introduction to relativistic quantum mechanics -...

Cambridge University Press978-0-521-76726-2 - Relativistic Quantum Physics: From Advanced Quantum Mechanics to IntroductoryQuantum Field TheoryTommy OhlssonExcerptMore information

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1

Introduction to relativistic quantum mechanics

1.1 Tensor notation

In this book, we will most often use so-called natural units, which means that wehave set c = 1 and � = 1. Furthermore, a general 4-vector will be written in termsof its contravariant index, i.e.

A = (Aμ) = (A0,A), (1.1)

where A0 is the time component and the 3-vector A contains the three spatial com-ponents such that A = (Ai ) = (A1, A2, A3).1 Thus, the contravariant componentsA1, A2, and A3 are the physical components, i.e. Ax , Ay , and Az , respectively,whereas the covariant components A1, A2, and A3 will be related to the contravari-ant components.2 Specifically, the 4-position vector (or spacetime point) is given by

x = (xμ) = (x0, x) = (x0, x1, x2, x3) = (ct, x, y, z) = {c = 1} = (t, x, y, z).(1.2)

Note the ‘abuse of notation’, which means that we will use the symbol x for bothrepresenting the 4-position vector as well as its first spatial component. In addition,we introduce the metric tensor as

g = (gμν) = diag(1,−1,−1,−1), (1.3)

which is called the Minkowski metric. In this case, the inverse of the metric tensoris trivially given by

g−1 = (gμν) = diag(1,−1,−1,−1). (1.4)

1 Note that we will use the convention that Greek indices take the values 0, 1, 2, or 3, whereas Latin (orRoman) indices take the values 1, 2, or 3.

2 In order for a vector to be invariant under transformations of coordinate systems, the components of thevector have to contra-vary (i.e. vary in the ‘opposite’ direction) with a change of basis to compensate for thatchange. Therefore, vectors (e.g. position, velocity, and acceleration) are contravariant, whereas so-calleddual vectors (e.g. gradients) are covariant.

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2 Introduction to relativistic quantum mechanics

Thus, for the Minkowski metric, we have that gμν = gμν , i.e. the covariant andcontravariant components are equal to each other, which does not hold for a generalmetric. Owing to the choice of the Minkowski metric, it also holds for the 4-vectorin Eq. (1.1) that A0 = A0, A1 = −A1 = Ax , A2 = −A2 = Ay , and A3 = −A3 =Az . In fact, in order to raise and lower indices of vectors and tensors, we can usethe Minkowski metric tensor and its inverse in the following way:

Aμ = gμν Aν, Aμ= gμν Aν, (1.5)

T νμ = gμλT

λν, Tμν = gμλgνωT λω, T μν = gμλTλν, T μν = gμλgνωTλω. (1.6)

Normally, in tensor notation à la Einstein,3 upper indices (or superscripts) of vec-tors and tensors are called contravariant indices, whereas lower indices (or sub-scripts) are called covariant indices. In addition, note that in Eqs. (1.5) and (1.6),we have used the so-called Einstein summation convention, which means that whenan index appears twice in a single term, once as an upper index and once as a lowerindex, it implies that all its possible values are to be summed over.

Using the Minkowski metric, we can also introduce the inner product betweentwo 4-vectors A and B such that

g(A, B) = AT gB = A · B = AμgμνBν = AμBμ = A0 B0 + Ai Bi = A0 B0 −A ·B,(1.7)

which is not positive definite.4 Therefore, the ‘length’ of a 4-vector A is given by

A2 = A · A = (A0)2 − A2, (1.8)

where we have again used an abuse of notation, since the symbol A2 denotes boththe second spatial contravariant component of the vector A and the ‘length’ ofthe vector A. Nevertheless, note that the ‘length’ is indefinite, i.e. it can be eitherpositive or negative. One says that A is time-like if A2 > 0, light-like if A2 = 0,and space-like if A2 < 0. In particular, it holds for a 4-position vector x that

x2 = (x0)2 − x2 = t2 − x2 − y2 − z2. (1.9)

Next, we introduce the Minkowski spacetime M such that M = (R4, g), whichis the set of all 4-position vectors [cf. Eq. (1.2)]. Note that the metric tensor g is abilinear form g : M × M → R such that g(x, y) = gμνxμyν , where gμν are strictlythe components of the metric g, which are usually identified with the tensor itself.

3 In 1921, A. Einstein was awarded the Nobel Prize in physics ‘for his services to Theoretical Physics, andespecially for his discovery of the law of the photoelectric effect’.

4 In general, note that the superscript T denotes the transpose of a matrix.






1.2 The Lorentz group 3

Finally, we introduce the totally antisymmetric Levi-Civita (pseudo)tensor inthree spatial dimensions

εi jk = εi jk =

⎧⎪⎨⎪⎩

1 if i jk are even permutations of 1,2,3

−1 if i jk are odd permutations of 1,2,3

0 if any two indices are equal

(1.10)

as well as in four spacetime dimensions

εμνλω =

⎧⎪⎨⎪⎩

1 if μνλω are even permutations of 0,1,2,3

−1 if μνλω are odd permutations of 0,1,2,3

0 if any two indices are equal

, (1.11)

which we define such that ε0123 = 1, which implies that ε0123 = −1. In addition,in three dimensions, the following contractions hold for the Levi-Civita tensor:

εi jkεi jk = 6, (1.12)

εi jkεk�m = δi�δ jm − δimδ j�, (1.13)

εi jkε�mn = δi�(δ jmδkn − δ jnδkm

)− δim(δ j�δkn − δ jnδk�

)+ δin

(δ j�δkm − δ jmδk�

), (1.14)

where δi j = δi j is the Kronecker delta such that δi j = 1 if i = j and δi j = 0 ifi �= j , whereas, in four dimensions, the corresponding relations are:

εαβγ δεαβγ δ = −24, (1.15)

εαβγμεαβγ ν = −6δμν , (1.16)

εαβμνεαβλω = −2(δμλ δνω − δμωδνλ

), (1.17)

εαμνλεαωρσ = −δμωδνρδλσ + δμρ δνωδλσ + δμωδνσ δλρ − δμρ δνσ δλω− δμσ δνωδλρ + δμσ δνρδλω, (1.18)

where δμν ≡ gμν = g μν such that δμν = 1 if μ = ν and δμν = 0 if μ �= ν.

1.2 The Lorentz group

A Lorentz transformation � is a linear mapping of M onto itself, � : M → M ,x �→ x ′ = �x ,5 which preserves the inner product, i.e. the inner product is invari-ant under Lorentz transformations:

x ′ · y′ = x · y, where x ′ = �x and y′ = �y. (1.19)

5 We will denote the set of Lorentz transformations by L, which is called the Lorentz group. See Appendix Afor the definition of a group as well as a short general discussion on group theory.







In component form, we have x ′μ = (�x)μ = �μνxν and y′μ = (�y)μ = � λ

μ yλ =�μλyλ, which means that

�μν�μλ = gνλ ⇔ (�T ) μν gμω�ωλ = gνλ. (1.20)

Thus, the Lorentz group is given by

L = {� : M → M : x ′ · y′ = x · y, where x ′ = �x , y′ = �y, and x, y ∈ M},(1.21)

i.e. it consists of real, linear transformations that leave the inner product invariant,and hence, one says that the Lorentz group is an invariance group.

An explicit example of a Lorentz transformation relating two inertial frames (orinertial coordinate systems)6 S and S′ (with 4-position vectors x and x ′, respec-tively) that move along the positive x1-axis is given by

x ′ =

⎛⎜⎜⎝

x ′0

x ′1

x ′2

x ′3

⎞⎟⎟⎠ =

⎛⎜⎜⎝

cosh ξ −sinh ξ 0 0−sinh ξ cosh ξ 0 0

0 0 1 00 0 0 1

⎞⎟⎟⎠⎛⎜⎜⎝

x0

x1

x2

x3

⎞⎟⎟⎠ = �(01)x, (1.22)

where ξ is any real number and �(01) = �(01)(ξ) denotes the Lorentz transforma-tion. Note that �(01) is often called the standard configuration Lorentz transfor-mation and constitutes an example of a boost (or standard transformation). Theparameter ξ is called the rapidity (or boost parameter). Using the hyperbolic iden-tity cosh2 ξ−sinh2 ξ = 1, one easily observes that indeed x ′2 = x2. By direct com-putation, one finds that�(01)(ξ+ξ ′) = �(01)(ξ)�(01)(ξ ′). Similar to Eq. (1.22), onecould define the Lorentz transformations �(02) and �(03). Another way of writingthe Lorentz transformation �(01) is

�(01) =

⎛⎜⎜⎝γ −βγ 0 0

−βγ γ 0 00 0 1 00 0 0 1

⎞⎟⎟⎠ , (1.23)

where the two parameters β and γ are introduced as

β ≡ v

c= {c = 1} = v, (1.24)

γ ≡ 1√1 − β2

= 1√1 − v2

= γ (v). (1.25)

6 An inertial frame is a reference system in which free particles (i.e. particles that are not influenced by anyforces) are moving with uniform velocity. Any reference system moving with constant velocity relative to aninertial frame is another inertial frame. Note that there are infinitely many inertial frames and that the laws ofphysics have to have the same form in all inertial frames, i.e. the laws of physics are so-called Lorentzcovariant.







Here v is the relative velocity of the two inertial frames. Note that, comparingEqs. (1.22) and (1.23), it holds that

cosh ξ = γ and sinh ξ = βγ = vγ. (1.26)

Thus, it follows that the rapidity is given by

ξ = artanh v ⇔ tanh ξ = v. (1.27)

The physical interpretation is as follows. A particle (or an observer K ) at rest inthe inertial frame S is represented by a world-line parallel to the time axis, i.e. thex0-axis. Without loss of generality, let us assume that K is at rest at the origin ofthe three-dimensional space in S. The same K can also be viewed from anotherinertial frame S′, which is related to S by the Lorentz transformation �(01). In theS′-coordinates, the world-line of K is given by

x ′0 = τ cosh ξ, x ′1 = −τ sinh ξ, x ′2 = x ′3 = 0,

where we have used x0 = τ and x1 = x2 = x3 = 0 for the S-coordinates. Thevelocity of K along the x ′1-axis is now

v′ = dx ′1

dt ′ = − tanh ξ ≤ 0,

whereas the velocity along the other axes is zero. Thus, we can interpret this resultas either (i) K (or the particle) is moving with velocity −v′ along the negative x ′1-axis or (ii) S′ is moving with velocity v = dx1/dt = tanh ξ ≥ 0 along the positivex1-axis (see Fig. 1.1). Note that v′ = −v.

Now, using the inner product g(�x,�y) = �x ·�y = x · y = g(x, y), wherex, y ∈ M and � ∈ L, we find that

(1) x · y = xT gy and

(2) �x ·�y = (�x)T g(�y) = xT�T g�y.

If conditions (1) and (2) are equivalent, which they are, since the inner product isinvariant under Lorentz transformations, they imply that

g = �T g�, (1.28)

which is basically the same equation as Eq. (1.20), but in matrix form. From thisequation, one obtains

• (det�)2 = 1 ⇒ det� = ±1 and

• (�0

0

)2 = 1 +∑3i=1

(�i

0

)2 ≥ 1 ⇒ �00 ≥ 1 or �0

0 ≤ −1.







� x1

�

×K

�−v′

S′S

� x ′1

�

� x1

�

×K

S

� x ′1

�

�v

S′

Figure 1.1 The two upper inertial frames show the first interpretation, i.e. S ismoving to the left relative to S′, whereas the two lower inertial frames show thesecond interpretation, i.e. S′ is moving to the right relative to S.

The four conditions det� = ±1,�00 ≥ 1, and�0

0 ≤ 1 can be used to classify theelements of the Lorentz group. Thus, we can divide the Lorentz group L [denotedSO(1, 3) in group theory] into the following subgroups (see Fig. 1.2):

L+ = {� ∈ L : det� = 1} pure (or proper) Lorentz group, (1.29)

L↑ = {� ∈ L : �0

0 ≥ 1}

orthochronous Lorentz group, (1.30)

L↑+ = L+ ∩ L↑ pure and orthochronous Lorentz group. (1.31)

Note that the three subsets

L↑− = {

� ∈ L : det� = −1,�00 ≥ 1

}, (1.32)

L↓− = {

� ∈ L : det� = −1,�00 ≤ 1

}, (1.33)

L↓+ = {

� ∈ L : det� = 1,�00 ≤ 1

}(1.34)

are not subgroups of L. However, L0 = L↑+ ∪ L↓

− is a subgroup of L. Hence, L+,L↑, L↑

+, and L0 are the invariant subgroups of L. The other subsets of L, which arenot subgroups, can be connected to L↑

+ by the three corresponding and followingLorentz transformations.

(1) Parity (or space inversion).

�P = diag(1,−1,−1,−1) ∈ L↑− (1.35)







� �00

�

det�

−1

−1

1

1

L↓− L↑

−

L↓+ L↑

+

Figure 1.2 Subgroups of the Lorentz group L. The intersection of the two ellipsescorresponds to the pure and orthochronous Lorentz group L↑

+, whereas the hori-zontal and vertical ellipses correspond to the subgroups L+ and L↑, respectively.

(2) Time reversal (or time inversion).

�T = diag(−1, 1, 1, 1) ∈ L↓− (1.36)

(3) Spacetime inversion.

�PT = �P�T = diag(−1,−1,−1,−1) ∈ L↓+ (1.37)

Thus, it holds that L↑− = �PL↑

+, L↓− = �TL↑

+, and L↓+ = �PTL↑

+, which areso-called cosets of L with respect to L↑

+. The four subsets (one subgroup and threecosets) L↑

+, L↑−, L↓

−, and L↓+ are disjoint and not continuously connected (see again

Fig. 1.2). Finally, we obtain

L = L↑+ ∪ L↑

− ∪ L↓− ∪ L↓

+ = L↑+ ∪�PL↑

+ ∪�TL↑+ ∪�PTL↑

+. (1.38)

The pure and orthochronous (or restricted) Lorentz group L↑+ [denoted SO+(1, 3)

in group theory] is a matrix Lie group (see Appendix A.2), which means thatevery matrix can be written in the form exp

(− i2ωμνMμν

), where ωμν ∈ R such

that ωμν = −ωνμ and Mμν are the so-called generators of L↑+. The number

of generators of L↑+ is six, since we define Mμν = −Mνμ, which implies that

M00 = M11 = M22 = M33 = 0. Thus, L↑+ is a six-parameter group. For example,

the infinitesimal generator for the ‘rotation’ in the x0x1-plane corresponding to theLorentz transformation �(01) is given by

M01 = id�(01)(ξ)

dξ

∣∣∣∣ξ=0

=

⎛⎜⎜⎝

0 −i 0 0−i 0 0 00 0 0 00 0 0 0

⎞⎟⎟⎠ . (1.39)







The other five infinitesimal generators M02, M03, M32, M13, and M21 can bederived in a similar way. Next, we introduce the infinitesimal generators J i (i =1, 2, 3) and K i (i = 1, 2, 3) for L↑

+, corresponding to rotations and boosts (or stan-dard transformations), respectively. Note that J i and K i are constructed in termsof Mμν , or vice versa, which we will investigate more in Sections 1.3 and 1.4.Nevertheless, it holds that

J i = −1

2εi jk M jk and K i = M0i . (1.40)

Then, one finds by simple computations that[J i , J j

] = iεi jk J k, (1.41)[J i , K j

] = iεi jk K k, (1.42)[K i , K j

] = −iεi jk J k, (1.43)

which form a Lie algebra (see Appendix A.3). Actually, defining the operators

j = 1

2(J + iK) and k = 1

2(J − iK) , (1.44)

i.e. linear combinations of the operators J and K, one obtains the following com-mutation relations [

j i , j j] = iεi jk j k, (1.45)[

j i , k j] = 0, (1.46)[

ki , k j] = iεi jkkk, (1.47)

which give an alternative basis for the Lie algebra. From the commutation relationsin Eqs. (1.45)–(1.47), we observe that the operators j and k are decoupled. This isdescribed by su(2) ⊕ su(2) called the Lorentz algebra,7 which is the Lie alge-bra of the Lie group SU(2) ⊗ SU(2) that can be represented as D j ⊗ D j ′ , whereD j

(j = 0, 1

2 , 1, . . .)

is an irreducible representation of SU(2). Note that D j isspanned by basis vectors | j,m〉, where m = − j,− j +1, . . . , j . Thus, we have thebasis vectors | j,m; j ′,m ′〉 = | j,m〉| j ′,m ′〉. In addition, we find the relations

J2 − K2 = 2(j2 + k2

), (1.48)

J · K = −i(j2 − k2

), (1.49)

which are invariant forms (i.e. they commute with the generators j and k) that aremultiplets of the unit operator 1 with the eigenvalues 2[ j ( j + 1)+ j ′( j ′ + 1)] and−i[ j ( j +1)− j ′( j ′ +1)], respectively. In fact, the invariant forms can be written as

7 Note that su(2) � so(3), but the Lie group generated by su(2) � so(3) is SU(2) and not SO(3).






1.3 The Poincaré group 9

J2 − K2 = 1

2MμνMμν, (1.50)

J · K = 1

8εμνρσMμνMρσ . (1.51)

The irreducible representations are denoted by D( j, j ′) ≡ D j ⊗ D j ′ , where j, j ′ =0, 1

2 , 1, . . . For example, an explicit representation for the generators of D(

12 ,0)

and

D(

0,12

)is given by

j = 1

2σ , k = 0 and j = 0, k = 1

2σ , (1.52)

respectively, where σ is the vector of Pauli matrices, i.e.

σ = (σ 1, σ 2, σ 3) =((

0 11 0

),

(0 −ii 0

),

(1 00 −1

)), (1.53)

which satisfy the commutation relation [σ i , σ j ] = 2iεi jkσ k . Thus, this leads to

D

(12 ,0

): J = 1

2σ , K = −i

1

2σ , (1.54)

D

(0,

12

): J = 1

2σ , K = i

1

2σ . (1.55)

1.3 The Poincaré group

The Poincaré group (or the inhomogeneous Lorentz group), which is a (linear) Liegroup, is given by

P = {(�, a) : xμ �→ x ′μ = �μνxν + aμ

}, (1.56)

where � ∈ L is a Lorentz transformation and a ∈ R4 is a translation. Thus, theLorentz group and the translation group are subgroups of the Poincaré group. Inaddition, note that one has P↑

+ if� ∈ L↑+, i.e. the pure and orthochronous Poincaré

group. The group multiplication law of the Poincaré group is

(�2, a2)(�1, a1) = (�2�1,�2a1 + a2), (1.57)

where (�1, a1) and (�2, a2) are two elements of the Poincaré group, i.e. �1,�2 ∈L and a1, a2 ∈ R4. In addition, the identity element is (14, 0) and the inverseis given by (�, a)−1 = (�−1,−�−1a), where (�, a) ∈ P . An element of thePoincaré group (�, a) can be represented by a 5 × 5 matrix in the following way:

(�, a) �→(� a0 1

), (1.58)







which means that the group multiplication law (1.57) simply corresponds to ordi-nary matrix multiplication of 5 × 5 matrices. The generators of the Poincaré groupare Mμν and Pμ, which give rise to the unitary operators that represent the elements(�, 0) and (14, a) of the Poincaré group, i.e.

U (�, 0) = exp(− i

2ωμνMμν), (1.59)

U (14, a) = exp(iaμPμ), (1.60)

where again ωμν and Mμν are the parameters and generators of the Lorentz sub-group, respectively, and aμ and Pμ are the parameters and generators of the trans-lation subgroup, respectively. Note that if the elements of the Poincaré group areclose to the identity or to first order in infinitesimal parameters of the Poincarégroup, we have

U (�, a) � exp(− i

2ωμνMμν + iaμPμ). (1.61)

The different generators of the Poincaré group satisfy the following commutationrelations:

[Mμν,Mρσ ] = i(gμρMνσ + gνσMμρ − gνρMμσ − gμσMνρ), (1.62)

[Mμν, Pσ ] = −i(gνσ Pμ − gμσ Pν), (1.63)

[Pμ, Pν] = 0, (1.64)

which are the commutation relations of the Poincaré algebra, i.e. the algebra thatcorresponds to the Poincaré group. Finally, returning to the example of the infinites-imal generator for the ‘rotation’ in the x0x1-plane, i.e. M01 in Eq. (1.39), the cor-responding generator of the Poincaré group is given by

M01 = id(�(01)(ξ), 0)

dξ

∣∣∣∣ξ=0

=

⎛⎜⎜⎜⎜⎝

0 −i 0 0 0−i 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

⎞⎟⎟⎟⎟⎠. (1.65)

In the case of the Lorentz group, the generator M01 is represented by a 4×4 matrix,whereas in the case of the Poincaré group, the generator M01 is represented by a5 × 5 matrix.

Exercise 1.1 Verify the commutation relations [J 1, J 2] = iJ 3, [K 1, K 2] = −iJ 3,and [J 1, K 2] = iK 3 using Eq. (1.62) as well as the definitions J = (J 1, J 2, J 3) =(M32,M13,M21) and K = (K 1, K 2, K 3) = (M01,M02,M03).




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