Who discovered the covering homomorphism between SU(2) and SO(3)?Why Feynman's integral is not taught today...
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Who discovered the covering homomorphism between SU(2) and SO(3)?
Why Feynman's integral is not taught today more widely and earlier in the academic physics curriculum?Who discovered the Rayleigh-Taylor instability?How did gyromagnetic ratio come up before quantum mechanics, and who introduced it?Who discovered diamagnetism first?Who discovered paramagnetism first?Who was first to differentiate between prime and irreducible elements?Was there any atomic model(s) that came between Bohr's and the actual beginning of Quantum Mechanics in early 20s?Who discovered L'Hôpital's rule?What new physics was discovered or needed as a result of the Manhattan Project?
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Who discovered this? It is quite nontrivial and very important in quantum mechanics.
mathematics physics
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Who discovered this? It is quite nontrivial and very important in quantum mechanics.
mathematics physics
edited 7 hours ago
Conifold
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asked 8 hours ago
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In old books on classical mechanics parametrization of $SO(3)$ by $SU(2)$ is called the Klein parametrization.
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– Alexandre Eremenko
6 hours ago
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$begingroup$
Who discovered this? It is quite nontrivial and very important in quantum mechanics.
mathematics physics
edited 7 hours ago
Conifold
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asked 8 hours ago
JohnJohn
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$endgroup$
Who discovered this? It is quite nontrivial and very important in quantum mechanics.
mathematics physics
mathematics physics
edited 7 hours ago
Conifold
40.5k2 gold badges64 silver badges145 bronze badges
asked 8 hours ago
JohnJohn
3311 silver badge8 bronze badges
edited 7 hours ago
Conifold
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asked 8 hours ago
JohnJohn
3311 silver badge8 bronze badges
edited 7 hours ago
Conifold
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edited 7 hours ago
Conifold
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edited 7 hours ago
Conifold
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40.5k2 gold badges64 silver badges145 bronze badges
asked 8 hours ago
JohnJohn
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asked 8 hours ago
JohnJohn
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JohnJohn
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2
$begingroup$
In old books on classical mechanics parametrization of $SO(3)$ by $SU(2)$ is called the Klein parametrization.
$endgroup$
– Alexandre Eremenko
6 hours ago
add a comment
|
2
$begingroup$
In old books on classical mechanics parametrization of $SO(3)$ by $SU(2)$ is called the Klein parametrization.
$endgroup$
– Alexandre Eremenko
6 hours ago
2
2
$begingroup$
In old books on classical mechanics parametrization of $SO(3)$ by $SU(2)$ is called the Klein parametrization.
$endgroup$
– Alexandre Eremenko
6 hours ago
$begingroup$
In old books on classical mechanics parametrization of $SO(3)$ by $SU(2)$ is called the Klein parametrization.
$endgroup$
– Alexandre Eremenko
6 hours ago
add a comment
|
1 Answer
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Hamilton and Klein, Klein was more explicit about it. Hamilton in Lectures on Quaternions (1853) realized that his representation of rotations of rigid bodies by the unit quaternions was not $1$-$1$, but $2$-$1$. Klein in Lectures on the Ikosahedron and the Solution of Equations
of the Fifth Degree (1888) replaced the unit quaternions by $2 × 2$ unitary matrices with the determinant $1$, now denoted $SU(2)$. He then more or less spelled out that the unit quaternions and $SU(2)$ are isomorphic groups, which are $2$-$1$ epimorphic onto the group of 3D rotations $SO(3)$.
Pauli proposed the "two-valuedness not describable classically", which was later identified with the electron spin, in 1924, and formalized it in the matrix form in 1927. In 1932 Heisenberg and Ivanenko guessed that the same effect regulates protons/neutrons as the states of a single particle, later dubbed nucleon, and incorporated it into their proton–neutron model of the nucleus.
Steiner cites this homomorphism as a prime example of "unreasonable effectiveness" of mathematics. Both times the mathematical machinery developed was not aimed, even indirectly, at the application it ended up being useful for. In the case of nucleus, any visible connection to rotations and 3D space is missing altogether.
answered 7 hours ago
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$begingroup$
Hamilton and Klein, Klein was more explicit about it. Hamilton in Lectures on Quaternions (1853) realized that his representation of rotations of rigid bodies by the unit quaternions was not $1$-$1$, but $2$-$1$. Klein in Lectures on the Ikosahedron and the Solution of Equations
of the Fifth Degree (1888) replaced the unit quaternions by $2 × 2$ unitary matrices with the determinant $1$, now denoted $SU(2)$. He then more or less spelled out that the unit quaternions and $SU(2)$ are isomorphic groups, which are $2$-$1$ epimorphic onto the group of 3D rotations $SO(3)$.
Pauli proposed the "two-valuedness not describable classically", which was later identified with the electron spin, in 1924, and formalized it in the matrix form in 1927. In 1932 Heisenberg and Ivanenko guessed that the same effect regulates protons/neutrons as the states of a single particle, later dubbed nucleon, and incorporated it into their proton–neutron model of the nucleus.
Steiner cites this homomorphism as a prime example of "unreasonable effectiveness" of mathematics. Both times the mathematical machinery developed was not aimed, even indirectly, at the application it ended up being useful for. In the case of nucleus, any visible connection to rotations and 3D space is missing altogether.
answered 7 hours ago
ConifoldConifold
40.5k2 gold badges64 silver badges145 bronze badges
$endgroup$
add a comment
|
$begingroup$
Hamilton and Klein, Klein was more explicit about it. Hamilton in Lectures on Quaternions (1853) realized that his representation of rotations of rigid bodies by the unit quaternions was not $1$-$1$, but $2$-$1$. Klein in Lectures on the Ikosahedron and the Solution of Equations
of the Fifth Degree (1888) replaced the unit quaternions by $2 × 2$ unitary matrices with the determinant $1$, now denoted $SU(2)$. He then more or less spelled out that the unit quaternions and $SU(2)$ are isomorphic groups, which are $2$-$1$ epimorphic onto the group of 3D rotations $SO(3)$.
Pauli proposed the "two-valuedness not describable classically", which was later identified with the electron spin, in 1924, and formalized it in the matrix form in 1927. In 1932 Heisenberg and Ivanenko guessed that the same effect regulates protons/neutrons as the states of a single particle, later dubbed nucleon, and incorporated it into their proton–neutron model of the nucleus.
Steiner cites this homomorphism as a prime example of "unreasonable effectiveness" of mathematics. Both times the mathematical machinery developed was not aimed, even indirectly, at the application it ended up being useful for. In the case of nucleus, any visible connection to rotations and 3D space is missing altogether.
answered 7 hours ago
ConifoldConifold
40.5k2 gold badges64 silver badges145 bronze badges
$endgroup$
add a comment
|
$begingroup$
Hamilton and Klein, Klein was more explicit about it. Hamilton in Lectures on Quaternions (1853) realized that his representation of rotations of rigid bodies by the unit quaternions was not $1$-$1$, but $2$-$1$. Klein in Lectures on the Ikosahedron and the Solution of Equations
of the Fifth Degree (1888) replaced the unit quaternions by $2 × 2$ unitary matrices with the determinant $1$, now denoted $SU(2)$. He then more or less spelled out that the unit quaternions and $SU(2)$ are isomorphic groups, which are $2$-$1$ epimorphic onto the group of 3D rotations $SO(3)$.
Pauli proposed the "two-valuedness not describable classically", which was later identified with the electron spin, in 1924, and formalized it in the matrix form in 1927. In 1932 Heisenberg and Ivanenko guessed that the same effect regulates protons/neutrons as the states of a single particle, later dubbed nucleon, and incorporated it into their proton–neutron model of the nucleus.
Steiner cites this homomorphism as a prime example of "unreasonable effectiveness" of mathematics. Both times the mathematical machinery developed was not aimed, even indirectly, at the application it ended up being useful for. In the case of nucleus, any visible connection to rotations and 3D space is missing altogether.
answered 7 hours ago
ConifoldConifold
40.5k2 gold badges64 silver badges145 bronze badges
$endgroup$
Hamilton and Klein, Klein was more explicit about it. Hamilton in Lectures on Quaternions (1853) realized that his representation of rotations of rigid bodies by the unit quaternions was not $1$-$1$, but $2$-$1$. Klein in Lectures on the Ikosahedron and the Solution of Equations
of the Fifth Degree (1888) replaced the unit quaternions by $2 × 2$ unitary matrices with the determinant $1$, now denoted $SU(2)$. He then more or less spelled out that the unit quaternions and $SU(2)$ are isomorphic groups, which are $2$-$1$ epimorphic onto the group of 3D rotations $SO(3)$.
Pauli proposed the "two-valuedness not describable classically", which was later identified with the electron spin, in 1924, and formalized it in the matrix form in 1927. In 1932 Heisenberg and Ivanenko guessed that the same effect regulates protons/neutrons as the states of a single particle, later dubbed nucleon, and incorporated it into their proton–neutron model of the nucleus.
Steiner cites this homomorphism as a prime example of "unreasonable effectiveness" of mathematics. Both times the mathematical machinery developed was not aimed, even indirectly, at the application it ended up being useful for. In the case of nucleus, any visible connection to rotations and 3D space is missing altogether.
answered 7 hours ago
ConifoldConifold
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edited 19 mins ago
answered 7 hours ago
ConifoldConifold
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answered 7 hours ago
ConifoldConifold
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answered 7 hours ago
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In old books on classical mechanics parametrization of $SO(3)$ by $SU(2)$ is called the Klein parametrization.
$endgroup$
– Alexandre Eremenko
6 hours ago