A. Appendix A.1 Theory of Special Relativity The theory of special relativity is based on the following axioms: 1. Constancy of the velocity of light: the vacuum velocity of light is the same in all uniformly moving systems, namely c ≈ 3 · 108 m/s. No information propagates faster than light. 2. Relativity principle: physical laws are equally valid in all inertial systems, i.e. there is no preferred reference frame. The relativity principle encompasses the homogeneity of space and time as well as the isotropy of space according to which there is no preferred point and direction in space and time. Due to these axioms, relativistic physical events are described mathemat- ically within a fourdimensional space where the product of the speed of light and time, x0 = ct, appears as an additional dimension on an equal footing with the three space dimensions x1,x2,x3. Minkowski space. The Minkowski space is a fourdimensional linear vector space over the body of real numbers. Its elements xμ are represented by four-component coordinate vectors, also called four-vectors, ⎛ ⎞ x0(t) ⎜ x1(t) ⎟ [xμ(t)] = ⎜ ⎟ ,x0(t)=ct . ⎝ x2(t) ⎠ x3(t) The scalar product of two four-vectors is defined as μ μ μ ν μ μν (x ) · (y )=x gμν y = x yμ = xμg yν =(xμ) · (yν ) , with the non-Euclidean metric tensor (1. index=row index, 2. index=column index) ⎛ ⎞ 1000 ⎜ 0 −100⎟ (g )=(gμν )=⎜ ⎟ ,gμαg = gμ = δμ . μν ⎝ 00−10⎠ αν ν ν 000−1 A. Wachter, Appendix. In: A. Wachter, Relativistic Quantum Mechanics, Theoretical and Mathematical Physics, pp. 349–361 (2011) DOI 10.1007/978-90-481-3645-2 c Springer Science+Business Media B.V. 2011 350 A. Appendix Notation. Additionally, there exist the following conventions: • Vectors with an upper index are called contravariant, and those with a lower index are called covariant. This can be transferred to tensors of higher μν rank. For example, T ρ is a twofold contravariant and onefold covariant tensor of rank 3. • Two indices, one of which is an upper and the other a lower index, are summed over so that the summation sign can be dropped (Einstein sum convention). • The metric tensor can be used to pull upper indices down and vice versa. For example: ν μνρ μα νρ μα νβ γ μα νβ ργ xμ = gμν x ,T = g Tα = g g Tαβ = g g g Tαβγ etc. μ Accordingly, the co- and contravariant vectors xμ and x differ solely by their spatial components. Contrary to the threedimensional Euclidean case, the norm of a four-vector is not positive definite. Instead, the following cases can occur: ⎧ ⎪ ⎨⎪ > 0 (time-like) μ 2 − 2 2 2 − 2 xμx = x0 x = c t x ⎪ = 0 (light-like) ⎩⎪ < 0 (space-like) , where the classification corresponds to the vector’s position relative to the μ light cone xμx =0. Lorentz transformations. The axioms of the theory of special relativity imply that the fourdimensional “distance” between two four-vectors xμ and yμ is preserved in any inertial system: μ μ (x − y)μ(x − y) =(x − y )μ(x − y ) . Lorentz transformations describe the relativistic transition from one inertial system to another. They are defined by the linear transformational equation μ μ μ ν μ μ x → x = Λ ν x + a , (a ) = space-time translation (A.1) of contravariant vectors. Along with the conservation of distance, one obtains from this the conditional equation μ ν Λ αgμν Λ β = gαβ (A.2) as well as the corresponding transformational behavior of covariant vectors ν ν α ν −1 β xμ = gμν x = gμν (Λ αx + a )=xβ[Λ ] μ + aμ , with the inverse transformation −1 β ν αβ β −1 β μ ν αβ μ β [Λ ] μ = gμν Λ αg = Λμ , [Λ ] μΛ γ = gμν Λ αg Λ γ = δγ .(A.3) A.1 Theory of Special Relativity 351 In matrix notation the condition (A.2) reads ΛT gΛ = g and corresponds to the property RT R = 1 of rotational matrices of threedimensional Euclidean geometry. Lorentz transformations with aμ = 0 constitute the homogeneous Lorentz group. Besides the distance, they also leave the scalar product of two μ μ μ four-vectors unchanged: x yμ = x yμ. For the general case, (a ) =0,one obtains the inhomogeneous Lorentz group or Poincar´e group which is not considered here. The homogeneous Lorentz group can be classified in the following way: 0 symbol det(Λ) Λ 0 group name L +1 > 0 proper P ·L −1 > 0 orthochronous R ·L −1 < 0 homogeneous P · R ·L +1 < 0 Contrary to all others, the transformations of the proper Lorentz group L can be interpreted as a series of infinitesimal transformations. Noninfinitesimal or discrete transformations are, for example, the parity transformation (space reflection) P : x0 → x0 = x0 ,xk → xk = −xk and Racah time reflection R : x0 → x0 = −x0 ,xk → xk = xk . The orthochronous group consists of the proper transformations L, the space reflection P , and their products P ·L. The homogeneous group is composed of the transformations L, P , R, and their products. Transformational behavior of differential operators. Taking into ac- ν −1 ν μ count x =[Λ ] μx [see (A.1)], we have ∂xν ∂ ∂ ∂xν ∂ =[Λ−1]ν =⇒ = = [Λ−1]ν . ∂xμ μ ∂xμ ∂xν ∂xμ ∂xν μ Therefore, ∂ = ∂ ∂xμ μ transforms as a covariant vector and, accordingly, ∂ = ∂μ ∂xμ as a contravariant vector. Furthermore, it follows that the d’Alembert oper- ator 352 A. Appendix 1 ∂2 ∂ ∂μ = − ∇2 μ c2 ∂t2 is Lorentz-invariant, i.e. a Lorentz scalar. Relativistic kinematics. In order to write down the equations of motions of relativistic mechanics Lorentz-covariantly1 (form invariantly), we must reformulate the Newtonian quantities of velocity, momentum, and force in such a way that they have a defined transformational behavior under Lorentz transformations. Contrary to dt, the eigentime differential 1 dx 2 dτ =dt 1 − c2 dt is a Lorentz scalar since, due to the conservation of the scalar product, we have 1 1 dτ 2 =dt2 − dx2 =dt2 − dx2 (dτ 2 > 0 for time-like vectors) . c2 c2 With this, the following quantities can be defined, of which the four-versions transform as xμ: • Four-velocity uμ: dxμ dt dxμ 1 c uμ = = , (uμ)= dτ dτ dt − v2 v 1 c2 dx v = = physical velocity. dt • Four-momentum pμ: μ μ μ cm m0 p = m0u , (p )= ,m= ,m0 = rest mass p − v2 1 c2 p = mv = physical momentum. • Four-force F μ: dpμ 1 dpμ 1 c dm F μ = = , (F μ)= dt (A.4) dτ − v2 dt − v2 F 1 c2 1 c2 dp F = = physical force. dt 1 In this book, as in many other textbooks, the transformational behavior of rela- tivistic four-quantities is called “Lorentz-covariant” regardless of the subtle dif- ference between “covariant” and “contravariant”. The same holds for relativistic equations whose form remains unchanged under Lorentz transformations. Quan- tities and equations are called “Lorentz-invariant” if their value is unchanged by Lorentz transformations (Lorentz scalar). A.1 Theory of Special Relativity 353 The first force equation also represents the Lorentz-covariant equation of mo- tion of relativistic mechanics, whereas the second equation is the threedimen- sional relativistic analogon of Newton’s equation. In the case of a conservative force field F = −∇V (x), the latter leads to the energy conservation E = mc2 + V (x) = const =⇒ E = mc2 for V =0 and, together with the definition of pμ, the energy-momentum relation for free particles μ 2 − 2 2 2 ⇐⇒ 2 2 2 2 4 pμp = p0 p = m0c E = p c + m0c . It is not always possible to formulate a given mechanical problem covariantly via (A.4) since not all types of forces allow a fourdimensional generalization. One example of this kind is the Coulomb force. As a static long range force, it requires an infinitely high propagational velocity, thus being in contradiction to the first axiom of special relativity. Lorentz-covariant electrodynamics. The basic equations of classical elec- trodynamics are the four Maxwell equations ⎫ ∇E(x,t)=4πρ(x,t) ⎪ ⎪ ⎪ 1 ∂B(x,t) ⎪ ∇ × E(x,t)+ =0 ⎬⎪ c ∂t ⎪ (A.5) ∇B(x,t)=0 ⎪ ⎪ ⎪ 1 ∂E(x,t) 4π ⎭⎪ ∇ × B(x,t) − = j(x,t) c ∂t c (in the Gaussian unit system), the continuity equation ∂ρ(x,t) + ∇j(x,t)=0 ∂t that follows from the first and fourth Maxwell equation, as well as the Lorentz force v F (x,t)=q E(x,t)+ × B(x,t) , (A.6) L c describing the force acting on a particle with charge q due to its movement through the fields E and B. That these equations can be formulated Lorentz-covariantly, i.e. that they are in accordance with special relativity, can be shown as follows: • First, the continuity equation can be immediately brought into the form cρ ∂ jμ(x)=0, (jμ)= μ j showing explicitly its Lorentz covariance, provided that jμ is a four-vector. However, this follows from the experimental fact that the charge q is a 354 A. Appendix Lorentz scalar: dq =d3xρ =d3xρ.Thus,dq transforms as a fourdimen- sional volume element d3xdx0 =d3xdx0 and, consequently, ρ as the 0-th component of a four-vector. • Introducing the scalar potential φ and the vector potential A, 1 ∂A(x,t) B(x,t)=∇ × A(x,t) , E(x,t)+ = −∇φ(x,t) , (A.7) c ∂t Maxwell’s equations can be rewritten as 1 ∂ ∇2φ + ∇A = −4πρ c ∂t 1 ∂2A 1 ∂φ 4π ∇2A − − ∇ ∇A + = − j . c2 ∂t2 c ∂t c From here and choosing the gauge 1 ∂φ ∇A = − (Lorentz gauge) , c ∂t one obtains the disentangled inhomogeneous wave equations 1 ∂2 ∇2 − φ = −4πρ ⇐⇒ ∂ ∂μφ =4πρ c2 ∂t2 μ 1 ∂2 4π 4π ∇2 − A = − j ⇐⇒ ∂ ∂μA = j c2 ∂t2 c μ c that are symmetric in A and φ.