Читать книгу Foundations of Quantum Field Theory - Klaus D Rothe - Страница 17

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 , and helicity . These states replace the states |s, σ 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 is the unitary operator corresponding to the Lorentz transformation (p) which takes our standard four-momentum ^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() as the rotation (say, in the plane containing and the z-axis) into the unit vector . The kinematic factor (μ/||)^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 invariant. Since the little group is a subgroup of the Lorentz group, the representation matrices 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

or

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()

⁴We have

⁵

⁶These commutation relations do not fix the sign of the generator Ki of boosts. We follow in (2.22) the convention adopted by S. Weinberg.

⁷The minus sign in the last commutation relations reflects the fact that the Lorentz group SO(3, 1) can be considered as the complexification of the rotation group in four dimensions, SO(4). The fact that the commutator of two generators of the boost is given as a linear combination of the generators of rotations is related to the phenomenon of the Thomas precession. From the group theoretic point of view it expresses the fact that, if we perform the sequence of infinitesimal boosts g(−δθ₂)g(−δθ₁)g(δθ₂)g(δθ₁) with , the effect is just an infinitesimal rotation in the frame we started from.

⁸We follow closely the notation of S. Weinberg, Phys. Rev. 133 (1964) B1318.

⁹We follow again closely the notation of S. Weinberg, Phys. Rev. 133 (1964) B1318 and Phys. Rev. 134 (1964) B882. It is to be kept in mind that, unlike us, S. Weinberg uses the metric gμν = (−1, 1, 1, 1).

¹⁰For a more detailed analysis see E. P. Wigner, Ann. Phys. Math. 40 (1939) 149.

¹¹E.P. Wigner, Theoretical Physics (International Atomic Energy Vienna, 1963) p. 59.

¹²We follow again the notation of S. Weinberg, Phys. Rev. 134 (1964) B882.