Galilean Algebra in the low velocity limit of Poincare Algebra (Weinberg vol 1)

In summary, Weinberg states that the J's constitute the SO(3) subgroup of Poincare which stays untouched, and the boosts will commute. He would say that K is of order 1/c, not 1/v.
  • #1
maverick280857
1,789
4
Hi,

Can someone please explain the following statement on page 62 of Weinberg's Vol 1 on QFT:

For a system of particles of typical mass m and typical velocity v, the (..) angular momentum operator is expected to be of order J ~ 1

(I understand the part for P ~ mv, so the "quote" is slightly distorted, intentionally).

Also how is

K of order 1/v
?

Thanks in advance!
 
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  • #2
Mmm, I would say that looks a little odd. The Galilean algebra can be obtained by a contraction of the Poincare algebra. In this contraction you scale, in the Poincare algebra, the spatial P_i and K_i by a factor epsilon. Then you calculate again the Lie algebra and take the limit epsilon --> 0. This epsilon can be seen as 1/c, and the contraction can be regarded as sending c to infinity.

The J's constitute the SO(3) subgroup of Poincare which stays untouched, and the boosts will commute. I would say that K is of order 1/c, not 1/v.
 
  • #3
haushofer said:
Mmm, I would say that looks a little odd. The Galilean algebra can be obtained by a contraction of the Poincare algebra. In this contraction you scale, in the Poincare algebra, the spatial P_i and K_i by a factor epsilon. Then you calculate again the Lie algebra and take the limit epsilon --> 0. This epsilon can be seen as 1/c, and the contraction can be regarded as sending c to infinity.

Are you referring to the so called Inonu-Wigner contraction here? (I don't know what it is, save a reference in Weinberg on the same page.)

The J's constitute the SO(3) subgroup of Poincare which stays untouched, and the boosts will commute. I would say that K is of order 1/c, not 1/v.

What about J being of order 1? How does that come about?
 
  • #4
maverick280857 said:
Are you referring to the so called Inonu-Wigner contraction here? (I don't know what it is, save a reference in Weinberg on the same page.)

Yes :) Would you ever find the tendency of wanting to know more about it in a pedagogical way: the notes of Robert Gilmore about Lie groups explain it quite well, in chapter 13.

What about J being of order 1? How does that come about?
Well, because in this particular contraction you leave the SO(3) untouched (because SO(3) is a subgroup of both the Poincare as the Galilei group, so you want to keep it), the generators J of SO(3) are not scaled by epsilon. I think that's what Weinberg means, but I'll check his statement.

But I assume you mean by J the [itex]J_i[/itex] which generate the rotations, not the [itex]J_{\mu\nu}[/itex], right?
 
  • #5
Ok, I have checked. My guess is that the 1/v of K is a typo, and should be read as 1/c. That would mean that K is of order [itex]\epsilon^{1}[/itex]. The J's are indeed the generators of SO(3), and because of the particular contraction you take they scale like [itex]\epsilon^{0}[/itex].

Otherwise it wouldn't make sense to me :)
 
  • #6
haushofer said:
Yes :) Would you ever find the tendency of wanting to know more about it in a pedagogical way: the notes of Robert Gilmore about Lie groups explain it quite well, in chapter 13.

Thanks. I'll check out the notes. Right now, I'm unable to reach his website.

But, Weinberg makes it look as if it is possible to get to the Galilean results 'by inspection' (and H = M + W).
 

1. What is Galilean Algebra and how does it relate to Poincare Algebra?

Galilean Algebra is a mathematical framework that describes the laws of motion in the Galilean spacetime, which is a simplified version of the more complex Poincare spacetime. Poincare Algebra is a mathematical framework that describes the symmetries of spacetime in special relativity. Galilean Algebra is considered to be the low velocity limit of Poincare Algebra because it is the approximation of Poincare Algebra at speeds much slower than the speed of light.

2. What are the fundamental principles of Galilean Algebra?

The fundamental principles of Galilean Algebra include the principles of Galilean relativity, which state that the laws of physics are the same in all inertial frames of reference, and Newton's laws of motion, which describe the relationship between an object's mass, velocity, and the forces acting upon it.

3. What are the key equations in Galilean Algebra?

The key equations in Galilean Algebra include the equations of motion, such as the equations for constant velocity and acceleration, and the equations for force and momentum.

4. What are the applications of Galilean Algebra?

Galilean Algebra is used in many areas of physics and engineering, particularly in classical mechanics, which deals with the motion of macroscopic objects at low speeds. It is also used in the study of fluid mechanics, thermodynamics, and electromagnetism.

5. How does Galilean Algebra differ from other mathematical frameworks?

Galilean Algebra differs from other mathematical frameworks, such as special relativity and general relativity, in that it does not take into account the effects of gravity and the finite speed of light. It is a simplified framework that is applicable only at low speeds, while other frameworks are more comprehensive and can describe a wider range of phenomena.

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