Relativistic Quantum Mechanics 1

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 SpecialRelativity 1 formalism was dictated by the fact that it needs to be covered in a few 1.2 One-particle states 7 lectures only. The emphasis is given to those elements of the formalism 1.3 The Klein–Gordon equation 8 which can be carried on to Relativistic Quantum Fields (RQF); the formalism which will be introduced and used later, at the graduate level. 1.4 The Diracequation 14 We will begin with a brief summary of special relativity, concentrating 1.5 Gaugesymmetry 33 on 4-vectors and spinors. One-particle states and their Lorentz transformations will be introduced, leading to the Klein–Gordon and the Dirac equations for probability amplitudes; i.e. Relativistic Quantum Mechan- ics (RQM). Readers who would like to get to the RQM quickly, without studying its foundation in the special relativity might skip the first sec- tions and start reading from the section 1.3. 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- µ 0 1 2 3 0 i tors are changing) not ”passive” (the 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 = gµν x y . This number is called a scalar product x y of x and µ3 · µ 3 y . The Lorentz transformation between two coordinate systems, Λ ν , A similar situation is taking place in an infinitely-dimensional vector space of states in quantum mechanics (complex numbers there). For every state, a vector known as a ket, for example |xi, there is a 1-form known as a bra, hx| which acting on a ket |yi gives a number hx|yi which is called a scalar product of two kets, |xi and |yi. 2 Relativistic Quantum Mechanics 0µ µ ν x =Λ ν x , leaves the scalar product unchanged which is equivalent µ ν to gρσ = gµν Λ ρΛ σ. In the standard configuration, the Lorentz transformation becomes the Lorentz boost along the first space coordinate direction and is given by 10 0 0 γ γβ 0 0 0 −1 0 0 − g = . γβ γ 0 0 v 1 0 0 −1 0 Λ= − and β = , γ = 0 0 0 −1 0 0 10 c v2 1 2 0 0 01 − c q where v is the velocity of the boost. 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 (rotations 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 this: what is a single space rotation which is equivalent to a combination of two arbitrary 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 2x2 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) · R is unitary; R† = R−1. or after some algebra R = exp[i(θ/2)(n σ)] (1.1) · 5Only two of the angles α, β, γ are in- where θ is the angle of rotation, α,β,γ5 are the angles between the dependent. 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) can be defined using four basis vectors, such 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 coordinates: 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 associate each 3-dimensional space (real numbers) vector x = (x1, x2, x3) 6No unit matrix, only three Pauli ma- with a corresponding spin matrix6 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 x0 and X is trans- 0 † 01 02 03 formed to X = RXR = x σx + x σy + x σz from which we can read coordinates of x0. 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 tanh(ρ) = β L = exp[( ρ + iθn) σ/2], (1.3) cosh(ρ)= γ, sinh(ρ)= βγ. − · where ρ = ρnρ is the rapidity. Now, combination of two Lorentz transformations is very transparent; just addition of real and imaginary parts in the exponent. Association of a 4-vector xµ with a spin matrix7 7Now, 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 X0 = LXL† = X is Hermitian; X = X†. 00 01 02 03 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 spinors8. 8Spinors are vectors in the sense of Spinors play an important role in RQM9 and in this section we will mathematical vector space but they are describe them in some detail. not vectors like a displacement x because 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 9 0 Spinors 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 0 † 01 02 0z X = RXR = x σx + x σy + x σz. Thus, a rotation of the coordinate system by θ =2π, R = 1 because of θ/2 in R, gives ξ0 = ξ and x0 = x. Continuing the rotation− by a further 2π, so all together by− 4π, results in ξ0 = 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. One of the two coherent neutron beams was going through a magnetic field of variable strength. In the magnetic field the neutrons’ magnetic moments were precessing with the Larmor frequency and the angle of the precession could be easily calculated as a function of the strength of the magnetic field. After passing through the magnetic field, the Fig. 1.1 The phase change of 4π is beam interfered with the second beam which followed a path outside needed to get the same intensity from the magnetic field. As demonstrated in fig 1.1, an angle of 4π was the interference of two neutron beams; needed for the neutron wave function to reproduce itself. A 2π rotation from 4. produced -1 in front of the original neutron wave function as predicted for spin 1/2 spinor. 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.

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