Sorry, for late reply: busy at new job and also busy doing X-mass shopping with my family.
Okay, for obvious reasons, this is not going to be a detailed study about group cohomology, central extensions or projective representations. Rather, we will try to learn something about these concepts from equations (1) and (2) in #3. Basically, we will use (1) to learn one piece of information then jump to (2) and learn another piece before introducing the poor-man version of the underline mathematics.
I will start by setting up my notations for the Galilei group G. The action of G on space and time (\vec{x},t) is given by \vec{x} \to R \vec{x} + \vec{\beta} t + \vec{a} ,t \to t + \tau , where R \in SO(3). General group element g is, therefore, parametrized by ten parameters g = (\tau , \vec{a} , \vec{\beta} , R). The above transformations allow us to deduce the group law g_{1} g_{2} = g_{12} \in G to be of the form \tau_{12} = \tau_{1} + \tau_{2} , \vec{a}_{12} = \vec{a}_{1} + R_{1}\vec{a}_{2} + \vec{\beta}_{1}\tau_{2} ,\vec{\beta}_{12} = \vec{\beta}_{1} + R_{1}\vec{\beta}_{2} , R_{12} = R_{1}R_{2} . The identity element is e = (0 , \vec{0} , \vec{0} , 1), and the inverse element is given by g^{-1} = \left( - \tau , - R^{-1}( \vec{a} - \vec{\beta} \tau ), - R^{-1} \vec{\beta} , R^{-1} \right). We will mostly ignore rotations by setting R = 1. When no confusion arises, I will put x = (t , \vec{x}) and write the coordinate transformations as x’ = gx. The quantum operators corresponding to the conjugate pair (\vec{x}, \vec{p}) will be denoted by (\vec{X},\vec{P}). The group G has three abelian subgroups: time translations g(\tau) = (\tau , \vec{0} , \vec{0} , 1), space translations g(\vec{a}) = (0 , \vec{a} , \vec{0} , 1) and boosts g(\vec{\beta}) = (0 , \vec{0} , \vec{\beta} , 1). Notice, in particular, that translations and boosts are commuting subgroups: g(\vec{a}) g(\vec{\beta}) = g(\vec{\beta}) g(\vec{a}). This, however, will no longer be true in Galilean-covariant physical theories, i.e., non-relativistic physics. Indeed, we will see that the Lie algebra of the generators does not follow from the Lie algebra of the group G, there will be an extension characterized by the mass.
Recall that under a finite Galilean boost g = (0 , \vec{0} , -\vec{\beta} , 1), the free particle Lagrangian L = \frac{1}{2} m \dot{x}^{2} changes by a total time derivative L’ - L = - \frac{d}{dt} \omega_{1}(t,\vec{x};\vec{\beta}) , where \omega_{1} (x ; \vec{\beta}) \equiv m \left( \vec{x} \cdot \vec{\beta} - \frac{1}{2} \beta^{2} t \right) . \ \ \ \ \ \ \ \ \ (1) Actually, \omega_{1} has a name in mathematics: A quantity depending on the variable x and n group elements, \omega_{n}(x ; g_{1}, \cdots , g_{n}), is called n-cochain, and when certain combination of n-cochains vanish, we then call it n-cocycle. We will come to that later but not bother ourselves too much with the names. Okay, infinitesimally (i.e., \beta^{2} \approx 0) we have \delta_{(\beta)} x_{i} = - \beta_{i}t , \ \ \ \delta_{(\beta)}L = - \frac{d}{dt}(m \vec{\beta} \cdot \vec{x}) . \ \ \ \ \ \ (2) Now, on actual trajectories, any variation of the Lagrangian is given by total derivative \delta L = \frac{d}{dt} (\vec{p} \cdot \delta \vec{x}) . \ \ \ \ \ \ \ \ \ \ \ (3) From (2) and (3) we obtain the (conserved) infinitesimal boost generator \mathcal{C} = \vec{\beta}\cdot \vec{C} = \vec{\beta} \cdot (m \vec{x} - t \vec{p}) . For the quantum boost operator, we will write C_{i} = m X_{i} - t P_{i} . \ \ \ \ \ \ \ \ (4) Using the commutation relations [X_{i} , P_{j}] = i\hbar \delta_{ij} (or the classical Poisson brackets) we can easily obtain the following commutation relation: [C_{i}, C_{j}] = 0 which confirms the abelian nature of boosts; [C_{i} , H] = [C_{i}, P^{2}/2m] = iP_{i} confirming that C_{i} is constant of motion; [C_{i}, L_{j} ] = [C_{i} , (\vec{X} \times \vec{P})_{j} ] = i\hbar \epsilon_{ijk}C_{k} showing that C_{i} is a 3-vector; and more importantly [C_{i}, P_{j}] = m i\hbar \delta_{ij} saying that we are now dealing with the (centrally) extended Galilei group \bar{G}_{m} by \mathbb{R}. \bar{G}_{m} is an 11-parameter Lie group \bar{g} = (\tau , \vec{a} , \vec{\beta} , R , \alpha) with composition law given by that of the original Galilei group G plus \alpha_{12} = \alpha_{1} + \alpha_{2} + \omega_{2}(g_{1} , g_{2}) , where, as we will explain later, \omega_{2}(g_{1},g_{2}) = m(\vec{\beta}_{1} \cdot R_{1}\vec{a}_{2} - \frac{1}{2}\beta_{1}^{2}\tau_{2} ), is the non-trivial 2-cocycle which defines the extension.
In QM, we are naturally interested in the unitary representations of symmetry groups on Hilbert space \mathscr{H} of the states, i.e., in unitary operators U(g) that implement transformations x \to gx of the coordinates on the states and wave functions \Psi^{’} (x) = \langle x | \Psi^{’} \rangle = \langle x | U(g) | \Psi \rangle . So for our boosts g = (0 , \vec{0} , -\vec{\beta} , 1), the unitary operator, which effects the finite transformation on the wave functions, is given by U(\vec{\beta}) = e^{\frac{i}{\hbar} \mathcal{C}} = e^{\frac{i}{\hbar} \ \vec{\beta} \cdot (m\vec{X} - t \vec{P})} . \ \ \ \ (5) For ease of notations I will from now on drop the arrows from the scalar products of vectors, for example, the expression a \cdot P will mean \vec{a} \cdot \vec{P}. So, instead of (5) I will simply write U(\vec{\beta}) = e^{\frac{i}{\hbar} \ \beta \cdot (m X - t P)} . \ \ \ \ (5')
Applying the identity e^{A + B} = e^{-\frac{1}{2}[A,B]} e^{A}e^{B} , to the RHS of Eq(5'), we get U(\vec{\beta}) = e^{- \frac{i}{2\hbar} m t \beta^{2}} \ e^{\frac{i}{\hbar} m \beta \cdot X} \ e^{-\frac{i}{\hbar} t \beta \cdot P} . \ \ \ \ \ (5'') Now, using the eigen-value equation \langle \vec{x}|e^{\frac{i}{\hbar}m \beta \cdot X} = \langle \vec{x}|e^{\frac{i}{\hbar}m \beta \cdot x}, and the definition of the translation operator e^{-\frac{i}{\hbar} a \cdot P} |\vec{x}\rangle = |\vec{x} + \vec{a}\rangle \ \Rightarrow \ \langle \vec{x}| e^{\frac{i}{\hbar} a \cdot P} = \langle \vec{x} - \vec{a}|, we can easily evaluate the action of U(\beta) on wave functions \Psi (t , \vec{x}) from Eq(5'')
\langle \vec{x}|\Psi^{’}(t)\rangle = \langle \vec{x} | U(\beta)|\Psi (t)\rangle = e^{\frac{i}{\hbar} m ( \beta \cdot x - \frac{1}{2} \beta^{2} t )} \langle \vec{x} - \vec{\beta}t | \Psi (t)\rangle , \ \ \ \ (6) or \Psi^{’} (t , \vec{x}) = U(\beta) \Psi (t , \vec{x}) = e^{\frac{i}{\hbar} \omega_{1} (t , \vec{x} ; \vec{\beta})} \Psi (t , \vec{x} - \vec{\beta}t) . \ \ \ \ \ (6’) Thus, the finite transformation of wave function contains the phase (or “1-cochain”) \omega_{1}(x ; \beta) = m(\beta \cdot x - \frac{1}{2} \beta^{2}t), that we have already seen in the finite change of the Lagrangian L under boosts. Notice that the above transformation law agrees with the fact (which was stressed by Weyl during the early days of QM) that physical (pure) states are described rays \mathscr{R}, not by vectors of a Hilbert space \mathscr{H}: \{\mathscr{R}\} = \mathscr{H} / \sim, where \sim is the equivalence relation which identifies vectors |\Psi \rangle and |\Psi^{’} \rangle of \mathscr{H} which differ in a phase.
Of course, boosts from an abelian group, we have actually shown that [\beta_{1}\cdot C , \beta_{2} \cdot C] = 0. Thus, the composition law for two successive boosts is simply given by U(\vec{\beta}_{2})U(\vec{\beta}_{1}) = U(\vec{\beta}_{1} + \vec{\beta}_{2} ) \equiv U(\vec{\beta}_{12}) . From this it follows that \langle \vec{x}|U(\vec{\beta}_{2})U(\vec{\beta}_{1})|\Psi (t)\rangle = \langle \vec{x}|U(\vec{\beta}_{12})|\Psi (t)\rangle . \ \ \ \ \ \ \ \ \ \ \ (7) Let us now use the transformation law (6’) to evaluate both sides of (7). We have already calculated the RHS, it is \langle \vec{x}|U(\vec{\beta}_{12})|\Psi (t)\rangle = e^{\frac{i}{\hbar}\omega_{1}(x ; \vec{\beta}_{12})} \langle \vec{x} - \vec{\beta}_{12}t | \Psi (t)\rangle , or
\Psi (t , \vec{x} - \vec{\beta}_{12}t ) = e^{- \frac{i}{\hbar}\omega_{1}(x ; \vec{\beta}_{12})} \langle \vec{x}|U(\vec{\beta}_{12})|\Psi (t)\rangle . \ \ \ \ \ \ \ \ \ (8) On the LHS of (7), we do the followings
<br />
\begin{align*}<br />
\langle \vec{x}|U(\vec{\beta}_{2})U(\vec{\beta}_{1})|\Psi (t)\rangle &= \int d^{3}y \ \langle \vec{x}|U(\vec{\beta}_{2}) | \vec{y}\rangle \langle \vec{y}|U(\vec{\beta}_{1})|\Psi (t)\rangle \\<br />
&= \int d^{3}y \ \langle \vec{x}|U(\vec{\beta}_{2}) | \vec{y}\rangle e^{\frac{i}{\hbar}\omega_{1} (t , \vec{y} ; \vec{\beta}_{1})} \Psi (t , \vec{y} - \vec{\beta}_{1} t ) \\<br />
&= \int d^{3}y \ e^{-\frac{i}{2\hbar} m t \beta_{2}^{2}} \ \langle \vec{x}| e^{\frac{i}{\hbar} m \beta_{2} \cdot X} \ e^{- \frac{i}{\hbar} t \beta_{2} \cdot P} | \vec{y} \rangle e^{\frac{i}{\hbar}\omega_{1} ( y ; \vec{\beta}_{1})} \Psi (t , \vec{y} - \vec{\beta}_{1} t ) \\<br />
&= e^{\frac{i}{\hbar}\omega_{1}(x ; \vec{\beta}_{2})} \ \int d^{3}y \ \langle \vec{x}| e^{- \frac{i}{\hbar} t \beta_{2} \cdot P} | \vec{y} \rangle \ e^{\frac{i}{\hbar}\omega_{1} ( y ; \vec{\beta}_{1})} \Psi (t , \vec{y} - \vec{\beta}_{1} t ) \\<br />
&= e^{\frac{i}{\hbar}\omega_{1}(x ; \vec{\beta}_{2})} \ \int d^{3}y \ \delta^{3}(x - y - \beta_{2}t ) e^{\frac{i}{\hbar}\omega_{1} ( y ; \vec{\beta}_{1})} \Psi (t , \vec{y} - \vec{\beta}_{1} t ) \\<br />
&= e^{\frac{i}{\hbar}\left[ \omega_{1}(x ; \beta_{2}) + \omega_{1}(x - \beta_{2}t ; \beta_{1}) \right]} \ \Psi (t , \vec{x} - \vec{\beta}_{12} t ) .<br />
\end{align*}<br />
Substituting Eq(8) in the last equality, we obtain
<br />
\langle \vec{x}|U(\vec{\beta}_{2})U(\vec{\beta}_{1})|\Psi \rangle = e^{\frac{i}{\hbar}\left[ \omega_{1}(x ; \beta_{2}) + \omega_{1}(x - \beta_{2}t ; \beta_{1}) - \omega_{1}(x ; \beta_{12})\right]} \langle \vec{x}|U(\vec{\beta}_{12})|\Psi \rangle .<br />
Comparing this result with Eq(7), we obtain the following algebraic condition on the 1-cochain \omega_{1}(x;\beta)
\Delta \omega_{1} \equiv \omega_{1}(x - \beta_{2}t ; \beta_{1}) + \omega_{1}(x ; \beta_{2}) - \omega_{1}(x ; \beta_{12}) = 0 \ \mbox{mod} (2 \pi \hbar \mathbb{Z}) . \ \ \ \ (9) When the 1-cochain \omega_{1}(x;\beta) satisfies (9), it is called 1-cocycle. And \Delta is the so-called co-boundary operator (I will say few more things about them later). Of course it is trivial thing to check that our phase function \omega_{1}(x;\beta) = m \beta \cdot x - \frac{1}{2} m \beta^{2}t does satisfy the 1-cocycle condition (9), after all, our derivation of the condition (9) was based on the explicit form of \omega_{1}(x;\beta). However, notice that the 1-cocycle \omega_{1}(x;\beta) can identically be written as \omega_{1}(t, \vec{x} ; \beta) = m \beta \cdot x - \frac{1}{2} m \beta^{2}t \equiv -\frac{m}{2t}(\vec{x} - \vec{\beta}t)^{2} + \frac{m}{2t} (\vec{x})^{2} . If we now introduce the function (i.e., 0-cochain) \alpha_{0}(t,\vec{x}) = -\frac{m}{2t} (\vec{x})^{2}, we obtain \omega_{1}(t, \vec{x} ; \beta) = \alpha_{0}(t , \vec{x} - \vec{\beta}t) - \alpha_{0}(t , \vec{x}) \equiv \Delta \alpha_{0} . \ \ \ \ (10) In general, a n-cocycle \omega_{n} is called trivial (i.e., it can be removed) if it can be written as \Delta of a (n-1)-cochain. Thus, (10) means that our boosts 1-cocycle \omega_{1} is a trivial one, i.e., we can removed it by adjusting the phase of the wave function. Indeed, using (10) we can rewrite the transformation law (6’) in the form \left(e^{\frac{i}{\hbar}\alpha_{0}(x)}U(\beta) e^{-\frac{i}{\hbar}\alpha_{0}(x)}\right) \left(e^{\frac{i}{\hbar}\alpha_{0}(x)} \Psi (x) \right) = e^{\frac{i}{\hbar}\alpha_{0}(x - \beta t)} \Psi (x - \beta t) . Thus, by defining new wave function \Phi (t , \vec{x}) = e^{\frac{i}{\hbar}\alpha_{0}(t , \vec{x})} \Psi (t, \vec{x}) , \ \ \ \ \ \ \ \ \ (11) and a new unitary operator V(\beta) = e^{\frac{i}{\hbar}\alpha_{0}(x)}U(\beta) e^{-\frac{i}{\hbar}\alpha_{0}(x)} , \ \ \ \ (12) we find the following simple transformation and composition laws \Phi^{’} (t , \vec{x}) = V(\beta) \Phi (t , \vec{x}) = \Phi (t , \vec{x} - \vec{\beta}t) , V(\beta_{1}) V(\beta_{2}) = V(\beta_{1} + \beta_{2}) \equiv V(\beta_{12}) . The function \alpha_{0}(t,\vec{x}) also removes \omega_{1} from our original Lagrangian L = \frac{1}{2}m \dot{x}^{2}. Indeed, we know that adding \frac{d}{dt}\alpha_{0}(t , \vec{x}) to L produces a new equivalent Lagrangian \hat{L} except that now \hat{L} is invariant under boosts: \hat{L} = L + \frac{d\alpha_{0}}{dt} \ \Rightarrow \ \Delta \hat{L} = \Delta L + \frac{d}{dt} \Delta \alpha_{0} . But, we know \Delta L = - \frac{d}{dt}\omega_{1}, then (10) leads to \Delta \hat{L} = \frac{d}{dt}(-\omega_{1} + \omega_{1} ) = 0. Okay, let’s finish the talk about 1-cocycle by showing that the phase-freedom in QM corresponds to the freedom of adding a total time-derivative to the Lagrangian. In other words, we would like to prove that a phase change in a wave function corresponds to a canonical transformation which changes the Lagrangian by a total time-derivative. In the Schrodinger equation for \Psi (t,q), we make the substitution (11) i\hbar \frac{\partial}{\partial t}\left( e^{-\frac{i}{\hbar}\alpha_{0}(t , Q)} \Phi (t , q)\right) = \frac{P^{2}}{2m}\left( e^{-\frac{i}{\hbar}\alpha_{0}(t , Q)} \Phi (t , q)\right) . Notice that I replaced the variable q by the operator Q, this is possible because in the coordinate representation we can use f(Q) \Phi (q) = f(q) \Phi (q). So, by doing the trivial algebra, we find
<br />
\begin{align*}<br />
i\hbar \frac{\partial \Phi}{\partial t} &= \frac{1}{2m} \left( e^{\frac{i}{\hbar}\alpha_{0}(t,Q)} \ P \ e^{-\frac{i}{\hbar}\alpha_{0}(t,Q)} \right)^{2} \Phi - \frac{\partial \alpha_{0}}{\partial t} \Phi \\<br />
&= \frac{1}{2m} \left( P + \frac{i}{\hbar}[\alpha_{0}(t,Q) , P] \right)^{2} \Phi - \frac{\partial \alpha_{0}}{\partial t} \Phi \\<br />
&= \left[ \frac{1}{2m} \left( P - \frac{\partial \alpha_{0}}{\partial q} \right)^{2} - \frac{\partial \alpha_{0}}{\partial t} \right] \Phi .<br />
\end{align*}<br />
So, the Hamiltonian relevant to \Phi is \hat{H} = \frac{1}{2m} \left( P - \frac{\partial \alpha_{0}}{\partial q} \right)^{2} - \frac{\partial \alpha_{0}}{\partial t} , and the corresponding Lagrangian is \hat{L} = \frac{1}{2}\dot{q}^{2} + \frac{d}{dt}\alpha_{0}(t,q) = L + \frac{d}{dt}\alpha_{0} . The important thing to notice is the following: by removing the trivial 1-cocycle \omega_{1}, the resulting pair ( \Phi , \hat{L}) are scalars with respect to boosts only, i.e., new phase and new time-derivative will reappear in ( \Phi , \hat{L}) when we consider Galilean transformations other than boosts, or compositions of boosts with translations. This is connected with the fact that G has intrinsic projective representations.
I am sorry because I have consumed all the free time I had. I will talk about the appearance of 2-cocycle (projective representations) and 3-cocycle in QM sometime soon.
