# Connection between the generators of the Galilean group and physical quantities

1. Jan 15, 2012

### kith

How can I see that the generators of the Galilean group correspond to energy, momentum, etc.?

References which cover the Galilean group and algebra as well as their realization in phase space are appreciated, especially if they are not too sophisticated.

Thanks
kith

2. Jan 16, 2012

### kith

Do I have to post this in the Quantum Physics Forum? ;-)

3. Jan 17, 2012

### vanhees71

I think this is a quite general topic and not only interesting in the context of quantum theory. In quantum theory, however, this approach becomes very important to define observables.

The basic idea behind the group theoretical approach is to start from a symmetry principle. Most fundamental are the symmetries of the space-time description. In the case of classical Newtonian physics the space-time structure is determined by the proper orthochronous Galilei group, which is a semidirect product of translations in space and time, rotations in space, and boosts, making up the 10-dimensional Lie group of space-time symmetries.

Each one-parameter subgroup leads, according to Noether's theorems, to a conserved quantity for closed systems. Thus we have 10 conserved quantities for any closed system, i.e., energy (time translations), 3 momentum components (spatial translations), 3 angular momentum components (spatial rotations) and the three coordinates of the center of mass (boosts).

In analyzing the properties of the Galilei group in terms of its Lie algebra, you find a little subtlety: There is a non-trivial central charge, the total mass of the system, which then can be interpreted as an 11th conservation law for mass. This is important for quantum theory, since there only the corresponding central extension of the Galilei group's covering group (substituting the rotation group SO(3) by its covering group, SU(2)) gives a physically meaningful dynamics of quantum systems.

All this is pretty nicely written up in

Ballentine, Quantum Mechanics, World Scientific 1998.

4. Jan 17, 2012

### kith

Thanks vanhees71! The corresponding chapter in Ballentine's book seems very interesting.

However, I wonder if there aren't any texts about classical mechanics which cover this. Phase space, Poisson brackets, Hamiltonian mechanics, Galilei transformations, Noether's theorem, etc. are treated in almost all textbooks. But I haven't seen any, where we start with a realization of the Galilean group acting on phase space, and derive Poisson brackets for the generators as well as their physical significance for example systems from that. This seems like the natural setting to derive Noether's theorem in. And it's propably the line of reasoning, Ballentine uses, too.

Well, maybe some textbooks about geometrical mechanics do a similar thing. But they seem very mathematical to me, while the basic approach, as just outlined, sounds quite simple.

Last edited: Jan 17, 2012
5. Jan 17, 2012

### vanhees71

I've tried something like this in a manuscript on classical mechanics. It's in German, but perhaps you get some idea, because the "formula density" is quite high :-).

http://theory.gsi.de/~vanhees/faq/mech/node47.html

6. Jan 18, 2012

### kith

Thanks again! This looks very promising. It will probably take a couple of days for me to read it.

7. Jan 18, 2012

### DrDu

Galilean symmetry has been studied intensively Jean-Marc Levy-Leblond

one citation which considers more the classical mechanics point of view is:

Reference Type: Journal Article
Author: Lévy-Leblond, Jean-Marc
Primary Title: Group-theoretical foundations of classical mechanics: The Lagrangian gauge problem
Journal Name: Communications in Mathematical Physics
Cover Date: 1969-03-01
Publisher: Springer Berlin / Heidelberg
Issn: 0010-3616
Subject: Physik und Astronomie
Start Page: 64
End Page: 79
Volume: 12
Issue: 1
Url: http://dx.doi.org/10.1007/BF01646436
Doi: 10.1007/BF01646436
Abstract: This paper is devoted to the study of the classical, single and free particle...