Mdthbs

How can we represent a spin states
$ lvert S_x:+rangle, lvert S_y:+rangle,lvert S_z:+rangle ,lvert S_x:-rangle, lvert S_y:-rangle $ and $lvert S_z:-rangle$ in position representation like
$ langle xlvert S_z:+rangle$ ?

Is there exist any position representation for spin operators $ S_z, S_y, S_z $.

Zettili says in his book of Quantum mechanics that " Spin cannot be described by a differential operator", Does it mean spin cannot have a position representation? He does not tell about it more.

Unlike the angular momentum operators which inherit their position representation from the position representation of $hat x$ and $hat p$:
$$
hat L_zto -ihbar (ypartial_x -xpartial_y)
$$
it is not possible to write $hat S_z$ in terms of the classical position and momenta because spin is an "intrinsic" rather than spatial degree of freedom: dimensional analysis shows that only products like $yp_x$ have units of angular momentum so, after accounting for the commutation relations to be respected, a position representation of spin would be more or less identical to the position representation of "ordinary" angular momentum.

Moreover, quantizing angular momentum necessarily leads to integer values of $ell$ through the use of (spherical) coordinates
see:

Gatland, Ian R. "Integer versus half-integer angular momentum." American Journal of Physics 74, no. 3 (2006): 191-192.

Thus a position-based representation for spin could not accommodate half-integer values.

Indeed, there is no position space representation of the spin states. That is because its Hilbert space is simply different from the Hilbert space of something like particle in a box. A very crude classical example would be think of a seesaw hinged at a place. Its configuration is given by two possible configurations: left-side-up or right-side-up (for simplicity assume you can't see an intermediate position).

If your seesaw wasn't fixed at a place but can move around in the park, then you will have something that needs to be described by both its position in the park as well as whether left or right side is up. In that case you will still have position as an additional information and you really can't write the configuration in terms its position (it is not like the seesaw being right-side-up is same as it being in the right-hand corner of the park). Such a description is what you will call a direct product of position space with spin space. An example of that would be a gas of spinning particles (say, electrons).

Alternatively, you can have an array of seesaws in the park. In that case you have to tell me which seesaw you are talking about in the park (described by its location in the park) and what's it configuration (left/right-side-up). Once again you don't have a position space representation but the position now becomes a label for the seesaws. Equivalent of that would be something like Heisenberg model where you have spins at different sites of a lattice.

I'll try to give a simple and rough answer.

There is no position representation for the spin.

Roughly, that is because spin is a different space. Spins are things that live in a different space, which is independent of Hilbert's space.

On the one hand, you've got wavefunctions, which live in Hilbert's space.

$psi(x) in mathcal{H}$

You can use Dirac notation: $|psi rangle$.

On the other hand, spins are objects living in $mathbb{C}^n$. With $s=½$, we've got $n=2$, so ½-spins live in $mathbb{C}^2$. We use a basis that is

$$lbrace left( begin{array}{c} 1 \ 0 end{array} right); left( begin{array}{c} 0 \ 1 end{array} right) rbrace$$

The first one corresponds to spin-up and the second one corresponds to spin down.

So, the full wavefunction is the tensor product of those tow things, that is

$psi(x)otimes left( begin{array}{c} a \ b end{array} right) $

The symbol $otimes$ is usually ommited. So basically, a spinor is

$$left( begin{array}{c} phi(x) \ varphi(x) end{array} right)$$

and it is the result of doign the tensor product of a wavefunction and a spin vector.

Only things in Hilbert space can have a position representation.

On the other hand, spins belong to an independent complex space $mathbb{C}^n$.

Maybe the OP is asking for a position-like representation of a particle's intrinsic spin, with the understanding that this is distinct from the particle's "orbital" angular momentum. Even with this understanding of the question, the answer is still no, as stated in the previously-posted answers.

Here, I'll approach the question from a slightly different perspective to highlight what goes wrong when we try to construct a position-like representation.

To have a position-like representation for spin, the wavefunction would need to depend on two coordinates, because we need two coordinates to specify the orientation of an axis in 3-d space. Specifically, the wavefunction would need to be defined on the surface of a sphere (say the unit sphere). Using a redundant system of coordinates $x,y,z$ subject to the constraint $x^2+y^2+z^2=1$, the generator of rotations about the $z$-axis can be written
$$
L_z = i(xpartial_y - ypartial_z),
$$
and the generators $L_x$ and $L_y$ can be written down by following the obvious pattern. With the natural rotation-invariant inner product, these operators are hermitian. They are the observables representing the components of the particle's intrinsic angular momentum about the canonical axes.

(Again, the $x,y,z$ coordinates here are not the coordinate's of the particle's position. This is an auxiliary coordinate system that we are using only to describe the orientation of the particle's intrinsic spin. This is what I mean by a position-like representation.)

It might look like we've succeeded in constructing a position-like representation for the spin operators, but we really haven't, because the thing we constructed doesn't constitute an irreducible representation of the rotation group. Instead, it includes an infinite list of irreducible representations of the rotation group, one for each integer value of the "total spin." Each of these represents a different species of particle, so we have failed to achieve the goal: we wanted a position-like representation for the spin of a given species of particle, say a particle with spin-$1$. Any such representation is finite-dimensional (three-dimensional in the spin-$1$ case, five-dimensional in the spin-$2$ case, and so on), and a position-like representation cannot be finite-dimensional.

If the goal was to construct a position-like representation of a spin-$1/2$ particle, then our failure is even more drastic, because the thing we constructed only contains integer-spin representations. as emphasized in ZeroTheHero's answer.

Here's the fifth answer, in a more mathematical fashion.

Using the standard terminology, for a massive non-specially relativistic particle with spin in three dimensions, the set of operators ${x,y,z,p_x,p_y,p_z, S_x,S_y, S_z}$ is irreducible, which means there is no linear or non linear functional dependence between them. This is a consequence of the representation theory of the Galilei algebra. We are then forced to the algebraic commutation relations which derive from the representation theory of the Galilei algebra (ignore i and hbar):

$$ [x_i,p_j] = delta_{ij}, [x_i, S_j] =0, [p_k, S_l] =0, [S_m,S_n]=epsilon_{mnq}S_q .$$

Let us call the vector space of this representation $ V$.
That $x$ and $p_x$ are irreducible and do not commute, it means that there are conditions to apply the famous uniqueness theorem of Stone and von Neumann to conclude that $ V$ must have $L^2 (mathbb R^3)$ as a subspace, so it is necessarily infinite-dimensional.

Let us now consider that the spin operators S form a representation of $text{so}(3)$ on $V$ by (essentially) self-adjoint operators. We know that $text{so}(3)$ is the compact Lie algebra of the compact group $text{SO(3)}$, so, by the theorem of Peter-Weyl, the representation space of the commutation relation must be finite-dimensional.
We are to conclude that the commutation relation for the spin operators should take place on a finite-dimensional subspace of $V$, we call $mathcal S_{fin}$. Of course, $mathcal S_{fin}$ is isomorphic as a vector space to $mathbb {C}^n$ for a particular finite $n$.

There are two options. The first:

$$ V = L^2 (mathbb R^3) otimes mathcal S_{fin},$$

which basically answers the question (the spin states do not have a position representation), because they live in different (rigged) Hilbert spaces and, by convention/definition, the scalar product between a spin state and a position (eigen)state is nil. The same applies to momentum and spin.

The second option would be $$ V= L^2 (mathbb R^3), mathcal S_{fin} < L^2 (mathbb R^3). $$

One can easily show that the three essentially self-adjoint operators $epsilon_{ijk}x_j p_k$ satisfy the $text{so(3)}$ commutation relations on $L^2 (S_2) <L^2 (mathbb R^3)$ whose dimension is necessarily an even integer. We have that, due to a coordinate change implemented unitarily, $L^2 (mathbb R^3) = L^2 (S^2) otimes L^2 ((0,infty), r^2 dr) $. The dimension of $mathcal S_{fin}$, however, can be any natural number. So, for integer spins, we have that the representation spaces of the same algebra are isomorphic. It follows that the generators are one the multiple of the other, hence the position, momentum, spin cannot be irreducible anymore.

The existence of spin comes purely from enforcing a projective unitary representation of the Galilei group on a (rigged) Hilbert space. Group theory shows that we are actually representing the universal covering group of which has a subgroup isomorphic to $text{SU(2)}$. Had we not been forced to use projection representations, but only vector ones, we would have had no way to exhibit a spin operator. The same goes for the specially relativistic case.