Foundations of Quantum Field Theory. Klaus D Rothe
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Название: Foundations of Quantum Field Theory

Автор: Klaus D Rothe

Издательство: Ingram

Жанр: Физика

Серия: World Scientific Lecture Notes In Physics

isbn: 9789811221941

isbn:

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      We then have in particular

      as well as

      For a pure rotation

      The case

      For the case of j = 1/2 it is easy to give an explicit expression for the matrices

and
representing a boost, since in that case the matrices representing
are just one half of the Pauli matrices (4.4):

      Using

      we have

      or recalling (2.14), we may write this also as

      We thus conclude that

      Defining

      we may thus write the matrices (2.34) in the compact form

      These matrices will play a central role in our discussion of the Dirac equation in Chapter 4. Their explicit form is most easily obtained by returning to (2.35) and noting that

      Now,

      

      Hence

      Similarly

      These explicit expressions will prove useful in our discussion of the Dirac equation in Chapter 4.

      For the rest of these lectures we set c = 1.

      In analogy to the Galilei transformations discussed in Chapter 1, we take U[L(

)] to be the unitary operator taking the state |s, σ > of a particle of spin s, sz = σ at rest into a 1-particle state of momentum
.9

      where

      with normalization

      Note that the spin of a particle at rest is a well defined quantity, whereas for a moving relativistic particle this is not the case. The kinematical factor introduced in (2.39) compensates for the non-covariant normalization of the 1-particle states:

      The form of this kinematical factor can be motivated in the following way: The normalization (2.42) of the 1-particle states corresponds to the completeness relation

      Now, d3p is not a relativistically invariant integration measure, whereas d3p/ω(

) is. Indeed, making use of the usual properties of the Dirac delta-function we have

      

      The delta-function insures the proper energy momentum relation for a free particle,

      while the theta-function insures that the vector pμ is time-like, that is, the particle has positive energy. Both properties are preserved by Lorentz transformations in

. Furthermore, d4p is a Lorentz-invariant measure since

      and

      We thus conclude that

      This explains roughly the origin of the kinematical factor in (2.39).10 Now let U[Λ] be the unitary operator inducing a Lorentz transformation on the 1-particle state |

, s, σ′
. Using the group property of Lorentz transformations СКАЧАТЬ