I always thought that the sum of angular momentum should be zero. I am wondering if this applies on a sub-atomic scale as well as a macroscopic scale.
Article Entanglement of arbitrary spin modes in expanding universe
spin geometry, spin representations etc are also commonly used in mathematics.
Here the term refers to the nontrivial representation of the spin group, the double covering group of the orthogonal group. i.e. they are projective representation of SO, or
spin representation of SO
I am talking about a physical phenomena. For istance a spinning fly wheel has angular momentum. Consider a universe that has just formed, There are two planets that both have spin. Before the formation of the two planets the angulr momentum was zero. After their formation the angular momentum will still be zero. This means one planet must have opposite spin to the other planet. If their raddii and masses and densities are equal then their rates of spin will be equal. If one is more massive than the other then the lighter will have to be spinning faster assuming they both have constant density and same radii. So if the angular momentum of the universe was zero prior to the big bang then it will have to be zero now. This means the sum of all the angular momentae in the entire universe will be zero. So this is a bit mathematical but it is referring to a physical phenomena.
Hello Donald,
A reasoning similar to yours implies that the total linear momentum of the Universe is zero. And furthermore, there should be a motionless center of mass of the Universe.
But how many technical details regarding these arguments can be worked out is not so clear. It may be worth a trial.
The term "spin" in Physics usually refers to angular momentum of atomic scale objects. Technically, for n >= 3 the special orthogonal group SO(n) is connected and has first homotopy group (also called fundamental group) isomorphic to Z_2, the integers modulo 2. Therefore its universal covering space, denoted Spin(n), is a well defined connected topological group (and a Lie group as well) with fibers consisting of two points, equivalently, it is a connected double cover.
If you consider instead the orthogonal group O(n), this is non-connected, has two connected components and the connected component of the identity is SO(n). The universal cover of O(n) is a topological space known as Pin(n), necessarily a non-connected double cover of O(n). But the group structure of Pin(n) is not unique. This subtlety is mentioned in http://en.wikipedia.org/wiki/Pin_group See references there.
In Classical Physics, the natural formalism for rotations in ordinary three dimensional space is based on SO(3), its tangent bundle, and the Lie algebra so(3) = tangent space to SO(3) at the identity. A good reference is the book Classical Mechanics by V. Arnold.
Then comes Quantum Mechanics and descendants. Schrödinger time dependent equation involves complex numbers. This forces the introduction of complex valued wave functions \psi. To go beyond the mathematical formalism of QM, the mathematical object \psi requires physical interpretation. The wave "amplitude" |\psi|^2 is interpreted "physically" as a probability distribution. This makes the "phase" disappear physically, because |\psi|^2 and |\exp(-i n t) \psi|^2 are one and the same physical state. But the phase is the natural way to consider rotations. Thus, rotations disappear in QM.
The quantistic way to recover something resembling rotation is to use Pauli matrices, equivalently, to use spin(3). I have been unable to make sense of these as rotations. Here the quantum dictum "Shut up and calculate" is acutely present. A nearby neon sign says "Abandon hope all ye who enter here".
In my opinion the unfortunate and mistaken choice of a unitary evolution equation ---that is, of Schrödinger evolution equation--- for the hydrogen atom made it impossible to understand microscopic rotational phenomena. Even worse, transitions themselves are contradicted by this equation.
On the other hand, Schrödinger eigenvalue equation is one of the most impressive wonders of science.
Most cordially and with best regards,
Daniel Crespin
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