Foundations of Quantum Field Theory. Klaus D Rothe

Foundations of Quantum Field Theory

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.

2.6Transformation properties of zero-mass 1-particle states

We are now in the position of discussing the Lorentz transformation properties of zero-mass particle states. The transformation rules have been completely worked out by E. Wigner.¹¹

In the case of zero mass particles we can no longer go into the rest frame of the particle to define a general state in terms of a Lorentz boost. In fact, it is well known that a massless particle of spin j is polarized either along or opposite to its direction of motion, corresponding to two possible helicity states. If parity is not conserved, there may exist but one helicity state, as is exemplified by the neutrino (anti-neutrino) with negative (positive) helicity. Correspondingly we expect these helicity states to transform under a one-dimensional representation, independent of the spin of the particle. Following Wigner, we choose for our “standard” state a particle moving in the positive z-direction with four-momentum

in the massive case. Whereas the states |s, σ

belong to a representation of the rotation group, the helicity states

furnish a representation of the little group, a subgroup of the Lorentz group consisting of all homogeneous proper Lorentz transformations leaving our standard 4-vector

invariant.

In analogy to the massive case, we define the state of a massless particle of arbitrary momentum

by “boosting” the standard state

> into the desired new state:

where

^μ into pμ,

and μ in (2.48) is an arbitrary parameter with the dimensions of a mass. There are various ways of defining

; we shall make the choice¹²

Here

is a “boost” along the z-axis with non-zero components (compare with (2.16)

To determine ϕ(|

|) we observe that

so that

We choose R(Скачать книгу