arXiv subjects

Kazuo Fujikawa

Publications and source records attributed to Kazuo Fujikawa.

At least 19 recordsLinked to original sources

arXiv2026

Neutrinoless double $\beta$ decay and leptogenesis in seesaw model

The fermion $\nu_{R}+C\overline{\nu_{R}}^{T}$ in the Type I seesaw model, which is commonly identified with a Majorana fermion, is not a Majorana fermion and thus no neutrinoless double $\beta$ decay (in the case of $\nu_{L}-C\overline{\nu_{L}}^{T}$) due to the theorem on the absence of a Majorana-Weyl fermion in the representation of the d=4 Lorentz group. The leptogenesis is realized naturally together with the neutrinoless double $\beta$ decay by first defining C symmetric Majorana fermions in the seesaw model using an analogue of the Bogoliubov transformation.

hep-ph↗

arXiv2025

Absence of Majorana-Weyl fermions in d=4 and the theory of Majorana fermions

It is customary to identify $\psi_{+}=\nu_{R} + C\overline{\nu_{R}}^{T}$ with a Majorana fermion on the basis of chirality changing charge conjugation $\tilde{C}: \nu_{R}\rightarrow C\overline{\nu_{R}}^{T}$ and parity $\tilde{P}: \nu_{R}\rightarrow i\gamma^{0}\nu_{R}$. The theorem on the absence of a Majorana-Weyl fermion in $d=4$ states $\tilde{C}\gamma_{5}\tilde{C}^{-1}= -\gamma_{5}$ with $\tilde{C}=C\gamma_{4}^{T}$, and thus the charge conjugation of the equivalent Majorana $\psi_{+}=(\frac{1+\gamma_{5}}{2})\nu_{R} + (\frac{1-\gamma_{5}}{2})C\overline{\nu_{R}}^{T}$ vanishes without subsidiary $\gamma_{5}\rightarrow - \gamma_{5}$, namely, not defined in field theory. To be consistent with the theorem, it is common to use a doublet representation of chirality preserving charge conjugation $\hat{C}:\nu_{R,L}\rightarrow C\overline{\nu_{L,R}}^{T}$ and parity $\hat{P}: \nu_{R,L}\rightarrow i\gamma^{0}\nu_{L,R}$ in theory containing both $\nu_{R,L}$. In the type I seesaw model, the latter formulation is applicable but $\psi_{+}=\nu_{R} + C\overline{\nu_{R}}^{T}$ is not a Majorana fermion. An analogue of the Bogoliubov transformation converts $\psi_{\pm}=\nu_{R, L} \pm C\overline{\nu_{R, L}}^{T}$, which are obtained by a precise diagonalization of the seesaw model, to Majorana fermions $\psi_{M_{1,2}}=(\psi \pm C\overline{\psi}^{T})/\sqrt{2}$ with a Dirac-type fermion $\psi$, as originally defined by Majorana. A chiral projection $[(1+\gamma_{5})/2] \psi_{M_{1}}$ of a Majorana fermion is not a chiral fermion, which ensures the presence of the neutrino-less double beta decay.

hep-ph↗

arXiv2024

Note on a BCS analogy of Majorana neutrinos

In this note, we discuss an analogy between the BCS theory and the seesaw model of neutrinos. We believe that the analogy indicates some fundamental aspects of Majorana neutrinos. A paper on the issue has been recently presented, and we would like to describe the background of the paper together with our personal views on the problem. In essence, the conventional construction of a Majorana neutrino from a chiral fermion is too simplified, and we argue that one would actually have to go through a Bogoliubov-type canonical transformation to generate two Majorana fermions from an effective single Dirac fermion in the seesaw model.

hep-ph↗

arXiv2024

Two classes of Majorana neutrinos in the seesaw model

The commonly used pseudo-C symmetry $(\nu_{L})^{c}=C\overline{\nu_{L}}^{T}$ is not defined in Lagrangian field theory. In general, there exist two classes of Majorana fermions; the first is associated with the Dirac-type fermion with the conventional C and P symmetries, and the second is associated with the Weyl-type fermion defined by CP symmetry only and formally characterised by the pseudo-C symmetry. Taking the seesaw model as an example, it is shown that a generalized Pauli--G\"{u}rsey (or Bogoliubov-type) canonical transformation converts the neutrino defined by the Weyl-type fermion to the neutrino defined by the Dirac-type fermion and thus to the conventional Majorana fermion, while preserving the canonical anti-commutation relations. The mixing angles in the weak lepton sector are not modified by this generalized Pauli--G\"{u}rsey transformation.

hep-ph↗

arXiv2024

Remark on neutrino oscillations

The oscillations of ultra-relativistic neutrinos are realized by the propagation of assumed zero-mass on-shell neutrinos with the speed of light in vacuum combined with the phase modulation by the small mass term $\exp[-i(m^{2}_{ν_{k}}/2|\vec{p}|)τ]$ with a time parameter $τ$. This picture is realized in the first quantization by the mass expansion and in field theory by the use of $δ(x^{0}-y^{0}-τ) \langle 0|T^{\star}ν_{L k}(x)\overline{ν_{L k}(y)}|0\rangle$ with the neutrino mass eigenstates $ν_{L k}$ and a finite positive $τ$ after the contour integral of the propagating neutrino energies. By noting that the conventional detectors are insensitive to neutrino masses, the measured energy-momenta of the initial and final states with assumed zero-mass neutrinos are conserved. The propagating neutrinos preserve the three-momentum in this sense but the energies of the massive neutrinos are conserved up to uncertainty relations and thus leading to oscillations. Conceptual complications in the case of Majorana neutrinos due to the charge conjugation in $d=4$ are also discussed.

hep-ph↗

arXiv2023

Berry's phase and quantum mechanical formulation of anomalous Hall effect

The canonical commutation relations in quantum mechanics are not maintained in the anomalous Hall effect described by Berry's phase in the presence of the electromagnetic vector potential. To define quantum mechanical formulation, one may avoid the electromagnetic vector potential but then the anomalous Nernst effect induced by Berry's curvature is not described using a modified phase space volume, although the anomalous Nernst effect by itself may be generated by the adiabatic Berry's curvature. We also comment on the Born-Oppenheimer approximation if it describes the anomalous Hall effect in the absence of the electromagnetic vector potential in quantum mechanics. An alternative view of the Bjorken-Johnson-Low prescription which is consistent with the principle of quantum mechanics and the existing Berry's phase theory of the anomalous Hall effect is also mentioned.

cond-mat.str-el↗

arXiv2022

Berry's phase and chiral anomalies

The basic materials of Berry's phase and chiral anomalies are presented to appreciate the phenomena related to those notions. As for Berry's phase, a general survey of the subject is presented using both Lagrangian and Hamiltonian formalisms. The canonical Hamiltonian formalism of the Born-Oppenheimer approximation, when applied to the anomalous Hall effect, can incorporate the gauge symmetry of Berry's connection but unable to incorporate the electromagnetic vector potential simultaneously. Transformed to the Lagrangian formalism with a time-derivative term allowed, the Born-Oppenheimer approximation can incorporate the electromagnetic vector potential simultaneously with Berry's connection, but the consistent canonical property is lost and thus becomes classical. The Lagrangian formalism can thus incorporate both gauge symmetries simultaneously but spoils the basic quantum symmetries, and thus results in classical anomalous Poisson brackets and the classical Nernst effect as in the conventional formalism. As for chiral anomalies, we present basic materials by the path integral formulation with an emphasis on fermions on the lattice. A chiral fermion defined by $γ_{5}$ on the lattice does not contain the chiral anomaly for the non-vanishing lattice spacing $a\neq0$. The idea of a spectral flow on the lattice does not lead to an anomaly for each species doubler separately but rather to a pair production in a general sense. We also mention that a specific construction called the Ginsparg-Wilson fermion, which is free of species doublers, may practically be useful. We discuss the representative applications of Berry's phase and chiral anomalies in nuclear physics and related fields to illustrate the use of these two basic notions.

nucl-th↗

arXiv2022

A path integral derivation of the equations of anomalous Hall effect

A path integral (Lagrangian formalism) is used to derive the effective equations of motion of the anomalous Hall effect with Berry's phase on the basis of the adiabatic condition $|E_{n\pm1}-E_{n}|\gg 2π\hbar/T$, where $T$ is the typical time scale of the slower system and $E_{n}$ is the energy level of the fast system. In the conventional definition of the adiabatic condition with $T\rightarrow {\rm large}$ and fixed energy eigenvalues, no commutation relations are defined for slower variables by the Bjorken-Johnson-Low prescription except for the starting canonical commutators. On the other hand, in a singular limit $|E_{n\pm1}-E_{n}|\rightarrow \infty$ with specific $E_{n}$ kept fixed for which any motions of the slower variables $X_{k}$ can be treated to be adiabatic, the non-canonical dynamical system with deformed commutators and the Nernst effect appear. In the Born-Oppenheimer approximation based on the canonical commutation relations, the equations of motion of the anomalous Hall effect is obtained if one uses an auxiliary variable $X_{k}^{(n)}=X_{k}+{\cal A}^{(n)}_{k}$ with Berry's connection ${\cal A}^{(n)}_{k}$ in the absence of the electromagnetic vector potential $eA_{k}(X)$ and thus without the Nernst effect. It is shown that the gauge symmetries associated with Berry's connection and the electromagnetic vector potential $eA_{k}(X)$ are incompatible in the canonical Hamiltonian formalism. The appearance of the non-canonical dynamical system with the Nernst effect is a consequence of the deformation of the quantum principle to incorporate the two incompatible gauge symmetries.

cond-mat.str-el↗

arXiv2021

Lensing of Dirac monopole in Berry's phase

Berry's phase, which is associated with the slow cyclic motion with a finite period, looks like a Dirac monopole when seen from far away but smoothly changes to a dipole near the level crossing point in the parameter space in an exactly solvable model. This topology change of Berry's phase is visualized as a result of lensing effect; the monopole supposed to be located at the level crossing point appears at the displaced point when the variables of the model deviate from the precisely adiabatic movement. The effective magnetic field generated by Berry's phase is determined by a simple geometrical consideration of the magnetic flux coming from the displaced Dirac monopole.

hep-th↗

arXiv2021

Parity of the neutron consistent with neutron-antineutron oscillations

In the analysis of neutron-antineutron oscillations, it has been recently argued in the literature that the use of the $iγ^{0}$ parity $n^{p}(t,-\vec{x})=iγ^{0}n(t,-\vec{x})$ which is consistent with the Majorana condition is mandatory and that the ordinary parity transformation of the neutron field $n^{p}(t,-\vec{x}) = γ^{0}n(t,-\vec{x})$ has a difficulty. We show that a careful treatment of the ordinary parity transformation of the neutron works in the analysis of neutron-antineutron oscillations. Technically, the CP symmetry in the mass diagonalization procedure is important and the two parity transformations, $iγ^{0}$ parity and $γ^{0}$ parity, are compensated for by the Pauli-Gürsey transformation. Our analysis shows that either choice of the parity gives the correct results of neutron-antineutron oscillations if carefully treated.

hep-ph↗

arXiv2020

Parity and CP operations for Majorana neutrinos

The parity transformation law of the fermion field $ψ(x)$ is usually defined by the "$γ^{0}$-parity" $ψ^{p}(t,-\vec{x}) = γ^{0}ψ(t,-\vec{x})$ with eigenvalues $\pm 1$, while the "$iγ^{0}$-parity" $ψ^{p}(t,-\vec{x})=iγ^{0}ψ(t,-\vec{x})$ is required for the Majorana fermion. The compatibility issues of these two parity laws arise in generic fermion number violating theories where a general class of Majorana fermions appear. In the case of Majorana neutrinos constructed from chiral neutrinos in an extension of the Standard Model, the Majorana neutrinos can be characterized by CP symmetry although C and P are separately broken. It is then shown that either choice of the parity operation, $γ^{0}$ or $iγ^{0}$, in the level of the starting fermions gives rise to the consistent and physically equivalent descriptions of emergent Majorana neutrinos both for Weinberg's model of neutrinos and for a general class of seesaw models. The mechanism of this equivalence is that the Majorana neutrino constructed from a chiral neutrino, which satisfies the classical Majorana condition $ψ(x)=C\overline{ψ(x)}^{T}$, allows the phase freedom $ψ(x)=e^{iα}ν_{L}(x) + e^{-iα}C\overline{ν_{L}(x)}^{T}$ with $α=0\ {\rm or}\ π/4$ that accounts for the phase coming from the different definitions of parity for $ν_{L}(x)$ and ensures the consistent definitions of CP symmetry $({\cal CP})ψ(x)({\cal CP})^{\dagger}= \pm iγ^{0}ψ(t,-\vec{x})$.

hep-ph↗

arXiv2020

Path integral of neutrino oscillations

We propose an idea of the constrained Feynman amplitude for the scattering of the charged lepton and the virtual W-boson, $l_β + W_ρ \rightarrow l_α + W_λ$, from which the conventional Pontecorvo oscillation formula of relativistic neutrinos is readily obtained using plane waves for all the particles involved. In a path integral picture, the neutrino propagates forward in time between the production and detection vertices, which are constrained respectively on the 3-dimensional spacelike hypersurfaces separated by a macroscopic positive time $τ$. The covariant Feynman amplitude is formally recovered if one sums over all possible values of $τ$ (including negative $τ$).

hep-ph↗

arXiv2020

No anomalous canonical commutators induced by Berry's phase

The monopole-like singularity of Berry's adiabatic phase in momentum space and associated anomalous Poisson brackets have been recently discussed in various fields. With the help of the results of an exactly solvable version of Berry's model, we show that Berry's phase does not lead to the deformation of the principle of quantum mechanics in the sense of anomalous canonical commutators. If one should assume Berry's phase of genuine Dirac monopole-type, which is assumed to hold not only in the adiabatic limit but also in the non-adiabatic limit, the deformation of the principle of quantum mechanics could take place. But Berry's phase of the genuine Dirac monopole-type is not supported by the exactly solvable version of Berry's model nor by a generic model of Berry's phase. Besides, the monopole-like Berry's phase in momentum space has a magnetic charge $e_{M}=2π\hbar$, for which the possible anomalous term in the canonical commutator $[x_{k},x_{l}]=i\hbarΩ_{kl}$ would become of the order $O(\hbar^{2})$.

hep-th↗

arXiv2020

Operatorial characterization of Majorana neutrinos

The Majorana neutrino $ψ_{M}(x)$ when constructed as a superposition of chiral fermions such as $ν_{L} + C\overline{ν_{L}}^{T}$ is characterized by $ ({\cal C}{\cal P}) ψ_{M}(x)({\cal C}{\cal P})^{\dagger} =iγ^{0}ψ_{M}(t,-\vec{x})$, and the CP symmetry describes the entire physics contents of Majorana neutrinos. Further specifications of C and P separately could lead to difficulties depending on the choice of C and P. The conventional $ {\cal C} ψ_{M}(x) {\cal C}^{\dagger} = ψ_{M}(x)$ with well-defined P is naturally defined when one constructs the Majorana neutrino from the Dirac-type fermion. In the seesaw model of Type I or Type I+II where the same number of left- and right-handed chiral fermions appear, it is possible to use the generalized Pauli-Gursey transformation to rewrite the seesaw Lagrangian in terms of Dirac-type fermions only; the conventional C symmetry then works to define Majorana neutrinos. In contrast, the "pseudo C-symmetry" $ν_{L,R}(x)\rightarrow C\overline{ν_{L,R}(x)}^{T}$ (and associated "pseudo P-symmetry"), that has been often used in both the seesaw model and Weinberg's model to describe Majorana neutrinos, attempts to assign a nontrivial charge conjugation transformation rule to each chiral fermion separately. But this common construction is known to be operatorially ill-defined and, for example, the amplitude of the neutrinoless double beta decay vanishes if the vacuum is assumed to be invariant under the pseudo C-symmetry.

hep-ph↗

arXiv2020

A new magnetic monopole inspired by Berry's phase

A new static and azimuthally symmetric magnetic monopolelike object, which looks like a Dirac monopole when seen from far away but smoothly changes to a dipole near the monopole position and vanishes at the origin, is discussed. This monopolelike object is inspired by an analysis of an exactly solvable model of Berry's phase in the parameter space. A salient feature of the monopolelike potential ${\cal A}_{k}(r,θ)$ with a magnetic charge $e_{M}$ is that the Dirac string is naturally described by the potential ${\cal A}_{k}(r,θ)$, and the origin of the Dirac string and the geometrical center of the monopole are displaced in the coordinate space. The smooth topology change from a monopole to a dipole takes place if the Dirac string, when coupled to the electron, becomes unobservable by satisfying the Dirac quantization condition. The electric charge is then quantized even if the monopole changes to a dipole near the origin. In the transitional region from a monopole to a dipole, a half-monopole with a magnetic charge $e_{M}/2$ appears.

hep-th↗

arXiv2019

Seesaw mechanism and pseudo C-symmetry

It is shown that the specific "charge conjugation" transformation used to define the Majorana fermions in the conventional seesaw mechanism, namely $(ν_{R})^{C}=C\bar{ν_{R}}^{T}$ for a chiral fermion $ν_{R}$ (and similarly for $ν_{L}$), is a hidden symmetry associated with CP symmetry, and thus it formally holds independently of the P- and C-violating terms in the CP invariant Lagrangian and it is in principle applicable to charged leptons and quarks as well. This hidden symmetry, however, is not supported by a consistent unitary operator and thus it leads to mathematical (operatorial) ambiguities. When carefully examined, it also fails as a classical transformation law in a Lorentz invariant field theory. To distinguish it from the standard charge conjugation symmetry, we suggest for it the name of pseudo C-symmetry. The pseudo C-symmetry is effective to identify Majorana neutrinos analogously to the classical Majorana condition. The analysis of CP breaking in weak interactions is performed using the conventional CP transformation, which is defined independently of the pseudo C-transformation, in the seesaw model after mass diagonalization. A way to ensure an operatorially consistent formulation of C-conjugation is to formulate the seesaw scheme by invoking a relativistic analogue of the Bogoliubov transformation.

hep-ph↗

arXiv2019

Topology change from a monopole to a dipole in Berry's phase

The smooth topology change of Berry's phase from a Dirac monopole-like configuration to a dipole configuration, when one approaches the monopole position in the parameter space, is analyzed in an exactly solvable model. A novel aspect of Berry's connection ${\cal A}_{k}$ is that the geometrical center of the monopole-like configuration and the origin of the Dirac string are displaced in the parameter space. Gauss' theorem $\int_{S}(\nabla\times {\cal A})\cdot d\vec{S}=\int_{V} \nabla\cdot (\nabla\times {\cal A}) dV=0$ for a volume $V$ which is free of singularities shows that a combination of the monopole-like configuration and the Dirac string is effectively a dipole. The smooth topology change from a dipole to a monopole with a quantized magnetic charge $e_{M}=2π\hbar$ takes place when one regards the Dirac string as unobservable if it satisfies the Wu-Yang gauge invariance condition. In the transitional region from a dipole to a monopole, a half-monopole appears with an observable Dirac string, which is analogous to the Aharonov-Bohm phase of an electron for the magnetic flux generated by the Cooper pair condensation. The main topological features of an exactly solvable model are shown to be supported by a generic model of Berry's phase.

hep-th↗

arXiv2019

A classical limit of Grover's algorithm induced by dephasing: Coherence vs entanglement

A new approach to the classical limit of Grover's algorithm is discussed by assuming a very rapid dephasing of a system between consecutive Grover's unitary operations, which drives pure quantum states to decohered mixed states. One can identify a specific element among $N$ unsorted elements by a probability of the order of unity after $k\sim N$ steps of classical amplification, which is realized by a combination of Grover's unitary operation and rapid dephasing, in contrast to $k\sim π\sqrt{N}/4$ steps in quantum mechanical amplification. The initial two-state system with enormously unbalanced existence probabilities, which is realized by a chosen specific state and a superposition of all the rest of states among $N$ unsorted states, is crucial in the present analysis of classical amplification. This analysis illustrates Grover's algorithm in extremely noisy circumstances. A similar increase from $k\sim \sqrt{N}$ to $k\sim N$ steps due to the loss of quantum coherence takes place in the {\em analog} model of Farhi and Gutmann where the entanglement does not play an obvious role. This supports a view that entanglement is crucial in quantum computation to describe quantum states by a set of qubits, but the actual speedup of the quantum computation is based on quantum coherence.

quant-ph↗