Spin direction and 4-spinor components

In summary, the physical interpretation of Dirac 4-spinors can be understood using the Weyl/Chiral representation and the electron as the particle. In the rest frame of the electron, the 4-spinor reduces to two identical 2-spinors, where the electron's spin is aligned along a unit vector "n" described in spherical coordinates. After a large boost in the z-direction, the 4-spinor becomes (0 b a 0), while a boost in the x-direction yields (a-b b-a a+b a+b) and a boost in the y-direction yields (a+ib b-ia a-ib b+ia). It is possible to visualize the spin direction of the electron using 4
  • #1
Zoot
9
0
I have questions regarding the physical interpretation of Dirac 4-spinors.
Using the Weyl/Chiral representation and the electron as the particle,
in the rest frame of the electron, the 4-spinor reduces to two identical
2-spinors, that is to say the 4-spinor takes on the form (a b a b). (see
for example 3.47 in Peskin/Schroeder). The 2-spinor (a b) where "a" and "b"
are generally complex numbers is the familiar
2-spinor from quantum mechanics and has a nice physical interpretation:
(a b) describes an electron whose spin is aligned along a unit vector "n"
whose direction is described in spherical coordinates with
θ = 2 arctan |b|/|a| and phi = phase difference between a and b when
expressed in polar form. In other words, if an electron described by (a b)
is measured along the direction of "n", it will be "spin-up" 100% of the time.
Now suppose I take this electron (a b a b) in its rest frame and give it a
LARGE boost in the z-direction. Using 3.49 or 3.50 from Peskin/Schroeder,
I get (a b a b) --> (0 b a 0). Or if instead I do a large boost in the
x-direction I get: (a b a b) --> (a-b b-a a+b a+b). A large boost in
the y-direction yields: (a b a b) --> (a+ib b-ia a-ib b+ia).

My questions are as follows:
(1) Is there a way to "picture" these 4-spinors in terms of the electron's
spin "pointing" along a certain spatial direction as we can do with 2-spinors?
(2) If the answer to (1) is yes, then which direction is the electron's
spin pointing after the large boosts described above?
(3) I have seen it mentioned that an electron's spin points at a velocity-
dependent angle from its momentum axis, with higher velocity making the
angle smaller. Is this true? If so, then for very large boosts (as
in my examples above) the spin should be pointing in the same direction
as the boost?

Sorry for the long post, but I'm trying to make a physical picture in my
mind of what happens to the spin direction of a boosted electron and how
it relates to the components of the 4-spinor, etc.
 
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  • #2
I have seen somewhere(incorrect?) that two of the upper components of 4-spinor are for electron and lower two are for positrons.the spin component along direction of momentum is defined by helicity.I don't think your third one is true.
 

1. What is spin direction?

Spin direction is a fundamental property of elementary particles that describes the direction of their intrinsic angular momentum. It can be either clockwise or counterclockwise, and is measured in units of spin.

2. What are 4-spinor components?

4-spinor components are mathematical quantities used to describe the spin state of a particle. They are represented by four complex numbers and are used in the Dirac equation to describe the behavior of spin-1/2 particles such as electrons.

3. How are spin direction and 4-spinor components related?

Spin direction and 4-spinor components are closely related. The spin direction of a particle is determined by its 4-spinor components, as these components represent the particle's spin state. Changes in spin direction can be described by changes in the 4-spinor components.

4. Can spin direction and 4-spinor components change?

Yes, spin direction and 4-spinor components can change. This can occur through interactions with other particles or through external forces. The change in spin direction can also result in a change in the 4-spinor components of a particle.

5. How is spin direction and 4-spinor components measured?

Spin direction and 4-spinor components are measured using experiments and mathematical calculations. For example, the Stern-Gerlach experiment can be used to measure the spin direction of particles, and mathematical equations such as the Dirac equation can be used to calculate the 4-spinor components.

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