1 note1 : November 6, 2016 Introduction can only be influenced by its immediate sur- roundings. In the context of particle physics Quantum Field (From this principle follows the finite speed of Theory is a theory of elementary particles and their information transmission.) interactions. It forms the basis for the Standard Model of elementary particles. Principle of covariance Elementary particles are the most fundamental The principle of covariance emphasizes formu- (structureless) particles — like electrons and pho- lation of physical laws using only those phys- tons — which exist in our universe. Elementary ical quantities the measurements of which the particles exhibit wave-particle duality: on the one observers in different frames of reference could hand they diffract and interfere as waves (fields), unambiguously correlate. on the other hand they appear and disappear as (Mathematically speaking, the physical quan- whole entities (called quanta). Hence the name of tities must transform covariantly, that is, un- the theory. der a certain representation of the group of co- A quantum field theory seeks i) to explain cer- ordinate transformations between admissible tain fundamental experimental observations — like frames of reference of the physical theory. This the existence of antiparticles; the spin-statistics re- group of coordinate transformations is referred lation; the CPT symmetry — and ii) too predict to as the covariance group of the theory.) the results of any given experiment, like the cross- section for the Compton scattering, or the value of In canonical quantum field theory the admissible the anomalous magnetic moment of the electron. frames of reference are the inertial frames of special There are two popular approaches to deal with relativity. The transformations between frames are quantum fields. One is the path integral formula- the velocity boosts, rotations, translations, and re- tion, where elementary particles have the property flections. Altogether they form the Poincar´egroup of being able to propagate simultaneously along all of coordinate transformations. Boosts and rota- possible trajectories with certain amplitudes. In tions together make up the Lorentz group. the other approach — called canonical quantization The covariant quantities are four-scalars, four- — elementary particles are field quanta: necessarily vectors etc. of the Minkowski space of special chunked ripples in the field. relativity (and also more complicated objects like In the end, the two formulations proved to be bispinors and others which we shall discuss later). equivalent. In this course we shall pursue the second ap- Covariant vectors of special relativity proach and build the quantum field theory in the Four-coordinates canonical way: as a classical Lagrangian field the- ory with the subsequent canonical quantization.1 An event3 in an inertial frame can be specified with four coordinates {t, r}, where t is the time of the Fundamental principles event, and r ≡ {x, y, z} are the three Euclidean spatial coordinates. Quantum field theory is built on several fundamen- The coordinates of the same event in different in- tal principles.2 ertial frames are connected by a linear transforma- tion from the Poincar´egroup. Rotations, transla- Principle of relativity tions and reflections do not couple time and spatial The principle of relativity is the requirement coordinates, but velocity boosts do. Therefore in that the laws of physics have the same form in special relativity time and spatial coordinates are all admissible frames of reference. inseparable components of one and the same ob- (In the absence of gravitation one can choose ject. It is only in the non-relativistic limit that to admit only inertial frames of reference as time separates from space. the laws of physics in the absence of gravita- The Lorentz transformation of coordinates under tion take particularly simple form in inertial a velocity boost v along the x-axis is given as frames.)  0   v    ct γ −γ c 0 0 ct Principle of locality  x0   −γ v γ 0 0   x    =  c    , (1) The principle of locality states that an object  y0   0 0 1 0   y  z0 0 0 0 1 z 1sometimes historically called “second quantization”. 2A “principle” is a physical law of more general — typi- 3An event in the theory of relativity is a physical pro- cally universal — applicability usually formulated as a sim- cess whose temporal duration and spatial extension can be ple and succinct statement. neglected in the present context. 2

where primes denote coordinates in the boosted where the diagonal tensor gab with the main di- frame; γ ≡ (1 − v2/c2)−1/2; and c is the speed of agonal {1, −1, −1, −1} is the metric tensor of the light in vacuum.4 Minkowski space of special relativity. The four-coordinates {t, r} are customarily de- Under Lorentz transformations the dual coordi- noted as xa, where a = 0, 1, 2, 3, such that nates transform with the Lorentz matrix where v is substituted with −v, which is actually the inverse x0 = t, x1 = x, x2 = y, x3 = z . (2) Lorentz matrix,

0 −1 b The Lorentz transformation (1) can then be conve- xa = (Λ )axb . (9) niently written as5

0a a b Four-gradient x = Λb x . (3) The partial derivatives of a scalar with respect to a where Λb is the 4×4 transformation matrix in equa- four-coordinates, tion (1). ∂  ∂ ∂ ∂ ∂  ∂ ∂a ≡ a = , , , ≡ , ∇ , (10) Invariant ∂x ∂t ∂x ∂y ∂z ∂t A direct calculation shows that velocity boosts to- apparently transform like dual coordinates, that is, gether with rotations, translations, and reflections via the inverse Lorentz matrix, — that is, all Lorentz transformations — conserve ∂ ∂xb ∂ ∂ the following form, = = (Λ−1)b . (11) ∂x0a ∂x0a ∂xb a ∂xb ∆s2 = ∆t2 − ∆r2 , (4) Covariant vectors and tensors where {∆t, ∆r} is the coordinate difference be- A contra-variant four-vector is a set of four objects, tween two events. This form is called the time-space Aa = {A0, A}, which transform from one inertial interval or invariant interval. frame to another in the same way as coordinates In its infinitesimal incarnation, in (3), A0a = ΛaAb . (12) ds2 = dt2 − dr2 , (5) b A co-variant four-vector is a set of four objects, Aa, the form determines the geometry of time-space which transform from one inertial frame to another and is called metric. The space with metric (5) in the same way as partial derivatives in (11), is called Minkowski space. 0 −1 b Aa = (Λ )aAb . (13) Dual coordinates A covariant tensor F ab is a set of 4×4 objects The Lorentz transformations conserve the form which transform between inertial frames as a prod- uct of two 4-vectors, t2 − r2 = t2 − x2 − y2 − z2 , (6) 0ab a b cd F = Λc ΛdF . (14) which can be conveniently written as There exist other covariant objects in special rel- 2 2 a t − r ≡ xax , (7) ativity, like bispinors, which cannot be built out of 4-vectors. They will be discussed later. where the four quantities xa — often called dual coordinates — are defined as

b xa ≡ {t, −r} = gabx , (8)

4in the following we shall mostly use the notation where

~ = c = 1 .

5note the “implicit summation” notation,

3 a b X a b Λb x ≡ Λb x . b=0