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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      invariant. Any 4-tuplet transforming like the coordinates in (2.1) is called a contravariant 4-vector. In particular, energy and momentum of a particle are components of a 4-vector

      with

      The same transformation law defines 4-vector fields at a given physical point:

      Note that the quadruple (Λx)μ referred to

is the same point as the quadruple xμ referred to
′. Thus, alternatively

      Notice that (2.4) and (2.5) represent inverse transformations of the reference frame, respectively. Examples are provided by the 4-vector current

and the vector potential
of electrodynamics in a Lorentz-covariant gauge.

      The differential element dxμ transforms like

      Hence it also transforms like a contravariant 4-vector, since

      The partial derivative

, on the other hand, transforms differently. The usual chain rule of differentiation gives

      From the inversion of (2.1) it follows that

      Hence for the partial derivative

we have the transformation law

      Four-tuples which transform like the partial derivative are called covariant 4-vectors. Contravariant and covariant 4-vectors are obtained from each other by raising and lowering the indices with the aid of the metric tensors gμν and gμν, defined by2

      respectively, in terms of which the invariant element of length (2.2) can be written in the form

      The requirement that ds2 be a Lorentz invariant

      now implies

      Thus the metric gμν is said to be a Lorentz-invariant tensor. It is convenient to write this equation in matrix notation by grouping the elements

into a matrix as follows:

      Defining the elements of the transpose matrix ΛT by

      we can write (2.8) as follows:

      From here we obtain for the inverse Λ−1,

      or in terms of components

      Define the dual to a contravariant 4-vector vμ by

      Thus gμν(gμν) serve to lower (raise) the Lorentz indices. In particular

. We have after a Lorentz transformation, upon using (2.9)

      or we conclude that vμ defined by (2.10) does indeed transform like a covariant 4-vector. In particular we see that the following 4-tuplets transform like covariant and contravariant 4-vectors, respectively:

      where

      The Lorentz invariance of the scalar product vμvμ = v2 allows us to divide the 4-vectors into three classes which cannot be transformed into each other by a Lorentz transformation:

      (a)vμ time-like (v2 > 0)

      (b)vμ space-like (v2 < 0)

      (c)vμ light-like (v2 = 0)

      This means in particular that space-time separates, as far as Lorentz transformations are concerned, into three disconnected regions referring to the interior and exterior of the light cone x2 = 0, as well as to the surface of the light cone itself. The trajectory of a point particle localized at the origin of the light cone at time t = 0 lies within the forward light cone; Moreover, if we attach a light cone to the particle at the point where it is momentarily localized, the tangent to the trajectory at that point does not intersect the surface of that light cone.

      The Lorentz invariant

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