# Spin and Curvature of Space

## Main Question or Discussion Point

According to contemporary ideas the spin of elementary particle is a certain mysterious inner moment of impulse for which it is impossible a somewhat real physical picture to create. The absence of spin visual picture, in opinion of a number of authors leaves the regrettable gap in quantum mechanics interpretation. On the other hand, there are highly developed geometrical disciplines which are difficult to apply to specific physical theories owing to the fact that it is not always possible to point out the
objects to which the geometrical notions could be corresponded.We point out to one interesting analogy which, in our view testifies to the geometrical interpretation of spin.

Let's recall that according Pauli principle the two identical
particles with half-integral spin (fermions) cannot be simultaneously in the same quantum state. The alternative of Pauli principle maintains that in one and the same quantum state any number of particles (bosons) with integral spin could be found (infinitely much in the limit). Thus, the two similar fermions can't be found in the same space point. For bosons the situation is quite
different.

The remarkable fact: when in one case in one and the same place of space one can't put more than one particle and in the other-infinitely much, which gives a hint that spin has a some-what geometrical sense. To speak in images the spin in one case creates very "tight", and in the other case - very "spacious" space. Why so? To this question we cannot now give an answer which speaks for necessity to find an answer in the geometrical notions.

That's why we proceed to the geometry and study some facts reminding us the situation with fermions and bosons.It is well-known that besides the Euclidean geometry there are other geometrical systems (Lobachevsky, Riemannian geometry).
According to Klein's interpretation, which is based on the
projective geometry, the Euclidean, Lobachevsky and Riemannian geometry’s are in the unified scheme. The most known indication toidentify the latter two geometry is: in the Riemannian geometry(elliptical) across given point can't draw a straight line which couldn't cross the given straight line (analogy with the fermion)and in the Lobachevsky geometry (hyperbolic) across every point the infinite set of straight lines is passing, not intersecting with the given hyperbolic straight line (the analogy with bosons).

The analogy yet proves nothing. But in this case this is the fact that requires close consideration, study and discussion.The suspicion arises that spin is the sign of elementary particle pointing out to its non-Euclidean nature. May be the zero curvature of our space develops from total positive and negative curvaturesof spaces created by fermions and bosons? Not this is a key tounderstand "the space-time foam", idea which was put forward by Wheeler and Hawking? Couldn't this approach help to solve the cosmological problems?

Yuri Danoyan

Related High Energy, Nuclear, Particle Physics News on Phys.org
Fermions---antysymmetric wave function
Bosonы--symmetric wave function

Elliptic--pozitive curvature(symmetric)
Hyperbolic--negative curvature(antisymmetric)

Summary:

symmetry(mathematical)+antisymmetriy(physical)

antisymmetry( mathematical)+symmetry(physical)

Nfer x Mfer +NbosxMbos=0 ???
Nfer x{K>0}+Nbos x{k<0}=0 ???

The analogy yet proves nothing.
True that nothing is proven, but it is interesting.
the two similar fermions can't be found in the same space point.
When you say similar, you must include spin and momentum. Two electrons with spin up can be in the same space point if they have different momenta.
Let's recall that according Pauli principle the two identical
particles with half-integral spin (fermions) cannot be simultaneously in the same quantum state. The alternative of Pauli principle maintains that in one and the same quantum state any number of particles (bosons) with integral spin could be found (infinitely much in the limit).
The link between spin and statistics is more rigorously understood in quantum field theory.

Supersymmetry is probably best suited for you purposes. It is the only extension the ordinary space-time symmetries to include fermion-boson symmetry. In superspace, you may give a more rigorous meaning to your intuitions.

See Gravitation of W.Misner,K.Thorn,J.Wheeler
Part IV, par.20 Conservation Laws for 4-Momentum and Angular Momentum
about connection between spin and curvature

Supersymmetry is probably best suited for you purposes

Best for me extended supersymmetry for n=3, cut by half, without graviton: 1 3 3 1, and 3 1 vice versa.

2. Only 3 elementary particles are stable with a half-integer spin (proton, electron, neutrino) and 1 is stable with an integer spin (photon),

2. Only 3 elementary particles are stable with a half-integer spin (proton, electron, neutrino) and 1 is stable with an integer spin (photon),
But why do you include the proton in this list, which is not elementary ? A proton is made up of quarks. If you include the proton, then why don't you include the neutron ? Then why don't you include heavier nuclei !?

Nobody observed decay of proton.
Nobody observed free quarks.
Longevity of neutron only 17 min.
Heaviest nuclei not stable.

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If you aren't prepared to accept the composite nature of protons, how do you explain the plethora of experimental data since we started doing the so-called "deep inelastic scattering" of protons?

The composite nature of protons not mean non-stable.