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Chapter 10
Relativistic Quantum Mechanics
In this Chapter we will address the issue that the laws of physics must be formulated in a form which is Lorentz–invariant, i.e., the description should not allow one to differentiate between frames of reference which are moving relative to each other with a constant uniform velocity ~v. The transformations beween such frames according to the Theory of Special Relativity are described by Lorentz transformations. In case that ~v is oriented along the x 1 –axis, i.e., ~v = v 1 ˆx 1 , these transformations are
x 1 ′ = x 1 − v 1 t √ 1 −
( (^) v 1 c
) 2 , t
′ (^) = t^ −^
v 1 √^ c^2 x^1 1 −
( (^) v 1 c
) 2 , x
′ 2 =^ x^2 ;^ x ′ 3 =^ x^3 (10.1)
which connect space time coordinates (x 1 , x 2 , x 3 , t) in one frame with space time coordinates (x′ 1 , x′ 2 , x′ 3 , t′) in another frame. Here c denotes the velocity of light. We will introduce below Lorentz-invariant differential equations which take the place of the Schr¨odinger equation of a par- ticle of mass m and charge q in an electromagnetic field [c.f. (refeq:ham2, 8.45)] described by an electrical potential V (~r, t) and a vector potential A~(~r, t)
iℏ
∂t
ψ(~r, t) =
[
2 m
i
q c
A~(~r, t)
]
ψ(~r, t) (10.2)
The replacement of (10.2) by Lorentz–invariant equations will have two surprising and extremely important consequences: some of the equations need to be formulated in a representation for which the wave functions ψ(~r, t) are vectors of dimension larger one, the components representing the spin attribute of particles and also representing together with a particle its anti-particle. We will find that actually several Lorentz–invariant equations which replace (10.2) will result, any of these equations being specific for certain classes of particles, e.g., spin–0 particles, spin– 12 particles, etc. As mentioned, some of the equations describe a particle together with its anti-particle. It is not possible to uncouple the equations to describe only a single type particle without affecting nega- tively the Lorentz invariance of the equations. Furthermore, the equations need to be interpreted as actually describing many–particle-systems: the equivalence of mass and energy in relativistic formulations of physics allows that energy converts into particles such that any particle described will have ‘companions’ which assume at least a virtual existence. Obviously, it will be necessary to begin this Chapter with an investigation of the group of Lorentz transformations and their representation in the space of position ~r and time t. The representation
288 Relativistic Quantum Mechanics
in Sect. 10.1 will be extended in Sect. 10.4 to cover fields, i.e., wave functions ψ(~r, t) and vectors with functions ψ(~r, t) as components. This will provide us with a general set of Lorentz–invariant equations which for various particles take the place of the Schr¨odinger equation. Before introduc- ing these general Lorentz–invariant field equations we will provide in Sects. 10.5, 10.7 a heuristic derivation of the two most widely used and best known Lorentz–invariant field equations, namely the Klein–Gordon (Sect. 10.5) and the Dirac (Sect. 10.7) equation.
10.1 Natural Representation of the Lorentz Group
In this Section we consider the natural representation of the Lorentz group L, i.e. the group of Lorentz transformations (10.1). Rather than starting from (10.1), however, we will provide a more basic definition of the transformations. We will find that this definition will lead us back to the transformation law (10.1), but in a setting of representation theory methods as applied in Secti. 5 to the groups SO(3) and SU(2) of rotation transformations of space coordinates and of spin. The elements L ∈ L act on 4–dimensional vectors of position– and time–coordinates. We will denote these vectors as follows
xμ^ def = (x^0 , x^1 , x^2 , x^3 ) (10.3)
where x^0 = ct describes the time coordinate and (x^1 , x^2 , x^3 ) = ~r describes the space coordinates. Note that the components of xμ^ all have the same dimension, namely that of length. We will, henceforth, assume new units for time such that the velocity of light c becomes c = 1. This choice implies dim(time) = dim(length).
Minkowski Space
Historically, the Lorentz transformations were formulated in a space in which the time component of xμ^ was chosen as a purely imaginary number and the space components real. This space is called the Minkowski space. The reason for this choice is that the transformations (10.1) leave the quantity s^2 = (x^0 )^2 − (x^1 )^2 − (x^2 )^2 − (x^3 )^2 (10.4)
invariant, i.e., for the transformed space-time–cordinates x′μ^ = (x′^0 , x′^1 , x′^2 , x′^3 ) holds
(x^0 )^2 − (x^1 )^2 − (x^2 )^2 − (x^3 )^2 = (x′^0 )^2 − (x′^1 )^2 − (x′^2 )^2 − (x′^3 )^2. (10.5)
One can interprete the quantity
−s^2 as a distance in a 4–dimensional Euclidean space if one chooses the time component purely imaginary. In such a space Lorentz transformations corre- spond to 4-dimensional rotations. Rather than following this avenue we will introduce Lorentz transformations within a setting which does not require real and imaginary coordinates.
The Group of Lorentz Transformations L = O(3,1)
The Lorentz transformations L describe the relationship between space-time coordinates xμ^ of two reference frames which move relative to each other with uniform fixed velocity ~v and which might be reoriented relative to each other by a rotation around a common origin. Denoting by xμ^ the
290 Relativistic Quantum Mechanics
elements of which satisfy this condition, is called O(3,1). This set is identical with the set of all Lorentz transformations L. We want to show now L = O(3,1) ⊂ GL(4, R) is a group. To simplify the following proof of the key group properties we like to adopt the conventional matrix notation for Lμν
L = ( Lμν ) =
L^00 L^01 L^02 L^03
L^10 L^11 L^12 L^13
L^20 L^21 L^22 L^33
L^30 L^31 L^32 L^33
.^ (10.13)
Using the definition (10.10) of g one can rewrite the invariance property (10.12)
LT^ gL = g. (10.14)
From this one can obtain using g^2 = 1 1 (10.15)
(gLT^ g)L = 11 and, hence, the inverse of L
L−^1 = g LT^ g =
L^00 −L^10 −L^20 −L^30
−L^01 L^11 L^21 L^31
−L^02 L^12 L^22 L^32
−L^03 L^13 L^23 L^33
.^ (10.16)
The corresponding expression for (LT^ )−^1 is obviously
(LT^ )−^1 = (L−^1 )T^ = g L g. (10.17)
To demonstrate the group property of O(3,1), i.e., of
O(3, 1) = { L, L ∈ GL(4, R), LT^ gL = g } , (10.18)
we note first that the identity matrix 11 is an element of O(3,1) since it satisfies (10.14). We consider then L 1 , L 2 ∈ O(3,1). For L 3 = L 1 L 2 holds
LT 3 g L 3 = LT 2 LT 1 g L 1 L 2 = LT 2 (LT 1 gL 1 ) L 2 = LT 2 g L 2 = g , (10.19)
i.e., L 3 ∈ O(3,1). One can also show that if L ∈ O(3,1), the associated inverse obeys (10.14), i.e., L−^1 ∈ O(3,1). In fact, employing expressions (10.16, 10.17) one obtains
(L−^1 )T^ g L−^1 = gLgggLT^ g = gLgLT^ g. (10.20)
Multiplying (10.14) from the right by gLT^ and using (10.15) one can derive LT^ gLgLT^ = LT^ and multiplying this from the left by by g(LT^ )−^1 yields
L g LT^ = g (10.21)
Using this result to simplify the r.h.s. of (10.20) results in the desired property
(L−^1 )T^ g L−^1 = g , (10.22)
i.e., property (10.14) holds for the inverse of L. This stipulates that O(3,1) is, in fact, a group.
10.1: Natural Representation of the Lorentz Group 291
Classification of Lorentz Transformations
We like to classify now the elements of L = O(3,1). For this purpose we consider first the value of det L. A statement on this value can be made on account of property (10.14). Using det AB = det A det B and det AT^ = det A yields (det L)^2 = 1 or
det L = ± 1. (10.23)
One can classify Lorentz transformations according to the value of the determinant into two distinct classes. A second class property follows from (10.14) which we employ in the formulation (10.12). Consid- ering in (10.12) the case ρ = 0, σ = 0 yields ( L^00
L^10
L^20
L^30
or since (L^10 )^2 + (L^20 )^2 + (L^30 )^2 ≥ 0 it holds (L^00 )^2 ≥ 1. From this we can conclude
L^00 ≥ 1 or L^00 ≤ − 1 , (10.25)
i.e., there exist two other distinct classes. Properties (10.23) and (10.25) can be stated as follows: The set of all Lorentz transformations L is given as the union
L = L↑ + ∪ L↓ + ∪ L↑− ∪ L↓− (10.26)
where L↑ +, L↓ +, L↑−, L↓− are disjunct sets defined as follows
L↑ + = { L, L ∈ O(3, 1), det L = 1, L^00 ≥ 1 } ; (10.27) L↓ + = { L, L ∈ O(3, 1), det L = 1, L^00 ≤ − 1 } ; (10.28) L↑− = { L, L ∈ O(3, 1), det L = − 1 , L^00 ≥ 1 } ; (10.29) L↓− = { L, L ∈ O(3, 1), det L = − 1 , L^00 ≤ − 1 }. (10.30)
It holds g ∈ L and − 11 ∈ L as one can readily verify testing for property (10.14). One can also verify that one can write
L↑− = gL↑ + = L↑ +g ; (10.31) L↓ + = −L↑ + ; (10.32) L↓− = − gL↑ + = − L↑ +g (10.33)
where we used the definition aM = {M 1 , ∃M 2 , M 2 ∈ M, M 1 = a M 2 }. The above shows that the set of proper Lorentz transformations L↑ + allows one to generate all Lorentz transformations, except for the trivial factors g and − 1 1. It is, hence, entirely suitable to investigate first only Lorentz transformations in L↑ +. We start our investigation by demonstrating that L↑ + forms a group. Obviously, L↑ + contains 11.
We can also demonstrate that for A, B ∈ L↑ + holds C = AB ∈ L↑ +. For this purpose we consider the value of C^00 = A^0 μBμ 0 =
j=1 A
(^0) j Bj (^0) + A (^00) B (^00). Schwartz’s inequality yields
∑^3
j=
A^0 j Bj^0
2 ≤
∑^3
j=
A^0 j
) 2 ∑^3
j=
Bj^0
10.1: Natural Representation of the Lorentz Group 293
Using (10.15) one can conclude T^ = − g g (10.41)
which reads explicitly
^00 ^10 ^20 ^30
^01 ^11 ^21 ^31
^02 ^12 ^22 ^32
^03 ^13 ^23 ^33
−^00 ^01 ^02 ^03
^10 −^11 −^12 −^13
^20 −^21 −^22 −^23
^30 −^31 −^32 −^33
.^ (10.42)
This relationship implies
μμ = 0 ^0 j = j^0 , j = 1, 2 , 3 j k = − kj , j, k = 1, 2 , 3 (10.43)
Inspection shows that the matrix has 6 independent elements and can be written
(ϑ 1 , ϑ 2 , ϑ 3 , w 1 , w 2 , w 3 ) =
0 −w 1 −w 2 −w 3 −w 1 0 −ϑ 3 ϑ 2 −w 2 ϑ 3 0 −ϑ 1 −w 3 −ϑ 2 ϑ 1 0
.^ (10.44)
This result allows us now to define six generators for the Lorentz transformations(k = 1, 2 , 3)
Jk = (ϑk = 1, other five parameters zero) (10.45)
Kk = (wk = 1, other five parameters zero). (10.46)
The generators are explicitly
J 1 =
;^ J^2 =
;^ J^3 =
K 1 =
;^ K^2 =
;^ K^3 =
These commutators obey the following commutation relationships
[ Jk, J] = km Jm (10.49) [ Kk, K] = − km Jm [ Jk, K] = km Km.
The operators also obey ~J · K~ = J 1 J 1 + J 2 J 2 + J 3 J 3 = 0 (10.50)
294 Relativistic Quantum Mechanics
as can be readily verified.
Exercise 7.1: Demonstrate the commutation relationships (10.49, 10.50).
The commutation relationships (10.49) define the Lie algebra associated with the Lie group L↑ +. The commutation relationships imply that the algebra of the generators Jk, Kk, k = 1, 2 , 3 is closed. Following the treatment of the rotation group SO(3) one can express the elements of L↑ + through the exponential operators
L(ϑ, ~~ w) = exp
ϑ~ · ~J + w~ · K~
; ϑ, ~~w ∈ R^3 (10.51)
where we have defined ϑ~ · ~J =
k=1 ϑkJk^ and^ w~^ ·^ K~^ =^
k=1 wkKk.^ One can readily show, following the algebra in Chapter 5, and using the relationship
Jk =
0 Lk
where the 3 × 3–matrices Lk are the generators of SO(3) defined in Chapter 5, that the transfor- mations (10.51) for w~ = 0 correspond to rotations of the spatial coordinates, i.e.,
L(~ϑ, ~w = 0) =
0 R(ϑ~)
Here R(ϑ~) are the 3 × 3–rotation matrices constructed in Chapter 5. For the parameters ϑk of the Lorentz transformations holds obviously
ϑk ∈ [0, 2 π[ , k = 1, 2 , 3 (10.54)
which, however, constitutes an overcomplete parametrization of the rotations (see Chapter 5). We consider now the Lorentz transformations for ~ϑ = 0 which are referred to as ‘boosts’. A boost in the x 1 –direction is L = exp(w 1 K 1 ). To determine the explicit form of this transformation we evaluate the exponential operator by Taylor expansion. In analogy to equation (5.35) it issufficient to consider in the present case the 2 × 2–matrix
L′^ = exp
w 1
∑^ ∞
n=
wn 1 n!
)n (10.55)
since
exp (w 1 K 1 ) = exp
L′^
.^ (10.56)
Using the idempotence property
( 0 − 1 − 1 0
296 Relativistic Quantum Mechanics
10.2 Scalars, 4–Vectors and Tensors
In this Section we define quantities according to their behaviour under Lorentz transformations. Such quantities appear in the description of physical systems and statements about transformation properties are often extremely helpful and usually provide important physical insight. We have encountered examples in connection with rotational transformations, namely, scalars like√ r = x^21 + x^22 + x^23 , vectors like ~r = (x 1 , x 2 , x 3 )T^ , spherical harmonics Ym(ˆr), total angular momentum states of composite systems like Ym(1 , 2 |ˆr 1 , rˆ 2 ) and, finally, tensor operators Tkm. Some of these quantities were actually defined with respect to representations of the rotation group in function spaces, not in the so-called natural representation associated with the 3–dimensional Euclidean space E^3. Presently, we have not yet defined representations of Lorentz transformations beyond the ‘natural’ representation acting in the 4–dimensional space of position– and time–coordinates. Hence, our definition of quantities with special properties under Lorentz transformations presently is confined to the natural representation. Nevertheless, we will encounter an impressive example of physical properties.
Scalars The quantities with the simplest transformation behaviour are so-called scalars f ∈ R which are invariant under transformations, i.e.,
f ′^ = f. (10.65)
An example is s^2 defined in (10.4), another example is the rest mass m of a particle. However, not any physical property f ∈ R is a scalar. Counterexamples are the energy, the charge density, the z–component x 3 of a particle, the square of the electric field | E~(~r, t)|^2 or the scalar product ~r 1 · ~r 2 of two particle positions. We will see below how true scalars under Lorentz transformations can be constructed.
4-Vectors The quantities with the transformation behaviour like that of the position–time vector xμ^ defined in (10.3) are the so-called 4–vectors aμ. These quantites always come as four components (a^0 , a^1 , a^2 , a^3 )T^ and transform according to
a′μ^ = Lμν aν^. (10.66)
Examples of 4-vectors beside xμ^ are the momentum 4-vector
pμ^ = (E, ~p) , E =
m √ 1 − ~v 2
, ~p =
m ~v √ 1 − ~v 2
the transformation behaviour of which we will demonstrate further below. A third 4-vector is the so-called current vector Jμ^ = (ρ, J~) (10.68)
where ρ(~r, t) and J~(~r, t) are the charge density and the current density, respectively, of a system of charges. Another example is the potential 4-vector
Aμ^ = (V, A~) (10.69)
where V (~r, t) and A~(~r, t) are the electrical and the vector potential of an electromagnetic field. The 4-vector character of Jμ^ and of Aμ^ will be demonstrated further below.
10.2: Scalars, 4–Vectors and Tensors 297
Scalar Product 4-vectors allow one to construct scalar quantities. If aμ^ and bμ^ are 4-vectors then aμgμν bν^ (10.70)
is a scalar. This property follows from (10.66) together with (10.12)
a′μgμν b′ν^ = Lμρgμν Lν σaρbσ^ = aρgρσbσ^ (10.71)
Contravariant and Covariant 4-Vectors It is convenient to define a second class of 4-vectors. The respective vectors aμ are associated with the 4-vectors aμ, the relationship being
aμ = gμν aν^ = (a^0 , −a^1 , −a^2 , −a^3 ) (10.72)
where aν^ is a vector with transformation behaviour as stated in (10.66). One calls 4-vectors aμ covariant and 4-vectors aμ^ contravariant. Covariant 4-vectors transform like
a′μ = gμν Lν ρgρσaσ (10.73)
where we defined gμν^ = gμν. (10.74)
We like to point out that from definition (10.72) of the covariant 4-vector follows aμ^ = gμν^ aν. In fact, one can employ the tensors gμν^ and gμν to raise and lower indices of Lμν as well. We do not establish here the consistency of the ensuing notation. In any case one can express (10.73)
a′ μ = Lμσaσ. (10.75)
Note that according to (10.17) Lμσ^ is the transformation inverse to Lσμ. In fact, one can express [(L−^1 )T^ ]μν = (L−^1 )ν μ and, accordingly, (10.17) can be written
(L−^1 )ν μ = Lμν^. (10.76)
The 4-Vector ∂μ An important example of a covariant 4-vector is the differential operator
∂μ =
∂xμ^
∂t
The transformed differential operator will be denoted by
∂′ μ^ def =
∂x′μ^
To prove the 4-vector property of ∂μ we will show that gμν^ ∂ν transforms like a contravariant 4- vector, i.e., gμν^ ∂′ ν = Lμρgρσ∂σ. We start from x′μ^ = Lμν xν^. Multiplication (and summation) of x′μ^ = Lμν xν^ by Lρσgρμ yields, using (10.12), gσν xν^ = Lρσgρμx′μ^ and gμσgσν = δμν ,
xν^ = gνσLρσgρμx′μ^. (10.79)
This is the inverse Lorentz transformation consistent with (10.16). We have duplicated the expres- sion for the inverse of Lμν to obtain the correct notation in terms of covariant, i.e., lower, and
10.3: Relativistic Electrodynamics 299
The operator m (^) dτd transforms like a scalar. Since xμ^ transforms like a contravariant 4-vector, the r.h.s. of (10.89) alltogether transforms like a contravariant 4-vector, and, hence, pμ^ on the l.h.s. of (10.89) must be a 4-vector. The momentum 4-vector allows us to construct a scalar quantity, namely
pμpμ = pμgμν pν^ = E^2 − ~p 2 (10.90)
Evaluation of the r.h.s. yields according to (10.67)
E^2 − ~p 2 = m^2 1 − ~v 2
m^2 ~v 2 1 − ~v 2
= m^2 (10.91)
or pμpμ = m^2 (10.92)
which, in fact, is a scalar. We like to rewrite the last result
E^2 = ~p 2 + m^2 (10.93)
or E = ±
~p 2 + m^2. (10.94)
In the non-relativistic limit the rest energy m is the dominant contribution to E. Expansion in (^) m^1 should then be rapidly convergent. One obtains
E = ±m ±
~p 2 2 m
(~p 2 )^2 4 m^3
+ O
(~p 2 )^3 4 m^5
This obviously describes the energy of a free particle with rest energy ±m, kinetic energy ± ~p^
2 2 m and relativistic corrections.
10.3 Relativistic Electrodynamics
In the following we summarize the Lorentz-invariant formulation of electrodynamics and demon- strate its connection to the conventional formulation as provided in Sect. 8.
Lorentz Gauge In our previous description of the electrodynamic field we had introduced the scalar and vector potential V (~r, t) and A~(~r, t), respectively, and had chosen the so-called Coulomb gauge (8.12), i.e., ∇ · A~ = 0, for these potentials. This gauge is not Lorentz-invariant and we will adopt here another gauge, namely,
∂tV (~r, t) + ∇ · A~(~r, t) = 0. (10.96)
The Lorentz-invariance of this gauge, the so-called Lorentz gauge, can be demonstrated readily using the 4-vector notation (10.69) for the electrodynamic potential and the 4-vector derivative (10.77) which allow one to express (10.96) in the form
∂μAμ^ = 0. (10.97)
We have proven already that ∂μ is a contravariant 4-vector. If we can show that Aμ^ defined in (10.69) is, in fact, a contravariant 4-vector then the l.h.s. in (10.97) and, equivalently, in (10.96) is a scalar and, hence, Lorentz-invariant. We will demonstrate now the 4-vector property of Aμ.
300 Relativistic Quantum Mechanics
Transformation Properties of Jμ^ and Aμ
The charge density ρ(~r, t) and current density J~(~r, t) are known to obey the continuity property
∂tρ(~r, t) + ∇ · J~(~r, t) = 0 (10.98)
which reflects the principle of charge conservation. This principle should hold in any frame of reference. Equation (10.98) can be written, using (10.77) and (10.68),
∂μJμ(xμ) = 0. (10.99)
Since this equation must be true in any frame of reference the right hand side must vanish in all frames, i.e., must be a scalar. Consequently, also the l.h.s. of (10.99) must be a scalar. Since ∂μ transforms like a covariant 4-vector, it follows that Jμ, in fact, has to transform like a contravariant 4-vector. We want to derive now the differential equations which determine the 4-potential Aμ^ in the Lorentz gauge (10.97) and, thereby, prove that Aμ^ is, in fact, a 4-vector. The respective equation for A^0 = V can be obtained from Eq. (8.13). Using ∇ · ∂t A~(~r, t) = ∂t∇ · A~(~r, t) together with (10.96), i.e., ∇ · A~(~r, t) = −∂tV (~r, t), one obtains
∂ t^2 V (~r, t) − ∇^2 V (~r, t) = 4πρ(~r, t). (10.100)
Similarly, one obtains for A~(~r, t) from (8.17) using the identity (8.18) and, according to (10.96), ∇ · A~(~r, t) = −∂tV (~r, t) ∂ t^2 A~(~r, t) − ∇^2 A~(~r, t) = 4 π J~(~r, t). (10.101)
Combining equations (10.100, 10.101), using (10.83) and (10.69), yields
∂μ∂μ^ Aν^ (xσ) = 4 π Jν^ (xσ). (10.102)
In this equation the r.h.s. transforms like a 4-vector. The l.h.s. must transform likewise. Since ∂μ∂μ^ transforms like a scalar one can conclude that Aν^ (xσ) must transform like a 4-vector.
The Field Tensor
The electric and magnetic fields can be collected into an anti-symmetric 4×4 tensor
F μν^ =
0 −Ex −Ey −Ez Ex 0 −Bz By Ey Bz 0 −Bx Ez −By Bx 0
.^ (10.103)
Alternatively, this can be stated
F k^0 = −F 0 k^ = Ek^ , F mn^ = −mnB^ , k, `, m, n = 1, 2 , 3 (10.104)
where mn^ = mn is the totally anti-symmetric three-dimensional tensor defined in (5.32). One can readily verify, using (8.6) and (8.9), that F μν^ can be expressed through the potential Aμ in (10.69) and ∂μ^ in (10.82) as follows
F μν^ = ∂ μAν^ − ∂ ν^ Aμ^. (10.105)
302 Relativistic Quantum Mechanics
Lorentz Force
One important property of the electromagnetic field is the Lorentz force acting on charged particles moving through the field. We want to express this force through the tensor F μν^. It holds for a particle with 4-momentum pμ^ as given by (10.67) and charge q
dpμ dτ
q m pν F μν^ (10.114)
where d/dτ is given by (10.86). We want to demonstrate now that this equation is equivalent to the equation of motion (8.5) where ~p = m~v/
1 − v^2. To avoid confusion we will employ in the following for the energy of the particle the notation E = m/
1 − v^2 [see (10.87)] and retain the definition E~ for the electric field. The μ = 0 component of (10.114) reads then, using (10.104),
dE dτ
q m ~p · E~ (10.115)
or with (10.86) dE dt
q E
~p · E .~ (10.116)
From this one can conclude, employing (10.93),
1 2
dE^2 dt
d~p 2 dt
= q ~p · E~ (10.117)
This equation follows, however, also from the equation of motion (8.5) taking the scalar product with ~p
~p · d~p dt
= q~p · E~ (10.118)
where we exploited the fact that according to ~p = m~v/
1 − v^2 holds ~p ‖ ~v. For the spatial components, e.g., for μ = 1, (10.114) reads using (10.103)
dpx dτ
q m
( EEx + pyBz − pz By ). (10.119)
Employing again (10.86) and (10.67), i.e., E = m/
1 − v^2 , yields
dpx dt = q
[
Ex + (~v × B~)x
]
which is the x-component of the equation of motion (8.5). We have, hence, demonstrated that (10.114) is, in fact, equivalent to (8.5). The term on the r.h.s. of (10.120) is referred to as the Lorentz force. Equation (10.114), hence, provides an alternative description of the action of the Lorentz force.
10.4: Function Space Representation of Lorentz Group 303
10.4 Function Space Representation of Lorentz Group
In the following it will be required to decribe the transformation of wave functions under Lorentz transformations. In this section we will investigate the transformation properties of scalar functions ψ(xμ), ψ ∈ C∞(4). For such functions holds in the transformed frame
ψ′(Lμν xν^ ) = ψ(xμ) (10.121)
which states that the function values ψ′(x′μ) at each point x′μ^ in the new frame are identical to the function values ψ(xμ) in the old frame taken at the same space–time point xμ, i.e., taken at the pairs of points (x′μ^ = Lμν xν^ , xμ). We need to emphasize that (10.121) covers solely the transformation behaviour of scalar functions. Functions which represent 4-vectorsor other non-scalar entities, e.g., the charge-current density in case of Sect. 10.3 or the bi-spinor wave function of electron-positron pairs in Sect. 10.7, obey a different transformation law. We like to express now ψ′(x′μ) in terms of the old coordinates xμ. For this purpose one replaces xμ^ in (10.121) by (L−^1 )μν xν^ and obtains
ψ′(xμ) = ψ((L−^1 )μν xν^ ). (10.122)
This result gives rise to the definition of the function space representation ρ(Lμν ) of the Lorentz group
(ρ(Lμν )ψ)(xμ) def = ψ((L−^1 )μν xν^ ). (10.123)
This definition corresponds closely to the function space representation (5.42) of SO(3). In analogy to the situation for SO(3) we seek an expression for ρ(Lμν ) in terms of an exponential operator and transformation parameters ϑ, ~~ w, i.e., we seek an expression which corresponds to (10.51) for the natural representation of the Lorentz group. The resulting expression should be a generalization of the function space representation (5.48) of SO(3), in as far as SO(3,1) is a generalization (rotation
- boosts) of the group SO(3). We will denote the intended representation by
L(~ϑ, ~w) def = ρ(Lμν (ϑ, ~~ w)) = ρ
e ~ϑ· J~ + w~· K~ ) (10.124)
which we present in the form
L(~ϑ, ~w) = exp
ϑ~ · J~ + w~ · K~
In this expression J~ = (J 1 , J 2 , J 3 ) and K~ = (K 1 , K 2 , K 3 ) are the generators of L(ϑ, ~~ w) which correspond to the generators Jk and Kk in (10.47), and which can be constructed following the procedure adopted for the function space representation of SO(3). However, in the present case we exclude the factor ‘−i’ [cf. (5.48) and (10.125)]. Accordingly, one can evaluate Jk as follows
Jk = lim ϑk → 0
ϑ 1
[
ρ
eϑk^ Jk
]
and Kk
Kk = lim wk → 0
w 1
[
ρ
ewk^ Kk^
]
10.4: Function Space Representation of Lorentz Group 305
[ K 1 , K 2 ] = [x^0 ∂ 1 + x^1 ∂ 0 , x^0 ∂ 2 + x^2 ∂ 0 ] = [x^0 ∂ 1 , x^2 ∂ 0 ] − [x^1 ∂ 0 , x^0 ∂ 2 ] = −x^2 ∂ 1 + x^1 ∂ 2 = − J 3 , (10.135)
[ J 1 , K 2 ] = [x^3 ∂ 2 − x^2 ∂ 3 , x^0 ∂ 2 + x^2 ∂ 0 ] = [x^3 ∂ 2 , x^2 ∂ 0 ] − [x^2 ∂ 3 , x^0 ∂ 2 ] = x^3 ∂ 0 + x^0 ∂ 3 = K 3. (10.136)
One-Dimensional Function Space Representation
The exponential operator (10.125) in the case of a one-dimensional transformation of the type
L(w^3 ) = exp
w^3 K 3
where K 3 is given in (10.128), can be simplified considerably. For this purpose one expresses K 3 in terms of hyperbolic coordinates R, Ω which are connected with x^0 , x^3 as follows
x^0 = R coshΩ , x^3 = R sinhΩ (10.138)
a relationship which can also be stated
R =
(x^0 )^2 − (x^3 )^2 if x^0 ≥ 0 −
(x^0 )^2 − (x^3 )^2 if x^0 < 0
and
tanhΩ = x^3 x^0
, cothΩ = x^0 x^3
The transformation to hyperbolic coordinates closely resembles the transformation to radial coordi- nates for the generators of SO(3) in the function space representation [cf. Eqs. (5.85-5.87)]. In both cases the radial coordinate is the quantity conserved under the transformations, i.e.,
x^21 + x^22 + x^23 in the case of SO(3) and
(x^0 )^2 − (x^3 )^2 in case of transformation (10.137). In the following we consider solely the case x^0 ≥ 0. The relationships (10.139, 10.140) allow one to express the derivatives ∂ 0 , ∂ 3 in terms of (^) ∂R∂ , (^) ∂∂Ω. We note
∂R ∂x^0
x^0 R
∂R
∂x^3
x^0 R
and
∂Ω ∂x^3
∂tanhΩ
∂tanhΩ ∂x^3
= cosh^2 Ω
x^0 ∂Ω ∂x^0
∂cothΩ
∂cothΩ ∂x^0 = − sinh^2 Ω
x^3
The chain rule yields then
∂ 0 =
∂R
∂x^0
∂R
∂x^0
x^0 R
∂R
− sinh^2 Ω
x^3
∂R
∂x^3
∂R
∂x^3
x^3 R
∂R
x^0
306 Relativistic Quantum Mechanics
Inserting these results into the definition of K 3 in (10.128) yields
K 3 = x^0 ∂ 3 + x^3 ∂ 0 =
The action of the exponential operator (10.137) on a function f (Ω) ∈ C∞(1) is then that of a shift operator
L(w^3 ) f (Ω) = exp
w^3
f (Ω) = f (Ω + w^3 ). (10.145)
10.5 Klein–Gordon Equation
In the following Sections we will provide a heuristic derivation of the two most widely used quan- tum mechanical descriptions in the relativistic regime, namely the Klein–Gordon and the Dirac equations. We will provide a ‘derivation’ of these two equations which stem from the historical de- velopment of relativistic quantum mechanics. The historic route to these two equations, however, is not very insightful, but certainly is short and, therefore, extremely useful. Further below we will provide a more systematic, representation theoretic treatment.
Free Particle Case
A quantum mechanical description of a relativistic free particle results from applying the correspon- dence principle, which allows one to replace classical observables by quantum mechanical operators acting on wave functions. In the position representation the correspondence principle states
E =⇒ Eˆ = −
i
∂t
p ~ =⇒ ˆ~p =
i
which, in 4-vector notation reads
pμ =⇒ pˆμ = iℏ(∂t, ∇) = iℏ∂μ ; pμ^ =⇒ pˆμ^ = i(∂t, −∇) = iℏ∂μ^. (10.147)
Applying the correspondence principle to (10.92) one obtains the wave equation
− ℏ^2 ∂μ∂μ ψ(xν^ ) = m^2 ψ(xν^ ) (10.148)
or (^) ( ℏ^2 ∂μ∂μ + m^2
ψ(xν^ ) = 0. (10.149)
where ψ(xμ) is a scalar, complex-valued function. The latter property implies that upon change of reference frame ψ(xμ) transforms according to (10.121, 10.122). The partial differential equation (10.151) is called the Klein-Gordon equation. In the following we will employ so-called natural units ℏ = c = 1. In these units the quantities energy, momentum, mass, (length)−^1 , and (time)−^1 all have the same dimension. In natural units the Klein–Gordon equation (10.151) reads
( ∂μ∂μ^ + m^2
ψ(xμ) = 0 (10.150)
