1 note1 : November 4, 2014 Introduction can only be influenced by its immediate sur- roundings. In the context of particle physics, Quantum Field Theory is a theory of elementary particles and their From this principle follows the finite speed of interactions. The Standard Model of elementary information transmission. particles is a quantum field theory. Principle of covariance By definition, elementary particles are the most The principle of covariance emphasizes formu- fundamental — structureless — particles (like elec- lation of physical laws using only those phys- trons and photons) which exist in our universe. Ele- ical quantities the measurements of which the mentary particles exhibit wave-particle duality: on observers in different frames of reference could the one hand they diffract and interfere as waves unambiguously correlate. (or fields), on the other hand they appear and dis- appear as whole entities (called quanta). Hence the Mathematically speaking, the physical quanti- name of the theory. ties must transform covariantly, that is, un- A quantum field theory seeks to explain certain der a certain representation of the group of fundamental experimental observations — like the coordinate transformations between admissi- existence of antiparticles; the spin-statistics rela- ble frames of reference of the physical theory. tion; the CPT symmetry — as well as to predict This group of coordinate transformations is re- the results of any given experiment, like the cross- ferred to as the covariance group of the theory. section for the Compton scattering, or the value of The principle of covariance does not require the anomalous magnetic moment of the electron. invariance of physical laws under the group of There are two popular approaches to deal with admissible transformations although in most quantum fields. One is the path integral formula- cases the equations are actually invariant. tion, where elementary particles have the property Only in the theory of weak interactions the of being able to propagate simultaneously along all equations are not invariant under reflections possible trajectories with certain amplitudes. In (but are, of course, still covariant). the other approach — called canonical quantization — elementary particles are field quanta: necessarily In canonical quantum field theory the admissible chunked ripples in the field. frames of reference are the inertial frames of special In the end, the two formulations proved to be relativity. The transformations between frames are equivalent. the velocity boosts, rotations, translations, and re- We shall pursue the second approach here and flections. Altogether they form the Poincar´egroup build the quantum field theory in the canonical of coordinate transformations. Boosts and rota- way: as a classical Lagrangian field theory with the tions together make up the Lorentz group. 1 subsequent canonical quantization . The covariant quantities are four-scalars, four- vectors etc. of the Minkowski space of special Fundamental principles relativity (and also more complicated objects like bispinors and others which we shall discuss later). Quantum field theory is built on several fundamen- tal principles. A “principle” is a physical law of more general — typically universal — applicability Covariant vectors of special relativity usually formulated as a simple and succinct state- ment. Four-coordinates An event in an inertial frame can be specified with Principle of relativity four coordinates {t, r}, where t is the time of the The principle of relativity is the requirement event, and r ≡ {x, y, z} are the three Euclidean that the laws of physics have the same form in spatial coordinates. all admissible frames of reference. The coordinates of the same event in different in- In the absence of gravitation one can choose ertial frames are connected by a linear transforma- to admit only inertial frames of reference. The tion from the Poincar´egroup. Rotations, transla- laws of physics take particularly simple form tions and reflections do not couple time and spatial in inertial frames. coordinates, but velocity boosts do. Therefore in special relativity time and spatial coordinates are Principle of locality inseparable components of one and the same ob- The principle of locality states that an object ject. It is only in the non-relativistic limit that 1sometimes historically called “second quantization”. time separates from space. 2 The Lorentz transformation of coordinates under Under Lorentz transformations the dual coordi- a velocity boost v along the x-axis is given as nates apparently transform with the Lorentz matrix where v is substituted by −v, which is actually the  0   v    ct γ −γ c 0 0 ct inverse Lorentz matrix (prove it), 0 v  x   −γ c γ 0 0   x   0  =     , (1) 0 −1 b  y   0 0 1 0   y  xa = (Λ )axb , (8) z0 0 0 0 1 z Four-gradient where primes denote coordinates in the boosted frame, γ ≡ (1 − v2/c2)−1/2, and c is the speed of The partial derivatives of a scalar with respect to light in vacuum2. four-coordinates, The four-coordinates {t, r} are customarily de- a ∂ ∂ ∂ ∂ ∂ ∂ noted as x , where a = 0, 1, 2, 3, such that ∂ ≡ =  , , , ≡  , ∇ , (9) a ∂xa ∂t ∂x ∂y ∂z ∂t x0 = t, x1 = x, x2 = y, x3 = z . (2) apparently transform like dual coordinates, that is, The Lorentz transformation (1) can then be conve- via the inverse Lorentz matrix, niently written as3 b ∂ ∂x ∂ −1 b ∂ 0a a b 0a = 0a b = (Λ )a b . (10) x = Λb x . (3) ∂x ∂x ∂x ∂x

a where Λb is the 4×4 transformation matrix in equa- Covariant vectors and tensors tion (1). A contra-variant four-vector is a set of four objects, Aa = {A0, A}, which transform from one inertial Invariant frame to another in the same way as coordinates A direct calculation shows that velocity boosts to- in (3), 0a a b gether with rotations, translations, and reflections A = Λb A . (11) — that is, all Lorentz transformations — conserve A co-variant four-vector is a set of four objects, A , the following form, a which transform from one inertial frame to another s2 = t2 − r2 , (4) in the same way as partial derivatives in (10), 0 −1 b which is then called the invariant interval. Aa = (Λ )aAb . (12) In its infinitesimal incarnation, A covariant tensor F ab is a set of 4×4 objects ds2 = dt2 − dr2 , (5) which transform between inertial frames as a prod- uct of two 4-vectors, the form determines the geometry of time-space 0ab a b cd and is called metric. The space with metric (5) F = Λc ΛdF . (13) is called Minkowski space. There exist other covariant objects in special rel- ativity, like bispinors, which cannot be built out of Dual coordinates 4-vectors. They will be discussed later. The invariant (4) can be conveniently written as

2 2 a t − r ≡ xax , (6) where xa are often called dual coordinates and are defined as

b xa ≡ {t, −r} = gabx , (7) where the diagonal tensor gab with the main di- agonal {1, −1, −1, −1} is the metric tensor of the Minkowski space of special relativity.

2 in the following the notation with ~ = c = 1 shall be mostly used. 3note the “implicit summation” notation, a b P3 a b Λb x ≡ b=0 Λb x .