Poincare group vs Galilean group

Question

• One can define the Poincare group as the group of isometries of the Minkowski space. Is its Lie algebra given either by the equations 2.4.12 to 2.4.14 (as also given in this page - https://en.wikipedia.org/wiki/Poincar%C3%A9_group) or equations 2.4.18 to 2.4.24 of Weinberg's volume 1 of his QFT books?

  What confuses me is that in deriving the commutation relations between $J^{\mu \nu}$ and $P^\mu$ he did use quantum theoretic arguments about the Hilbert space operator $U$ but I guess there is nothing quantum about the Lie algebra he derives in the aforementioned equations. Is that right?

• This quantum confusion steepens when one looks at the $K_i$ (..relativistic boost along the $i^{th}$ spatial direction..), $P_j$ (..linear momentum along the $j^{th}$ spatial direction..) commutator being non-zero. This is justified by saying that the exponential action of the boosts and the translations on the Hilbert space states do not commute and that is being reflected here. (..they pick up an extra phase proportional to the mass and the dot product of the boost velocity and the displacement vector..)

But if the afore mentioned equations are really the Lie algebra of the isometry group of the Minkowski spacetime then in the Galilean limit shouldn't they be instead reflecting the fact that Galilean boosts and translations when acting on the spacetime coordinates do infact commute? But the $K_i$ and $P_j$ commutation continues to be non-zero even when the Galilean limit is taken on page 62.

This makes me strongly suspicious that the equations 2.4.12 to 2.4.14 are not the Lie algebra of the isometry group of the Minkowski spacetime but are the Lie algebra of the group whose elements are $U(\Lambda, a)$ (..using Weinberg's notation..) …right?

• So is the "low velocity" limit taken on page 62 recovering non-relativistic quantum theory ? (and not Newtonian physics)

• On page 89 of the same book he derives the topology of the inhomogeneous Lorentz group as being $R^4 \times R^3 \times S^3/Z_2$. Since this is a connected manifold, I guess that by the term ``inhomogeneous Lorentz group" he is meaning only the proper orthochronous component of the full relativistic symmetry group. right?

• I can't see how the above topology matches with the semidirect product structure for posibly the same thing as given on this Wikipedia page -http://en.wikipedia.org/wiki/Poincaré_group?

When people talk of the Poincare group is the full symmetry group of relativity is what is being referred to or is it just its proper orthochronous component (and not the other 3 components) ?

• I am familiar with the notion of "central charge" as in the "first" term on the RHS of the TT OPE of CFT… what also has the interpretation as the zero-point energy when doing the plane-minus-point<->cylinder conformal transformation.

In this light it is not clear to me as to what is meant when one says that one can add in the "mass" as a central charge to the Galilean group... an extra generator which commutes with all the rest so that with this "central extension" the free particles will lie in the unitary representations of the Galilean group rather than the projective representations before the extension.

I would be grateful if someone can shed light on this issue and help reconcile the two "different" notions of central charge.

• When one takes a "low velocity" limit of the Poincare algebra to get the Galilean algebra then is one taking just a non-reltivistic limit or is one also taking a non-quantum limit?

(... I guess this will depend on my first query about whether what Weinberg calls as the Poincare algebra in the quoted equations has any quantum effect encoded in it {as it seems to be!} or is it just the Lie algebra of the isometry group of the Minkowski spacetime...)

Answer 1

It is important to distinguish between three group actions that are named "Galilean":

-The Galilean transformation group of the Eucledian space (as an automorphism group).

-The Galilean transformation group of the classical phase space (whose Lie algebra constitute a Lie subalgebra of the Poisson algebra of the phase space). This is the classical action.

-The Galilean transformations of the wavefunctions (which are infinite dimensional irreducible representations). This is the quantum action.

Only the first group action is free from the central extension. Both classical and quantum actions include the central extension (which is sometimes called the Bargmann group). Thus, the central extension is not purely quantum mechanical, however, it is true that most textbooks describe the central extension for the quantum case. I'll explain first the quantum case, then I'll return to the classical case and compare oth cases to the Poincare group.

In quantum mechanics, a wavefunction in general is not a function on the configuration manifold, but rather a section of a complex line bundle over the phase space. In general the lift of a symmetry (an automorphism of the phase space) is an automorphism of the line bundle which is therefore a $\mathbb{C}$ extension of the automorphism of the base space. In the case of a unitary symmetry, this will be a $U(1)$ extension. Sometimes, this extension is trivial as in the case of the Poincare group. Now, the central extensions of a Lie group $G$ are classified by the group cohomology group $H^2(G, U(1))$. In general, it is not trivial to compute these cohomology groups, but the case of the Galilean and Poincare groups can be heuristically understood as follows:

The application of the Galilean group action $\dot{q} \rightarrow \dot{q}+v$ to the non-relativistic action of a free particle: $S = \int_{t_1}^{t_2}\frac{m }{2}\dot{q}^2dt$, produces a total derivative leading to $S \rightarrow S + \frac{m}{2}v^2(t_2-t_1) + mvq(t_2) - mvq(t_1)$: Now Since the propagator $G(t_1, t_2)$ transforms as $ exp(\frac{iS}{\hbar})$ and the inner product $\psi(t_1)^{\dagger} G(t_1, t_2) \psi(t_2)$ must be invariant, we get that the wavefunction must transform as:

$\psi(t,q) \rightarrow exp(\frac{i m}{2\hbar}(v^2 t+2vq) \psi(t,q) $

Now, no application of a smooth canonical transformation can romove the total derivative from the transformation law of the action, this is the indication that the central extension is non-trivial.

The case of the Poincare group is trivial. The relativistic free particle action is invariant under the action of the Poincare group, thus the transformation of the wavefunction doen not acquire additional phases and the group extension is trivial.

Classically, the phase space is $T^{*}R^3$ and the action of the boosts on the momenta is given by: $p \rightarrow p + mv$, thus the generators of the boosts must have the form $K = mvq$, then the action is easily obtained using the Poisson brackets{q, p} = 1, and the Poisson bracket of a Boost and a translation is non-trivial {K, p} = m.

The reason that the Lie algebra action acquires the central extension in the classical case is that the action is Hamiltonian, thus realized by Hamiltonian vector fields and vector fields do not commute in general.

The Iwasawa decomposition of the Lorentz group provides the answer to your second question:

$SO^{+}(3,1) = SO(3) A N$ where $A$ is generated by the Boost $M_{01}$ and $N$ is the Abelian group generated by $M_{0j}+M_{1j}$, $j>1$. Now both subgroups $A$ and $N$ are homeomorphic as manifolds to $R$ and $R^2$ respectively.

To your third question: The limiting process which produces the Galilean group from the Poincare group is called the Wingne-Inonu contraction. This contraction produces the non-relativistic limit. Its relation to quantum mechanics is that there is a notion of contarction of Lie groups unitary representations, however not a trivial one.

Update

In classical mechanics, observables are expressed as functions on the phase space. see for example chapter 3 of Ballentine's book for the explicit classical realization of the generators of the Galilean group.

This is a case where the full geometric quantization recepie can be carried out. See the following two articles for a review. (The full proof appears in page 95 of the second article. The technical computations are more readable in pages 8-9 of the first article).

The central extensions appear in the process of prequantization.

• First please notice that the Hamiltonian vector fields $X_f$ corresponding to the Galilean Lie algebra generators close to the non-centrally extended algebra, (because, the hamiltonian vector field of constant functions vanish).

• However, the prequantized operators

$\hat{f} = f - i\hbar(X_f - \frac{i}{\hbar}i_{X_f} \theta)$, ($\theta$ is a symplectic potential whose exterior derivative equals the symplectic form) close to the centrally extended algebra because their action is isomorphic to the action of the Poisson algebra.

The prequantized operators are used as operators over the Hilbert space of the square integrble polarized sections, thus they provide a quantum realization of the centrallly extended Lie algebra.

Regarding your second question, the Wingne-Inonu contraction acts on the level of the abstract Lie algebra and not for its specific realizations. A given realization is termed "Quantum", if it refers to a realization on a Hilbert space (in contrast to realization by means of Poisson brackets, which is the classical one).

Answer 2

You are still asking way too many questions at once. So again, consider splitting them next time. I will address only the topology part here.

As topological spaces we have $$SO(3) = {\mathbb R \mathbb P}^3 = S^3 /\sim ,$$ $$SO(4) = S^3 \times S^3 / \sim, $$ $$SO^+(1,3) = {\mathbb R}^3 \times S^3 / \sim $$ (in all of these cases $\sim$ is an identification of the antipodal points on the relevant spheres). One can get these results by noting that the double cover of $SO(3)$ is $SU(2)$, double cover of $SO(4)$ is $SU(2) \times SU(2)$ and the double cover of $SO^+(1,3)$ is $SL(2, {\mathbb C})$. (Note that the above factorizations can be also written as $X/({\mathbb Z} / 2{\mathbb Z})$ which is a factorization of a space by the action of the group on the space; in this case the action is sending a point to its antipodal point).

Adding translations only means using semidirect product on the level of groups, or direct product on the level of topological spaces. In total, we have that the connected component of the Poincaré group is ${\mathbb R}^4 \times {\mathbb R}^3 \times S^3 / \sim$. As for the wikipedia page, I am not sure about your confusion. It defines Poincaré group as ${\mathbb R}^4 \rtimes O(1,3)$ and topologically this is disjoint union of four copies of ${\mathbb R}^4 \times {\mathbb R}^3 \times S^3 / \sim$.