Lagrangian Mechanics

Appendix A Lagrangian Mechanics

Lagrangian mechanics is not only a very beautiful and powerful formulation of mechanics, but it is also needed as a preparation for a deeper understanding of all fundamental interactions in physics. All fundamental equations of motion in physics are encoded in Lagrangian field theory, which is a generalization of Lagrangian mechanics for fields. Furthermore, the connection between symmetries and conservation laws of physical systems is best explored in the framework of the Lagrangian formulation of dynamics, and we also need Lagrangian field theory as a basis for field quantization. Suppose we consider a particle with coordinates x(t) moving in a potential V(x). Then Newton’s equation of motion

mx¨ =−∇V(x) (A.1) is equivalent to the following statement (Hamilton’s principle, 1834): The action integral t1 t1 m S[x]= dt L(x, x˙ ) = dt x˙ 2 − V(x) (A.2) t0 t0 2 is in first order stationary under arbitrary perturbations x(t) → x(t) + δx(t) of the path of the particle between fixed endpoints x(t0) and x(t1) (i.e. the perturbation is only restricted by the requirement of fixed endpoints: δx(t0) = 0 and δx(t1) = 0). This is demonstrated by straightforward calculation of the first order variation of S, t1 δS[x]=S[x + δx]−S[x]= dt [mx˙ · δx˙ − δx · ∇V(x)] t0 t1 =− dt δx · (mx¨ + ∇V(x)). (A.3) t0

© The Editor(s) (if applicable) and The Author(s), under exclusive license to 677 Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1 678 A Lagrangian Mechanics

Partial integration and δx(t0) = 0, δx(t1) = 0 were used in the last step. Equation (A.3) tells us that δS[x]=0 holds for arbitrary path variation with ﬁxed endpoints if and only if the path x(t) satisﬁes Newton’s equations,

mx¨ + ∇V(x) = 0. (A.4)

This generalizes to arbitrary numbers of particles (x(t) → xI (t),1≤ I ≤ N). Application of Hamilton’s principle to the N-particle action t1 S[x1, ··· , xN ]= dt L(x, x˙ ) (A.5) t0 with the N-particle Lagrange function

N m L(x, x˙ ) = I x˙ 2 − V(x , ··· , x ) (A.6) 2 I 1 N I=1 yields the Newton equations of motion for the N-particle system

∂ mI x¨ I =− V(x1, ··· , xN ). (A.7) ∂xI

On the level of unconstrained particle motion in a potential V , Hamilton’s principle does not seem to offer any obvious advantage over the study of the corresponding Newton equations. However, Hamilton’s principle becomes a particularly powerful tool when we are dealing with constrained mechanical systems, where the constraints can be expressed in the form

ζc(x1, ··· , xN ,t)= 0, 1 ≤ c ≤ C. (A.8)

Here C is the number of constraints, and the explicit time dependence indicates that the constraints themselves can change with time. For example, we could have a two-particle system with the constraint that the two particles must maintain a certain distance, where the distance may be a given function of time R(t),

ζ(x1, x2,t)=|x1(t) − x2(t)|−R(t) = 0, (A.9) whereas the constraint to maintain constant distance R would be written as ζ(x1, x2) =|x1(t) − x2(t)|−R = 0. Hamilton’s principle becomes very powerful because it still applies if we solve the C constraints (A.8) through introduction of 3N − C generalized coordinates qa which parametrize all the remaining allowed motions of the system under the constraints, i.e. we express the coordinates xI of the system through the allowed motions qa(t) under the constraints (A.9), A Lagrangian Mechanics 679

j j xI = xI (q, t). (A.10)

j The explicit time-dependence in xI (q, t) will arise if the constraints are time- j j dependent, and the full time-dependence xI (t) = xI (q(t), t) of the particle coordinates during the motion of the system will also involve an implicit time- dependence from the motions qa(t) of the generalized coordinates. The velocity components are therefore j ∂ j ∂ j x˙I (t) =˙qa(t) xI (q, t) + xI (q, t) . (A.11) ∂qa q=q(t) ∂t q=q(t)

Here the substitutions qa = qa(t) are performed after evaluating the partial derivatives. For example, for the 2-particle system with the constraint (A.9) we could use the = a ≤ ≤ = = ﬁve generalized coordinates qa X1 ,1 a 3, q4 ϑ, q5 ϕ, where ⎛ ⎞ sin ϑ cos ϕ m1x1 + m2x2 X = , x − x = R(t)⎝ sin ϑ sin ϕ ⎠. (A.12) m + m 1 2 1 2 cos ϑ

The mappings (A.10) are then ⎛ ⎞ sin ϑ cos ϕ m2 x = X + R(t)⎝ sin ϑ sin ϕ ⎠, (A.13) 1 m + m 1 2 cos ϑ ⎛ ⎞ sin ϑ cos ϕ m1 x = X − R(t)⎝ sin ϑ sin ϕ ⎠. (A.14) 2 m + m 1 2 cos ϑ

Performing the transformation (A.10) and (A.11) to the actual degrees of freedom qa in the Lagrange function (A.6) of the unconstrained N-particle system yields the Lagrange function L(q, q,t)˙ of the constrained N-particle system and the corresponding action t1 S[q]= dt L(q,q,t).˙ (A.15) t0

The nontrivial and very useful observation is that Hamilton’s principle also applies to the constrained system: The constrained mechanical system will still evolve in such a way that the action S[q] is stationary under arbitrary ﬁrst order perturbations qa(t) → qa(t) + δqa(t) subject only to the conditions that the initial and ﬁnal state are not perturbed: δqa(t) = 0, δqa(t1) = 0. 680 A Lagrangian Mechanics

First order variation of the action with fixed endpoints (i.e. δq(t0) = 0, δq(t1) = 0) yields after partial integration t1 ∂L ∂L δS[q]=S[q + δq]−S[q]= dt δq + δq˙ a ∂q a ∂q˙ t0 a a a t1 ∂L d ∂L = dt δq − , (A.16) a ∂q dt ∂q˙ t0 a a a where again the fixation of the endpoints eliminated the boundary terms. δS[q]=0 for arbitrary path variation qa(t) → qa(t) + δqa(t) with fixed endpoints then immediately tells us the equations of motion in terms of the generalized coordinates,

∂L d ∂L − = 0. (A.17) ∂qa dt ∂q˙a

These equations of motion are called Lagrange equations of the second kind or Euler–Lagrange equations or simply Lagrange equations. The quantity

∂L pa = (A.18) ∂q˙a is denoted as the conjugate momentum to the coordinate qa. The conjugate momentum is conserved if the Lagrange function depends only on the generalized velocity component q˙a but not on qa, dpa/dt = 0. Furthermore, if the Lagrange function does not explicitly depend on time, we have

dL ∂L = p q¨ + q˙ . (A.19) dt a a ∂q a a a

The Euler–Lagrange equation then implies that the Hamilton function1

1The Hamilton function is always conserved if there is no explicit time-dependence in the Lagrange function, and generically it corresponds to the energy of the system. However, if there are explicitly time-dependent constraints such that L = L(q, q)˙ is not explicitly time-dependent, then the conserved Hamilton function may not be the energy of the system. An example for this is provided e.g. by a particle which is tied to an upright circular wire while the wire is rotating around the z-axis: x(t) = R[ex cos ϑ(t) · cos(ωt) + ey cos ϑ(t) · sin(ωt) + ez sin ϑ(t)], L(ϑ, ϑ)˙ = mR2(ϑ˙ 2 + ω2 cos2 ϑ)/2, H = mR2(ϑ˙ 2 − ω2 cos2 ϑ)/2. H is conserved, but the energy of the particle is the non-conserved kinetic energy T = L. One can show in this system that the constraint force which forces the particle to rotate with angular velocity ω around the z-axis performs work. A Lagrangian Mechanics 681 H = paq˙a − L (A.20) a is conserved, dH/dt = 0. For a simple example, consider a particle of mass m in a gravitational ﬁeld g =−gez. The particle is constrained so that it can only move on a sphere of radius r. An example of generalized coordinates are angles ϑ, ϕ on the sphere, and the Cartesian coordinates {X, Y, Z} of the particle are related to the generalized coordinates through

X(t) = r sin ϑ(t) · cos ϕ(t), (A.21) Y(t) = r sin ϑ(t) · sin ϕ(t), (A.22) Z(t) = r cos ϑ(t). (A.23)

The kinetic energy of the particle can be expressed in terms of the generalized coordinates, m m m K = r˙2 = X˙ 2 + Y˙ 2 + Z˙ 2 = r2 ϑ˙ 2 +˙ϕ2 sin2 ϑ , (A.24) 2 2 2 and the potential energy is

V = mgZ = mgr cos ϑ. (A.25)

This yields the Lagrange function in the generalized coordinates, m m L = r˙2 − mgZ = r2 ϑ˙ 2 +˙ϕ2 sin2 ϑ − mgr cos ϑ, (A.26) 2 2 and the Euler–Lagrange equations yield the equations of motion of the particle, g ϑ¨ =˙ϕ2 sin ϑ cos ϑ + sin ϑ, (A.27) r d ϕ˙ sin2 ϑ = 0. (A.28) dt The conjugate momenta

∂L 2 ˙ pϑ = = mr ϑ (A.29) ∂ϑ˙ and ∂L p = = mr2ϕ˙ sin2 ϑ (A.30) ϕ ∂ϕ˙ 682 A Lagrangian Mechanics are just the angular momenta for rotation in ϑ or ϕ direction. The Hamilton function is the conserved energy

2 p2 ˙ pϑ ϕ H = pϑ ϑ + pϕϕ˙ − L = + + mgr cos ϑ = K + U. (A.31) 2mr2 2mr2 sin2 ϑ The immediately apparent advantage of this formalism is that it directly yields the correct equations of motion (A.27) and (A.28) for the system without ever having to worry about finding the force that keeps the particle on the sphere. Beyond that the formalism also provides a systematic way to identify conservation laws in mechanical systems, and if one actually wants to know the force that keeps the particle on the sphere (which is actually trivial here, but more complicated e.g. for a system of two particles which have to maintain constant distance), a simple extension of the formalism to the Lagrange equations of the first kind can yield that, too. The Lagrange function is not simply the difference between kinetic and potential energy if the forces are velocity dependent. This is the case for the Lorentz force. The Lagrange function for a non-relativistic charged particle in electromagnetic fields is m L = x˙ 2 + qx˙ · A − q. (A.32) 2 This yields the correct Lorentz force law mx¨ = q(E + v × B) for the particle, cf. Sect. 15.1. The relativistic versions of the Lagrange function for the particle can be found in Eqs (B.91) and (B.93).

Derivation of the Lagrange Equations for the Generalized Coordinates qa from d’Alembert’s Principle

A derivation of the Lagrange equations for the generalized coordinates of a constrained N-particle system from Newton’s equations works in the following way: On a microscopic level, the equations of motion of the constrained N-particle j system with coordinates xI ,1≤ i ≤ N,1≤ j ≤ 3, should be

d ˙ j + ∂ = j (mI xI ) j V(x1...N ) CI . (A.33) dt ∂xI

j j Here FI =−∂V(x1...N )/∂xI are the forces which the particles could respond j to through virtual (i.e. instantaneous) displacements δxI without violating the j constraints, whereas CI are the forces which enforce the constraints. Constraints prevent particles or systems of particles to move in certain directions, i.e. the constraint forces must act opposite to the directions of motion which are prevented A Lagrangian Mechanics 683 by the constraints. This implies that all the allowed instantaneous displacements δxI of the particles must be orthogonal to the constraint forces:2 δxI · CI = 0. (A.34) I

However, all the allowed movements under the constraints where encoded in the generalized coordinates qa in the form (A.10), and the condition on the allowed virtual displacements of the system therefore amounts to

∂x δq I · C = 0. (A.35) a ∂q I I a

We can also understand this equation in a complementary way: The constraint forces CI restrict the motions of the N-particle system to the allowed motions δxI = (∂xI /∂qa)δqa, but they cannot interfere in any way with those allowed motions. Both arguments lead to Eq. (A.35): The constraint forces must be orthogonal to the allowed motions.Thisisd’Alembert’s principle. Furthermore, since δqa is an otherwise unconstrained permissible shift of the system, d’Alembert’s principle can be expressed without it,

∂x I · C = 0. (A.36) ∂q I I a

Due to the Newton equations (A.33), d’Alembert’s principle (A.36) implies

∂x d ∂ I · (m x˙ j ) + V(x ) = 0. (A.37) ∂q dt I I ∂x j 1...N I a I

Note that this is not a trivial consequence of the Newton equations for the N particles, because the term in the bracket is not the Newton equation for the particle with mass mI . The Newton equation for that particle is Eq. (A.33). With the deﬁnition (A.6) of the Lagrange function of the N-particle system, we can write (A.37) in the form

∂x j d ∂L ∂L I − = 0. (A.38) ∂q dt ∂x˙ j ∂x j I,j a I I

2Equation (A.34) only limits the allowed virtual displacements of the system. It does not prevent the constraint forces from performing work on the system during the actual time evolution xI (t) = xI (q(t), t) of the system. 684 A Lagrangian Mechanics

To analyze the implications of (A.38), we repeat Eq. (A.11) for the velocity components in shorthand notation,

dx j ∂x j ∂x j x˙ j = I = q˙ I + I . (A.39) I dt a ∂q ∂t a a

This implies in particular the two equations

∂x˙ j ∂x j I = I (A.40) ∂q˙a ∂qa and

∂x˙ j ∂2x j ∂2x j I = q˙ I + I . (A.41) ∂q b ∂q ∂q ∂t∂q a b a b a

Substitution of (A.40)into(A.41) yields

∂x˙ j ∂2x˙ j ∂2x˙ j I = q˙ I + I . (A.42) ∂q b ∂q ∂q˙ ∂t∂q˙ a b b a a

Equation (A.40) also yields

∂2x˙ j I = 0, (A.43) ∂q˙a∂q˙b and therefore

d ∂x˙ j ∂2x˙ j ∂2x˙ j I = q˙ I + I , (A.44) dt ∂q˙ b ∂q ∂q˙ ∂t∂q˙ a b b a a and this implies with (A.42)

d ∂x˙ j ∂x˙ j I = I . (A.45) dt ∂q˙a ∂qa

With these preliminaries we can now revisit Eqs. (A.38). Substitution of Eqs. (A.40) and (A.45)into(A.38) yields ∂x˙ j d ∂L ∂x j ∂L d ∂x˙ j ∂x˙ j ∂L I − I + I − I = 0, (A.46) ∂q˙ dt ∂x˙ j ∂q ∂x j dt ∂q˙ ∂q ∂x˙ j I,j a I a I a a I or after combining terms: A Lagrangian Mechanics 685

d ∂x˙ j ∂L ∂L I − = 0. (A.47) dt ∂q˙ ∂x˙ j ∂q I,j a I a

j However, the coordinates xI are independent of the generalized velocities q˙a, and therefore Eq. (A.47) is just the Euler–Lagrange equation (A.17),

d ∂L(q,q)˙ ∂L(q,q)˙ − = 0. (A.48) dt ∂q˙a ∂qa

The Euler–Lagrange equations for a constrained particle system therefore follow from the Newton equations if we take into account d’Alembert’s principle. On the other hand, Newton’s equation for a particle is a special case of the Euler– Lagrange equations in the case of unconstrained motion. Classical mechanics as a theory of both constrained and unconstrained mechanical systems can therefore be based either on Hamilton’s principle with action (A.15), which only incorporates the actual mechanical degrees of freedom qa, or on the Newton equations (A.33) combined with d’Alembert’s principle. However, once this is realized, Hamilton’s principle provides the most efﬁcient way to derive the equations of motion and the conservation laws of a mechanical system.

Symmetries and Conservation Laws in Classical Mechanics

We call a set of ﬁrst order transformations

→ = − → = + t t t (t), qa(t) qa(t ) qa(t) δqa(t) (A.49) a symmetry of a mechanical system with action t1 S[q]= dt L(q(t),q(t),t)˙ (A.50) to if it changes the form dt L(q(t),q(t),t)˙ in ﬁrst order of (t) and δqa(t) at most by a term of the form dB = dt(dB/dt):

δ(dt L(q,q,t))˙ ≡ dt L(q (t ), q˙ (t ), t ) − dt L(q(t),q(t),t)˙ d = dt B (q(t), t). (A.51) dt δq, This is equivalent to the statement that the new coordinates qa(t ) satisfy the same differential equations with respect to t as the old coordinates qa(t) satisfy with respect to t. 686 A Lagrangian Mechanics

To see how this implies conservation laws in the mechanical system, we have to evaluate δ(dt L(q,q,t))˙ for the transformations (A.49). We have to take into account that (A.49) implies

d d dt = dt(1 −˙ (t)), = (1 +˙ (t)) , (A.52) dt dt and therefore also

d d d d δq˙ (t) = q (t ) − q (t) =˙ (t) q (t) + δq (t). (A.53) a dt a dt a dt a dt a The ﬁrst order change in dt L is therefore (with summation convention for the index a)

δ(dt L) = dt L(q (t ), q˙(t ), t ) − dt L(q(t),q(t),t)˙ ∂L d d ∂L ∂L = dt δqa + ˙ qa + δqa −˙ L − . (A.54) ∂qa dt dt ∂q˙a ∂t

Now we substitute

∂L d ∂L d ∂L δq˙a = δqa − δqa (A.55) ∂q˙a dt ∂q˙a dt ∂q˙a and

∂L d ∂L ∂L d ∂L dL ˙ q˙a − L = q˙a − L − q¨a − q˙a + ∂q˙a dt ∂q˙a ∂q˙a dt ∂q˙a dt

d ∂L ∂L d ∂L ∂L = q˙a − L + q˙a − + . (A.56) dt ∂q˙a ∂qa dt ∂q˙a ∂t

This yields

∂L d ∂L δ(dt L) = dt(δqa + q˙a) − ∂qa dt ∂q˙a

d ∂L + dt (δqa + q˙a) − L . (A.57) dt ∂q˙a

Comparison of Eqs. (A.51) and (A.57) implies an on-shell conservation law

d Q = 0 (A.58) dt δq, with the conserved charge A Lagrangian Mechanics 687

∂L ∂L Qδq, = L −˙qa − δqa + Bδq, . (A.59) ∂q˙a ∂q˙a

μ Bδq, is the one-dimensional version of the current K in Lagrangian ﬁeld theory μ and Qδq, is the one-dimensional version of the conserved current J ,seethe paragraph after Eq. (16.27). Bδq, = 0 in most cases. However, a noticeable exception are Galilei boosts in nonrelativistic N-particle mechanics (where I enumerates the particles). The Lagrange function

1 L = m x˙ 2 − V (|x − x |) (A.60) 2 I I IJ I J I I

∂L 1 H =−Q/ = x˙ · − L = m x˙ 2 + V (|x − x |), (A.62) I ∂x˙ 2 I I IJ I J I I I I

∂Q P = = m x˙ , (A.63) ∂ I I I and rotations δxI (t) =−ϕ × xI (t), which yields conservation of total angular momentum,

∂Q M = = m x × x˙ . (A.64) ∂ϕ I I I I

The corresponding conserved charges from the symmetries of the Lagrange function L =−mc c2 − x˙ 2(t) of a free relativistic particle are

∂L mx˙ p = = , (A.65) ∂x˙ 1 − (x˙ /c)2 688 A Lagrangian Mechanics

mc2 H = cp 0 = x˙ · p − L = = c p2 + m2c2, (A.66) 1 − (x˙ /c)2 M = x × p, (A.67)

μ μν and the conserved charges which follow from Lorentz invariance δx = ω xν, ωμν =−ωνμ,are

Mμν = xμpν − xνpμ, (A.68)

jk which of course includes the charges Mi = ij k M /2 from rotations. Appendix B The Covariant Formulation of Electrodynamics

Electrodynamics is a relativistic ﬁeld theory for every frequency or energy of electromagnetic waves because photons are massless. Understanding of electro- magnetism and of photon-matter interactions therefore requires an understanding of special relativity. Furthermore, we also want to understand the quantum mechanics of relativistic electrons and other relativistic particles, and the covariant formulation of electrodynamics is also very helpful as a preparation for relativistic wave equations like the Klein–Gordon and Dirac equations.

Lorentz Transformations

The scientiﬁc community faced several puzzling problems around 1900. Some of these problems led to the development of quantum mechanics, but two of the problems motivated Einstein’s Special Theory of Relativity: • In 1881 and 1887 Michelson had demonstrated that light from a terrestrial light source always moves with the same speed c in each direction, irrespective of Earth’s motion. • The basic equation of Newtonian mechanics, F = d(mu)/dt, is invariant under Galilei transformations of the coordinates:

t = t, x = x − vt. (B.1)

Therefore any two observers who use coordinates related through a Galilei transformation are physically equivalent in Newtonian mechanics. However, in 1887 (at the latest) it was realized that Galilei transformations do not leave Maxwell’s equations invariant, i.e. if Maxwell’s equations describe electromagnetic phenomena for one observer, they would not hold for another

© The Editor(s) (if applicable) and The Author(s), under exclusive license to 689 Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1 690 B The Covariant Formulation of Electrodynamics observer moving with constant velocity v relative to the ﬁrst observer (because it was assumed that the coordinates of these two observers are related through the Galilei transformation (B.1)). Instead, Voigt (1887) and Lorentz (1892–1904) realized that Maxwell’s equations would hold for the two observers if their coordinates would be related e.g. through a transformation of the form

ct − (v/c)x x − vt ct = ,x= ,y= y, z = z, (B.2) 1 − (v2/c2) 1 − (v2/c2) and they also realized that coordinate transformations of this kind imply that light would move with the same speed c in both coordinate systems,

x2 + y2 + z2 − c2 t2 = x 2 + y 2 + z 2 − c2 t 2. (B.3)

Voigt was interested in the most general symmetry transformation of the wave equations for electromagnetic ﬁelds, while Lorentz tried to explain the results of the Michelson experiment. In 1905 Einstein took the bold step to propose that then the coordinates measured by two observers with constant relative speed v must be described by transformations like (B.2), but not by Galilei transformations.1 This was a radical step, because it implies that two observers with non-vanishing relative speed assign different time coordinates to one and the same event, and they also have different notions of simultaneity of events. The same statement in another formulation: Two different observers with non-vanishing relative speed slice the four-dimensional universe differently into three-dimensional regions of simultaneity, or into three- dimensional universes. Einstein abandoned the common prejudice that everybody always assigns the same time coordinate to one and the same event. Time is not universal. The speed of light in vacuum is universal. In the following we use the abbreviations

v 1 β = ,γ= . (B.4) c 1 − β2

The transformation (B.2) and its inversion then read

ct = γ(ct − βx), x = γ(x− βct), y = y, z = z, (B.5)

ct = γ(ct + βx ), x = γ(x + βct ). (B.6)

The spatial origin x = y = z = 0ofthe(ct,x,y,z) system satisﬁes x = βct = vt (use x = 0inx = γ(x− βct)), and therefore moves with velocity vex relative

1This idea was also enunciated by Poincaré in 1904, but Einstein went beyond the statement of the idea and also worked out the consequences. B The Covariant Formulation of Electrodynamics 691 to the (ct,x,y,z) system. In the same way one ﬁnds that the spatial origin of the − (ct,x,y,z) system moves with velocity vex through the (ct ,x ,y ,z) system. Therefore this is the special Lorentz transformation between two coordinate frames with a relative motion with speed v in x-direction as seen from the unprimed frame, or in (−x)-direction as seen from the primed frame. Equation (B.2) tells us that for motion in a certain direction (x-direction in (B.2)), the coordinate in that direction is affected non-trivially by the transformation, while any orthogonal coordinate does not change its value. This immediately allows for a generalization of (B.2) in the case that the relative velocity v points in an arbitrary direction. It is convenient to introduce a rescaled velocity vector β = v/c and the corresponding unit vector βˆ = β/β = v/v.The(3 × 3)-matrix βˆ ⊗ βˆ T projects any vector x onto its component parallel to β,

ˆ ˆ T xβ = β ⊗ β · x, (B.7) while the component orthogonal to β is

ˆ ˆ T x⊥β = x − xβ = (1 − β ⊗ β ) · x. (B.8)

From the form of the special Lorentz transformation (B.2) we know that the coordinate |xβ | parallel to v will be rescaled by a factor

1 1 γ = = , (B.9) 1 − (v2/c2) 1 − β2 and be shifted by an amount −γvt =−γβct. Similarly, the time coordinate ct will be rescaled by the factor γ and be shifted by an amount −γβ|xβ |. Finally, nothing will happen to the transverse component x⊥β . We can collect these observations in a (4 × 4)-matrix equation relating the two four-dimensional coordinate vectors,

ct γ − γ βT ct = · (B.10) x − γ β 1 − βˆ ⊗ βˆ T + γ βˆ ⊗ βˆ T x

This is the general Lorentz transformation between two observers if the spatial sections of their coordinate frames were coincident at t = 0. The most general transformation of this kind also allows for constant shifts of the coordinates and for a rotation of the spatial axes,

ct γ − γ βT 1 0T ct − cT = · · , (B.11) x − γ β 1 − βˆ ⊗ βˆ T + γ βˆ ⊗ βˆ T 0 R x − X where R is a 3 × 3 rotation matrix. Without the coordinate shifts this is the most general proper orthochronous Lorentz transformation, where orthochronous refers 692 B The Covariant Formulation of Electrodynamics to the fact that we did not include a possible reversal of the time axis, and the designation proper excludes a reversal of an odd number of spatial axes2 in R, i.e. we require det(R) = 1. With the coordinate shifts included, (B.11) is denoted as an inhomogeneous Lorentz transformation or a Poincaré transformation. The Poincaré transformations (B.11) and the subset of Lorentz transformations (T = 0, X = 0) form the groups of proper orthochronous Poincaré transformations or proper orthochronous Lorentz transformations, respectively. Inclusion of time or spatial reﬂections yields the Poincaré group or the Lorentz group. The Lorentz group is apparently a subgroup of the Poincaré group, and the rotation group is a subgroup of the Lorentz group. In four-dimensional notation the 4-vector of coordinates is xμ = (ct, x), and the 4-vector shorthand for the transformation (B.11)is

μ μ ν ν x = ν(x − X ). (B.12) where the (4 × 4) transformation matrix is

T T μ γ − γ β 1 0 ={ ν}= · . (B.13) − γ β 1 − βˆ ⊗ βˆ T + γ βˆ ⊗ βˆ T 0 R

We will see below that it plays a role where we attach the indices for the explicit numerical representation of the matrix in terms of matrix elements. Usually, ifamatrixisgivenfor without explicitly deﬁning index positions, the default convention is that it refers to a superscript row index and a subscript column index, μ as above, ={ ν}. This is important, because as soon as a boost is involved (i.e. β = 0), we will ﬁnd that

μ μ μ ν(β) = ν (− β) = ν (β). (B.14)

Up to a possible reﬂection of t or of a spatial coordinate, the transformation equation (B.11) provides the general solution to the following problem: Find the most general coordinate transformation {ct, x}={ct,x,y,z}→{ct, x}={ct,x,y,z} which leaves the expression x2 − c2 t2 invariant, i.e. such that for arbitrary coordinate differentials c t, x we have

x2 − c2 t2 = x 2 − c2 t 2. (B.15)

This equation implies in particular that if one of our observers sees a light wave moving at speed c, then this light wave will also move with speed c for the second observer,

x2 − c2 t2 = 0 ⇔ x 2 − c2 t 2 = 0. (B.16)

2Reversal of two spatial axes corresponds to a proper rotation by π. B The Covariant Formulation of Electrodynamics 693

In fact it sufﬁces to require only that anything moving with speed c will also have speed c in the new coordinates, and that the spatial coordinates are Cartesian in both frames. Up to rescalings of the coordinates the most general coordinate transformation is then the general inhomogeneous Lorentz transformation (B.11). Any constant offset Xμ between coordinate systems vanishes for differences of coordinates. Equation (B.12) therefore implies the following equation for Lorentz transformation of coordinate differentials,

μ μ α dx = αdx . (B.17)

The condition (B.15),

dx2 − c2dt2 = dx 2 − c2dt 2 (B.18) can also be written as

μ ν α β ημνdx dx = ηαβ dx dx (B.19) with the special (4 × 4)-matrix (1 is the 3 × 3 unit matrix)

−1 0T η = , (B.20) μν 0 1

Equation (B.15) also implies

μ ν α β ημνdx dy = ηαβ dx dy (B.21) for any pair of Lorentz transformed 4-vectors dx and dy (simply insert the 4-vector dx+dy into (B.19)). This implies that Lorentz transformations leave the Minkowski metric ημν invariant:

μ ν ημν α β = ηαβ . (B.22)

If we multiply this equation with the components ηβγ of the inverse Minkowski tensor, we ﬁnd

μ ν βγ γ γ ημν α β η = δα ≡ ηα . (B.23)

This tells us a relation between the (4 × 4)-matrix with “pulled indices” and its inverse −1:

γ ν βγ −1 γ μ ≡ ημν β η = ( ) μ. (B.24)

Explicitly, if 694 B The Covariant Formulation of Electrodynamics

T T μ γ − γ β 1 0 { ν}= · , (B.25) − γ β 1 − βˆ ⊗ βˆ T + γ βˆ ⊗ βˆ T 0 R then

T T ν ρ σν γγβ 1 0 {μ }≡{ημρ σ η }= · . γ β 1 − βˆ ⊗ βˆ T + γ βˆ ⊗ βˆ T 0 R

β We can also “pull” or “draw” indices on four-vectors, e.g. dxα ≡ ηαβ dx = (−cdt, dx). Let us ﬁgure out how this 4-vector transforms under the Lorentz transformation (B.17):

= ν = ν α = ν αβ = β = −1 β dxμ ημνdx ημν αdx ημν αη dxβ μ dxβ dxβ ( ) μ.

Four-vectors with this kind of transformation behavior are denoted as covariant 4- vectors, while dxμ is an example of a contravariant 4-vector. Another example of a covariant 4-vector is the vector of partial derivatives

∂ 1 ∂ ∂ ≡ = , ∇ . (B.26) μ ∂xμ c ∂t

We can check that this is really a covariant 4-vector by calculating how it transforms under Lorentz transformations. According to the chain rule of differentiation we ﬁnd

α ∂ ∂x ∂ ∂ ≡ = . (B.27) μ ∂xμ ∂xμ ∂xα However, we have

α − ∂x − dxα = ( 1)α dx ν ⇒ = ( 1)α (B.28) ν ∂xμ μ and therefore

= −1 α = α ∂μ ( ) μ∂α μ ∂α. (B.29)

Pairs of co- and contravariant indices do not transform if they are summed over. Assume e.g. that F αβ are the components of a 4 × 4 matrix which transform according to

αβ μν μ ν αβ F → F = α β F . (B.30)

αβ The combination ∂αF then transforms under Lorentz transformations according to B The Covariant Formulation of Electrodynamics 695

μν = α μ ν βγ = ν α βγ = ν αγ ∂μF μ ∂α β γ F γ η β ∂αF γ ∂αF , (B.31) i.e. the summed index pair does not contribute to the transformation. Only “free” indices (i.e. indices which are not paired and summed from 0 to 3) transform under Lorentz transformations.

The Manifestly Covariant Formulation of Electrodynamics

Electrodynamics is a Lorentz invariant theory, i.e. all equations have the same form in all coordinate systems which are related by Poincaré transformations. However, this property is hardly recognizable if one looks at Maxwell’s equations in traditional notation,

1 ∂ ∇ · E = , ∇ × E + B = 0, (B.32) 0 ∂t

1 ∂ ∇ · B = 0, ∇ × B − E = μ j. (B.33) c2 ∂t 0 “Lorentz invariance” seems far from obvious: How, e.g. would the electric and magnetic ﬁelds transform under a Lorentz transformation of the coordinates? Apparently we seem to have three three-dimensional vectors and one scalar in the equations. We can combine the current density j and the charge density into a current 4-vector

j ν = (c, j). (B.34)

For the ﬁeld strengths it helps to recall that the homogeneous Maxwell’s equations are solved through potentials , A,

∂ B = ∇ × A, E =− A − ∇. (B.35) ∂t If one combines the potentials into a 4-vector,

Aμ = (−/c, A), (B.36) it is possible to realize that the electromagnetic ﬁeld strengths Ei, Bi are related to antisymmetric combinations of the 4-vectors ∂μ, Aν, 696 B The Covariant Formulation of Electrodynamics

⎛ ⎞ 0 −E1/c −E2/c −E3/c ⎜ ⎟ ⎜ E1/c 0 B3 −B2 ⎟ Fμν = ∂μAν − ∂νAμ = ⎝ ⎠. (B.37) E2/c −B3 0 B1 E3/c B2 −B1 0

This electromagnetic ﬁeld strength tensor F was introduced by Minkowski in 3 1907. The matrix elements Fμν are its covariant components. The contravariant components of F are ⎛ ⎞ 0 E1/c E2/c E3/c ⎜ ⎟ μν μα νβ μ ν ν μ ⎜ −E1/c 0 B3 −B2 ⎟ F = η η Fαβ = ∂ A − ∂ A = ⎝ ⎠. (B.38) −E2/c −B3 0 B1 −E3/c B2 −B1 0

From this one can easily read off the transformation behavior of the ﬁelds under Lorentz transformations,

μ μ μ α x → x = αx , (B.39)

→ = α ∂μ ∂μ μ ∂α, (B.40)

→ = α Aμ(x) Aμ(x ) μ Aα(x) (B.41)

→ = − = α β − Fμν(x) Fμν(x ) ∂μAν(x ) ∂νAμ(x ) μ ν (∂αAβ (x) ∂β Aα(x)) α β = μ ν Fαβ (x). (B.42)

Evaluation of Fμν(x ) for a boost

T μ γ − γ β { ν}= (B.43) − γ β 1 − βˆ ⊗ βˆ T + γ βˆ ⊗ βˆ T yields with β = v/c E (x ,t ) = γ E(x,t)+ v × B(x,t) − (γ − 1)βˆ βˆ · E(x,t) γ 2 = γ E(x,t)+ v × B(x,t) − v v · E(x,t) , (γ + 1)c2

3Minkowski [115]. A translation of his results for dielectric materials into contemporary tensor notation can be found in Dick [35]. B The Covariant Formulation of Electrodynamics 697

1 B (x ,t ) = γ B(x,t)− v × E(x,t) − (γ − 1)βˆ βˆ · B(x,t) c2 1 γ 2 = γ B(x,t)− v × E(x,t) − v v · B(x,t) , c2 (γ + 1)c2 or expressed in terms of the ﬁeld strength components parallel and perpendicular to the relative velocity v between the two observers,

E(x ,t ) = E(x,t), B(x ,t ) = B(x,t), (B.44) E⊥(x ,t ) = γ E⊥(x,t)+ v × B⊥(x,t) , (B.45)

1 B⊥(x ,t ) = γ B⊥(x,t)− v × E⊥(x,t) . (B.46) c2

Electric and magnetic ﬁelds mix under Lorentz transformations, i.e. the distinction between electric and magnetic ﬁelds depends on the observer. The equations

μν ν ∂μF =−μ0j (B.47) are the inhomogeneous Maxwell’s equations

1 1 ∂ ∇ · E = , ∇ × B − E = μ j, 2 0 (B.48) 0 c ∂t while the identities (with the four-dimensional -tensor, 0123 =−1)

κλμν κλμν ∂λFμν = 2 ∂λ∂μAν ≡ 0 (B.49) are the homogeneous Maxwell’s equations

∂ ∇ · B = 0, ∇ × E + B = 0. (B.50) ∂t These identities can also written in terms of the dual ﬁeld strength tensor ⎛ ⎞ 0 −B1 −B2 −B3 ⎜ ⎟ ˜ μν 1 μναβ ⎜ B1 0 E3/c −E2/c ⎟ F = Fαβ = ⎝ ⎠ (B.51) 2 B2 −E3/c 0 E1/c B3 E2/c −E1/c 0 as

˜ μν ∂μF = 0. (B.52) 698 B The Covariant Formulation of Electrodynamics

→ = + The gauge freedom Aμ(x) Aμ(x) Aμ(x) ∂μf(x)apparently leaves the ﬁeld strength tensor Fμν invariant. In conventional terms this is

(x) = (x) − f(x),˙ A (x) = A(x) + ∇f(x). (B.53)

We have written Maxwell’s equations explicitly as equations between 4-vectors,

μν ν ˜ μν ∂μF =−μ0j ,∂μF = 0, (B.54) and this ensures that they hold in this form for every inertial observer. This is the form invariance (or simply “invariance”) of Maxwell’s equations under Lorentz transformations. With the identiﬁcation of the current 4-vector j μ = (c, j), the local conservation law for charges can also be written in manifestly Lorentz invariant form,

∂ + ∇ · j = ∂ j μ = 0. (B.55) ∂t μ

Relativistic Mechanics

In special relativity it is better to express everything in quantities which transform linearly with combinations of the matrices and −1. As a consequence of the transformation law

μ μ ν dx = νdx , (B.56) ordinary velocities dx/dt and accelerations d2x/dt2 transform nonlinearly under Lorentz boosts, due to the transformation of the time coordinates in the denomina- tors. Therefore it is convenient to substitute the physical velocities and accelerations with “proper" velocities and accelerations, which do not require division by a transforming time parameter t. Suppose the x-frame is the frame of a moving object. In its own frame the trajectory of the object is x = 0. However, we know that the Lorentz transformation μ (B.56) leaves the product dx dxμ invariant,

μ = 2 − 2 2 = μ = 2 − 2 2 dx dxμ dx c dt dx dxμ dx c dt . (B.57)

Therefore we have in particular for the time dt ≡ dτ measured by the moving object along its own path x = 0

1 v2 dτ2 = dt2 − dx2 = 1 − dt2, (B.58) c2 c2 B The Covariant Formulation of Electrodynamics 699 i.e. up to a constant dt τ = dt 1 − (v2/c2) = . (B.59) γ

This is an invariant, i.e. it has the same value for each observer. Every observer will measure their own specific time interval t between any two events happening to the moving object, but all observers agree on the same value t τ = dt 1 − (v2/c2) (B.60) 0 which elapsed on a clock moving with the object. The time τ measured by an object between any two events happening to the object is denoted as the proper time or eigentime of the object. The definition of eigentime entails a corresponding definition of the proper velocity or eigenvelocity of an object in an observer’s frame: Divide the change in the object’s coordinates dx in the observer’s frame by the time interval dτ elapsed for the object itself while it was moving by dx:

dx dx U = = γ = γ v. (B.61) dτ dt This is a hybrid construction in the sense that a set of coordinate intervals dx measured in the observer’s frame is divided by a coordinate interval dτ measured in the object’s frame.4 The notion of proper velocity may seem a little artiﬁcial, but it is useful because it can be extended to a 4-vector using the fact that {dxμ}=(dx0,dx) = (cdt, dx) is a 4-vector under Lorentz transformations. If we deﬁne

dx0 cdt U 0 = = = γc, (B.62) dτ dτ then

U μ = dxμ/dτ = (U 0, U) = γ(c,v) (B.63)

μ μ μ α is a 4-vector which transforms according to U → U = αU under Lorentz transformations. This 4-velocity vector satisﬁes

2 μ μ ν 2 0 2 2 U ≡ U Uμ ≡ ημνU U = U − (U ) =−c . (B.64)

4There is a limit v ≤ c on the physical speed v =|v| of moving objects. No such limit holds for the “eigenspeed" |U|, but the speed of signal transmission relative to an observer is v, not |U|. 700 B The Covariant Formulation of Electrodynamics

The conservation laws

N N N N (in) = (out) (in) = (out) pi pi , Ei Ei (B.65) i=1 i=1 i=1 i=1 for momentum and energy in a collision would not be preserved under Lorentz transformations if the nonrelativistic definitions for momentum and energy would be employed, due to the nonlinear transformations of the particle velocities. This would mean that if momentum and energy conservation would hold for one observer, they would not hold for another observer with different velocity! However, the conservation laws are preserved if energy and momentum transform linearly, like a 4-vector, under Lorentz transformations. We have already μ identified 4-velocities {U }=γ(c,v) with the property limβ→0 U = v.This motivates the definition of the 4-momentum

p0 = mU 0, p = mU, (B.66) i.e. the relativistic deﬁnition of the spatial momentum of a particle of mass m and physical velocity v is mv p = mU = γmv = . (B.67) 1 − (v2/c2)

The physical meaning of the fourth component mc p0 = mU 0 = γmc = (B.68) 1 − (v2/c2) can be inferred from the nonrelativistic limit: v c yields

v2 p0 mc 1 + . (B.69) 2c2

This motivates the identiﬁcation of cp 0 with the energy of a particle of mass m and speed v:

mc2 E = cp 0 = γmc2 = . (B.70) 1 − (v2/c2)

Division of the two Eqs. (B.67) and (B.70) yields p v = c2 , (B.71) E and subtracting squares yields the relativistic dispersion relation B The Covariant Formulation of Electrodynamics 701

E2 − c2p2 = m2c4. (B.72)

μ 2 2 This is usually written as pμp =−m c . Equations (B.72) and (B.71) imply in particular for massless particles the relations E = cp and v = c. For the formulation of the relativistic version of Newton’s law, we observe that the rate of change of 4-momentum with eigentime deﬁnes a 4-vector with the units of force,

d dxμ dpμ d f μ = m = = γ pμ. (B.73) dτ dτ dτ dt It transforms linearly under Lorentz transformations because we divided a 4-vector dxμ or dpμ by invariants dτ2 or dτ, respectively. Conventionally, one still deﬁnes three-dimensional forces according to

d 1 F = p = f , (B.74) dt γ i.e. F is not the spatial component of a 4-vector, but f = γ F is. For the 0-component f 0 we ﬁnd with the relativistic dispersion relation E = c p2 + m2c2, d dx0 d dx0 d d E d f 0 = m = mγ = (γ mc) = = p2 + m2c2 dτ dτ dτ dt dτ dτ c dτ p d v d v = · p = · p = · f . (B.75) p2 + m2c2 dτ c dτ c

The 4-vector of the force is therefore

(f 0, f ) = (β · f , f ) = (γ β · F ,γF ). (B.76)

Multiplication of (cf. (B.75))

d E v = · f (B.77) dτ c c with c/γ gives energy balance in conventional form,

d c d d E = p · p = v · p = v · F . (B.78) dt p2 + m2c2 dt dt

The nonrelativistic Newton equation for motion of a charged particle in electromagnetic ﬁelds contains the Lorentz force 702 B The Covariant Formulation of Electrodynamics

F = qE + qv × B. (B.79)

We can get a hint at how the relativistic equation has to look like by expressing this combination of ﬁelds in terms of the ﬁeld strength tensor (B.37),

dx0 Ei = cF i = F i ,εi Bk = F i . (B.80) 0 0 dt jk j The latter equation implies

i i j k i j (v × B) = ε jkv B = F j v , (B.81) and therefore

dx0 dxj dxν F i = qEi + qεi vj Bk = qFi + qFi = qFi . (B.82) jk 0 dt j dt ν dt This would be a spatial part of a 4-vector if we would not derive with respect to the laboratory time t, but with respect to the eigentime τ of the charged particle:

dxν f i = γFi = qFi . (B.83) ν dτ The time component is then

dxi 1 dxi f 0 = qF0 = qγ E = γqβ · E (B.84) i dτ c i dt and the electromagnetic force 4-vector is

dxν f μ = qFμ = (γ qβ · E,γq(E + v × B)). (B.85) ν dτ The equation of motion of the charged particle in 4-vector notation is therefore

d2xμ dxν m = qFμ , (B.86) dτ2 ν dτ or

μ μ ν mx¨ (τ) = qF ν(x(τ))x˙ (τ). (B.87)

The time component yields after rescaling with c/γ again the energy balance equation

dE = qv · E. (B.88) dt B The Covariant Formulation of Electrodynamics 703

The spatial part is after rescaling with γ −1:

d p = q(E + v × B), (B.89) dt The only changes in (B.88) and (B.89) with respect to the nonrelativistic equations are the velocity dependences of E and p:

mv mc2 p = ,E= . (B.90) 1 − (v2/c2) 1 − (v2/c2)

Equations (B.87) are completely equivalent to Eqs. (B.89) and (B.88). Note that Eq. (B.88) is a consequence of (B.89) just like Eq. (B.87) with μ = 0isalsoa consequence of the other three equations with spatial values for μ. The virtue of Eqs. (B.87)isthemanifest covariance of these equations, since linearly transforming equations between 4-vectors must hold in every inertial frame. Contrary to this, covariance is not apparent in Eqs. (B.88) and (B.89), but since they are equivalent to the manifestly covariant Eqs. (B.87), they also must hold in every inertial frame. Covariance is only hidden in the nonlinear transformation behavior of Eqs. (B.88) and (B.89). However, for practical purposes Eqs. (B.88) and (B.89) are often more useful. The relativistic Lagrange function for a charged particle in terms of the laboratory time t is 2 2 L(t) =−mc c − x˙ (t) + qx˙ (t) · A(x(t), t) − q(x(t), t). (B.91)

This yields the canonical momentum

v = ∂L(t) = √ mc + = + pcan qA p qA, (B.92) ∂x˙ c2 − v2 and the equations of motion in the form (B.89). The relativistic action is 2 2 2 S = dt L(t) = − mc c dt − dx + qdx · A − qdt dxμ dxν dxμ = dτ − mc −η + qA = dτ L . (B.93) μν dτ dτ μ dτ (τ)

The formulation in terms of the eigentime τ of the particle yields the canonical μ ν 2 2 2 momentum (use ημν(dx /dτ)(dx /dτ) =−c from the equation c dτ = μ ν − ημνdx dx after the derivative)

∂L dxν p = (τ) = mη + qA = p + qA , (B.94) can,μ ∂x˙μ μν dτ μ μ μ 704 B The Covariant Formulation of Electrodynamics and the Lagrange equation is the manifestly covariant formulation (B.87)ofthe equations of motion. The action for a system of charged particles is with xa ≡ x(τa) 1 S = dτ L + d4x L = dτ − m c −˙x2 + q A (x ) ·˙x a a c F a a a a μ a a a a 1 4 μν − d xFμν(x)F (x). (B.95) 4μ0c

μ μ μ This action is invariant under constant spacetime translations δx = δxa =− , and we should have local conservation laws in terms of an energy-momentum tensor μ Tν . We can calculate this tensor using our general results from Sect. 16.2 also for the superposition (B.95) of one-dimensional and four-dimensional terms in the action. Equation (16.23) yields

0 = δS = δS|on−shell d ∂L 1 ∂L = dτ − ν a + d4x∂ ν∂ A · F − μL a dτ ∂x˙ν c μ ν ρ ∂ A F a a a μ ρ d ∂L = d4x dτ δ(x − x ) − ν a a a dτ ∂x˙ν a a a L 1 4 ν ∂ F μ + d x∂μ ∂νAρ · − LF . (B.96) c ∂μAρ

μ We can recast the contribution from the particle terms using (note ∂μ ≡ ∂/∂x = ∂/∂yμ(τ)) d ∂L d4x dτ δ(x − y(τ)) − ν dτ ∂y˙ν(τ) ∂L d = d4x dτ ν δ(x − y(τ)) ∂y˙ν(τ) dτ ∂L ∂ = d4x dτ ν y˙μ(τ) δ(x − y(τ)) ∂y˙ν(τ) ∂yμ(τ) ∂L = d4x∂ dτ δ(x − y(τ)) − ν y˙μ(τ). (B.97) μ ∂y˙ν(τ)

We can substitute (B.97)into(B.96) to ﬁnd 1 4 ν μ δS| − =− d x ∂ (B.98) on shell c μ ν B The Covariant Formulation of Electrodynamics 705 with the energy-momentum tensor (written for simplicity only for a single charged particle) L μ = μL − · ∂ F + − ∂L ˙μ ν ην F ∂νAρ c dτ δ(x x(τ)) ν x (τ) ∂μAρ ∂x˙ (τ) 1 1 = ∂ A · F μρ − F F μρ + A j μ μ ν ρ 4μ νρ ν 0 0 x˙ (τ)x˙μ(τ) + dτ mc2 ν δ(x − x(τ), (B.99) −˙x2(τ) where j μ(x) = qc dτ x˙μ(τ)δ(x − x(τ)). (B.100)

To make the tensor (B.99) gauge invariant, we add the identically conserved improvement term

μ 1 μρ 1 μρ μ δν =− ∂ρ(AνF ) =− ∂ρAν · F − Aνj . (B.101) μ0 μ0

This yields in eigentime parametrization (x˙2(τ) =−c2) the symmetric gauge invariant energy-momentum tensor μ 1 μρ 1 μ ρσ μ Tν = Fνρ F − ην Fρσ F + dτ mcx˙ν(τ)x˙ (τ)δ(x − x(τ)) μ0 4

μ 1 μρ 1 μ ρσ vνv = Fνρ F − ην Fρσ F + mc √ δ(x − x(t)). (B.102) μ0 4 c2 − v2 √ We used cdτ = c2 − v2dt in the ﬁnal step. Contrary to the 4-velocity x˙μ = U μ, the four quantities vμ = dxμ/dt = U μ/γ = (c, v) are not components of a 4-vector, but still convenient for the representation of the classical energy-momentum tensor after integration over the eigentime of the particle. We also note that we can write the current density (B.100) in the form

j μ(x) = qvμ(t)δ(x − x(t)). (B.103)

The results for the energy density, energy current density and momentum density of the classical particles plus ﬁelds system are

3 0 00 0 2 1 2 mac H = cP = T = E + B + δ(x − xa(t)), (B.104) 2 2μ 2 − 2 0 a c va 706 B The Covariant Formulation of Electrodynamics

3 0i 1 mac va S = ceiT = E × B + δ(x − xa(t)), (B.105) μ 2 − 2 0 a c va 1 i0 macva 1 P = eiT = 0E × B + δ(x − xa(t)) = S, (B.106) c 2 − 2 c2 a c va and the stress tensor is

0 2 1 2 1 T = E + B 1 − 0E ⊗ E − B ⊗ B 2 2μ0 μ0 va ⊗ va + mac δ(x − xa(t)). (B.107) 2 − 2 a c va

Classical Electromagnetic Hamiltonian in Coulomb Gauge

We can use spatial integration by parts in (B.104) to derive the classical Hamiltonian in Coulomb gauge, q ∇ · A(x,t)= 0,(x,t)= a . (B.108) 4π |x − x (t)| a 0 a

2 We note the following two contributions to 0E /2:

∂ ∂ ∇ · A ⇒− ∇ · A = 0, (B.109) 0 ∂t 0 ∂t and q 0 (∇(x,t))2 ⇒− 0 (x,t) (x,t)= (x,t) a δ(x − x (t)) 2 2 2 a a q q = a b δ(x − x (t)), (B.110) 8π |x (t) − x (t)| a a =b 0 a b where we left out the divergent self-energy contributions in the last term. The square of the magnetic ﬁeld can be written as

B2 = (∇ × A)2 = tr[(∇ ⊗ A) · (∇ ⊗ A)T ]−tr[(∇ ⊗ A) · (∇ ⊗ A)] ⇒ tr[(∇ ⊗ A) · (∇ ⊗ A)T ]. (B.111)

The classical electromagnetic Hamiltonian in Coulomb gauge is therefore B The Covariant Formulation of Electrodynamics 707 3 0 ˙ 2 1 H = d x A (x,t)+ ∂iAj (x,t)· ∂iAj (x,t) 2 2μ0 q q m c3 + a b + a . (B.112) 8π |x (t) − x (t)| 2 − 2 a =b 0 a b a c va(t)

The different contributions depend on time, but the time translation invariance of the system implies that H is constant if the system is closed.

Classical Electromagnetic Hamiltonian in Lorentz Gauge

We have in Lorentz gauge

1 ∂ ∇ · A + = 0,∂2Aμ =−μ j μ. (B.113) c2 ∂t 0

2 This implies the following contribution to 0E /2:

∂ ∂ 0 (∇)2 + ∇ · A ⇒ 0 (∇)2 − ∇ · A 2 0 ∂t 2 0 ∂t ∂2 = 0 (∇)2 + 0 = 0 (∇)2 + ( + ρ) ⇒ ρ − 0 (∇)2, 2 c2 ∂t2 2 0 2 where ρ(x,t)= qaδ(x − xa(t)). (B.114) a

On the other hand, the contribution from the magnetic ﬁeld is

B2 = (∇ × A)2 = tr[(∇ ⊗ A) · (∇ ⊗ A)T ]−tr[(∇ ⊗ A) · (∇ ⊗ A)] ⇒ tr[(∇ ⊗ A) · (∇ ⊗ A)T ]−(∇ · A)2 (B.115)

1 ∂ 2 = tr[(∇ ⊗ A) · (∇ ⊗ A)T ]− . (B.116) c4 ∂t

The classical electromagnetic Hamiltonian in Lorentz gauge is therefore 2 3 0 ∂A(x,t) 1 H = d x + ∂iAj (x,t)· ∂iAj (x,t) 2 ∂t 2μ0 708 B The Covariant Formulation of Electrodynamics

∂(x,t) 2 − 0 − 0 (∇(x,t))2 2c2 ∂t 2

3 mac + qa(xa(t), t) + . (B.117) 2 − 2 a a c va(t)

Relativistic Center of Mass Frame

Considerations of two-particle systems in relativistic quantum mechanics also require the relativistic notion of center of mass frames. If two particles have momenta p1 and p2 in an inertial frame, every inertial frame with the property = + that the total momentum P p1 p2 of the two-particle system in that frame vanishes is traditionally denoted as a center of mass frame for the two-particle system, although “zero total momentum frame” would be a more appropriate name. We will nevertheless continue to use the traditional name “center of mass frame”. Two center of mass frames for the system then differ at most by a combination of a translation and a rotation and possibly an inversion of the time axis. According to Eq. (B.11) the task to actually transform into a center of mass frame then amounts to ﬁnd a Lorentz boost into a frame moving with velocity v = cβ such that with E = E1 + E2

E P = γ P − γ β = 0 (B.118) c and

P ⊥ = P ⊥ = 0. (B.119)

The condition (B.119) implies that we have to choose βP , and that P = P , P = P . The condition (B.118) is then solved by

P β = c , (B.120) E E γ = . (B.121) E2 − c2P 2

The momentum vectors of the two particles in the center of mass frame are therefore

= =− =− p1,⊥ p1,⊥ p2,⊥ p2,⊥, (B.122) and B The Covariant Formulation of Electrodynamics 709

E p − E p = E − P = 2 1, 1 2, =− p1, p1, E1 p2,. E2 − c2P 2 E E2 − c2P 2 (B.123) The corresponding energies in the center of mass frame are

E2 − c2p2 + E E − c2p · p = E − 2 P · = 1 1 1 2 1 2 E1 E1 c p1, , E2 − c2P 2 E E2 − c2P 2

E2 − c2p2 + E E − c2p · p = 2 2 1 2 1 2 E2 . (B.124) E2 − c2P 2

For consistency we notice = + = 2 − 2 2 E E1 E2 E c P , (B.125) as also implied by P 2 = P 2 with P = 0. We also note that if we deﬁne the center of energy of the two-particle system,

x E + x E R = 1 1 2 2 , (B.126) E1 + E2 and if the particles do not interact, then the center of energy velocity is exactly the velocity (B.120) of the center of mass frame,

p + p R˙ = c2 1 2 = cβ. (B.127) E1 + E2

If the particles interact, we need to also take into account the contributions from the ﬁelds which mediate the interactions to the center of energy and to the energy and momentum balances of the system, see (18.227) and (18.228). Appendix C Completeness of Sturm–Liouville Eigenfunctions

Completeness of eigenfunctions of self-adjoint operators is very important in quantum mechanics. Formulating exact theorems and proofs in general situations is a demanding mathematical problem. However, the setting of Sturm–Liouville problems with homogeneous boundary conditions in one dimension is sufﬁciently simple to be treated in a single appendix.

Sturm–Liouville Problems

Sturm–Liouville problems are linear boundary value problems consisting of a second order differential equation

d dψ(x) g(x) − V(x)ψ(x) + E(x)ψ(x) = 0(C.1) dx dx in an interval a ≤ x ≤ b and homogeneous boundary conditions1 (Sturm 1836, Liouville 1837)

ψ(a) = 0,ψ(b)= 0. (C.2)

The functions g(x), V(x)and (x) are real and continuous in a ≤ x ≤ b, and we also assume that the functions g(x) and (x) are positive in a ≤ x ≤ b.

1General Sturm–Liouville boundary conditions would only require linear combinations of ψ(x) and ψ(x) to vanish at the boundaries, but for our purposes it is sufﬁcient to impose the special conditions ψ(a) = 0, ψ(b) = 0.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to 711 Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1 712 C Completeness of Sturm–Liouville Eigenfunctions

In ket notation without reference to a particular representation, we would write Eq. (C.1)as

1 E(x)|ψ(E)= pg(x)p|ψ(E)+V(x)|ψ(E). (C.3) h¯ 2

We can assume ψ(x) ≡x|ψ(E) to be a real function, and in this appendix we always assume that a and b are finite. We also require first order differentiability of g(x) and continuity of V(x)and (x). We can also assume ψ(a) > 0. We know that ψ(a) = 0 because ψ(a) = 0 together with ψ(a) = 0 and the Sturm–Liouville equation (C.1) would imply ψ(x) = 0. Furthermore, linearity of the Sturm–Liouville equation implies that we can always change the sign of ψ(x) to ensure ψ(a) > 0. Multiplication of Eq. (C.1) with ψ(x) and integration yields b b H [ψ]≡ dx g(x)ψ 2(x) + V(x)ψ2(x) = E dx (x)ψ2(x) ≡ Eψ|ψ, a a where the last equation defines the scalar product b φ|ψ= dx (x)φ(x)ψ(x). (C.4) a

It is easy to prove that (C.4) deﬁnes a scalar product since ψ|ψ≥0 and ψ|ψ= 0 ⇔ ψ(x) = 0, and

0 ≤ψ + λφ|ψ + λφ=ψ|ψ+2λψ|φ+λ2φ|φ (C.5) has a minimum for ψ|φ λ =− , (C.6) φ|φ which after substitution in (C.5) yields the Schwarz inequality

ψ|φ2 ≤ψ|ψφ|φ. (C.7)

The Sturm–Liouville equation (C.1) arises as an Euler–Lagrange equation from variation of the action

S[ψ]=Eψ|ψ−H[ψ] b = dx E(x)ψ2(x) − g(x)ψ 2(x) − V(x)ψ2(x) (C.8) a with ﬁxed endpoints ψ(a) and ψ(b). C Completeness of Sturm–Liouville Eigenfunctions 713

The stationary values of S[ψ] for arbitrary fixed endpoints ψ(a) and ψ(b) are S[ψ] = g(a)ψ(a)ψ (a) − g(b)ψ(b)ψ (b), (C.9) on−shell where the designation “on-shell” means that ψ(x) satisfies the Euler–Lagrange equation (C.1)ofS[ψ]. If we think of the Sturm–Liouville problem as a one-dimensional scalar field theory, G(x) = 1/4g2(x) would play the role of a metric in a ≤ x ≤ b and H[ψ] would be the energy of the field ψ(x) if ψ(x) is normalized, ψ|ψ=1. Suppose ψi(x) and ψj (x) are solutions of the Sturm–Liouville problem (C.1) and (C.2) with eigenvalues Ei and Ej , respectively. Use of the Sturm–Liouville equation (C.1) and partial integration yields x x d d Ei dξ (ξ)ψj (ξ)ψi(ξ) = dξ ψj (ξ) V(ξ)ψi(ξ) − g(ξ) ψi(ξ) a a dξ dξ x = + − d dξ V(ξ)ψj (ξ)ψi(ξ) g(ξ)ψj (ξ)ψi (ξ) g(x)ψj (x) ψi(x), a dx (C.10) and after another integration by parts we find x (Ei − Ej ) dξ (ξ)ψi(ξ)ψj (ξ) a d d = g(x) ψ (x) ψ (x) − ψ (x) ψ (x) . (C.11) i dx j j dx i

This equation implies for Ei = Ej

d d ln ψ (x) = ln ψ (x), (C.12) dx i dx j i.e. ψi(x) has to be proportional to ψj (x): There is no degeneracy of eigenvalues in the one-dimensional Sturm–Liouville problem. For x = b,Eq.(C.11) implies the orthogonality property

(Ei − Ej )ψi|ψj =0 (C.13) and taking into account the absence of degeneracy yields

ψi|ψj ∝δij . (C.14) 714 C Completeness of Sturm–Liouville Eigenfunctions

Liouville’s Normal Form of Sturm’s Equation

We can gauge the functions g(x) and (x) away through a transformation of variables x (ξ) x → X = dξ ,ψ(x)→ (X) = ((x)g(x))1/4 ψ(x). (C.15) a g(ξ)

This yields b (x) 0 ≤ X ≤ B = dx ,(0) = 0,(B)= 0, (C.16) a g(x) and the Sturm–Liouville equation (C.1) assumes the form of a one-dimensional Schrödinger equation,

d2 (X) − V (X)(X) + E(X) = 0 (C.17) dX2 with

V(x) g(x)(x) + (x)g(x) 5g(x)2(x) g2(x) V(X)= + − − (x) 42(x) 163(x) 16g(x)(x) g(x)(x) + . (C.18) 82(x)

Second order differentiability of (x) and g(x) is usually assumed. However, we only have to require continuity of the positive functions (x) and g(x) since we can deal with δ-function singularities in one-dimensional potentials. Equation (C.17)isLiouville’s normal form of the Sturm–Liouville equation.

Nodes of Sturm–Liouville Eigenfunctions

For the following reasoning we assume that we have smoothly continued the functions V(x), (x) > 0 and g(x) > 0 for all values of x ∈ R. It does not matter how we do that. To learn more about the nodes of the eigenfunctions ψi(x) of the Sturm– Liouville boundary value problem, let us now assume that ψ(x,λ) and ψ(x,μ) are solutions of the incomplete initial value problems C Completeness of Sturm–Liouville Eigenfunctions 715

d dψ(x,λ) λ(x)ψ(x, λ) = V(x)ψ(x,λ) − g(x) ,ψ(a,λ)= 0, (C.19) dx dx

d dψ(x,μ) μ(x)ψ(x, μ) = V(x)ψ(x,μ) − g(x) ,ψ(a,μ)= 0, (C.20) dx dx with λ>μ, but contrary to the boundary value problem (C.1) and (C.2)wedonot impose any conditions at x = b. In that case there exist solutions to the Sturm– Liouville equations for arbitrary values of the parameters λ, μ, and we can again require dψ(x,λ) dψ(x,μ) > 0, > 0. (C.21) dx x=a dx x=a

We recall the following facts from the theory of differential equations: The solution ψ(x,λ) to the initial value problem (C.19) is unique up to a multiplicative constant, and ψ(x,λ) depends continuously on the parameter λ. The last fact is important, because it implies that the nodes y(λ) of ψ(x,λ), ψ(y(λ),λ) = 0, depend continuously on λ. Continuity of y(λ) is used in the demonstration below that the boundary value problem (C.1) and (C.2) has a solution for every value of b. Multiplication of Eq. (C.19) with ψ(x,μ) and Eq. (C.20) with ψ(x,λ),integration from a to x>a, and subtraction of the equations yields x (λ − μ) dξ (ξ)ψ(ξ,λ)ψ(ξ,μ) a x d dψ(ξ,μ) d dψ(ξ,λ) = dξ ψ(ξ,λ) g(ξ) − ψ(ξ,μ) g(ξ) a dξ dξ dξ dξ

dψ(x,μ) dψ(x,λ) = g(x) ψ(x,λ) − ψ(x,μ) . (C.22) dx dx

Now assume that y(μ) is the ﬁrst node of ψ(x,μ) larger than a:

ψ(y(μ),μ) = 0,y(μ)>a. (C.23)

Substituting x = y(μ) in (C.22) yields y(μ) dψ(x,μ) (λ − μ) dx (x)ψ(x,λ)ψ(x,μ) = g(y(μ))ψ(y(μ), λ) . a dx x=y(μ)

We know that

(λ − μ)(x)ψ(x,μ) > 0 (C.24) 716 C Completeness of Sturm–Liouville Eigenfunctions for a

This implies ψ(x,λ) must change its sign at least once for a a of the function ψ(x,λ) moves closer to a if λ increases. We are not really concerned with differentiability properties of the leftmost node y(λ), but we can express the previous observation also as

dy(λ) y(λ) > a, < 0. (C.26) dλ

Now assume that λ is small enough2 so that even y(λ) > b. Then we can increase the parameter λ until we reach a value λ = E1 such that y(E1) = b. This is then the lowest eigenvalue of our original Sturm–Liouville boundary value problem (C.1), and the corresponding eigenfunction is

ψ1(x) = ψ(x,λ = E1). (C.27)

The eigenfunction ψ1(x) for the lowest eigenvalue E1 has no nodes in aa,

2The alert reader might worry that all y(λ) might be smaller than b, so that there is no ﬁnite small value λ with y(λ)>b, or otherwise that all y(λ) might be larger than b, so that no ﬁnite value E1 with y(E1) = b would exist. These cases can be excluded through Sturm’s comparison theorem, to be discussed later. C Completeness of Sturm–Liouville Eigenfunctions 717

We know

(λ − μ)(x)ψ(x,μ) < 0 (C.30) for y1(μ) < x < y2(μ), and dψ(x,μ) dψ(x,μ) g(y1(μ)) < 0,g(y2(μ)) > 0. (C.31) dx dx x=y1(μ) x=y2(μ)

This tells us that ψ(x,λ) has to change sign in the interval y1(μ) < x < y2(μ), i.e. it must have at least one node there. We know that the ﬁrst node y1(λ) < y1(μ) is outside of this interval. Therefore we can infer that at least the second node y2(λ) of ψ(x,λ) must be smaller than y2(μ): y2(λ) < y2(μ). We can repeat this reasoning for the pair of adjacent nodes yn−1(μ), yn(μ) of ψ(x,μ), and we always ﬁnd for λ>μthat yn(λ) < yn(μ),

dy (λ) a

All nodes of the function ψ(x,λ) on the right-hand side of x = a move closer to a if λ increases. Therefore we can repeat the reasoning above which had let us to the ﬁrst solution ψ1(x) with eigenvalue E1 of our Sturm–Liouville problem. To ﬁnd the second eigenfunction, we increase λ>E1 until we hit a value λ = E2 such that y2(E2) = b, and the corresponding eigenfunction

ψ2(x) = ψ(x,E2) (C.33) will have exactly one node y1(E2) in the interval, a

ψn(x) = ψ(x,En), (C.34) and this function will have n−1 nodes a

Sturm’s Comparison Theorem and Estimates for the Locations of the Nodes yn(λ)

Sturm’s comparison theorem makes a statement about the change of the nodes yn > a of the solution ψ(x,λ) of 718 C Completeness of Sturm–Liouville Eigenfunctions

d dψ(x,λ) g(x) + (λ(x) − V(x)) ψ(x,λ) = 0,ψ(a,λ)= 0, (C.35) dx dx if the functions g(x), (x) and V(x)change. To prove the comparison theorem, we do not use Liouville’s normal form, but perform the following simple transformation of variables, x = dx = X ,(X,λ) ψ(x,λ). (C.36) a g(x )

This transforms (C.35) into the following form,

d2(X,λ) + (λR(X) − V(X)) (X,λ) = 0,(0,λ)= 0, (C.37) dX2

R(X) = g(x)(x) > 0,V(X)= g(x)V(x), (C.38) and the nodes Yn > 0of(X,λ) are related to the nodes yn >aof ψ(x,λ)through yn dx Yn = . (C.39) a g(x)

Now we consider another Sturm–Liouville problem of the form (C.37), but with different functions

λS(X) − W(X) > λR(X) − V(X), (C.40)

d2(X, λ) + (λS(X) − W(X)) (X, λ) = 0,(0,λ)= 0, (C.41) dX2 and we denote the positive nodes of (X, λ) with Zn. We also require again (0)>0, (0)>0. Equations (C.37) and (C.41)imply Yn dX[V(X)− W(X)− λ (R(X) − S(X))] (X, λ)(X, λ) Y − n 1 d(X,λ) d(X,λ) = (Yn,λ) − (Yn−1,λ) . (C.42) dX dX X=Yn X=Yn−1

The following terms in (C.42) have all the same sign, [V(X)− W(X)− λ (R(X) − S(X))] (X,λ) , (C.43) Yn−1

This implies that (X, λ) must change its sign in Yn−1

Increasing λR(X) − V(X)moves the nodes Yn > 0 of the function (X,λ) to the left. From this we can ﬁrst derive bounds for the nodes Yn > 0 which arise from the nodes of the solutions of

+ − min(X, λ) (λRmax Vmin) min(X, λ) = + − = min(X, λ) gmax (λmax Umin) min(X, λ) 0, (C.46)

min(0,λ)= 0, (C.47) and

+ − max(X, λ) (λRmin Vmax) max(X, λ) = + − = max(X, λ) gmin (λmin Umax) max(X, λ) 0, (C.48)

max(0,λ)= 0. (C.49)

Here we use the bounds of the continuous functions g(x), V(x), (x) on a ≤ x ≤ b,

0 <min ≤ (x) ≤ max. (C.51)

The solutions of both Eqs. (C.46) and (C.48) have nodes if (recall that both g(x) > 0 and (x) > 0)

λ>Umin/max, (C.52) and the two solutions are min(X, λ) ∝ sin gmax (λmax − Umin)X , (C.53) max(X, λ) ∝ sin gmin (λmin − Umax)X . (C.54)

This yields bounds for the nodes Yn > 0of(X,λ), 720 C Completeness of Sturm–Liouville Eigenfunctions

nπ nπ √ ≤ Yn ≤ √ . (C.55) gmax (λmax − Umin) gmin (λmin − Umax)

However, we also know from Eq. (C.39) that gminYn ≤ yn − a ≤ gmaxYn, and therefore3

gminnπ gmaxnπ a + √ ≤ yn ≤ a + √ . (C.56) gmax (λmax − Umin) gmin (λmin − Umax)

This implies in particular that there is no accumulation point for the nodes yn of ψ(x,λ), and yn must grow like n for large n. For our previous proof that ψ1(x) (C.27) has its first node at y1 = b, we needed the assumption that there are small enough values of λ such that the first node y1(λ) of ψ(x,λ) satisfies y1(λ) > b. We can now confirm that from the lower bound in (C.56). It will suffice to choose

U U g2 π 2 min <λ< min + min . 2 (C.57) max max maxgmax(b − a)

We also needed the assumption that for large enough λ the first node y1(λ) > a would be smaller than b. This is easily confirmed from the upper bound in (C.56). It is sufficient to choose

U g2 π 2 λ> max + max . 2 (C.58) min mingmin(b − a)

Eigenvalue Estimates for the Sturm–Liouville Problem

We have found that the Sturm–Liouville boundary value problem (C.1) and (C.2) has an increasing, non-degenerate set of eigenvalues

S[ψ]=Eψ|ψ−H[ψ] b = dx E(x)ψ2(x) − g(x)ψ 2(x) − V(x)ψ2(x) . (C.60) a

For every continuous function ψ(x)in a ≤ x ≤ b we deﬁne the normalized function

3These bounds can be strengthened by a longer proof, but the present result is completely sufﬁcient for our purposes. C Completeness of Sturm–Liouville Eigenfunctions 721

ψ(x) ψ(x)ˆ = √ . (C.61) ψ|ψ

Since S[ψ] is homogeneous in ψ, ψ(x) is a stationary point of S[ψ] if and only if ψ(x)ˆ is a stationary point of

S[ψˆ ]=E − H [ψˆ ], (C.62) which implies also that ψ(x)ˆ is a stationary point of the functional b H [ψ] dx g(x)ψ 2(x) + V(x)ψ2(x) H[ψˆ ]= = a . (C.63) ψ|ψ b 2 a dx (x)ψ (x) ˆ We have already found that there is a discrete subset ψn(x), n ∈ N, of stationary ˆ ˆ ˆ points of H[ψ] which satisfy the boundary conditions ψn(a) = 0, ψn(b) = 0, and are mutually orthogonal, ˆ ˆ ψm|ψn=δmn. (C.64)

Use of the Sturm–Liouville equation and the boundary conditions yields the values ˆ ˆ of the functional H [ψ] at the stationary points ψn(x), ˆ H [ψn]=En. (C.65)

We already know E1

Fa,b = {ψ(x),a ≤ x ≤ b|ψ(a) = 0,ψ(b)= 0, ψ|ψ=1} , (C.67) and in general we have a minimum ˆ H [ψn]=En (C.68) on the space of functions

F (n) = { ≤ ≤ | = = | = | = a,b ψ(x),a x b ψ(a) 0,ψ(b) 0, ψ ψ 1, ψi ψ 0, 1 ≤ i ≤ n − 1} . (C.69) 722 C Completeness of Sturm–Liouville Eigenfunctions

ˆ The explicit form of H[ψ] in Eq. (C.63) shows that all the eigenvalues En increase if g(x) increases or V(x)increases or (x) decreases. However, those continuous functions must be bounded on the ﬁnite interval a ≤ x ≤ b,

0 <min ≤ (x) ≤ max. (C.71)

Therefore we can replace those functions with their extremal values to derive estimates for the eigenvalues En. The Sturm–Liouville problems for the extremal values are + − = gmin/maxψn (x) En,min/maxmax/min Umin/max ψn(x) 0, (C.72)

ψn(a) = 0,ψn(b) = 0, (C.73) with solutions

x − a ψ (x) ∝ sin nπ (C.74) n b − a and corresponding eigenvalues

1 n2π 2 E = U + g . n,min/max min/max min/max 2 (C.75) max/min (b − a)

This implies the bounds

2 2 2 2 1 + n π ≤ ≤ 1 + n π Umin gmin 2 En Umax gmax 2 . max (b − a) min (b − a)

In particular, at most a ﬁnite number of the lowest eigenvalues En can be negative, and the eigenvalues for large n must grow like n2. Both of these observations are crucial for the proof that the set ψn(x) of eigenfunctions of the Sturm–Liouville problem (C.1) and (C.2) provide a complete basis for the expansion of piecewise continuous functions in a ≤ x ≤ b.

Completeness of Sturm–Liouville Eigenstates

We now assume that the Sturm–Liouville eigenstates are normalized,

ψi|ψj =δij . (C.76) C Completeness of Sturm–Liouville Eigenfunctions 723

Let φ(x) be an arbitrary smooth function on a ≤ x ≤ b with φ(a) = 0 and φ(b) = 0, and deﬁne

n ϕn(x) = φ(x) − ψi(x)ψi|φ. (C.77) i=1

Then we have

n 2 0 ≤ϕn|ϕn=φ|φ− ψi|φ , (C.78) i=1 i.e. for all n we have a Bessel inequality

n 2 φ|φ≥ ψi|φ . (C.79) i=1

We also have ϕn|ψi=0, 1 ≤ i ≤ n, and ϕn(a) = 0, ϕn(b) = 0, i.e.

∈ F (n+1) ϕn(x) a,b . (C.80)

Therefore the minimum property of the eigenvalue En+1 implies

H [ϕn] En+1 ≤ . (C.81) ϕn|ϕn

We have n b [ ]= [ ]− | + H ϕn H φ 2 ψi φ dx g(x)φ (x)ψi (x) V(x)φ(x)ψi(x) i=1 a n b + | | + ψi φ ψj φ dx g(x)ψi (x)ψj (x) V(x)ψiψj (x) . i,j=1 a (C.82)

In the ﬁrst sum, partial integration and use of the Sturm–Liouville equation yields b b + = dx g(x)φ (x)ψi (x) V(x)φ(x)ψi(x) Ei dx (x)φ(x)ψi(x) a a

= Eiψi|φ. (C.83)

In the double sum, partial integration and use of the Sturm–Liouville equation yields 724 C Completeness of Sturm–Liouville Eigenfunctions b b + = dx g(x)ψi (x)ψj (x) V(x)ψiψj (x) Ei dx (x)ψi(x)ψj (x) a a

= Eiδij . (C.84)

This implies

n 2 H [ϕn]=H [φ]− Eiψi|φ . (C.85) i=1

Since at most ﬁnitely many of the eigenvalues Ei can be negative, Eq. (C.85) tells us that the functional H [ϕn] must remain bounded from above for n →∞,e.g.for

N 2 H[ϕn]≤H [φ]+ |Ei|ψi|φ . (C.87) i=1

On the other hand, Eq. (C.81) yields for n>N(to ensure En+1 > 0),

n [ ] 2 H ϕn ϕn|ϕn=φ|φ− ψi|φ ≤ (C.88) En+1 i=1

2 and since En+1 grows like n for large n while H [ϕn] must remain bounded, we ﬁnd the completeness relation 2 b n lim ϕn|ϕn= lim dx (x) φ(x) − ψi(x)ψi|φ = 0 (C.89) n→∞ n→∞ a i=1 or equivalently,

n φ|φ= lim φ|ψiψi|φ. (C.90) n→∞ i=1

Completeness of the series

∞ ψi(x)ψi|φ∼φ(x) (C.91) i=1 C Completeness of Sturm–Liouville Eigenfunctions 725 in the sense of Eq. (C.89) is denoted as completeness in the mean, and is sometimes also expressed as

n l.i.m.n→∞ ψi(x)ψi|φ=φ(x), (C.92) i=1 where l.i.m. stands for “limit in the mean”. Completeness in the mean says that the ∞ | series i=1 ψi(x) ψi φ approximates φ(x) in the least squares sense. Completeness in the mean also implies for the two piecewise continuous functions f and g

∞ ∞ f(x)± g(x) ∼ ψi(x)ψi|f ± ψi(x)ψi|g (C.93) i=1 i=1 and therefore

n 1 = lim |ψiψi| (C.95) n→∞ i=1 in the calculation of matrix elements between sufﬁciently smooth functions (where “sufﬁciently smooth = continuously differentiable to a required order” depends on the operators we use). This is all that is really needed in quantum mechanics. However, for piecewise smooth functions, the relation also holds pointwise almost everywhere (see remark 3 below). I would like to add a few remarks: 1. The completeness property (C.89) also applies to piecewise continuous functions in a ≤ x ≤ b and functions which do not vanish at the boundary points, because every piecewise continuous function can be approximated in the mean by a smooth function which vanishes at the boundaries. 2. If φ(x)is a smooth function satisfying the Sturm–Liouville boundary conditions, as we have assumed in the derivation of (C.89), the series under the integral sign will even converge uniformly to φ(x),

n lim ψi(x)ψi|φ=φ(x), (C.96) n→∞ i=1 726 C Completeness of Sturm–Liouville Eigenfunctions

i.e. for all a ≤ x ≤ b and all values >0, there exists an n( ) such that n − | ≥ φ(x) ψi(x) ψi φ < if n n( ). (C.97) i=1

Uniformity of the convergence refers to the fact that the same n( ) ensures (C.97) for all a ≤ x ≤ b. 3. If φ(x) is piecewise smooth in a ≤ x ≤ b, it can still be expanded pointwise in Sturm–Liouville eigenstates. Except for points of discontinuity of φ(x), and except for the boundary points if φ(x) does not satisfy the same boundary conditions as the eigenfunctions ψi(x), the expansion

n φ(x) = lim ψi(x)ψi|φ (C.98) n→∞ i=1

holds pointwise, and the series converges uniformly to φ(x) in every closed interval which excludes discontinuities of φ(x) (and the series converges to the arithmetic mean in the points of discontinuity). The boundary points must also be excluded if φ(x) does not satisfy the Sturm–Liouville boundary conditions. For example, the functions 2 nπx x|s =s (x) = sin ,n∈ N, (C.99) n n a a

provide a complete set of eigenfunctions on 0 ≤ x ≤ a for the Sturm–Liouville problem

d2 + k2 s (x) = 0,s(0) = s (a) = 0. (C.100) dx2 n n n n

On the other hand, the functions 2 − δ nπx x|c =c (x) = n,0 cos ,n∈ N , (C.101) n n a a 0

provide a complete set of eigenfunctions on 0 ≤ x ≤ a for the Sturm–Liouville problem

2 d + k2 c (x) = 0,c(0) = c (a) = 0. (C.102) dx2 n n n n | = | | The expansion cm n sn sn cm of the basis (C.101) in terms of the basis (C.99) follows from C Completeness of Sturm–Liouville Eigenfunctions 727

2 1 − (−)n−m n s |c = . (C.103) n m 2 − 2 1 + δm,0 n m π

On the level of the unrenormalized cosine functions, this yields

∞ πx 4 2n + 1 πx cos 2m = sin (2n + 1) , (C.104) a π (2n + 1)2 − 4m2 a n=0

∞ πx 8 n πx cos (2m + 1) = sin 2n , (C.105) a π 4n2 − (2m + 1)2 a n=1

pointwise in 0

We use the following equation as a deﬁnition of Hermite polynomials,

1 d n 1 H (x) = exp x2 x − exp − x2 , (D.1) n 2 dx 2 because we initially encountered them in this form in the solution of the harmonic oscillator in Chap. 6. We can use the identity d 1 d 1 x + f(x)= exp − x2 exp x2 f(x) (D.2) dx 2 dx 2 to rewrite Eq. (D.1) in the form 1 1 d 1 n 1 H (x) = exp x2 2x − exp − x2 exp x2 exp − x2 n 2 2 dx 2 2 1 1 d 1 1 n = exp x2 2x − exp − x2 exp x2 exp − x2 2 2 dx 2 2

d n = 2x − 1, (D.3) dx or we can use the identity d 1 d 1 x − f(x)=−exp x2 exp − x2 f(x) (D.4) dx 2 dx 2 to rewrite Eq. (D.1) in the Rodrigues form

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d n H (x) = exp x2 − exp −x2 . (D.5) n dx

The Rodrigues formula implies ∞ n ∞ n n z = 2 ∂ − − 2 z Hn(x) exp x n exp (x z) n! ∂z = n! n=0 n=0 z 0 = exp x2 exp −(x − z)2 = exp 2xz − z2 . (D.6)

The residue theorem then also yields the representation ! n! exp 2xz − z2 H (x) = dz , (D.7) n 2πi zn+1 where the integration contour encloses z = 0 in the positive sense of direction, i.e. counter clockwise. Another useful integral representation for the Hermite polynomials follows from (D.3) and the equation ∞ ∞ ∂ n du(2u)n exp −(u + v)2 = du − 2v − exp −(u + v)2 −∞ −∞ ∂v

∂ n √ = − 2v − π. (D.8) ∂v

This yields in particular for v =−ix, ∞ √ n 2 n du(2u) exp −(u − ix) = i πHn(x). (D.9) −∞

Combination of Eqs. (D.6) and (D.9) yields Mehler’s formula [111], ∞ n ∞ ∞ z 1 n 2 Hn(x)Hn(x ) = Hn(x)√ du(−2iuz) exp −(u − ix ) n! πn! −∞ n=0 n=0 ∞ 1 = √ du exp −4ixuz + 4u2z2 exp −(u − ix )2 π −∞ 1 z x2 + x 2 − xx = √ exp −4z . (D.10) 1 − 4z2 1 − 4z2

This requires |z| < 1/2 for convergence. In Sects. 6.3 and 13.1 we need this in the form for |z| < 1, D Properties of Hermite Polynomials 731

∞ n 2 2 z x + x H (x)H (x ) exp − (D.11) n n 2nn! 2 n=0 1 1 + z2 x2 + x 2 − 4zxx = √ exp − . (D.12) 1 − z2 2 1 − z2

Indeed, applications of this equation for the harmonic oscillator are usually in the framework of distributions and require the limit |z|→1. In principle we should therefore replace the corresponding phase factors z in Sects. 6.3 and 13.1 with z exp(− ), and take the limit →+0 after applying any distributions which are derived from (D.12). Appendix E The Baker–Campbell–Hausdorff Formula

The Baker–Campbell–Hausdorff formula explains how to combine the product of operator exponentials exp(A) · exp(B) into a single operator exponential exp[(A, B)], if the series expansion for (A, B) provided by the Baker– Campbell–Hausdorff formula converges. We try to determine (A, B) as a power series in a parameter λ,

∞ n exp[λA]·exp[λB]=exp[(λA, λB)],(λA,λB)= λ cn(A, B). n=1

We also use the notation of the adjoint action of an operator A on an operator B,

A(ad) ◦ B =−[A, B]. (E.1)

We start with

exp[αA]·exp[βB]=exp[(αA, βB)]. (E.2)

This implies with Lemma (6.46) the equations

∞ ∂ (−)n n B = exp[−(αA, βB)] exp[(αA, βB)]= [(αA,βB),∂ ] ∂β n! β n=1 ∞ (−)n n−1 =− [ (αA,βB),∂ (αA, βB)] n! β n=1 ∞ 1 n−1 = (αA, βB)(ad) ◦ ∂ (αA, βB) n! β n=1

© The Editor(s) (if applicable) and The Author(s), under exclusive license to 733 Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1 734 E The Baker–Campbell–Hausdorff Formula

exp[(αA, βB)(ad)]−1 = ◦ ∂β (αA, βB) (E.3) (αA, βB)(ad) and

∞ ∂ 1 n A =−exp[(αA, βB)] exp[−(αA, βB)]=− [(αA,βB),∂ ] ∂α n! α n=1 ∞ 1 n−1 = [ (αA,βB),∂ (αA, βB)] n! α n=1 ∞ 1 n−1 = −(αA, βB)(ad) ◦ ∂ (αA, βB) n! α n=1 1 − exp[−(αA, βB)(ad)] = ◦ ∂α(αA, βB). (E.4) (αA, βB)(ad)

For the inversion of these equations, we note

− exp(z) − 1 1 z exp(−z/2) = = z z exp(z) − 1 exp(z/2) − exp(−z/2) + − = z exp(z/2) exp( z/2) − z 2 exp(z/2) − exp(−z/2) 2 ∞ z z z (−)n+1 z = coth − = 1 + B z2n − , (E.5) 2 2 2 (2n)! n 2 n=1 − 1 − exp(−z) 1 z exp(z/2) = = z z 1 − exp(−z) exp(z/2) − exp(−z/2) + − = z exp(z/2) exp( z/2) + z 2 exp(z/2) − exp(−z/2) 2 ∞ z z z (−)n+1 z = coth + = 1 + B z2n + , (E.6) 2 2 2 (2n)! n 2 n=1 where the coefﬁcients Bn are Bernoulli numbers. The previous equations yield (with (αA, βB)(ad) ◦ A =−[(αA,βB),A])

(αA, βB)(ad) (αA, βB)(ad) ∂ (αA, βB) = coth ◦ A α 2 2 1 − [(αA,βB),A], (E.7) 2 E The Baker–Campbell–Hausdorff Formula 735

(αA, βB)(ad) (αA, βB)(ad) ∂ (αA, βB) = coth ◦ B β 2 2 1 + [(αA,βB),B], (E.8) 2

∂λ(λA, λB) = ∂α(αA, βB) + ∂β (αA, βB) α=β=λ (λA, λB)(ad) (λA, λB)(ad) = coth ◦ (A + B) 2 2 1 + [A − B,(λA,λB)], (E.9) 2 i.e.

∞ (−)n+1 ∂ (λA, λB) = A + B + B ·[(λA, λB)(ad)]2n ◦ (A + B) λ (2n)! n n=1 1 + [A − B,(λA,λB)]. (E.10) 2 Equation (E.10) provides us with a recursion relation for the n-th order coefﬁcient functions cn(A, B),

n/2 1 (−)m+1 (n + 1)c + (A, B) = [A − B,c (A, B)]+ B n 1 2 n (2m)! m m=1 × [ [ [ [ + ]] ]] ck2m (A, B), ..., ck2 (A, B), ck1 (A, B), A B ... , (E.11) 1≤k1,k2,...k2m, k1+...+k2m=n with

c0(A, B) = 0,c1(A, B) = A + B. (E.12)

The ﬂoor function x maps to the next lowest integer smaller or equal to x, i.e. n/2=n/2ifn is even, n/2=(n − 1)/2ifn is odd. Equation (E.11) yields for the next three terms

1 c (A, B) = [A, B], (E.13) 2 2 1 1 c (A, B) = [A − B,[A, B]] + B [A + B,[A + B,A + B]] 3 12 6 1 1 1 = [A, [A, B]] + [B,[B,A]], (E.14) 12 12 736 E The Baker–Campbell–Hausdorff Formula

1 c (A, B) = [A − B,[A, [A, B]] + [B,[B,A]]] 4 96 1 + B [A + B,[[A, B],A+ B]] 16 1 1 = [A, [A, [A, B]]] − [B,[B,[B,A]]] + [A, [B,[B,A]]] 96 −[B,[A, [A, B]]] − [A, [A, [A, B]]] + [B,[B,[B,A]]] +[A, [B,[B,A]]] − [B,[A, [A, B]]]

1 1 1 = [A, [B,[B,A]]] − [B,[A, [A, B]]] = [A, [B,[B,A]]]. 48 48 24 The Jacobi identity

[A, [B,C]] + [B,[C, A]] + [C, [A, B]] = 0 (E.15) was used in the last step for c4. Appendix F The Logarithm of a Matrix

= = ∞ n ! Exponentials of square matrices G, M exp G n=0 G /n , are frequently used for the representation of symmetry transformations. Indeed, the properties of continuous symmetry transformations are often discussed in terms of their ﬁrst order approximations 1 + G, where it is assumed that continuity of the symmetries allows for parameter choices such that max |Gij | 1. It is therefore of interest that the logarithm G = ln M of invertible square matrices can also be deﬁned. Suppose M is an invertible square matrix which is related to its Jordan canonical form through

= −1 ·⊕ · M T nJ n T . (F.1)

Each of the smaller square matrices J n has the form

J = λ1 (F.2) or the form ⎛ ⎞ λ 100... 000 ⎜ ⎟ ⎜ 0 λ 10... 000⎟ ⎜ ⎟ ⎜ ...... ⎟ ⎜ ...... ⎟ J = ⎜ ...... ⎟, (F.3) ⎜ ⎟ ⎜ 0000... λ 10⎟ ⎝ 0000... 0 λ 1 ⎠ 0000... 00λ and det(M) = 0 implies that none of the eigenvalues λ can vanish. We do not presume whether the matrix M is real or complex. However, the Jordan canonical form may require that we allow for complex eigenvalues λn and complex

© The Editor(s) (if applicable) and The Author(s), under exclusive license to 737 Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1 738 F The Logarithm of a Matrix transformation matrices T to ensure that the characteristic equation det(M−λ1) = 0 for the eigenvalues and the corresponding eigenvector conditions can be solved. In the case (F.2)wehave J = exp ln λ1 , ln J = ln λ1. (F.4)

However, it is also possible to construct the logarithm of a Jordan block matrix (F.3). The direct sum of the logarithms of all the matrices J n then yields the logarithm of the matrix M,

= −1 ·⊕ · = −1 ·⊕ · M exp(T n ln J n T ), ln M T n ln J n T . (F.5)

Suppose the Jordan matrix (F.3)isa(ν + 1) × (ν + 1) matrix. We deﬁne (ν + × + ≤ ≤ = 1) (ν 1) matrices N n,0 n ν, according to (N n)ij δi+n,j , i.e. N 0 is + × + the (ν 1) (ν 1) unit matrix and N 1≤n≤ν has non-vanishing entries 1 only in the n-th diagonal above the main diagonal. These matrices satisfy the multiplication · = − − + = n law N m N n (ν m n )Nm+n, which also implies N n (N 1) . + × + = + Each (ν 1) (ν 1) Jordan block can be written as J λN0 N 1, and its logarithm can be deﬁned through ⎛ ⎞ ln λλ−1 −λ−2/2 λ−3/3 ... (−)ν−1λ−ν/ν ⎜ ⎟ ⎜ 0lnλλ−1 −λ−2/2 ... (−)ν−2λ−(ν−1)/(ν − 1) ⎟ ⎜ ⎟ ⎜ ...... ⎟ ⎜ ...... ⎟ X = ln J = ⎜ ...... ⎟ ⎜ − −2 ⎟ ⎜ 00 0 0 ... λ /2 ⎟ ⎝ 00 0 0 ... λ−1 ⎠ 00 0 0 ... ln λ

ν (−λ)−n = N ln λ − N . (F.6) 0 n n n=1

We can prove exp(X) = J in the following way. The N-th power of X is ν+1−N N N = N + − N + − X N 0(ln λ) ( ) ν ni 1≤n1,n2...nN i=1 − − − − (−λ) n1 n2 ... nN × N . n1+n2+...+nN n1 · n2 · ...· nN ν+2−N N−1 N − (−) N ln λ ν + − ni 1≤n1,n2...nN−1 i=1 − − − − (−λ) n1 n2 ... nN−1 × N n1+n2+...+nN−1 n1 · n2 · ...· nN−1 F The Logarithm of a Matrix 739

ν+3−N N−2 N + (−)N (ln λ)2 ν + − n 2 i 1≤n1,n2...nN−2 i=1 − − − − (−λ) n1 n2 ... nN−2 × N + ... n1+n2+...+nN−2 n1 · n2 · ...· nN−2 ν −n − (−λ) − N(ln λ)N 1 N . (F.7) n n n=1

The parameter 0 < <1 is introduced in this equation to avoid the ambiguity of (x) at x = 0. We can combine terms in Eq. (F.7)intheform

N N − XN = N (ln λ)N + (−)m (ln λ)N m 0 m m=1 ν+1−m m − − − − (−λ) n1 n2 ... nm × ν + − n N i n1+n2+...+nm n1 · n2 · ...· nm 1≤n1,n2...nm i=1 ν min(N,M) − N − = N (ln λ)N + (−λ) M N (−)m (ln λ)N m 0 M m M=1 m=1 + − M1 m 1 × . (F.8) n1 · n2 · ...· nm 1≤n1,n2...nm n1+n2+...+nm=M

In the next step we isolate the term with M = 1inthesum,

(ln λ)N−1 XN = N (ln λ)N + N N + (ν − 2 + ) 0 λ 1 ν min(N,M) − N − × (−λ) M N (−)m (ln λ)N m M m M=2 m=1 + − M1 m 1 × . (F.9) n1 · n2 · ...· nm 1≤n1,n2...nm n1+n2+...+nm=M

If we sum these expressions to calculate exp(X), only the ﬁrst two terms survive in

∞ XN exp(X) = 1 + = λN + N = J , (F.10) N! 0 1 N=1 740 F The Logarithm of a Matrix because the sum over N in the term of order M reduces to

∞ (ln λ)N−m = λ, (F.11) (N − m)! N=m and the remaining sums yield for M ≥ 1theresult

+ − M (−)m M1 m 1 ! · · · = m n1 n2 ... nm m 1 1≤n1,n2...nm n1+n2+...+nm=M ! M ∞ + + + − − 1 (−)m zn1 n2 ... nm M 1 = dz 2πi |z|<1 m! n1 · n2 · ...· nm m=1 n1,n2...nm=1 ! ∞ ∞ + + + − − 1 (−)m zn1 n2 ... nm M 1 = dz 2πi |z|<1 m! n1 · n2 · ...· nm m=1 n1,n2...nm=1 ! m ∞ m ∞ n 1 (−) z − − = dz z M 1 2πi | | m! n z <1 m=1 n=1 ! ∞ m 1 [ln(1 − z)] − − = dz z M 1 2πi | | m! z <1 m=1 ! 1 −M−1 −M = dz z − z =−δM,1. (F.12) 2πi |z|<1

Equation (F.6) is a special case of a general procedure to deﬁne functions M → ∈ Z + f(M) of square matrices [60, 80], and for every n , the matrix X 2πinN 0 is also a logarithm of J . However, this construction cannot be used to infer that every element of a matrix group M can be generated through single Lie algebra exponentials, M = = i ∈ M = i| exp(G), G α Y i Lie , Y i ∂ ln M/∂α α=0. See e.g. Z. Qiao, R. Dick, J. Phys. Commun. 3, 075008 (2019), for discussions of the cases when ln(M)/∈ sl(2, R) or ln(M)/∈ sl(2, C),ifM ∈ SL(2, R) or M ∈ SL(2, C), respectively. On the other hand, positive deﬁnite Hermitian matrices and unitary matrices can be generated through single exponentials with real parameters αi, and for a general invertible matrix M we can use its polar decomposition1

1For the ordering of factors in Eq. (F.14), note that due to the Hermiticity of M · M+,

+ − + + − + + − + − [(M · M ) 1/2 · M] · (M · M ) 1/2 · M = M · (M · M ) 1/2 · (M · M ) 1/2 · M + + − = M · (M · M ) 1 · M = 1. (F.13) F The Logarithm of a Matrix 741

+ + − M = H · U = (M · M )1/2 ·[(M · M ) 1/2 · M] (F.14) in terms of a positive deﬁnite Hermitian and a unitary factor (or a symmetric and an orthogonal factor if M is real) to show that M can be generated from at most two exponential factors. This is the case e.g. for the standard representation of Lorentz transformations = H · U, where the symmetric matrix H = (β) is the pure boost part and the orthogonal factor U = (ϕ) is the rotation, see Eq. (H.1), where ˆ u = artanh(β)β (H.12) and ij = ij k ϕk (H.6). However, for the Lorentz group SO(1, 3) it is possible to ﬁnd X = ln for every proper orthochronous Lorentz transformation in such a way that X = u(u, ϕ) · K + ϕ(u, ϕ) · L ∈ so(1, 3) is an element of the Lie algebra of the Lorentz group (See e.g. [129, 139, 182]). The advantage of the factorized representation in the product of a boost and a rotation is that v = c·tanh(u)uˆ is actually a velocity of relative motion between two frames and ϕ is an actual rotation angle times rotation axis, whereas these direct interpretations do not apply to the combined parameters u(u, ϕ) and ϕ(u, ϕ). Appendix G Dirac γ Matrices

It is useful for the understanding and explicit construction of γ matrices to discuss their properties in a general number d of spacetime dimensions. γ matrices in more than four spacetime dimensions are regularly used in theories which hypothesize the existence of extra spacetime dimensions. On the other hand, variants of the Dirac equation in two space dimensions or three spacetime dimensions have also become relevant in materials science for the description of electrons in Graphene and other two-dimensional materials.

γ -Matrices in d Dimensions

The condition (22.66), {γμ,γν}=−2ημν, implies that any product of n gamma coefficients γα · γβ · ...· γω can be reduced to a product of n − 2 coefficients if two indices have the same value. We can also re-order any product such that the indices have increasing values. These observations imply that the d coefficients γμ can produce at most 2d linearly independent combinations

1,γ0,γ1, ..., γd−1,γ0 · γ1,γ0 · γ2, ..., γ0 · γ1 · ...· γd−1. (G.1)

We are actually interested in matrix representations of the algebra generated by (22.66), and consider the coefficients γ μ and the objects in (G.1) as matrices in the following. We first discuss the case that d is an even number of spacetime dimensions, and we define multi-indices J through

= · · · = J γμ1 γμ2 ... γμn ,μ1 <μ2 <...<μn,n(J) n. (G.2)

It is easy to prove that

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tr(J ) = 0. (G.3)

For even n(J ) this follows from the anti-commutativity of the γ -matrices and the cyclic invariance of the trace. For odd n(J ) this follows from the fact that there is at least one γ -matrix not contained in J ,e.g.γ1, and therefore

=− 2 · =− · · = 2 · =− = tr(J ) tr(γ1 J ) tr(γ1 J γ1) tr(γ1 J ) tr(J ) 0. (G.4)

The product I · J reduces either to a -matrix K if I = J , or otherwise

2 =± I 1, (G.5) and this implies orthogonality of all the -matrices and 1,

tr(I · J ) ∝ δIJ. (G.6)

For even number of spacetime dimensions d this implies that all the 2d matrices in (G.1) are indeed linearly independent, and therefore a minimal matrix representation of (22.66) requires at least (2d/2 × 2d/2)-matrices. We will see by explicit construction that such a representation exists, and because 2d/2 is the minimal dimension, the representation must be irreducible, i.e. cannot split into smaller matrices acting in spaces of lower dimensions. The representation also turns out to be unique up to similarity transformations

→ = · · −1 γμ γμ A γμ A . (G.7)

For odd number of spacetime dimensions d, we also deﬁne the matrices J according to (G.1), but now the previous proof of tr(J ) = 0 only goes through for all the matrices J except for the last matrix in the list,

0,1,...d−1 = γ0 · γ1 · ...· γd−1. (G.8)

For odd d, this matrix contains an odd number of γ -matrices, and it contains all γ -matrices, such that the previous proof of vanishing trace for odd n(J ) does not go through for this particular matrix. Furthermore, this matrix has the properties

[0,1,...d−1,J ]=0, (G.9)

2 = − (d+2)(d−1)/2 = − (d−1)/2 0,1,...d−1 ( ) 1 ( ) 1. (G.10)

Commutativity with all other matrices implies that in every irreducible representation

(d−1)/4 0,1,...d−1 =±(−) 1, (G.11) see the following subsection for the proof. GDiracγ Matrices 745

This also implies that every product J of n(J ) ≥ (d + 1)/2 γ matrices is up to a numerical factor a product I of n(I) = d − n(J ) ≤ (d − 1)/2 γ matrices, J = 1 · J ∝ 0,1,...d−1 · J n(J )≥(d+1)/2 n(J )≥(d+1)/2 n(J )≥(d+1)/2 ∝ I . (G.12) n(I)≤(d−1)/2

Therefore there are only 2d−1 linearly independent matrices in (G.1) for odd d, and the minimal possible dimension of the representation is only 2(d−1)/2.The explicit construction later on conﬁrms that the minimal dimension also works for odd number of spacetime dimensions. There are two different equivalence classes of matrix representations with dimension 2(d−1)/2. We can summarize the results on the dimensions of γ matrices in d space(-time) dimensions by the statement the irreducible representations of the Dirac algebra are provided by 2d/2 × 2d/2 matrices, where the ﬂoor function in the exponents rounds to the next lowest integer and is also often written in terms of Gauss brackets: d/2≡[d/2]G = d/2ifd is even, d/2≡[d/2]G = (d − 1)/2ifd is odd.

Proof that in Irreducible Representations 0,1,...d−1 ∝ 1for Odd Spacetime Dimension d

0,1,...d−1 commutes with all J . Suppose that we have an irreducible matrix representation of (G.1) in a vector space V of dimension dimV .Ifλ is an eigenvalue of 0,1,...d−1 ∝ 1,

det(0,1,...d−1 − λ1) = 0, (G.13) we have dim (0,1,...d−1 − λ1) · V ≤ dimV − 1, (G.14) and

J · (0,1,...d−1 − λ1) · V = (0,1,...d−1 − λ1) · J · V. (G.15)

The last equation would imply that (0,1,...d−1 − λ1) · V , if non-empty, would be an invariant subspace under the action of the γ -matrices, in contradiction to the irreducibility of V . Therefore we must have

(0,1,...d−1 − λ1) · V =∅,0,1,...d−1 = λ1 (G.16) 746 G Dirac γ Matrices in every irreducible representation. Equation (G.10) tells us that

− λ =±(−)(d 1)/4. (G.17)

The proof is simply an adaptation of the proof of Schur’s lemma from group theory.

Recursive Construction of γ -Matrices in Different Dimensions

We will use the following conventions for the explicit construction of γ -matrices: Up to similarity transformations, the γ -matrices in two spacetime dimensions are

01 01 γ = ,γ= . (G.18) 0 10 1 −10

For the recursive construction in higher dimensions d ≥ 3wenowassumethatγμ, 0 ≤ μ ≤ d − 2, are γ -matrices in d − 1 dimensions. For the construction of γ -matrices in an odd number d of spacetime dimensions there are two inequivalent choices,

(d−3)/2 −1 0 =±i γ γ ...γ − =± ,= γ , 1 ≤ i ≤ d − 2, 0 0 1 d 2 0 1 i i

d−1 =−iγ0. (G.19)

For the construction of γ -matrices in an even number d ≥ 4 of spacetime dimensions there is only one equivalence class of representations,

0 1 0 −γ0γi 0 = ,i = , 1 ≤ i ≤ d − 2, 1 0 γ0γi 0

0 −γ0 d−1 = . (G.20) γ0 0

Note that it does not matter from which of the two possible representations ±γ0 in the odd number d − 1 of lower dimensions we start since 0 intertwines the two possibilities,

0i0 =−i, 1 ≤ i ≤ d − 1. (G.21)

The possibility of similarity transformations implies that there are inﬁnitely many equivalent possibilities to construct these bases of γ -matrices. The construction described here was motivated from the desire to have Weyl bases (i.e. all γ μ have only off-diagonal non-vanishing (2(d/2)−1 × 2(d/2)−1) blocks) in even dimensions, GDiracγ Matrices 747 and to have the next best solution, viz. Dirac bases (i.e. γ 0 =±diag(1, −1),allγ i like in a Weyl basis), in odd dimensions. Note that all the representations (G.19) and (G.20)oftheγ -matrices in odd or even dimensions fulﬁll

+ = + =− γ0 γ0,γi γi, (G.22) or equivalently

+ = γμ γ0γμγ0. (G.23)

Every set of γ -matrices is equivalent to a set satisfying Eq. (G.23). We will prove this in the following subsection.

Proof That Every Set of γ -Matrices is Equivalent to a Set Which Satisﬁes Eq. (G.23)

In this section we do not use summation convention but spell out all summations explicitly. d/2 d/2 We deﬁne 2 × 2 matrices X0 = γ0, Xi = iγi,

{Xμ,Xν}=2δμν (G.24) and prove that the matrices Xμ are equivalent to a set of unitary matrices Yμ. Since −1 = the matrices Yμ also satisfy Yμ Yμ, unitarity also implies Hermiticity of Yμ.We use the abbreviation N = 2d/2, and consider the set S of N × N matrices " = · · ≤ˆ= d, d even 1,XI Xμ1 ... Xμn ,n n d−1 (G.25) 2 ,dodd

This set does not form a group, but only a group modulo Z2. But this is sufﬁcient for the standard argument for equivalence to a set of unitary matrices. The N × N matrix = + + · = + H 1 XI XI H (G.26) I is invariant under right translations in the set S (i.e. right multiplication of all elements by some ﬁxed element Z), because that just permutes the elements, up to possible additional minus signs which cancel in H, + + H = Z · Z + (XI · Z) · (XI · Z). (G.27) I 748 G Dirac γ Matrices

H also has N positive eigenvalues, because

· = + · = H ψα hαψα,ψα ψβ δαβ , (G.28) implies = + · · = + | · |2 hα ψα H ψα 1 XI ψα > 0. (G.29) I

If we deﬁne the matrix with columns ψα,Eqs.(G.28) and (G.29)imply

+ + diag(h1,...hN ) = · H · , H = · diag(h1,...hN ) · . (G.30)

Now deﬁne + + −1 Yμ = · diag( h1,... hN ) · · Xμ · ( · diag( h1,... hN ) · ) .

These matrices are indeed unitary, −1 + · = · · + · + Yμ Yμ diag h1,... hN Xμ + 2 × · diag h1,... hN · · Xμ

− + 1 × · diag h1,... hN · −1 = · · + · + · · diag h1,... hN Xμ H Xμ − + 1 × · diag h1,... hN ·

− + 1 = · diag h1,... hN · ×[ + · + · + · · ] Xμ Xμ I (ZI Xμ) (ZI Xμ) − + 1 × · diag h1,... hN ·

− + 1 = · diag h1,... hN · · H

− + 1 × · diag h1,... hN · = 1, (G.31) which concludes the proof of equivalence of the matrices Xμ to a set of matrices Yμ which are both unitary and Hermitian. GDiracγ Matrices 749

Equivalence of the γ matrices to Hermitian or anti-Hermitian matrices also implies that every reducible representation of γ matrices is fully reducible.

Uniqueness Theorem for γ Matrices

Every irreducible matrix representation of the algebra generated by (22.66) is equivalent to one of the representations constructed in the previous section. We ﬁrst consider the case of even number of dimensions d. The theorem says that in this case every irreducible matrix representation of (22.66) is equivalent to the representation in terms of 2d/2 × 2d/2 matrices constructed in Eq. (G.20).

Proof Suppose the N1 × N1 matrices γ1,μ and the N2 × N2 matrices γ2,μ,0≤ μ ≤ d − 1, are two sets of matrices which satisfy the conditions (22.66). V1 is the N1- dimensional vector space in which the matrices γ1,μ act. We use the representations from the previous section, Eq. (G.20), for the matrices γ2,μ. This implies N1 ≥ d/2 N2 = 2 . a We denote the components of the matrices γ1,μ and γ2,μ with γ1,μ b and α γ2,μ β , respectively, and deﬁne again multi-indices J for the two sets of γ matrices (cf. Eq. (G.2)),

= · · · ≤ ≤ = r,J γr,μ1 γr,μ2 ... γr,μn , 1 r 2,μ1 <μ2 <...<μn,n(J) n.

The squares of these matrices satisfy

2 r,I =±1 = sI 1, (G.32) where the sign factor

= − n(I)[n(I)+1]/2 sI ( ) ημ1μ1 (G.33) arises as the product of the factor (−)n(I)[n(I)−1]/2 from the permutations of γ matrices times a factor (−)n(I) from the sign on the right-hand side of (22.66). Only

ημ1μ1 appears on the right-hand side of (G.33) because we have only one timelike direction. For the case of general spacetime signature one could simply include the product ημ1μ1 ημ2μ2 ...ημnμn . The results from the previous section for even d tell us that a set r,I with ﬁxed d r, after augmentation with the Nr × Nr unit matrix r,0 = 1, contains 2 linearly independent matrices. α We deﬁne the N1 · N2 different N1 × N2 matrices Ea with components

α b b α (Ea ) β = δa δ β . (G.34)

We use these matrices to form the N1 × N2 matrices 750 G Dirac γ Matrices α α α a = Ea + sJ 1,J · Ea · 2,J , (G.35) J i.e. in components, α b b α b α ( a ) β = δa δ β + sJ (1,J ) a(2,J ) β . (G.36) J

Suppose I = J . The conditions (22.66) imply that there is always a multi-index K = I such that

r,I · r,J =±r,K , (G.37) and inversion of this equation yields

sI sJ r,J · r,I =±sK r,K . (G.38)

This implies α α α α 1,I · a = 1,I · Ea + Ea · 2,I + sJ 1,I · 1,J · Ea · 2,J J =I α α = sI 1,I · Ea · 2,I + Ea α + sI sJ 1,I · 1,J · Ea · 2,J · 2,I · 2,I J =I α α α = sI 1,I · Ea · 2,I + Ea + sK 1,K · Ea · 2,K · 2,I K =I α = a · 2,I . (G.39)

α We know that the matrices (G.36) are not null matrices, a = 0, because we know d that the 2 matrices {1,2,J } are linearly independent, b b δa 1 + sJ (1,J ) a2,J = 0. (G.40) J

This implies that the N1-dimensional vector space V1 with basis vectors e1,b,1≤ d/2 b ≤ N1, contains non-vanishing sets of N2 = 2 ≤ N1 basis vectors

α b d/2 e1,β = e1,b( a ) β , 1 ≤ β ≤ 2 ≤ N1, (G.41) which are invariant under the action of the γ matrices, GDiracγ Matrices 751

b α c α b β e1,b(γ1,μ) c( a ) δ = e1,b( a ) β (γ2,μ) δ. (G.42)

Therefore the representation of γ matrices in V1 is either reducible into invariant d/2 d/2 subspaces of dimension 2 ,orwehaveN1 = 2 . In the latter case we must have

α det( a ) = 0, (G.43) because representations spaces of dimension 2d/2 are irreducible, and therefore

α α −1 γ1,μ = a · γ2,μ · ( a ) (G.44) is equivalent to the representation from the previous section for even d. Thus concludes the proof for even d. For odd d we observe that the matrices γμ,0 ≤ μ ≤ d − 2, form a set of γ matrices for a (d − 1)-dimensional Minkowski space, which according to the previous result is either reducible or equivalent to the corresponding representation (G.20) from the previous section. However, using those matrices, the missing matrix γd−1 can easily be constructed according to the prescription

(d−1)(d−2)/4 γd−1 =±(−) γ0 · γ1 · ...· γd−2. (G.45)

(d−1)/2 (d−1)/2 Now assume that the matrices γμ,0≤ μ ≤ d−2, are 2 ×2 matrices, i.e. they form an irreducible representation of γ matrices for a (d − 1)-dimensional Minkowski space. In that case completeness of the set

= · · · ≤ ≤ − J γμ1 γμ2 ... γμn , 0 μ1 <μ2 <...<μn d 2 (G.46) in GL(2(d−1)/2) implies that (G.45) are the only options for the construction of γd−1. Completeness of the set (G.46) also implies that the two options for the sign in (G.45) correspond to two inequivalent representations. On the other hand, if the matrices γμ,0 ≤ μ ≤ d − 2, form a reducible representation of γ matrices for a (d − 1)-dimensional Minkowski space, they (d−1)/2 (d−1)/2 must be equivalent to matrices with irreducible 2 × 2 matrices γˆμ, 0 ≤ μ ≤ d − 2, in diagonal blocks. Then one can easily prove from the anti- commutation relations and the completeness of the set

ˆ =ˆ ·ˆ · ·ˆ ≤ ≤ − J γμ1 γμ2 ... γμn , 0 μ1 <μ2 <...<μn d 2 (G.47)

(d−1)/2 (d−1)/2 in 2 -dimensional subspaces that the matrix γd−1 must consist of 2 × (d−1)/2 ˆ ·ˆ · ·ˆ 2 =− 2 blocks which are proportional to γ0 γ1 ... γd−2. The property γd−1 1 can then be used to demonstrate that γd−1 must be equivalent to a matrix which only has matrices

(d−1)(d−2)/4 γˆd−1 =±(−) γˆ0 ·ˆγ1 · ...·ˆγd−2 (G.48) 752 G Dirac γ Matrices in diagonal 2(d−1)/2 × 2(d−1)/2 blocks, i.e. a representation of γ matrices for odd number d of dimensions is either equivalent to one of the two irreducible 2(d−1)/2- dimensional representations distinguished by the sign in (G.45), or it is a reducible representation. In the recursive construction of γ matrices described above, I separated the two equivalence classes of irreducible representations through the sign of γ0 instead of γd−1,cf.(G.19). We can cast the sign from γd−1 to γ0 through the similarity transformation

γ0 → γ0 · γd−1 · γ0 · γ0 · γd−1 =−γ0, (G.49)

γd−1 → γ0 · γd−1 · γd−1 · γ0 · γd−1 =−γd−1, (G.50)

γi → γ0 · γd−1 · γi · γ0 · γd−1 = γi, 1 ≤ i ≤ d − 2. (G.51)

Contraction and Trace Theorems for γ Matrices

Here we explicitly refer to four spacetime dimensions again. The generalizations to any number of spacetime dimensions are trivial. Equation (22.66) implies

σ γ γσ =−4. (G.52)

The higher order contraction theorems then follow from (22.66) and application of the next lower order contraction theorem, e.g.

σ μ σ μ μ σ μ γ γ γσ ={γ ,γ }γσ − γ γ γσ = 2γ , (G.53)

σ μ ν μν σ μ ν ρ ρ ν μ γ γ γ γσ = 4η ,γγ γ γ γσ = 2γ γ γ . (G.54)

The trace of a product of an odd number of γ matrices vanishes. The trace of a product of two γ matrices is determined by their basic anti-commutation property, tr γμγν =−4ημν. (G.55)

The trace of a product of four γ matrices is easily evaluated using their anti- commutation properties and cyclic invariance of the trace tr γκ γλγμγν = 8ηκλημν − tr γκ γλγνγμ = 8ηκλημν − tr γμγκ γλγν = 8ηκλημν − 8ημκ ηλν + tr γκ γμγλγν = 8ηκλημν − 8ημκ ηλν + 8ημληκν − tr γκ γλγμγν , GDiracγ Matrices 753 i.e. tr γκ γλγμγν = 4ηκλημν − 4ημκ ηλν + 4ημληκν. (G.56)

For yet higher orders we observe tr γα ...γα γμγν =−2ημνtr γα ...γα − tr γμγα ...γα γν 1 2n 1 2n 1 2n =−2ημνtr γα ...γα + tr γα γμγα ...γα γν 1 2n 1 2 2n + 2ημα tr γα ...γα γν 1 2 2n =− − 2ημνtr γα1 ...γα2n tr γα1 ...γα2n γμγν 2n − − i 2 ( ) ημαi tr γα1 ...γαi−1 γαi+1 ...γα2n γν , i=1 i.e. we have a recursion relation =− tr γα1 ...γα2n γμγν ημνtr γα1 ...γα2n 2n − − i ( ) ημαi tr γα1 ...γαi−1 γαi+1 ...γα2n γν . i=1

This yields for products of six γ matrices tr γργσ γκ γλγμγν =−4ηρσ ηκλημν + 4ηρκησλημν − 4ηρληκσημν + 4ησκηλνημρ

− 4ησληκνημρ + 4ησνηλκ ημρ − 4ηρκηλνημσ + 4ηρληκνημσ

− 4ηρνηλκ ημσ + 4ηρσ ηλνημκ − 4ηρλησνημκ + 4ηρνηλσ ημκ

− 4ηρσ ηκνημλ + 4ηρκησνημλ − 4ηρνηκσημλ, (G.57) and the trace of the product of eight γ matrices contains 105 terms, tr γαγβ γργσ γκ γλγμγν = 4ηρσ ηκλημνηαβ − 4ηρκησλημνηαβ + 4ηρληκσημνηαβ

− 4ησκηλνημρ ηαβ + 4ησληκνημρ ηαβ − 4ησνηλκ ημρ ηαβ + 4ηρκηλνημσ ηαβ

− 4ηρληκνημσ ηαβ + 4ηρνηλκ ημσ ηαβ − 4ηρσ ηλνημκ ηαβ + 4ηρλησνημκ ηαβ

− 4ηρνηλσ ημκ ηαβ + 4ηρσ ηκνημληαβ − 4ηρκησνημληαβ + 4ηρνηκσημληαβ

− 4ησκηλμηνβ ηαρ + 4ησληκμηνβ ηαρ − 4ησμηλκ ηνβ ηαρ + 4ηκλημβ ηνσ ηαρ

− 4ηκμηλβ ηνσ ηαρ + 4ηκβημληνσ ηαρ − 4ησλημβ ηνκηαρ + 4ησμηλβ ηνκηαρ

− 4ησβημληνκηαρ + 4ησκημβ ηνληαρ − 4ησμηκβηνληαρ + 4ησβημκ ηνληαρ

− 4ησκηλβ ηνμηαρ + 4ησληκβηνμηαρ − 4ησβηλκ ηνμηαρ + 4ηρκηλμηνβ ηασ 754 G Dirac γ Matrices

− 4ηρληκμηνβ ηασ + 4ηρμηλκ ηνβ ηασ − 4ηκλημβ ηνρ ηασ + 4ηκμηλβ ηνρ ηασ

− 4ηκβημληνρ ηασ + 4ηρλημβ ηνκηασ − 4ηρμηλβ ηνκηασ + 4ηρβ ημληνκηασ

− 4ηρκημβ ηνληασ + 4ηρμηκβηνληασ − 4ηρβ ημκ ηνληασ + 4ηρκηλβ ηνμηασ

− 4ηρληκβηνμηασ + 4ηρβ ηλκ ηνμηασ − 4ηρσ ηλμηνβ ηακ + 4ηρλησμηνβ ηακ

− 4ηρμηλσ ηνβ ηακ + 4ησλημβ ηνρ ηακ − 4ησμηλβ ηνρ ηακ + 4ησβημληνρ ηακ

− 4ηρλημβ ηνσ ηακ + 4ηρμηλβ ηνσ ηακ − 4ηρβ ημληνσ ηακ + 4ηρσ ημβ ηνληακ

− 4ηρμησβηνληακ + 4ηρβ ημσ ηνληακ − 4ηρσ ηλβ ηνμηακ + 4ηρλησβηνμηακ

− 4ηρβ ηλσ ηνμηακ + 4ηρσ ηκμηνβ ηαλ − 4ηρκησμηνβ ηαλ + 4ηρμηκσηνβ ηαλ

− 4ησκημβ ηνρ ηαλ + 4ησμηκβηνρ ηαλ − 4ησβημκ ηνρ ηαλ + 4ηρκημβ ηνσ ηαλ

− 4ηρμηκβηνσ ηαλ + 4ηρβ ημκ ηνσ ηαλ − 4ηρσ ημβ ηνκηαλ + 4ηρμησβηνκηαλ

− 4ηρβ ημσ ηνκηαλ + 4ηρσ ηκβηνμηαλ − 4ηρκησβηνμηαλ + 4ηρβ ηκσηνμηαλ

− 4ηρσ ηκληνβ ηαμ + 4ηρκησληνβ ηαμ − 4ηρληκσηνβ ηαμ + 4ησκηλβ ηνρ ηαμ

− 4ησληκβηνρ ηαμ + 4ησβηλκ ηνρ ηαμ − 4ηρκηλβ ηνσ ηαμ + 4ηρληκβηνσ ηαμ

− 4ηρβ ηλκ ηνσ ηαμ + 4ηρσ ηλβ ηνκηαμ − 4ηρλησβηνκηαμ + 4ηρβ ηλσ ηνκηαμ

− 4ηρσ ηκβηνληαμ + 4ηρκησβηνληαμ − 4ηρβ ηκσηνληαμ + 4ηρσ ηκλημβ ηαν

− 4ηρκησλημβ ηαν + 4ηρληκσημβ ηαν − 4ησκηλβ ημρ ηαν + 4ησληκβημρ ηαν

− 4ησβηλκ ημρ ηαν + 4ηρκηλβ ημσ ηαν − 4ηρληκβημσ ηαν + 4ηρβ ηλκ ημσ ηαν

− 4ηρσ ηλβ ημκ ηαν + 4ηρλησβημκ ηαν − 4ηρβ ηλσ ημκ ηαν + 4ηρσ ηκβημληαν

− 4ηρκησβημληαν + 4ηρβ ηκσημληαν. (G.58) Appendix H Spinor Representations of the Lorentz Group

The explicit form of the Lagrange density (22.120) for the Dirac ﬁeld and the appearance of the factor = + · γ 0 are determined by the requirement of Lorentz invariance of L and the transformation properties of spinors under Lorentz transformations. However, before we can elaborate on these points, we have to revisit the Lorentz transformation (B.13), which is also denoted as the vector representation because it acts on spacetime vectors. We can discuss this in general numbers n of spatial dimensions and d = n + 1 of spacetime dimensions.

Generators of Proper Orthochronous Lorentz Transformations in the Vector and Spinor Representations

We can write the two factors of a proper orthochronous Lorentz transformation (B.13) as exponentials of Lie algebra elements,

1 (u, ) = (u) · ( ) = exp(u · K) · exp ij L . (H.1) 2 ij

For the boost part we use explicit construction to prove that every proper Lorentz boost can be written in the form exp(u · K). For the rotation part we can use the general result that every element of a compact Lie group can be written as a single exponential of a corresponding Lie algebra element, or we can use the fact that a general n × n rotation matrix consists of n orthonormal row vectors, which ﬁxes the general form in terms of n(n − 1)/2 parameters, and then demonstrate that the n(n − 1)/2 parameters ij of ij exp( Lij /2) provide a general parametrization of n orthonormal row vectors.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to 755 Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1 756 H Spinor Representations of the Lorentz Group

Alternatively, we can consider (H.1) as an example for the polar decomposition (F.14) and infer the representation in terms of matrix exponentials from the results on matrix logarithms in Appendix F. The boost part is i0 i0 (u) = exp(u · K) = exp Li0 = exp i Mi0 (H.2) and the spatial rotation is

1 i ( ) = exp ij L = exp ij M , (H.3) 2 ij 2 ij where ij is the rotation angle in the ij plane. The generators are (in the vector representation), ρ ρ ρ ρ Lμν σ = i Mμν σ = η μηνσ − η νημσ . (H.4)

These matrices generate the Lie algebra so(1,d − 1),

[Lμν,Lρσ ]=ηνρ Lμσ + ημσ Lνρ − ημρ Lνσ − ηνσ Lμρ λ λ =−(Lμν)ρ Lλσ − (Lμν)σ Lρλ. (H.5)

In four-dimensional Minkowski space, the angles ij are related to the rotation i angles ϕi around the x -axis according to

1 ϕ = , = ϕ . (H.6) i 2 ij k jk ij ij k k To see how the boost vector u is related to the velocity v = cβ, we will explicitly calculate the boost matrix (u). We have with a contravariant row index and a covariant column index, as in (B.13), ⎛ ⎞ 0 −u1 ... −ud−1 ⎜ ⎟ −u 0 ... 0 T i i ⎜ 1 ⎟ 0 −u u · K = u Li0 = iu Mi0 = ⎜ . . . ⎟ = , ⎝ . . . ⎠ −u 0 −ud−1 0 ... 0

u2 0T 1 0T (u · K)2 = ,(u · K)2n = u2n , (H.7) 0 u ⊗ uT 0 uˆ ⊗ uˆ T

T + + 0 −uˆ (u · K)2n 1 = u2n 1 . (H.8) −uˆ 0 H Spinor Representations of the Lorentz Group 757

For the interpretation of the (d − 1) × (d − 1) matrices uˆ ⊗ uˆ T and 1 − uˆ ⊗ uˆ T , note that for every (d − 1)-dimensional spatial vector r

T T r = uˆ (uˆ · r) = (uˆ ⊗ uˆ ) · r (H.9) is the part r of the vector which is parallel to u, and

T r⊥ = r − r = (1 − uˆ ⊗ uˆ ) · r (H.10) is the part of the vector which is orthogonal to u. Substitution of the results (H.7) and (H.8)into(H.2) yields for the boost in the direction uˆ = βˆ

cosh(u) 0T 0 −uˆ T (u) = + sinh(u) 0 1 + uˆ ⊗ uˆ T [cosh(u) − 1] −uˆ 0

γ − γ βT = , (H.11) − γ β 1 − uˆ ⊗ uˆ T + γ uˆ ⊗ uˆ T i.e.

1 1 + β γ = cosh(u), β = tanh(u), u = artanh(β) = ln . (H.12) 2 1 − β

The parameter u is usually denoted as the boost parameter or rapidity of the Lorentz transformation. It may also be worthwhile to write down the corresponding rotation matrix in four-dimensional Minkowski space. If we use the 3 × 3 matrices from Sect. 7.4 for 1 the spatial subsections of the rotation matrices Lmn,

1 (L ) = (L ) = , (H.13) i jk 2 imn mn jk ij k the rotation matrices take the following form,

1 0T 1 0T ( ) = exp = , (H.14) 0 ϕ · L 0 exp(ϕ · L) with the 3 × 3 rotation matrix exp(ϕ · L) = ϕˆ ⊗ ϕˆ T + 1 − ϕˆ ⊗ ϕˆ T cos ϕ + ϕˆ · L sin ϕ. (H.15)

1In this Appendix we use underscore only for 2 × 2 matrices. 758 H Spinor Representations of the Lorentz Group

Application of the matrix ϕˆ · L generates a vector product,

(ϕˆ · L) · r =−ϕˆ × r. (H.16)

The commutation relations of the Lorentz generators in the Ki, Li notation are

[Li,Lj ]=− ij k Lk, [Li,Kj ]=− ij l Kl, [Ki,Kj ]= ij k Lk. (H.17)

The anticommutation relations (22.66) imply that the properly normalized commutators of γ -matrices,

i S = [γ ,γ ] (H.18) μν 4 μ ν also provide a representation of the Lie algebra so(1,d-1) (H.5), [Sμν,Sρσ ]=i ημρ Sνσ + ηνσ Sμρ − ηνρ Sμσ − ημσ Sνρ λ λ = i(Lμν)ρ Sλσ + i(Lμν)σ Sρλ. (H.19)

See Eqs. (H.24)–(H.26) for the proof. This representation of the Lorentz group is realized in the transformation of Dirac spinors ψ(x) under Lorentz transformations

1 x = ( ) · x = exp μνL · x, (H.20) 2 μν

i ψ (x ) = U()· ψ(x) = exp μνS · ψ(x). (H.21) 2 μν

The anticommutation relations (22.66) also imply invariance of the γ -matrices under Lorentz transformations x = ( ) · x,

i i γ μ = μ ( ) exp κλS · γ ν · exp − ρσ S = γ μ, (H.22) ν 2 κλ 2 ρσ see Eq. (H.27). This invariance property of the γ -matrices also implies form invariance of the Dirac equation under Lorentz transformations, i ihγ¯ μ∂ ψ (x ) − mcψ (x ) = exp κλS · ihγ¯ μ∂ ψ(x) − mcψ(x) , μ 2 κλ μ (H.23) i.e. all inertial observers can use the same set of γ -matrices, and the Dirac equation has the same form for all of them. H Spinor Representations of the Lorentz Group 759

Veriﬁcation of the Lorentz Commutation Relations for the Spinor Representations

The anti-commutation relations (22.66) imply

[γμγν,γρ]=γμ{γν,γρ}−{γμ,γρ}γν = 2ημρ γν − 2ηνρ γμ σ = 2(Lμν)ρ γσ , (H.24) where the matrices Lμν were given in (H.4). Equation (H.24) also implies

σ [Sμν,γρ]=i(Lμν)ρ γσ (H.25) and i i i [S ,S ]= [S , [γ ,γ ]] = [[S ,γ ],γ ]− [[S ,γ ],γ ] μν ρσ 4 μν ρ σ 4 μν ρ σ 4 μν σ ρ 1 1 =− (L ) λ[γ ,γ ]+ (L ) λ[γ ,γ ] 4 μν ρ λ σ 4 μν σ λ ρ λ λ = i(Lμν)ρ Sλσ + i(Lμν)σ Sρλ. (H.26)

Equation (H.25) implies the Lorentz invariance of the γ -matrices,

i i exp μνS · γ · exp − κλS = exp − μνL /2 σ γ 2 μν ρ 2 κλ μν ρ σ −1 σ = ( )ρ γσ . (H.27)

Scalar Products of Spinors and the Lagrangian for the Dirac Equation

The Hermiticity relation (G.23) implies the following Hermiticity property of the Lorentz generators,

+ = 0 0 Sμν γ Sμνγ , (H.28) and therefore

+ + i ψ (x ) = ψ (x) · γ 0 exp − μνS γ 0. (H.29) 2 μν

The adjoint spinor 760 H Spinor Representations of the Lorentz Group

+ ψ(x) = ψ (x) · γ 0 (H.30) therefore transforms inversely to the spinor ψ(x),

i ψ (x) = ψ(x) · exp − μνS , (H.31) 2 μν and the product of spinors

+ ψ(x) · φ(x) = ψ (x) · γ 0 · φ(x) (H.32) is Lorentz invariant. This yields a Lorentz invariant Lagrangian for the Dirac equation, ihc¯ L = ψ(x)· γ μ · ∂ ψ(x)− ∂ ψ(x)· γ μ · ψ(x) − mc2ψ(x)· ψ(x). (H.33) 2 μ μ

The Spinor Representation in the Weyl and Dirac Bases of γ -Matrices

In even dimensions, the construction (G.20) yields γ -matrices of the form

= 01 = 0 σ i γ0 ,γi − , (H.34) 1 0 σ i 0

(d/2)−1 × (d/2)−1 with Hermitian (2 2 ) matrices σ i, which satisfy { }= σ i,σj 2δij . (H.35)

The spinor representation of the Lorentz generators in this Weyl basis is

i i = = σ i 0 Si0 γiγ0 − , (H.36) 2 2 0 σ i [ ] = i [ ]=−i σ i,σj 0 Sij γi,γj [ ] . (H.37) 4 4 0 σ i,σj

This is the advantage of a Weyl basis: The 2d/2 components of a spinor explicitly split into two Weyl spinors with 2(d/2)−1 components. The two Weyl spinors transform separately under proper orthochronous Lorentz transformations. A Dirac spinor representation in even dimensions is therefore reducible under the group of proper orthochronous Lorentz transformations. However, the form of Si0 tells us that H Spinor Representations of the Lorentz Group 761 the two Weyl spinors are transformed into each other under time or space inversions. Therefore the representation of the full Lorentz group really requires the full 2d/2- dimensional Dirac spinor. The rotation generators in the Dirac representation in even dimensions are the same as in the Weyl basis, but the boost generators become

i 0 σ i Si0 = . (H.38) 2 σ i 0

For an odd number of spacetime dimensions our construction provides γ -matrices of the form,

− =± 1 0 = 0 σ i ≤ ≤ − γ0 ,γi − , 1 i d 2, (H.39) 01 σ i 0

01 γ − =−i . (H.40) d 1 1 0

The rotation generators Sij ,1≤ i, j ≤ d − 2, are the same as in d − 1 dimensions, but rotations of the (i, d − 1) plane are generated by

1 = σ i 0 Si,d−1 − , (H.41) 2 0 σ i and the boost generators are off-diagonal,

i 1 =± 0 σ i =± 01 Si0 ,Sd−1,0 − . (H.42) 2 σ i 0 2 1 0

The proper orthochronous Lorentz group therefore mixes all the 2(d−1)/2 components of a Dirac spinor in odd dimensions. We also remark that the general boost matrix acting on the spinors is in every representation 1 u u U(u) = exp iuiS = exp uiγ γ = cosh + uˆ · γ γ sinh i0 2 0 i 2 0 2 + 1/4 − 1/4 = 1 1 β + 1 β 2 1 − β 1 + β 1 1 + β 1/4 1 − β 1/4 + βˆ · γ γ − 2 0 1 − β 1 + β 762 H Spinor Representations of the Lorentz Group γ + 1 γ − 1 = + βˆ · γ γ , (H.43) 2 2 0

2 i i U (u) = exp 2iu Si0 = exp u γ0γi = cosh(u) + uˆ · γ0γ sinh(u)

= γ + γ β · γ0γ . (H.44)

Construction of the Vector Representation from the Spinor Representation

Equation (22.66) implies

d/2 tr(γμγν) =−2 ημν. (H.45)

This and the invariance of the γ -matrices (H.22) can be used to reconstruct the vector representation of a proper orthochronous Lorentz transformation from the corresponding spinor representation, − i i μ ( ) =−2 d/2 tr exp − κλS · γ μ · exp ρσ S · γ . (H.46) ν 2 κλ 2 ρσ ν

We can also use Eq. (H.45) to transform every vector into a spinor of order 2 (or every tensor of order n into a spinor of order 2n), μ μ −d/2 μ x(γ) = x γμ,x=−2 tr γ · x(γ) , (H.47) and the invariance of the γ -matrices implies

i i x μ = μ ( )xν ⇔ x (γ ) = exp κλS · x(γ) · exp − ρσ S . ν 2 κλ 2 ρσ

Construction of the Free Dirac Spinors from Spinors at Rest

We use c = 1 and d = 4 in this section. The Dirac equation in momentum space (22.77) is for a Dirac spinor ψ(E,0) at rest

(m − Eγ 0)ψ(E, 0) = 0. (H.48)

The Hermitian 4 × 4matrixγ 0 can only have eigenvalues ±1, which each must be two-fold degenerate because γ 0 is traceless. Therefore Dirac spinors at rest must H Spinor Representations of the Lorentz Group 763 correspond to energy eigenvalues E =±m. To construct the free Dirac spinors for arbitrary on-shell momentum 4-vector we can then use a boost into a frame where the fermion has on-shell momentum 4-vector ±p, ±E ± p2 + m2 ±m → = · , (H.49) 0 ±p 0 and Eq. (H.21) then implies ψ(± p2 + m2, ±p) = U()· ψ(±m, 0). (H.50)

The Lorentz boost which takes us from the rest frame of the fermion into a frame where the fermion has on-shell momentum 4-vector ±p is

γ − γ βT (u) ={μ (u)}= ν − γ β 1 − uˆ ⊗ uˆ T + γ uˆ ⊗ uˆ T 1 p2 + m2 pT = , (H.51) m p m1 − mpˆ ⊗ pˆ T + p2 + m2pˆ ⊗ pˆ T i.e. with E(p) ≡ p2 + m2, 1 E(p) p γ = cosh(u) = p2 + m2 = ,γβ = uˆ sinh(u) =− , (H.52) m m m p v = β =− . (H.53) p2 + m2

The minus sign makes perfect sense: We have to transform from the particle’s rest frame into a frame which moves with speed v =−vparticle relative to the particle to observe the particle with speed vparticle = p/E(p). The rapidity parameter of the particle is 1 1 + β 1 p2 + m2 +|p| u = artanh(β) = ln = ln 2 1 − β 2 p2 + m2 −|p| p2 + m2 +|p| = ln . (H.54) m

General equations for the boost matrix on spinors were given in (H.43) and (H.44). In the present case we have 1 U 2(u) = p2 + m2 − p · γ γ , (H.55) m 0 764 H Spinor Representations of the Lorentz Group i.e. we can also write 1 2 2 U(u) = √ p + m − p · γ0γ . (H.56) m

The corresponding boost matrices in the Dirac representation (22.67) are

− = 0 σ i γ0γi − , (H.57) σ i 0 u − ˆ T · u 1 · 1/2 = cosh 2 u σ sinh 2 = √ E(p) p σ U(u) − ˆ · u u · . u σ sinh 2 cosh 2 m p σ E(p)

For the evaluation of the hyperbolic functions, we note u cosh(u) + 1 E(p) + m cosh = = , (H.58) 2 2 2m u cosh(u) − 1 E(p) − m |p| sinh = = = √ . (H.59) 2 2 2m 2m(E(p) + m)

This yields

1 E(p) + m p · σ U(u) = √ . (H.60) 2m(E(p) + m) p · σ E(p) + m

The rest frame spinors satisfying Eq. (H.48) in the Dirac representation are ⎛ √ ⎞ ⎛ ⎞ 2m √0 ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 2m ⎟ u(0, 1 ) = ⎝ ⎠,u(0, − 1 ) = ⎝ ⎠, (H.61) 2 0 2 0 0 0 ⎛ ⎞ ⎛ ⎞ 0 0 ⎜ 0 ⎟ ⎜ 0 ⎟ − 1 = ⎜ √ ⎟ 1 = ⎜ ⎟ v(0, 2 ) ⎝ ⎠,v(0, 2 ) ⎝ ⎠, (H.62) 2m √0 0 2m

± 1 and application of the spinor boost matrix (H.60) yields the spinors u(p, 2 ) and ± 1 v(p, 2 ) in agreement with Eqs. (22.83)–(22.86). The initial construction there − · − ± 1 from m γ p gave us the negative energy solutions v( p, 2 ) for momentum 4-vector (−E(p), p), whereas the construction from Eq. (H.49) gave us directly the H Spinor Representations of the Lorentz Group 765

± 1 − = − − negative energy solutions v(p, 2 ) for momentum 4-vector p ( E(p), p), which in either derivation are ﬁnally used in the general free solution (22.92).

Lorentz Covariance of Charge Conjugation

Finally, we should also demonstrate that charge conjugation (22.130) is Lorentz covariant in the sense that the charge conjugate spinor c(x) transforms exactly like the original spinor (x),

i (x) → (x ) = exp μνS · (x), (H.63) 2 μν under the Lorentz transformation

1 x → x = exp μνL · x. (H.64) 2 μν

Recall that in the derivation of (22.130), which holds in the Dirac and the Weyl ∗ = representations of γ matrices, we have used that in those representations γμ 2 =− γ2γμγ2. This and γ2 1 implies

∗ i S =− [γ γ γ ,γ γ γ ]=γ S γ , (H.65) μν 2 2 μ 2 2 ν 2 2 μν 2 and therefore the complex conjugate Dirac spinor transforms according to

∗ i ∗ (x ) = exp − μνγ S γ · (x) 2 2 μν 2

n ∗ 1 i ∗ = (x) + − μνγ S γ n · (x) n! 2 2 μν 2 n=1

n ∗ 1 i ∗ = (x) − γ · μνS n · γ · (x) 2 n! 2 μν 2 n=1

i ∗ =−γ · exp μνS · γ · (x). (H.66) 2 2 μν 2

Multiplying both sides of the equation with iγ2 yields the result that charge conjugate Dirac spinors transform exactly like Dirac spinors under Lorentz transformations,

i c (x ) = exp μνS · c(x). (H.67) 2 μν 766 H Spinor Representations of the Lorentz Group

Another way to express this is to say that the complex conjugate spinor representation is unitarily equivalent to the spinor representation through the unitary + −1 transformation matrix iγ2 = (iγ2) = (iγ2) . This unitary equivalence of the representations makes charge conjugate spinors indistinguishable from Dirac spinors under Lorentz transformations. Appendix I Transformation of Fields Under Reﬂections

In this Appendix we will assume d = 4 for the number of spacetime dimensions. The proper orthochronous Lorentz transformations were introduced in Appendix B and we have discussed exponential representations of boosts and rotations in 2 μ ν Eqs. (H.1)–(H.42). However, the relativistic line element ds =−ημνdx dx is also invariant under reﬂections1

μ μ ν ν Pμ : dx →−dx ,dx→ dx (ν = μ). (I.1)

The product of any two spatial reﬂections is a rotation of the corresponding spatial plane by π,cf.(H.14) and (H.15),

PiPj = exp(iπMij ), (I.2) and this implies that we can write any particular spatial reﬂection as a combination of the reﬂection P = P1P2P3 of all spatial directions with a rotation by π,

i P = Pexp π M . (I.3) i 2 ij k jk

1The reflections dt →−dt or dxi →−dxi (or up to constant shifts, t →−t or xi →−xi )are usually denoted as time or space inversions. This convention likely originated from the fact that in algebraic fields (here “field” refers to the mathematical definition of a set which allows for addition, subtraction, multiplication, and division where possible) x →−x is the inversion operation with respect to addition. However, the operations Pμ are reversals of time or spatial directions which arise from reflections at four-dimensional hyperplanes located at some coordinate value Xμ: xμ → 2Xμ − xμ. Therefore we prefer the designation reflections for these transformations.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to 767 Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1 768 I Transformation of Fields Under Reﬂections

Therefore it is sufﬁcient to discuss the two discrete Lorentz transformations T = P0 (reversal of time direction) and P. The spatial inversion P is also denoted as a parity transformation. We can determine the transformation properties of ﬁelds under P and T from the requirement that electrodynamics should be invariant under these transformations, i.e. we postulate that the equations

2 2 2 μ [h∂¯ − iQA] φ − m c φ = 0,γ[ih∂¯ μ + qAμ] − mc = 0, (I.4) and (cf. Eq. (22.257) for the scalar particle contribution to the current density) 1 μν ν ν + + ν Q + ν − ∂μF = qγ + iQc ∂ φ · φ − φ · ∂ φ + 2i φ A φ (I.5) μ0c h¯ hold in this form also for an observer that uses reﬂected spatial axes or uses decreasing values of t to label the future. We know already from classical electrodynamics how electromagnetic ﬁelds and charge distributions transform under P and T, see e.g. [88],

: =− = = − =− − T t t, x x,j0(x,t) j0(x, t), j (x,t) j(x, t), E (x,t)= E(x, − t), B (x,t)=−B(x, − t), (I.6)

: = =− = − =− − P t t, x x,j0(x,t) j0( x,t), j (x,t) j( x,t), E (x,t)=−E(− x,t), B (x,t)= B(− x,t). (I.7)

The transformation properties of the electromagnetic ﬁelds imply that (up to gauge transformations) the vector potentials transform according to

: = − =− − T A0(x,t) A0(x, t), A (x,t) A(x, t), (I.8) : = − =− − P A0(x,t) A0( x,t), A (x,t) A( x,t). (I.9)

The components of Aμ(x) transform under P like the derivative operators ∂μ, such = =− that the covariant derivatives transform like D0 D0, Di Di. We can therefore get the correct transformation behavior of the currents on the right-hand side of Maxwell’s equations (I.5) and preserve the matter equations (I.4)ifwetransform the matter ﬁelds (up to gauge transformations) according to

P: φ (x,t)= φ(− x,t), (x,t)= γ 0(− x,t). (I.10)

On the other hand, the partial derivatives and vector potentials pick up relative minus signs under time reversal T: I Transformation of Fields Under Reﬂections 769

− =− − h∂¯ 0 iqA0(x,t) h∂¯ 0 iqA0(x,t ), (I.11)

h¯∇ − iqA(x,t)= h¯∇ + iqA (x,t ). (I.12)

The transformation properties of scalar and spinor ﬁelds under time reversal therefore need to invoke complex conjugations to preserve the matter equations of motion (I.4), and they need to reverse the signs of some of the derivatives of the Dirac ﬁeld after complex conjugation while leaving the other derivative terms unchanged. In a Dirac of Weyl basis of γ matrices (22.67) and (22.68), this can be achieved (up to gauge transformations) through the transformation laws

∗ ∗ T: φ (x,t)= φ (x, − t), (x,t)= γ1γ3 (x, − t). (I.13)

Relativistic quantum electrodynamics is invariant under P and T and exchanges the role of particles and anti-particles under charge conjugation C (22.5) and (22.130): q →−q,Q →−Q,

c ∗ c ∗ C: φ (x) = φ (x), (x) = iγ2 (x). (I.14)

However, there is a subtle difference between C and T in the treatment of products of fermionic operators. Both C and T involve complex conjugations of the fermion fields, and to write e.g. mass terms and currents in the Lagrangian again in standard form requires a transposition. For charge conjugation C, the transposition catches an extra minus sign (the fermion fields are transposed as “q”-numbers, taking into account anti-commutation), whereas for time reversal T, no extra minus sign results from the transpositions (the fermion fields are transposed as “c”-numbers, i.e. like ordinary complex numbers). This lack of extra minus sign for time reversal is sometimes expressed by saying that time reversal acts as

+ + T: φ(x,t)⇒ φ (x, − t), (x,t)⇒ (x, − t)γ3γ1. (I.15)

In the conventions outlined here, the combination CPT of all three transformations acts on scalar and spinor ﬁelds and real vector potentials according to

: = − = − = − CPT φ (x) φ( x), (x) γ5( x), Aμ(x) Aμ( x), (I.16) with extra minus signs in the transformation of fermion products from the different transposition properties under C and T, which can also be stated in the rule

T CPT: (x) ⇒ (− x)γ5 (I.17) for the transformation of the fermions, with subsequent q-number commutation to recover the standard representation for bilinear products · · . The γ5 matrix in Eq. (I.16) is deﬁned as 770 I Transformation of Fields Under Reﬂections

0 1 2 3 γ5 = iγ γ γ γ . (I.18)

μ It anti-commutes with the basic γ -matrices, {γ5,γ }=0, and it takes the following forms in the Dirac or Weyl representations:

0 1 1 0 γ (D) = ,γ(W) = . (I.19) 5 1 0 5 0 −1

The extra minus sign from the different transposition properties under C and T (or equivalently the rule (I.17)) implies invariance of mass terms under CPT,

(x)(x) ⇒ (x) (x). (I.20)

If we do not care about the direct mapping of solutions of the Dirac equation under time reversal, we can replace the transformation (I.13) and (I.15) with the time reversal transformation

T: (x,t)= γ1γ3(x, − t). (I.21) on the spinor ﬁelds. This has exactly the same effect as the transformation (I.13) and (I.15) on bilinear products of spinor ﬁelds of the form · · , which is what we really care about. The CPT transformation (I.16) on the spinors then becomes

∗ CPT: (x) = γ5 (− x), (I.22) and the invariance (I.20) of the mass term then simply follows from the q-number (i.e. anti-commutative) transposition under charge conjugation. The γ5 matrix also deﬁnes left-handed and right-handed chiral components of spinors through the decomposition of spinors into eigenspinors of γ5 with eigenvalues ±1,

1 − γ 1 + γ (x) = (x) + (x), (x) = 5 (x), (x) = 5 (x), L R L 2 R 2

γ5L(x) =−L(x), γ5R(x) = R(x). (I.23)

The effects of the transformations C, P, and T can be illustrated in chiral gauge theories, where e.g. only left-handed particles of charge q couple to a gauge ﬁeld.2

2This is standard and convenient terminology in particle physics, where this case appears in electroweak interactions. However, the language is slightly misleading, as there are no special creation operators e.g. for left-handed or right-handed electrons or neutrinos. There are only the electron and neutrino operators accompanied by chiral projections of the corresponding u and v spinors. Therefore it would be more precise to say that certain gauge ﬁelds couple only to the left-handed components of electrons and neutrinos. I Transformation of Fields Under Reﬂections 771

The transformation rules described above imply (with h¯ = c = 1 and for any convention (I.13), (I.15), and (I.21) for time reversal) the transformations ⎧ 1 + γ ⎪ −→C μ − μ 5 − ⎪ iγ ∂μ qγ Aμ m , ⎪ 2 ⎪ ⎪ ⎪ + ⎪ P μ μ 1 γ5 ⎪ −→ iγ ∂μ + qγ Aμ − m , − ⎨ 2 μ μ 1 γ5 iγ ∂μ + qγ Aμ − m 2 ⎪ ⎪ T 1 − γ5 ⎪ −→ iγ μ∂ + qγμA − m , ⎪ μ μ 2 ⎪ ⎪ ⎪ ⎪ CPT 1 − γ ⎩ −→ iγ μ∂ − qγμA 5 − m , μ μ 2 i.e. a theory of left-handed particles in a (conventionally) right-handed {x,y,z} frame is a theory of right-handed anti-particles in that frame, and is also a theory of right-handed particles in the left-handed frame {−x,−y,−z}. Chiral gauge theories are invariant under time reversal in the sense that forward evolution (calculation of a ﬁnal state from an initial state) and backward evolution (calculation of the initial state from a ﬁnal state) are described by exactly the same equation: There is only one time-evolution operator. Finally, CPT maps a left-handed chiral gauge theory for particles in right-handed frames into the corresponding left-handed chiral gauge theory for the anti-particles in left-handed frames. Appendix J Green’s Functions in d Dimensions

We denote the number of spatial dimensions with d in this appendix, and we suspend the use of summation convention until we reach (J.118). Green’s functions are solutions of linear differential equations with δ function source terms. Basic one-dimensional examples are provided by the conditions

d d2 S(x) − κS(x) =−δ(x), G(x) − κ2G(x) =−δ(x), (J.1) dx dx2 with solutions a a − 1 G(x) = exp(−κ|x|) + exp(κ|x|) + A exp(κx) + B exp(−κx), (J.2) 2κ 2κ and d S(x) = G(x) + κG(x) dx = a(−x)exp(κx) + (a − 1)(x) exp(κx) + 2κAexp(κx) = C exp(κx) + (−x)exp(κx) = C exp(κx) − (x) exp(κx), (J.3)

C = C + 1 = 2κA + a. (J.4)

That the functions (J.2) and (J.3) satisfy the conditions (J.1) is easily conﬁrmed by using

d d |x|=(x) − (−x), (±x) =±δ(x). (J.5) dx dx The solutions of the conditions in the limit κ → 0are

© The Editor(s) (if applicable) and The Author(s), under exclusive license to 773 Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1 774 J Green’s Functions in d Dimensions

|x| d (−x) − (x) G(x) = αx + β − ,S(x)= G(x) = α + . (J.6) 2 dx 2 The appearance of integration constants signals that we can impose boundary conditions on the Green’s functions. An important example for this is the requirement of vanishing Green’s functions at spatial infinity, which can be imposed if the real part of κ does not vanish. For positive real κ this implies the one-dimensional Green’s functions 1 G(x) = exp(−κ|x|), S(x) = (−x)exp(κx). (J.7) 2κ However, in one dimension we cannot satisfy the boundary condition of vanishing Green’s functions at infinity if κ = 0, and we will find the same result for the scalar Green’s function G(x) in two dimensions. We can satisfy conditions that the Green’s functions (J.6) should vanish on a semiaxis x<0orx>0forκ = 0by choosing α =∓1/2, β = 0. On the other hand, if κ = ik is imaginary with k>0, the Green’s function i G(x) = exp(ik|x|) (J.8) 2k describes the spatial factor of outgoing waves exp[i(k|x|−ωt)], i.e. the one- dimensional version of outgoing spherical waves.

Green’s Functions for the Schrödinger Equation

We are mostly concerned with Green’s functions associated with time-independent Hamilton operators p2 h¯ 2 H = + V(x) = dd x |x − + V(x) x|. (J.9) 2m 2m

Note that the number of spatial dimensions d is left as a discrete variable. The inversion condition for the energy-dependent Schrödinger operator,

(E − H) Gd,V (E) = 1, (J.10) in x representation is the condition 2 h¯ E + − V(x) x|G (E)|x =δ(x − x ). (J.11) 2m d,V J Green’s Functions in d Dimensions 775

Equations (J.10) and (J.11) show that we should rather talk about a Green’s operator Gd,V (E) (or a resolvent in mathematical terms), with matrix elements x|Gd,V (E)|x . We will instead continue to use the designation Green’s function both for Gd,V (E) and the Fourier transformed operator Gd,V (t) and for all their representations in x or k space variables (or their matrix elements with respect to any other quantum states). The designation Green’s function originated from the matrix elements Gd,V (x, x ; E) ≡x|Gd,V (E)|x . These functions preceded the resolvent Gd,V (E) because the inception of differential equations preceded the discovery of abstract operator concepts and Bra-ket notation. The Green’s function Gd,V (E) can eventually be calculated perturbatively in terms of the free Green’s function Gd (E) ≡ Gd,V=0(E). The equations

(E − H0) Gd,V (E) = 1 + V Gd,V (E), (E − H0) Gd (E) = 1, (J.12) yield

Gd,V (E) = Gd (E) + Gd (E)V Gd,V (E)

= Gd (E) + Gd (E)V Gd (E) + Gd (E)V Gd (E)V Gd,V (E) ∞ ∞ n n = Gd (E) (V Gd (E)) = (Gd (E)V ) Gd (E). (J.13) n=0 n=0

From the geometric series appearing in (J.13) we can also ﬁnd the representations

1 1 Gd,V (E) = Gd (E) = Gd (E) (J.14) 1 − V Gd (E) 1 − Gd (E)V which are of course equivalent to the original condition (E − H)Gd,V (E) = 1 through −1 −1 −1 (E − H0 − V ) = (E − H0) 1 − (E − H0) V −1 −1 −1 = 1 − (E − H0) V (E − H0) (J.15) and the corresponding relation with E − H0 extracted on the right-hand side of V . Whether the formal iteration (J.13) yields a sensible numerical approximation depends on the potential V , the energy E, and on the states for which we wish to 2 calculate the corresponding matrix element of Gd,V (E). We deﬁned H0 = p /2m as the free Hamiltonian, and we have used the ﬁrst two terms of (J.13) in potential scattering theory in the Born approximation. Other applications of series like (J.13) in perturbation theory would include a solvable part V0 of the potential in H0 and use only a perturbation V = V − V0 for the iterative solution. However, our main concern in the following will be the free Green’s function Gd (E). 776 J Green’s Functions in d Dimensions

The variable E in (J.10) can be complex, but Gd,V (E) will become singular for values of E in the spectrum of H . It is therefore useful to explicitly add a small imaginary part if E is constrained to be real, which is the most relevant case for us. To discuss the implications of a small imaginary addition to E, consider Fourier transformation of (J.10) into the time domain. Substitution of

∞ Gd,V (E) = dt Gd,V (t) exp(iEt/h),¯ (J.16) −∞ 1 ∞ Gd,V (t) = dE Gd,V (E) exp(− iEt/h),¯ (J.17) 2πh¯ −∞ yields

d ih¯ − H G (t) = δ(t). (J.18) dt d,V

We can solve this equation in the form

a a − 1 G (t) = (t)K (t) + (− t)K (t) d,V ih¯ d,V ih¯ d,V a − (− t) = K (t), (J.19) ih¯ d,V if Kd,V (t) is the solution of the time-dependent Schrödinger equation

d ih¯ − H K (t) = 0 (J.20) dt d,V with initial condition Kd,V (0) = 1. Indeed, we have found this solution and used it extensively in Chap. 13. It is the time evolution operator

i K (t) = U(t) = exp − Ht . (J.21) d,V h¯

Equations (J.19) and (J.21) imply that the Green’s function in the energy representation is a ∞ Gd,V (E) = dt exp[i(E − H + i )t/h¯] ih¯ 0 1 − a 0 − dt exp[i(E − H − i )t/h¯] ih¯ −∞ a 1 − a = + , (J.22) E − H + i E − H − i with a small shift >0. J Green’s Functions in d Dimensions 777

The time-dependent Green’s function (J.19) solves the inhomogeneous Schrödinger equation with a state |φ(t) driving the evolution of the state |ψ(t),

d ih¯ − H |ψ (t)=0. (J.25) dt 0

The Green’s function (J.19) and (J.22) with a = 1istheretarded Green’s function, because the solution (J.24) receives only contributions from |φ(t) at times t

(t) it G (t) = exp − p2 , (J.26) d ih¯ 2mh¯

2m 1 Gd (E) =− Gd (E) = . (J.27) h¯ 2 E + i − (p2/2m)

The rescaled Green’s function Gd (E) is an inverse Poincaré operator

2mE + x|Gd (E)|x =−δ(x − x ), (J.28) h¯ 2 and has been introduced to make the connection with electromagnetic Green’s functions and potentials more visible. Equations (J.26) and (J.27) do not generate any spectacular dependence on the number d of spatial dimensions in the k-space representation of the retarded free Green’s functions, 778 J Green’s Functions in d Dimensions

(t) ht¯ k|G (t)|k = exp − i k2 δ(k − k ) ≡ G (k,t)δ(k − k ), (J.29) d ih¯ 2m d

δ(k − k ) k|Gd (E)|k = ≡ Gd (k,E)δ(k − k ), (J.30) k2 − (2mE/h¯ 2) − i and also the d-dependence of the mixed representations is not particularly notewor- thy, e.g.

¯ |G | = | G = (t) · − ht 2 x d (t) k x k d (k,t) √ d exp ik x i k , (J.31) ih¯ 2π 2m

1 exp(ik · x) x|G (E)|k=x|kG (k,E)= √ . (J.32) d d d 2 2 2π k − (2mE/h¯ ) − i

The x-representation of the time-dependent Green’s function, 1 x|G (t)|x = dd k G (k,t)exp[ik · (x − x )]≡G (x − x ,t), (J.33) d (2π)d d d is (t) ht¯ G (x,t) = dd k exp i k · x − k2 d ih(¯ 2π)d 2m (t) m d mx2 = exp i . (J.34) ih¯ 2πiht¯ 2ht¯

This equation holds in the sense that Gd (x − x ,t − t ) has to be integrated with an absolutely or square integrable function x|φ(t) to yield a solution x|ψ(t) (J.24) of an inhomogeneous Schrödinger equation. The representation of the retarded free Green’s function in the time-domain is interesting in its own right, but in terms of dependence on the number d of dimensions, the operator ih¯Gd (t) and its representations ih¯Gd (k,t) and ih¯Gd (x,t) are simply products of d copies of the corresponding one-dimensional Green’s function ih¯G1(t) and its representations. Free propagation in time separates completely in spatial dimensions.1 The interesting dimensional aspects of the Green’s function appear if we represent it in the energy domain and in x-space,

1 2 This is a consequence of the separation of the free non-relativistic Hamiltonian H0 = p /2m. However, this property does not hold in relativistic quantum mechanics, and therefore the free time-dependent Green’s function in the relativistic case is not a product of one-dimensional Green’s functions, see (J.94). J Green’s Functions in d Dimensions 779 1 x|G (E)|x = dd k G (k,E)exp[ik · (x − x )] d (2π)d d ≡ Gd (x − x ,E). (J.35)

This requires a little extra preparation.

Polar Coordinates in d Dimensions

Evaluation of the d dimensional Fourier transformation in (J.35) involves polar coordinates in d dimensional k space. Furthermore, it is also instructive to derive the zero energy Green’s function Gd (0) directly in x space, which is also conveniently done in polar coordinates. Therefore we use x space as a paradigm for the discussion of polar√ coordinates in d dimensions with the understanding that in k space, r = x2 is replaced with k = k2. We deﬁne polar coordinates r, θ1,...θd−1 in d dimensions through π x = r sin θ · sin θ · ...· sin θ − · sin θ − ,ϕ= − θ − 1 1 2 d 2 d 1 2 d 1 x2 = r sin θ1 · sin θ2 · ...· sin θd−2 · cos θd−1, . .

xd−1 = r sin θ1 · cos θ2,

xd = r cos θ1. (J.36)

This yields corresponding tangent vectors along the radial coordinate lines, cf. (5.51),

∂x a = = e , (J.37) r ∂r r and along the θi coordinate lines

∂x ai = = r sin θ1 · sin θ2 · ...sin θi−1ei. (J.38) ∂θi

Here we deﬁned the unit tangent vector along the θi coordinate line

ai ei = . (J.39) |ai|

This should not be confused with Cartesian basis vectors since we do not use any Cartesian basis vector in this section. 780 J Green’s Functions in d Dimensions

The induced metric is gμν = aμ ·aν, see Sect. 5.4. This yields in the present case gμν = 0,grr = 1, (J.40) μ =ν and

2 2 2 2 gii = r sin θ1 · sin θ2 · ...· sin θi−1, 1 ≤ i ≤ d − 1. (J.41)

The Jacobian determinant (5.76) of the transformation from polar to Cartesian coordinates and the related volume measure (5.74) are then √ d−1 d−2 d−3 g = r sin θ1 · sin θ2 · ...· sin θd−2 (J.42) and

d d−1 d−2 d−3 d x = drdθ1 ...dθd−1 r sin θ1 · sin θ2 · ...· sin θd−2. (J.43)

In particular, the hypersurface area of the (d − 1)-dimensional unit sphere is √ d'−2 π d n 2 π S − = 2π dθ sin θ = . (J.44) d 1 (d/2) n=1 0 ∇ = μ The gradient operator μ a ∂μ is

d−1 ∂ ei ∂ ∇ = er + . (J.45) ∂r r sin θ1 · sin θ2 · ...· sin θi−1 ∂θi i=1

For the calculation of the Laplace operator, we need the derivatives (recall that we do not use summation convention in this appendix)

∂er ej · = sin θ1 · sin θ2 · ...sin θj−1 (J.46) ∂θj and

∂ei ej · = δj,i+1 cos θi + (j > i + 1) cos θi · sin θi+1 · ...· sin θj−1. (J.47) ∂θj

This yields

− ∂2 d − 1 ∂ 1 d 1 1 ∂2 = + + ∂r2 r ∂r r2 sin2 θ · sin2 θ · ...· sin2 θ − ∂θ2 i=1 1 2 i 1 i J Green’s Functions in d Dimensions 781

− − 1 d 2 d 1 cot θ ∂ + i . 2 2 2 2 (J.48) r sin θ · sin θ · ...· sin θ − ∂θi i=1 j=i+1 1 2 i 1

We only need the radial part of the Laplace operator for the direct calculation of the zero energy Green’s function Gd (x,E = 0) ≡ Gd (r). The condition

1 d − d G (r) = rd 1 G (r) =−δ(x) (J.49) d rd−1 dr dr d implies after integration over a spherical volume with radius r,

√ d d−1 d 2 π d−1 d S − r G (r) = r G (r) =−1. (J.50) d 1 dr d (d/2) dr d

This yields ⎧ ⎪ (a − r)/2,d= 1, ⎨⎪ − −1 = G (r) = (2π) ln(r/a), d 2, (J.51) d ⎪ ⎪ √ −1 ⎩ d−2 d d−2 ≥ 2 4 π r ,d3.

The integration constant determines for d = 1 and d = 2 at which distance a the Green’s function vanishes. For d ≥ 3 the vanishing integration constant ∝ a2−d is imposed by the usual boundary condition limr→∞ Gd≥3(r) = 0. The free Green’s function in the x-representation with full energy dependence is still translation invariant and isotropic, x|Gd (E)|x ≡Gd (x − x ,E)= Gd (|x − x|,E), and can be gotten from integration of the condition

2m Gd (x,E)+ EGd (x,E)=−δ(x). (J.52) h¯ 2

The result (J.51) motivates an ansatz

Gd (x,E)= fd (r, E)Gd (r). (J.53)

This will solve (J.52) if the factor fd (r, E) satisﬁes

d2 3 − d d 2m fd (r, E) + fd (r, E) + Efd (r, E) = 0,fd (0,E)= 1. (J.54) dr2 r dr h¯ 2

This yields together√ with the requirement√ Gd (x,E)|E<0 ∈ R and analyticity in E (and the convention −E|E>0 =−i E), 782 J Green’s Functions in d Dimensions

√ d−2 (−E) −2mE 2 √ r = − − Gd (x,E) √ d K d 2 2mE 2π hr¯ 2 h¯ √ d−2 2 √ + π (E) 2mE (1) r i √ d H d−2 2mE , (J.55) 2 2π hr¯ 2 h¯ where the conventions and deﬁnitions from [1] were used for the modiﬁed Bessel and Hankel functions. The result (J.55) tells us that outgoing spherical waves of energy E>0ind dimensions are given by Hankel functions,

√ d−2 2 √ = iπ 2mE (1) r Gd (x,E >0) √ d H d−2 2mE , (J.56) 2 2π hr¯ 2 h¯ with asymptotic form

d−1 1 k 2 d − 3 Gd (x,E >0) exp ikr − i π , (J.57) kr 1 2k 2πr 4 while d dimensional Yukawa potentials of range a are described by modiﬁed Bessel functions,

d−2 1 r 2 r = − Vd (r) √ d K d 2 , (J.58) 2π rd−2 a 2 a with asymptotic form − exp( r/a) Vd (r a) √ − √ d−1 . (J.59) 2 ad 3 2πr

The result (J.55) can also be derived through Fourier transformation (J.35)fromthe energy-dependent retarded Green’s function in k space, 1 G (x,E) = dd k G (k,E)exp(ik · x), (J.60) d (2π)d d 1 Gd (k,E) = , (J.61) k2 − (2mE/h¯ 2) − i or in terms of poles in the complex k plane (where k ≡|k| for d>1), J Green’s Functions in d Dimensions 783

(E) Gd (k,E) = (k − 2mE/h¯ 2 − i )(k + 2mE/h¯ 2 + i ) (−E) + . (J.62) (k − i −2mE/h¯ 2)(k + i −2mE/h¯ 2)

This yields for d>1 (for the ϑ integral see [137, p. 457, no. 6]) ∞ π d−1 S − k exp(ikr cos ϑ) − G (x,E) = d 2 dk dϑ sind 2 ϑ d d 2 2 (2π) 0 0 k − (2mE/h¯ ) − i ∞ π d−1 1 k exp(ikr cos ϑ) − = dk dϑ sind 2 ϑ √ + − 2 2 d−1 d 1 d 1 0 0 k − (2mE/h¯ ) − i 2 π 2 √ d 1 ∞ k = √ dk J d−2 (kr) d √ d− 2 2 2π r 2 0 k − (2mE/h¯ ) − i 2 √ d−2 (−E) −2mE 2 √ r = − − √ d K d 2 2mE 2π hr¯ 2 h¯ √ d−2 2 √ + π (E) 2mE (1) r i √ d H d−2 2mE . (J.63) 2 2π hr¯ 2 h¯

For the k integral for E>0see[138, p. 179, no. 28]. The real part of the integral for E<0 is given on p. 179, no. 35. The integrals can also be performed with symbolic computation programs, of course. The k integral actually diverges for d ≥ 5, but recall that we have found the same solution for arbitrary d from the ansatz (J.53). Fourier transformation of (J.62)ford = 1 directly yields the result (21.7), which also coincides with (J.63)ford = 1.

The Time Evolution Operator in Various Representations

We have seen that the Green’s function Gd,V (t) in the time domain is intimately connected to the time evolution operator

U(t) = exp(− iHt/h¯) (J.64) through Eqs. (J.19) and (J.21). We can also deﬁne an energy representation for the time evolution operator in analogy to Eqs. (J.16) and (J.17), 784 J Green’s Functions in d Dimensions

∞ U(E) = dt U(t)exp(iEt/h)¯ = 2πhδ(E¯ − H). (J.65) −∞

Indeed, we have encountered this representation of the time evolution operator already in the frequency decomposition (5.37) of states, √ |ψ(ω)= 2πU(hω)¯ |ψ(t = 0). (J.66)

The free d dimensional evolution operator in the time domain is simply the product of d one-dimensional evolution operators,

it U (t) = exp − p2 , (J.67) 0 2mh¯

ht¯ k|U (t)|k =U (k,t)δ(k − k ), U (k,t)= exp − i k2 , (J.68) 0 0 0 2m

¯ | | = 1 · − ht 2 x U0(t) k √ d exp ik x i k , (J.69) 2π 2m and 1 x|U (t)|x = d3k exp ik · (x − x ) U (k,t)= U (x − x ,t), 0 (2π)d 0 0 m d mx2 U (x,t)= exp i . (J.70) 0 2πiht¯ 2ht¯

Just like for the free Green’s functions, the dependence on d becomes more interesting in the energy domain. The equation √ h¯ 2k2 2m 2mE U (k,E)= 2πhδ¯ E − = π δ |k|− (J.71) 0 2m E h¯ yields 1 U (x,E) = dd k exp(ik · x)U (k,E) 0 (2π)d 0 √ √ − π √ = (E)Sd−2 d d 2 r d−2 d d−1 2m E dϑ exp i 2mE cos ϑ sin ϑ 2 (πh)¯ 0 h¯ d−2 m 1 mE 2 √ r = (E) J d−2 2mE . (J.72) h¯ πhr¯ 2 2 h¯ J Green’s Functions in d Dimensions 785

We have encountered several incarnations of the time evolution equation

|ψ(t)=U0(t − t )|ψ(t ), (J.73) e.g. with k|ψ≡k|ψ(0), d x|ψ(t)= d k x|U0(t)|kk|ψ ¯ = 1 d · − ht 2 √ d d k exp i k x k ψ(k) 2π 2m d = d x U0(x − x ,t − t )x |ψ(t ). (J.74)

Equation (J.72) implies with |ψ≡|ψ(t = 0) the (x,ω) representation for free states in terms of their (x,t = 0) representations, 1 x|ψ(ω)=√ dt exp(iωt) x|ψ(t) 2π 1 d = √ d x dt exp(iωt) x|U0(t)|x x |ψ 2π 1 d = √ d x x|U0(hω)¯ |x x |ψ 2π d−2 m 1 mω 2 = (ω)√ 2πh¯ π 2h¯ ⎛ ⎞ | × d ⎝ 2mω | − |⎠ x ψ d x J d−2 x x √ − . (J.75) 2 h¯ |x − x|d 2

In turn, Eq. (5.35) implies for the initial state 1 |ψ=√ dω|ψ(ω), (J.76) 2π and therefore the kernel in (J.75) must yield a d dimensional δ function,

⎛ ⎞ d−2 ⎛ ⎞ ∞ 2 m ⎝ 2mω 1 ⎠ ⎝ 2mω ⎠ √ dω J d−2 |x| = δ(x), (J.77) d | | 2π h¯ 0 h¯ x 2 h¯ or in terms of magnitude of wave number, 786 J Green’s Functions in d Dimensions

∞ d−2 1 k 2 − = √ d dk k J d 2 (kr) δ(x). (J.78) 2π 0 r 2

However, this is just the familiar relation 1 dd k exp(ik · x) = δ(x) (J.79) (2π)d after evaluation of the angular integrals in polar coordinates in k space. In particular, for d = 1 we ﬁnd ⎛ ⎞ m 2mω x|ψ(ω)=(ω) dx cos⎝ (x − x )⎠ x |ψ. (J.80) πhω¯ h¯

In two dimensions we ﬁnd ⎛ ⎞ m 2 ⎝ 2mω ⎠ x|ψ(ω)=(ω)√ d x J0 |x − x | x |ψ, (J.81) 2πh¯ h¯ and in three dimensions we ﬁnd ⎛ ⎞ x| | = √ m 3 ⎝ 2mω | − |⎠ ψ x ψ(ω) (ω) d x sin x x . (J.82) 2π 3h¯ h¯ |x − x |

The free states in the (x,ω)representation are superpositions of stationary radial waves, where each point contributes with weight x|ψ.

Relativistic Green’s Functions in d Spatial Dimensions

Although the theory in this subsection is relativistic, we will not use manifestly Lorentz covariant 4-vector notation like x = (ct, x), k = (ω/c, k) for the spacetime and momentum variables because we are also interested in mixed representations of the Green’s functions like G(x,ω). For the manifestly covariant notation see the note at the end of this Appendix. The relativistic free scalar Green’s function in the time domain must satisfy

2 2 2 1 ∂ m c − − Gd (x,t; x ,t ) =−δ(x − x )δ(t − t ). (J.83) c2 ∂t2 h¯ 2

This yields after transformation into (k,ω)space J Green’s Functions in d Dimensions 787

Gd (k,ω; k ,ω) = Gd (k,ω)δ(k − k )δ(ω − ω ), (J.84) where the factor Gd (k,ω)is

= 1 Gd (k,ω) 2 2 2 . (J.85) k2 − ω + m c − i c2 h¯ 2

The shift −i , >0, into the complex plane is such that this reproduces the retarded non-relativistic Green’s function (J.30) in the non-relativistic limit

mc2 + E ω ⇒ , (J.86) h¯ when terms of order O(E2) are neglected. However, in the relativistic case this yields both retarded and advanced contributions in the time domain. This convention for the poles in the relativistic theory was introduced by Feynman [52] and yields the Green’s functions of Stückelberg and Feynman. The solution in x = (ct, x) space is then

Gd (x,t; x ,t ) = Gd (x − x ,t − t ), (J.87) 1 G (x,t)= dωG (x,ω)exp(− iωt), (J.88) d 2π d 1 G (x,ω)= dd k G (k,ω)exp(ik · x). (J.89) d (2π)d d

The integral is the same as in (J.63) with the substitution

2m ω2 m2c2 E → − , (J.90) h¯ 2 c2 h¯ 2 i.e.

d−2 2 − ¯| | 2 4 − ¯ 2 2 2 = (mc h ω ) m c h ω Gd (x,ω) √ d 2π hcr¯ 2 4 2 2 r ×K d−2 m c − h¯ ω 2 hc¯

d−2 ¯| |− 2 ¯ 2 2 − 2 4 2 + π (h ω mc ) h ω m c i √ d 2 2π hcr¯ r × (1) 2 2 − 2 4 H d−2 h¯ ω m c . (J.91) 2 hc¯ 788 J Green’s Functions in d Dimensions

The ω = 0 Green’s functions

d−2 m2c2 1 mc 2 mc − x =− x x = d−2 2 Gd ( ) δ( ), Gd ( ) √ d K r , h¯ 2π hr¯ 2 h¯ yield again the results (J.51) in the limit m → 0, albeit with diverging integration constants in low dimensions,

h¯ 2h¯ a = = ,a= = exp(−γ). (J.92) d 1 mc d 2 mc In terms of poles in the complex k plane, the complex shift in (J.85) implies

c2(hω¯ − mc2) Gd (k,ω) = ck − ω2 − (mc2/h)¯ 2 − i ck + ω2 − (mc2/h)¯ 2 + i

c2(mc2 − hω)¯ + . (J.93) ck − i (mc2/h)¯ 2 − ω2 ck − i (mc2/h)¯ 2 − ω2

However, in terms of poles in the complex ω plane, Eq. (J.85) implies

c2 Gd (k,ω)=− . ω − c k2 + (mc/h)¯ 2 + i ω + c k2 + (mc/h)¯ 2 − i

Fourier transformation to the time domain therefore yields a representation of the relativistic free Green’s function which explicitly shows the combination of retarded positive frequency and advanced negative frequency components, 1 G (k,t) = dωG (k,ω)exp(− iωt) d 2π d exp − i k2 + (mc/h)¯ 2 ct = ic(t) 2 k2 + (mc/h)¯ 2 exp i k2 + (mc/h)¯ 2 ct + ic(− t) 2 k2 + (mc/h)¯ 2 exp − i k2 + (mc/h)¯ 2 c|t| = ic . (J.94) 2 k2 + (mc/h)¯ 2 J Green’s Functions in d Dimensions 789

On the other hand, shifting both poles into the lower complex ω plane,

2 (r) =− c Gd (k,ω) , ω − c k2 + (mc/h)¯ 2 + i ω + c k2 + (mc/h)¯ 2 + i yields the retarded relativistic Green’s function 1 G(r)(k,t) = dωG(r)(k,ω)exp(− iωt) d 2π d sin k2 + (mc/h)¯ 2 ct 2 = c(t) = c (t)Kd (k,t), (J.95) k2 + (mc/h)¯ 2 cf. Eq. (22.15). If Kd (x,t)exists, then one can easily verify that the properties

1 ∂2 m2c2 − − Kd (x,t)= 0, (J.96) c2 ∂t2 h¯ 2 ∂ Kd (x, 0) = 0, Kd (x,t) = δ(x) (J.97) ∂t t=0

(r) = 2 K imply that Gd (x,t) c (t) d (x,t) is a retarded Green’s function. Further- more, shifting both poles into the upper complex ω plane,

2 (a) =− c Gd (k,ω) , ω − c k2 + (mc/h)¯ 2 − i ω + c k2 + (mc/h)¯ 2 − i yields the advanced relativistic free Green’s function 1 G(a)(k,t) = dωG(a)(k,ω)exp(− iωt) d 2π d sin k2 + (mc/h)¯ 2 ct =−c(− t) k2 + (mc/h)¯ 2

=− 2 − K = (r) − c ( t) d (k,t) Gd (k, t). (J.98) 790 J Green’s Functions in d Dimensions

We have been cautious in our terminology to separate the propagators, which are time evolution kernels and propagate initial conditions, from the Green’s functions, which propagate perturbations or driving terms in a differential equation. However, denoting the Green’s functions with Feynman conventions for the poles as “Feynman Green’s functions” is awkward. At the expense of precision, we will therefore adopt the convention to denote those Green’s functions as Feynman propagators.

Retarded Relativistic Green’s Functions in (x, t) Representation

(r) Evaluation of the Green’s functions Gd (x,t)and Gd (x,t)for the massive Klein– Gordon equation is very cumbersome if one uses standard Fourier transformation between time and frequency. It is much more convenient to use Fourier transformation with imaginary frequency, which is known as Laplace transformation. We will demonstrate this for the retarded Green’s function. We try a Laplace transform of (r) 2 Gd (x,t)in the form ∞ = − (r) " ≥ gd (x,w) dt exp( wt)Gd (x,t), w 0. (J.99) 0

The completeness relation for Fourier monomials, 1 ∞ 1 i∞ δ(t) = dω exp(− iωt) = dw exp(wt) (J.100) 2π −∞ 2πi −i∞ then yields the inversion of (J.99), i∞ (r) = 1 Gd (x,t) dw exp(wt)gd (x,w). (J.101) 2πi −i∞

The condition (J.83)onthed dimensional scalar Green’s functions then implies

2Assuming only "w ≥ 0 assumes that the retarded Green’s functions are integrable along the time axis. This makes physical sense since the impact of a perturbation which occured at time t = 0 at the point x = 0 that is felt at the point x should decrease with time. The assumption can also be justiﬁed a posteriori from the explicit results (J.113)–(J.115), which show that the Green’s functions for d ≤ 3 oscillate and decay for t →∞. For Laplace transforms of less well behaved functions G(x,t) one can require "w>νif exp(−νt)G(x,t) is bounded for t →∞. The vertical integration contour for the inverse transformation (J.101) must then be to the right of − ∞→ + ∞ (r) ν i ν i . However, this would not save the day for the non-existent functions Gd≥4(x,t), although we can ﬁnd functions gd (x,w)(J.109)and(J.110) for every number d of dimensions. J Green’s Functions in d Dimensions 791

w2 m2c2 − − gd (x,w)=−δ(x) (J.102) c2 h¯ 2 with solution · 1 d exp(ik x) gd (x,w)= d k . (J.103) (2π)d k2 + (w/c)2 + (mc/h)¯ 2

In one dimension this yields c exp − w2 + (mc2/h)¯ 2 |x|/c g1(x, w) = . (J.104) 2 w2 + (mc2/h)¯ 2

In higher dimensions, we need to calculate ∞ π S − − − exp(ikr cos ϑ) g (x,w) = d 2 dk dϑ kd 1 d 2 ϑ d d sin 2 2 2 (2π) 0 0 k + (w/c) + (mc/h)¯ 1 ∞ kd−1 1 = √ dk √ J d−2 (kr). (J.105) d 2 + 2 + 2 d−2 2π 0 k (w/c) (mc/h)¯ kr 2

We can formally reduce (J.105)ford ≥ 3 to the corresponding integrals in lower dimensions by using the relation

n 1 d J (x) J + (x) − ν = ν n , (J.106) x dx xν xν+n see number 9.1.30, p. 361 in [1]. This yields for d = 2n + 1

n √ 1 − 1 ∂ − = 2n − √ d−2 J d 2 (kr) k krJ− 1 (kr) kr 2 r ∂r 2 n 2 − 1 ∂ = k 2n − cos(kr), (J.107) π r ∂r and for d = 2n + 2,

n 1 −2n 1 ∂ J d−2 (kr) = k − J (kr). √ d−2 0 (J.108) kr 2 r ∂r

The resulting relations for the Green’s functions in the (x,w) representations are then 792 J Green’s Functions in d Dimensions 1 ∂ n 1 ∞ cos(kr) + x = − g2n 1( ,w) dk 2 2 2 2πr ∂r π 0 k + (w/c) + (mc/h)¯ − 2 + 2 2 1 ∂ n c exp w (mc /h)¯ r/c = − , (J.109) 2πr ∂r 2 w2 + (mc2/h)¯ 2 and

n ∞ 1 ∂ 1 kJ0(kr) + x = − g2n 2( ,w) dk 2 2 2 2πr ∂r π 0 k + (w/c) + (mc/h)¯ 1 ∂ n 1 r = − K w2 + (mc2/h)¯ 2 . (J.110) 2πr ∂r 2π 0 c

(r) Inverse Laplace transformation yields the retarded Green’s functions Gd (x,t), 1 ∂ n c G(r) (x,t) = − (ct − r)J mc c2t2 − r2/h¯ , (J.111) 2n+1 2πr ∂r 2 0

n − (r) 1 ∂ c (ct r) 2 2 2 G + (x,t) = − √ cos mc c t − r /h¯ . (J.112) 2n 2 2πr ∂r 2π c2t2 − r2

Except for m = 0 and d = 1, the terms proportional to (ct − r), i.e. the components inside the light cone, to the retarded Green’s functions decrease like t−d/2 for t →∞and r ﬁxed. However, the δ function singularities on the light cone become unacceptable for d ≥ 4. The retarded relativistic Green’s functions in one, two and three dimensions are therefore c G(r)(x, t) = (ct −|x|)J mc c2t2 − x2/h¯ , (J.113) 1 2 0 c (ct − r) G(r)(x,t)= √ cos mc c2t2 − r2/h¯ , (J.114) 2 2π c2t2 − r2 and

2 − (r) c mc (ct r) 2 2 2 G (x,t)= δ(r − ct) − √ J1 mc c t − r /h¯ . (J.115) 3 4πr 4πh¯ c2t2 − r2

The (x,t) representations of the corresponding advanced Green’s functions then follow from (J.98)as

(a) = (r) − Gd (x,t) Gd (x, t). (J.116) J Green’s Functions in d Dimensions 793

The propagator function Kd (x,t) for the free Klein–Gordon ﬁelds follows from (J.95) and (J.98)as

2K = (r) − (a) c d (x,t) Gd (x,t) Gd (x,t). (J.117)

(r) K The functions Gd≥4(x,t) and d≥4(x,t) do not exist, but the corresponding (r) = 2 K (r) functions Gd (k,t) c (t) d (k,t)(J.95) and Gd (k,ω)exist in any number of dimensions.

Green’s Functions for Dirac Operators in d Dimensions

We now restore summation convention. The Green’s functions for the free Dirac operator must satisfy

mc iγ μ∂ − S (x,t)=−δ(x)δ(t). (J.118) μ h¯ d

Since the Dirac operator is a factor of the Klein–Gordon operator, the solutions of Eqs (J.118) and (J.83) are related by

mc S (x,t)= iγ μ∂ + G (x,t) (J.119) d μ h¯ d and d Gd (x,t) = d x dt Sd (x − x,t − t) · Sd (x ,t ) d = d x dt Sd (x ,t ) · Sd (x + x,t + t). (J.120)

2 μ The free Dirac Green’s function in wave number representation is (here k ≡ k kμ)

(mc/h)¯ − γ · k Sd (k) = , (J.121) k2 + (mc/h)¯ 2 − i where the pole shifts again correspond to the Feynman propagator with retarded and advanced components. The corresponding retarded Green’s function (both poles in the lower complex ω plane) and the advanced Green’s function (both poles in the upper half-plane) are related to the time evolution kernel of the Dirac ﬁeld (cf. Eq. (22.108)), 794 J Green’s Functions in d Dimensions mc − γ · p W(k,t)= exp − icp 0t/h¯ · γ 0, (J.122) 2p0 p0=±E(k)/c through

(r) = W · 0 (a) =− − W · 0 Sd (k,t) ic(t) (k,t)γ ,Sd (k,t) ic( t) (k,t)γ . (J.123)

Green’s Functions in Covariant Notation

The relativistic free scalar Green’s function satisﬁes

p2 + m2c2 Gd = 1, (J.124) h¯ 2 i.e. in the k = (ω/c, k) domain

k|Gd |k =Gd (k)δ(k − k ), (J.125) where the factor Gd (k) is

1 Gd (k) = . (J.126) k2 + (m2c2/h¯ 2) − i

This yields after transformation into x = (ct, x) space (D = d + 1), 1 x|G |x = dDk dDk k|G |k exp[i(k · x − k · x )], d (2π)D d 1 = dDkG (k) exp[ik · (x − x )]=G (x − x ), (J.127) (2π)D d d which satisﬁes

2 2 2 2 2 m c 2 m c ∂ − x|Gd |x = ∂ − x|Gd |x =−δ(x − x ). (J.128) h¯ 2 h¯ 2

Therelationto(J.83)–(J.115)is

1 x|G |x = G (x,t; x ,t ), k|G |k =cG (k,ω; k ,ω), (J.129) d c d d d

Gd (k) = Gd (k,ω). (J.130) J Green’s Functions in d Dimensions 795

Translation invariance (J.125) and (J.127) implies that the Green’s function in mixed representation is proportional to plane waves, x|Gd |k=Gd (k)x|k. The fermionic Green’s function satisﬁes γ · p + mc mc − γ · p S = 1,S= G , (J.131) h¯ d d h¯ d or in various representations,

mc iγ · ∂ − x|S |x =−δ(x − x ), (J.132) h¯ d

k|Sd |k =Sd (k)δ(k − k ), x|Sd |k=Sd (k)x|k, (J.133)

mc (mc/h)¯ − γ · k Sd (k) = − γ · k Gd (k) = , (J.134) h¯ k2 + (m2c2/h¯ 2) − i

1 x|S |x =S (x − x ) = S (x − x ,t − t ) d d c d 1 = dDkS (k) exp[ik · (x − x )] (2π)D d

mc = iγ · ∂ + G (x − x ). (J.135) h¯ d

The pole shifts in (J.126) and (J.134) correspond to the Feynman conventions. This yields ic dd k G (x) = (t) exp(ik · x) + (− t)exp(− ik · x) ,(J.136) d (2π)d 2ω(k) ic dd k mc S (x) = (t) − γ · k exp(ik · x) (J.137) d (2π)d 2ω(k) h¯ mc + (− t) + γ · k exp(− ik · x) , (J.138) h¯ where k0 = ω(k)/c. These Green’s functions are related to matrix elements of the Dirac picture operators (or free Heisenberg picture operators) for the Klein–Gordon or Dirac ﬁelds through

+ 1 i0|T φ(x)φ (x )|0 = x|G |x , (J.139) c c d 796 J Green’s Functions in d Dimensions

i0|Tcψ(x)ψ(x )|0 =x|Sd |x , (J.140) where Tc is the chronological time ordering operator (17.205). (r) (r) For the retarded Green’s functions Gd and Sd both poles have to be shifted into the lower k0 plane. Note that as a consequence of (J.135) the free fermionic Green’s functions also satisfy

mc i∂ S (x − x )γ μ + S (x − x ) = δ(x − x ). (J.141) μ d h¯ d

Green’s Functions as Reproducing Kernels

Suppose that V is a (d + 1)-dimensional spacetime volume with boundary ∂V. Equation (J.128) and the Klein–Gordon equation imply for a free ﬁeld φ(x) and for x in V the relation 2 2 D m c 2 φ(x) = d x φ(x ) − ∂ Gd (x − x ) V h¯ 2 ↔ D μ = d x ∂ Gd (x − x ) ∂μ φ(x ) . (J.142) V

The Gauss theorem in D spacetime dimensions then yields a representation for all values of φ(x) inside of V in terms of the values of the Klein–Gordon ﬁeld on the boundary ∂V,

↔ d μ φ(x) = d x n Gd (x − x ) ∂μ φ(x ) , (J.143) ∂V

μ where n is an outward bound normal vector with n0 = 1 on spacelike boundaries t = constant, t >t, and n0 =−1ont = constant, t

(r) − = − K − − Gd (x x ) c(t t ) d (x x ,t t ), (J.144) or the advanced Green’s function,

(a) − = − K − − Gd (x x ) c(t t) d (x x ,t t), (J.145) J Green’s Functions in d Dimensions 797 and ∂V contains only the spacelike surface t t, then (J.143) is the solution (22.14) of the initial value problem (t t) for the Klein–Gordon ﬁeld. For free Dirac ﬁelds the Dirac equation and (J.141) implies for x ∈ V the equation d μ ψ(x) = i d x nμSd (x − x )γ ψ(x ). (J.146) ∂V

This yields again the initial/ﬁnal value solution (22.107) if ∂V contains only the spacelike surface t t, since the retarded and advanced Green’s functions are related to the time evolution kernel (22.108) through

(r) − = − W − − · 0 Sd (x x ) i(t t ) d (x x ,t t ) γ , (J.147)

(a) − =− − W − − · 0 Sd (x x ) i(t t) d (x x ,t t ) γ , (J.148) see also Eqs. (J.135), (J.144), (J.145), and (22.110).

Liénard–Wiechert Potentials in Low Dimensions

The massless retarded Green’s functions solve the basic electromagnetic wave equation for the electromagnetic potentials in Lorentz gauge,

μ ν ν μ ∂μ∂ A (x) =−μ0j (x), ∂μA (x) = 0, (J.149) μ d+1 (r) μ A (x) = μ0 d x G (x − x ) j (x ). (J.150) d m=0

In three dimensions this yields the familiar Liénard–Wiechert potentials from the contributions of the currents on the backward light cone of the spacetime point x, μ 1 1 Aμ (x,t)= 0 d3x j μ x ,t − |x − x | . (J.151) d=3 4π |x − x| c

However, in one and two dimensions, the Liénard–Wiechert potentials sample charges and currents from the complete region inside the backward light cone,

3The future value problem or backwards evolution problem asks the question: Which field configuration φ(x) at time t yields the prescribed field configuration φ(x) at time t >tthrough time evolution with the equations of motion? 798 J Green’s Functions in d Dimensions

∞ t−(|x−x|/c) μ = μ0c μ Ad=1(x, t) dx dt j (x ,t ), (J.152) 2 −∞ −∞ t−(|x−x|/c) μ μ = μ0c 2 j (x ,t ) Ad=2(x,t) d x dt . (J.153) 2π −∞ c2(t − t)2 − (x − x)2

Stated differently, a δ function type charge-current fluctuation in the spacetime point x generates an outwards traveling spherical electromagnetic perturbation on the forward light cone starting in x if we are in three spatial dimensions. However, in one dimension the same kind of perturbation fills the whole forward light cone uniformly with electromagnetic fields, and in two dimensions the forward light cone is filled with a weight factor [c2(t − t)2 − (x − x)2]−1/2.How can that be? The electrostatic potentials (J.51)ford = 1 and d = 2 hold the answer to this. Those potentials indicate linear or logarithmic confinement of electric charges in low dimensions. Therefore a positive charge fluctuation in a point x must be compensated by a corresponding negative charge fluctuation nearby. Both fluctuations fill their overlapping forward light cones with opposite electromagnetic fields, but those fields will exactly compensate in the overlapping parts in one dimension, and largely compensate in two dimensions. The net effect of these opposite charge fluctuations at a distance a is then electromagnetic fields along a forward light cone of thickness a, i.e. electromagnetic confinement in low dimensions effectively ensures again that electromagnetic fields propagate along light cones. This is illustrated in Fig. J.1.

+ − x

Fig. J.1 The contributions of nearby opposite charge ﬂuctuations at time t = 0inonespatial dimension generate net electromagnetic ﬁelds in the hatched “thick” light cone region References

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Symbols Bethe sum rule, 351 2-spinor, 178 Bloch factor, 206 Bloch function, 206 A Bloch operators, 534 Absorption coefficient, 465 Bloch state, 206 Absorption cross section, 292 Bohmian mechanics, 172 for photons, 459 Bohr magneton, 611 Active transformation, 83 Bohr radius, 154 Adjoint operator, 78 for nuclear charge Ze, 160 Amplitude vector, 539 Boost parameter, 757 Angular momentum, 134 Born approximation, 232 in relative motion, 130 Born-Oppenheimer approximation, 516 Angular momentum density Boson number operator, 628 for electromagnetic fields, 482 Bra-ket notation, 63, 73 in relativistic field theory, 629 in linear algebra, 73 for scalar Schrödinger field, 362 in quantum mechanics, 74 for spin 1/2 Schrödinger field, 363 Bremsstrahlung, 489 Angular momentum operator, 134 Brillouin zone, 206, 530 addition, 181 commutation relations, 136 C in polar coordinates, 134 Canonical quantization, 367 Annihilation operator, 107 Capture cross section, 291 for non-relativistic particles, 375 due to spontaneous photon emission, 488 for photons, 439 Ceiling function, 257 for relativistic fermions, 604 Center of mass frame for relativistic scalar particles, 587 relativistic, 708 Anti-commutator, 369 Center of mass motion, 129 Attenuation coefficient, 465 Charge conjugation, 584 Auger process, 288 Dirac field, 605 for general representations of Dirac B matrices, 636 Baker–Campbell–Hausdorff formula, 113, 733 Klein-Gordon field, 584 γ matrices, 743 Lorentz covariance, 765 Berry phase, 308 on products of fermion fields, 636

© The Editor(s) (if applicable) and The Author(s), under exclusive license to 805 Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1 806 Index

Charge operator domination for low-energy interactions, for the Dirac field, 603 653 for the Klein-Gordon field, 588 Coulomb waves, 162 for the Schrödinger field, 370 Covalent bonding, 520 in terms of Schrödinger picture Covariant derivative, 334 operators, 374 Covariant transformation behavior, 66 Chirality, 770 for non-orthogonal transformations, 82 Christoffel symbols, 98 Creation operator, 107 in terms of metric, 104 for non-relativistic particles, 375 Classical electron radius, 476 for photons, 439 Clebsch-Gordan coefficients, 183 for relativistic fermions, 604 Coherent states, 115 for relativistic scalar particles, 587 for the electromagnetic field, 441 overcompleteness, 119 Collapse of the wave function, 507 D Color center, 59 Darwin term, 618 spherical model, 167 Decay rate, 281 Commutator, 88 Decoherence Commutators involving operator products, 371 induced by environment, 492 Completeness in the mean, 725 Degeneracy pressure, 426 Completeness of eigenstates, 30, 711 δ function, 25 Completeness relations, 34 Density of states, 249 in cubic quantum wire, 92 in d dimensions, 257 example with continuous and discrete in the energy scale, 255 states, 50 inter-dimensional, 574 for Fourier monomials, 204 non-relativistic particles, 257 for free spherical waves, 149 in radiation, 257 for hydrogen eigenstates, 165 relation to non-relativistic Green’s function, in linear algebra, 69 568 for spherical harmonics, 144 relation to relativistic Green’s function, 580 for Sturm–Liouville eigenfunctions, 724 Density of states operator, 259 for time-dependent Wannier states, 211 Derivative operator for transformation matrices for photon alternating, 446 wave functions, 437 Differential scattering cross section, 293 for Wannier states, 209, 534 for photons, 474 Compton scattering, 654 Dihydrogen cation, 516 non-relativistic limit, 664 Dimensions of states, 95 scattering cross section, 664 Dipole approximation, 338 Compton wavelength, 16 Dipole line strength, 348 Confluent hypergeometric function, 154 Dipole moment Conjugate momentum, 680 induced, 342 Conservation laws, 356 intrinsic, 340 Conserved charge, 358 Dipole selection rules, 338 and local phase invariance, 633 Dirac equation, 595 as symmetry generator, 428 free solution, 600 Conserved current, 358 with minimal coupling, 597 and local phase invariance, 633 non-relativistic limit, 610 Contravariant transformation behavior, 66 relativistic covariance, 758 for non-orthogonal transformations, 82 Dirac γ matrices, see γ matrices Copenhagen interpretation, 493 Dirac picture, 266, 272 Correlation function, 313 in quantum field theory, 384 Coulomb gauge, 432 Dirac’s δ function, 25 Coulomb potential, 448 Dual bases, 67 Index 807

E between frequency and time domain, 92 Eddington tensor, 134 Fourier transforms, 25 Edelstein effect, 641 Frequency-time Fourier transformation, 92 Effective mass, 216 f -sum rule, 347 Ehrenfest theorem, 24 for N particles, 402 in second quantization, 410 G Einstein-Podolsky-Rosen paradox, 497 γ5 matrix, 769 Electric dipole line strength, 348 γ matrices, 596 Electromagnetic coupling, 331 construction in d dimensions, 746 Electron-electron scattering, 666 Dirac basis, 596 Electron interference invariance under Lorentz transformations, biprism experiments, 23 758 Electron-nucleus scattering, 649 Weyl basis, 597 Electron-photon scattering, 654 Gauss bracket, 745 Energy-momentum tensor, 359 Gaussian wave packet, 53 for classical charged particle in free evolution, 54 electromagnetic fields, 706 width, 53 for the Maxwell field, 437 Generators, 138 in quantum optics, 446 from charge operators, 428 for scalar quantum electrodynamics, 625 Golden Rule, 286 for spinor quantum electrodynamics, 605 Golden Rule #1, 288 Energy-time uncertainty, 57, 91, 300, 459 for scattering cross sections, 298 Environmentally induced decoherence, 492 Golden Rule #2, 290 Epistemic collapse, 493 for scattering cross sections, 298 Epistemic reduction, 493 Green’s function, 53, 227 tensor, 134 advanced, 247, 777 Euler–Lagrange equations, 354, 680 in d dimensions, 773 for field theory, 355 for Dirac operator, 793 Ewald construction, 72 for harmonic oscillator, 271 Exchange energy, 405 relations between scalar and spinor Green’s Exchange hole, 425 functions, 793 Exchange integral, 406 inter-dimensional, 573 Exchange interaction, 406 retarded, 53, 228, 568, 573, 777 Exchange term as vacuum expectation value, 409 kinetic, 404 Group, 132 Group theory, 133

F Fermi momentum, 260 H Fermion number operator, 628 Hamiltonian density Fermi’s trick, 287 in Coulomb gauge, 609 improved derivation, 302 for the Dirac field, 603 Feynman propagators, 790 for the Klein-Gordon field, 588 Field operator, 368 for the Maxwell field, 438 Field quantization, 367 in scalar QED, 625 Fine structure constant, 154 for the Schrödinger field, 360 Floor function, 257 for spinor QED, 607 Fluorescence Hard sphere, 237 is different from scattering, 476 Harmonic oscillator, 105 Fock space, 376 coherent states, 115 Foldy-Wouthuysen transformation, 617 eigenstates, 107 Form factors, 244 in k-representation, 111 Fourier transformation in x-representation, 109 808 Index

Harmonic oscillator (cont.) for the Maxwell field, 432 isotropic, 105 for the Dirac field, 603, 760 in polar coordinates, 171 for the Klein-Gordon field, 587 solution by the operator method, 106 for the Schrödinger field, 355 Hartree-Fock equations, 415 Lagrange function, 679 orthogonalized, 404, 417 for particle in electromagnetic fields, 331 Heisenberg evolution equation, 271 for small oscillations in N particle system, Heisenberg Hamiltonian, 407 537 for spin-1 bosons, 415 Lagrangian field theory, 353 Heisenberg picture, 266, 271 Larmor frequency, 306 Heisenberg uncertainty relations, 87 Laue conditions, 71 Hellmann-Feynman theorem, 85 Length dimension of states, 95 Hermite polynomials, 110, 729 Length form Hermitian operator, 32 matrix elements, 454 Higher order commutator, 113 oscillator strength, 347 Hubbard model, 534 Length gauge, 486 Hydrogen atom, 152 Liénard–Wiechert potentials, 797 bound states, 152 Lippmann-Schwinger equation, 227 ionized states, 162 in terms of the full Green’s function, 246 radial expectation values, 159 Local density of states, 259 Waller’s method, 168 Lorentz group, 133, 692 Hydrogen molecule ion, 520 generators, 756 spinor representation, 758 in Weyl and Dirac bases, 760 I vector representation, 756 Interaction picture, see Dirac picture construction from the spinor Inverse Edelstein effect, 641 representation, 762 Ionization rate, 281 Lorentzian absorption line, 462 Lowering operator, 107

J Junction conditions for wave functions, 48 M Matrix logarithm, 737 Maxwell field, 431 K covariant quantization, 619 Kato cusp condition, 529 quantization in Coulomb gauge, 438 Klein-Gordon equation, 584 quantization in Lorentz gauge, 621 with minimal coupling, 584 Mehler formula, 111, 271, 730 non-relativistic limit, 590 Minimal coupling, 334 Klein–Nishina cross section, 663 in the Dirac equation, 597 Klein’s paradox, 592 in the Klein-Gordon equation, 584 oblique incidence, 627 to relative motion in two-body problems, Kramers-Heisenberg formula, 475 443 Kramers-Pasternak relation, 160 in the Schrödinger equation, 332 Kronig-Penney model, 212 Møller operators, 279, 411 density of states, 262 Møller scattering, 666 Van Hove singularities, 262 Mollwo’s law, 59 Kummer’s function, 154 spherical model, 167 Kummer’s identity, 163 Momentum density in Coulomb gauge, 609 for the Dirac field, 603 L for the Klein-Gordon field, 588 Ladder operators, 107 for the Maxwell field, 438 Lagrange density, 353 in scalar QED, 625 Index 809

for the Schrödinger field, 360 to Dirac field, 597 in spinor QED, 607 to Klein-Gordon field, 584 Mott-Gordon states Photon coupling to relative motion, 443 in parabolic coordinates, 245 Photon emission, 450 in polar coordinates, 166 Photon flux, 461 Mott scattering, 653 Photon scattering, 469 Planck’s radiation laws, 7 Poincaré group, 133, 692 N Polar coordinates in d dimensions, 779 Noether’s theorem, 358 Polarizability tensor Nonlinear Schrödinger equation, 402 dynamic, 343 Non-relativistic limit frequency dependent, 344 Dirac equation, 610 static, 341 Klein-Gordon equation, 590 Potential scattering, 225 Normal coordinates, 539 Potential well, see Quantum well Normal modes, 539 Probability density, 20 Normal ordering, 440 Projection postulate, 493 for Dirac fields, 603 Propagator, 52 for Klein-Gordon fields, 588 for harmonic oscillator, 271 for photons, 440 as vacuum expectation value, 410 N-point function, 313 Number operator, 108 for the Schrödinger field, 370 Q in terms of Schrödinger picture Quantization operators, 374 of the Dirac field, 603 of the Klein-Gordon field, 585 of the Maxwell field, 438 O phonons, 548 Occupation number operator, 108 of the Schrödinger field, 368 Ontological collapse, 493 Quantum dot, 44 Optical theorem, 233 Quantum field, 368 from partial wave analysis, 237 Quantum state Oscillator strength, 344 uniqueness, 109 Quantum virial theorem, 82 Quantum well, 44 P Quantum wire, 44 Parabolic coordinates, 103, 242 Parity transformation, 768 Passive transformation, 66 Path integrals, 311 R Pauli equation, 610 Raising operator, 107 Pauli matrices, 177 Rapidity, 757 Pauli term, 463 Rashba term, 619 Perturbation theory, 189, 265 Rayleigh-Jeans law, 6 and effective mass, 217 Rayleigh scattering, 477 time-dependent, 265 Reduced mass, 129 time-independent, 189 Reflection coefficient δ with degeneracy, 195, 200 for function potential, 50 without degeneracy, 189, 194 for Klein’s paradox, 594 Phonons, 548 for a square barrier, 42 Photoelectric effect, 15 Relative motion, 129 Photon, 438 Relativistic spinor momentum eigenstates, 603 Photon absorption, 459 Relativistic spinor plane wave states, 603 Photon coupling Reproducing kernels, 796 810 Index

Rotation group, 133 Symmetric operator, 32 defining representation, 136 Symmetries and conservation laws, 356 generators, 138 Symmetry group, 132 matrix representations, 136 Rutherford scattering, 241 T Temporal gauge, 479 S Tensor product, 65 Scalar, 179 Thomas-Reiche-Kuhn sum rule, 347 under translations, 125 Thomson cross section, 476, 664 Scattering Time evolution kernel Coulomb potential, 241 for free Dirac fields, 601 hard sphere, 237 for free Klein-Gordon fields, 586 Scattering amplitude, 294 Time evolution operator, 52, 267 in potential scattering, 232 composition property, 269 Scattering cross section, 226 for harmonic oscillator, 270 for two particle collisions, 643 as solution of initial value problem, 269 Scattering matrix, 279, 388 on states in the interaction picture, 273 for δ potential in one dimension, 49 two evolution operators in the interaction in higher order, 298 picture, 274 with vacuum processes divided out, unitarity, 270 388 Time ordering operator, 267 Scattering phase shifts, 152 Transition frequency, 278 and cross sections, 235 Transition probability, 278 for the hard sphere, 248 Transmission coefficient SSchrödinger equation for δ function potential, 50 time-dependent, 18 for Klein’s paradox, 594 time-independent, 39 for a square barrier, 42 Schrödinger picture, 266 Transverse δ function, 437 Schrödinger’s cat, 505 Triplet states, 397 SSchrödinger’s equation, 17 Tunnel effect, 42 Second quantization, 367 Two-particle state, 393 Self-adjoint operator, 30, 79 Two-particle system, 129 Separation of variables, 100 Shift operator, 112 Singlet state, 397 U Sokhotsky-Plemelj relations, 29 Uncertainty relations, 87 Sommerfeld’s fine structure constant, Unitary operator, 79 154 Spherical Coulomb waves, 162 Spherical harmonics, 141 V Spin, 389 Vector, 179 Spinor, 179 Vector addition coefficients, 183 left-handed, 770 Velocity form right-handed, 770 matrix elements, 454 Spin-orbit coupling, 181, 613, 618 oscillator strength, 347 vector model, 365 Velocity gauge, 486 Squeezed states, 121 Virial theorem, 81 Stark effect, 339 Stefan-Boltzmann constant, 10 Stimulated emission, 467 W Stress-energy tensor, see Energy-momentum Wannier functions, 208 tensor, 360 Wannier operators, 534 Summation convention, 64 Wannier states Index 811

completeness relations, 209 Y time-dependent, 210 Yukawa potentials, 565 Wave function collapse, 507 Wave packets, 51 Wave-particle duality, 15 Z Weyl gauge, 479 Zeeman term, 176 Wien’s displacement law, 5 from minimal coupling, 349