(p.637) Appendix B Special Relativity
(p.637) Appendix B Special Relativity
(p.637) Appendix B
Physical laws are expressed in terms of functions of the spatial coordinatesand 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
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,, with. Then, under the Lorentz transformation in eqn (B.1) a four-vector V μ transforms as
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, the four-gradient transforms as
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 rankand, respectively. For example, T μ v is a tensor of rank two or, more precisely, of rank, which transforms as
The scalar product of two four-vectors V μ and W μ is defined 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 μgiven by
Let us introduce the infinitesimal space-time interval dτ, also called the infinitesimal proper-time interval109, which is given by
The invariance of the velocity of light implies that dτ2 is Lorentz-invariant (for light rays dτ2 = 0 in all reference frames). Hence, the matrixis subject to the condition
Taking ρ = σ = 0 in eqn (B.14), we obtain
Therefore, the set of all Lorentz transformations is divided into four subsets according to the signs of DetΛ and. The subgroupof 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
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) ∈
Time inversion: T = diag (–1, 1, 1, 1) ∈,
Total inversion: PT = diag(–1, –1, –1, –1) ∈.
Spatial rotations have
A Lorentz transformation which connects two coordinate systems with different velocities is called a boost. The elements of the matrixof a Lorentz boost that connects a coordinate system x with a coordinate system x′ moving with velocityrelative to x are
(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
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
The condition in eqn (B.43) implies that the generators J μv must satisfy the commutation relations
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
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
For a rotation by an angle θ around the x k axis we have
(p.645) B.2.1 Fields
Under a Lorentz transformation in eqn (B.1), a multicomponent field
For an infinitesimal Lorentz transformation in eqn (B.22) let us write S(1 +εω) as
Taking into account the transformation of coordinates, the full transformation of ψ(x) under a Lorentz transformation is given by
The space part of the angular momentum operator is
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
A general representation of the Poincaré group is constituted by objects Ψ which transform as
For an infinitesimal Poincaré transformation
The condition in eqn (B.75) leads to the following Lie algebra of the Poincaré group generators:
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
The second Casimir operator of the Poincaré group is
In three-vector notation, taking into account that, we have
We can evaluate the Lorentz-invariant W 2 in the rest frame of a particle with mass m, where
A complete set of commuting observables is composed of P 2, the three components of, 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 ofand one of the four components of W μ distinguish different states of the same particle. It is convenient to use the helicity operatordefined by
(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, 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, cε 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.