# What's the difference between spinor and spin?

Some related information might be found here: What is the difference between a spinor and a vector or a tensor? and Wikipedia seemed to have an explanation but was not very clear.

From what I read, in Dirac equation, spinor seemed to be a method of factorize the coefficient of wave function, while spin was the component of the spinor?

Could you tell me what's the difference between spinor and spin?

• For simplicity, one may consider $\Psi=(\psi_L,\psi_R)$ for massless Dirac in the simplest case. But a general discussion is very welcomed. Nov 24 '19 at 21:33
• A spinor is a representation of the Lorentz group. Spin is one of the labels of a representation of the little group of massive particles, to wit, the rotation group (cf. this PSE post). Do these words mean anything to you?4 Nov 24 '19 at 21:38
• @AccidentalFourierTransform Sort of. But I read that mass was used for coupling interaction, and spin was in another symmetry group(decomposition) of its own? Nov 24 '19 at 21:49
• It would help if you provided some information about whether you are studying physics, math, or neither, and at what level. Nov 24 '19 at 21:54
• @G.Smith advanced QM was done and math was fine, but I'm new to particle. and recently struggling with QFT version of language. Nov 24 '19 at 21:58

The spin of a particle is its intrinsic angular momentum, a vector quantity unrelated to any actual rotation of the particle. As you learned in quantum mechanics its magnitude is quantized as $$\sqrt{s(s+1)}\hbar$$ and any of its three components as $$m_s\hbar$$. Here $$s=0,1/2,1,3/2,2,...$$ is the principal spin quantum number and $$m_s$$, which ranges from $$+s$$ to $$-s$$ in steps of 1, is the secondary spin quantum number.

When talking about a particle or field, often the value of $$s$$ is called “the spin”. But sometimes, if one is focused on, say, the $$z$$-component of the angular momentum, $$m_s$$ is called “the spin”; it should really be called “the $$z$$-component of the spin”, but that gets tedious.

So “the spin” can mean (1) the intrinsic angular momentum; (2) the quantum number $$s$$ that specifies the magnitude of that angular momentum; (3) a component (usually the $$z$$-component) of that angular momentum; (4) the quantum number $$m_s$$ that specifies that component.

In quantum field theory, the quanta of various kinds of fields have various spins. A scalar field is said to be “spin 0” because its quanta have $$s=0$$. A spinor field is said to be “spin 1/2“ because its quanta have $$s=1/2$$. A vector field is said to he “spin 1“ because its quanta have $$s=1$$. A tensor field with two indices is said to be “spin 2” because its quanta have $$s=2$$.

In QFT, you can also think about spin more abstractly in terms of how the field transforms under spatial rotations rather than how much intrinsic angular momentum its quanta have. This connection should not be too surprising, because the conservation of angular momentum is related to invariance under rotations.

Under rotations of the coordinate system, the fields have to transform according to a “representation” of the rotation group. The relevant mathematics is the theory of Lie groups (like the rotation group $$SO(3)$$ and the larger Lorentz and Poincaré groups) and their representations. A representation is a set of linear transformations in an abstract vector space (often a complex one) of arbitrary dimension that compose in the same way that the abstract group elements do. (“In the same way” means their composition is homomorphic to the group composition.)

Scalar, spinor, vector, and tensor fields are different representations of the same rotation group, in different dimensions. Here the dimensions are not physical dimensions like height and width but “field dimensions”. For example, a Dirac spinor has four components. Under rotation, they mix together linearly, similarly to how $$x$$, $$y$$, and $$z$$ spatial coordinates mix together linearly under a rotation. This four-dimensional complex vector space is an abstract representation space, not a geometric space like spacetime. Weyl and Majorana spinors have two components and “live” in a two-dimensional abstract representation space.

The word “spinor” is used to mean several different but closely related things: (1) A particular representation with $$s=1/2$$; (2) An element in the representation space, i.e., a multi-component field that transforms according to this representation; (3) A particle that is a quantum of such a field.

A fuller discussion of spinors would get into projective representations, covering groups, and other arcana of group theory. Hopefully this intro will get you oriented to the simpler ideas first.