Tensor Algebra
To understand the concept of a tensor, it's essential to first review fundamental ideas related to vector spaces and linear maps We will assume the reader has a basic familiarity with vectors and vector spaces, avoiding excessive detail Let's begin by examining a finite-dimensional vector space \( V^n \) defined over the real numbers \( \mathbb{R} \) With an arbitrary basis \( \{e_i\}_{i=1}^n \), any vector \( v \) in \( V^n \) can be expressed as \( v = v_i e_i \).
One is probably used to see a vector written in the following form v= n i = 1 v i e i (1.2)
In this article, we will omit the bold vector symbol and use the Einstein summation convention for repeated indices, leading to a more concise expression We will examine an invertible change of basis represented by the matrix A, where the determinant of A is non-zero This allows us to establish a connection between the new basis and the original basis.
This article primarily focuses on concepts that do not pertain to complex spaces, although a general definition over C exists Once readers grasp the ideas presented, they can easily explore the broader applications in complex spaces.
In the context of vectors, tensors, and manifolds within special relativity, the relationship between basis vectors can be expressed as e_j = A_{ij} e_i, where the upper index denotes the row and the lower index denotes the column, with summation over repeated indices implied Conversely, the inverse relationship can be represented as e_j = (A^{-1})_{ij} e_i.
Invariance of a quantity simplifies the transformation law for vector components, expressed as \( v = v_j e_j = v_j (A^{-1})_{ij} e_i = v_i e_i \) Under an invertible change of basis, the vector components transform according to the equation \( v_i = (A^{-1})_{ij} v_j \), which can also be represented as \( v_i = A_{ij} v_j \).
The dual space ofV n , denoted asV n ∗ is defined as the space of all the linear maps (applications) fromV n toR: β: V n → R β: v → β(v), (1.7) with the following property β(λ 1 u 1+λ 2 u 2 )=λ 1 β(u 1 )+λ 2 β(u 2 ), (1.8)
∀u 1 ,u 2 ∈V n and∀λ 1 ,λ 2 ∈R The spaceV n ∗ is also a vector space of dimension n and its elements are usually called covectors Using an arbitrary basis{ω i } n i = 1 an elementβ ∈V n ∗ can be written as β=β i ω i (1.9)
Therefore, given an elementv∈V n the linear mapβ(v)∈Rcan be explicitly writ- ten as β(v)=β i ω i (v j e j )=β i v j ω i (e j ) (1.10)
The quantity ω_i(e_j) is influenced by the selected bases {ω_i} and {e_i} However, a specific basis known as the dual basis of V_n* exhibits a unique property where ω_i(e_j) equals the Kronecker delta, δ_ij, which is defined as δ_ij = 0 if i ≠ j and δ_ij = 1 if i = j Moving forward, we will utilize the dual basis and denote its elements as {e_j} Consequently, the relationship (1.11) transforms to e_i(e_j) = δ_ij, leading to a simplified expression for β(v), which can be represented as β(v) = β_i e_i(v_j e_j) = β_i v_j e_i(e_j) = β_i v_j δ_ij = β_i v_i.
Let’s now deduce how the elements{e i }must transform under a change of the basis {e j } → {e j }in order to maintain the duality condition: e i (e j )=δ i j =e i (e j ) (1.14)
Let’s suppose that the transformation{e i } → {e i }is given by an invertible matrix
Inserting (1.3) and (1.15) into (1.14) we easily get e i (e j )=B l i e l (A k j e k )=B l i A k j e l (e k )=B l i A k j δ k l =δ i j (1.16) Therefore, we obtain the following relation between the matricesAandB
B k i A k j =δ i j ⇒ B A=I ⇒ A=B − 1 (1.17) where Iis then×nidentity matrix In conclusion, the components of the covector and the basis ofV n ∗ obey the following transformation rules β i = A i j β j , e i =(A − 1 ) i j e j (1.18)
4 1 Vectors, Tensors, Manifolds and Special Relativity
1.1.2 Covariant and Contravariant Laws of Transformation
Summing up, givenv ∈V n a vector (v=v i e i ) andβ ∈V n ∗ a covector (β=β i e i ), we have the following law of transformation for{e i }and{β i } e i =A i j e j , β i =A i j β j (1.19)
We shall call this, thecovariantlaw of transformation For{e i }and{v i }we have found e i =(A − 1 ) i j e j , v i =(A − 1 ) i j v j (1.20)
The contravariant law of transformation distinguishes between covariant and contravariant quantities using upper and lower indices It is essential to note that not all elements with these indices are necessarily covariant or contravariant An explicit example will be provided in the following sections.
For a finite dimensional vector space V n ,the dual space of its dual space V n ∗ ,(called double dual space, denoted as V n ∗∗ )is isomorphic to V n
In this discussion of general algebra, we focus on the one-to-one correspondence between the vector space \( V^n \) and its double dual \( V^{n**} \), indicating that no new information is gained from \( V^{n**} \) Consequently, we can equate the vector space \( V^n \) with its double dual \( V^{n**} \) This equivalence allows us to interpret \( V^n \) as the collection of all linear maps from \( V^n* \) to \( \mathbb{R} \), represented as \( v: V^{n*} \to \mathbb{R} \) By identifying \( V^n \) with the dual space of \( V^{n*} \), we define \( e_j(e_i) \equiv e_i(e_j) = \delta_{ij} \) and introduce a useful notation for linear maps: \( v(\beta) \equiv \beta(v) = \beta_i v_i \).
In this article, we will define tensors, a fundamental concept in algebra that extends the ideas of vectors, covectors, and linear maps To begin, we will introduce the tensor product, which serves as the foundation for understanding tensors.
Given two vector spacesV n andV m of finite dimensionsnandm, the tensor product is a map of the form:
∀v, v 1 , v 2 ,∈V n ,∀w, w 1 , w 2 ,∈V m and∀λ∈R Note that the commutative property doesn’t hold for the tensor product by definition.
The productV n ⊗V m is a vector space of dimensionnãmand its elements are called tensors Ifv = v i e i ∈ V n andw =w j e j ∈ V m thenq ≡ v⊗wcan be written as: q =v⊗w=v i w j e i ⊗e j ≡q i j e i ⊗e j (1.25)
The tensor product can be defined over any finite sequence of vector spaces, dual spaces or both We can define for example 2 :
Tensors are multilinear maps that act on vector spaces and their duals.
2 We shall only be concerned with identical copies of vector spaces and their duals, therefore all spaces considered from now on will be of dimension n.
6 1 Vectors, Tensors, Manifolds and Special Relativity
The quantityu⊗w (α,β)can be expressed using dual bases as u⊗w (α,β)=u(α) v(β)
However, not all tensors defined overV n ∗ ×V n ∗ are of the formu⊗v The general way of defining a tensor will be given in the following sections.
A rank two contravariant tensor, or a(2,0)tensor is a linear map of the form: t: V n ∗ ×V n ∗ → R t: (α,β) → t(α,β), (1.29) with the following properties:
∀α,β,α 1 ,α 2 ,β 1 ,β 2 ,∈ V n ∗ and∀λ 1 ,λ 2 ∈R It is straightforward to deduce that the following property also holds t(λ 1 α,λ 2 β)=λ 1 λ 2 t(α,β), (1.31)
For any vectors α and β in the vector space V n ∗, and for real numbers λ1 and λ2, the relationship between these vectors can be expressed as t(α, β) = t(α i e i, β j e j) = α i β j t(e i, e j), where t(e i, e j) is defined as the tensor components t i j associated with a specific basis Consequently, the tensor t can be represented using a basis and the tensor product as t = t i j e i ⊗ e j Therefore, the expression for t(α, β) can be rewritten as t(α, β) = t i j e i ⊗ e j (α k e k, β l e l).
Whenever t can be separated ast i j = v i w j withu, w ∈ V n (meaning that t =v⊗w, as in the previous section) it is said thatt is a separable tensor.
A rank two covariant tensor, or a(0,2)tensor, is a linear map of the form: t : V n ×V n → R t : (u, v) → t(u, v), (1.35) with the same properties 1, 2 as in the previous case Thus, we can expresstusing a basis as follows t=t i j e i ⊗e j (1.36)
We have to be somewhat careful when we defining mixed tensors For example, we can define a (1,1) mixed tensor in two different ways The first one
8 1 Vectors, Tensors, Manifolds and Special Relativity t : V n ∗ ×V n → R t : (α, v) → t(α, v), (1.38) withtfirst acting onV n ∗ and afterwards onV n It must be written as t=t i j e i ⊗e j (1.39)
The other way of defining a (1,1) tensor is t : V n ×V n ∗ → R t : (v,α) → t(v,α) (1.40)
In this caset must be written as t =t i j e i ⊗e j (1.41)
To prevent confusion in tensor notation, blank spaces are often used between indices to clarify the order of application For instance, consider the mapping \( t: V^n \times V^{n*} \times V^{n*} \times V \to \mathbb{R} \) It is essential to express \( t \) as \( t = t^{ijkl} e_i \otimes e_j \otimes e_k \otimes e_l \).
In practical calculations, it is common to overlook blank spaces, particularly in fields like physics, where tensor components are often represented without them, such as "il j k."
1.1.9 Tensor Transformation Under a Change of Basis
Let’s consider a(2,0)tensor Under a change of basis of the form (1.3) we have the following t=t kl e k ⊗e l =t kl e i ⊗e j (A − 1 ) i k (A − 1 ) l j =t i j e i ⊗e j (1.44) Thus, the law of transformation of a rank two contravariant tensor is: t i j =t kl (A − 1 ) i k (A − 1 ) l j (1.45)
Obviously, for a rank two covariant tensor the law of transformation is: t i j =t kl A k i A l j (1.46) and for a(1,1)mixed tensor we have: t i j =t k l (A −1 ) i k A l j t j i =t l k (A −1 ) i k A l j
The generalization to(r,s)tensors (r-times contravariant ands-times covariant) is straightforward A (0,0) tensor is called a scalar (remains invariant under a change of basis) A vector is a(1,0)tensor and a covector is(0,1)tensor.
The transformation laws (1.45), (1.46), and (1.47) illustrate the fundamental definition of tensors To establish that a quantity qualifies as a tensor, it is enough to verify that it adheres to these tensor transformation laws.
The tensor product is a method for creating tensors from higher rank tensors, extending beyond the previous constructions using only vectors or covectors If \( t \) is a \( (r,s) \) tensor and \( b \) is a \( (m,n) \) tensor, then the tensor product \( t \otimes b \) results in a \( (r+m,s+n) \) tensor For instance, when \( t = t^{ij} e_i \otimes e_j \) and \( b = b^{kl} e_k \otimes e_l \) are both \( (1,1) \) tensors, the product \( q = t \otimes b \) forms a \( (2,2) \) tensor, explicitly expressed as \( q = t^{ij} b^{kl} e_i \otimes e_j \otimes e_k \otimes e_l \).
The Kronecker delta is easily demonstrated to be a rank two mixed tensor, characterized by its symmetry, where δij = δji Although mixed tensors require careful attention to index order and may necessitate leaving blank spaces, we will adopt the simplified notation δij ≡ δji for most calculations, as the specific order is generally inconsequential in practice.
10 1 Vectors, Tensors, Manifolds and Special Relativity
Tensor Calculus
A vector field is defined as a continuous function that assigns a vector to every point in space A ∈ E n, where the vector depends on parameters x i ∈ R n The transformation of this vector field is influenced by these parameters To illustrate this, consider two points A and B in E n with coordinates x i and y i in different reference frames, which may be related by a translation By defining Δx i as the difference between these coordinates, we can observe that the interval Δs 2, represented by the equation Δs 2 = g i j Δx i Δx j, remains invariant under transformation.
Taking x i and y i to be infinitesimally close (thus x i and y i ) our scalar interval becomesdifferential ds 2 =g i j d x i d x j =ds 2 =g i j d x i d x j (1.62)
In the context of vector fields, it is common to associate the components denoted by \( x^i \) with the coordinates When a change of coordinates occurs, represented as \( x^i \rightarrow x^i \), the transformation of the vector field components adheres to the chain rule, resulting in the equation \( d x^i = \partial x^i \).
The contravariant law of transformation, represented by the equation ∂x j d x j (1.63), can be extended to any tensor A tensor that acts as a continuous function of parameters x i ∈ R n is referred to as a tensor field For instance, (2,0) and (0,2) tensor fields can be expressed in the forms t = t i j (x)e i (x)⊗e j (x) and l = l i j (x)e i (x)⊗e j (x) (1.64).
The value of a tensor field evaluated at a point A, represented as t A, must remain consistent regardless of the reference frame used, as it depends on specific parameters x i For instance, in the case of a (2,0) tensor field, this relationship is expressed as t A = t i j (x A)e i (x A)⊗e j (x A), demonstrating that the tensor's value is invariant across different coordinate systems.
A very familiar example where the componentse i of the basis depend on the coor- dinatesx j are vectors expressed incurvilinear coordinates The basis is given by e 1 (x)≡ ˆu φ , e 2 (x)≡ ˆu θ , e 3 (x)≡ ˆu r (1.66) withx j =(r,θ,φ).
In conclusion, taking quick look at (1.62) and (1.63), we identify thecontravari- antlaw of transformation with
3 Here we will use the short-hand notation f ( x i ) ≡ f ( x )
14 1 Vectors, Tensors, Manifolds and Special Relativity e i (x ) = ∂x i
Thus, thecovariantlaw of transformation will be given by e i (x )= ∂x l
The law of transformation for mixed tensor fields is obviously given by t i j (x)= ∂x i
The generalization to(r,s)tensor fields is straightforward A(0,0)field is called a scalar field and it obeys: φ(x)=φ (x ) (1.70)
The intrinsic definition of tensor fields, as outlined in expressions (1.67–1.70), is commonly found in many physics textbooks However, to achieve a comprehensive understanding of tensors, it is essential to explore the foundational concepts leading up to this definition.
Using the the intrinsic definition of tensor fields it is straightforward to introduce another object which is called tensor density We say that t i 1 i r j
1 j s (x)are the components of a(r,s)tensor density ofweight W if under a change of coordinates x i →x i they obey the following transformation law t i 1 i r j 1 j s (x ) det
∂x j s t l 1 l r m 1 m s (x) (1.71) where det(∂x i /∂x j )is the determinant of the Jacobian matrix of the given transfor- mation Same definition is valid for any(r,s)type tensor i.e.,t i 1 i j l i l + 1 i r
The totally antisymmetric Levi-Civita symbol, denoted as i 1 i k i m i n, exhibits a relationship with permutations defined by the equation i 1 i k i m i n = (−1) p i 1 i m i k i n, where p indicates the parity of the permutation (with p=1 for odd permutations and p=2 for even permutations) This symbol is classified as a tensor density with a weight of W = −1.
We will now move on to the next section and generalize everything to non-Euclidean spaces and introduce properly the concept ofmanifold.
Manifolds
A manifold is a mathematical structure that generalizes the concept of Euclidean space, defined as a set of points with a one-to-one correspondence with \( R^n \) However, the definition of a manifold extends beyond Euclidean spaces, encompassing a broader range of geometric structures Hobson offers an insightful and intuitive definition of manifolds that captures their essence, regardless of whether they are Euclidean or not.
A manifold is a set that can be continuously parametrized, with its dimension defined by the number of independent parameters needed to uniquely identify any point within it In its simplest form, a manifold consists of a collection of points, but in physics, the focus is on "differential manifolds," which are both continuous and differentiable A manifold is considered continuous if, near any point P, there exist other points with coordinates that differ only infinitesimally from those of P It is differentiable if a scalar field can be defined and differentiated at every point within the manifold An N-dimensional manifold requires N independent real coordinates to fully specify any point.
In the study of manifolds, we can establish local mappings between a manifold and R^n, but global mappings are often elusive A surface, for instance, requires two parameters for its description, categorizing it as a two-dimensional manifold When considering a flat surface, a global one-to-one mapping with R^2 is possible However, for a sphere, a global mapping to R^2 cannot be achieved, highlighting the complexities of manifold topology.
4 M.P Hobson, G.P Efstathiou and A.N Lanseby, General Relativity, An Introduction for Physicists
16 1 Vectors, Tensors, Manifolds and Special Relativity
In R², we can represent the entire surface, but we can only describe specific areas using local charts or regions This concept can be extended to apply to any manifold, illustrating the importance of localized descriptions in understanding complex geometric structures.
An N-dimensional differential manifold Mis a set of elements (points) P, together with a collection of subsets{O α }ofMthat satisfy the following three conditions:
1.∀ P∈Mthere is at least a subsetO α , so thatP∈O α Equivalently:
2.For eachO α there is a diffeomorphism (differential with its inverse also differen- tial)Ψ α : Ψ α :O α → Ψ α (O α )≡U α ⊆ R n
3.IfO α ∩O β = {ỉ}thenΨ α (O α ∩O β )andΨ β (O α ∩O β )are open sets ofR n and the application: Ψ β ◦Ψ α − 1 : Ψ α (O α ∩O β )→Ψ β (O α ∩O β ) is a diffeomorphism Here we have defined the composed operatorΨ β ◦Ψ α − 1 (X)≡ Ψ β (Ψ α − 1 (X)) This application is called achange of coordinates.
The sets(Ψ α ,O α )are calledcharts The set formed of all charts is called anatlas.
Thus, for a given manifoldMof dimension n, if we can find an atlas that contains only one chart, then we shall say that the manifold isflator that it hastrivial topology.
A surface exhibits two types of curvature: intrinsic (Gauss curvature) and extrinsic Surfaces with no intrinsic curvature, such as a cylinder, can be flattened, while those with intrinsic curvature, like a sphere, cannot be unfolded into a flat surface, indicating that their topology is non-trivial.
In geometry, a curve is defined as a one-dimensional manifold, while a surface represents a two-dimensional manifold, and a solid volume is classified as a three-dimensional manifold These concepts are commonly encountered in various contexts For instance, to describe a curve, we require a single parameter, denoted as t In three-dimensional space, this curve can be represented by the function r(t) = (x(t), y(t), z(t)).
To effectively describe a surface in R³, two parameters, u and v, are required This can be represented mathematically as s(u, v) = (x(u, v), y(u, v), z(u, v)) For instance, specific regions of the surface can be expressed as s(u, v) = (u, v, f(u, v)).
In the previous examples we have described (and visualised) these objects in
In the context of R³, the term "embedding" refers to the process of visualizing curves, surfaces, or volumes within three-dimensional space However, not all one, two, and three-dimensional manifolds conform to the typical definitions of curves, surfaces, or solid volumes For instance, certain two-dimensional manifolds cannot be embedded in R³ but can exist in higher-dimensional spaces, Rⁿ, where n is greater than three Consequently, these objects do not represent conventional surfaces.
In the context of General Relativity, our four-dimensional manifold, composed of space-time points known as events, features a complex topology that cannot be easily embedded in a higher-dimensional space, as we lack evidence for such dimensions To illustrate this concept, consider a two-dimensional civilization living on a surface; its inhabitants can only perceive intrinsic curvature, while extrinsic curvature remains accessible only to a three-dimensional observer Similarly, we, as four-dimensional beings, are confined to measuring the intrinsic curvature of our space-time manifold, which directly relates to the gravitational force, without any knowledge of a potential higher-dimensional space.
18 1 Vectors, Tensors, Manifolds and Special Relativity
Consider the velocity vector of a moving body This vector is tangent to the curve that describes the trajectory of the moving body for every regular point of the curve.
A vector field defined over a surface is tangent to the surface at every regular point, indicating that it resides within the tangent plane of that surface Extending this concept to a general n-dimensional manifold, we assert that vector fields are associated with the tangent space of the manifold M For each regular point P within the manifold, we define the tangent space as T P (M) The collective union of all these tangent spaces is represented as T(M) Thus, we express a vector v in T(M) as v = v i (x)e i (x), and at a specific point P, it can be denoted as v P = v i (x P)e i (x P).
In our discussion, we refer to a vector field as 'v' and a vector as 'P.' This distinction emphasizes the differences between vector spaces Vn, En, and Rn By extending our analysis to generic manifolds, these differences become even more apparent.
These three spaces are, no doubt, different.
As we have already seen, under a change of coordinatesx i → x i , the bases of the vector fields obey the covariant law: e i (x )= ∂x l
In differential geometry, a common practice is to define a basis composed of partial derivatives that adhere to a specific transformation Utilizing this basis, a vector field can be expressed in the form v P = v i (x) ∂.
Thus we can interpret a vectorvas map of the form: v: F(M)→ R v: f →v(f), (1.79) withF(M)being the space of the all differentiable applications f : M→ R, (1.80) and withvsatisfying the following two conditions:
∀a,b ∈ R, ∀ f, g ∈ F(M) The composed function operator◦ is defined as
(f ◦g)(X)≡ f(g(X)) Therefore, given a function f ∈F(M)and a regular point
The cotangent space is defined as the dual space of the tangent space, and a suitable basis for this dual space is formed by the functions \(dx^i\) Consequently, a covector field can be expressed as \(\beta_P = \beta_i(x)dx^i\).
P ∈T P ∗ (M); β=β i (x)d x i ∈T ∗ (M) , (1.83) where T ∗ (M)is defined as the union of allT P ∗ (M) Using this formalism, given v∈T P (M)andβ∈T P ∗ (M),β(v) P ∈Ris defined as: β(v) P =β j d x j v i ∂
A common inquiry in differential geometry is why the expression \( \delta_{ij} = \frac{\partial}{\partial x^j} \bigg|_P \) holds true To clarify, we define the differential of a function \( f \in F(M) \) at a point \( P \in M \) as a mapping from the tangent space \( T_P(M) \) to the real numbers \( \mathbb{R} \) Specifically, this differential, denoted as \( df \), takes a vector \( v_P \) and evaluates it at point \( P \), yielding \( df_P(v) \equiv v_P(f) \).
20 1 Vectors, Tensors, Manifolds and Special Relativity
Thus,d f P (v)can be written as d f P (v)=d f v i ∂
=δ i j , (1.88) which is exactly the answer we were looking for As we have seen previously, ifvis a vector andβa covectorv(β)=β(v)=v i β i withe j (e i )≡e i (e j )=δ i j , therefore it is legitimate to also make the following definition:
Thus, with this choice of bases we can write any tensor field overM A few examples are: l=l i 1 i m (x)d x i 1 ⊗ ã ã ã ⊗d x i m , (1.90) t =t i 1 i n (x) ∂
From now on we shall start using the simplified notation:∂ i ≡ ∂
∂x i Consider the partial derivative of a scalar field∂ j φ(x) Under a change of coordinates x i →x i :
The components ∂ j φ(x) represent a rank one covariant tensor field When examining the derivative of a vector field, denoted as ∂ i v j, we can observe how it transforms under a change of basis, specifically as x i transitions to x i.
Comments on Special Relativity
Before we get to discuss a few Special Relativity topics we need to introduce the Minkowski space The Minkowski spaceM 4is a four dimensional flat manifold.
5 However, there are extensions of the theory which also include torsion.
Its tangent spaceM 4is a four dimensional vector space together with a metric tensor g μν = diag{1,−1,−1,−1} Our coordinates inR 4 are the space-time coordinates 7 x μ ≡(x 0 ,x 1 ,x 2 ,x 3 )≡(t,x i )≡(t, x ) (1.105)
In this article, we denote space-time coordinates using Greek letters μ and ν, ranging from 0 to 3, while Roman letters i and j are reserved for spatial coordinates from 1 to 3 Some authors define the coordinates as x μ ≡ g μν x ν = (x 0, x 1, x 2, x 3) = (t, x i) = (t, -x i) = (t, -x), but it is crucial to recognize that x μ represents coordinates rather than vector or covector components A well-defined vector is expressed as d x μ = (d x 0, d x 1, d x 2, d x 3) = (dt, d x i) = (dt, d x) Consequently, the covector d x μ is properly defined as d x μ ≡ g μν d x ν = (d x 0, -d x 1, -d x 2, -d x 3) = (dt, -d x i) = (dt, -d x).
After all this being said, the consequences of the postulates of the Special Theory of Relativity can be easily translated mathematically into the following sentence.
The Quantity ds 2 =g μν d x μ d x ν =dt 2 −d x 2 =dt 2 −(d x 1 ) 2 −(d x 2 ) 2 −(d x 3 ) 2 (1.109) must be invariant for any inertial observer.This means ds 2 =g μν d x μ d x ν =ds 2 =g αβ d x α d x β (1.110)
In the context of Special Relativity, it's important to note that the metric tensor does not transform, which is why we refer to it as gαβ instead of αβ Consequently, not all transformations are permissible; only those that preserve invariance are allowed.
In Quantum Field Theory, it is conventional to use natural units where the speed of light, c, is set to 1, leading to the temporal coordinate being expressed as ct.
24 1 Vectors, Tensors, Manifolds and Special Relativity the metric tensor These transformations are the ones that belong to the Poincaré Group, which can be aLorentz TransformationΛ μ ν plus aspace-time translation a μ
(wherea μ are constants) x μ → x μ =Λ μ ν x ν +a μ (1.111) The interesting thing about Lorentz transformation is that they don’t depend onx μ :
In Cartesian coordinates, the property of partial derivatives allows them to function as covariant derivatives, eliminating the need to consider Christoffel symbols However, this property does not extend to curvilinear coordinates, where the relationship breaks down While it is possible to generalize to arbitrary coordinates, such an approach proves ineffective in the context of pure Special Relativity This generalization is seamlessly integrated into General Relativity, where the tensor field g μν is defined as a well-behaved function of coordinates, g μν = g μν (x).
(1.111) and (1.112) we find that the Lorentz transformation is also the contravariant law of transformation for tensors
The equations of motion can be expressed in a straightforward, Lorentz-invariant form using Cartesian coordinates, represented as \( \frac{d p^\mu}{d\tau} = f^\mu \) Here, the momentum four-vector is defined as \( p^\mu = m u^\mu = m \frac{d x^\mu}{d s} = m \frac{d x^\mu}{d \tau} = (m\gamma, m\gamma v) \), with the proper time interval \( d\tau = \frac{ds}{c} \), noting that \( c \) is set to 1.
Any other inertial observer (related to the original one by a Lorentz transformation, or a space-time translation) will describe the equations of motion in the same way d p μ dτ = f μ (1.116)
Let us quickly review the most important properties of the Lorentz transformations. From (1.110) we obtain g μν d x μ d x ν =g αβ d x α d x β
From the previous equation we straightforwardly obtain
Contracting with the metric tensorg μρ both sides of the equation, we get to
(Λ − 1 ) ρ σ = g μρ g ασ Λ α μ (1.120) This justifies why in QFT literature many authors use the following notation: Λ μ ν ≡ Λ μ ν , Λ ν μ ≡ (Λ − 1 ) μ ν (1.121)
With this notation, due to property (1.120) one can relate a Lorentz transformation with its inverse bye f f ectivel yraising and lowering indicesas if Λ μ ν wasa tensor.
We will use this notation from now on.
The relation (1.118) can further give as information on the Lorentz transforma- tions Using the matrix notation, it reads Λ T gΛ = g (1.122)
Taking the matrix determinant on both sides of the previous equation we obtain
Taking into consideration the sign ofΛ 0 0 and the value±1 of the determinant one can define four sets of Lorentz transformations Aproper orthochronousLorentz
26 1 Vectors, Tensors, Manifolds and Special Relativity transformation (which can be a rotation or a boost) 8 satisfies det(Λ)=1 andΛ 0 0 ≥1.
An infinitesimal proper orthochronous Lorentz transformation can be expressed as a continuous transformation from unity, represented by the equation Λ μ ν = δ ν μ + Δω μ ν + O(Δω²) Here, δ ν μ denotes the Kronecker delta, which plays a crucial role in the upcoming chapter where we will explore the Lagrangian density formalism and Noether’s theorem By analyzing the previous equation, one can infer that Δω μ ν = -Δω ν μ, leading to the approximation g μν ≈ g αβ (δ α μ + Δω α μ)(δ β ν + Δω β ν).
Thusg βμ Δω β ν = −g αν Δω α μ Defining Δω μν ≡ g μα Δω α ν , (1.126) we finally obtainΔω μν = −Δω νμ , or equivalentlyΔω μ ν = −Δω ν μ , which is the relation we wanted to prove.
The Levi-Civita tensor density in four dimensions behaves like a tensor under proper orthochronous Lorentz transformations, as indicated by the determinant condition det(Λ) = 1 In contrast, other Lorentz transformations, such as parity (Λ μ ν (P) = diag{1, −1, −1,−1}) and time reversal (Λ μ ν (T) = diag{−1, 1, 1,1}), do not maintain this property.
Obviously under these two transformations the Levi-Civita tensor density changes sign.
Any arbitrary Lorentz boost can be broken down into rotations and a boost along a single axis Furthermore, any Lorentz transformation can be expressed in terms of boosts, rotations, parity, and time reversal.
8 More on boosts and relativistic kinematics will be seen in Chap 3.
9 More on the Levi-Civita tensor density in four dimensions will be discussed in Chap 5.
C.W Misner, K.S Thorne y John Archibald Wheeler, Gravitation, W H Freeman, New York
M.P Hobson, G.P Efstathiou, A.N Lanseby, General Relativity, An Introduction for Physicists,
J.N Salas, J.A de Azcarraga, Class notes
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R Abraham, J.E Marsden, T.S Ratiu, Manifolds, Tensor Analysis, and Applications
J.B Hartle, Gravity: An Introduction to Einstein’s General Relativity
This chapter revisits the fundamental concepts of Lagrangian and Hamiltonian mechanics, along with Noether’s theorem We begin by examining a discrete system with N degrees of freedom, where we state and prove Noether’s theorem Subsequently, we extend these concepts to continuous systems and present the generalized formulation of Noether’s theorem Finally, we apply these principles to derive several established results in Quantum Field Theory.
Lagragian Formalism
In this section, we will omit the super-index notation for coordinates or vectors previously discussed, as it is not applicable to the discrete case The action for a discrete system with N degrees of freedom is defined as follows:
The Lagrangian of the system, denoted as L(q_i, q̇_i, t), is expressed as S(q_i) = ∫ t_1^t_2 L(q_i, q̇_i, t) dt, where {q_i} from i = 1 to N represent the generalized coordinates, and q̇_i is the derivative of q_i with respect to time To derive the Euler-Lagrange equations of motion, we consider small variations in the generalized coordinates, maintaining fixed endpoints, represented by δq_i(t_1) = δq_i(t_2) = 0 This leads to the application of the first-order Taylor expansion of the Lagrangian.
≡L(q i ,q˙ i ,t)+δL, (2.3) © Springer International Publishing Switzerland 2016 29 where summation over repeated indices is also understood It is straightforward to demonstrate that thevariationand thedifferentiationoperators commute: δq i (t)=q i (t)−q i (t) ⇒ d dt(δq i )= ˙q i (t)− ˙q i (t)=δq˙ i (t) (2.4) Thus, we obtain the following expression forδL: δL = ∂L
In order to obtain the equations of motion we apply theStationary Action Principle:
For the physical paths, the action must be a maximum, a minimum or an inflexion point This translates mathematically into: δS=δ t 2 t 1 dt L t 2 t 1 dtδL =0 (2.6)
Becauseδq i (t 1 )=δq i (t 2 )=0 the second integral vanishes: t 2 t 1 dt d dt
Therefore, we are left with δS t 2 t 1 dt
∂q˙ i δq i =0, (2.9) for arbitraryδq i Thus the following equations must hold
∀q i These equations are called the Euler-Lagrange equations of motion.
From (2.8) we can also deduce an important aspect of Lagrangians, that they are not uniquely defined:
The Lagrangian L(q i ,q˙ i ,t) and its modified form L(q i ,q˙ i ,t) + d F(q i ,t) dt (2.11) yield identical equations of motion To verify that incorporating a term like d F(q i ,t)/dt into the Lagrangian does not affect the equations of motion, we can apply equation (2.10) to the term d F(q i ,t)/dt.
Next we will present one of the most important theorems of analytical mechanics, a powerful tool that allows us to relate the symmetries of a system with conserved quantities.
Noether ’ s Theorem
There is a conserved quantity associated with every symmetry of the Lagrangian of a system.
Let’s consider a transformation of the type q i →q i =q i +δq i , (2.13) so that the variation of the Lagrangian can be written as the exact differential of some functionF:
Note that here we allowFto also depend onq˙ i (that was not the case for (2.11)) On the other hand, we know that we can writeδL as: δL ∂L
Using the equations of motion, we express the infinitesimal variation δq i as q i = q i + δq i = q i + f i, where |f| is a constant and f is a smooth function As we approach the limit of δq i → 0, it follows that lim δq i → 0 results in q i = q i.
Thus, necessarilyFmust be of the formF=F, and d dt
∂q˙ i f i =F(q i ,q˙ i ,t)+C, (2.19) withC an integration constant We therefore conclude, that the conserved quantity associated to our infinitesimal symmetry is:
Examples
Next, we are going to apply this simple formula to a few interesting cases and reproduce some typical results such as energy and momentum conservation, angular momentum conservation, etc.
Let’s consider an infinitesimal time shift:t →t+ The first order Taylor expansion ofq i andq˙ i is given by: δq i =q i (t+)−q i (t)=q˙ i (t)+O( 2 ), δq˙ i = ˙q i (t+)− ˙q i (t)=q¨ i (t)+O( 2 )= d dt(δq i ) (2.21)
If the Lagrangian does not exhibit an explicit time dependence(∂L/∂t =0)then δL =∂L
2.3 Examples 33 Thus, the conserved quantity is given by the following
∂q˙ i q˙ i −L =E , (2.23) where Eis the associated energy of the system.
Let’s consider a Lagrangian of the formL =T −V, whereT is the kinetic energy of the system and V a central potential In this case the canonical momentum p i defined as p i ≡ ∂L
, (2.24) obeys p i =∂T/∂q˙ i Due to the fact that the potential is central andT =T(q i )the Lagrangian obeys
L( r α + n , v α )=L( r α , v α ), (2.25) withr α the coordinates of the particleαandn an arbitrary spatial direction with
|n| =1 We conclude thatδL =0 Under this spatial translation the coordinates of the particleαtransform the following way: r α →r α =r α + n , (2.26) that is r α j → r α j =r α j +n j , (2.27) with j = 1,2,3 Therefore f j = n j The conserved quantity is straightforwardly obtained
∂q˙ α j n j α p α j n j α p α n=P n , (2.28) for an arbitraryn Thus, the constant associated to this transformations is the total momentumPof the system.
In this analysis, we examine a Lagrangian exhibiting similar characteristics to the previous example During an infinitesimal rotation, the transformation is defined as \( r^\alpha = r^\alpha - n \times r^\alpha \) and \( r^\alpha_j = r^\alpha_j + j^{km} n_m r^\alpha_k \) As established, the variation in the Lagrangian, denoted as \( \delta L \), remains zero It is evident that the force component can be expressed as \( f^j = j^{km} n_m r^\alpha_k \).
(where j km is the totally antisymmetric three-dimensional Levi-Civita tensor den- sity) The conserved quantity is therefore (remember that summation over all repeated indices is understood):
Again, this holds for an arbitraryn, thus, the conserved quantity is the total angular momentumLof the system.
For this last example we shall consider the same type of Lagrangian as in the previous cases A Galileo transformation reads r α →r α =r α +v t, (2.31) withva constant velocity vector, therefore: ˙ r α → ˙r α = ˙r α +v (2.32)
Under these transformationsδL =δT Let’s calculateT explicitly:
Considering an infinitesimal transformationv= nwith| | 1 and ignoring terms ofO( 2 )we have δL =δT = d dt(M R n ) (2.34)
2.3 Examples 35 The conserved quantity is then given by:
C α p α j n j t−M R n=( P t−M R ) n , (2.35) for an arbitraryn The conserved quantity associated to this transformation is then
Hamiltonian Formalism
We define the Hamiltonian functional of a physical system as
H(q i ,p i ,t)≡p i q˙ i −L, (2.36) where p i is called the canonical conjugated momentum p i ≡ ∂L
, (2.37) as it was already introduced in (2.24) If the Euler-Lagrange equations (2.10) are satisfied then: ˙ p i = ∂L
The Hamiltonian equations of motion are obtained just as before by applying the principle of the stationary action: δS t 2 t 1 dtδL t 2 t 1 dtδ(p i q˙ i −H) t 2 t 1 dt δp i q˙ i +p i δq˙ i −∂H
This must hold for arbitraryδp i yδq i , therefore, the Hamiltonian equations of motion are simply given by: ˙ q i = ∂H
If the Hamiltonian exhibits an explicit time dependence, it can be easily related to the time dependence of the Lagrangian d H dt = d dt(p i q˙ i −L)
Therefore we get to the following simple relation in partial derivatives:
Continuous Systems
In our previous discussions, we focused on discrete systems with a finite number of degrees of freedom (N) Now, we shift our attention to systems characterized by an infinite number of degrees of freedom (N → ∞), which necessitates a change in our approach Instead of using discrete coordinates (q i), we must adopt a continuous field defined at every point in space and varying over time, represented as q i (t) → φ(x, t) ≡ φ(x μ) ≡ φ(x) This transition is essential as it incorporates both spatial and temporal dependencies in our analysis.
In this article, we utilize the compact relativistic notation introduced in Chapter 1, assuming that the partial derivatives of the fields are Lorentz (or Poincaré) covariant quantities, represented as ∂μφ(x).
It will also be useful to define the following contravariant quantity
We focus on Lagrangians that remain invariant under both space-time translations and Lorentz transformations, adhering to the principles of the Poincaré group Consequently, these Lagrangians do not explicitly depend on the space-time coordinates \( x^\mu \) The most general form of such a Lagrangian, which incorporates these characteristics, can be expressed accordingly.
The Lagrangian density, denoted as L, is a functional that depends on multiple fields, represented by {φ i } with i ranging from 1 to M Consequently, the action can be expressed as an integral of this Lagrangian density, encapsulating the dynamics of the system.
Just as in the discrete case, in order to obtain the Euler-Lagrange equations of motion we will consider small variations of the fields, keeping the extremes fixed φ (x)=φ i (x)+δφ i (x); δφ i (x 1 )=δφ i (x 2 )=0 (2.49)
While this may not encompass all scenarios, our focus on applying field theory specifically to Special Relativity necessitates a concentration on this particular case.
Under these variations, we define δL φ i (x),∂ μ φ i (x)
, (2.50) thus, we obtain the following δL= ∂L
In the context of continuous systems, the variation and derivation operators commute, similar to previous findings By applying the principle of Stationary Action, we derive the Euler-Lagrange equations, which are fundamental in understanding the dynamics of such systems.
∂[∂ μ φ i (x)] δφ i (x)=0, (2.52) for arbitraryδφ i (x), therefore, the equations we are looking for take the form
∀φ i ,i = 1, ,M Let’s now take another look at (2.52) Because δφ i (x 1 ) δφ i (x 2 )=0, we have found that x 2 x 1
Thus, if we consider an arbitrary functional of the formb μ φ i (x)
We conclude that a Lagrangian density is not uniquely defined Similar to the discrete case, one can always add a functional of the form∂ μ b μ φ i (x) without altering the equations of motion Therefore
, (2.56) render the same equations of motion.
Hamiltonian Formalism
We define the Hamiltonian density as
H(π i (x),φ i (x), ∇ φ i (x)) ≡ ˙φ i (x)π i (x)−L, (2.57) whereφ˙ i (x)≡∂ t φ i (x)andπ i (x)is the canonical momentum associated to the field φ i (x): π i (x)≡ ∂L
The action can be written in terms of the Hamiltonian density as
When applying the principle of the stationary action we obtain δS x 2 x 1 d 4 x δπ i φ˙ i −∂H
=0, (2.60) for arbitraryδπ i andδφ i Thus the equations of motion simply read φ˙(x)= ∂H
∂(∂ k φ(x)) (2.61) where∂ k are the spatial derivatives (k=1,2,3).
Noether ’ s Theorem (The General Formulation)
Until now we have only introduced a global variation of a field, which is defined as the variation of the shape of the field without changing the space-time coordinates x μ : δφ i (x)≡φ i (x)−φ i (x) (2.62)
In addition to the previously discussed variations, we introduce the concept of local variation, which refers to the differences observed in fields at the same space-time point but within two distinct coordinate systems This is mathematically expressed as δφ i (x)≡φ i (x)−φ i (x) We then consider a continuous space-time transformation, represented as x μ → x μ = x μ + x μ, which can either be a proper orthochronous Lorentz transformation or a space-time translation At first order in x, the local variation δφ i (x) can be simplified to δφ i (x) = φ i (x)−φ i (x).
We therefore, have found the following relation between δφ(x)andδφ¯ (x)for an infinitesimal transformation of the type (2.64): δφ i (x)=δφ i (x)+ ∂ μ φ i (x) x μ (2.66)
We can draw the following conclusion Ifφ i (x )=φ i (x)(which is in general the case for a scalar field; it is also the case for spinor fields under space-time translations) then δφ i (x)= − ∂ μ φ i (x) x μ (2.67)
Noether's Theorem presents an equivalent transformation method for fields, allowing for a transformation of the coordinates to be expressed as φ i (x) → φ i (x) = φ i (x - x) This formulation highlights the relationship between symmetries and conservation laws in physics.
Let us now deduce how the Lagrangian transforms under these type of variations.
In order to keep the notation short, we shall introduce the following short-hand notations:
∂x μ Keeping only terms up toO(x)we can calculateδL(x)under (2.64): δL(x)=L (x )−L(x)
≈δL(x)+ ∂ μ L(x) x μ , (2.70) where we have introduced the following notation
∂[∂ ν φ i (x)]∂ μ ∂ ν φ i (x) (2.71) Also we have used the following approximation: δ[∂ μ φ i (x)] ≈∂ μ [δφ i (x)] − ∂ ν φ i (x)∂x ν
For the last equality we have used that for a Lorentz (Poincaré) transformation
The variation operator δ¯ commutes with the derivation operator under Lorentz (Poincaré) continuous transformations, as previously noted in Chapter 1 Consequently, we can express δ¯L in a manner analogous to equation (2.66), resulting in the following expression: δL(x) = δL(x) + ∂μL(x) xμ.
The transformation of the action is crucial in understanding symmetries within a system When a transformation preserves the equations of motion, it signifies a symmetry of the system In this context, the action S undergoes a transformation represented as S → S, highlighting the invariance of the action under such symmetries.
=S+ d 4 x∂ μ b μ (x), (2.74) so thatδS = δS (thus generating the same equations of motion) Introducing the Jacobian matrix we have
This must hold for all space-time volumes , therefore:
The determinant of the Jacobian matrix is equal to 1 for a proper orthocronous Lorentz transformation or a space-time translation, thus δL(x)−∂ μ b μ (x)=0 (2.77)
Inserting the explicit form ofδLfrom (2.51) in the last expression, we obtain ∂L(x)
Using the Euler-Lagrange equations of motion we finally get to the conservation law we were looking for
Noether's Theorem provides a general formulation that applies to continuous space-time transformations, as well as transformations that involve only field variations without altering the space-time configuration In scenarios where only field variations are considered, the conserved Noether current can be expressed by setting x μ = 0 in the relevant equations.
As it is usual, we can also define aconserved charge Qassociated to the conserved current j μ as:
Q d 3 x j 0 , dQ dt d 3 x∂ 0 j 0 = − d 3 x ∇j=0 (2.81)Next, we shall take a few illustrative examples.
Examples
In the context of infinitesimal space-time translations, we consider the transformation \( x^\mu \rightarrow x^\mu = x^\mu - \mu \), where \( \mu \) represents real constants For both scalar and spinor fields, the fields remain invariant, expressed as \( \phi_i(x) = \phi_i(x) \), leading to \( \delta \phi_i(x) = 0 \) Consequently, our Lagrangians also remain unchanged, indicated by the relation \( \mathcal{L}(x) = \mathcal{L}(x) \) This analysis allows us to conclude that \( \partial_\mu b^\mu = 0 \).
We can thus, eliminate theb μ term from (2.80) and the conserved current is simply given by: j μ = ∂L
The conservation law∂ μ j μ =0 holds for any arbitrary constants ν , therefore we actually have four conserved currents:
∂(∂ μ φ i )∂ ν φ i −Lg μν , (2.84) withT μν the four-momentum tensor The conserved Noether charges are then given by
=(H, P ), (2.85) where we have used (2.46) As we can see, the conserved charges are the Hamiltonian and three-momentum operators.
Consider a Lagrangian that depends on the fieldsφ 1andφ 2withφ 1=φandφ 2=φ †
By applying an infinitesimal global phase redefinition to the field φ(x) as φ(x) = e^(-iθ)φ(x), where θ is constant and does not vary with space-time coordinates, we derive the variations δφ(x) = -iθφ(x) and δφ†(x) = iθφ†(x).
In the context of this transformation, we can simplify the equation by setting x μ = 0, leading to δL = ¯δL = ∂ μ b μ When focusing solely on the free Dirac or Klein-Gordon Lagrangians, we find that δL = 0 = ∂ μ b μ, allowing us to eliminate b μ from the equation Consequently, the conserved current is expressed as j μ = ∂L.
∂(∂ μ φ † )iφ † θ, (2.88) for an arbitraryθ Thus, redefining the current without theθ multiplying term we find
In particular, for the free Dirac LagrangianL D = ¯ψ(x)(iγ μ ∂ μ −m)ψ(x)we obtain the well known result:
Consider the following infinitesimal proper orthochronous 3 Lorentz transformation: x μ → x μ =x μ +ω μ ν x ν , (2.91) withω μ ν = −ω ν μ real constants Definingω μν ≡g μα ω α ν , it is easy to show that the field transformation reads φ i (x )=φ i (x)+1
4[γ μ ,γ ν ]for spinorial 4 fields and, ( μν i ) =0 for scalar fields (no summation over the “i” index must be understood in (2.92) nor in the following expression) Using (2.66) we easily find: δφ i (x)= 1
Our Lagrangians maintain Lorentz invariance, leading to δ¯L = 0, which allows us to eliminate b μ in equation (2.80) similarly to prior examples Consequently, the expression for the conserved current is given by j μ = ∂L.
2L(g μα x β −g μβ x α )ω αβ , (2.94) for arbitraryω αβ Thus, we obtain
3 See Chap.1 for more details.
Chapter 5 provides detailed information on spinor algebra and includes the proof of the conservation law for the angular momentum pseudo tensor \( J^{\mu}_{\alpha\beta} \) It is important to note that summation over all repeated indices is implicitly understood in the previous expression.
A Pich, Class Notes on Quantum Field Theory http://eeemaster.uv.es/course/view.php?id=6
W Greiner, J Reinhardt, D.A Bromley (Foreword), Field Quantization
E.L Hill, Hamilton’s principle and the conservation theorems of mathematical physics Rev Mod. Phys 23, 253
J.A de Azcárraga, J.M Izquierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics Cambridge Monographs in Mathematical Physics
J.A Oller, Mecnica Terica, http://www.um.es/oller/docencia/versionmteor.pdf
M Kaksu, Quantum Field Theory: A Modern Introduction
D.V Galtsov, Iu.V Grats, Ch Zhukovski, Campos Clásicos
5 I am calling it pseudo tensor because it is obviously not invariant under translations!.
Relativistic Kinematics and Phase Space
This article provides a comprehensive list of essential formulas for calculating relativistic collisions and decays, covering one-to-two and one-to-three body decays, as well as two-to-two scattering processes in both center of mass and laboratory frames It also includes simplified general formulas for one, two, and three-body Lorentz invariant phase space While no explicit calculations are performed, readers are encouraged to replicate the results presented.
Conventions and Notations
For all the calculations in this book we will adopt themostly minusMinkowski metric g= diag{1,−1,−1,−1} If a particle has a relativistic three-momentump=γm v, then we define the contravariant four-momentum vector as: p μ =(E, p )=(γm,γm v ) (3.1)
Thusp 2 ≡p μ p μ =m 2 ; here we have takenc=1 as usual We will also be needing the Kallen lambda function defined as λ(x,y,z)≡x 2 +y 2 +z 2 −2x y−2yz−2x z (3.2)
An essential aspect of understanding Lorentz boosts is to correctly address the signs in the transformations We adopt a passive transformation approach, considering two inertial reference frames, O and O', with parallel axes Frame O' moves with a constant velocity v = vẑ (where v > 0) along the positive z-axis relative to frame O Consequently, an observer in frame O perceives frame O' as moving with a velocity of -v This relationship can be mathematically expressed as xμ = Λμν xν.
Fig 3.1 Two inertial reference frames O and O with parallel axes and relative constant velocity
A moving object with four-momentum \( p^\mu \) in reference frame \( O \) is represented as \( p^\mu = \Lambda^{\mu\nu} p_\nu \) by an observer in frame \( O' \) This transformation is crucial for analyzing the relationship between the center of mass collision angle and the laboratory frame angle in the process \( a + b \rightarrow 1 + 2 \) Additionally, it will be applied in Chapter 4, where we explore three and four-body kinematics and phase-space through angular observables The inverse transformation can be derived by substituting \( -\gamma v \) with \( +\gamma v \) in equation (3.4).
Process: a ! 1 þ 2
In the center of mass (CM) reference frame we have the following configuration:
1 −p a p 2 p a μ =(m a , 0 ), p 1 μ =(E 1 ,−p ), p 2 μ =(E 2 , p ) (3.5) where E 1,E 2and|p|are given by
2m a λ 1 / 2 (m 2 a ,m 2 1 ,m 2 2 ) (3.6) The threshold value ofs(minimum value ofsfor the on-shell production of particles
Process: a ! 1 þ 2 þ 3
In the CM frame we have the following configuration: a 2 3 p 2 p 1 p 3
Becausep 1 +p 2 +p 3 = 0 the process takes place in the same plane Using the angles shown in the previous figure:
The standard approach for the definition of the Lorentz invariant kinematical variables is given by: t 1≡s 23 ≡(p a −p 1 ) 2 =(p 2+p 3 ) 2 , t 2≡s 13 ≡(p a −p 2 ) 2 =(p 1+p 3 ) 2 , t 3≡s 12 ≡(p a −p 3 ) 2 =(p 1+p 2 ) 2 (3.9)
Thet i invariants satisfy the following relation: i t i =m 2 a + i m 2 i (3.10)
It is easy to show that they also satisfy: t i =(p a −p i ) 2 = p a 2 +p 2 i −2p a ãp i =m 2 a +m 2 i −2m a E i (3.11) Therefore we can expressE i and|p i |in terms of Lorentz invariant quantities as
The threshold values forsands i j are denoted ass t h ands i j t h and: s≡p 2 a =m 2 a s t h =(m 1+m 2+m 3 ) 2 , s i j =t k s i j t h =(m i +m j ) 2 =t k t h , (i = j =k) (3.13)
From (3.12) and (3.13) is easy to deduce that the maximum value of the energy of the particleiin the CM frame is
Process: 1 þ 2 ! 3 þ 4
In the CM reference frame we have the following configuration:
We can define the following Lorentz invariant quantities: s≡(p 1+p 2 ) 2 =(p 3+p 4 ) 2 , t≡(p 1−p 3 ) 2 =(p 2−p 4 ) 2 , u≡(p 1−p 4 ) 2 =(p 2−p 3 ) 2 (3.17) One can easily check that: s+t+u i m 2 i (3.18)
3.4 Process: 1 + 2 → 3 + 4 51 The CM energies and momenta in terms ofsandm i are given by:
The CM collision angle, shown in the previous figure, in terms of Lorentz invariant quantities is given by cosθ= s(t−u)+(m 2 1 −m 2 2 )(m 2 3 −m 2 4 ) λ 1 / 2 (s,m 2 1 ,m 2 2 )λ 1 / 2 (s,m 2 3 ,m 2 4 ) (3.20)
In the laboratory (L) reference frame we consider that the particle 2 is at rest, thus we have the following configuration:
1 p 1 L 2 θ L p 3 L p 4 L p μ 1 , L =(E 1 L , p L 1 ), p 2 μ , L =(m 2 , 0 ), p μ 3 , L =(E 3 L , p L 3 ), p 4 μ , L =(E 4 L , p 4 L ) (3.21) The energies and momenta in the L reference frame are given by:
The expression forθ L scattering angle can be easily related to the CM one with the following expression tanθ L = m 2|p |sinθ
|p|E 3+ |p |E 2cosθ, (3.23) where all the quantities from the RHS of the equation are expressed in the CM reference frame.
Next we present simple expressions for the Lorentz invariant phase space for one,two and three final state particles in terms of the previously discussed kinematical variables.
Lorentz Invariant Phase Space
The standard definition for the Lorentz invariant phase space forNfinal state particles is given by d Q N ≡ S n
2E l δ ( 4 ) (P i −P f ), (3.24) where S n =1/n!is the corresponding symmetry factor (withnthe number of final state identical particles) and whereP i ,P f are the initial and final state total momenta of the system.
We have the following simple expression: d Q 1 = 2π d 3 p a
We can easily integrate the previous expression and obtain d Q 1 = 2π δ(s−m a ) (3.26)
We have the following expression: d Q 2 = S n
If we choose to work in the CM reference frame we simply obtain: d Q 2 = S n
In the context of scattering processes, the final state center-of-mass momentum is represented by |p|, as indicated in equation (3.19) The differential solid angle is defined as d = dcosθ dφ When the scattering matrix is independent of the phase space, we can simplify our calculations by performing the angular integration.
In certain situations, it is more straightforward to represent the scattering matrix using the variables s, t, and the associated masses, while also expressing dcosθ in relation to these variables Notably, when examining equation (3.18) for a collision at a constant s, which is commonly encountered in collider experiments (where ds = 0), we observe that dt is equal to -du (3.30).
Therefore, using (3.20) we can expressdcosθas: dcosθ= 2s dt λ 1 / 2 (s,m 2 1 ,m 2 2 )λ 1 / 2 (s,m 2 3 ,m 2 4 ) (3.31) The integration limits are easy to obtain using the same expression (3.20):
(cosθ) min =s(2t min +s− m 2 i )+(m 2 1 −m 2 2 )(m 2 3 −m 2 4 ) λ 1 / 2 (s,m 2 1 ,m 2 2 )λ 1 / 2 (s,m 2 3 ,m 2 4 ) = −1 (3.33) Thus, we simply get: t max = 1
Ifm 1=m 2orm 3=m 4these expressions get a lot simpler If we are dealing with a decay 1→3+4, then we must setm 2=0 ands=m 2 1
The Lorentz invariant phase space corresponding to three final state particles is given by: d Q 3 = S n
In the CM reference frame we obtain the simple and compact expression d Q 3 = S n
128π 3 s ds 23 ds 13 (3.37) The integration limits are given by: s 13 mi n =(m 1+m 3 ) 2 , s 13 max =(√ s−m 2 ) 2 , (3.38) and s mi n 23 = 1
, (3.40) and where we have defineds≡(p 1+p 2+p 3 ) 2 =(p a +p b ) 2 as the CM invariant energy (fixed for colliders) If it’s a decay i.e.,a→1+2+3 then we must simply sets=m 2 a
In conclusion, we will present the decay rate and cross-section formulas based on our established conventions For a specific process represented by the transition matrix M, the decay rate can be calculated using these formulas.
In the equation σ(a+b→1+ããã+N) = 1, the parameters (2j_a + 1) λ_a, λ_1, , λ_N represent the spin and polarization of the initial state particle and the polarizations of the final state particles, respectively When analyzing the average over initial state polarizations and summing over final state polarizations, we must consider specific cases: for a polarized initial state, the summation over λ_a and the averaging factor 1/(2j_a + 1) are omitted, while for polarized final states, the summation over λ_1, λ_2, etc., should be excluded.
A Pich, Class Notes on Quantum Field Theory http://eeemaster.uv.es/course/view.php?id=6
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In this chapter, we explore an alternative method for defining kinematical variables and phase space, focusing on invariant masses and angles of particle pairs in their center of mass reference frame This approach is particularly useful for analyzing rare decays, such as B → K∗ We will detail the kinematics and phase space considerations for one-to-three and one-to-four body decays.
Three Body Angular Distributions
In the decay process represented by A(k) → b(p) + c(p1) + d(p2), the momenta of the particles involved are denoted as k, p, p1, and p2 By selecting a specific pair of particles, namely c and d, we can analyze the decay as two sequential processes The initial decay is defined by this separation, allowing for a clearer understanding of the interactions involved.
In the analysis of particle decay, the process A(p) decays into a real particle b(p) and a virtual set of particles cd(q), where q is defined as the sum of momenta p1 and p2 The invariant mass (q²) of the system is determined by the two real particles c(p1) and d(p2) In the center of mass (CM) reference frame of particle A, we can observe a specific distribution of momenta pμ.
A = (q 0 , 0, 0, |q|), (4.2) where we have supposed thatbmoves along the negativezˆaxis and wherep 0 ,q 0 and|q|are given by (see Chap.3for details) © Springer International Publishing Switzerland 2016 57
It is straightforward to obtain the pãq invariant product pãq = 1
Going to the CM reference frame of thecd system we have the following configu- ration ˆ x c d p 1 p 2 θ z ˆ
In this reference framepandq are given by p μ cd = (p 0 , 0, 0, −|p |), q μ cd = ( q 2 , 0, 0, 0) (4.7)
Using the previously calculatedpãqproduct and the expressionp 2 =m 2 b =(p 0 ) 2 −
The four momenta p 1andp 2are given by p 2 μ cd = (p 2 0 , |p 2 |sinθ, 0, |p 2 |cosθ), (4.10) p 1 μ cd = (p 1 0 , −|p 2 |sinθ, 0, −|p 2 |cosθ), (4.11) where p 0 2 = 1
It is straightforward to obtain the following invariant products: p 1 ãq = 1
For the productpãp 2we simply havepãp 2= p 0 p 2 0 +|p | |p 2 |cosθwhich explicitly reads pã p 2 = 1
To calculate the squared transition matrix for a specific process, one can utilize the invariant products to express all variables in terms of the masses \( m_i \), the invariant squared mass of the \( cd \) system \( q^2 \), and \( \cos \theta \) It is essential to accurately define the corresponding phase space using these variables A particularly useful formula in this context is \( d^4 k \delta(k^2 - m^2) d^3 k \).
2k 0 , (4.17) where k 0 > 0 The generic expression for the three-body phase space is given by(see Chap.3)
2π × A × B, (4.18) where we have definedAandBthe following way
We will now manipulate these expressions to obtain the appropriate results in terms of the desired variables For the first term we have
To derive the final expression, we start with the equation \(8\pi m^2 A \lambda^{1/2} m^2 A, m^2 b, q^2 dq^2\) (4.21) We applied equation (3.28) and performed integration over the solid angle, simplifying the notation by omitting the integral symbol in the last line The next step involves computing \(B\), where we again utilize equation (3.28) and integrate over \(\phi\), maintaining generality in our approach.
Finally, the expression of the differential three body phase space that we were looking for has the form d Q 3 = S n
Fig 4.1 Higgs-like scalar particle φ decaying into a pair of W gauge bosons k
In the analysis of the studied process, if it resembles the scenario depicted in Fig 4.1, then \( q^2 \) represents the momentum of the virtual W boson Conversely, if the mass \( m_\phi \) exceeds \( 2m_W \), we encounter two on-shell particles instead of one real and one virtual W boson, leading to \( q^2 = m_W^2 \) Consequently, the squared transition matrix will incorporate a squared propagator of this nature.
1 q 2 −m 2 W 2 +m 2 W Γ W 2 (4.24) which is regulated by a Breit-Wigner term, as a first order approximation (for more details see Chaps.7and12) Working in the narrow width approximation we have Γ lim W → 0
The Dirac delta function can be integrated back into the phase space definition, allowing us to simplify the expression for dQ3 Consequently, we find that dQ3 equals 1.
Four Body Angular Distributions
Imagine that in the previous processbis not a real particle, but represents a virtual set of two particles (c˜d˜) i.e., the process is given by
We can select two pairs of particles, represented by \(c\) and \(d\), and another pair denoted as \(c\) and \(d\), allowing us to decompose our decay process into sequential decays This approach mirrors the previous case, with the initial decay defined by the equation provided.
The decay of particle A into two virtual sets of particles, cd(q) and c˜d(˜p), is described by the equations q = p1 + p2 and p = ˜p1 + ˜p2 These particles are characterized by their invariant squared masses, q² and p² In the center of mass (CM) reference frame of particle A, the four-momenta p and q are represented as pμ.
A = (q 0 , 0, 0, |q|), (4.28) where, again, we have supposed thebmoves along the negativezˆaxis and where, this time p 0 = 1 2m A m 2 A +p 2 −q 2
Thus, the invariant product pãq is simply pãq = 1
Figure 4.2 illustrates the complete distribution of angles and momenta in the center of mass (CM) reference frame of the system In this frame, the four-momenta are expressed as p μ cd = (p 0, 0, 0, −|p|) and q μ cd = (q 2, 0, 0, 0), where p 0 is equal to 1.
Fig 4.2 Four body decay angular distribution
The four-momenta \( p_1 \) and \( p_2 \) are expressed as \( p^\mu_2 = (p^0_2, |p_2| \sin \theta_d \cos \chi, |p_2| \sin \theta_d \sin \chi, |p_2| \cos \theta_d) \) and \( p^\mu_1 = (p^0_1, -|p_2| \sin \theta_d \cos \chi, -|p_2| \sin \theta_d \sin \chi, -|p_2| \cos \theta_d) \), with \( p^0_2 \), \( p^0_1 \), and \( |p_2| \) derived from the same equations as in the previous section Consequently, the scalar products yield consistent results, confirming that \( p^\mu_1 p_\mu^2 = 1 \).
Boosting 1 p μ 2 into the CM reference frame of Awe obtain p 2 μ
|p 2 |sinθ d sinχ, γvp 0 2 +γ|p 2 |cosθ d ), (4.39) with v = |q| q 0 , γ = q 0 q 2 (4.40)
1 See Chap.3 for the explicit expression for Lorentz boosts.
A similar expression can be found forp μ 1 , however it won’t be needed.
We now move on and analyse the kinematics in the CM reference frame of the ˜ cd˜system We have p μ ˜ c d ˜ = ( p 2 ,0,0,0), q μ ˜ c d ˜ = (q 0 ,0,0,|q |), (4.41) with q 0 = 1
Forp˜1andp˜2we simply have the following expressions ˜ p 2 μ ˜ c d ˜ = (p˜ 0 2 ,| ˜p 2 |sinθ d ˜ ,0,| ˜p 2 |cosθ d ˜ ), (4.44) ˜ p 1 μ ˜ c d ˜ = (p˜ 0 1 ,−| ˜p 2 |sinθ d ˜ ,0,−| ˜p 2 |cosθ d ˜ ) (4.45) The components of the four-momenta are again straightforward to obtain ˜ p 0 2 = 1
We can now calculate, as in thecd system, similar invariant scalar products ˜ p 1ãp = 1
2(p 2 −m 2 c ˜ −m 2 ˜ d ) qã ˜p 1 = pãq−qã ˜p 2 (4.49) with qã ˜p 2 = 1
Using momentum conservation we can also calculate products of the type p i ã ˜p j
We find p 1ã ˜p 2 = qã ˜p 2−p 2ã ˜p 2 p 2ã ˜p 1 = pã p 2−p 2ã ˜p 2 p 1ã ˜p 1 = pã p 1−qã ˜p 2+p 2ã ˜p 2 (4.51)
The remaining product we need to calculate is p 2ã ˜p 2 Boosting p˜ 2 μ into the CM frame of Awe obtain ˜ p μ 2
Using the expressions (4.39) and (4.52) we can now evaluate the p 2ã ˜p 2invariant product in the CM reference system ofA p 2ã ˜p 2 = (γp 0 2 +γv|p 2|cosθ d )(γ˜ p˜ 0 2 − ˜γv˜| ˜p 2 |cosθ d ˜ )
The squared transition matrix can be expressed using squared invariant masses, p², q², the masses mᵢ, and the three angles θₗ, θₗ˜, and χ Additionally, employing the helicity amplitudes approach allows for a systematic grouping of the transition matrix terms, leading to the derivation of angular coefficients.
2 For more details check Furhter Reading.
Last we will calculate the needed phase space According to the definition given in the previous chapter, the generic expression for the four-body phase space is given by d Q 4 = S n
(2π) 2 × A × B × C, (4.55) where we have definedA,BandCthe following way
2p˜ 0 2 δ ( 4 ) (p− ˜p 1− ˜p 2 ) (4.58) Using the same techniques as in the previous section we can manipulateA
We have utilized equation (3.28) and integrated it over the solid angle for simplicity, omitting the integral symbols Additionally, for variable B, we can integrate over the azimuthal angle without losing generality, leading to our results.
Finally, forCwe must keep both angles
Thus, our final expression of the four-body phase space in terms of angular variables is given by d Q 4 = S n
In scenarios akin to the previous section, where p2 or q2, or both, represent the momenta of virtual particles that may enter the on-shell region, we can apply the narrow width approximation This allows us to reabsorb the Dirac delta function(s) into dQ4 and perform the integration without additional complications, mirroring the approach taken earlier.
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This chapter covers the fundamental aspects of Dirac spinor algebra essential for fermion calculations, including the commuting and anti-commuting relations of Dirac matrices It provides a foundation for calculating spinor traces, which can be applied to more complex trace calculations involving these matrices Additionally, the transformation of spinors under Lorentz transformations is discussed, along with bilinear covariants The chapter concludes with a brief comparison of Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD) in the context of a simple process.
Dirac Matrices
We shall start this chapter by introducing the well known Dirac equation It reads i∂/−m ψ(x) = 0 , (5.1) where we have used the Feynman slashed notation∂/≡γ μ ∂ μ Theγ μ Dirac matrices obey the Clifford algebra
Thus, it is straightforward to deduce γ 0 2
In the context of quantum field theory, the identity matrix in four dimensions, denoted as I₄, plays a crucial role in the formulation of gamma matrices The gamma matrices, represented as γ₀ and γᵢ, satisfy specific relations, including γ₄ = γ₀ and γᵢ = -γᵢ Additionally, one can derive a new matrix, γ₅, defined as γ₅ ≡ iγ₀γ₁γ₂γ₃, which is equal to -i These matrices are essential for understanding the algebraic structure of Dirac equations and the behavior of fermions in high-energy physics.
4 μναβ γ μ γ ν γ α γ β , (5.4) © Springer International Publishing Switzerland 2016 69 with 0123 = − 0123=1, and the second one σ μν ≡ i
It is easy to check that γ 5 σ μν = i
In the previous expression we have definedσ αβ ≡g ρ α g δβ σ ρδ Using the Clifford algebra one can easily prove γ 5 † = γ 5 , σ μν† = γ 0 σ μν γ 0 (5.7) and
Thus, using the Clifford algebra and the previously introducedσ μν matrix, one can writeγ μ γ ν as γ μ γ ν = 2g μν −γ ν γ μ = g μν −iσ μν (5.9)
With all the previously introduced concepts one can calculate any tensor contraction i.e., γ μ γ μ =4I 4 , γ μ γ ν γ μ = −2γ ν , γ μ /pγ μ = −2/p, (5.10) etc., or products of the type
/ pq/ = pãq−iσ μν p μ q ν , (5.11) or tensor identities like γ μ γ ν γ ρ +γ ρ γ ν γ μ = 2 g μν γ ρ −g μρ γ ν +g νρ γ μ
The following results involving the Levi-Civita tensor density 1 will also turn out to be useful: αβμν αβσρ = 2 δ ρ μ δ σ ν −δ μ σ δ ρ ν
1 See Chap.1 for the definition of tensor density.
Some authors prefer using the notation μ ν in place of the traditional Kronecker delta tensor δ ν μ, viewing it as a mnemonic device for adjusting tensor indices This notation serves to indicate that raising an index of the Kronecker delta tensor effectively transforms it into the metric tensor, expressed as g μα δ α ν = g μν.
Dirac Traces
When computing transition matrix elements involving fermions, one usually needs to calculate traces of products of Dirac matrices One could find useful the following generic trace properties
Tr{A} = Tr{A T }, Tr{A+B} = Tr{A} +Tr{B}, (5.16) whereT stands for transposed, and
The cyclic property of traces allows for the calculation of any trace involving Dirac matrices, as expressed by the formula Tr{A N − 1 A N A 1 A N − 2} Utilizing previous findings, we can derive various interesting and straightforward results related to these matrices.
Tr{γ 5} = Tr{γ μ } = Tr{σ μν } = 0 (5.18) For an odd number of Dirac matrices one finds (k∈N)
Tr{γ μ 1 γ μ 2k + 1 } = 0, (5.19) therefore, the following equality also holds
Tr{γ 5 γ μ 1 γ μ 2k + 1 } = 0 (5.20) Other interesting results are
With the tools given here, one can calculate more lengthy and complicated traces involving Dirac matrices with no further complication.
Spinors and Lorentz Transformations
In this section we shall study the transformation of a spinor fieldψ(x)under a Lorentz transformation x μ → x μ = μ ν x ν (5.26)
In the context of spinor fields, the transformation can be expressed as ψ(x) → ψ'(x) = S(Λ)ψ(x), where S(Λ) is a linear operator that implements the Lorentz transformation on the spinor field ψ(x) To further explore the characteristics of this operator, we note that the transformation can also be represented as ψ'(x) = S(Λ)ψ(x) = S(Λ)ψ(-x) = S(Λ)S(-Λ)ψ(x).
We have established that S(−1) = S − 1, and now we will explore the implications of this operator on the Dirac matrices A fundamental principle of both Special and General Relativity is that all equations of motion must remain consistent across all reference frames Consequently, the Dirac equation, expressed as iγ μ ∂ μ − m ψ(x) = 0, must be reformulated using primed variables, resulting in iγ ν ∂ ν − m ψ(x) = 0, where ∂ ν ≡ ∂/∂x ν.
Thus, under a Lorentz transformation (5.26) the Dirac gamma matrices are related through γ ν = S() ν μ γ μ
Let’s now try to find and explicit expression for S()for a proper orthochronous Lorentz transformation (1.124) μ ν = δ ν μ + ω μ ν +O(ω 2 ) (5.33)
We write down the ansatz
(withω μν ≡g αν ω μ α ) whereb μν must be 4×4 antisymmetric matrices (in the μ,ν indices due to the fact that ω μν is also antisymmetric) Thus, the relation (5.32) takes the form γ ν I 4+b σρ ω σρ γ ν +ω ν μ γ μ I 4−b αβ ω αβ
Keeping only terms up toO(ω)we obtain ω ν μ γ μ = γ ν b αβ ω αβ −b σρ ω σρ γ ν
On the other hand we can writeω ν μ γ μ as ω ν μ γ μ = 1
Comparing the last two equations (for arbitraryω αβ ) we find
It is easy to check that this relation is satisfied for b αβ = −i
Thus, the operator that implements an infinitesimal proper orthochronus Lorentz transformation on the spinor field is given by
4σ μν ω μν + , (5.41) with its inverse given by
4σ μν ω μν + ã ã ã = γ 0 S † γ 0 (5.42) For a parity transformation (1.127) we simply find
Definingψ(x)≡ψ † (x)γ 0 , we obtain that under a Lorentz transformation this spinor transforms as ψ (x ) = ψ(x)S − 1 () (5.44)
Restricting our transformations to proper orthochronous Lorentz boosts and parity we can construct the following bilinear covariants ψ (x )ψ (x )=ψ(x)ψ(x) → scalar, (5.45) ψ (x )γ 5 ψ (x )()ψ(x)γ 5 ψ(x) → scalar density, (5.46) ψ (x )γ μ ψ (x )= μ ν ψ(x)γ ν ψ(x) → vector, (5.47) ψ (x )γ 5 γ μ ψ (x )() μ ν ψ(x)γ 5 γ ν ψ(x) → vector density, (5.48) ψ (x )σ μν ψ (x )= μ α ν β ψ(x)σ αβ ψ(x) → tensor (5.49)