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I'm trying to understand the $\ast$-algebra approach to QM and QFT, and so I have decided to first try to understand how this works in one of the simplest systems: a particle with spin 1/2.

This is usually one of the first examples in traditional QM books, after the postulates are stated, like Cohen's book.

So we need one algebra of observables. The idea is that we think of what are the observables needed for the theory. For a particle with spin 1/2 we need the observables $S_x,S_y,S_z$ satisfying angular momentum commutation relations:

$$[S_i,S_j]=i\hbar\epsilon_{ijk}S_k$$

It is well known though that these are the commutation relations of the Lie algebra $\mathfrak{su}(2)$. We have one $\ast$ operation in that algebra which is the usual

$$a^\ast=(\bar{a})^T,$$

where $\bar{}$ indicates complex conjugation.

So it seems that to deal with the spin 1/2 system in the algebraic approach, the appropriate $\ast$-algebra is $\mathscr{A}=\mathfrak{su}(2)$.

The next step QM books would take are: (1) ask about states of definite spin angular momentum and (2) predict probabilities for a general state.

Now immediately we see one problem. The $\mathfrak{su}(2)$ algebra is not tied to spin 1/2, but rather, to any angular momentum! Thus it already seems we have something weird: we have more than we would like.

Second, it is not clear on this approach how we would describe the states of definite $S_z$ for example. In the Hilbert space it is obvious: we have the states $|+\rangle,|-\rangle$.

On the algebraic approach it is far less clear. We should consider a state as a functional $\omega : \mathscr{A}\to \mathbb{C}$ with $\omega(1)=1$ and $\omega(a^\ast a)\geq 0$ and it is not clear how we would build one such $\omega$ associated to $|+\rangle$ and $|-\rangle$. Furthermore it is not clear how superposition works in this setting: a linear combination of states gives one statistical mixture, not a quantum superposition.

So this invites a few questions:

  1. What should we do about the fact that the algebra describes more than we expected (i.e., describes any spin, while our system is spin 1/2)? Does this end up being one "constraint on the physically allowed states", i.e., we have states compatible with the system and those which are not?

  2. How do we describe states of definite spin projection? Please, here lets forget that $\mathfrak{su}(2)$ naturaly acts on $2$-tuples of complex numbers, since this would be, IMHO, going back to the usual approach. I want to think with the algebraic framework here. I believe the idea is to set $\omega(S_x)=\omega(S_y)=0$ and $\omega(S_z)=1$, but I can't see how this works in practice.

  3. What about quantum superposition? If I want the analogue of a state $|\psi\rangle = c_1 |+\rangle + c_2 |-\rangle$ what I would do? It is obviously not $\omega = c_1 \omega_+ + c_2 + \omega_-$ - If I understood this is one statistical mixture and not even makes sense with $c_1,c_2\in \mathbb{C}$.

In summary, how can we set up the spin 1/2 system given as example in most QM books with the algebraic approach?

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  • $\begingroup$ I am preparing a detailed answer for this construction (for a general angular momentum $j$). I hope to finish it soon. $\endgroup$ – David Bar Moshe Mar 14 '18 at 10:10
  • $\begingroup$ Thanks for the help @DavidBarMoshe! I'm self studying AQFT on curved spacetimes, and it is being a bit difficult to get started (no one in my department knows the algebraic approach, so there's no one to discuss). In the last days I've been trying to carry out this example myself. I believe that for a system with spin $j$ we should find one state such that $\pi_\omega(S^2)=j(j+1)\hbar^2\mathbf{1}$. I believe then that a state for spin $j$ and another for spin $j'$ are disjoint (the representations seem totally different, starting from dimensionality). I hope I'm on the right track. $\endgroup$ – user1620696 Mar 15 '18 at 0:40
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First, your unital $^*$-algebra $A$ has to be defined as a unital algebra finitely generated by three self-adjoint elements $S_k$ satisfying the commutation relations of $su(2)$. ($A$ can be viewed as isomorphic to the complex universal enveloping algebra of the Lie algebra $su(2)$) As a consequence $S^2 := S^2_1+ S_2^2 + S_3^2$ belongs to the center of the algebra.

Here, you can distinguish among the possible values of the spin: the physical requirement for spin $s$ system is just $S^2 = \hbar^2 s(s+1)I$. Where $I$ is the unit element of the algebra.

Next, states are positive normalized linear functional $\omega : A \to \mathbb C$. The standard representations of QM are therefore provided by algebraic pure states, i.e., states $\omega : A \to \mathbb C$ whose GNS representation is irreducible.

As is well-known, these GNS representations $\pi_\omega(A)$ are (unitarily) isomorphic to $B(\mathbb C^{2s+1})$, the whole algebra of (bounded) linear operators over $\mathbb C^{2s+1}$, and, there, the cyclic vector $\Psi_\omega$ is any unit vector of $\mathbb C^{2s+1}$ and $\pi_\omega(S_k)$ are the standard self-adjoint matrices representing the spin components. In particular you can fix $\omega$ such that $\Psi_\omega$ coincides with $|s_z=1/2\rangle$, but every other choice of a unit vector of $\mathbb C^{2s+1}$ is also possible.

Coherent superpositions of states can be constructed as usual using pure states in the folium of a fixed pure state.

It is clear that the algebraic formalism is quite useless here...

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