Formalism and representation in Quantum Mechanics

Question

I am just curious about the formalism of basic Quantum Mechanics. Lets take for instance the system of a spin-$\frac{1}{2}$ particle. The state of the particle is described by a vector in an abstract Hilbert space that is two dimensional (say $\mathcal{H}$). The set of Endomorphisms on $\mathcal{H}$ form a group (which i hope will be the $SU(2)$ group). Now I will just define an abstract Endomorphic map in $\mathcal H$, such that $$ \hat\sigma_z : \left|+\right> \rightarrow \left|+\right> \qquad || \qquad \left|-\right> \rightarrow -\left|-\right>$$ where $\left|+\right>,\left|-\right> \in \mathcal H$

Clearly, the operator $\hat \sigma_z$ is Hermitian and the eigenvectors are orthonormal and hence can be chosen as a basis set. Hence any arbitrary vector can be expanded about this. $$ \left|\psi\right> = c_+\left|+\right> + c_-\left|-\right> \qquad ~~{where}~~\qquad \mathbf C \ni c_\pm = \left<\pm|\psi\right> $$ Now from what I have learnt so far, I sort of see that I can construct an a map called Representation $\mathcal R$ such all the elements for $\mathcal H$ gets mapped to $\mathbf C^2 $

$$ \mathcal R : \mathcal H \rightarrow \mathbf C^2 \qquad | \qquad \mathcal R\big(\left|\psi\right> \big) = \begin{pmatrix} c_+\\ c_-\\ \end{pmatrix} $$

This representation map preserves the inner product also I believe. For instance,

$$ \left<\phi|\psi\right> \rightarrow \begin{pmatrix} d_+ & d_-\\ \end{pmatrix} \begin{pmatrix} c_+\\ c_-\\ \end{pmatrix} \in \textbf C $$

Further the operators can also be mapped by this representation map, where the abstract operators get mapped to square matrices.

$$ \mathcal R : \text{End}(\mathcal H) \rightarrow \text{End}(\mathbf C^2) \quad|\quad \mathcal R(\hat A) = \begin{pmatrix} \left<+\right|\hat A\left|+\right> & \left<+\right|\hat A\left|-\right>\\ \left<-\right|\hat A\left|+\right> & \left<-\right|\hat A\left|-\right>\\ \end{pmatrix} $$

With this setup, the Pauli matrices and the vector's 2-D irrep all correspond this map $\mathcal R$ right ? So all those things correspond to a representation constructed using the eigen vectors of $\sigma_z$ ?

I also wish to know how would one make this kind of a connection in the cases of position basis, especially between $\left|x\right>$ and $L_2$ spaces.

PS: I know this question is of least use to any particular community of research or even people learning, but this is just out of my curiosity. Pardon me if this is a very ridiculous question.

Answer 1

For the relation between the abstract position basis and the $L^2$ spaces, I refer you to my answer here (read the other answers too, they're good ;) )

You are quite close with your understanding of the representations, but not quite there:

First of all, for the 2-dim spin-$\frac{1}{2}$ Hilbert space $\mathcal{H}_{\uparrow\downarrow}$, the set of endomorphisms $\mathrm{End}(\mathcal{H}_{\uparrow\downarrow})$ is not $\mathrm{SU}(2)$, but the whole of the 2D matrices, i.e. $\mathbb{C}^{2\times2}$. This is because every finite-dimensional Hilbert space of dimension $n$ is first and foremost a complex vector space, and all these are isomorphic to $\mathbb{C}^n$.

Now, a representation of a given group $G$ on any space $V$ is just a homomorphism $\rho : G \rightarrow \mathrm{Aut}(V)$. Since we have the inclusion $\mathrm{SU}(2) \subset \mathbb{C}^{2\times2}$, the space $\mathcal{H}_{\uparrow\downarrow}$ comes prequipped with a representation of $\mathrm{SU}(2)$. Since $\mathrm{SU}(2)$ is a Lie group, it has generators which lie in its Lie algebra, and every representation $\rho$ of the Lie group induces a representation $\mathrm{d}\rho : \mathrm{LieAlg}(G) \rightarrow \mathrm{End}(V)$ of the algebra (and vice versa, with a few caveats).

[Lie groups are amazing things, and very fundamental to theoretical physics, especially the understanding of symmetries. I advise you try to learn more things about them than I will say here.]

The three generators of $\mathrm{SU}(2)$ are canonically denoted $\sigma_{x},\sigma_y,\sigma_z$. You may now pick eigenvectors of (e.g.) $\mathrm{d}\rho(\mathrm{\sigma_z})$ on $\mathcal{H}_{\uparrow\downarrow}$ and use them as your basis. If you call these eigenvectors $|\pm\rangle$ (it is no accident that the eigenvalues of $\sigma_{x,y,z}$ in the fundamental representation (that is what this is) are $+1$ and $-1$), you have defined the same basis you have in your OP.

Of course, in this concrete example where the target space $\mathcal{H}_{\uparrow\downarrow}$ is just isomorphic to $\mathbb{C}^2$, the space on which $\mathrm{SU}(2)$ is natively defined, $\mathrm{d}\rho$ and $\rho$ are just identity (more precisely: inclusion) maps.

Now, everything else you called a representation in your question is just "ordinary" change of basis, this is what it means to choose the eigenvectors of $\sigma_z$ as a new basis for the vector space $\mathbb{C}^2$.

Feel free to ask for clarifications/additions if I have missed the point of your question or you didn't understand something.