## Carlo Giunti and Chung W. Kim

Print publication date: 2007

Print ISBN-13: 9780198508717

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198508717.001.0001

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# (p.637) Appendix B Special Relativity

Source:
Fundamentals of Neutrino Physics and Astrophysics
Publisher:
Oxford University Press

(p.637) Appendix B

Special Relativity

Physical laws are expressed in terms of functions of the spatial coordinates$x→$and the temporal coordinate t. However, physical laws describe events which occur independently of the system of coordinates, also called the reference frame, which is adopted by an observer. Therefore, the transformations between different reference frames and the quantities which are independent of the reference frame play a crucial role in the formulation of physical theories.

According to the principle of relativity, the physical laws are equally valid in all reference frames. In other words, there is no privileged frame of reference. Therefore, the structure of the equations representing physical laws must be independent of the adopted reference frame. This is achieved by expressing physical laws through covariant equations, i.e. equations written in tensorial form. This is called the principle of covariance.

The theory of special relativity treats the relations between different inertial reference frames, which are systems of coordinates in which free particles propagate with uniform motion, in the absence of gravity. The relative velocity of different inertial reference frames is constant. They are related by Lorentz transformations, which are a consequence of the observed invariance of the velocity of light (see Refs. [183, 951]).

# B.1 The Lorentz group

A Lorentz transformation is a linear homogeneous coordinate transformation

(B.1)
from a reference frame S with coordinates x μ = (x 0, x 1, x 2, x 3) to a reference frame S′ with coordinates xμ = (x0, x1, x2, x3). The constant 4 × 4 matrix$∧vμ$must satisfy the constraints in eqn (B.14). In each reference frame x 0 = t is the time coordinate and$x→=(x1,x2,x3)$is the three-vector of spatial coordinates. Hence, the coordinates of x μ are often indicated as$xμ=(x0,x→)$or$xμ=(t,x→)$. The indices µ and ν in eqn (B.1) are called Lorentz indices. In eqn (B.1) and in the following equations, Lorentz indices run from 0 to 3 and repeated indices, one low and one high, are summed from 0 to 3.

Four-vectors are geometrical objects which form a vector space. They are represented in each reference frame by four real numbers called components. The components of a four-vector transform as the coordinates under Lorentz transformations. We denote four-vectors through their four components in a given reference (p.638) frame: for example,$Vμ=(V0,V1,V2,V3)=(V0,V→)$, with$V→=(V1,V2,V3)$. Then, under the Lorentz transformation in eqn (B.1) a four-vector V μ transforms as

(B.2)
In particular, x μ is the coordinate four-vector. Note that the index of a four-vector is always placed in the upper position.

Covariant four-vectors108, denoted with a low index, e.g. V μ = (V 0,V 1, V 2,V 3), are geometrical objects analogous to four-vectors, whose components transform as the components of the four-gradient ∂μ ≡ ∂/∂x μ = (∂0, ∂1, ∂2, ∂3). Since$∂μxv=δμv=∂′μx′v$, the four-gradient transforms as

(B.3)
High indices are called contravariant and low indices are called covariant. Thus, sometimes four-vectors are also called contravariant four-vectors in order to distinguish them from the covariant four-vectors.

Tensors are geometrical objects whose components have one or more indices and transform as products of contravariant and/or covariant four-vectors. The number of indices is the rank of the tensor. Contravariant and covariant four-vectors are tensors of rank one, or, more precisely, of rank$(01)$and$(10)$, respectively. For example, T μ v is a tensor of rank two or, more precisely, of rank$(11)$, which transforms as

(B.4)

The scalar product of two four-vectors V μ and W μ is defined by

(B.5)
where g μv is the metric tensor. The norm of a four-vector V μ is given by
(B.6)

The metric tensor allows the lowering of the indices of vectors and tensors. For example, for each four-vector V μ there is a corresponding covariant four-vector V μgiven by

(B.7)
A tensor can be represented in contravariant (e.g. T μv), covariant (e.g. T μv = g μρ g T ρσ), or mixed form (e.g.$Tvμ=gvρTμρ$). The contravariant metric tensor (p.639) g μv, which allows the raising of indices, is defined by
(B.8)
For example,
(B.9)
Using contravariant and covariant indices, the scalar product in eqn (B.5) can be written in the alternative forms
(B.10)

Let us introduce the infinitesimal space-time interval dτ, also called the infinitesimal proper-time interval109, which is given by

(B.11)
with the metric tensor
(B.12)
Space-time with this metric is called Minkowski space-time. Equation (B.8) implies that the contravariant and covariant metric tensors are equal:
(B.13)

The invariance of the velocity of light implies that dτ2 is Lorentz-invariant (for light rays dτ2 = 0 in all reference frames). Hence, the matrix$Λvμ$is subject to the condition

(B.14)
which is the fundamental relation defining the Lorentz transformations. This condition implies the Lorentz-invariance of the metric tensor and of the scalar product. It can also be written as
(B.15)
or in the compact matrix form
(B.16)

Equation (B.8) allows one to write eqn (B.15) as

(B.17)
which shows that
(B.18)

(p.640) The Lorentz transformations form the Lie group110 ℒ. Taking the determinant of eqn (B.16) and taking into account that DetM T = DetM for any matrix M, one finds

(B.19)
The Lorentz transformations with DetΛ = +1 are called proper and form the subgroup ℒ+ of the Lorentz group. The Lorentz transformations with DetΛ = –1 are called improper and form the set ℒ, which is not a group.

Taking ρ = σ = 0 in eqn (B.14), we obtain

(B.20)
which implies that
(B.21)
The subgroup ℒ↑ of Lorentz transformations with$Λ00≥+1$is called orthochronous. The set ℒ↓ of antichronous Lorentz transformations with$Λ00≥−1$do not form a group.

Therefore, the set of all Lorentz transformations is divided into four subsets according to the signs of DetΛ and$Λ00$. The subgroup$L+↑$of proper orthochronous Lorentz transformations is called the restricted Lorentz group. The restricted Lorentz group is a six-parameter continuous group. This can be seen by considering an infinitesimal Lorentz transformation

(B.22)
with infinitesimal ε. The condition in eqn (B.14), which defines Lorentz transformations, implies that ω is an antisymmetric matrix:
(B.23)
Since ω is a 4 × 4 matrix, it contains six independent components111. A finite restricted Lorentz transformation matrix Λ can be written as an infinite product of infinitesimal transformations, leading to the exponential form
(p.641)
(B.24)

Any Lorentz transformation can be written as a product of a restricted Lorentz transformation times one of the following three discrete transformations:

Space inversion: P = diag (1, –1, –1, –1) ∈$L—↑$

Time inversion: T = diag (–1, 1, 1, 1) ∈,$L_↓$

Total inversion: PT = diag(–1, –1, –1, –1) ∈$L+↓$.

Spatial rotations have

(B.25)
where R is a unimodular (Det R = +1) orthogonal (R T = R –1) matrix. Spatial rotations form a subgroup of the restricted Lorentz group. For a rotation by an angle θ (0 ≤ θ ≤ π) around the unit vector$n→$we have
(B.26)
Given a unimodular orthogonal matrix R, the angle θ and the unit vector$n→$are given by
(B.27)

A Lorentz transformation which connects two coordinate systems with different velocities is called a boost. The elements of the matrix$Λvμ$of a Lorentz boost that connects a coordinate system x with a coordinate system x′ moving with velocity$v→$relative to x are

(B.28)
with
(B.29)
For a boost with velocity v in the direction of the x k axis we have
(B.30)
with the so-called rapidity φ, given by
(B.31)
Rapidities are quite useful because they add in successive boosts along the same axis. Indeed, two successive boosts along the same axis with velocities and rapidities$v→1,ϕ1$and$v→2,ϕ2$are equivalent to a boost along the same axis with velocity$v→$and rapidity φ given by
(B.32)

In general a orthochronous Lorentz transformation can always be decomposed into a rotation times a boost (see Ref. [951]). To show this, let us write the (p.642) transformation matrix$Λvμ$as

(B.33)
where a and b are 3 × 1 column matrices and M is a 3 × 3 matrix, and γ ≥ 0. From eqn (B.18), we get
(B.34)
The relations Λ–1Λ = 1 and ΛΛ–1 = 1 lead, respectively, to the constraints
(B.35)
(B.36)
The origin of the reference frame S′, with coordinates xμ = (x0, 0, 0, 0), in the reference frame S has coordinates given by$x=Λ−1x′:x0=γx′0,x→=ax′0$. Hence, in the reference frame S the origin of the reference frame S′ moves with constant velocity
(B.37)
From eqn (B.28) we can write the corresponding boost as
(B.38)
Let us now consider$ΛΛv→−1$, which should correspond to a rotation. Indeed, since$Λv→−1=Λ−v→$, we obtain
(B.39)
and 0 is the zero 3 × 1 column matrix. ΛR represents a rotation, because the relations in eqns (B.35) and (B.36) imply that RT = R–1. Hence, the general orthochronous Lorentz transformation Λ in eqn (B.33) can be written as the product of the rotation ΛR and the boost Λ[neq]:
(B.40)
Using the relations in eqns (B.35) and (B.36) one can easily show that Λ can also be written as the product of the boost$ΛRv→$and the rotation ΛR:
(B.41)

# (p.643) B.2 Representations of the Lorentz group

The Lorentz transformation of four-vectors in eqns (B.1) and (B.2) is the defining representation of the Lorentz group. A general representation of the Lorentz group is made by objects Ψ that transform as

(B.42)
with operators D(Λ) that satisfy the multiplication rule of the Lorentz group: for two Lorentz transformations Λ1 and Λ2
(B.43)

For an infinitesimal Lorentz transformation in eqn (B.22) the operator D(Λ) must be infinitesimally close to the identity and can be written as

(B.44)
where J μv = –J μv is a set of six operators which are the generators of the Lorentz group in the representation under consideration. If the operators J μv are Hermitian, the representation is unitary (D = D –1).

The condition in eqn (B.43) implies that the generators J μv must satisfy the commutation relations

(B.45)
The problem of finding all the representations of the restricted Lorentz group is reduced to finding all the sets of operators that satisfy the commutation relations in eqn (B.45).

The defining four-vector representation is a matrix representation of the Lorentz group. Comparing eqns (B.22) and (B.44), we obtain the generators

(B.46)
These matrices are Hermitian and generate a unitary representation.

In general, the expression of D(e ω) for a finite restricted Lorentz transformation Λ = e ω in eqn (B.24) can be written as an infinite product of infinitesimal transformations by using the multiplication property in eqn (B.43), leading to the exponential form

(B.47)

(p.644) The six generators J µν can be divided into the angular momentum three-vector operators112

(B.49)
and the boost three-vector operator
(B.50)
that satisfy the commutation relations
(B.51)
(B.52)
(B.53)
The commutation relations in eqn (B.51) are those of angular momentum operators, which generate three-dimensional rotations. Since the angular momentum operators satisfy a closed algebra, rotations form a group. On the other hand, eqn (B.53) shows that Lorentz boosts do not form a group.

The operator D(1 + εω) in eqn (B.44) can be written as

(B.54)
with
(B.55)
(B.56)

For a boost with rapidity φ in the direction of the x k axis, we have

(B.57)
In the case of a boost with finite rapidity φ in the direction of the x k axis we have
(B.58)

For a rotation by an angle θ around the x k axis we have

(B.59)
For a rotation by a finite angle θ around the x k axis we have
(B.60)

## (p.645) B.2.1 Fields

Under a Lorentz transformation in eqn (B.1), a multicomponent field

(B.61)
transforms as
(B.62)
where S(Λ) is a n × n matrix that depends on the spin of the particle. For example, for a scalar field n = 1 and S(Λ) = 1 and for a vector field n = 4 and S(Λ) = Λ.

For an infinitesimal Lorentz transformation in eqn (B.22) let us write S(1 +εω) as

(B.63)
where S μv is the spin part of the generators of the Lorentz group in the representation under consideration.

Taking into account the transformation of coordinates, the full transformation of ψ(x) under a Lorentz transformation is given by

(B.64)
Using eqn (B.63) and taking into account the antisymmetric character of ωμv (see eqn (B.23)), for the infinitesimal Lorentz transformation in eqn (B.22), we have
(B.65)
Therefore, the operator D(1 + εω) that implements the infinitesimal Lorentz transformation in eqn (B.22) on ψ(x) is given by
(B.66)
Comparing this expression with eqn (B.44), we obtain the generators of the Lorentz group:
(B.67)
These generators can be written as
(B.68)
with the space-time part
(B.69)
where P μ = iμ (see eqn (B.86)), which is the same for all fields. Since L μv satisfy the commutation relations in eqn (B.45) and commute with S μv, in general, (p.646) S μv must satisfy the commutation relations in eqn (B.45) of the Lorentz group generators.

The space part of the angular momentum operator is

(B.70)
implying that$L→$is the usual spatial angular momentum operator$L→=x→×P→$. Denoting with S k the spin part of the angular momentum operator,
(B.71)
the total angular momentum operator$J→$can be written in general as the sum of its spatial and spin parts:
(B.72)

# B.3 The Poincaré group and its representations

The Poincaré group (also known as the inhomogeneous Lorentz group) is the Lorentz group augmented with space-time translations. A Poincaré transformation is a coordinate transformation

(B.73)
with four arbitrary real constants a μ representing space-time translations.

A general representation of the Poincaré group is constituted by objects Ψ which transform as

(B.74)
with operators D(Λ, a) which satisfy the multiplication rule of the Poincaré group:
(B.75)

For an infinitesimal Poincaré transformation

(B.76)
with infinitesimal ε, the operator D(1 + εω, εb) can be written as
(B.77)
where J μv = –J μv are the six generators of the Lorentz group and P μ are the four generators of space-time translations.

The condition in eqn (B.75) leads to the following Lie algebra of the Poincaré group generators:

(B.78)
(p.647)
(B.79)
(B.80)
Furthermore, the angular momentum operators in eqn (B.49) and boost operators in eqn (B.50) satisfy the following commutation relations with the energy and momentum operators:
(B.81)
(B.82)
(B.83)
(B.84)

Analogously to eqn (B.64), the transformation of a field ψ(x) under a Poincaré transformation in eqn (B.73) is given by

(B.85)
In this case, the generators J μv of the Lorentz group are given by eqn (B.67) and the generators P μ of space-time translations are given by
(B.86)

Covariance of physical laws under Poincaré transformations imply that all quantities defined in Minkowski space-time must belong to a representation of the Poincaré group. By definition, the states that describe elementary particles belong to irreducible representations of the Poincaré group. These representations can be classified by the eigenvalues of the Casimir operators, which are the functions of the generators that commute with all the generators. This property implies that the eigenvalues of the Casimir operators remain invariant under group transformations.

There are two Casimir operators of the Poincaré group. The first one is

(B.87)
The corresponding eigenvalues
(B.88)
are the squared masses of particles. In the real world we observe only time-like or light-like four-momenta, i.e. particles with positive or zero mass. Furthermore, the temporal components p 0 of the four-momenta of physical particles, which correspond to energy and whose sign is a Lorentz invariant, are always positive.

The second Casimir operator of the Poincaré group is

(B.89)
where W μ is the Pauli–Lubanski four-vector
(B.90)
(p.648) such that
(B.91)

In three-vector notation, taking into account that$L→⋅P→=0,$, we have

(B.92)
Hence, W 0 depends only on$P→$and the spin operator$S→$.

We can evaluate the Lorentz-invariant W 2 in the rest frame of a particle with mass m, where

(B.93)
(B.94)
Since from the nonrelativistic quantum theory of angular momentum we know that the eigenvalue of$S→2$is s(s + 1), where s is the spin of the particle (half-integer or integer), we see that the eigenvalue of the relativistic invariant W 2,
(B.95)
gives the spin of the particle.

A complete set of commuting observables is composed of P 2, the three components of$P→$, W 2 and one of the four components of W μ. The eigenvalues of P 2 and W 2 or, equivalently, the mass m and the spin s, distinguish (possibly together with other quantum numbers) different particles. Given a particle with mass m and spin s, the eigenvalues of$P→$and one of the four components of W μ distinguish different states of the same particle. It is convenient to use the helicity operator$h^$defined by

(B.96)
which is proportional to W 0. Therefore the states of a particle with mass m and spin s can be distinguished by the eigenvalues of$P→$and$h^$.

## Notes:

(108) In differential geometry, covariant four-vectors are called dual four-vectors or one-forms (see Refs. [808, 1047, 941]). They form the dual vector space of linear functions of four-vectors into the real numbers.

(109) In the rest frame, where dx 1 = dx 2 = dx 3 = 0, we have dτ = dx 0.

(110) A group G is a set of objects, called elements of G, which can be combined to form an operation called the product, denoted by ab, for a, bG. The product must satisfy the following four conditions:

1. (1.) ab ε G, if a, b ε G.

2. (2.) (ab)c = a(bc), if a, b, cε G (associativity).

3. (3.) There is a unit element I ε G such that Ia = aI = a for all a ε G.

4. (4.) Each a ε G has an inverse a –1 ε G such that aa –1 = a –1 a = I.

A Lie group is a group whose elements can be parameterized and the parameters of a product are analytic functions of the parameters of the factors.

(111) The number of independent components of a N×N antisymmetric matrix is N(N–1)/2.

(112) The inverse relation is

((B.48))