This is the first draft of my notes. It needs to be translated into English, de- bugged and edited. I plan to do it (with a help of my English speaking friends) once all my lectures are at this level, hopefully by TT12. I hope that even in this very preliminary form, it will help my students in their studies of the sub- ject. Relativistic Quantum Mechanics 1 The aim of this chapter is to introduce a relativistic formalism which can be used to describe particles and their interactions. The choice of the 1.1 Special Relativity 1 formalism was dictated by the fact that it needs to be covered in a few 1.2 One-particle states 8 lectures only. The emphasis is given to those elements of the formalism 1.3 The Klein–Gordon equation which can be carried on to Relativistic Quantum Fields (RQF); the 10 formalism which will be introduced and used later, at the graduate level. 1.4 The Dirac equation 14 We will begin with a brief summary of special relativity, concentrating 1.5 Gauge symmetry 17 on 4-vectors and spinors. One-particle states and their Lorentz transfor- mations will be introduced, leading to the Klein-Gordon and the Dirac equations for probability amplitudes; i.e. Relativistic Quantum Mechan- ics (RQM). Intrinsic problems of RQM will be discussed and a region of applicability of RQM will be defined. Free particle wave functions will be constructed and particle interactions will be described using their probability currents. A gauge symmetry will be introduced to derive a particle interaction with a classical gauge field. 1.1 Special Relativity Einstein’s special relativity is a necessary and fundamental part of any Albert Einstein formalism of particle physics. We will begin with its brief summary. For a full account, please refer to specialized books, like for example (1) or (2). A chapter about spinors in (3) is recommended1. 1Please note that in that chapter, The basic elements of special relativity are 4-vectors (or contravariant transformations are in ”active”, (the 4-vectors) like a 4-displacement2 xµ = (t, x) = (x0, x1, x2, x3) = (x0, xi) coordinate system is not changing, vec- tors are changing) not ”passive” (the µ 0 1 2 3 0 i or a 4-momentum p = (E, p) = (p , p , p , p ) = (p , p ). 4-vectors coordinate system is changing but vec- have real components and form a vector space. There is a metric tensor tors don’t) sense as in this book µν gµν = g which is used to form a dual space to the space of 4-vectors. 2µ, ν = 0, 1, 2, 3 and i, j = 1, 2, 3. This dual space is a vector space of linear functions, known as 1-forms (or covariant 4-vectors), which act on 4-vectors. For every 4-vector xµ, ν there is an associated with it 1-form xµ = gµν x . Such a 1-form is a linear function which acting on a 4-vector yµ gives a real number ν µ ν µ3 3 = gµν x y . This number is called a scalar product x y of x and y . A similar situation is taking place in · µ infinitely-dimensional vector space of The Lorentz transformation between two coordinate systems, Λ ν leaves µ ν states in quantum mechanics (complex the scalar product unchanged which is equivalent to gρσ = gµν Λ ρΛ σ. numbers there). For every state, a vec- In the standard configuration, the Lorentz transformation becomes the tor known as a ket, for example x , there is a 1-form known as a bra, | x which acting on a ket y gives a num- | ber x y which is called| a scalar prod- uct of | two kets, x and y . | | 2 Relativistic Quantum Mechanics Lorentz boost along the first space coordinate direction and is given by Given Given Given Given Lorentz boost matrix, beta and gamma and the metric tensor on the margin. Two Lorentz boosts along different directions are equivalent to a single boost and a space rotation. This means that Lorentz transformations which can be seen as space-time rotations, include Lorentz boosts (ro- tations by a pure imaginary angle) as well as space rotations (by a pure real angle). Representing Lorentz transformations by 4-dimensional real matrices acting on 4-vectors is not well suited to combine Lorentz boosts and space rotations in a transparent way. Even a simple question like what a single space rotation is, equivalent to a combination of two ar- bitrary space rotations, is hard to answer. A better way is to represent Lorentz transformations by 2-dimensional complex matrices. First we consider a 3-dimensional real space and rotations. With every rotation in that 3-dimensional real space we can associate a 2 x 2 4also known as Hamilton’s quaternion complex spin matrix4 or spinor transformation or rotation op- erator. R = cos(θ/2) + isin(θ/2)(σxcos(α) + σycos(β) + σzcos(γ)) or R = cos(θ/2) + isin(θ/2)(n σ) · 1 R is unitary; R† = R− . or after some algebra R = exp[i(θ/2)(n σ)] (1.1) · where θ is the angle of rotation, α, β, γ are the angles between the axis of rotation n and the coordinate axes and σ = (σx, σy, σz) are Pauli matrices. The vector space of spin matrices (a subspace of all 2 x 2 complex matrices) has four basis vectors. They can be chosen as the unit matrix and three basis vectors formed using Pauli matrices: iσx, iσy, iσz. In this basis, the spin matrix R has the following coor- dinates: cos(θ/2), sin(θ/2)cos(α), sin(θ/2)cos(β) and sin(θ/2)cos(γ). Combining two rotations, one multiplies corresponding spin matrices and describes the outcome using the above basis; thus getting all the parameters of the equivalent single rotation. The next step is to asso- ciate each 3-dimensional space (real numbers) vector x = (x1, x2, x3) 5No unit matrix, only three Pauli ma- with a corresponding spin matrix5 trices as the basis. 1 2 3 X = x σx + x σy + x σz. (1.2) Then, under the space rotation, x is transformed to x and X is trans- 1 2 3 formed to X = RXR† = x σx + x σy + x σz from which we can read coordinates of x. 1.1 Special Relativity 3 The beauty of the above approach is that it extends seamlessly to space-time rotations; i.e. to the Lorentz transformations. The spin matrix R of 1.1 becomes the Lorentz transformation L = exp[( ρ + iθn) σ/2], (1.3) − · where ρ = ρnρ is rapidity. Now, combination of two Lorentz transfor- mations is very transparent; just addition of real and imaginary parts in the exponent. Association of a 4-vector xµ with a spin matrix6 6Now, four matrices as the basis. 0 1 2 3 X = x + x σx + x σy + x σz (1.4) allows to get its Lorentz transformed coordinates from X = LXL† = X is Hermitian; X = X†. 0 1 2 3 x + x σx + x σy + x σz. And finally, the Lorentz boost alone (θ = 0) along nρ is L = exp[ ρ σ/2] = cosh(ρ/2) n σ sinh(ρ/2). (1.5) − · − ρ · 1.1.1 Spinors Spin matrices act on two component complex vectors called spinors7. 7Spinors are vectors in the sense of Spinors play an important role in RQM8 and in this section we will mathematical vector space but they are not vectors like a displacement x be- describe them in some detail. cause they transform (for example un- α Under the space rotation, a spinor ξ (ξ to be more precise) transforms der rotation) differently. the following way 8Spinors like vectors or tensors are used ξ = Rξ. in different parts of physics, including In a comparison, coordinates of a vector x transform under space rota- classical mechanics. tion as 1 2 z X = RXR† = x σx + x σy + x σ3. Thus, a rotation of the coordinate system by θ = 2π, R = 1 because of − θ/2 in R, gives ξ = ξ and x = x. Continuing the rotation by a further − 2π, so all together by 4π, results in ξ = ξ. Is that counter intuitive minus sign because of that 2π rotation good for any physics? Yes it is, as was demonstrated in a beautiful experiment (4) using neutrons. description of the neutron interference experiment and the first rela- tion between spinors and spin. So the counter intuitive minus sign was needed for the neutron to relate to the outside world, to be entangled with it the way quantum states can be entangled. How to visualize a spinor9? Quoting from (3): ”Aim the laser, pull 9i.e. going from complex numbers to the trigger, and send a megajoule pulse from here and now (event ) to 3-dimensional real space. the there and then (event : center of the crater Aristarchus, 400,000O km from in space, and 400,000P km from in light-travel time). The laser hasO been designed to produce, not aO mere spot of light, but an 4 Relativistic Quantum Mechanics Fig. 1.1 Spinor represented by (1) ”flagpole” [Penrose terminology; track of pulse of light; null vector ] plus (2) ”flag” [arrow ( ) flashed onto surface of moon by laser pulse fromOP earth or, in expanded viewP −→in the insert above, a flag itself, substituted for the arrow] plus (3) the orientation-entanglement relation between the flag and its surroundings [symbolized by strings drawn from corners of flag to surroundings].