Re: How Many Distinct Invariants of the Poincaré Group

I do not know if a group has invariants a priori. Only when you assume
that Physics is invariant under the transformations of a group, then the
group and the Physics yield invariants. Each observable corresponds to a
Unitary operator that represents the transformations for that operator.
For example if you assume Poincare Invariance in a four dimentional
space, then you get 10 operators. Four of them correspond to the four
translations, and the corresponding invariance is the conservation of
the four momentum. The other six operators are the six rotations in the
four space. These represent the angular momentum conservation, and some
boosts representing rotations around the time axis.

All the above is common knowledge.

Poincare group is a Lie Group. Interesting Physics might come out if we
discretize the Poincare group somehow. Weinberg in his Quantum Fields
Theory Volume 2 shows how the Lorentz Group can be mapped to a discrete
SL(2,C) group. He constructs four vectors from the 2x2 Mobius
Transformations in SL(2,C).

Lorentz group invariance is completely equivalent to Special Relativity.
Poincare group is larger.

"Shubee" <e.Shubee@xxxxxxxxx> wrote in message
news:92bb58d3-8058-4c00-9b2e-d5e39d08cedc@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Poincaré lists 8 distinct but elementary invariants in his paper, ON
THE DYNAMICS OF THE ELECTRON. See the equation number 5 and 7 in
http://www.univ-nancy2.fr/poincare/bhp/pdf/hp2007gg.pdf
How many invariants in special relativity are you aware of? How many
distinct invariants of the Poincaré group exist? And how many distinct
invariants of the Poincaré group can you derive?
This is how mathematicians measure the understanding of physicists in
spacetime.

I quote:

"Every geometry is defined by a group of transformations, and the goal
of every geometry is to study invariants of this group." Klein,
Erlanger Program.

"Each type of geometry is the study of the invariants of a group of
transformations; that is, the symmetry transformation of some chosen
space." Stewart and Golubitsky 1993, p. 44.

"A geometry is defined by a group of transformations, and investigates
everything that is invariant under the transformations of this given
group." Weyl 1952, p. 133.

"The geometry of Minkowski space is defined by the Poincaré group."
http://www.everythingimportant.org/relativity/generalized.htm

Shubee


.

Relevant Pages

  • Re: Are *observed* SR effects real?
    ... special relativity is the study of frame-dependent quantities. ... "Every geometry is defined by a group of transformations, ... of every geometry is to study invariants of this group." ...
    (sci.physics.relativity)
  • Re: The Essential History of Special Relativity
    ... transformations in the Cartesian system of coordinates ... particular choice of the Cartesian coordinates, ... theory of invariants, which has to do with the laws that ... equations of a straight line can be expressed be saying ...
    (sci.math)
  • Re: Are *observed* SR effects real?
    ... According to Klein, Weyl and Wikipedia, geometry is the study of ... Sorry, I was just talking about physics, not geometry. ... driven by the invariants of the geometry of spacetime. ... "Every geometry is defined by a group of transformations, ...
    (sci.physics.relativity)
  • Re: Relativity is not a theory about relative motion ?
    ... > Galilean relativity contains the same invariants as special relativity, ... > namely energy, momentum and angular momentum, plus one additional invar- ... Are you conflating invariant (under a group of transformations) with a ...
    (sci.physics.relativity)
  • Re: The Traditional Superficial Explanation of Relativity
    ... how 'beauty' anyone believes his theory is but it fails to explain data. ... lists 8 distinct but elementary invariants in his paper. ... "Every geometry is defined by a group of transformations, ...
    (sci.physics.relativity)