Название: Foundations of Quantum Field Theory
Автор: Klaus D Rothe
Издательство: Ingram
Жанр: Физика
Серия: World Scientific Lecture Notes In Physics
isbn: 9789811221941
isbn:
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 byor
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).
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