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 style="font-size:15px;">      The gauge covariance of the equation of motion (3.11) allows us to choose in particular the covariant Lorentz gauge ∂·A = 0. In this gauge the 4-tuplet Aμ = (A0,
) transforms like a 4-vector. This demonstrates the manifest Lorentz covariance of the equation of motion (3.16) in the Lorentz gauge.

       Negative energy solutions and antiparticles

      Consider the case where the vector potential is independent of time. In that case there exist stationary solutions

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      suggesting again a two-particle interpretation for a general wave packet. Labelling these solutions by the charge q appearing in the covariant derivative (3.12), substitution into Eq. (3.11) leads to the equations

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      where we have set Φ = A0. From here we see that

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      This suggests the identification of the “negative energy” solution ϕ(−) with the respective antiparticle. The transformation (3.18) for scalar fields is referred to as charge conjugation.

       Probability interpretation

      In the free-particle case, we attempted to identify the probability density with the time component of a conserved 4-vector-current. It was found to satisfy all requirements, provided we restricted ourselves to positive energy solutions from the outset. As we now show, this will no longer be possible in the presence of an electromagnetic interaction, which will invariably lead to transitions to states involving negative energy solutions.

      Following the general line of approach adopted in the free particle case, we note to begin with that we can define again a conserved current by

figure

      This current is gauge-invariant

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      and thus defines a Lorentz covariant observable.

      With the aid of the equation of motion (3.11) one easily checks that this current is conserved:

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      Since jμ transforms like a 4-vector density, we take its zero component to define the probability density:

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      Unfortunately this definition of the probability density already violates positivity for “positive energy” solutions; indeed, consider the stationary wave function

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      Substitution into (3.19) yields

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      Hence, even if E > 0, this density is not positive semi-definite, since the sign of the Coulomb potential can be either positive or negative.

      The above considerations lead us to abandon at this stage our search for a relativistic scalar wave equation conforming to the principles of non-relativistic quantum mechanics. We shall, however, return to the field equation (3.11) after having learned in Chapter 7 to interpret ϕ(x) as an operator-valued field acting on a Hilbert space of Fock states.

       The Dirac Equation

      We begin this chapter by obtaining the relativistic “Schroödinger” equation for a free spin-1/2 field by following first the historical approach, and then presenting a derivation based on Lorentz covariance and space-time parity alone. This leads us to a four-component wave equation which is first-order in space and time coordinates. We present the solution of this equation for three choices of basis: The Dirac, Weyl (or chiral), and Majorana representations. The latter representation is shown to be particularly useful for the case of Majorana fermions, i.e. fermions which are their own anti-particles. We show that the Dirac equation allows for the notion of a probability density after suitable interpretation of the negative energy states.

      In this section we present the derivation of the Dirac equation by following the historical path, as well as a purely group-theoretical approach relying on Lorentz transformation properties alone. We then obtain the general solution of these equations in terms of the four independent Dirac spinors.

       Dirac equation: historical derivation

      Since in a manifestly Lorentz-covariant wave equation, space and time variables should appear on equal footing, Dirac demanded that the hamiltonian in the equation

figure

      should depend linearly on the momentum

canonically conjugate to figure. This led him to the Ansatz1

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      The triple figure and β in this equation cannot be just numbers, since this would already be inconsistent with rotational covariance. Hence they are expected to be given by matrices. These matrices must be hermitian, in order to warrant the hermiticity of the Hamilton operator. Furthermore, Eq. (4.2) should lead to the correct relation between energy and momentum for free particles. In order to see what this implies, we differentiate Eq. (4.2) with respect to time, thus obtaining

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      Here the bracket {A, B} denotes the anticommutator of two objects:

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      In order to get the desired energy momentum relation, this equation has to reduce to the Klein–Gordon equation, which is the case if

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