A Explicit construction of Galilean-invariant space

Adgorn
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I'm trying to explicitly find a projective unitary scalar representation of the Galilean group. I'll denote a generic element of the group by ##(a, {\bf b},R, {\bf v})##, corresponding respectively to time translation, space translation, rotation and boosts. In a representation with central charge ##M##, the commutation relations of the boost and space translation genrations are:

$$\tag{1} [K_i,P_j]=-iM\delta_{ij}.$$

The candidate I'm considering is the most obvious one. Take the space of square integrable functions on ##\mathbb R^3## (momentum space) and give it the action:

$$\tag {2}(a, {\bf b},R, {\bf v})\phi({\bf p})=e^{i[-a(\frac {p^2} {2M}+E_0)+{\bf b}\cdot{\bf p}]}\phi(R{\bf p}+M{\bf v}).$$

Another candidate which I saw in a paper was the inverse transformation:

$$\tag {3}(a, {\bf b},R, {\bf v})\phi({\bf p})=e^{i(aE-{\bf b}\cdot{\bf p})}\phi(R^{-1}({\bf p}-M{\bf v})).$$

At any rate, I'm having a hard time reconciling any option for the phase with ##(1)##. The Baker-Campbell-Hausdorff formula along with ##(1)## implies that for a composition of a translation and a boost:

$$e^{-i{\bf v} \cdot {\bf K}}e^{-i{\bf b} \cdot {\bf P}}=e^{iM\frac {{\bf v}\cdot {\bf b}} 2}e^{-i({\bf v} \cdot {\bf K}+{\bf b} \cdot {\bf P})}.$$

But if I actually compose these operations in either ##(2)##or ##(3)##, I don't get the ##\frac 1 2## factor. So are these really projective representations? If not, what choice of phase factor is appropriate?

The papers I'm refering to are:

1.https://doi.org/10.1007/BF01646020

2.https://doi.org/10.1007/BF01645427
 
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Adgorn said:
[...]

The papers I'm refering to are:

1.https://doi.org/10.1007/BF01646020

2.https://doi.org/10.1007/BF01645427

These are classical papers. I believe CMP was freely available on the NumDam server.

1. Lévy-Leblond, J.-M. (1967). Nonrelativistic particles and wave equations. Communications in Mathematical Physics, 6(4), 286–311. doi:10.1007/bf01646020

2. Lévy-Leblond, J.-M. (1967). Galilean quantum field theories and a ghostless Lee model. Communications in Mathematical Physics, 4(3), 157–176. doi:10.1007/bf01645427
 
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