Can mass be defined using projective Hilbert space formalism instead of Hilbert space formalism?

Question

The origin of mass in quantum mechanics was clarified by Bargmann in his famous paper, On Unitary Ray Representations of Continuous Groups (Annals of Mathematics, Second Series, Vol. 59, No. 1 (Jan., 1954), pp. 1-46). Bargmann showed that mass is connected to the fact that a group homomorphism from the Galilei group $G$ into the automorphisms of the projective Hilbert space $PH$ generally cannot be lifted to a continuous homomorphism between $G$ and the unitary group $U(H)$, only up to a cocycle $\omega:G\times G\to U(1)$. This cocycle determines a central extension of $G$ by $U(1)$. Bargmann showed that the second cohomology class $H^2(G,U(1))$ is one-dimensional, that is, the equivalence classes of the central extensions of $G$ by $U(1)$ can be parametrized with real numbers. This real number is the mass of the quantum mechanical system.

It's interesting to me that mass enters into the description of a quantum mechanical system only when we switch from the automorphisms of the projective Hilbert space $PH$ to the unitary group $U(H)$ of $H$. However, theoretically, the projective Hilbert space carries all information about a quantum mechanical system. Except its mass? Then mass isn't a property of a quantum mechanical system? Is it a property only of our description (reflecting our convenience, instead of the properties of a quantum mechanical system)? Or could we define mass somehow also using a projective Hilbert space instead of a Hilbert space?

Answer 1

However, theoretically, the projective Hilbert space carries all information about a quantum mechanical system.

Well, it is not correct. This is just the standard situation when the physical system is not affected by superselection rules or gauge symmetries. In that case, the pure states are one-to-one with rays in the projective space. But this is not the general case.

More generally speaking, some physical information may be embodied in central charges and to see them you have to go in the Hilbert space.

The classical mass is a property of the quantum system and it cannot be defined in the pure projective space. It explicitly shows up when writing down the action of the continuous projective-unitary representation of the Galilean boost on some important observables, as the total momentum (even if the concrete representation of the latter depends on the representative chosen in the equivalence class).

In practice, consider an equivalence class $c$ of unitary-projective representation of the Galileo group. Pick out a continuous unitary projective rep $U\in c$ and consider, in that representation, the subrepresentation of spatial translations. Let $P^U_k$ be the selfadjoint generator of these translations along $x_k$: The momentum alnog $x_k$. That is an observable. In principle you may have different similar concrete momentum observables (all with the same spectrum and the same commutation relations with the other generators) depending on the choice of $U\in c$. However, in all cases, if $\{V_k^U(\chi)\}_{\chi \in \mathbb{R}}$ is the representation of the boost along $x_k$ contained in $U \in c$, $$V_k^U(\chi) P^U_k V_k^U(\chi)^\dagger = m_c\chi I + P_k^U\:.\tag{1}$$ The real number $m_c$ depends only on $c$ and not on $U$. $m$ turns out to be a central charge as soon as one passes from (1) to the infinitiesimal version of it.

The left-hand side of (1) is the momentum observable in a reference frame connected with the initial one by a boost transformation. Hence (1) can be experimentally tested, for instance comparing expectation values, in that sense, the value of $m$ can be experimentally grasped (it would be as killing a fly with a gun, though). This proves that the mass is not

a property only of our description (reflecting our convenience, instead of the properties of a quantum mechanical system)

As a final comment, I stress that a very similar situation is proper of Hamiltonian mechanics, where the Galileian group needs the information of the mass of the system in order to be implemented in terms of canonical transformations. Again it pops out in therms of a central charge.

The common basic reason of all that buisness is that the Galileo group does not include an information about "the mass of the system". The fact that the mass appears into an easy form (affine term) is physically necessary to implement the (classical) law on mass additivity.