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I'm reading an introductory review on quantum walks and at some point it incorporates spin into the translation operator in a way that I don't follow.

Initially it states that the translation by distance $l$ is defined as

$$ U_l|\psi_x\rangle=|\psi_{x-l}\rangle $$

$$ U_l=\exp(-iPl) $$

where $|\psi_x\rangle$ is the position wavefunction and $P$ is the momentum operator.

The discussion then shifts to an object with a position and spin parts to its wavefunction written as $|\Psi\rangle=\alpha^\uparrow|\uparrow\rangle\otimes|\psi_x\rangle+\alpha^\downarrow|\downarrow\rangle\otimes|\psi_x\rangle$. The article then goes on to say that the translation of this object is now described by

$$ U_l=\exp(-2iS_z\otimes Pl) $$

meaning that

$$ U_l|\uparrow\rangle\otimes|\psi_x\rangle=|\uparrow\rangle\otimes|\psi_{x-l}\rangle $$ $$ U_l|\downarrow\rangle\otimes|\psi_x\rangle=|\downarrow\rangle\otimes|\psi_{x+l}\rangle. $$

Now I understand why this new operator behaves the way it does but I don't get why it's used in the first place. Why has the $S_z$ operator suddenly become part of the generator of translation? Why does spin affect this at all? Also where did that factor of 2 come from in the exponent?

This is the review article: http://arxiv.org/pdf/quant-ph/0303081v1.pdf

The relevant section begins on page 2.

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Why has the $S_z$ operator suddenly become part of the generator of translation?

Because the authors want their $U$ operation to make the spin and the position interact. The process that moved the $S_z$ into the generator wasn't anything physical or simulated, it was just the authors defining a useful operation.

Also where did that factor of 2 come from in the exponent?

Think of it as the strength or duration of the interaction. It's the $t$ in $U(t) = e^{-itH}$ that you end up computing when converting from a Hamiltonian to a unitary matrix (more generally, see Schrödinger equation).

I don't know why the authers picked 2 specifically, but there's not anything stopping them from doing it. Maybe it's just to undo the factor of $\frac{1}{2}$ they introduced in the definition of $S_z$?

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