Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

Majorana Formula

Alvin Meckler

Published Sep 15, 1958• 16 citations
Publisher

Abstract

The Majorana formula for the transition probability of a general spin is here rederived and brought into a more compact form. The method is based on the use of the projection operators which characterize the spin vector as definitely quantized along a specified unit vector. If the Hamiltonian contains only terms linear in the spin, it is possible to define a moving instantaneous axis along which the moving spin vector maintains its quantization. The Majorana problem is reduced to the calculation of the joint probability of quantization along two different axes.The usual expression of the Majorana formula contains a factor which is the transition probability of a spin-\textonehalf{} vector. It is shown here how this factor is related to the angle through which the general spin is turned.

DOI
10.1103/physrev.111.1447

Related papers

Majorana Treatment of Spin-1 System

Tenzin Kalden

The objective of this project is to use the Majorana description of a spin-1 system to examine many of its properties. In the Majorana approach, an arbitrary pure state of a spin-1 system is represented by a pair of points on the unit sphere, or a pair of unit vectors. An expression is derived for the squared overlap of two states in terms of their Majorana (or M-) vectors and used to work out all the orthogonalities and bases that are possible for the three geometrically distinct types of spin-1 states. Finally, a geometrical construction is given of a set of mutually unbiased bases for a spin-1 system. The applications of this result to problems in quantum information are mentioned.

•Jan 1, 2015•0 citations

Spin-½ Probabilities from the Ordinary Dot and Cross Product: A Real Three-Vector Picture of the Spinor Inner Product

Srinivasa Mugeraya

The standard formalism treats spin-½ states as elements of a complex vector space, and the geometrical content of that description is not easy to state simply. We show that the transition probability between two spin-½ pure states can be computed using only the ordinary dot and cross products of real three-vectors. Replacing the ket |-⟩ by the quaternion unit j and applying an overall phase factor of i carries a normalised state to a real unit three-vector. Writing A∘B for the sum of the dot product and the i-component of the cross product, we prove that square of |A∘B|2 equals the standard quantum probability exactly (Theorem 2). This operation is the projection of the quaternion product of conjugate of \( \overline{a}b \) onto the subalgebra spanned by 1 and i (Proposition 2), and by Lagrange’s identity its four component contributions partition unit probability. The construction does not remove the complex structure from the description of a spin-½ state: it relocates it from an algebraic scalar into a distinguished spatial axis, and we state the resulting limitations precisely. The calculation requires nothing beyond vector algebra, whereas the standard route requires a complex Hilbert space.

•Sep 24, 2026•0 citations

The Berry phase for spin in the Majorana representation

J H Hannay

A continuous cyclic sequence of quantum states has an associated geometric, or Berry, phase . For spin J , such a sequence is described by a cyclic change in the 2J+1 coefficients of the basis states . The Berry phase is analysed here for the general case - that is, the coefficients are allowed to vary in an arbitrary cyclic manner. The result is expressed in geometric terms, specifically in the democratic representation due to Majorana. This uniquely characterizes the spin state , up to overall phase, by the positions of 2 J dots on the unit sphere of directions in real space. If the positions are denoted by unit vectors , where , each traces out a parametrized loop on the sphere, and the Berry phase is given by an integral of combinations of these vectors.

Journal of Physics A Mathematical and General•Jan 16, 1998•65 citations

Majorana representation for mixed states

Eduardo Serrano-Ensástiga, Daniel Braun

We generalize the Majorana stellar representation of spin-$s$ pure states to mixed states, and in general to any Hermitian operator, defining a bijective correspondence between three spaces: the spin-density matrices, a projective space of homogeneous polynomials of four variables, and a set of equivalence classes of points (constellations) on spheres of different radii. The representation behaves well under rotations by construction, and also under partial traces where the reduced density matrices inherit their constellation classes from the original state $\ensuremath{\rho}$. We express several concepts and operations related to density matrices in terms of the corresponding polynomials, such as the anticoherence criterion and the tensor representation of spin-$s$ states described in Giraud et al. [Phys. Rev. Lett. 114, 080401 (2015)].

arXiv:1909.07740•Physical Review A•Feb 26, 2020•25 citations

A baby Majorana quantum formalism

Hervé Bergeron, Evaldo M. F. Curado, Jean‐Pierre Gazeau +1 more

The aim of the present paper is to introduce and to discuss the most basic fundamental concepts of quantum physics by means of a simple and pedagogical example. An appreciable part of its content presents original results. We start with the Euclidean plane which is certainly a paradigmatic example of a Hilbert space. The pure states form the unit circle (actually a half of it), the mixed states form the unit disk (actually a half of it), and rotations in the plane rule time evolution through Majorana-like equations involving only real quantities for closed and open systems. The set of pure states or a set of mixed states solve the identity and they are used for understanding the concept of integral quantization of functions on the unit circle and to give a semi-classical portrait of quantum observables. Interesting probabilistic aspects are developed. Since the tensor product of two planes, their direct sum, their cartesian product, are isomorphic (2 is the unique solution to x^x= x\times x = x+x), and they are also isomorphic to C^2, and to the quaternion field H (as a vector space), we describe an interesting relation between entanglement of real states, one-half spin cat states, and unit-norm quaternions which form the group SU$(2)$. We explain the most general form of the Hamiltonian in the real plane by considering the integral quantization of a magnetic-like interaction potential viewed as a classical observable on the unit $2$-sphere. Finally, we present an example of quantum measurement with pointer states lying also in the Euclidean plane.

arXiv:1701.04026•Jan 15, 2017•4 citations

Berry phases and unitary transformations

Thomas F. Jordan

Berry phases for spin are defined for any closed loop made by a vector changing direction in three-dimensional space. A sequence of rotations moves the vector along the loop. Each rotation is around the axis perpendicular to the moving vector. The Berry phases come from the eigenvalues of the unitary operator for the product of these rotations. The angle of the product rotation is shown to be the solid angle enclosed by the loop. The proof uses the ordinary language of quantum mechanics. The product is calculated from the commutation relations for spin. A general framework is set up to define Berry phases for other transformations and states like those for rotations and spin. The integral formula is derived. Alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics. An example is used to show how one Hamiltonian may be simpler than others. Adiabatic evolution is obtained in the limit as a coupling constant goes to zero, so the adiabatic changes are made by a weak perturbation that acts over a long time. The Berry phase is the same whether the coupling constant is large or small. A rationale for the definition of Berry phases is obtained from any of the alternatives for dynamics. It is particularly clear in both the limit of adiabatic evolution, where the Hamiltonian does not contain the operator that generates Berry phases, and the opposite extreme where the Hamiltonian is the operator that generates Berry phases. The latter is illustrated with an example for spin. The general definition of Berry phases and the method of calculating them are illustrated by obtaining Berry phases from Lorentz transformations. They are similar to those obtained from rotations. A vector traces a loop on a unit hyperboloid instead of the unit sphere. In place of the solid angle, the Berry phases contain the analogous measure of the area enclosed by the loop on the hyperboloid. The sign is opposite what it is for rotations. The general definition is shown to fit any unitary representation of a semisimple Lie group. The complete set of commuting operators is chosen to contain a basis for a Cartan subalgebra of the Lie algebra of generators. If a sequence of transformations by unitary operators in the group representation takes each operator in the Cartan subalgebra around a loop in the Lie algebra back to the same operator, it can be made by unitary operators in the group representation that have all the properties required for the definition of Berry phases. The loops are made by vectors moving in a real space, the Lie algebra. The motion of the vectors must maintain the lengths and angles defined by the Cartan metric of the Lie algebra. The Berry phases are determined by one loop made by one suitably chosen operator from the Cartan subalgebra.

Journal of Mathematical Physics•Sep 1, 1988•56 citations
The Majorana formula for the transition probability of a general spin is here rederived and brought into a more compact form. The method is based on the use of the projection operators which characterize the spin vector as definitely quantized along a specified unit vector. If the Hamiltonian contains only terms linear in the spin, it is possible to define a moving instantaneous axis along which the moving spin vector maintains its quantization. The Majorana problem is reduced to the calculation of the joint probability of quantization along two different axes.The usual expression of the Majorana formula contains a factor which is the transition probability of a spin-\textonehalf{} vector. It is shown here how this factor is related to the angle through which the general spin is turned.
The objective of this project is to use the Majorana description of a spin-1 system to examine many of its properties. In the Majorana approach, an arbitrary pure state of a spin-1 system is represented by a pair of points on the unit sphere, or a pair of unit vectors. An expression is derived for the squared overlap of two states in terms of their Majorana (or M-) vectors and used to work out all the orthogonalities and bases that are possible for the three geometrically distinct types of spin-1 states. Finally, a geometrical construction is given of a set of mutually unbiased bases for a spin-1 system. The applications of this result to problems in quantum information are mentioned.
The standard formalism treats spin-½ states as elements of a complex vector space, and the geometrical content of that description is not easy to state simply. We show that the transition probability between two spin-½ pure states can be computed using only the ordinary dot and cross products of real three-vectors. Replacing the ket |-⟩ by the quaternion unit j and applying an overall phase factor of i carries a normalised state to a real unit three-vector. Writing A∘B for the sum of the dot product and the i-component of the cross product, we prove that square of |A∘B|2 equals the standard quantum probability exactly (Theorem 2). This operation is the projection of the quaternion product of conjugate of a‾b \overline{a}b ab onto the subalgebra spanned by 1 and i (Proposition 2), and by Lagrange’s identity its four component contributions partition unit probability. The construction does not remove the complex structure from the description of a spin-½ state: it relocates it from an algebraic scalar into a distinguished spatial axis, and we state the resulting limitations precisely. The calculation requires nothing beyond vector algebra, whereas the standard route requires a complex Hilbert space.
A continuous cyclic sequence of quantum states has an associated geometric, or Berry, phase . For spin J , such a sequence is described by a cyclic change in the 2J+1 coefficients of the basis states . The Berry phase is analysed here for the general case - that is, the coefficients are allowed to vary in an arbitrary cyclic manner. The result is expressed in geometric terms, specifically in the democratic representation due to Majorana. This uniquely characterizes the spin state , up to overall phase, by the positions of 2 J dots on the unit sphere of directions in real space. If the positions are denoted by unit vectors , where , each traces out a parametrized loop on the sphere, and the Berry phase is given by an integral of combinations of these vectors.
We generalize the Majorana stellar representation of spin-sss pure states to mixed states, and in general to any Hermitian operator, defining a bijective correspondence between three spaces: the spin-density matrices, a projective space of homogeneous polynomials of four variables, and a set of equivalence classes of points (constellations) on spheres of different radii. The representation behaves well under rotations by construction, and also under partial traces where the reduced density matrices inherit their constellation classes from the original state $\ensuremath{\rho}$. We express several concepts and operations related to density matrices in terms of the corresponding polynomials, such as the anticoherence criterion and the tensor representation of spin-sss states described in Giraud et al. [Phys. Rev. Lett. 114, 080401 (2015)].
The aim of the present paper is to introduce and to discuss the most basic fundamental concepts of quantum physics by means of a simple and pedagogical example. An appreciable part of its content presents original results. We start with the Euclidean plane which is certainly a paradigmatic example of a Hilbert space. The pure states form the unit circle (actually a half of it), the mixed states form the unit disk (actually a half of it), and rotations in the plane rule time evolution through Majorana-like equations involving only real quantities for closed and open systems. The set of pure states or a set of mixed states solve the identity and they are used for understanding the concept of integral quantization of functions on the unit circle and to give a semi-classical portrait of quantum observables. Interesting probabilistic aspects are developed. Since the tensor product of two planes, their direct sum, their cartesian product, are isomorphic (2 is the unique solution to x^x= x\times x = x+x), and they are also isomorphic to C^2, and to the quaternion field H (as a vector space), we describe an interesting relation between entanglement of real states, one-half spin cat states, and unit-norm quaternions which form the group SU(2)(2)(2). We explain the most general form of the Hamiltonian in the real plane by considering the integral quantization of a magnetic-like interaction potential viewed as a classical observable on the unit 222-sphere. Finally, we present an example of quantum measurement with pointer states lying also in the Euclidean plane.
Berry phases for spin are defined for any closed loop made by a vector changing direction in three-dimensional space. A sequence of rotations moves the vector along the loop. Each rotation is around the axis perpendicular to the moving vector. The Berry phases come from the eigenvalues of the unitary operator for the product of these rotations. The angle of the product rotation is shown to be the solid angle enclosed by the loop. The proof uses the ordinary language of quantum mechanics. The product is calculated from the commutation relations for spin. A general framework is set up to define Berry phases for other transformations and states like those for rotations and spin. The integral formula is derived. Alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics. An example is used to show how one Hamiltonian may be simpler than others. Adiabatic evolution is obtained in the limit as a coupling constant goes to zero, so the adiabatic changes are made by a weak perturbation that acts over a long time. The Berry phase is the same whether the coupling constant is large or small. A rationale for the definition of Berry phases is obtained from any of the alternatives for dynamics. It is particularly clear in both the limit of adiabatic evolution, where the Hamiltonian does not contain the operator that generates Berry phases, and the opposite extreme where the Hamiltonian is the operator that generates Berry phases. The latter is illustrated with an example for spin. The general definition of Berry phases and the method of calculating them are illustrated by obtaining Berry phases from Lorentz transformations. They are similar to those obtained from rotations. A vector traces a loop on a unit hyperboloid instead of the unit sphere. In place of the solid angle, the Berry phases contain the analogous measure of the area enclosed by the loop on the hyperboloid. The sign is opposite what it is for rotations. The general definition is shown to fit any unitary representation of a semisimple Lie group. The complete set of commuting operators is chosen to contain a basis for a Cartan subalgebra of the Lie algebra of generators. If a sequence of transformations by unitary operators in the group representation takes each operator in the Cartan subalgebra around a loop in the Lie algebra back to the same operator, it can be made by unitary operators in the group representation that have all the properties required for the definition of Berry phases. The loops are made by vectors moving in a real space, the Lie algebra. The motion of the vectors must maintain the lengths and angles defined by the Cartan metric of the Lie algebra. The Berry phases are determined by one loop made by one suitably chosen operator from the Cartan subalgebra.