Foundations of Quantum Field Theory. Klaus D Rothe
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Название: Foundations of Quantum Field Theory

Автор: Klaus D Rothe

Издательство: Ingram

Жанр: Физика

Серия: World Scientific Lecture Notes In Physics

isbn: 9789811221941

isbn:

СКАЧАТЬ matrices9

figure

      As one easily convinces oneself, one has (from here on we follow the convention of Itzykson and Zuber and of most other authors, and choose ϵ0123 = 1)

figure

      This expression defines the γ5 matrix in both representations.

       (d)Lorentz transformation properties of γ 5

      For Λ a Lorentz transformation, we have the algebraic property

figure

      Now

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      Hence we conclude that γ5 “transforms” in particular like a pseudoscalar under space reflections, and in general as

figure

       (e)Traces involving γ 5

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      Here the first relation follows from the fact that there exists no Levi–Civita tensor with two indices in four dimensions. The second relation follows from the fact that the right-hand side should be a Lorentz invariant pseudotensor of rank four, for which ϵμνλρ is the only candidate, and choosing the indices as in (4.49) to fix the constant.

      In the extreme relativistic limit we expect the mass of the fermion to be negligible. For m = 0 the Dirac Hamiltonian operator commutes with γ5 . Hence we may classify the eigenfunctions of the Hamiltonian by the eigenvalues of γ5 . It is thus desirable to work in the Weyl representation, where γ5 is diagonal. In this representation the Dirac operator becomes off-diagonal in the large momentum limit, and the Weyl equations (4.27) and (4.28) reduce to the form

figure

      or

figure

      The 2 × 2 matrix figure represents the projection of the angular momentum operator on the direction of motion of the particle and is called the helicity operator. Correspondingly one refers to its eigenvalues ±1/2 as helicity.

      Equations (4.52) and (4.53) are just the Weyl equations for a massless particle. If parity is not conserved, we may confine ourselves to either one of the two equations, that is to either particles polarized in the direction of motion (positive helicity) or opposite to the direction of motion (negative helicity). This is the case for neutrinos (antineutrinos) participating in the parity-violating weak interactions, which carry helicity −1/2 (+1/2). If parity is conserved, both helicity states must exist.

      The fact that the massive Dirac equation turns into Weyl equations in the “infinite momentum frame” shows that at high energies massive particles are polarized “parallel” or “anti-parallel” to the direction of motion. However, whereas the helicity of a massless particle is a Lorentz invariant, this is not the case for a massive particle: If a massive particle is polarized in the direction of motion in one inertial frame, its polarization will be a superposition of all possible spin projections in a different inertial system. Phrased in a different way: If the particle is massive one can always catch up with it and ultrapass it, so that the particle appears moving “backwards”, while continuing to be polarized in the original direction. With a zero mass particle you can never catch up since it is moving at the speed of light.

       Chirality

      As the last argument above shows, the m = 0 case has to be treated separately, and cannot be obtained as the zero-mass limit of massive case discussed so far, which was based on the existence of a rest frame of the particle. According to our discussion in the chapter on Lorentz transformations, zero-mass 1-particle states indeed transform quite differently from the massive ones.

      In the zero mass case, the Dirac equations for the U and V spinors reduce to one and the same equation:

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      Let us define the “spin” operator

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      In terms of the Gamma matrices (Dirac or Weyl basis) this operator reads

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      The Dirac equation may then be written in the form

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      where figure. Thus figure is just the helicity operator in the 4-component representation.

      The helicity operator (4.56) commutes with the free Dirac Hamiltonian. The same applies to γ5, if the mass of the particle is zero. Since furthermore

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      we may classify the eigenstates of the zero-mass Dirac Hamiltonian according to their helicity and chirality, the latter being defined as the corresponding eigenvalue ±1 of γ5. Such states are obtained from the solutions U to the Dirac equation with the aid of the projection operator

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      We have

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      where

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      Recalling that in the Weyl representation

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      we have

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      The eigenvalue of γ5 thus coincides with twice the eigenvalue of the helicity operator: particles of positive (negative) chirality, carry helicity СКАЧАТЬ