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

American Journal of Physics

Representation of Rotations in Three-Space by Complex 2×2 Matrices

Marshall J. Walker

Published Apr 1, 1956• 0 citations
Publisher

Abstract

A comparison is made between the representation of rotations in three dimensional space by real 3×3 matrices and complex 2×2 matrices. The relationship is used as a means of visualizing first-rank spinors.
DOI
10.1119/1.1934194

Related papers

ON REPRESENTATIONS OF THE ROTATION GROUP WITH COMPLEX SPIN

S.S. Sannikov

•Sep 1, 1965•0 citations

Invariant spinor representations of finite rotation matrices

Nikolai L. Manakov, Alexei V. Meremianin, Anthony F. Starace

Our recent results [N. L. Manakov et al., Phys. Rev. A 57, 3233 (1998); 61, 022103 (2000)] on the invariant representations of finite rotation matrices (FRM's) of integer rank j (in terms of tensor products of vectors connected with a space-fixed reference frame) are generalized here for the general case of arbitrary (i.e., integer or half-integer) rank j. This extension is carried out by using new spinor representations of FRM's in terms of specially introduced spinor-annihilation operators. We demonstrate that all widely used, standard representations of FRM's follow as special cases of our invariant representation for particular parametrizations of the rotation parameters. As the simplest application of invariant spinor representations of FRM's, the factorized form of Wigner ${\mathbf{d}}^{j}(\ensuremath{\beta})$ matrices with an arbitrary rank j is obtained as a product of two triangular matrices composed of various powers of $\mathrm{cos}(\ensuremath{\beta}/2).$

Physical Review A•Aug 14, 2001•2 citations

Rotations in Three-Dimensional Space

Масуд Чайчиан, Rolf Hagedorn

There are unfortunately a large variety of possibilities for writing down a rotation; also there are two different interpretations of the word ‘rotation’. Obviously, if one rotates first the coordinate frame and then, by the same rotation, the space, all points will obtain again the same coordinate as before these rotations; the same is true in the reverse order. Rotational invariance means then that the result of this rotation, seen by an observer who did not participate in the rotation, is again a possible state of the unrotated system. One says the rotations in the physical three-dimensional space induce unitary transformations in the Hilbert space, whose matrices give rise to (infinitely many independent) representations of the rotation group. The active interpretation means carrying out the sequence of rotations using a fixed system of rotation axes; the passive interpretation means carrying out the rotations using a system of rotation axes which participates in the rotation.

CRC Press eBooks•Jul 14, 2023•0 citations

Representation of spinors by real and complex tensor aggregates

Vladimir A. Zhelnorovich

Theoretical and Mathematical Physics•Jan 1, 1970•6 citations

Analytical Representation of General Spin

Takao Yamamoto

The spin operators and their eigenstates are usu:tlly represented by matrices.But it is also pos.sible to represent them analytically in several ways.I) An interesting example is shown as follows.The spin operators are assumed to be the following dif• ferential operators of the two real variables u and v J.,=1/2(uv--fJ2/8u8v), J y=z/2 (v.8/8u-u.8/8v) , Jz=I/4(U2_82/8u2_V2+82/8v2) ,

Progress of Theoretical Physics•Nov 1, 1953•0 citations

Spinors and Spatial Vectors Are Part of the Number System; They Are Extensions of Complex Numbers into Three-Dimensional Space

Srinivasa Mugeraya

Spinors are used for the computation of probability in quantum mechanics. They are treated as elements of a complex vector space in contrast to a real vector space. Spinor theory is abstract mathematics with an ambiguous interpretation. The overall phase of spinors does not affect the computation of probability in quantum mechanics. If two spinors have an overall phase of imaginary number i, then they can be treated like vectors in a real vector space. The square of the magnitude of the number, which is the sum of dot product and the term having basis vector i of cross product of two such vectors, is equal to the probability computed using the corresponding spinors. Therefore, the geometry of such spinors can be easily depicted in a three-dimensional space like vectors in a real vector space. Spinors are not isotropic vectors in Hilbert space. The sum of dot product and cross product of two complex numbers is equivalent to the quotient of the division of one complex number by another. Similarly, the sum of dot product and cross product of two vectors is the quotient of the division of one vector by another. Using the rules of division of vectors, we can find the rules of multiplication of vectors. The rules of multiplication and division of basis vector i match the rules of multiplication and division of imaginary number i. Therefore, basis vector i is nothing but imaginary number i. Multiplication of dot and cross product of two vectors to the second vector will give us the components of the first vector that are parallel and orthogonal to the second vector. Vectors are also made up of complex numbers like spinors. The reason for finding dot product of complex numbers in the process of computing the probability is to ignore the overall phase in contrast to the phase difference. This is misconstrued as complex vector space in quantum mechanics. 3-D number which is nothing but a spinor with new notation is an extension of vector algebra. It can have a real number as a term in addition to the terms i, j and k. In all other respects, it is a vector. 3-D numbers are part of the number system like real numbers and complex numbers. The real numbers are one dimensional numbers, the complex numbers are two-dimensional numbers and the 3-D numbers as well as vectors and spinors are three-dimensional numbers. As polarisation and spin are one and the same, we can straight away apply 3-D numbers to polarisation of light. 3-D numbers will greatly simplify the study of areas of science where three-dimensional space is involved.

•Aug 30, 2023•1 citation
Our recent results [N. L. Manakov et al., Phys. Rev. A 57, 3233 (1998); 61, 022103 (2000)] on the invariant representations of finite rotation matrices (FRM's) of integer rank j (in terms of tensor products of vectors connected with a space-fixed reference frame) are generalized here for the general case of arbitrary (i.e., integer or half-integer) rank j. This extension is carried out by using new spinor representations of FRM's in terms of specially introduced spinor-annihilation operators. We demonstrate that all widely used, standard representations of FRM's follow as special cases of our invariant representation for particular parametrizations of the rotation parameters. As the simplest application of invariant spinor representations of FRM's, the factorized form of Wigner ${\mathbf{d}}^{j}(\ensuremath{\beta})$ matrices with an arbitrary rank j is obtained as a product of two triangular matrices composed of various powers of $\mathrm{cos}(\ensuremath{\beta}/2).$
There are unfortunately a large variety of possibilities for writing down a rotation; also there are two different interpretations of the word ‘rotation’. Obviously, if one rotates first the coordinate frame and then, by the same rotation, the space, all points will obtain again the same coordinate as before these rotations; the same is true in the reverse order. Rotational invariance means then that the result of this rotation, seen by an observer who did not participate in the rotation, is again a possible state of the unrotated system. One says the rotations in the physical three-dimensional space induce unitary transformations in the Hilbert space, whose matrices give rise to (infinitely many independent) representations of the rotation group. The active interpretation means carrying out the sequence of rotations using a fixed system of rotation axes; the passive interpretation means carrying out the rotations using a system of rotation axes which participates in the rotation.
Spinors are used for the computation of probability in quantum mechanics. They are treated as elements of a complex vector space in contrast to a real vector space. Spinor theory is abstract mathematics with an ambiguous interpretation. The overall phase of spinors does not affect the computation of probability in quantum mechanics. If two spinors have an overall phase of imaginary number i, then they can be treated like vectors in a real vector space. The square of the magnitude of the number, which is the sum of dot product and the term having basis vector i of cross product of two such vectors, is equal to the probability computed using the corresponding spinors. Therefore, the geometry of such spinors can be easily depicted in a three-dimensional space like vectors in a real vector space. Spinors are not isotropic vectors in Hilbert space. The sum of dot product and cross product of two complex numbers is equivalent to the quotient of the division of one complex number by another. Similarly, the sum of dot product and cross product of two vectors is the quotient of the division of one vector by another. Using the rules of division of vectors, we can find the rules of multiplication of vectors. The rules of multiplication and division of basis vector i match the rules of multiplication and division of imaginary number i. Therefore, basis vector i is nothing but imaginary number i. Multiplication of dot and cross product of two vectors to the second vector will give us the components of the first vector that are parallel and orthogonal to the second vector. Vectors are also made up of complex numbers like spinors. The reason for finding dot product of complex numbers in the process of computing the probability is to ignore the overall phase in contrast to the phase difference. This is misconstrued as complex vector space in quantum mechanics. 3-D number which is nothing but a spinor with new notation is an extension of vector algebra. It can have a real number as a term in addition to the terms i, j and k. In all other respects, it is a vector. 3-D numbers are part of the number system like real numbers and complex numbers. The real numbers are one dimensional numbers, the complex numbers are two-dimensional numbers and the 3-D numbers as well as vectors and spinors are three-dimensional numbers. As polarisation and spin are one and the same, we can straight away apply 3-D numbers to polarisation of light. 3-D numbers will greatly simplify the study of areas of science where three-dimensional space is involved.