How Many Distinct Invariants of the Poincaré Group

Shubee

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/r...eneralized.htm

Shubee

John Eristu

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@gmail.com> wrote in message
news:92bb58d3-8058-4c00-9b2e-d5e39d08cedc@59g2000hsb.googlegroups.com...
> 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/r...eneralized.htm
>
> Shubee
>

paulaireilly

On Apr 4, 12:44 pm, Shubee <e.Shu...@gmail.com> wrote:
> Poincaré lists 8 distinct but elementary invariants in his paper, ON
> THE DYNAMICS OF THE ELECTRON. See the equation number 5 and 7 inhttp://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?

Ten. It's a ten dimensional group - four translations, three rotations,
three boosts. For readers who aren't relativists, a 'boost' is a
translation in momentum space - a change in the 'velocity of the frame'
(or better, the rapidity.)

Boosts commute with rotations around the direction of the boost (a
'screw transformation') but not with other rotations. So boosts and
rotations mix, so take the 'types' of invariant listed below with a
grain of salt.

The invariants are: energy-momentum (four components, derived from
translations) Angular momentum (three components, derived from rotations
- but note that boosts and rotations mix) "Initial position of the
center of mass" (three components,derived from boosts, but see above)

>And how many can you derive?

You mean derive from the generators of the transformations? All of
them... I'm not sure if you're asking if they /can be/ derived, or if I
personally can derive them - English has an ambiguity here. Do you mean
you want to see the derivations from the infinitesimal generators?

Rock Brentwood

John Eristu wrote:
> I do not know if a group has invariants a priori.

Poincare' has 2 invariants.

The Lie algebra is given by
{J(u), J(v) + K(t) + P(b)} = J(uxv) + K(uxt) + P(uxb)
{K(s), K(t) + P(b)} = -(1/c)^2 J(sxt) + (s.b) H/c^2
{P(a), P(b)} = 0
{J(u) + K(s) + P(a), H} = P(s)
where components are denoted as J(u) = J.u = J_i u^i, for convenience.

The invariants are M^2 - P.P/c^2 and |MJ + P x K|^2 - (1/c)^2 (P.J)^2,
where M = H/c^2.

The non-linear (smooth) functional invariants are those residing in
the functional algebra C^{infinity}(L**), where L is the Lie algebra
of the Poincare' group and L* its dual and L** its double dual.

This is a Poisson manifold endowed with the bracket
{f,g} = sum e_c f^c_{ab} df/de_a dg/de_b
where (e_a: a=1,...,10) is the basis of L** and the structure
coefficients are given by [Y_a,Y_b] = f^c_{ab} Y_c, where (Y_a: a =
1,..,10) is the basis of L.

An invariant f is then whatever satisfies the relation {e_a, f} = 0
for all the e_a. This gives you a set of linear differential
equations. Solving them yields the invariants.

If you restrict focus to the subset of equations
{p(v), f} = 0; {h, f} = 0
where p(v) and h are the elements of L** corresponding respectively to
P(v) and H in L, then you obtain the functional subspace that commutes
with {p_1,p_2,p_3,h}. This is just functional subspace spanned by
{p_1,p_2,p_3,h} themselves, as well as the components of the vector w
= mj + pxk (m = h/c^2) and w_0 = p.j.

A Poincare' invariant, f, can therefore only be a function of p, h, w
and w_0. It's whatever functions of these 8 arguments that have 0
Poisson bracket with j(v) and k(s) (i.e., the elements of L** that
correspond respectively to J(v) and K(s)).

That's 2 sets of differential equations (3 each, 6 in all). They're
not hard to write out and solve. I essentially did the whole exercise
here a couple years back -- not only for Poincare' but for an entire
family of groups that simultaneously contains Poincare', Galilei and
(4-D) Euclid as subgroups.

Except for special representation families, the only solutions to the
equations {j(u},f} = 0 and {k(s),f} = 0 are |p|^2 - (h/c)^2 and |w|^2
- (w_0/c)^2.

See
The Wigner Classification for Galilei/Poincare'/Euclid
http://federation.g3z.com/Physics/in...eralizedWigner

The differential equations are listed there.