What is the meaning of quantum spin axes and directions? I just completed watching all 24 lectures of the Quantum Mechanics course on "The Great Courses". The course was excellent overall, but I am a bit confused about a few things. For example, they use the terms "spin-up" and "spin-down", as well as spin axes x, y, and z. What does the "up" in "spin-up" mean? Does it mean "up" in the normal sense of the local gravitational field? If not, what? If two quantum particles are far apart and are both "spin-up", does that mean their spin axes are both in the same direction in inertial space? Nothing was mentioned in the course about this that I recall. Also, how are the x, y, and z axes defined? Are they common for all particles? In what sense?
 A: The directions are arbitrary, and depends entirely on the direction you define as "up" (or down, left, right etc.). If for example you chose the positive y-axis as being the "up" direction, then if you were to measure spin and find it to be pointing in the same direction as the positive y-axis, then the particle has a spin "up". If you measure it to be spinning in the direction of the negative y-axis, then you have measured it as having a spin "down".
These directions depend completely upon your choice of what direction is "up", i.e., on your choice of axes. It has nothing to do with the direction of gravity, but there is a natural inclination to chose “up” as pointing in the positive y-direction of the standard Cartesian axes. It’s intuitive and you will always find your lecturers doing the same.
If you were to pass electrons through a magnetic field so that they emerge with their spins pointing along a particular direction, you may define this direction as being "up" (or left, right etc.) and then detect particles who's spins point in different directions. Again, you define which directions are up or down, left and right etc.
