Dirac Hydrodynamics in 19 Forms

We start this first section by recalling the general ideas of the conversion of the Dirac theory in its hydrodynamic formulation. As stated in the introduction, for us, Dirac hydrodynamics, or relativistic spinning quantum mechanics in hydrodynamic form, is just the relativistic extension with a spin of quantum mechanics in hydrodynamic form as it was initially conceived by Madelung. As such, it is also taken as a synonym to spinor theory in the polar form.

We begin the presentation by recalling the notations. First of all, we define the Clifford matrices ${\mathsf{\gamma }}^a$ as the set of matrices verifying $\{{\mathsf{\gamma }}_a,{\mathsf{\gamma }}_b\}=2\mathbb{I}{\eta }_{ab}$, where ${\eta }_{ab}$ is the Minkowski matrix. From them, we define $[{\mathsf{\gamma }}_a,{\mathsf{\gamma }}_b]/4={\mathsf{\sigma }}_{ab}$ verifying $2i{\mathsf{\sigma }}_{ab}={\epsilon }_{abcd}\mathsf{\pi }{\mathsf{\sigma }}^{cd}$ where ${\epsilon }_{abcd}$ is the completely antisymmetric pseudo-tensor and in which the $\mathsf{\pi }$ matrix is defined. This matrix is what is traditionally indicated by ${\mathsf{\gamma }}^5$ as the fifth matrix after ${\mathsf{\gamma }}^1$, ${\mathsf{\gamma }}^2$, ${\mathsf{\gamma }}^3$, ${\mathsf{\gamma }}^4$ in Kaluza–Klein theories, at the time when the temporal coordinate was still designated as the fourth coordinate. Nowadays, we use a zero to indicate the temporal gamma matrix, and thus we should use a four for the fifth gamma matrix, or better still, we should just drop an index that no longer corresponds to any actual dimension. But even worse, in some signatures, the position of the index five, whether upper or lower, changes the sign of the gamma matrix, with ensuing risks of errors that become possible. To avoid the potential for mistakes, we prefer to employ a notation in which there is no index at all. The choice of $\mathsf{\pi }$ instead of $\mathsf{\gamma }$ is explained by the fact that the matrix is parity-odd as opposed to the usual gammas being parity-even. This is not dissimilar to the normally accepted choice of using $\mathsf{\sigma }$ for the commutators of gamma matrices. Identity ${\mathsf{\gamma }}_i{\mathsf{\gamma }}_j{\mathsf{\gamma }}_k={\mathsf{\gamma }}_i{\eta }_{jk}-{\mathsf{\gamma }}_j{\eta }_{ik}+{\mathsf{\gamma }}_k{\eta }_{ij}+i{\epsilon }_{ijkq}\mathsf{\pi }{\mathsf{\gamma }}^q$ shows that any product of more than two matrices can always be reduced to two matrices, therefore suggesting that the set $(\mathbb{I},{\mathsf{\gamma }}^a,{\mathsf{\sigma }}_{ab},{\mathsf{\gamma }}^a\mathsf{\pi },\mathsf{\pi })$ is complete, and so it is also a basis for the space of $4\times 4$ complex matrices. From these definitions, it is straightforward that ${\mathsf{\sigma }}_{ab}$ are the generators of the Lorentz algebra. Exponentiation of the generators offers $\mathsf{\Lambda }$ as the element of the Lorentz group. We notice that $\mathsf{\Lambda }$ is a complex Lorentz transformation, as opposed to ${\mathsf{\Lambda }}_b^a$ being the real Lorentz transformation. The two are linked by the relation $\mathsf{\Lambda }{\mathsf{\gamma }}^b{\mathsf{\Lambda }}^{-1}{\mathsf{\Lambda }}_b^a={\mathsf{\gamma }}^a$ which, in turn, also specifies that the Clifford matrices are all constant, as is expected and well known. With the complex Lorentz transformation $\mathsf{\Lambda }$ and a general unitary phase $e^{iq\alpha }$, we define $\mathit{S}=\mathsf{\Lambda }{e}^{iq\alpha }$ as the spinorial transformation. The spinorial transformation is therefore just the product of boosts and rotations as well as a gauge shift of charge q as it is supposed to be to account for both space-time and electrodynamic transformations.

So far, nothing is new, and, aside from the specific convention we employ, all can be found in textbooks. With these tools, we define spinor fields as objects that, under spinorial transformations, transform according to where $\overline{\psi }={\psi }^{†}{\mathsf{\gamma }}^0$ is the (unique) adjunction procedure. With these adjoint spinors, we can construct the bi-linears, which are all real tensors. We defined six of them for reasons of symmetry and simplicity, but as it is clear from the linear independence of the Clifford matrices, they are not all linearly independent. In fact, showing that the two antisymmetric tensors are just the Hodge dual of one another. By taking only one of them, $M_{ab}$, for example, together with the other four bi-linears, we may form a set of bi-linears that are all linearly independent, although not independent. Indeed, showing that if ${\mathsf{\Phi }}^2+{\mathsf{\Theta }}^2\ne 0$; then, $M_{ab}$ can also be dropped in favour of the two vectors and the two scalars. The axial vector and the vector with the pseudo-scalar and scalar are also not independent since and, in the case in which ${\mathsf{\Phi }}^2+{\mathsf{\Theta }}^2\ne 0$, we can see that the axial vector is space-like, while the vector is time-like. Finally, to complete the list of identities between the bi-linears that we use below.

Now, a result that can be proven is that under the general assumption ${\mathsf{\Phi }}^2+{\mathsf{\Theta }}^2\ne 0$, it is always possible to write any spinor field in the polar form, which in a chiral representation is given according to for a pair of functions $\varphi$ and $\beta$ and for some $\mathit{L}$ having the structure of a spinorial transformation and such that the polar form is unique up to the discrete transformation $\beta \to \beta +\pi$ and up to the reversal of the third axis [20,21]. The three elements $\mathit{L}$, $\varphi$ and $\beta$ in (11) need some explanation. After that, (11) is substituted in (4); one obtains showing that $\varphi$ and $\beta$ are a real scalar and a real pseudo-scalar called the module and the chiral angle. Equipped with the polar form, we can also normalize where $u^a$ and $s^a$ are the velocity vector and the spin axial vector. Identities (7) and (8) reduce to showing that the velocity has only three independent components given by the three components of its spatial part (indeed, constraint $u_a{u}^a=1$ fixes the temporal component), whereas the spin has only two independent components, assigned by the two angles that, in the rest frame, its spatial part forms with the third axis (in fact, because of constraint $u_a{s}^a=0$, the temporal component of the spin is zero in the rest frame, and due to constraint $s_a{s}^a=-1$, one of the spatial components is fixed). The physical interpretation of this mathematical fact is that, in the frame at rest, there remains no spatial component of the velocity and its temporal component is unity, and because this also implies the vanishing of the temporal component of the spin, its spatial component, when aligned along the third axis, selects this axis as the axis of symmetry of the system. Hence, any rotation around this axis would have to be an irrelevant rotation, as it would be unable to have any effect on the spinor itself. In the following sections, we still refer to rotations around the third axis because we chose this axis for all explicit computations, although it is clear that from a covariant perspective, we would refer to them as rotations around the spin axis. The remaining identities (9) and (10) are which are useful later on. As for $\mathit{L}$, we know that it has the structure of a spinorial transformation, and thus it is a product of a gauge times a Lorentz transformation. Its interpretation is of straightforward reading. It is the specific transformation that takes an assigned spinor into its rest frame with the spin aligned along the third axis, however general the initial spinor is. As such, $\mathit{L}$ should not be confused with $\mathit{S}$ because even if mathematically they are the same and both are a spinorial transformation, from a physical perspective, they are very different, with $\mathit{S}$ indicating the way in which spinors behave under transformations and $\mathit{L}$ indicating how any given spinor can always be seen as a suitable deformation of its simplest rest frame spin-eigenstate form. Metaphorically, if the spinor were to be a top spinning on a table, then $\mathit{S}$ would indicate 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 $\mathit{L}$ would indicate how the top is spinning. Therefore, the advantage of writing spinor fields in the polar form is that in their four complex components, the eight real functions are re-organized in such a way that the two real scalars, that is, the two true degrees of freedom ($\varphi$ and $\beta$), remain isolated from the six real parameters, which can always be transferred into the frame (encoded within $\mathit{L}$). The fact that $\mathit{L}$ is the product of a gauge times a Lorentz transformation might in principle lead us to think that it has, in total, 1 + 6 = 7 parameters, and this appears to be in contradiction with the fact that it effectively has only six parameters as we have just explained. The resolution of this paradox is that, as is clear from (11), the action of a gauge transformation is indistinguishable from the action of a rotation around the third axis. The consequence is that there is a redundancy between the phase and the angle of rotation around the third axis, so that the a priori seven parameters are in fact reduced to six parameters alone. If the total parameters of the combined gauge and Lorentz transformations are essentially six and one is encoded by the gauge transformation, then there can be no more than five parameters that remain in the Lorentz transformation, and this is precisely what happens. The five parameters are given by the three rapidities of the velocity vector and the two Euler angles of the spin axial vector, as we discussed above. This redundancy should not be surprising, and, indeed, it is rather common. An identical situation happens in the Standard Model, where the $\mathrm{U}\left(1\right)\times \mathrm{SU}\left(2\right)$ group has four parameters, but the hypercharge and the third component of the isospin combine to form a single parameter, so that the entire group does not have four but three Goldstone bosons solely. We notice that this is more than an example; it is a true mathematical analogy. Indeed, the fact that the parameters of $\mathit{L}$ can always be transferred into the frame means that they are the Goldstone bosons associated to the spinor field by definition. The Goldstone bosons we have here play for the spinor field exactly the same role that the Goldstone bosons in the Standard Model play for the Higgs field as demonstrated in reference [22]. We return back to this after discussing what happens at the differential level.

For now, we notice that the preferred frame mentioned in [19] is the frame in which $\mathit{L}=\mathbb{I}$ with the spinor at rest and in the spin eigenstate. The generality achieved in [20,21] is hence due to the $\mathit{L}$ operator.

For general spinor fields, the spinorial covariant derivative is defined according to in terms of the spinorial connection ${\mathit{C}}_{\mu }$, which is itself defined by its transformation law where $\mathit{S}$ is the spinorial transformation law. This spinorial connection can be decomposed according to where $C_{\phantom{ab}\mu }^{ab}$ is the spin connection of the tetrads of the space-time and $A_{\mu }$ is the gauge potential. We do not provide, in the present work, the details of the explicit form of the spin connection written in terms of the tetrad fields because this is very standard material of any introductory course of General Relativity in the tetradic formalism [19].

Now, when in the spinorial covariant derivative the spinor field is written in the polar form, something interesting starts to emerge. To see what, recall that one can always write for some ${\partial }_{\mu }\xi$ and ${\partial }_{\mu }{\xi }_{ij}$ which are in fact the Goldstone fields of the spinor. In (21), the redundancy between phase and angle of rotation around the third axis shows that the apparent seven Goldstone fields are effectively six Goldstone fields only, since ${\partial }_{\mu }{\xi }_{12}$ can always be re-absorbed in a redefinition of $q{\partial }_{\mu }\xi$ in general (in fact, component ${\partial }_{\mu }{\xi }_{12}$ multiplies ${\mathsf{\sigma }}^{12}$ while $q{\partial }_{\mu }\xi$ multiplies the identity matrix, and because for rest frame spin-eigenstates the action of ${\mathsf{\sigma }}^{12}$ is equivalent to that of the identity matrix by definition, we have components ${\partial }_{\mu }{\xi }_{12}$ and $q{\partial }_{\mu }\xi$ always displaying some redundancy, as we also discussed before). In $q{\partial }_{\mu }\xi$, the presence of the charge has been made explicit so as to simplify the definition of quantities since now q can be collected in the definition of $P_{\mu }$ in (22). It is now possible to see that, after that the Goldstone fields are transferred into the frame, they combine with gauge potential and spin connection to become the longitudinal components of the $P_{\mu }$ and $R_{ij\mu }$ objects, which then turn into a real vector and a real tensor called gauge and space-time tensorial connections, as it has been demonstrated in [22]. With (22) and (23), we finally have as the polar form of the spinor field covariant derivative. From it, we can deduce that are valid as general identities. Recall that rotations around the spin axis are unable to have effects on any component of $s_i$ and $u_i$ and therefore on ${\nabla }_{\nu }{s}_i$ and ${\nabla }_{\nu }{u}_i$ in general. Because of (25), this means that the components of $R_{ab\nu }$ that correspond to the rotations around the spin axis remain undetermined. By using (25), we can write and therefore one can always write for some vector $V_{\mu }=\frac{1}{4}{R}_{ij\mu }{\epsilon }^{ijcd}{u}_c{s}_d$ that is no more specified. This vector is precisely what represents the components of $R_{ab\nu }$ that correspond to rotations around the spin axis, as is clear from the fact that $2V_{\mu }=R_{12\mu }$ whenever the spinor field is in its rest frame with the spin aligned along the third axis. As we commented in the previous sub-section, a general spinor field can always be seen as the simplest spinor field at rest and with the spin aligned along the third axis after a suitable deformation, and as we see now, such a deformation $\mathit{L}$ has the structure of a spinorial transformation in which generator $ij$ has a parameter whose derivative with respect to the $\mu$th coordinate is given by component $R_{ij\mu }$ of the tensorial connection. As deformation $\mathit{L}$ can be interpreted like a strain, tensorial connection $R_{ij\mu }$ is interpretable like the strain-rate tensor. This analogy is very strong if we consider that $R_{ij\mu }$ in (27) can be written in terms of derivatives of spin and velocity, and that the derivative of velocity is the same object with which the strain rate tensor is constructed in continuum mechanics. Together, gauge and space-time tensorial connections $R_{ij\mu }$ and $P_{\mu }$ have a total of $24+4=28$ components, so that once we subtract the four components that cannot be determined, we remain with an effective total of 24 components. These 24 components, added to the $4\times 2=8$ components of ${\nabla }_{\mu }\varphi$ and ${\nabla }_{\mu }\beta$, make up for the full 32 components of the spinor field covariant derivative. The full counting of components is perfectly matched. As for the parallel with the Standard Model that we discussed in the previous sub-section, we can say that the tensorial connections $R_{ij\mu }$ and $P_{\mu }$ are just the geometric and electrodynamic analog of vector bosons $W_{\nu }^{\pm }$ and $Z_{\nu }$ and the frame in which the spinor field is at rest and with spin along the third axis is the analog of the unitary gauge [22]. As we said, these analogies are due to a strict mathematical parallel between the polar form of spinor fields and the Higgs field in the Standard Model, a parallel holding up until symmetry breaking.

To complete the comparison with previous works, we can finally say that it is precisely the definition of the tensorial connection that allowed the results of Jakobi and Lochak [20,21] on the polar form of the spinor field to be extended to its differential structures [22]. And from this, we can now move to the dynamics.