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