interest [27], since they are a class that contains also Majorana and Weyl spinors). In this case, it is always possible to write any Dirac spinor in polar form which, in chiral representation, is given as
$$\begin{aligned} \psi =\phi \ e^{-\frac{i}{2}\beta \varvec{\pi }} \ \varvec{L}^{-1}\left( \begin{array}{l} 1\\ 0\\ 1\\ 0 \end{array}\right) \end{aligned}$$
(18)
for a pair of functions \(\phi \) and \(\beta \) and for some \(\varvec{L}\) having the structure of a spinorial transformation (that is, \(\varvec{L}\) has mathematically the same structure of \(\varvec{S}\)) [8, 9]. Then
$$\begin{aligned} \Theta \!=\!2\phi ^{2}\sin {\beta }\ \ \ \ \ \ \ \ \ \ \ \ \Phi \!=\!2\phi ^{2}\cos {\beta } \end{aligned}$$
(19)
showing that \(\phi \) and \(\beta \) are a real scalar and a real pseudo-scalar, called module and chiral angle. We can also normalize
$$\begin{aligned} S^{a}\!=\!2\phi ^{2}s^{a}\ \ \ \ \ \ \ \ \ \ \ \ U^{a}\!=\!2\phi ^{2}u^{a} \end{aligned}$$
(20)
where \(u^{a}\) and \(s^{a}\) are the normalized velocity vector and spin axial-vector. Then (15) and (16–17) reduce to
$$\begin{aligned} u_{[\mu }s_{\nu ]}\varvec{\sigma }^{\mu \nu }\varvec{\pi }\psi \!+\!\psi =0 \end{aligned}$$
(21)
and
$$\begin{aligned} u_{a}u^{a}= & {} -s_{a}s^{a}\!=\!1 \end{aligned}$$
(22)
$$\begin{aligned} u_{a}s^{a}= & {} 0 \end{aligned}$$
(23)
showing that the velocity has only 3 independent components, which could be identified with the 3 components of its spatial part, whereas the spin has only 2 independent components, which could be identified with the 2 angles that, in the rest-frame, its spatial part forms with the third axis. As for the matrix \(\varvec{L}\) we can read its meaning as that of the specific transformation that takes a given spinor into its rest frame with spin aligned along the third axis. We have already said that \(\varvec{L}\) has mathematically the same structure of \(\varvec{S}\), but it is also important to specify that from a physical perspective they are very different. In fact, while \(\varvec{S}\) denotes the most general spinorial transformation, \(\varvec{L}\) is that special spinorial transformation that takes a generic spinor in its simplest rest-frame spin-eigenstate form. Metaphorically, if the spinor were to be a top spinning on a table, then \(\varvec{S}\) would tell how to move from the fixed system of reference in which the table is at rest to the rotating system of reference in which the top is at rest, while \(\varvec{L}\) would tell how the top is spinning. For spinors in polar form, their 4 complex components, or 8 real functions, are re-organized in such a way that the 2 real scalars \(\phi \) and \(\beta \) remain isolated from the 6 parameters of \(\varvec{L}\) that can always be transferred into the frame and which are thus the Goldstone fields of the spinor. In fact, the Goldstone fields we have here for the spinor play the same role played by the Goldstone bosons in the Standard Model for the Higgs field. The 3 velocities and 2 angles amount to a total of 5 parameters in the Lorentz transformation, and since the phase adds 1 parameter, the full spinorial transformation has a total of 6 parameters. Or in alternative, one could count 1 parameter for the phase plus 6 parameters that Lorentz transformations have in general, for a total of 7 parameters, then subtract one parameter that is redundant, due to the fact that the gauge transformation and the rotation around the third axis have the same effect on the spinor, as is clear from (18). To continue the parallel with the Standard Model, recall that the \(\mathrm {U(1)\!\times \!SU(2)}\) gauge group has a total of 4 parameters, but the hypercharge and the third component of the isospin combine to form a single parameter. Finally, to conclude this parallel, remark that here the frame in which the spinor is at rest and with spin aligned along the third axis is analogous to what in the Standard Model is for the Higgs field the unitary gauge [10].
Now, in general, the spinorial covariant derivative is defined according to
$$\begin{aligned} \varvec{\nabla }_{\mu }\psi \!=\!\partial _{\mu }\psi \!+\!\varvec{C}_{\mu }\psi \end{aligned}$$
(24)
in terms of the spinorial connection \(\varvec{C}_{\mu }\) which is itself defined by its transformation
$$\begin{aligned} \varvec{C}_{\mu }\!\rightarrow \!\varvec{S}\left( \varvec{C}_{\mu } \!-\!\varvec{S}^{-1}\partial _{\mu }\varvec{S}\right) \varvec{S}^{-1} \end{aligned}$$
(25)
where \(\varvec{S}\) is the spinorial transformation. This spinorial connection can be decomposed according to
$$\begin{aligned} \varvec{C}_{\mu }\!=\!\frac{1}{2}C^{ab}_{\mu }\varvec{\sigma }_{ab} \!+\!iqA_{\mu }\varvec{\mathbb {I}} \end{aligned}$$
(26)
where \(C^{ab}_{\mu }\) is the spin connection of the space-time and \(A_{\mu }\) is the gauge potential. Because in general
$$\begin{aligned} \varvec{L}^{-1}\partial _{\mu }\varvec{L}\!=\!iq\partial _{\mu }\zeta \mathbb {I} \!+\!\frac{1}{2}\partial _{\mu }\zeta _{ij}\varvec{\sigma }^{ij} \end{aligned}$$
(27)
for some \(\zeta \) and \(\zeta _{ij}\) that are precisely the Goldstone fields of the spinor. Then we can define the quantities
$$\begin{aligned} P_{\mu }:= & {} q(\partial _{\mu }\zeta \!-\!A_{\mu }) \end{aligned}$$
(28)
$$\begin{aligned} F_{ij\mu }\!:= & {} \!\partial _{\mu }\zeta _{ij}\!-\!C_{ij\mu } \end{aligned}$$
(29)
which are proven to be real tensors. From (18) and (28, 29) we get
$$\begin{aligned} \varvec{\nabla }_{\mu }\psi \!=\!(-\frac{i}{2}\nabla _{\mu }\beta \varvec{\pi } \!+\!\nabla _{\mu }\ln {\phi }\mathbb {I} \!-\!iP_{\mu }\mathbb {I}\!-\!\frac{1}{2}F_{ij\mu }\varvec{\sigma }^{ij})\psi \end{aligned}$$
(30)
as the polar form of the covariant derivative. By taking this form and contracting on the left with \(\overline{\psi }\varvec{\gamma }^{k}\) we get
$$\begin{aligned} \overline{\psi }\varvec{\gamma }^{k}\varvec{\nabla }_{\mu }\psi \!=\!U^{k}\nabla _{\mu }\ln {\phi }\!-\!\frac{i}{2}\nabla _{\mu }\beta S^{k}\!-\!iP_{\mu }U^{k} \!+\!\frac{i}{4}F_{ij\mu }\varepsilon ^{kijq}S_{q}-\frac{1}{2}F_{ij\mu }\eta ^{ki}U^{j} \end{aligned}$$
(31)
whose real part is simply
$$\begin{aligned} \nabla _{\mu }U_{k}\!=\!2U_{k}\nabla _{\mu }\ln {\phi }+U^{j}F_{jk\mu }. \end{aligned}$$
(32)
This can be written as
$$\begin{aligned} \nabla _{\mu }(2\phi ^{2}u_{k})\!=\!2\phi ^{2}u_{k}\nabla _{\mu }\ln {\phi ^{2}} +2\phi ^{2}u^{j}F_{jk\mu } \end{aligned}$$
(33)
and so
$$\begin{aligned} 2\nabla _{\mu }\phi ^{2}u_{k}+2\phi ^{2}\nabla _{\mu }u_{k} \!=\!2\phi ^{2}u_{k}\nabla _{\mu }\ln {\phi ^{2}}+2\phi ^{2}u^{j}F_{jk\mu } \end{aligned}$$
(34)
where the derivatives of the module eventually cancel. Therefore
$$\begin{aligned} \nabla _{\mu }u_{k}\!=\!u^{j}F_{jk\mu } \end{aligned}$$
(35)
as a general identity. The same is true for the axial-vector. In conclusion, we have
$$\begin{aligned} \nabla _{\mu }s_{i}\!=\!F_{ji\mu }s^{j}\ \ \ \ \ \ \ \ \ \ \ \ \nabla _{\mu }u_{i}\!=\!F_{ji\mu }u^{j} \end{aligned}$$
(36)
which are valid as general identities. These last two identities imply in turn the validity of the relation
$$\begin{aligned} F_{ab\mu }\equiv & {} u_{a}\nabla _{\mu }u_{b}\!-\!u_{b}\nabla _{\mu }u_{a} \!+\!s_{b}\nabla _{\mu }s_{a}\!-\!s_{a}\nabla _{\mu }s_{b} \!+\!(u_{a}s_{b}\!-\!u_{b}s_{a})\nabla _{\mu }u_{k}s^{k}\nonumber \\{} & {} +\frac{1}{2}F_{ij\mu }\varepsilon ^{ijcd}\varepsilon _{abpq}s_{c}u_{d}s^{p}u^{q} \end{aligned}$$
(37)
which can be written as
$$\begin{aligned} F_{ab\mu }= & {} u_{a}\nabla _{\mu }u_{b}\!-\!u_{b}\nabla _{\mu }u_{a} \!+\!s_{b}\nabla _{\mu }s_{a}\!-\!s_{a}\nabla _{\mu }s_{b} \!+\!(u_{a}s_{b}\!-\!u_{b}s_{a})\nabla _{\mu }u_{k}s^{k}\nonumber \\{} & {} \quad +2\varepsilon _{abij}u^{i}s^{j}V_{\mu } \end{aligned}$$
(38)
for some vector
$$\begin{aligned} V_{\mu }:=\frac{1}{4}F_{ij\mu }\varepsilon ^{ijcd}u_{c}s_{d} \end{aligned}$$
(39)
that cannot be specified in terms of covariant derivatives of velocity and spin. Finally, we can write (30) with (38) as
$$\begin{aligned} \varvec{\nabla }_{\mu }\psi= & {} [-\frac{i}{2}\nabla _{\mu }\beta \varvec{\pi } \!+\!\nabla _{\mu }\ln {\phi }\mathbb {I} \!-\!i(P_{\mu }\!-\!V_{\mu })\mathbb {I}\!-\!(u_{a}\nabla _{\mu }u_{b}\nonumber \\{} & {} +s_{b}\nabla _{\mu }s_{a} \!+\!u_{a}s_{b}\nabla _{\mu }u_{k}s^{k})\varvec{\sigma }^{ab}]\psi \end{aligned}$$
(40)
having used \(2i\varvec{\sigma }_{ab}\!=\!\varepsilon _{abcd}\varvec{\pi }\varvec{\sigma }^{cd}\) and (21). After that the Goldstone fields are transferred into the frame, they combine with spin connection and gauge potential to become the longitudinal components of the \(P_{\mu }\) and \(F_{ij\mu }\) tensors. The \(P_{\mu }\) and \(F_{ij\mu }\) objects, therefore, have the same information content of spin connection and gauge potential while being real tensors, and it is for this reason that they are called space-time and gauge tensorial connections. In this, the tensorial connections we have here are the geometric and electrodynamic analog of the weak bosons \(W_{\nu }^{\pm }\) and \(Z_{\nu }\) of the Standard Model. Velocity and spin can only determine 5 parameters of \(\varvec{L}\), and so their derivatives can only determine 20 components of \(F_{ij\mu }\) with the 4 missing components corresponding to rotations around the third axis encoded by the \(F_{12\mu }\) component. Because for rest-frame spin-eigenstate spinors it is \(u^{0}\!=\!1\) and \(s^{3}\!=\!1\) it follows that \(2V_{\mu }\!=\!F_{12\mu }\) and thus the missing components are encoded in the \(V_{\mu }\) vector. Alternatively, one can count the 24 components of the space-time tensorial connection plus the 4 components of the gauge tensorial connection, totalling 28 components, and subtract the four components that are redundant, due to the fact that \(V_{\mu }\) and \(P_{\mu }\) act identically on the spinor field, as is clear from the structure of (40) [10].
It is also important to notice that the relations
$$\begin{aligned} R_{\alpha \rho \mu \nu }= & {} -(\nabla _{\mu }F_{\alpha \rho \nu }\!-\!\nabla _{\nu }F_{\alpha \rho \mu } \!+\!F_{\alpha \kappa \mu }F_{\eta \rho \nu }g^{\kappa \eta } \!-\!F_{\alpha \kappa \nu }F_{\eta \rho \mu }g^{\kappa \eta }) \end{aligned}$$
(41)
$$\begin{aligned} qF_{\mu \nu }= & {} -(\nabla _{\mu }P_{\nu }\!-\!\nabla _{\nu }P_{\mu }) \end{aligned}$$
(42)
are valid as geometric identities, which means that the \(F_{ab\mu }\) and \(P_{\mu }\) tensors can be, respectively, seen as gauge-invariant and covariant potentials of the Riemann curvature \(F_{\alpha \rho \mu \nu }\) and the Maxwell strength \(F_{\mu \nu }\) in general.
Due to the importance of the tensorial connection, and specifically the space-time tensorial connection \(F_{ab\mu }\) we now give further comments about its geometrical interpretation. To this end, let us suppose to perform the pointwise tetrad transformation given by
$$\begin{aligned} \hat{e}^{i}\!=\!\Lambda ^{i}_{a}e^{a}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \hat{e}_{a}\!=\!\Lambda _{a}^{i}e_{i} \end{aligned}$$
(43)
where \(\Lambda _{a}^{\;\;i}\) denotes a suitable real Lorentz transformation and \(\Lambda ^i_{\;\;a}\!=\!(\Lambda ^{-1})_{a}^{\;\;i}\) is its inverse. Real Lorentz transformations are tied to the corresponding complex Lorentz transformation (or spinorial transformation) \(\varvec{L}\) by the requirement
$$\begin{aligned} \varvec{L}\varvec{\gamma }^{j}\varvec{L}^{-1}\Lambda ^i_{\;\;j}=\varvec{\gamma }^{i} \end{aligned}$$
(44)
where both real and complex (or spinorial) transformations have the same parameters, which are in general functions of the space-time coordinates. Because of the change of trivialization induced by the real Lorentz transformation, the spinor field \(\psi \) undergoes the corresponding spinorial transformation
$$\begin{aligned} \hat{\psi }\!=\!\varvec{L}(\Lambda ^{-1})\psi \end{aligned}$$
(45)
by construction. In the same way, a spin connection \(\Omega ^{ij}_{\;\;\;\mu }\) on the space-time transforms according to
$$\begin{aligned} \hat{\Omega }^{ij}_{\;\;\;\mu } = \Omega ^{hk}_{\;\;\;\mu }\Lambda ^i_{\;\;h}\Lambda ^j_{\;\;k} - \Lambda ^{js}\frac{\partial }{\partial x^\mu }\Lambda ^i_{\;\;s} \end{aligned}$$
(46)
in which we have denoted \(\Lambda ^{js}=\Lambda ^{j}_{\;\;t}\eta ^{ts}\) for brevity. In particular, we consider the flat spin connection whose coefficients \(\Omega ^{ij}_{\;\;\;\mu }\) are zero in the frame where the spinor is at rest and with spin aligned along the third axis, which in the parallel to the Standard Model we will call the unitary frame. Correspondingly, we will call such spin connection the Goldstone connection. After transformation (43), in the new tetrad \(\hat{e}_i\) the Goldstone connection is
$$\begin{aligned} \hat{\Omega }^{ij}_{\;\;\;\mu }\!=\!-\Lambda ^{js}\frac{\partial }{\partial x^\mu }\Lambda ^i_{\;\;s} \end{aligned}$$
(47)
and identity (27) can be written as
$$\begin{aligned} \frac{1}{2}\Lambda ^{js}\partial _{\mu }\Lambda ^i_{\;\;s}\varvec{\sigma }_{ij}\varvec{L} \!=\!\partial _{\mu }\varvec{L} \end{aligned}$$
(48)
where \(\varvec{L}=\varvec{L}(\Lambda ^{-1})\) is the matrix appearing in equation (45). Indeed one has
$$\begin{aligned} \frac{1}{2}\Lambda ^{js}\partial _{\mu }\Lambda ^i_{\;\;s}\varvec{\sigma }_{ij}\varvec{L}= & {} \frac{1}{8}\left[ \partial _{\mu }(\Lambda ^i_{\;\;s}\varvec{\gamma }_{i}), \Lambda ^{js}\varvec{\gamma }_{j}\right] \varvec{L}\nonumber \\= & {} \frac{1}{8}\left[ \partial _{\mu }(\varvec{L}\varvec{\gamma }_{s}\varvec{L}^{-1}), \eta ^{st}\varvec{L}\varvec{\gamma }_{t}\varvec{L}^{-1}\right] \varvec{L} \nonumber \\= & {} \frac{1}{4}\partial _{\mu }(\varvec{L}\varvec{\gamma }_{s}\varvec{L}^{-1}) (\varvec{L}\varvec{\gamma }^{s}\varvec{L}^{-1})\varvec{L}\nonumber \\= & {} \partial _{\mu }\varvec{L}\!+\!\frac{1}{4}\varvec{L}\varvec{\gamma }_{s} \partial _{\mu }\varvec{L}^{-1}\varvec{L}\varvec{\gamma }^{s} \!=\!\partial _{\mu }\varvec{L} \end{aligned}$$
(49)
in view of identities \(\varvec{\gamma }_s\left[ \varvec{\gamma }_i,\varvec{\gamma }_j\right] \varvec{\gamma }^s\!=\!0\) holding in 4 dimensions. From (48) we derive the relation
$$\begin{aligned} \Lambda ^{js}\frac{\partial }{\partial x^\mu }\Lambda ^i_{\;\;s}\frac{1}{2}\varvec{\sigma }_{ij} \!=\!\frac{\partial \varvec{L}}{\partial x^\mu }\varvec{L}^{-1} \!=\!-\frac{1}{2}\frac{\partial \zeta ^{ij}}{\partial x^\mu }\varvec{\sigma }_{ij} \end{aligned}$$
(50)
and because of the linear independence of the \(\varvec{\sigma }_{ij}\) matrices we conclude that
$$\begin{aligned} \frac{\partial \zeta ^{ij}}{\partial x^\mu } = - \Lambda ^{js}\frac{\partial }{\partial x^\mu }\Lambda ^i_{\;\;s} \end{aligned}$$
(51)
as a general relation. The space-time tensorial connection is defined as
$$\begin{aligned} F^{ij}_{\;\;\;\mu } = \frac{\partial \zeta ^{ij}}{\partial x^\mu }-C^{ij}_{\;\;\;\mu } \end{aligned}$$
(52)
where \(C^{ij}_{\;\;\;\mu }\) denotes the Levi-Civita spin connection. Making use of equations (47) and (51) we can re-write (52) as
$$\begin{aligned} F^{ij}_{\;\;\;\mu } = \hat{\Omega }^{ij}_{\;\;\;\mu }\!-\!C^{ij}_{\;\;\;\mu } \end{aligned}$$
(53)
which represents the difference between the Goldstone connection and the Levi-Civita spin connection, clarifying the tensor nature of the quantity \(F^{ij}_{\;\;\;\mu }\) and its meaning. More precisely, it is evident that the tensor \(F_{ab\mu }\) identifies with the contorsion tensor of the Goldstone connection. In such a circumstance, the conditions that the vectors \(u^i\) and \(s^i\) must satisfy (36) are the same as requiring that both vectors \(u^i\) and \(s^i\) verify
$$\begin{aligned} \hat{\nabla }_\mu u^i = 0\ \ \ \ \ \ \ \ \hat{\nabla }_\mu s^i = 0 \end{aligned}$$
(54)
that is they are constant with respect to the Goldstone covariant derivative.
Having constructed all kinematic objects we need, we are able to outline the dynamics of Dirac fields in polar form.
It is seen that the Dirac equation
$$\begin{aligned} i\varvec{\gamma }^{\mu }\varvec{\nabla }_{\mu }\psi \!-\!m\psi \!=\!0 \end{aligned}$$
(55)
can be written equivalently in polar form as [10]
$$\begin{aligned}{} & {} \nabla _{\mu }\ln {\phi ^{2}}\!+\!F_{\mu \nu }^{\nu } \!-\!2P^{\rho }u^{\nu }s^{\alpha }\varepsilon _{\mu \rho \nu \alpha }\!+\!2ms_{\mu }\sin {\beta }\!=\!0 \end{aligned}$$
(56)
$$\begin{aligned}{} & {} \nabla _{\mu }\beta \!+\!\frac{1}{2}\varepsilon _{\mu \alpha \nu \iota }F^{\alpha \nu \iota } \!-\!2P^{\iota }u_{[\iota }s_{\mu ]}\!+\!2ms_{\mu }\cos {\beta }\!=\!0 . \end{aligned}$$
(57)
Making use of expression (38), (56, 57) become
$$\begin{aligned}{} & {} \nabla _{\mu }\ln {\phi ^{2}}\!+\!u_{\mu }\nabla _{i}u^{i}\!-\!s_{\mu }\nabla _{i}s^{i} \!-\!u^{\nu }\nabla _{\nu }u_{\mu }\!+\!s^{\nu }\nabla _{\nu }s_{\mu } \!+\!(u_{\mu }s^{\nu }\!-\!u^{\nu }s_{\mu })\nabla _{\nu }u_{i}s^{i}\nonumber \\{} & {} \quad -2(P^{\rho }\!-\!V^{\rho })u^{\nu }s^{\alpha }\varepsilon _{\mu \rho \nu \alpha } \!+\!2ms_{\mu }\sin {\beta }\!=\!0 \end{aligned}$$
(58)
$$\begin{aligned}{} & {} \nabla _{\mu }\beta \!-\!\varepsilon _{\mu \alpha \nu \beta }u^{\alpha }\nabla ^{\nu }u^{\beta } \!+\!\varepsilon _{\mu \beta \nu \alpha }s^{\beta }\nabla ^{\nu }s^{\alpha }\nonumber \\{} & {} \quad +\varepsilon _{\mu \alpha \beta \nu }u^{\alpha }s^{\beta }\nabla ^{\nu }u^{i}s_{i} \!-\!2(P^{\nu }\!-\!V^{\nu })u_{[\nu }s_{\mu ]} \!+\!2ms_{\mu }\cos {\beta }\!=\!0. \end{aligned}$$
(59)
For simplicity, let us introduce the notations
$$\begin{aligned} 2Z_{\mu }:= & {} \nabla _{\mu }\ln {\phi ^{2}}\!+\!u_{\mu }\nabla _{i}u^{i}\!-\!s_{\mu }\nabla _{i}s^{i} \!-\!u^{\nu }\nabla _{\nu }u_{\mu }\!+\!s^{\nu }\nabla _{\nu }s_{\mu }\nonumber \\{} & {} +(u_{\mu }s^{\nu }\!-\!u^{\nu }s_{\mu })\nabla _{\nu }u_{i}s^{i} \end{aligned}$$
(60)
$$\begin{aligned} 2Y_{\mu }:= & {} \nabla _{\mu }\beta \!-\!\varepsilon _{\mu \alpha \nu \beta }u^{\alpha }\nabla ^{\nu }u^{\beta } \!+\!\varepsilon _{\mu \beta \nu \alpha }s^{\beta }\nabla ^{\nu }s^{\alpha } \!+\!\varepsilon _{\mu \alpha \beta \nu }u^{\alpha }s^{\beta }\nabla ^{\nu }u^{i}s_{i} \end{aligned}$$
(61)
so that the Dirac equations in polar form can be expressed in the more compact form
$$\begin{aligned}{} & {} Z_{\mu }\!-\!(P^{\rho }\!-\!V^{\rho })u^{\nu }s^{\alpha }\varepsilon _{\mu \rho \nu \alpha } \!+\!ms_{\mu }\sin {\beta }\!=\!0 \end{aligned}$$
(62)
$$\begin{aligned}{} & {} Y_{\mu }\!-\!(P^{\nu }\!-\!V^{\nu })u_{[\nu }s_{\mu ]} \!+\!ms_{\mu }\cos {\beta }\!=\!0. \end{aligned}$$
(63)
Taking (62) multiplied by \(\varepsilon ^{\mu \eta \pi \tau }u_{\pi }s_{\tau }\) plus (63) multiplied by \(u^{[\mu }s^{\eta ]}\) gives
$$\begin{aligned} P^{\eta }\!-\!V^{\eta }\!=\!m\cos {\beta }u^{\eta }\!+\!Y_{\mu }u^{[\mu }s^{\eta ]} \!+\!Z_{\mu }u_{\pi }s_{\tau }\varepsilon ^{\mu \pi \tau \eta } \end{aligned}$$
(64)
in which the vector \(P^{\eta }\!-\!V^{\eta }\) has been made explicit. Because \(\nabla _{i}S^{i}\!=\!4m\phi ^{2}\sin {\beta }\) [11], the classical limit \(S^{i}\!\rightarrow \!0\) implies also \(\beta \!\rightarrow \!0\) so that (64) reduces to \(P^{\eta }\!=\!mu^{\eta }\) showing that \(P_{\eta }\) is the momentum of the particle. General expression (64) is therefore an extension of the momentum so to include the spin. Hence, the vector \(P_{\mu }\!-\!V_{\mu }\) can be interpreted as the most general momentum after the redundancy between gauge and rotation around the third axis is removed.
The Dirac field has energy tensor given by
$$\begin{aligned} T^{\rho \sigma } \!=\!\frac{i}{4}(\overline{\psi }\varvec{\gamma }^{\rho }\varvec{\nabla }^{\sigma }\psi \!-\!\varvec{\nabla }^{\sigma }\overline{\psi }\varvec{\gamma }^{\rho }\psi \!+\!\overline{\psi }\varvec{\gamma }^{\sigma }\varvec{\nabla }^{\rho }\psi \!-\!\varvec{\nabla }^{\rho }\overline{\psi }\varvec{\gamma }^{\sigma }\psi ) \end{aligned}$$
(65)
which in polar variables becomes
$$\begin{aligned} T^{\rho \sigma }= & {} \phi ^{2}(P^{\rho }u^{\sigma }\!+\!P^{\sigma }u^{\rho } \!+\!s^{\sigma }\nabla ^{\rho }\beta /2 \!+\!s^{\rho }\nabla ^{\sigma }\beta /2 \!-\!\frac{1}{4}F_{\alpha \nu }^{\sigma }s_{\kappa } \varepsilon ^{\rho \alpha \nu \kappa }\nonumber \\{} & {} -\frac{1}{4}F_{\alpha \nu }^{\rho }s_{\kappa } \varepsilon ^{\sigma \alpha \nu \kappa }) \end{aligned}$$
(66)
where (30) was used. By plugging now (38) we obtain
$$\begin{aligned} T^{\rho \sigma }= & {} \phi ^{2}[(P^{\rho }\!-\!V^{\rho })u^{\sigma } \!+\!(P^{\sigma }\!-\!V^{\sigma })u^{\rho } \!+\!s^{\sigma }\nabla ^{\rho }\beta /2 \!+\!s^{\rho }\nabla ^{\sigma }\beta /2\nonumber \\{} & {} -\frac{1}{2}s_{\kappa }u_{\alpha }\nabla ^{\sigma }u_{\nu }\varepsilon ^{\rho \nu \kappa \alpha } \!-\!\frac{1}{2}s_{\kappa }u_{\alpha }\nabla ^{\rho }u_{\nu }\varepsilon ^{\sigma \nu \kappa \alpha }]. \end{aligned}$$
(67)
As is clear from the treatment of the dynamics in polar form, the vector \(P_{\mu }\!-\!V_{\mu }\) has a very special role and such a special role will become even more prominent when the Lie derivative of spinor fields will be considered.
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