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Introduction note1 : August 29, 2012 Introduction Principle of locality The principle of locality states that an object can Quantum Field Theory—in the context of parti- only be influenced by its immediate surroundings. cle physics—is a theory of elementary particles and From this principle follows the finite speed of in- their interactions. The Standard Model of elemen- formation transmission. tary particles is a quantum field theory. By definition, elementary particles are the most Principle of covariance fundamental—structureless—particles (like elec- trons and photons). They exhibit wave-particle du- The principle of covariance emphasizes formulation ality: on the one hand they diffract and interfere as of physical laws using only those physical quanti- waves (fields), on the other hand they appear and ties the measurements of which the observers in disappear as whole entities, called quanta. Hence different frames of reference could unambiguously the name of the theory. correlate. A quantum field theory seeks to explain certain Mathematically speaking, the physical quantities fundamental experimental observations—the exis- must transform covariantly, that is, under a certain tence of antiparticles, the spin-statistics relation, representation of the group of coordinate transfor- the CPT symmetry—as well as predict the results mations between admissible frames of reference of of any given experiment, like the cross-section for the physical theory. This group of coordinate trans- the Compton scattering, or the value of the anoma- formations is referred to as the covariance group of lous magnetic moment of the electron. the theory. There are two popular approaches to deal with The principle of covariance does not require in- quantum fields. One is the path integral formu- variance of physical laws under the group of admis- lation, where elementary particles have the prop- sible transformations although in most cases the erty of being able to propogate simultaneously equations are actually invariant. Only in the the- along all possible trajectories with certain apm- ory of weak interactions the equations are not in- plitudes. In the other approach—called canon- variant under reflections (but are, of course, still ical quantization—elementary particles are field covariant). quanta: necessarily chunked ripples in the field. In canonical quantum field theory the admissible In the end, the two formulations proved to be frames of reference are the inertial frames of special equivalent. relativity. The transformations between frames are 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. subsequent canonical quantization1. The convariant quantities are four-scalars, four- vectors etc. of the Minkowski space of special relativity (and also more complicated objects like Fundamental principles 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 Covariant vectors of special relativity general—typically universal— applicability usually Four-coordinates formulated as a simple and succinct statement. An event in an inertial frame can be specified with four coordinates {t, r}, where t is the time of the Principle of relativity event, and r ≡ {x,y,z} are the three Euclidean spatial coordinates. The principle of relativity is the requirement that The coordinates of the same event in different in- the laws of physics have the same form in all ad- ertial frames are connected by a linear transforma- missible frames of reference. tion from the Poincar´egroup. Rotations, transla- In the absence of gravitation one can choose to tions and reflections do not couple time and spatial admit only inertial frames of reference. The laws coordinates, but velocity boosts do. Therefore in of physics take particularly simple form in inertial special relativity time and spatial coordinates are frames. inseparable components of one and the same ob- ject. It is only in the non-relativistic limit that 1sometimes historically called “second quantization”. time separates from space. 1 note1 : August 29, 2012 The Lorentz transformation of coordinates under Four-gradient a velocity boost v along the x-axis is given as The partial derivatives of a scalar with respect to ′ v ct γ −γ c 0 0 ct four-coordinates, x′ −γ v γ 0 0 x ′ = c , (1) y 0 0 10 y ∂ ∂ ∂ ∂ ∂ ∂ ′ ∂a ≡ = , , , ≡ , ∇ , (9) z 0 0 01 z ∂xa ∂t ∂x ∂y ∂z ∂t where primes denote coordinates in the boosted apparently transform like dual coordinates, that is, 2 2 −1 2 frame, γ ≡ (1 − v /c ) / , and c is the speed of via the inverse Lorentz matrix, light in vacuum2. b The four-coordinates {t, r} are customarily de- ∂ ∂x ∂ −1 ∂ = = (Λ )b . (10) noted as xa, where a =0 ... 3, such that ∂x′a ∂x′a ∂xb a ∂xb 0 1 2 3 x = t, x = x, x = y, x = z . (2) Covariant vectors and tensors The Lorentz transformation (1) can then be conve- 3 A contra-variant four-vector is a set of four objects, niently written as 0 Aa = {A , A}, which transform from one inertial ′a a b x =Λb x . (3) frame to another in the same way as coordinates a in (3), where Λ is the 4×4 transformation matrix in equa- ′ b A a =ΛaAb . (11) tion (1). b A co-variant four-vector is a set of four objects, Aa, Invariant which transform from one inertial frame to another in the same way as partial derivatives in (10), A direct calculation shows that velocity boosts (to- gether with rotations, translations, and reflections) ′ −1 b Aa = (Λ )aAb . (12) conserve the following form, 2 2 2 A covariant tensor F ab is a set of 4×4 objects s = t − r , (4) which transform between inertial frames as a prod- which is then called the invariant interval. uct of two 4-vectors, In its infinitesimal incarnation, ′ab a b cd 2 2 2 F =Λ Λ F . (13) ds = dt − dr , (5) c d the form determines the geometry of time-space There exist other covariant objects in special rel- and is called metric. The space with metric (5) ativity, like bispinors, which cannot be built out of is called Minkowski space. 4-vectors. They will be discussed later. Dual coordinates 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. Under Lorentz transformations the dual coordi- nates apparently transform with the Lorentz matrix where v is substituted by −v, which is actually the inverse Lorentz matrix (prove it), ′ −1 b xa = (Λ )axb , (8) 2in the following the notation with ~ = c = 1 shall be mostly used. 3note the “implicit summation” notation, a b 3 a b Λb x ≡ Pb=0 Λb x . 2.
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