# (p.637) Appendix B Special Relativity

# (p.637) Appendix B Special Relativity

(p.637) Appendix B

Special Relativity

Physical laws are expressed in terms of functions of the spatial coordinates$\overrightarrow{x}$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

*S*with coordinates

*x*

^{μ}= (

*x*

^{0},

*x*

^{1},

*x*

^{2},

*x*

^{3}) to a reference frame

*S*′ with coordinates

*x*′

^{μ}= (

*x*′

^{0},

*x*′

^{1},

*x*′

^{2},

*x*′

^{3}). The constant 4 × 4 matrix${\wedge}_{v}^{\mu}$must satisfy the constraints in eqn (B.14). In each reference frame

*x*

^{0}=

*t*is the time coordinate and$\overrightarrow{x}=({x}^{1},{x}^{2},{x}^{3})$is the three-vector of spatial coordinates. Hence, the coordinates of

*x*

^{μ}are often indicated as${x}^{\mu}=({x}^{0},\overrightarrow{x})$or${x}^{\mu}=(t,\overrightarrow{x})$. 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,${V}^{\mu}=({V}^{0},{V}^{1},{V}^{2},{V}^{3})=({V}^{0},\overrightarrow{V})$, with$\overrightarrow{V}=({V}^{1},{V}^{2},{V}^{3})$. Then, under the Lorentz transformation in eqn (B.1) a four-vector *V* ^{μ} transforms as

*x*

^{μ}is the coordinate four-vector. Note that the index of a four-vector is always placed in the upper position.

*Covariant four-vectors*^{108}, 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${\partial}_{\mu}{x}^{v}={\delta}_{\mu}^{v}={{\partial}^{\prime}}_{\mu}{{x}^{\prime}}^{v}$, the four-gradient transforms as

*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${(}_{0}^{1})$and${(}_{1}^{0})$, respectively. For example, *T* ^{μ} *v* is a tensor of rank two or, more precisely, of rank${(}_{1}^{1})$, which transforms as

The scalar product of two four-vectors *V* ^{μ} and *W* ^{μ} is defined by

*g*

_{μv}is the

*metric tensor*. The norm of a four-vector

*V*

^{μ}is given by

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* _{μ}given by

*T*

^{μv}), covariant (e.g.

*T*

_{μv}=

*g*

_{μρ}

*g*

_{vσ}

*T*

^{ρσ}), or mixed form (e.g.${T}_{v}^{\mu}={g}_{v\rho}{T}^{\mu \rho}$). The contravariant metric tensor (p.639)

*g*

^{μv}, which allows the raising of indices, is defined by

Let us introduce the *infinitesimal space-time interval* dτ, also called the *infinitesimal proper-time interval*^{109}, which is given by

*Minkowski space-time*. Equation (B.8) implies that the contravariant and covariant metric tensors are equal:

The invariance of the velocity of light implies that dτ^{2} is Lorentz-invariant (for light rays dτ^{2} = 0 in all reference frames). Hence, the matrix${\Lambda}_{v}^{\mu}$is subject to the condition

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

(p.640)
The Lorentz transformations form the Lie group^{110} ℒ. Taking the determinant of eqn (B.16) and taking into account that Det*M* ^{T} = Det*M* for any matrix *M*, one finds

*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

*orthochronous*. The set ℒ↓ of antichronous Lorentz transformations with${\Lambda}_{0}^{0}\ge -1$do not form a group.

Therefore, the set of all Lorentz transformations is divided into four subsets according to the signs of DetΛ and${\Lambda}_{0}^{0}$. The subgroup${L}_{+}^{\uparrow}$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

^{111}. A finite restricted Lorentz transformation matrix Λ can be written as an infinite product of infinitesimal transformations, leading to the exponential form (p.641)

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) ∈${L}_{\u2014}^{\uparrow}$

**Time inversion**: *T* = diag (–1, 1, 1, 1) ∈,${L}_{\_}^{\downarrow}$

**Total inversion**: *PT* = diag(–1, –1, –1, –1) ∈${L}_{+}^{\downarrow}$.

Spatial rotations have

*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$\overrightarrow{n}$we have

*R*, the angle θ and the unit vector$\overrightarrow{n}$are given by

A Lorentz transformation which connects two coordinate systems with different velocities is called a *boost*. The elements of the matrix${\Lambda}_{v}^{\mu}$of a Lorentz boost that connects a coordinate system *x* with a coordinate system *x*′ moving with velocity$\overrightarrow{v}$relative to *x* are

*v*in the direction of the

*x*

^{k}axis we have

*rapidity*φ, given by

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${\Lambda}_{v}^{\mu}$as

^{–1}Λ = 1 and ΛΛ

^{–1}= 1 lead, respectively, to the constraints

*S*′, with coordinates

*x*′

^{μ}= (

*x*′

^{0}, 0, 0, 0), in the reference frame

*S*has coordinates given by$x={\Lambda}^{-1}{x}^{\prime}:{x}^{0}=\gamma {{x}^{\prime}}^{0},\overrightarrow{x}=a{{x}^{\prime}}^{0}$. Hence, in the reference frame

*S*the origin of the reference frame

*S*′ moves with constant velocity

_{R}represents a rotation, because the relations in eqns (B.35) and (B.36) imply that R

^{T}= R

^{–1}. Hence, the general orthochronous Lorentz transformation Λ in eqn (B.33) can be written as the product of the rotation Λ

_{R}and the boost Λ[neq]:

_{R}:

# (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

*D*(Λ) that satisfy the multiplication rule of the Lorentz group: for two Lorentz transformations Λ1 and Λ

_{2}

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

*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

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

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

(p.644)
The six generators *J* ^{µν} can be divided into the angular momentum three-vector operators^{112}

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

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

*x*

^{k}axis we have

For a rotation by an angle θ around the *x* ^{k} axis we have

*x*

^{k}axis we have

## (p.645) B.2.1 Fields

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

*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

*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

^{μv}(see eqn (B.23)), for the infinitesimal Lorentz transformation in eqn (B.22), we have

*D*(

**1**+ εω) that implements the infinitesimal Lorentz transformation in eqn (B.22) on ψ(

*x*) is given by

*P*

^{μ}=

*i*∂

^{μ}(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

*S*

^{k}the spin part of the angular momentum operator,

# B.3 The Poincaré group and its representations

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

*a*

^{μ}representing space-time translations.

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

*D*(Λ,

*a*) which satisfy the multiplication rule of the Poincaré group:

For an infinitesimal Poincaré transformation

*D*(

**1**+

*εω, εb*) can be written as

*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:

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

*J*

^{μv}of the Lorentz group are given by eqn (B.67) and the generators

*P*

^{μ}of space-time translations are given by

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

*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

*W*

_{μ}is the Pauli–Lubanski four-vector

In three-vector notation, taking into account that$\overrightarrow{L}\cdot \overrightarrow{P}=0,$, we have

*W*

^{0}depends only on$\overrightarrow{P}$and the spin operator$\overrightarrow{S}$.

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

*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},

A complete set of commuting observables is composed of *P* ^{2}, the three components of$\overrightarrow{P}$, *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$\overrightarrow{P}$and one of the four components of *W* ^{μ} distinguish different states of the same particle. It is convenient to use the helicity operator$\widehat{h}$defined by

*W*

^{0}. Therefore the states of a particle with mass

*m*and spin

*s*can be distinguished by the eigenvalues of$\overrightarrow{P}$and$\widehat{h}$.

## 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 d*x* ^{1} = d*x* ^{2} = d*x* ^{3} = 0, we have dτ = d*x* ^{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, b* ∈ *G*. The product must satisfy the following four conditions:

(1.)

*ab*ε*G*, if*a, b*ε*G*.(2.) (

*ab)c = a(bc*), if*a, b, c*ε*G*(associativity).(3.) There is a unit element

*I*ε*G*such that*Ia = aI = a*for all*a*ε*G*.(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.