Majorana Spinors Research Articles

The 1932 Majorana Equation: A Forgotten but Surprisingly Modern Particle Theory

The Standard Model is an up-to-date theory that best summarizes current knowledge in particle physics. Although some problems still remain open, it represents the leading model which all physicists refer to. One of the pillars which underpin the Standard Model is represented by the Lorentz invariance of the equations that form its backbone. These equations made it possible to predict the existence of particles and phenomena that experimental physics had not yet been able to detect. The first hint of formulating a fundamental theory of particles can be found in the 1932 Majorana equation, formulated when electrons and protons were the only known particles. Today we know that parts of the hypotheses set by Majorana were not correct, but his equation hid concepts that are found in the Standard Model. In this study, the Majorana equation is revisited and solved for free particles. The time-like, light-like and space-like solutions, represented by infinite-component wave functions, are discussed.

Nilpotent local fermionic symmetry with generalized self-duality in D = n + n dimensions embedding supersymmetric integrable models

We present local nilpotent fermionic symmetry with a vector-spinor as its gauge field in D=2+2 space-time dimensions. Our field-content is (AμI,ψμI,χI), where I=1,2,⋯,g≡dim G is the adjoint-index of a gauge group G, while ψμI and χI are Majorana spinors. The field strengths FμνI of AμI and RμνI of ψμI satisfy the self-duality conditions FμνI=+(1/2)ϵμνρσFρσI and RμνI=+(1/2)ϵμνρσRρσI. The lowest flow of super-KdV equations and super-KP equations are embedded into this system in ways more natural than the conventional supersymmetric self-dual theories in D=2+2. This indicates that our theory even without supersymmetry in D=2+2 can serve as the “Master Theory” of all supersymmetric integrable systems in lower dimensions. Our D=2+2 result is further generalized to D=n+n(n=3,4,⋯) with the field content (AμI,ψμI,χI,Bμ1⋯μn−3), with the (n−3)-rank auxiliary Abelian tensor Bμ1⋯μn−3. The generalized self-dual conditions are ▪ and ▪. We conjecture that our theory in D=n+n is the “Grand-Master Theory” of all supersymmetric integrable systems in lower dimensions.

Hodge duality transformations in tensor-hierarchy formulations

We show in arbitrary space-time dimensions that duality-transformations are possible in general tensor-hierarchy with field-strengths of arbitrary ranks. We first consider the non-Abelian tensor field-strength of the potential , and its compensator field-strength of the potential , where is the adjoint index. We next perform duality transformations into their respective Hodge-dual field-strength of the potential and the field-strength of the potential . As a typical application, we show this to take place in dimensions with an N = 1 supersymmetric set consisting of a Yang–Mills vector-multiplet , an extra vector-multiplet , and a compensating chiral multiplet . The λ I and χ I are Majorana spinors, and is an extra vector with the Proca–Stückelberg scalar CI , while EI is a pseudo-scalar. We perform duality-transformations on and to their Hodge-dual fields and , respectively, ending up with the new set of multiplets and . Another example is the set of multiplets and in . Even though has the maximal 4th rank field-strength, its supersymmetric dual system exists with and after a duality-transformation. Our duality-transformations provide previously-unknown-links among supersymmetric tensor-hierarchy formulations.

Meson spectrum of SU(2) QCD1+1 with quarks in Large representations

We consider SU(2) quantum chromodynamics in 1 + 1 dimensions with a single quark in the spin J representation of the gauge group and study the theory in the large J limit where the gauge coupling g2 → 0 and J → ∞ with λ = g2J2 fixed. We work with a Dirac spinor field for arbitrary J, and with a Majorana spinor for integer J since the integer spin representations of SU(2) are real, and analyze the two cases separately.The theory is reformulated in terms of global colour non-singlet fermion bilocal operators which satisfy a W∞ × U(2J + 1) algebra. In the large J limit, the dynamics of the bilocal fields is captured by fluctuations along a particular coadjoint orbit of the W∞ algebra. We show that the global colour-singlet sector of the bilocal field fluctuations satisfy the same integral equation for meson wavefunctions that appears in the ’t Hooft model. For Majorana spinors in the integer spin J representation, the Majorana condition projects out half of the meson spectrum, as a result of which the linear spacing of the asymptotic meson spectrum for Majorana fermions is double that of Dirac fermions. The Majorana condition also projects out the zero mass bound state that is present for the Dirac quark at zero quark mass.We also consider the formulation of the model in terms of local charge densities and compute the quark spectral function in the large J limit: we see evidence for the absence of a pole in the quark propagator.

Octonion Internal Space Algebra for the Standard Model

This paper surveys recent progress in our search for an appropriate internal space algebra for the standard model (SM) of particle physics. After a brief review of the existing approaches, we start with the Clifford algebras involving operators of left multiplication by octonions. A central role is played by a distinguished complex structure that implements the splitting of the octonions O=C⊕C3, which reflect the lepton-quark symmetry. Such a complex structure on the 32-dimensional space S of Cℓ10 Majorana spinors is generated by the Cℓ6(⊂Cℓ10) volume form, ω6=γ1⋯γ6, and is left invariant by the Pati–Salam subgroup of Spin(10), GPS=Spin(4)×Spin(6)/Z2. While the Spin(10) invariant volume form ω10=γ1…γ10 of Cℓ10 is known to split S on a complex basis into left and right chiral (semi)spinors, P=12(1−iω6) is interpreted as the projector on the 16-dimensional particle subspace (which annihilates the antiparticles).The standard model gauge group appears as the subgroup of GPS that preserves the sterile neutrino (which is identified with the Fock vacuum). The Z2-graded internal space algebra A is then included in the projected tensor product A⊂PCℓ10P=Cℓ4⊗Cℓ60. The Higgs field appears as the scalar term of a superconnection, an element of the odd part Cℓ41 of the first factor. The fact that the projection of Cℓ10 only involves the even part Cℓ60 of the second factor guarantees that the color symmetry remains unbroken. As an application, we express the ratio mHmW of the Higgs to the W boson masses in terms of the cosine of the theoretical Weinberg angle.

C−P−T fractionalization

Discrete spacetime symmetries of parity P or reflection R, and time-reversal T, act naively as $\mathbb{Z}_2$-involutions in the passive transformation on the spacetime coordinates; but together with a charge conjugation C, the total C-P-R-T symmetries have enriched active transformations on fields in representations of the spacetime-internal symmetry groups of quantum field theories (QFTs). In this work, we derive that these symmetries can be further fractionalized, especially in the presence of the fermion parity $(-1)^{\rm F}$. We elaborate on examples including relativistic Lorentz invariant QFTs (e.g., spin-1/2 Dirac or Majorana spinor fermion theories) and nonrelativistic quantum many-body systems (involving Majorana zero modes), and comment on applications to spin-1 Maxwell electromagnetism (QED) or interacting Yang-Mills (QCD) gauge theories. We discover various C-P-R-T-$(-1)^{\rm F}$ group structures, e.g., Dirac spinor is in a projective representation of $\mathbb{Z}_2^{\rm C}\times \mathbb{Z}_2^{\rm P} \times \mathbb{Z}_2^{\rm T}$ but in an (anti)linear representation of an order-16 nonabelian finite group, as the central product between an order-8 dihedral (generated by C and P) or quaternion group and an order-4 group generated by T with T$^2=(-1)^{\rm F}$. The general theme may be coined as C-P-T or C-R-T fractionalization.

Generalized Mathisson–Papapetrou–Tulczyjew–Dixon equations

We derive two generalizations of Mathisson–Papapetrou–Tulczyjew–Dixon equations from Casalbuoni–Brink–Schwarz type pseudoclassical Lagrangians of Majorana spinors on a Riemann–Cartan spacetime. One has an additional gauge freedom, which makes the equations of motion also be a generalization of Wong’s equations. The other is a spinor model coupled with a Rarita–Schwinger field to preserve the supersymmetry. The coupling to the torsion is modified due to the existence of the Rarita–Schwinger field. In both extensions, the Tulczyjew condition is automatically satisfied.

Dark Range of ω In Brans-Dicke Gravity

The variational field equations of Brans-Dicke scalar-tensor theory of gravitation that is coupled to a mass-varying sterile neutrino field are presented in Riemannian and non-Riemannian setting in the language of exterior differential forms over four-dimensional spacetime. In a gravitational plane wave geometry in Rosen coordinates, with the scalar field set equal to a constant, the field equations admit ghost neutrino solutions. These correspond to Majorana spinor configurations for which the total neutrino stress-energy-momentum tensor vanishes. We show here that in the absence of a sterile neutrino in the gravitational wave geometry, there are scalar field configurations for which the total scalar field stress-energy-momentum tensor vanishes for negative values of the Brans-Dicke parameter —3/2 < ω < 0.

Variational field equations of a Majorana neutrino coupled to Einstein's theory of general relativity

A consistent variational derivation of the Majorana 4-spinor field equations coupled to Einstein's theory of gravitation is given. The equivalence of the first and the second order variational field equations is explicitly demonstrated. The Lagrange multiplier 2-forms we use turn out to be precisely the Belinfante-Rosenfeld 2-forms that are needed to symmetrize the canonical energy-momentum tensor of the Majorana spinor.

Weyl and Majorana Spinors as Pure Goldstone Bosons

General spinors in polar form display a structure that is made up by real scalar degrees of freedom plus six components which can be recognized as Goldstone bosons: in the present paper we show that of all singular spinors, Weyl and Majorana spinors have no real degree of freedom and so that they can be interpreted as pure Goldstone states.

Majorana quanta, string scattering, curved spacetimes and the Riemann Hypothesis

The Riemann Hypothesis states that the Riemann zeta function ζ(z) admits a set of ‘non-trivial’ zeros that are complex numbers supposed to have real part 1/2. Their distribution on the complex plane is thought to be the key to determine the number of prime numbers before a given number. Hilbert and Pólya suggested that the Riemann Hypothesis could be solved through the mathematical tools of physics, finding a suitable Hermitian or unitary operator that describe classical or quantum systems, whose eigenvalues distribute like the zeros of ζ(z). A different approach is that of finding a correspondence between the distribution of the ζ(z) zeros and the poles of the scattering matrix S of a physical system. Our contribution is articulated in two parts: in the first we apply the infinite-components Majorana equation in a Rindler spacetime and compare the results with those obtained with a Dirac particle following the Hilbert-Pólya approach showing that the Majorana solution has a behavior similar to that of massless Dirac particles and finding a relationship between the zeros of zeta end the energy states. Then, we focus on the S-matrix approach describing the bosonic open string scattering for tachyonic states with the Majorana equation. Here we find that, thanks to the relationship between the angular momentum and energy/mass eigenvalues of the Majorana solution, one can explain the still unclear point for which the poles and zeros of the S-matrix of an ideal system that can satisfy the Riemann Hypothesis, exist always in pairs and are related via complex conjugation. As claimed in the literature, if this occurs and the claim is correct, then the Riemann Hypothesis could be in principle satisfied, tracing a route to a proof.

Novel 3D supersymmetric massive Yang-Mills theory

We construct three-dimensional, $\mathcal{N}=1$ off-shell supersymmetric massive Yang-Mills (YM) theory whose YM equation is ``third-way'' consistent. This means that the field equations of this model do not come from variation of a local action without additional fields, yet the gauge-covariant divergence of the YM equation still vanishes on shell. To achieve this, we modify the massive Majorana spinor equation so that its supersymmetry variation gives a modified YM equation whose bosonic part coincides with the third-way consistent pure massive YM model.

Supersymmetry algebras in arbitrary signature and their R-symmetry groups

String theory, specifically type-II superstring theory, can be formulated in any ten-dimensional signature. To facilitate the study of supergravity and superstring theories in this setting, we present a uniform construction of supersymmetry algebras in arbitrary dimension and signature, which generalizes the ideas underlying symplectic Majorana spinors. In our formalism R-symmetry acts on an auxiliary multiplicity space which makes its action manifest. This allows us to provide extensive tables which list the R-symmetry groups of extended supersymmetry algebras for all signatures together with other useful information. Twisted (‘type-*’) supersymmetry algebras in Lorentz signature with non-compact R-symmetry groups are shown to be part of a general pattern resulting from the interplay between complex superbrackets and reality conditions. As an application we show how the relations between type-II string theories in ten and nine dimensions can be extracted from their supersymmetry algebras. We also use our results to determine the special geometry of vector and hypermultiplet scalar manifolds of four-dimensional mathcal{N} = 2 and three-dimensional mathcal{N} = 4 supergravity theories for all signatures.

Scale-invariant supergravity theory in component formulation

We formulate N=1 supergravity theory in four-dimensions with local scale invariance in semi-on-shell component formulation in four dimensions. The algebra we adopt has the generators (Pm,Mmn,Qα,S) consisting of the generators in super-Poincaré algebra: (Pm,Mmn,Qα) with 4+6+4 components, and that of local scale-transformation S with one additional component. Our field content consists of the on-shell multiplet of supergravity (eμm,ψμα), the tensor multiplet (Bμν,χα,φ) and the ‘Weylon multiplet’ (Sμ,ρα,Cμνρ), where Sμ is the Weylon that gauges local scale symmetry, ψμ,χ and ρ are Majorana spinors, while Cμνρ is a tensor auxiliary field. Similar to the non-supersymmetric case that has no minimal coupling between the Sμ-field and spin 1/2 field, the Weylon does not minimally couple to spin 3/2 gravitino field in our locally supersymmetric generalization. By going to the Einstein frame from the original Jordan frame, we show that our system is equivalent to a system with the gauged dilaton-shift symmetry. This implies that our system bypasses persistent dilaton-related problems, avoiding a long-distance force for a massless dilaton, a minimum value of its potential, or a run-away vacuum.

Statistical properties of linear Majorana fermions

AbstractA Majorana fermion is the single fermionic particle that is its own antiparticle. Its dynamics is determined by the Majorana equation, where the spinor field is by definition equal to its charge‐conjugate field. In this paper, we investigated Shannon's entropy of linear Majorana fermions to understand how this quantity is modified due to an external potential of the linear type linear. Subsequently, we turn our attention to the construction of an ensemble of these Majorana particles to study the thermodynamic properties of the model. Finally, we show how Shannon's entropy and thermodynamic properties are modified under the linear potential action.

The Variational Field Equations of Cosmological Topologically Massive Supergravity

AbstractWe discuss the formulation of cosmological topologically massive (simple) supergravity theory in three‐dimensional Riemann‐Cartan space‐times. We use the language of exterior differential forms and the properties of Majorana spinors on 3‐dimensional space‐times to explicitly demonstrate the local supersymmetry of the action density involved. Exact coupled field equations that are complete in both of their bosonic and fermionic sectors are derived by a first order variational principle subject to a torsion‐constraint imposed by the method of Lagrange multipliers. Cotton and Cottino 2‐forms that are complete to all orders in the gravitino field are derived and their properties such as trace are investigated.

Polar form of spinor fields from regular to singular: the flag-dipoles

In this paper, we perform the polar analysis of the spinorial fields, starting from the regular cases and up to the singular cases: we will give for the first time the polar form of the spinorial field equations for the singular cases constituted by the flag-dipole spinor fields. Comments on the role of further spinor sub-classes containing Majorana and Weyl spinors will be sketched.

Dirac, Majorana, Weyl in 4D

This is a review of some elementary properties of Dirac, Weyl and Majorana spinors in 4D. We focus in particular on the differences between massless Dirac and Majorana fermions, on one side, and Weyl fermions, on the other. We review in detail the definition of their effective actions, when coupled to (vector and axial) gauge fields, and revisit the corresponding anomalies using the Feynman diagram method with different regularisations. Among various well known results we stress in particular the regularisation independence in perturbative approaches, while not all the regularisations fit the non-perturbative ones. As for anomalies, we highlight in particular one perhaps not so well known feature: the rigid relation between chiral and trace anomalies.

Biquaternionic Dirac Equation Predicts Zero Mass for Majorana Fermions

A biquaternionic version of the Dirac Equation is introduced, with a procedure for converting four-component spinors to elements of the Pauli algebra. In this version, mass appears as a coefficient between the 4-gradient of a spinor and its image under an outer automorphism of the Pauli algebra. The charge conjugation operator takes a particulary simple form in this formulation and switches the sign of the mass coefficient, so that for a solution invariant under charge conjugation the mass has to equal zero. The multiple of the charge conjugation operator by the imaginary unit turns out to be a complex Lorentz transformation. It commutes with the outer automorphism, while the charge conjugation operator itself anticommutes with it, providing a second more algebraic proof of the main theorem. Considering the Majorana equation, it is shown that non-zero mass of its solution is imaginary.

Trajectory construction of Dirac evolution

We extend our programme of representing the quantum state through exact stand-alone trajectory models to the Dirac equation. We show that the free Dirac equation in the angular coordinate representation is a continuity equation for which the real and imaginary parts of the wave function, angular versions of Majorana spinors, define conserved densities. We hence deduce an exact formula for the propagation of the Dirac spinor derived from the self-contained first-order dynamics of two sets of trajectories in 3-space together with a mass-dependent evolution operator. The Lorentz covariance of the trajectory equations is established by invoking the ‘relativity of the trajectory label'. We show how these results extend to the inclusion of external potentials. We further show that the angular version of Dirac's equation implies continuity equations for currents with non-negative densities, for which the Dirac current defines the mean flow. This provides an alternative trajectory construction of free evolution. Finally, we examine the polar representation of the Dirac equation, which also implies a non-negative conserved density but does not map into a stand-alone trajectory theory. It reveals how the quantum potential is tacit in the Dirac equation.