Foundations of Quantum Field Theory. Klaus D Rothe

Foundations of Quantum Field Theory

as the rotation (say, in the plane containing

and the z-axis) into the unit vector

|)^1/2 in (2.48) is inserted because of our choice of non-relativistic normalization of the states

In order to obtain the transformation law of the states (2.48) under a general Lorentz transformation, we now proceed in a way analogous to that followed in Section 2.5. We thus have

But the transformation

leaves our standard vector (κ, 0, 0, κ) invariant and hence belongs to the little group. Indeed,

Hence the four-by-four matrix

belongs to the “little group” of the Lorentz Group. The corresponding unitary operator

thus also does not change the momentum of the state

, and must induce the following linear transformation on the massless 1-particle state

where

is an irreducible representation of the little group (compare with (2.46)). Correspondingly we have from (2.52)

In order to obtain the representation matrices of the little group, we must examine the nature of the transformations, leaving our standard vector

are obtained as a special case of the representation matrices for a general Lorentz transformation, discussed in Section 2.4.

It suffices to look at an infinitesimal transformation of the form

where the infinitesimal parameters are now required to satisfy

Inspection of (2.56) shows that

is in general a function of three parameters θ, χ₁ and χ₂ with the non-zero components given by

The unitary operator acting on the states is correspondingly given by

where

μν are the generators of Lorentz transformations satisfying the Lie algebra (2.21). Using (2.22) we then have for an infinitesimal transformation,

or using (2.27),

where J₃ = A₃ + B₃, and where A₋ and B₊ stand for

Since

we see that B₊ and A₋ act as raising and lowering operators for the eigenvalues of J₃, respectively. The eigenvalues of the helicity operator J₃ = A₃ + B₃ of a general state |A, a; B, b > are λ = a + b. In nature a massless spin j particle only exists in two helicity states with helicity λ = +j (right handed) or λ = −j (left handed). For a transformation of the little group not to change this helicity we therefore demand for such an helicity state

This leads to the identification

A spin-j massless particle thus transforms under the

(right handed) or

(left handed) representation of the little group. It then follows from (2.59) by exponentiation, that (A₋ and B₊ annihilate the state)

the phase Θ(θ, _{χ1, χ2}) being some more or less complicated function of the little group parameters, which reduces to θ in (2.57) for infinitesimal transformations. It must satisfy the group property

Hence finally we have from (2.55),

This result will play an important role when we proceed to discuss the quantization of the electromagnetic field in Chapter 7.

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¹The inhomogeneous Lorentz transformations including the space-time translations will be discussed in Chapter 9. We adopt the convention that repeated upper and lower indices are to be summed over.

²This chapter is largely based on the papers by S. Weinberg in Physical Review. Note that Weinberg uses the metric gμν = (−1, 1, 1, 1).

³Lorentz transformations representing a boost to momentum

we denote by L(Скачать книгу