Spinor Boosts: Weinberg's QFT Book Insights on Rotation and Momentum Effects

[mahdiarnt@yahoo.com]

Weinberg shows in his book on QFT that the effect of a rotation on a
massive particle's spin is the same as applying exp(-i \theta J.n) on
the state, regardless of particle's momentum. In other words just
rotate the spinor in its own representation. This makes calculations
very simple. But what about boosts? There is of course a general
approach described in Weinberg's book using little group elements; but
it is too difficult to apply it each time. Is there any general theorem
for boost like that Weinberg presented for rotations? And also what
about massless particles?

 

[EGoldstein]

I´m finding the exact same problem. In fact I´m trying to solve the first problem in Weinberg Vol I. Chpater 2 and while all the other items I managed to find out, I still hhave many problems in getting the Wigner rotation for the Boost.

 

[sir-pinski]

In his book on quantum field theory, Weinberg highlights the simplicity and elegance of spinor boosts in comparison to rotation effects on a massive particle's spin. He shows that the effect of a rotation on a particle's spin can be represented by a simple rotation of the spinor in its own representation, regardless of the particle's momentum. This simplifies calculations and makes the concept of spinor boosts very intuitive.

However, when it comes to boosts, Weinberg's approach is more complex. While he presents a general approach using little group elements, it can be difficult to apply in practice. This raises the question of whether there is a general theorem for boosts, similar to the one Weinberg presented for rotations.

Unfortunately, there is no such general theorem for boosts that is as simple and intuitive as the one for rotations. Boosts are more complicated than rotations because they involve both spatial and temporal components. This makes it challenging to find a simple representation for the effect of a boost on a particle's spin.

Furthermore, the concept of spin becomes more complicated for massless particles, as they do not have a well-defined rest frame. In this case, the little group approach for boosts becomes even more difficult to apply. However, there are other techniques, such as spinor helicity formalism, that can be used to study the spin properties of massless particles.

In conclusion, while Weinberg's approach for rotations using spinor boosts is elegant and intuitive, the same cannot be said for boosts. Boosts are more complex and there is no general theorem that simplifies their effects on a particle's spin. Furthermore, the concept of spin becomes more complicated for massless particles, making the little group approach even more challenging to apply.