Читать книгу Foundations of Quantum Field Theory - Klaus D Rothe - Страница 16

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 , we have

It is easy to see that the matrix

is not equal to one unless Λ represents a pure boost colinear with . In general RW represents a pure rotation — the so-called Wigner rotation — in the rest frame of the particle. We may thus make use of the completeness relation

valid in the rest frame of the particle in order to write (2.44) in the form

where we have made the identification

with D^(s)[RW] a (2s + 1)-dimensional irreducible representation of the rotation group. We thus finally have

At this point we can now firmly establish the correctness of our choice of normalization factor in (2.46). To this end we start from the completeness relation

and multiply this relation from the left with U[Λ], and from the right with U⁻¹[Λ]:

We now make use of (2.46) in order to rewrite this relation as

Making use of the unitarity of the matrix representation of the rotation group, we have

Hence we obtain from above

Recalling the transformation property of the integration measure, Eq. (2.43), the above expression reduces to

showing that our choice of normalization is consistent with the Lorentz covariance of the completeness relation.