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Zoot
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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.
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.