formulation, analogous to the one of the spherical harmonics, as discussed in
the subsection 4.4. More explicitly, let us consider that the physical
components of the wave function have been realized via a function on the
hyperboloid $dS_{D-2}$ of radius $\mu>0$ embedded in ${\mathbb{R}}^{D-2,1}$
with some set of auxiliary commuting vectors of ${\mathbb{R}}^{D-2,1}$ (for
the spin degrees of freedom) and the corresponding $O(D-2,1)$-covariant wave
equations of the UIR are known explicitly. The step 1 is therefore assumed to
be performed.

Step 2. In order to have manifest Lorentz invariance, all auxiliary variables
are lifted to $D$-vectors: the coordinates of the internal de Sitter spacetime
are denoted by $\xi^{\mu}$ and the auxiliary variables by $u_{A}^{\mu}\,$. The
wave function is taken to be $\Phi(p,\xi,u_{A})\,$, where the internal vector
$\xi$ plays a role similar to the one in the infinite-spin representations. An
important distinction is that in the ambient space formulation, one would
evaluate the wave function on the hypersurface $\xi^{2}=\mu^{2}$ instead of
imposing this relation on the wave function, as in (73). The
$O(D-2,1)$-covariant wave equations for the UIR of the little group $O(D-2,1)$
must be $O(D-1,1)$-covariantized accordingly. Concretely, this implies that
the components of the covariant wave function carry an (infinite-dimensional)
irrep of the Lorentz group.

Step 3. These covariantized wave equations and the tachyonic Klein-Gordon
equation $(p^{2}-m^{2})\psi=0$ must be supplemented by two equations: say the
orthogonality condition (71), similarly to the infinite spin representation,
and the transversality condition (47), similarly to the massive
representation. The orthogonality condition (71) may be replaced by another
transversality equation for the vector $\xi\,$.

Step 4. Now, the equation (71) implies that the internal vector belongs to the
hyperplane ${\mathbb{R}}^{D-2,1}$ orthogonal to the momentum $p\,$. Its
intersection with the hypersurface $\xi^{2}=\mu^{2}$ restricts $\xi$ to the
internal de Sitter space $dS_{D-2}\subset{\mathbb{R}}^{D-2,1}$. Moreover, the
condition (47) sets to zero all components of the wave function along the
momentum. Therefore, the remaining components are physical and carry an UIR of
the little group $O(D-2,1)$ by construction (see step 2). ∎

Example: The simplest non-trivial example corresponds to a tachyonic
representation of the inhomogeneous Lorentz group $IO(D-1,1)$ induced by a
representation of the little group $O(D-2,1)$ corresponding to “massive scalar
field” on the “internal de Sitter spacetime” $dS_{D-2}$ with $D\geqslant 4\,$.
This UIR belongs to the principal continuous series of UIR of the group
$O(D-2,1)$ and it may be realized as the space of harmonic functions on
${\mathbb{R}}^{D-2,1}$ of (complex) homogeneity degree $s$ equal to
$\frac{3-D}{2}+i\,\sigma$ (with $\sigma$ a positive real parameter) evaluated
on the unit one-sheeted hyperboloid $dS_{D-2}\subset{\mathbb{R}}^{D-2,1}\,$.
They can be regarded as a generalization of the spherical harmonics in the
Lorentzian case, where the degree is a complex number. The eigenvalue of the
quadratic Casimir operator (4) of the little group $O(D-2,1)$ on this
representation is equal to

$\displaystyle{\cal
C}_{2}\Big{(}\mathfrak{so}(D-2,1)\Big{)}\,=\,\left(\frac{D-3}{2}\right)^{2}+\sigma^{2}\,.$
(80)

The d’Alembertian on the unit hyperboloid evaluated on such functions is
precisely equal to the former eigenvalue (as is true for the Laplacian on the
unit sphere evaluated on spherical harmonics) so the corresponding fields on
the internal spacetime $dS_{D-2}$ are indeed “massive”. Inserting the above
result in (79), one sees that the quartic Casimir operator is negative for the
corresponding tachyonic representation. In four-dimensional Minkowski
spacetime, this implies that the Pauli-Lubanski vector is time-like. The
Lorentz-covariant wave function is taken to be $\Phi(p,\xi)$ evaluated on
$\xi^{2}=1$ and the corresponding relativistic equations for the induced
tachyonic representation may be chosen as

$\displaystyle\left(p^{2}-m^{2}\right)\,\Phi$ $\displaystyle=$ $\displaystyle
0\,,$ (81)
$\displaystyle\left(p\cdot\frac{\partial}{\partial\xi}\right)\,\Phi$
$\displaystyle=$ $\displaystyle 0\,,$ (82)
$\displaystyle\left(\frac{\partial}{\partial\xi}\cdot\frac{\partial}{\partial\xi}\right)\Phi$
$\displaystyle=$ $\displaystyle 0\,,$ (83)
$\displaystyle\left(\xi\cdot\frac{\partial}{\partial\xi}-s\right)\Phi$
$\displaystyle=$ $\displaystyle 0\,,$ (84)

where one should remember that $s=\frac{3-D}{2}+i\,\sigma\,$. Notice the
formal analogy with the equations (48) for the massive scalar field.

Remark: There might be sometimes confusion in the folklore surrounding the
tachyons. We would like to insist on the fact that the tachyonic
representations are indeed unitary (by definition). Still, their physical
interpretation is problematic because they are not causal in the sense that
one may show that the support of their propagator requires superluminal
propagation. Roughly speaking, the acausality is obvious because the momentum
is space-like, $p^{2}=+m^{2}\,$. The confusing point is that one may try to
circumvent this problem in the following way: solving $p^{2}-m^{2}=0$ by
$p^{\mu}=(im,\overrightarrow{0})$ enforces causality, but the price to pay is
the loss of unitarity. Indeed, the energy is pure imaginary, hence a naive
plane-wave $e^{\pm i\,p_{{}_{0}}\,x^{0}}$ is actually a non-integrable
exponential $e^{\pm mx^{0}}\,$. These remarks are summarized in the following
table:

$E=p_{0}$ | $|\overrightarrow{p}|$ | Unitarity | Causality
---|---|---|---
$0$ | $m$ | OK | KO
$\pm im$ | $0$ | KO | OK

Nevertheless, the tachyonic representations should not be discarded too
quickly on such physical grounds. Actually, if tachyonic representations
appear in the spectrum of a theory, then it merely signals a local instability
of the field theory in the sense that the perturbation theory is performed
around an unstable vacuum, and the tachyon might roll to a stable vacuum (if
any). For instance, the Higgs particle is described by nothing but a tachyonic
scalar field (induced by the trivial representation of the little group). By
analogy, one may wonder if some infinite-component tachyonic field (induced by
a non-trivial representation of the little group) could not play a role in
some huge Brout–Englert–Higgs mechanism providing mass to an infinite tower of
gauge fields in various massless irreps.

## Acknowledgments

X. B. is especially grateful to his collaborator J. Mourad for numerous
fruitful discussions on group theory and relativistic wave equations. He also
thanks K. Noui for stimulating exchanges on the unitary representations in the
case of three-dimensional spacetimes.

The authors acknowledge the Institut des Hautes Études Scientifiques de Bures-
sur-Yvette and the Université de Mons-Hainaut for hospitality during the first
stage of this work.

## References

  * [1] B. C. Hall, “An elementary introduction to groups and representations” math-ph/0005032. Expanded version published as Graduate Texts in Mathematics 222 (Springer-Verlag, 2003).
  * [2] A. O. Barut and R. Raczka, Theory of group representations and applications (Polish Scientific Publishers, 1977).
  * [3] X. Bekaert, N. Boulanger, S. de Buyl, F. Dolan and J.-H. Park, “Lie algebras: representation and applications,” to be published in the proceedings of the First Modave Summer School in Mathematical Physics (Modave, June 2005).
  * [4] V. Bargmann and E. P. Wigner, “Group theoretical discussion of relativistic wave equations,” Proc. Nat. Acad. Sci. 34 (1948) 211.
  * [5] S. Weinberg, The quantum theory of fields, Vol. 1: Foundations (Cambridge University Press, 1995) Chapter 2.
  * [6] E. P. Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Annals Math. 40 (1939) 149.
  * [7] E. P. Wigner, “Invariant quantum mechanical equations of motion,” in International Atomic Energy Agency (Vienna, 1963) 59.
  * [8] L. Brink, A. M. Khan, P. Ramond and X. Z. Xiong, “Continuous spin representations of the Poincaré and super-Poincaré groups,” J. Math. Phys. 43 (2002) 6279 [hep-th/0205145].
  * [9] B. Binegar, “Relativistic field theories in three dimensions,” J. Math. Phys. 23 (1982) 1511.
  * [10] M. Hamermesh, Group theory and its application to physical problems (Addison-Wesley, 1962) Chapter 10;
H. Bacry, Leçons sur la théorie des groupes et les symétries des particules
élémentaires (Dunod, 1967) Chapter 4;
W. Fulton and J. Harris, Representation theory: a first course (Springer,
2004) Chapters 4, 6, 15 and 19.

  * [11] S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins, “Normal forms for tensor polynomials. 1: The Riemann tensor,” Class. Quantum Grav. 9 (1992) 1151.
  * [12] R.C. King, “Branching rules for classical Lie groups using tensor and spinor methods,” J. Phys. A: Math. Gen. 8 (1975) 429\.
  * [13] R. C. King, “The dimensions of irreducible tensor representations of the orthogonal and symplectic groups,” Canad. Jour. Math. 23 (1971) 176.
  * [14] E. Inönü and E. P. Wigner, “On the contraction of groups and their representations,” Proc. Nat. Acad. Sci. 39 (1953) 510.
  * [15] X. Bekaert and J. Mourad, “The continuous spin limit of higher spin field equations,” JHEP 0601 (2006) 115 [hep-th/0509092].
  * [16] I. L. Buchbinder and S. M. Kuzenko, Ideas and methods of supersymmetry and supergravity: Or a walk through superspace (Institute of Physics Publishing, 1998) Chapters 1 and 6.
  * [17] X. Bekaert and N. Boulanger, “Tensor gauge fields in arbitrary representations of $GL(D,\mathbb{R})$,” Commun. Math. Phys. 245 (2004) 27 [hep-th/0208058]; [hep-th/0606198].
  * [18] I. Bandos, X. Bekaert, J. A. de Azcarraga, D. Sorokin and M. Tsulaia, “Dynamics of higher spin fields and tensorial space,” JHEP 0505 (2005) 031 [hep-th/0501113].
  * [19] R. Jackiw and V. P. Nair, “Relativistic wave equations for anyons,” Phys. Rev. D 43 (1991) 1933;
J. L. Cortes and M. S. Plyushchay, “Anyons as spinning particles,” Int. J.
Mod. Phys. A 11 (1996) 3331 [hep-th/9505117].

  * [20] M. Fierz, “Über die relativistische theorie krafterfreier teilchen mit beliebigem spin,” Helv. Phys. Acta XII (1939) 3.
  * [21] P. A. M. Dirac, “Relativistic wave equations,” Proc. Roy. Soc. Lond. 155A (1936) 447.
  * [22] S. Weinberg, “Photons and gravitons in perturbation theory: derivation of Maxwell’s and Einstein’s equations,” Phys. Rev. 138 (1965) B988.
  * [23] E.P. Wigner, “Relativistische Wellengleichungen” Z. Physik 124 (1947) 665\.
  * [24] A. M. Khan and P. Ramond, “Continuous spin representations from group contraction,” J. Math. Phys. 46 (2005) 053515 [Erratum-ibid. 46 (2005) 079901] [hep-th/0410107].
  * [25] V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Annals Math. 48 (1947) 568;
E. A. Thieleker, “The unitary representations of the generalized Lorentz
groups,” Trans. Amer. Math. Soc. 199 (1974) 327.


aainstitutetext:  Departamento de Estadística, Matemática e Informática and
Centre of Operations Research (CIO), Universidad Miguel Hernandez, E-03202
Elche, Alicante, Spain bbinstitutetext: Institut de Matemàtica
Multidisciplinar and Departament de Matemàtica Aplicada,
Universitat Politècnica de València, E-46022 València, Spain. ccinstitutetext:
Departament d’Astronomia i Astrofísica, Universitat de València, E-46100
Burjassot, València, Spain; and Observatori Astronòmic, Universitat de
València, E-46980 Paterna, València, Spain.

# Essential Fierz identities for a fermionic field

Roberto Dale b,1 Alicia Herrero 111Corresponding author. c and Juan Antonio
Morales-Lladosa<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

For a single fermionic field, an interpretation of the Fierz identities (which
establish relations between the bilinear field observables) is given. They
appear closely related to the algebraic class (regular or singular) of the
spin 2-form $S$ associated to the spinor field. If $S\neq 0$, the Fierz
identities follow from the 3+1 decomposition of the eigenvector equations for
$S$ with respect to an inertial laboratory, which makes this interpretation
suitable for fermionic particle physics models. When $S=0$, the Fierz
identities reduce to three constraints on the current densities associated
with the spinor field, saying that they are orthogonal, equimodular, the
vector current being timelike and the axial one being spacelike.

##  1 Introduction

A fermionic field is usually described by a four-component spinor $\Psi$,
$\Psi(x)\in{\mathbb{C}}^{4}$ at each space-time event $x$. Given a basis of
the space-time exterior algebra, say $\\{{\Gamma}^{A}\\}_{A=1}^{16}$, a set of
sixteen bilinear forms, $\Psi^{\dagger}\Gamma^{A}\Psi$, can be constructed
from $\Psi$, where $\Psi^{\dagger}$ is the hermitian conjugate spinor of
$\Psi$. These bilinear forms reveal physical properties of the field and
behave in a specific tensorial manner under Lorentz transformations.
Substituting the hermitian conjugate spinor by the Dirac adjoint
$\bar{\Psi}=\Psi^{\dagger}\gamma^{0}$, these bilinear forms are called
bilinear Dirac covariants (or local electron observables), where $\gamma^{0}$
is the Dirac conjugation matrix (see, for instance, Sakurai ). Reciprocally,
$\Psi$ can be obtained (up to a global phase factor) from its bilinear
concomitants. This is the spinor reconstruction theorem (see Refs.
Kaempffer-1981 ; Takahashi-1982 ; Takahashi-1983a ; Takahashi-1983b ;
Crawford-1985 ; Holland-1986 ; Crawford-1990 ).

The 16 bilinear covariants are not generically independent Laporte-
Uhlenbeck-1931a ; Uhlenbeck-Laporte-1931b ; deBroglie-1934 ; Pauli-1936 ;
Kofink-1937 ; Takabayasi-1957 . In fact, in every space-time event, the
bilinear forms are algebraic quantities, in the sense that their definition
does not depend on the dynamics (Dirac, Klein-Gordon equations) associated
with the field. Moreover, the products of any two bilinear forms, when
expressed in terms of linear combinations of all of them, satisfy certain
algebraic relations, which are usually referred as Fierz identities. They are
derived from the completeness relations that give the canonical basis of the
exterior algebra in terms of the basis under consideration
$\\{{\Gamma}^{A}\\}_{A=1}^{16}$ (see, for instance, Nishi-2005 ; Minogin-2011
; Minogin-2014 ). The Fierz identities are also known as Pauli-Fierz
identities Zhelnorovich-2019 .

A similar type of relations involving two different spinors, in particular a
field and its derivative Minogin-2012 , appears when the energy tensor of the
field is analyzed Inglis-Jarvis-2016 , or in the formulation the Maxwell-Dirac
equations based on the bilinear covariants Inglis-Jarvis-2014 . For a
compendium of historical references on this topic, see the introduction in
Minogin-2011  and Takabayasi-1957 , where a continuous media model governed by
the Dirac equation is studied.

Originally, in the case of the $\beta$-decay, the Fierz identitities involved
four interacting fermionic fields Fermi-1934 ; Fierz-1937 . An exhaustive
treatment of the generalized Fierz identities, that is, for a quadruple of
fermionic fields in an arbitrary dimension —with application beyond the
Standard Model of particle physics— is given in Appendix B of Ref. TOrtin-2004
. Applications of some generalized Fierz relations for fermion interaction
processes, including numerical implementations by hand, are reported in Ref.
Nieves-Pal-2004 .

The geometry attached to a spin 2-form is tacitly used in particle physics
theories Bilenky ; Lesov . In fact, the Hamiltonian constructed from four
interacting fermions takes into account the null currents when the symmetry
under parity is broken by the interaction. The null currents of the spin
2-form associated with a charged lepton and its associated neutrino enter in
the Hamiltonian by means of the operators
$C_{\pm}\equiv\frac{1}{2}(I\pm\gamma^{5}$). A linear combination of products
of two bilinear covariants with the form
$\bar{\Phi}_{1}\gamma^{\mu}C_{\pm}\Phi_{2}$ is proportional to the sum or the
difference of the timelike current $J^{\mu}$ and the spacelike current
$K^{\mu}$. Therefore, the null leptonic currents, $l$ and $n$, enter in the
Hamiltonian of the interaction. These currents $l$ and $n$ are associated with
the fundamental directions of a regular spin 2-form, in this case constructed
from a pair of leptons (for instance, an electron, $\Psi_{e}$, and its
associated neutrino $\Psi_{\nu_{e}}$). The notation used is introduced in next
section.

This paper is devoted to establish the essential set of relations between the
bilinear covariants attached to a single spinor field $\Psi$. By ‘essential’
we means non redundant, mathematically manageable, and able to retain the
whole physical information of the Fierz identities. In Minogin-2011 , Minogin
obtained a reduced set of 21 Fierz type relations, providing six different
geometric representations for the 16 electron observables, depending on the
chosen laboratory frame and the spinor field parameterization. The 21
relations in Ref. Minogin-2011  are not exactly the same as the 21 relations
deduced directly from the algebraic structure of a regular spin 2-form $S$ in
this work. Here, both sets of 21 identities are compared, establishing that
are linearly related between them and deduced from the sixteen essential
relations given by the algebraic structure of $S$. Refs. Minogin-2011 ;
Minogin-2014 ; Minogin-2012  are written in a clear 3-dimensional Euclidean
notation and goes beyond earlier works.

A 4-dimensional Lorentzian representation of the Fierz identities is done in
Refs. Crawford-1985 ; Holland-1986 ; Zhelnorovich-2019 , where an overcomplete
set of Fierz type relations is reported. This Lorentzian approach will be here
revisited and reinterpreted in terms of the invariant algebraic elements of
the spin 2-form $S$.

At this point, we would like to add a historical comment. In Ref. Laporte-
Uhlenbeck-1931a  (page 1396, footnote 17), Uhlenberg and Laporte acknowledge a
private communication by Rainich concerning some results and a “rigorous proof
of the fact that the Dirac equations possess only two algebraic quadratic
invariants”. In Ref. Uhlenbeck-Laporte-1931b , this idea was further
developed. It seems that Rainich considered the possibility of describing the
electron spin observables constraints on the basis of the algebraic
classification of electromagnetic fields in Minkowski space-time. This work
goes on this Rainich’s pioneering idea Rainich-1925 .

The paper is distributed as follows. To begin with, in Sec. 2 the necessary
terminology to read the paper and some previous results related to algebraic
properties of a Lorentzian antisymmetric covariant two-tensor (2-form) and to
the bilinear covariants associated to a spinor are summarized. In Sec. 3, the
eigenvector equations for the spin 2-form and its (star or Hodge) dual 2-form
are decomposed with respect to an arbitrary observer. In Sec. 4, the complete
independent set of identities between the local electron observables is
written in a covariant form and its equivalence to the eigenvector equations
for the spin 2-form is established. In Sec. 5, our result is compared with the
set of relations presented in Ref. Minogin-2011 , where the standard
3-dimensional Euclidean notation is used. In Sec. 6, the algebraic
classification of a spinor field is presented, laying stress on the role
played by the character (regular or singular) of the spin 2-form and the Fierz
identities for this classification. A discussion about the results concerns to
Sec. 7, containing our conclusions.

##  2 Terminology and Preliminaries

The main sign conventions and notation adopted in this paper are as follows:

(i) $g$ is the metric of the the Minkowski space-time, with signature
$(-,+,+,+)$. Let $\\{e_{\mu}\\}_{\mu=0}^{3}$ be a space-time basis, and
$\\{\theta^{\mu}\\}_{\mu=0}^{3}$ its algebraic dual,
$\theta^{\mu}(e_{\nu})=\delta^{\mu}_{\nu}$, with $\delta^{\mu}_{\nu}$ the
Kronecker delta. In biunivocal correspondence with
$\\{\theta^{\mu}\\}_{\mu=0}^{3}$, there exist four $\gamma$-matrices,
$\\{\gamma^{\mu}\\}_{\mu=0}^{3}$, that satisfy the Clifford algebra anti-
commutation relations:

$\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=-2g^{\mu\nu}I,$ (1)

where $g^{\mu\nu}=g(\theta^{\mu},\theta^{\nu})$ and $I$ is the $4\times 4$
unit matrix.222Notice that there is no limitation on the causal character of
the vectors in the basis. Dirac electron theory was originally presented in a
$\gamma$ matrix representation related to an orthonormal basis Dirac-1928 .
Later, Derrick Derrick-1981  analyzed the Dirac equation in some unusual basis
constituted by four metrically symmetric vectors (or symmetric frame), and he
used their attached $\gamma$-matrix representations. A frame
$\\{e_{A}\\}_{A=1}^{4}$ is said to be a symmetric frame (for the metric $g$)
if $g_{AA}=g(e_{A},e_{A})=\mu$ and $g_{AB}=g(e_{A},e_{B})=\nu$, $A\neq B$.
Derrick’s work would be further developed due to Derrick basis are
representative of seven symmetric causal classes of relativistic frames
RS-1992 . Thus, starting from a Derrick’s symmetric frame Derrick-1981 ;
RS-1992 , one obtains four $\gamma$-matrices, which are metrically
indistinguishable. From a causal point of view, the relativistic space-time
frames (and coordinate systems) have been classified in 199 causal classes 199
; 199b . Then, in accordance with the Clifford anti-conmmutation relations
given by Eq. (1), there exists 199 causal classes of $\gamma$-matrix
representations. Such an abundance of $\gamma$ matrix representations could be
used to describe fermion processes in non inertial frames, in order to develop
further the research presented in Ref. Chapman-Leiter-1976 .

(ii) The bilinear covariants associated with a spinor $\Psi$ and its Dirac
adjoint $\bar{\Psi}=\Psi^{\dagger}\gamma^{0}$ are defined by (cf. Minogin-2011
; Lounesto ),

$\begin{array}[]{l}\Omega_{1}=\bar{\Psi}\Psi,\\\\[5.69046pt]
J^{\mu}=\bar{\Psi}\gamma^{\mu}\Psi,\\\\[5.69046pt]
S^{\mu\nu}=\frac{\rm{i}}{2}\bar{\Psi}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})\Psi,\\\\[5.69046pt]
K^{\mu}=\bar{\Psi}\gamma^{\mu}\gamma^{5}\Psi,\\\\[5.69046pt] \Omega_{2}={\rm
i}\,\bar{\Psi}\gamma^{5}\Psi,\end{array}$ (2)

where $\rm{i}=\sqrt{-1}$, and
$\gamma^{5}=-\rm{i}\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$. Here, $\Psi$ and
its adjoint $\Psi^{\dagger}$ are represented by a column and a row complex
matrix, respectively.

The bilinear covariants (2) are phase independent and represent physical
observables. In an orthonormal inertial frame,333Accordingly with Eq. (1) and
the chosen signature $(-,+,+,+)$, the Dirac representation for the
$\gamma$-matrices associated with an orthonormal basis is taken as
$\gamma^{0}_{D}=\left(\begin{array}[]{cc}I_{2}&0_{2}\\\
0_{2}&-I_{2}\end{array}\right)\quad,\quad\vec{\gamma}_{D}=\left(\begin{array}[]{cc}0_{2}&\vec{\sigma}\\\
-\vec{\sigma}&0_{2}\end{array}\right)\,,$ where $I_{2}$ ($0_{2}$) is the
$2\times 2$ unit (zero) matrix, and
$\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ stands for the Pauli
matrices,  $\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\quad\sigma_{2}=\left(\begin{array}[]{cr}0&-i\\\
i&0\end{array}\right),\quad\sigma_{3}=\left(\begin{array}[]{cr}1&0\\\
0&-1\end{array}\right).\quad$ they are interpreted as density quantities that
transform in a specific tensorial way under the action of the Lorentz group.
They are called: the scalar $\Omega_{1}$, the vector current
$J=J_{\mu}\theta^{\mu}$, the spin 2-form
$S=S_{\mu\nu}\theta^{\mu}\otimes\theta^{\nu}=\frac{1}{2}S_{\mu\nu}\theta^{\mu}\wedge\theta^{\nu}$,
the axial current $K=K_{\mu}\theta^{\mu}$ and the pseudoscalar $\Omega_{2}$,
densities. The metric $g$ is used to lower and rise indices.

(iii) $\eta$ is the metric volume element of $g$, defined by
$\eta_{\alpha\beta\gamma\delta}=-\sqrt{-{\rm
det}\,g}\,\epsilon_{\alpha\beta\gamma\delta}$, where
$\epsilon_{\alpha\beta\gamma\delta}$ stands for the Levi-Civita permutation
symbol, $\epsilon_{0123}=1$. The Hodge (or star) dual operator associated with
$\eta$ is denoted by an asterisk $*$. For instance, the dual spin 2-form $*S$
has components
$*(S)_{\mu\nu}=\frac{1}{2}\eta_{\mu\nu\lambda\rho}S^{\lambda\rho}$, and if
$x,y,z$ are space-time vectors, one has that

$[*(x\wedge y\wedge
z)]_{\alpha}=\eta_{\alpha\beta\gamma\delta}x^{\beta}y^{\gamma}z^{\delta},$ (3)

where $\wedge$ stands for the wedge or exterior product (antisymmetrized
tensorial product of totally antisymmetric tensors).

(iv) Given $P$ and $Q$ second order tensors, the tensor $P\times Q$ denotes
its matrix product, or contraction of adjacent indices, that is

$(P\times Q)_{\mu}\,^{\nu}=P_{\mu\rho}\,Q^{\rho\nu}.$

The trace of $P$ is $\mathop{\rm tr}\nolimits(P)\equiv P^{\mu}_{\mu}$. Then,
for the spin 2-form $S$ one has

$\mathop{\rm tr}\nolimits(S^{2})=\mathop{\rm tr}\nolimits(S\times
S)=-S_{\mu\nu}S^{\mu\nu},$ (4) $\mathop{\rm
tr}\nolimits(S\times*S)=-S_{\mu\nu}(*S)^{\mu\nu}.$ (5)

(v) For an observer of unit velocity $u$, $u^{2}\equiv g(u,u)=-1$, any vector
$x$ splits as:

$x=x^{0}u+\vec{x}=(x^{0},\vec{x}),$ (6)

where $x^{0}=-x\cdot u\equiv-g(x,u)$ and $\vec{x}\in E_{\perp}$ are the time-
like and space-like components of $x$ relative to $u$, respectively, and
$E_{\perp}$ denotes the three-space orthogonal to $u$. Given two vectors
$\vec{x}$, $\vec{y}\in E_{\perp}$, their vector or cross product is expressed
as

$\vec{x}\times\vec{y}=*(u\wedge\vec{x}\wedge\vec{y}).$ (7)

The interior or contracted product by $u$ is denoted by $i(u)$. For instance,
if $S$ is a covariant $2$-tensor, one has that
$[i(u)S]_{\nu}=u^{\mu}S_{\mu\nu}$ is a covector but if $S$ is a mixed
$2$-tensor, then $[i(u)S]^{\nu}=u^{\mu}S_{\mu}^{\;\,\nu}$ is a vector.

A 2-form $S$ is decomposed as

$S=u\wedge e-*(u\wedge h),$ (8)

where $e=-i(u)S$ and $h=-i(u)*S$ are the electric and magnetic part of $S$
with respect to $u$, respectively. Then,444For any space-time 2-form $F$,
$*(*F)=-F$ . An extensive treatment on the algebraic properties of a space-
time 2-form can be seen in Ref. Fradkin-1978 ; Tolo-Joan-1988 . In Ref. Tolo-
Joan-1988 , the covariant determination of the eigendirections of $F$ is
applied to characterize the differential conditions allowing the permanence of
the null character of Maxwell fields.

$*S=u\wedge h+*(u\wedge e).$ (9)

Note that to change $S$ by $*S$, $S\hookrightarrow*S$, is equivalent to change
the electric and magnetic parts of $S$ as $(e,h)\hookrightarrow(h,-e)$.

(vi) The characteristic equation of a space-time 2-form $S$ is

$\lambda^{4}-A\lambda^{2}-\frac{B^{2}}{4}=0,$ (10)

where $A$ and $B$ are quadratic algebraic scalars defined as:

$A\equiv\frac{1}{2}\mathop{\rm tr}\nolimits(S^{2})=e^{2}-h^{2},\quad
B\equiv\frac{1}{2}\mathop{\rm tr}\nolimits(S\times*S)=2\ e\cdot h.$ (11)

From (10), the eigenvalues of $S$ are $\pm\alpha$ and $\pm{\rm i}\beta$, with

$|\alpha|={1\over\sqrt{2}}\sqrt{\sqrt{A^{2}+B^{2}}+A}$ (12)

and

$|\beta|={1\over\sqrt{2}}\sqrt{\sqrt{A^{2}+B^{2}}-A}\ .$ (13)

Consequently,

$A=\alpha^{2}-\beta^{2},\quad B=2\alpha\beta\,.$ (14)

(vii) A 2-form $S$ is said to be regular if $A^{2}+B^{2}>0$. Otherwise,
$A=B=0$, and $S$ is said to be singular. In terms of the eigenstructure of $S$
one has the following characterization:

(a) A 2-form $S$ is regular if, and only if, there exist two vectors $l,n$ and
two algebraic invariants $\alpha,\beta$, with $\alpha^{2}+\beta^{2}\not=0$
such that555Vectors and covectors that are metrically equivalent are denoted
with the same symbol without chance of confusion. For instance,
$l_{\mu}=g_{\mu\nu}l^{\nu}$ is the covector associated to the vector $l$ by
the metric. So, in the equation $i(l)S=-\alpha l$, the $l$ of the left side of
the equation is a vector while the $l$ of the right side is a covector, when
we consider $S$ as an antisymmetric covariant $2$-tensor.

$\begin{array}[]{cl}i(l)S=-\alpha l\ ,&i(n)S=\alpha n\ ,\\\\[8.5359pt]
i(l)*S=\beta l\ ,&i(n)*S=-\beta n.\end{array}$ (15)

Then $l$ and $n$ are necessary null (light-like) vectors, $l^{2}=n^{2}=0$, and
are called the principal directions of $S$. Moreover, $S$ and $*S$ admit the
following canonical expressions

$S=\alpha\,n\wedge l+\beta*(n\wedge l),$ (16) $*S=\beta\,l\wedge
n-\alpha*(l\wedge n),$ (17)

where, without loss of generality, we have chosen a parameterization of $l$
and $n$ such that $l\cdot n=-1$.

(b) A 2-form $S$ is singular if, and only if, there exists a vector $l$ such
that

$i(l)S=i(l)*S=0.$ (18)

Then $l$ is necessarily null, $l^{2}=0$, and defines the fundamental direction
of $S$, which can be expressed as

$S=l\wedge p,$ (19)

where $p$ is a determined space-like vector, up to the transformation $\
p\hookrightarrow p+\mu l$.

##  3 Spin eigenvector equations: relative formulation

In this section we express the eigenvector equations of a spin 2-form $S$ with
respect to a given inertial observer $u$. For a regular $S$, Eqs. (15) give
the eigenstructure of $S$ and $*S$, where $\alpha$ and $\beta$ satisfy
$\alpha^{2}+\beta^{2}>0$ and $l$ and $n$ are the principal directions of $S$.
For the observer $u$, $l$ and $n$ decompose as

$l=l^{0}u+\vec{l},\quad n=n^{0}u+\vec{n}\ .$ (20)

Then, the relation $i(l)S=-\alpha l$ is equivalent to:

$\alpha
l^{0}=\vec{l}\cdot\vec{e}\quad\textrm{and}\quad\alpha\vec{l}=l^{0}\vec{e}+\vec{l}\times\vec{h}\,,$
(21)

and the relation $i(n)S=\alpha n$ can be written as

$\alpha
n^{0}=-\vec{n}\cdot\vec{e}\quad\textrm{and}\quad\alpha\vec{n}=-n^{0}\vec{e}-\vec{n}\times\vec{h}\,,$
(22)

$\vec{e}$ and $\vec{h}$ being the electric and magnetic part of $S$ with
respect to $u$, respectively.

Replacing $\vec{e}$ by $\vec{h}$ and $\vec{h}$ by $-\vec{e}$ by means of the
Hodge duality, the relations $i(l)*S=\beta l$, $i(n)*S=-\beta n$ in (15) lead
to

$\beta
l^{0}=\vec{l}\cdot\vec{h}\quad\textrm{and}\quad\beta\vec{l}=l^{0}\vec{h}-\vec{l}\times\vec{e}\,,$
(23) $\beta
n^{0}=-\vec{n}\cdot\vec{h}\quad\textrm{and}\quad\beta\vec{n}=-n^{0}\vec{h}+\vec{n}\times\vec{e},$
(24)

respectively. Thus, the principal directions, $l$ and $n$, of a regular spin
2-form $S$ satisfy Eqs. (21)-(24).

For a singular $S$, the relations given in (18) lead to

$\vec{l}\cdot\vec{e}=0\quad\textrm{and}\quad
l^{0}\vec{e}+\vec{l}\times\vec{h}=0\,,$ (25)
$\vec{l}\cdot\vec{h}=0\quad\textrm{and}\quad
l^{0}\vec{h}-\vec{l}\times\vec{e}=0\,,$ (26)

which can also be obtained just making $\alpha=\beta=0$ in (21) and (23).
Thus, the fundamental direction $l$ of a spin 2-form $S$ satisfies Eqs.
(25)-(26).

Moreover, the eigenvalues, $\pm\alpha$ and $\pm\rm{i}\,\beta$, of the spin
2-form $S$ have a direct relation with the bilinears $\Omega_{1}$ and
$\Omega_{2}$ defined in (2). In fact, from the definition of $S$ in (2) one
gets that 666To obtain these results we use the fact that the matrix
$Z=4\Psi\bar{\Psi}$ can be decomposed as
$Z=\Omega_{1}I_{4}+J_{\mu}\gamma^{\mu}+\frac{\rm{i}}{2}S_{\mu\nu}\gamma^{\mu}\gamma^{\nu}+K_{\mu}\gamma^{\mu}\gamma^{5}-\rm{i}\
\Omega_{2}\gamma^{5},$ since $Z$ is not a general multivector but comes from
the spinor $\Psi$.

$\mathop{\rm
tr}\nolimits(S^{2})=2(\Omega_{2}^{2}-\Omega_{1}^{2}),\quad\mathop{\rm
tr}\nolimits(S\times*S)=4\Omega_{1}\Omega_{2}.$

Then, from Eqs. (11)-(14), one realizes that

$\alpha=\Omega_{2}\quad{\rm and}\quad\beta=\Omega_{1}.$ (27)

These relations (27) are consistent with the common apellation used for
$\Omega_{1}$ and $\Omega_{2}$ as scalar and pseudo-scalar quantities,
respectively. Let $\\{e_{\mu}\\}_{\mu=0}^{3}$ an orthonormal tetrad adapted to
the geometry of a regular 2-form $S$, that is,

$l=\frac{1}{\sqrt{2}}(e_{0}+e_{1}),\qquad n=\frac{|}{\sqrt{2}}(e_{0}-e_{1}).$
(28)

The parity transformation $(e_{0},e_{1})\hookrightarrow(e_{0},-e_{1})$ changes
$l$ by $n$ and the space-time volume element changes its sign,
$\eta\hookrightarrow-\eta$. Then, $(S,*S)\hookrightarrow(S,-*S)$ and,
according to Eqs. (16) and (17),
$(\alpha,\beta)\hookrightarrow(-\alpha,\beta)$, that is,

$(\Omega_{1},\Omega_{2})\hookrightarrow(\Omega_{1},-\Omega_{2})\,,$ (29)

under the considered spatial reflection.

In the next section the algebraic structure of the spin 2-form $S$ is
connected to the bilinear covariants associated with the fermionic field
$\Psi$ and the Fierz identities they satisfy.

##  4 Algebraic interpretation of the Fierz identities

In Ref. Crawford-1985 , an overcomplete set of Fierz identities between the
bilinear covariants for any $S\neq 0$ is reported. Using the notation
introduced in Sec 2, these identities are written as:777They are referred as
Eq. (1.3) in Crawford-1985 , and apply for any (regular or singular) spin
$2$-form. A similar set of relations for these identities has been also
considered in Holland-1986 , and Zhelnorovich-2019  (pages 136, 137).

$i(J)S=-\alpha K,\quad i(K)S=-\alpha J,$ (30) $i(J)*S=\beta K,\quad
i(K)*S=\beta J,$ (31) $\alpha S-\beta*S=J\wedge K,\quad\beta
S+\alpha*S=*(J\wedge K),$ (32) $\mathop{\rm tr}\nolimits
S^{2}=2(\alpha^{2}-\beta^{2}),\quad\mathop{\rm
tr}\nolimits(S\times*S)=4\alpha\beta.$ (33)

These overcomplete set can be reduced to an equivalent set of independent
identities according with the following statements.

  1. (a)

For a regular spin 2-form, $S\neq 0$, the Fierz identities for the
corresponding fermionic field are the eigenvector equations for $S$ and $*S$
given by (15); they are a set of 16 one-component relations. Then, the spin
2-form can be written as

$S=\alpha\,n\wedge l+\beta*(n\wedge l)\,,$ (34)

where $l$ and $n$ are the principal null directions of $S$, and $\alpha$ and
$\beta$ provide the eigenvalues of $S$ and $*S$.

  2. (b)

For a singular spin 2-form, $S\neq 0$, the Fierz identities for a fermionic
field are the eigenvector equations for $S$ and $*S$, $i(l)S=i(l)*S=0$; they
are a set of 8 one-component relations. Then, the spin 2-form is given by

$S=l\wedge p\,,$ (35)

where $l$ is the fundamental direction of $S$ and $p$ a space-like vector
orthogonal to $l$.

  3. (c)

For a zero spin 2-form, $S=0$, the Fierz identities for a fermionic field are
equivalent to the existence of two non-collinear density spinor currents, $l$
and $n$, that are null, $l^{2}=n^{2}=0$, and may be parametrized by taking
$l\cdot n=-1$.

Note that Eq. (15) exclusively involves intrinsic algebraic elements
associated with $S$: its eigenvalues and eigendirections. Moreover,
expressions in (15) can be applied for any spin 2-form $S\neq 0$, that is,
when $S$ is regular or singular. The singular case corresponds to take
$\alpha=\beta=0$. On the other hand, when $S=0$, the Fierz identities reduce
to three algebraic scalar relations that constrain the causal character of the
spinor current densities.

In the following we present the proof of these statements in separated
subsections and comment on some of their consequences.

###  4.1 Regular spin 2-form

In order to justify the statement (a), let us consider a regular spin 2-form
$S$. From the principal directions $l$ and $n$ and the invariants $\alpha$ and
$\beta$ appearing in Eqs. (15), let us define

$J_{\xi}\equiv\sqrt{\frac{1}{2}(\alpha^{2}+\beta^{2})}\,(e^{\xi}l+e^{-\xi}{n})$
(36)

and

$K_{\xi}\equiv\sqrt{\frac{1}{2}(\alpha^{2}+\beta^{2})}\,(e^{\xi}l-e^{-\xi}{n})\
,$ (37)

where $\xi$ is a real parameter. Since $l$ and $n$ are null vectors,
$l^{2}=n^{2}=0$, and taking $l\cdot n=-1$ one has

$-J_{\xi}^{2}=K_{\xi}^{2}=\alpha^{2}+\beta^{2}>0,\qquad J_{\xi}\cdot
K_{\xi}=0,$ (38)

for all $\xi\in\mathbb{R}$. Moreover, a direct calculation leads to that for
every real value $\xi$, the double one-parametric family of currents
$\\{(J_{\xi},K_{\xi})\\}_{\xi\in\mathbb{R}}$ satisfies the relations
(30)-(33). Therefore, a regular spin 2-form $S$ determines the spinor currents
$J$ and $K$, up to a boost in the timelike plane $\\{l,n\\}$ expanded by the
principal directions of $S$, that is

$\begin{array}[]{l}J_{\xi}=\cosh\xi\,J+\sinh\xi\,K\ ,\\\ \\\
K_{\xi}=\sinh\xi\,J+\cosh\xi\,K\ .\end{array}$ (39)

For $\xi=0$, the spinors currents $J_{0}\equiv J$ and $K_{0}\equiv K$ are the
associated to the spinor $\Psi_{0}\equiv\Psi$ as given in (2); and the
relations (38) are called the bilinear Pauli identities (see Ref.
Kaempffer-1981 , and references therein). In addition, the pairs
$(J_{\xi},K_{\xi})$ satisfy Eqs. (38) and are the currents for a family of
spinors $\\{\Psi_{\xi}\\}_{\xi\in\mathbb{R}}$, which share the same spin
2-form $S$ for any value of $\xi$.

Note that, at each space-time event, for an observer $e_{0}$ in the 2-plane
$\\{l,n\\}$ and a unit space-like vector of this plane, $e_{1}$ , orthogonal
to $e_{0}$, the parity transformation
$(e_{0},e_{1})\hookrightarrow(e_{0},-e_{1})$ interchanges the null currents
$l$ and $n$ each in another and tranforms the functions $(\alpha,\beta)$ into
$(-\alpha,\beta)$. Consequently, from Eqs. (34) and (35), the family of
currents $\\{(J_{\xi},K_{\xi})\\}_{\xi\in{\mathbb{R}}}$ remains invariant
under the above parity transformation, but its individual currents transform
as $J_{\xi}\hookrightarrow J_{-\xi}$ and $K_{\xi}\hookrightarrow-K_{-\xi}$. On
the other hand, from the time inversion
$(e_{0},e_{1})\hookrightarrow(-e_{0},e_{1})$ one has
$(l,n)\hookrightarrow(-n,-l)$, and then
$(J_{\xi},K_{\xi})\hookrightarrow(-J_{-\xi},K_{-\xi})$.

The spinor reconstruction theorem could be reformulated (for the regular case)
in terms of a set of seven elements (the 6 quantities of $S$ and a real boost
parameter $\xi$). Moreover, this reformulation could be expressed explicitly
in terms of the eigenstructure of $S$ and the $\xi$ parameter. In fact, the
eigenvalues of $S$ are explicitly obtained in terms of $S$ from (12) and (13).
Nevertheless, the explicit obtention of $l$ and $n$ in terms of $S$ requires
to apply the projection method (used in Fradkin-1978 ; Tolo-Joan-1988 ), which
is based on the minimal polynomial equation that $S$ satisfies.

####  4.1.1 Relative formulation of the Fierz identities

From now on, the index $\xi$ will be omitted in the spinor currents without
loss of generality, since the relations (30)-(33) are satisfied for any
$\xi\in{\mathbb{R}}$, for the given regular spin 2-form $S$.

Note that the statement (a) also allows to express the Fierz identities in a
3-dimensional formulation. In order to do it, let us split $J$ and $K$ as
$J=(j^{0},\vec{j})$ and $K=(k^{0},\vec{k})$, for a given inertial observer
$u$. Then, by addition and subtraction of the scalar equations (time-like
parts) in Eqs. (21), (22), (23) and (24) between them, one gets:

$\displaystyle\alpha j^{0}$ $\displaystyle=$
$\displaystyle\vec{k}\cdot\vec{e}\ ,$ (40) $\displaystyle\alpha k^{0}$
$\displaystyle=$ $\displaystyle\vec{j}\cdot\vec{e}\ ,$ (41)
$\displaystyle\beta j^{0}$ $\displaystyle=$ $\displaystyle\vec{k}\cdot\vec{h}\
,$ (42) $\displaystyle\beta k^{0}$ $\displaystyle=$
$\displaystyle\vec{j}\cdot\vec{h}\ ,$ (43)

where we have taken into account that $\alpha^{2}+\beta^{2}>0$. Similarly, for
the space-like parts of the same Eqs. (21), (22), (23) and (24), one obtains

$\displaystyle\alpha\vec{j}$ $\displaystyle=$ $\displaystyle
k^{0}\vec{e}+\vec{k}\times\vec{h}\ ,$ (44) $\displaystyle\alpha\vec{k}$
$\displaystyle=$ $\displaystyle j^{0}\vec{e}+\vec{j}\times\vec{h}\ ,$ (45)
$\displaystyle\beta\vec{j}$ $\displaystyle=$ $\displaystyle
k^{0}\vec{h}-\vec{k}\times\vec{e}\ ,$ (46) $\displaystyle\beta\vec{k}$
$\displaystyle=$ $\displaystyle j^{0}\vec{h}-\vec{j}\times\vec{e}\ .$ (47)

Then, Eqs. (40)-(47) provide the local physical interpretation (i. e. the
relative formulation with respect a local space-time observer) of the
eigenvector equations for a regular spin 2-form. In Sec. 5, this result is
compared with the one obtained in Ref. Minogin-2011 .

###  4.2 Singular spin 2-form

In order to justify the statement (b), let us consider a singular spin 2-form
$S$ and its fundamental direction $l$, which satisfies Eqs. (25)-(26). Now,
$l$ is the unique fermionic null density current. Then, the singular case may
be understood as a limit situation of the regular case when
$\alpha^{2}+\beta^{2}\to 0$. These can be interpreted in two different ways
from expressions (38):

(b1) The two fermionic currents are light-like and colinear. This is
equivalent to consider that $\xi\to\pm\infty$ in expressions (39) and
corresponds to take a boost in the 2-plane $\\{l,n\\}$ whose velocity goes to
the light velocity ($\tanh\xi\to\pm 1$).

(b2) One of the currents becomes ligth-like and the other one goes to zero,
that is, $J_{\xi}\equiv l$ and $K_{\xi}=0$, or reciprocally. In fact, Eq. (25)
is equivalent to Eqs. (43) and (47) by taking $\alpha=0$, or
$(k^{0},\vec{k})=0$, and replacing
$(j^{0},\vec{j})\hookrightarrow(l^{0},\vec{l})$. In a similar way, Eq. (26) is
equivalent to Eqs. (43) and (47) by taking $\beta=0$ and the above
replacement, $(j^{0},\vec{j})\hookrightarrow(l^{0},\vec{l})$.

Thus, for the singular case, only the 8 relations given in (25)-(26) are non-
trivial and significantly encode the Fierz identities.

###  4.3 Zero spin 2-form, $S=0$

Statement (c) says that there exist two density equimodular and orthogonal
four-currents, $J$ and $K$, one being time-like and the other one being space-
like, that is $J^{2}=-K^{2}<0$ and $J\cdot K=0$. In fact, one can realize that
the 21 relations obtained in Ref. Minogin-2011  reduce to the three
constraining relations on the vector and the axial density currents associated
with the spinor field (see Eqs. (54) and (55) later, in the next section).
Actually, there exist a one-parameter family of spinors having $S=0$ and two
currents (39) defining the same timelike 2-plane.

##  5 Fierz identities and electron local observables constraints

In Ref. Minogin-2011  a set of $21$ algebraic equations relating the $16$
electron local observables (there denoted as $j_{0}$, $\vec{j}$, $f_{0}$,
$\vec{f}$, $b$, $g$, $\vec{c}$ and $\vec{d}$) is given. In this section, this
set of equations is compared with the ones obtained in the previous section.
To make the comparison clearer, we first report the correspondence between our
notation and the one used in Minogin-2011 :

$\displaystyle\alpha$ $\displaystyle\to$ $\displaystyle g\ ,$ (48)
$\displaystyle\beta$ $\displaystyle\to$ $\displaystyle b\ ,$ (49)
$\displaystyle\vec{e}$ $\displaystyle\to$ $\displaystyle-\vec{d}\ ,$ (50)
$\displaystyle\vec{h}$ $\displaystyle\to$ $\displaystyle-\vec{c}\ ,$ (51)
$\displaystyle(j^{0},\vec{j})$ $\displaystyle\to$
$\displaystyle(j^{0},\vec{j})\ ,$ (52) $\displaystyle(k^{0},\vec{k})$
$\displaystyle\to$ $\displaystyle(f^{0},\vec{f})\ .$ (53)

Next, the set of 21 relations established in Sec. 4 is rewritten using this
correspondence in notation and conveniently ordered as:

  1. 1.

Currents constraining relations (3 equations):

$j_{0}^{2}-\vec{j}^{\,2}=b^{2}+g^{2}=-f_{0}^{2}+\vec{f}^{\,2},$ (54)
$j_{0}f_{0}=\vec{j}\cdot\vec{f},$ (55)

which correspond to Eqs. (38) since the scalar products are referred to an
inertial frame, as well as $j_{0}=-j^{0}$ and $f_{0}=-f^{0}$, according with
the chosen Minkowski metric signature. These three equations correspond to Eq.
(31) in Ref. Minogin-2011  (here underlined).

  2. 2.

The spin algebraic relations derived from $S$ (2 equations):

$\displaystyle bg$ $\displaystyle=$ $\displaystyle\vec{c}\cdot\vec{d}\,$ (56)
$\displaystyle b^{2}-g^{2}$ $\displaystyle=$
$\displaystyle\vec{c}^{\,2}-\vec{d}^{\,2},$ (57)

which are Eqs. (11) and (14). The first equation, Eq. (56), corresponds to Eq.
(26.2b), and the second one, Eq. (57), results by subtracting (27.3) from
(27.2) in Ref. Minogin-2011 .

  3. 3.

The scalar relations among the 16 bilinear forms (4 equations)

$\displaystyle gj^{0}$ $\displaystyle=$ $\displaystyle-\vec{f}\cdot\vec{d}\,$
(58) $\displaystyle gf^{0}$ $\displaystyle=$
$\displaystyle-\vec{j}\cdot\vec{d}\,$ (59) $\displaystyle bj^{0}$
$\displaystyle=$ $\displaystyle-\vec{f}\cdot\vec{c}\,$ (60) $\displaystyle
bf^{0}$ $\displaystyle=$ $\displaystyle-\vec{j}\cdot\vec{c}\,$ (61)

that is, Eqs. (40), (41), (42) and (43), which, in increasing numbered order,
correspond to Eqs. (26.3b), (26.1b), (26.3a) (26.1a) in Minogin-2011 ,
respectively.

  4. 4.

The 3-vectorial Euclidian relations among the 16 bilinear forms (12 equations)

$\displaystyle g\vec{j}$ $\displaystyle=$
$\displaystyle-f^{0}\vec{d}-\vec{f}\times\vec{c}\ ,$ (62) $\displaystyle
g\vec{f}$ $\displaystyle=$ $\displaystyle-j^{0}\vec{d}-\vec{j}\times\vec{c}\
,$ (63) $\displaystyle b\vec{j}$ $\displaystyle=$
$\displaystyle-f^{0}\vec{c}+\vec{f}\times\vec{d}\ ,$ (64) $\displaystyle
b\vec{f}$ $\displaystyle=$ $\displaystyle-j^{0}\vec{c}+\vec{j}\times\vec{d}\
,$ (65)

that is, Eqs. (44), (45), (46) and (47), which, in increasing numbered order,
correspond to Eqs. (28.6), (28.2), (28.7), (28.3) in Minogin-2011 ,
respectively.

Therefore, we have shown that the set of 21 equations given in Minogin-2011
is equivalent to the specification of the algebraic structure of $S$ and $*S$.
Furthermore, up to linear combinations, this set transforms into Eqs. (36),
(37) and (38) in Ref. Minogin-2011 , which are, respectively, 6 escalar type
equations, 9 vector type equations and 6 tensor type one component relations.

##  6 The role of the spin 2-form in the spinor classification

The algebraic classification of a four component space-time spinor (Lounesto
classification) Lounesto  is made by means of its bilinear covariants, given
by (2), taking into account the Fierz identities. This classification splits
in six algebraic types of spinors, which are characterized according to the
possible nullities of the algebraic invariants $\alpha$ and $\beta$, the spin
2-form $S$, and the current $K$ (the current $J$ is taken nonzero) Lounesto :

  1. (i)

$\alpha\neq 0,\beta\neq 0$.

  2. (ii)

$\alpha=0,\beta\neq 0$.

  3. (iii)

$\alpha\neq 0,\beta=0$.

  4. (iv)

$\alpha=\beta=0,S\neq 0,K\neq 0$.

  5. (v)

$\alpha=\beta=0,S\neq 0,K=0$.

  6. (vi)

$\alpha=\beta=0,S=0,K\neq 0$.

The Lounesto classification of a spinor points out the role played by the spin
2-form $S$ in distinguishing the general class to which a fermionic field
belongs. From an algebraic point of view, the generic class $S\neq 0$ splits
into two subclasses with different algebraic character (regular or singular)
of $S$.

The regular class splits in the types (i), (ii) and (iii) of the Lounesto
classification, and corresponds to Dirac fermions fields while the singular
class splits in the type (iv) of flag dipole spinors and the type (v) of flag
pole-spinors. Concretely, types (iv) and (v) are identified with cases (b1)
and (b2) obtained as a limit situation of the regular case in Subsection 4.2,
respectively.888This two types have been further analyzed in Cavalcanti-2014 .
Moreover, type (iv) has also been recently studied in Refs. CoVillalobos-
HdaSilva-daRocha-2015 ; CoVillalobosBAB-2020  where the possibility $J=0$ is
also contemplated, and type (v) contains the Majorana fermions and the Elko
fermions, both being eigenspinors of the charge conjugation operator (see
Refs. Ahluwalia-2019 ; Ah-Gru-2005-PRD ; Ah-Gru-2005 ; Ro-Ro-2006 ; daRocha-
HdaSilva-2007 ; HdaSilva-daRocha-2013 ; HdaSilva-CoVillalobos-BRS-2016 ).

The non generic class $S=0$ is the Lounesto type (vi) and contains the Weyl
(or massless) neutrino field.

The null eigendirections of $S$ and $*S$ does not change under a rotation of
dualitity,

$\begin{array}[]{rl}\tilde{S}&=(\cos\theta)S+(\sin\theta)*S,\\\\[5.69046pt]
*\tilde{S}&=-(\sin\theta)S+(\cos\theta)*S,\end{array}$ (66)

but $\alpha$ and $\beta$ transform according with:

$\tilde{\alpha}=\alpha\cos\theta-\beta\sin\theta,\quad\tilde{\beta}=\alpha\sin\theta+\beta\cos\theta,$
(67)

and keeping that $\tilde{\alpha}^{2}+\tilde{\beta}^{2}=\alpha^{2}+\beta^{2}$.
Consequently, the current family $\\{J_{\xi},K_{\xi}\\}_{\xi\in\mathbb{R}}$
given by (36) and (37) is invariant under the transformation (66). But, the
quadratic scalars $A$ and $B$, given by (11), transform as a rotation of angle
$2\theta$,

$\tilde{A}=A\cos 2\theta-B\sin 2\theta,\quad\tilde{B}=A\sin 2\theta+B\cos
2\theta\ ,$ (68)

or equivalently,

$\begin{array}[]{rcl}\tilde{\alpha}^{2}-\tilde{\beta}^{2}&=&(\alpha^{2}-\beta^{2})\cos
2\theta-2\alpha\beta\sin 2\theta,\\\\[5.69046pt]
2\tilde{\alpha}\tilde{\beta}&=&(\alpha^{2}-\beta^{2})\sin
2\theta+2\alpha\beta\cos 2\theta.\end{array}$ (69)

For this reason, one has that a duality rotation does not change the regular
or singular character of the 2-form $S$ but it can move the spinor from one
spinor type to another one inside the same subclass (regular or singular) in
the spinor classification.

##  7 Conclusions

In particle physics, for example in the $\beta$-decay theory or similar
situations in electroweak interactions, Fierz identities involve algebraic
constraints between interacting fermionic fields. To deepen in the essential
information that such identities encode, we have considered the case of a
single fermionic field. Then, expressing the eigendirections of a Lorentzian
2-form with respect to an inertial observer (Sec. 3), the Fierz identities are
interpreted in terms of the eigenstructure of the spin density tensor $S$,
both in the regular and the singular cases (Sec. 4). This study displays how
large Fierz identities appear closely related to the space-time algebra.

In Ref. Minogin-2011  by Minogin, a set of 21 identities that the bilinear
electron observables satisfy were obtained and written in a suitable
3-dimensional Euclidean notation (see Sec. 5, which summarizes Sections 4 and
5 of Ref. Minogin-2011 ). In this paper, a geometric interpretation of an
equivalent set of identities based on the eigenvector relations fulfilled by
$S$ and $*S$ has been performed. The differences between both sets have been
analyzed. In Section 6, the relation of the Fierz identities and the duality
rotation with the algebraic types of the spinor classification has been
established.

The algebraic interpretation of the essential Fierz identities between the
bilinear covariants constructed with two spinors in terms of their spin
2-forms requires further investigation. The existence or not of any connection
between the eigenstructures of a pair of interacting spin 2-forms should be
the starting point to extend forward the present research: (i) the
interpretation of the Fierz identities for a pair of fermion fields, and (ii)
the algebraic classification of the energy tensor of a sole fermionic field
Inglis-Jarvis-2016 , which involves the bilinear covariants constructed from
the field and its first derivatives.

Originally, the Fierz identities come from the restrictions that a change of
basis in the space-time induces on the basis of the Minkowski exterior algebra
(specified by four vectors, six 2-planes and four 3-planes). This geometric
vision of the whole Fierz identities is related with the causal classification
of the the space-time frames 199 . In fact, this classification results from
the analysis of the whole set of constraints between the causal characters of
the 14 geometric elements of a space-time frame: its 4 directions, its 6 two-
planes and its 4 three-planes. This consideration will help to explore whether
there exists a deeper connection between the Lorentzian structure of the
space-time geometry and the fundamental interactions between elementary
particles.

###### Acknowledgements.

This work has been supported by the Spanish Ministerio de Ciencia, Innovación
y Universidades, Projects PID2019-109753GB-C21 and PID2019-109753GB-C22.

## References

  * (1) J. J. Sakurai, Advanced Quantum Mechanics,Addison-Wesley Pub. Comp., New York, 1967, page 105.
  * (2) F. A. Kaempffer, Spinor electrodynamics as a dynamics of currents, Phys. Rev. D 23 (1981) 918–921. https://doi.org/10.1103/PhysRevD.23.918
  * (3) Y. Takahashi, Reconstruction of a spinor via Fierz identities, Phys. Rev. D 26 (1982) 2169–2171. https://doi.org/10.1103/PhysRevD.26.2169
  * (4) Y. Takahashi, A Spinor Reconstruction Theorem, Prog. Theor. Phys. 69 (1983) 369-–371. https://doi.org/10.1143/PTP.69.369
  * (5) Y. Takahashi, The Fierz identities—A passage between spinors and tensors, J. Math. Phys. 24 7 (1983) 1783–1790. https://doi.org/10.1063/1.525896
  * (6) J. P. Crawford, On the algebra of Dirac bispinor densities: factorization and inversion theorems, J. Math. Phys. 26 7 (1985) 1439–1441. https://doi.org/10.1063/1.526906
  * (7) P. R. Holland, Relativistic algebraic spinors and quantum motions in phase space, Found. Phys. 16, (1986) 701–719. https://doi.org/10.1007/BF00735377
  * (8) J. P. Crawford, Bispinor geometry for even-dimensional space-time, J. Math. Phys. 31 8 (1990) 1991–1997. https://doi.org/10.1063/1.528648
  * (9) O. Laporte, G. H. Uhlenbeck, Application of spinor analysis to the Maxwell and Dirac equations, Phys. Rev. 37 11 (1931) 1380–1397. https://doi.org/10.1103/PhysRev.37.1380
  * (10) G. H. Uhlenbeck, O. Laporte, New Covariant Relations Following from the Dirac Equations, Phys. Rev. 37 11 (1931) 1552–1554. https://doi.org/10.1103/PhysRev.37.1552.2
  * (11) L. de Broglie, L’Electron Magnétic. Théorie de Dirac, Hermann et Cie. Editors, Paris, 1934.
  * (12) W. Pauli, Contributions mathématiques à la théorie des matrices de Dirac, Annales Inst. H. Poincaré 6 2 (1936) 109–136. `http://www.numdam.org/item/AIHP_1936__6_2_109_0/`
  * (13) W. Kofink,Über das magnetische und elektrische Moment des Elektrons nach der Diracschen Theorie, Annalen der Physik 442 (1937) 91–98. https://doi.org/10.1002/andp.19374220106
  * (14) T. Takabayasi, Relativistic Hydrodynamics of the Dirac Matter. Part I. General Theory, Theor. Phys. Suppl. 4 (1957) 1–80. https://doi.org/10.1143/PTPS.4.2
  * (15) C. C. Nishi, Simple derivation of general Fierz-type identities, Am. J. Phys. 73 (2005) 1160–1163. https://doi.org/10.1119/1.2074087
  * (16) V. G. Minogin, Geometry of the electron local observables, Annals of Physics 326 (2011) 2773–2792. https://doi.org/10.1016/j.aop.2011.06.002
  * (17) V. G. Minogin, Natural geometric representation for electron local observables, Annals of Physics 342 (2014) 1–10. https://doi.org/10.1016/j.aop.2013.12.006
  * (18) V. A. Zhelnorovich, Theory of Spinors and Its Application in Physics and Mechanics, Springer, Switzerland, 2019.
  * (19) V. G. Minogin, Identities for the electron forms 2 and their 3D representation, Annals of Physics 327 (2012) 823–840. https://doi.org/10.1016/j.aop.2011.12.001
  * (20) S. Inglis, P. Jarvis, Maxwell–Dirac stress–energy tensor in terms of Fierz bilinear currents, Annals of Physics 366 (2016) 57–75. https://doi.org/10.1016/j.aop.2016.01.007
  * (21) S. Inglis, P. Jarvis, Fierz bilinear formulation of the Maxwell–Dirac equations and symmetry reductions, Annals of Physics 348 (2014) 176–222. https://doi.org/10.1016/j.aop.2014.05.017
  * (22) E. Fermi, Zeitschrift für Physik, 88 (1934) 161. English translation by F.L. Wilson, Fermi’s theory of beta decay, Am. J. Phys. 36 (1968) 1150–1160. https://doi.org/10.1119/1.197438
  * (23) M. Fierz, Zur fermischen theorie des $\beta$-zerfalls, Zeitschrift für Physik, 104 (1937) 553–565. https://doi.org/10.1007/BF01330070
  * (24) T. Ortín, Gravity and Strings, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  * (25) J. F. Nieves, P. B. Pal, Generalized Fierz identities, Am. J. Phys. 72 (2004) 1100–1108. https://doi.org/10.1119/1.1757445
  * (26) S. Bilenky, Introduction to the Physic of Massive and Mixed Neutrino, Springer-Verlag, Berlin, Heidelberg, 2010.
  * (27) A. Lesov, The Weak Force: From Fermi to Feynman, Senior Thesis, University of South Carolina, 2009, arXiv:0911.0058[phyisc.hist-ph].
  * (28) G. Y. Rainich, Electrodynamics in the general relativity theory, Trams. Am. Math. Soc. 27 (1925) 106–136. https://doi.org/10.2307/1989168
  * (29) P. A. M. Dirac, The Quantum Theory of the Electron, Proc. Roy. Soc. London A, 117 (1928) 610–624. https://doi.org/10.1098/rspa.1928.0023
  * (30) G. H. Derrick, On a completely symmetric choice of space–time coordinates, J. Math. Phys. 22 12 (1981) 2896–2902. https://doi.org/10.1063/1.525170
  * (31) B. Coll, J. A. Morales, Symmetric frames on Lorentzian spaces, J. Math. Phys. 32, (1991) 2450–2455. https://doi.org/10.1063/1.529173
  * (32) B. Coll, J. A. Morales, 199 Causal classes of space-time frames, Int. Jour. Theor. Phys. 31 (1992) 1045–1062. https://doi.org/10.1007/BF00675095
  * (33) B. Coll, J. J. Ferrando, J. A. Morales–Lladosa, Newtonian and relativistic emission coordinates. Phys. Rev. D 80 (2009) 064038. https://doi.org/10.1103/PhysRevD.80.064038
  * (34) T. C. Chapman, D. J. Leiter, On the generally covariant Dirac equation, Am. J. Phys. 44 9 (1976) 858–862. https://doi.org/10.1119/1.10256
  * (35) P. Lounesto, Clifford Algebras and Spinors, second ed., London Mathematical Society Lecture Note Series 286, Cambridge University Press, Cambridge, 2001.
  * (36) D. M. Fradkin, Covariant electromagnetic projection operators and a covariant description of charged particle guiding centre motion, J. Phys A: Math. Gen. 11 6 (1978) 1069–1086. https://doi.org/10.1088/0305-4470/11/6/010
  * (37) B. Coll, J. J. Ferrando, On the permanence of the null character of Maxwell fields, Gen. Relativ. Gravit. 20 1 (1988) 51–64. https://doi.org/10.1007/BF00759255
  * (38) R. T. Cavalcanti, Classification of singular spinor fields and other mass dimension one fermions. Int. J. Mod. Phys. D 23 (2014) 1444002. https://doi.org/10.1142/S0218271814440027
  * (39) C. H. Coronado Villalobos, J. M. Hoff da Silva, Rold$\tilde{\rm a}$o da Rocha, Questing mass dimension 1 spinor fields. Eur. Phys. J. C 75 (2015) 266. https://doi.org/10.1140/epjc/s10052-015-3498-2
  * (40) C. H. Coronado Villalobos, R. G. Bueno Rogerio, A. R. Aguirre, B. Beghetto, On the generalized spinor classification: beyond the Lounesto’s classification. Eur. Phys. J. C 80 (2020) 228. https://doi.org/10.1140/epjc/s10052-020-7801-5
  * (41) D.V. Ahluwalia, Mass dimension one fermions, Cambridge monograph on mathematical physics, Cambridge University Press, Cambridge, 2019. e-print arXiv:2007.15098v2 [hep-th].
  * (42) D.V. Ahluwalia-Khalilova, D. Grumiller, Dark matter: A spin one-half fermion field with mass dimension one?. Phys. Rev. 72 (2005) 067701. https://doi.org/10.1103/PhysRevD.72.067701
  * (43) D.V. Ahluwalia-Khalilova. D. Grumiller, Spin-half fermions with mass dimension one: theory, phenomenology, and dark matter. J. Cosmol. Astropart. Phys. 2005 (2005) 012. https://doi.org/10.1088/1475-7516/2005/07/012
  * (44) R. da Rocha, W. A. Rodrigues, JR, Where are ELKO spinor fields in Lounesto spinor field classification?, Modern Physics Letters A 21 01 (2006) 65–74. https://doi.org/10.1142/S0217732306018482
  * (45) R. da Rocha, J. M. Hoff da Silva, From Dirac spinor fields to eigenspinoren des ladungskonjugationsoperators. J. Math. Phys. 48 (2007) 123517. https://doi.org/10.1063/1.2825840
  * (46) J. M. Hoff da Silva, R. da Rocha, Unfolding physics from the algebraic classification of spinor fields, Physics Letters B 718 (2013) 1519–1523. https://doi.org/10.1016/j.physletb.2012.12.026
  * (47) J. M. Hoff da Silva, C. H. Coronado Villalobos, R. J. Bueno Rogerio, E. Scatena, On the bilinear covariants associated to mass dimension one spinors. Eur. Phys. J. C 76 (2016) 563. https://doi.org/10.1140/epjc/s10052-016-4408-y


Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

1st Workshop on Complex Data Challenges in Earth Observation (CDCEO) 2021,
November 01, 2021

[orcid=0000-0002-6560-6316<EMAIL_ADDRESS>url=https://jleinonen.github.io/, ]

# Spatiotemporal Weather Data Predictions with Shortcut Recurrent-
Convolutional Networks: A Solution for the Weather4cast challenge

Jussi Leinonen  Federal Office of Meteorology and Climatology MeteoSwiss,
Locarno-Monti, Switzerland

(2021)

###### Abstract

This paper presents the neural network model that was used by the author in
the Weather4cast 2021 Challenge Stage 1, where the objective was to predict
the time evolution of satellite-based weather data images. The network is
based on an encoder-forecaster architecture making use of gated recurrent
units (GRU), residual blocks and a contracting/expanding architecture with
shortcuts similar to U-Net. A GRU variant utilizing residual blocks in place
of convolutions is also introduced. Example predictions and evaluation metrics
for the model are presented. These demonstrate that the model can retain sharp
features of the input for the first predictions, while the later predictions
become more blurred to reflect the increasing uncertainty.

###### keywords:

weather satellite data neural networks gated recurrent units

##  1 Introduction

Weather forecasting requires the prediction of spatiotemporal changes in
atmospheric fields. Most commonly, this prediction is carried out using
numerical weather prediction (NWP, e.g. [1]), which models the evolution of
the atmosphere by computationally evaluating approximate physical laws, such
as forward-integrating the partial differential equations of fluid dynamics.

Although the NWP approach to forecasting is firmly grounded in first
principles, NWP models are computationally expensive: state-of-the-art models
are run on powerful supercomputers and yet take a significant amount of time
to complete a model run, which limits their usefulness in short-term
predictions. These issues have motivated considerable research on using
artificial intelligence (AI) and machine learning (ML) methods for weather
prediction, particularly after the prominent progress in convolutional neural
networks (ConvNets) in the last decade. ConvNets can learn to recognize
complex spatial structures, which make them well-suited for modeling
atmospheric fields. They have been used as components of weather models, for
example, to speed up computationally expensive modules of NWP codes. Recurrent
neural networks have also been used to model the time evolution of atmospheric
fields. Numerous applications and potential uses of ML/AI in weather in
climate sciences have been discussed by, e.g., [2, 3, 4, 5]. Thus, there is
considerable interest in further developing ML model architectures for these
applications.

To support the development of ML/AI for weather prediction, the Weather4cast
2021 Challenge had the objective to predict the time evolution of atmospheric
satellite data products using machine-learning methods. This paper presents
the ML solution submitted by the author.

##  2 Challenge Data and Objectives

The Weather4cast 2021 Challenge Stage 1 presented the competitors with a data
problem of predicting the time evolution of satellite-based measurements of
four different atmospheric fields: temperature, representing either the
surface temperature (where no cloud is present) or the cloud top temperature
(in cloudy areas); convective rainfall rate (crr_intensity); probability of
tropopause folding (asii_turb_trop_prob); and cloud mask (cma). These are data
products created by the Nowcasting Satellite Application Facility (NWC SAF)
processing from satellite data from the EUMETSAT-operated MeteoSat satellites.
The products were given for six regions (numbered R1–R6) in Europe, Northern
Africa and the Middle East, each $256\times 256$ pixels of approximately $4\
\mathrm{km}\times 4\ \mathrm{km}$ each. A few other variables also were
available such as cloud type, as well as static data with the elevation of the
terrain, the latitude and the longitude.

The goal of the challenge was to predict, in each area, the next $32$ frames
of a sequence given the previous $4$ frames, with frames separated by $15\
\mathrm{min}$. Training and validation datasets, with complete $4+32$-frame
samples, as well as a test dataset with only the $4$-frame inputs, were
provided for regions R1–R3, constituting the “Core” competition. Meanwhile,
R4–R6 only had test data available, meaning that they had to be evaluated
using models trained on R1–R3; this was called the “Transfer Learning”
competition. Furthermore, all regions had a set of “held-out” data which were
made available only during the final week of the competition; the final
results were based on the performance with these data.

The performance of the models was evaluated using the mean-square error (MSE)
for each variable. However, some adjustments were made to the MSE to account
for the particular needs of each variable, except for crr_intensity. First,
the loss for temperature was modified to account for varying amounts of
missing data in each region. Second, asii_turb_trop_prob is a probabilistic
variable and the output of the model was passed through a truncated and
normalized logit transform before the evaluation of the MSE. Third, although
cma is technically evaluated using the MSE, the variable in the output data
file is required to be quantized such that the value is either $0$ or $1$;
therefore, model output values $<0.5$ are rounded to $0$ and outputs $\geq
0.5$ are rounded to $1$ before evaluation. The details of the metrics can be
found in [6].

##  3 Solution

###  3.1 Models

The model presented here is a neural network combining recurrent-convolutional
layers and shortcut connections in an encoder-forecaster architecture. The
architecture is presented in Fig. 1. It is based on that developed in [7] for
precipitation nowcasting and adopted by [8], as well as similar to that of
[9], with some differences that are described below.

Figure 1: Illustration of the network architecture.

The encoder section consists of four recurrent downsampling stages. Each stage
first passes the sequence through a residual block [10], with each frame
processed using the same convolutional filters. A strided convolution in the
residual block is used to downsample the input by a factor of 2. Then, the
sequence is processed by a gated recurrent unit (GRU) layer [11]; a tensor of
zeros is passed as the initial state of the GRU. The number of channels in the
convolutions is increased with increasing depth in the encoder.

The forecaster section is approximately a mirror image of the encoder section.
Each stage consists of a GRU layer which is followed by bilinear upsampling
and a residual block. A shortcut similar to U-Net [12] is utilized: The final
state of each GRU in the encoder is passed through a convolution and then used
as the initial state of the GRU of corresponding depth in the forecaster. This
allows the high-resolution features of the recent frames to be passed through,
preventing the first predictions from being blurry. A final projection and a
sigmoid activation produce the output as a single variable constrained between
$0$ and $1$.

The main difference of the architecture presented here to that of [7] is that
the use of Trajectory GRU (TrajGRU) is rejected as TrajGRU was found to cause
training instability. Two variants are considered instead. The first utilizes
the Convolutional GRU (ConvGRU) layer adopted by e.g. [9, 13, 14]. In the
second variant, the convolution in the ConvGRU is replaced by a residual block
modified to be used for this purpose. The use of the residual block increases
the depth of the operations in the GRU and is thus expected to allow it to
better process nonlinear transformations and also to increase the distance at
which pixels can influence each other at each step of the ConvGRU. The latter
effect may recover some of the advantages of TrajGRU over ConvGRU that [7]
found. The author is unaware of previously published instances of a residual
layer being used in place of the convolution in GRU. In this paper, this
variant is called “ResGRU”, although the same abbreviation was used for a
different combination of GRUs and residual connections in [15].

The models were implemented using TensorFlow/Keras [16] version 2.4. The
source code and the pre-trained models can be found through the links in
Appendix A.

###  3.2 Training

Since the scores for the target variables were evaluated independently from
each other, a separate instance of the model was trained for each target
variable, but using all variables as inputs for each model. The models were
trained on the training dataset of R1–R3 such that every available gapless
sequence of $36$ frames was used for training, resulting in $72192$ different
sequences (albeit with considerable overlap). The training was performed with
combined data from all regions R1–R3 in order to increase the training dataset
size and improve the ability of the model to generalize; specializing the
model to single regions was not attempted. The static data (latitude,
longitude and elevation) were also used for training. Data augmentation by
random rotation in $90\degree$ increments as well as random top-down and left-
right mirroring was used to further increase the effective number of training
samples. The model for asii_turb_trop_prob was trained using a custom logit
loss corresponding to the metric specified in [6], while the other variables
were trained using the standard MSE loss. The Adam optimizer [17] was used to
train the models with a batch size of $32$.

The progress of the training was evaluated using the provided validation
dataset for R1–R3. After each training epoch, the evaluation metric was
computed on the validation set and then:

  1. 1.

If the metric improved upon the best evaluation result, the model weights were
saved.

  2. 2.

If the metric had not improved in $3$ epochs, the learning rate was reduced by
a factor of $5$.

  3. 3.

If the metric had not improved in $10$ epochs, the training was stopped early.

In practice, condition 3 was never activated as the model continued to achieve
marginal gains on the validation data at least every few epochs until the
maximum training time of $12\ \mathrm{h}$ or $24\ \mathrm{h}$ (depending on
the training run) was reached. This suggests that the model did not suffer
significantly from overfitting, which typically causes the validation loss to
start increasing even as the training loss keeps decreasing. This is perhaps
due to the relatively modest number of weights in the models by standards of
modern ConvNets, approximately $12.1$ million weights in the ConvGRU variant
and $18.6$ million in the ResGRU variant. The loss over the validation set was
used as the metric for each variable except cma for which a rounded MSE that
takes the ${0,1}$ quantization into account was used.

A parallel setup of eight Nvidia Tesla V100 GPUs was used to train the models.
Training for one epoch took approximately 20 minutes with this hardware. The
eight parallel GPUs only provided a speedup of a factor of approximately $3$
compared to training on a single GPU, suggesting that single-GPU training of
the models should be feasible, although the batch size would likely have to be
reduced as the models require rather large amounts of GPU memory.

##  4 Results

Both the ConvGRU and ResGRU variants of the model were trained for each target
variable. The evaluation results for the validation dataset are shown in Table
1. Comparisons to TrajGRU were found impractical as the models using TrajGRU
would not converge properly due to the training instability mentioned in Sect.
3.1.

Table 1: Evaluation metrics for the validation dataset. | temperature | crr_intensity | asii_turb_trop_prob | cma
---|---|---|---|---
ConvGRU | $0.004564$ | $\mathbf{0.0001259}$ | $0.002250$ | $0.1393$
ResGRU | $\mathbf{0.004356}$ | $0.0001278$ | $\mathbf{0.002161}$ | $\mathbf{0.1376}$

Based on the evaluation results, three submissions were made to the final
leaderboards of Weather4cast Stage 1: one using the ConvGRU variant for all
variables (codenamed V4c), another using ResGRU (V4rc), and a third using the
best model for each variable based on the validation metrics (V4pc). It was
indeed this last model that produced the best results also on the leaderboards
for both the Core and Transfer Learning competitions, as shown in Table 2.

Table 2: Evaluation metrics for the held-out test dataset, as computed by the Weather4cast website (https://www.iarai.ac.at/weather4cast/). | Core | Transfer learning
---|---|---
ConvGRU | $0.5051$ | $0.4658$
ResGRU | $0.5014$ | $0.4626$
Best combination | $\mathbf{0.4987}$ | $\mathbf{0.4607}$

Figures 2–5 show examples of the predictions using the validation dataset.
These are all shown for the same scene except for Fig. 3, where a different
scene was chosen because the one used for the others did not contain
precipitation. It is clear that the predictions start relatively sharp and get
blurrier over time, reflecting the increasing uncertainty. The blurriness is
likely exacerbated by the use of the MSE metric, specified in the data
challenge, which is prone to regression to the mean. Especially in Fig. 4, one
can also see that the model can predict the motion of features in the images.

Figure 2: An example of predictions for the temperature variable. The frames
on the left correspond to past temperature, while the frames on the right show
the real future temperature (top row) and the predicted temperature (bottom
row). The $T$ coordinate refers to the index of the frame in the sequence,
with $T=0$ represents the last input data point and $T=1$ the first
prediction. The model output normalized to the range $(0,1)$ is shown. Figure
3: As Fig. 2, but for crr_intensity. A different case is shown as the case of
Fig. 2 does not contain precipitation. Figure 4: As Fig. 2, but for
asii_turb_trop_prob. Figure 5: As Fig. 2, but for cma. The white contours in
the predictions indicate $0.5$, the threshold of the cloud mask in the output.

##  5 Conclusions

The model presented here reached the top of the final leaderboards in both the
Core and the Transfer Learning categories of the Weather4cast 2021 Challenge
Stage 1. It is a versatile solution to the problem of predicting the evolution
of atmospheric fields, producing sharp predictions for the near term and
increasing the uncertainty for longer lead times. The architecture can be
easily adapted to other tasks such as probabilistic predictions or outputs
that are different from the inputs. Further research is needed to handle, for
instance, different spatial and temporal resolutions of inputs and data
available for future time steps.

###### Acknowledgements.

This project benefited from parallel development in the fellowship “Seamless
Artificially Intelligent Thunderstorm Nowcasts” from the European Organisation
for the Exploitation of Meteorological Satellites (EUMETSAT). The hosting
institution of this fellowship is MeteoSwiss in Switzerland. The author thanks
U. Hamann and A. Rigazzi for discussions regarding the model and training.

## References

  * Bauer et al. [2015] P. Bauer, A. Thorpe, G. Brunet,  The quiet revolution of numerical weather prediction,  Nature 525 (2015) 47–55. doi:doi.org/10.1038/nature14956.
  * McGovern et al. [2017] A. McGovern, K. L. Elmore, D. J. Gagne, II, S. E. Haupt, C. D. Karstens, R. Lagerquist, T. Smith, J. K. Williams,  Using artificial intelligence to improve real-time decision-making for high-impact weather,  Bull. Amer. Meteor. Soc. 98 (2017) 2073–2090. doi:10.1175/BAMS-D-16-0123.1.
  * Reichstein et al. [2019] M. Reichstein, G. Camps-Valls, B. Stevens, M. Jung, J. Denzler, N. Carvalhais, Prabhat,  Deep learning and process understanding for data-driven earth system science,  Nature 566 (2019) 195–204. doi:10.1038/s41586-019-0912-1.
  * Huntingford et al. [2019] C. Huntingford, E. S. Jeffers, M. B. Bonsall, H. M. Christensen, T. Lees, H. Yang,  Machine learning and artificial intelligence to aid climate change research and preparedness,  Environmental Research Letters 14 (2019) 124007. doi:10.1088/1748-9326/ab4e55.
  * Haupt et al. [2021] S. E. Haupt, W. Chapman, S. V. Adams, C. Kirkwood, J. S. Hosking, N. H. Robinson, S. Lerch, A. C. Subramanian,  Towards implementing artificial intelligence post-processing in weather and climate: proposed actions from the oxford 2019 workshop,  Philos. Trans. R. Soc. London, Ser. A 379 (2021) 20200091. doi:10.1098/rsta.2020.0091.
  * IARAI [2021] IARAI, Weather4cast 2021: Competition metrics, 2021. URL: https://www.iarai.ac.at/weather4cast/wp-content/uploads/sites/3/2021/04/w4c.pdf.
  * Shi et al. [2017] X. Shi, Z. Gao, L. Lausen, H. Wang, D.-Y. Yeung, W.-k. Wong, W.-c. WOO,  Deep learning for precipitation nowcasting: A benchmark and a new model,  in: I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, R. Garnett (Eds.), Advances in Neural Information Processing Systems, volume 30, Curran Associates, Inc., 2017. URL: https://proceedings.neurips.cc/paper/2017/file/a6db4ed04f1621a119799fd3d7545d3d-Paper.pdf.
  * Franch et al. [2020] G. Franch, D. Nerini, M. Pendesini, L. Coviello, G. Jurman, C. Furlanello,  Precipitation nowcasting with orographic enhanced stacked generalization: Improving deep learning predictions on extreme events,  Atmosphere 11 (2020). doi:10.3390/atmos11030267.
  * Ravuri et al. [2021] S. Ravuri, K. Lenc, M. Willson, D. Kangin, R. Lam, P. Mirowski, M. Fitzsimons, M. Athanassiadou, S. Kashem, S. Madge, R. Prudden, A. Mandhane, A. Clark, A. Brock, K. Simonyan, R. Hadsell, N. Robinson, E. Clancy, A. Arribas, S. Mohamed, Skillful precipitation nowcasting using deep generative models of radar, 2021. arXiv:2104.00954.
  * He et al. [2016] K. He, X. Zhang, S. Ren, J. Sun,  Deep residual learning for image recognition,  in: The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016. doi:10.1109/CVPR.2016.90.
  * Cho et al. [2014] K. Cho, B. van Merrienboer, D. Bahdanau, Y. Bengio,  On the properties of neural machine translation: Encoder-decoder approaches,  in: Proceedings of SSST-8, Eighth Workshop on Syntax, Semantics and Structure in Statistical Translation, 2014, pp. 103–111.
  * Ronneberger et al. [2015] O. Ronneberger, P. Fischer, T. Brox,  U-net: Convolutional networks for biomedical image segmentation,  in: N. Navab, J. Hornegger, W. M. Wells, A. F. Frangi (Eds.), Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015, 2015, pp. 234–241. doi:10.1007/978-3-319-24574-4_28.
  * Tian et al. [2020] L. Tian, X. Li, Y. Ye, P. Xie, Y. Li,  A generative adversarial gated recurrent unit model for precipitation nowcasting 17 (2020) 601–605. doi:10.1109/LGRS.2019.2926776.
  * Leinonen et al. [2020] J. Leinonen, D. Nerini, A. Berne,  Stochastic super-resolution for downscaling time-evolving atmospheric fields with a generative adversarial network,  IEEE Trans. Geosci. Remote Sens. (2020). doi:10.1109/TGRS.2020.3032790.
  * Gao and Wai [2020] W. Gao, R.-J. Wai,  A novel fault identification method for photovoltaic array via convolutional neural network and residual gated recurrent unit,  IEEE Access 8 (2020) 159493–159510. doi:10.1109/ACCESS.2020.3020296.
  * Chollet et al. [2015] F. Chollet, et al., Keras, https://keras.io, 2015.
  * Kingma and Ba [2014] D. P. Kingma, J. Ba,  Adam: A method for stochastic optimization,  in: 3rd International Conference for Learning Representations, San Diego, California, USA, 2014. URL: https://arxiv.org/abs/1412.6980.
  * Leinonen [2021] J. Leinonen, Model weights for a Weather4cast 2021 Challenge Stage 1 solution, 2021. doi:10.5281/zenodo.5101213.

##  Appendix A Online Resources

The source code with instructions to replicate the results presented in this
paper can be found at https://github.com/jleinonen/weather4cast-stage1. The
model weights used in the challenge submissions can be downloaded at [18].