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:

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      Summarizing we have in the Dirac, Weyl and Majorana representations

figure

      where

figure

       Majorana spinors

      Applying the unitary operator (4.65) to the Dirac spinors in the Dirac representation (4.21) (we use c−1 = iσ2)

figure

      we obtain for the corresponding spinors in the Majorana representation

figure

      Recalling that

figure

      we see from (4.70) and (4.71) that

figure

      In the case of Majorana fermions, it is convenient to redefine the phase of the Dirac spinors via the replacements

figure

      For this new choice of phase, Eq. (4.72) is replaced by

figure

       Self-conjugate Dirac fields

      Fields describing fermions which are their own anti-particles are said to be self-conjugate and have the Fourier representation

figure

      with the property (4.73). These fields are real,

figure

      and play the role of real scalar fields in the case of spin 1/2 fields.

      ________________________

      1P.A.M. Dirac, The Principles of Quantum Mechanics, 4th edn. (Oxford University Press, Oxford 1958).

      2Here and in what follows: in formulae which hold generally, without reference to a particular basis such as the Dirac representation, we omit the subscript D.

      3The reason for introducing c will become clear in Chapter 7, Eq. (7.15).

      4The minus sign is a consequence of our Dirac scalar product.

      5We follow in general the notation of S. Weinberg, Phys. Rev. 133 (1964) B1318.

      6Substituting (4.27) into (4.28) yields figure. Noting that with our definitions (4.24) for tμ and figure, we have figure, we recover the Klein–Gordon equation, as desired.

      7H. Weyl, The Theory of Groups and Quantum Mechanics (Dover Publications, Inc. New York, 1931).

      8See Chapter 5 for this terminology.

      9Note that

figure

      where S is given by (4.40). This agrees with the usual V − A coupling of neutrinos in the weak interactions.

       The Free Maxwell Field

      As is well known, the electromagnetic field can be interpreted on the quantum level as a flux of quanta, called photons. In fact, this interpretation first arose in connection with Planck’s formula describing the spectrum of black-body radiation. As Maxwell’s equations show, these quanta propagate with the velocity of light in all inertial frames, so there exists no rest frame we can associate with them. Photons can thus be viewed as “massless” particles. According to our discussion in Section 6 of Chapter 2, the respective 1-particle states should thus transform according to a one-dimensional representation of the little group.

      The Aharonov Bohm effect shows that it is the vector potential Aμ which plays the fundamental role in quantum mechanics. This vector potential however transforms under the figure representation of the Lorentz group and involves a priori four degrees of freedom. Of these, A0 is associated with the “Coulomb potential” and thus corresponds to non-radiative degrees of freedom which are only present if there is a matter source. This leaves us with three degrees of freedom. One of these is not observable (on classical level) as a result of the underlying gauge invariance of physical quantities. For a pure radiation field one is thus left with only two degrees of freedom, corresponding to the two helicity states of a photon. These statements become obvious in the Coulomb gauge, which is thus also called the “physical” gauge. We are, however, not limited to this choice of gauge which in practical calculations complicates matter considerably, due to the fact that it breaks manifest Lorentz covariance. We shall thus review the solutions of Maxwell’s equations in two different gauges — the Lorentz gauge and the (non-covariant) Coulomb gauge.

      In the absence of a source, the Maxwell equations (Coulomb’s and Ampère’s law) become

figure

      with the usual identification

figure

      for the magnetic and electric fields, respectively.

      The electromagnetic field tensor Fμν can be written in the form

figure

      and is evidently invariant under the gauge transformation

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