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

Home , Bispinor, Spinor

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 transformations 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, vectors 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 product 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 (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 what a single space rotation 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 2 x 2 4also known as Hamilton’s quaternion complex spin matrix4 or spinor transformation or rotation operator. 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 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) 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 transformations 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 relation 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]. When the spinor itself is multiplied by a factor ρeiσ, the components of the null vector (flagpole) are multiplied by the factor ρ2 and the flag is rotated through the angle 2σ about the flagpole. Taken from GRAVITATION.

illuminated arrow. Following Roger Penrose, speak of the null vector as a ”flagpole”, and of the illuminated arrow as a”flag”. A spinor (fig.OP 1.1) consists of this combination of (1) null flagpole plus (2) flag plus (3) the orientation-entanglement relation between the flag and its surroundings... Two numbers, such as the familiar polar angles θ (angle with the z-axis) and φ (azimuth around z-axis from x-axis) tell the direction of its (the laser pulse) flight. A third number, r, gives the distance to the moon and also the travel time for light to reach the moon. A fourth number, an angle ψ, tells the azimuth of the illuminated arrow shot onto the surface of the moon, this azimuth to be measured from the eθ direction (where ψ = 0), around the flagpole in a righthanded sense. Than the spinor associated with the flagpole plus flag ... is

ξ1 cos(θ/2)exp( iφ/2 + iψ/2) = (2r)1/2 ξ2 sin(θ/2)exp(−iφ/2 + iψ/2)     and the 4-vector of the flagpole (the null vector associated with the spinor) is x0 r x1 rsin(θ)cos(φ) = (2r)1/2 .”  x2   rsin(θ)sin(φ)   x3   rcos(θ)      To visualize the minus sign appearing after 2π rotation, it is probably better to consider an ”active” instead of a ”passive” transformation for a 1.1 Special Relativity 5 moment, like in the experiment with neutrons, described above: so the coordinate system and the environment is stationary and the flagpole and the flag move around like the precessing neutron. One can think about the flag resting on the top surface of the Möbius strip like in fig. 1.2. During the rotation, the flag surface follows the strip surface and after 2π the flag ends up below the strip; + to be above the strip and - to be below the strip. So far, one could think about spinors as being identical with Pauli spinors10 of non-relativistic quantum mechanics11. This is wrong. In order to see that, we will look at spin matrices X of eqn 1.4 from a different angle, seeing them as tensors created by a tensor product12 of 2-dimensional spinors, Weyl spinors rooted in space-time, not in space only like Pauli spinors. Let’s consider the Lorentz transformation of a a c spin matrix X built from spinors ξ = and η = : b d     a a X = c d = L c d L†. b b     �  �  We can see that ξ = Lξ but η = L η (after taking transpose, L T = Fig. 1.2 Rotating a flag along the   ∗ † Möbiusstrip; every 2π the flag is on L∗). There are two different types of spinors, transforming differently. a different side of the strip. Those which transform with the complex conjugation are called dotted 10 α Eigenstates of spin operators, like the Weyl spinors, distinguished from the undotted ξ by a dot written above spin projection on the z axis 1/2¯hσz for α˙ α the index: η ; for example, (ξ )∗ is a dotted spinor. The spin matrix X a spin 1/2 particle in non-relativistic is then written as Xαβ˙ . There is a metric tensor quantum mechanics. 11 0 1 giving components of spin matrices  = αβ = X of eqn 1.2 αβ 1 0  −  12A tensor product or outer product or (and an identical one for the dotted spinors) to create a dual space of dyadic product; something like this: ˙ 1-forms; ξ =  ξβ (and η =  ηβ). The scalar product ξαζ = a α αβ α˙ α˙ β˙ α c d . α β b αβξ ζ (and similarly for the dotted spinors) is invariant with respect   to the Lorentz transformation. Because undotted Weyl spinors and dot- α = 1, 2 andα ˙ = 1�, 2.  ted Weyl spinors are different objects, the scalar product, or in general 2 1 ξ1 = ξ and ξ2 = ξ . any contraction, can only be performed on the same type of spinors; an − α undotted index is contracted with another undotted or a dotted index ξ ξα = 0 α α is contracted with another undotted one; one can’t contract a dotted ξ ζα = ξαζ index with an undotted one. − In order to get more insight, we will now go beyond Lorentz trans- Hermann Weyl. formations and consider a space inversion P ; P (x0, x) = (x0, x). The space inversion P commutes with space rotations but it doesn’t− commute with Lorentz transformations because Lorentz transformations affect the time component and P doesn’t. One can see that, considering P Λ, the Lorentz boost followed by the space inversion in the 4-dimensional space- time: P Λ = ΛP and that Λ = Λ. If Λ is the boost with a velocity v,  Λ is the boost with the velocity v. Thus [P, Λ] = 0 and therefore P is not proportional to the identity− operator which commutes with every operator. Back to Weyl spinors. Because the space inversion is not proportional to the identity operator, the space inversion is not transforming ξα into 6 Relativistic Quantum Mechanics

ξα times a number. It transforms ξα into a spinor of a different type, which transforms under the Lorentz transformation differently than ξα. As Pauli spinors represent spin in non-relativistic quantum mechanics, Weyl spinors are going to represent spin in RQM. If so, we know that the space inversion leaves spin unaffected and therefore under P , ξα needs to be transformed to a spinor representing the same spin state, i.e. the space inversion of ξα needs to transform under space rotations the same α way as ξ . Out of all 3 possibilities, only 1-form ηα˙ transforms the same α α For a better grasp of the material pre- way. So under the space inversion ξ ηα˙ and ηα˙ ξ . sented in this part of the section on α → → As spinors ξ and ηα˙ play very important role in RQM, here is the spinors, it is recommended to read the chapter on fermions in (5). In the dis- summary of how do they transform under cussion on the space inversion, P 2 = 1 α α is assumed. This is fine for all particles rotation R ξ Rξ ηα˙ Rηα˙ , (1.6) except Majorana particles. No Majo- → → rana particle has been discovered so far but a neutrino could be one. In case Lorentz transfor- α α 1 2 ξ Lξ ηα˙ (L†)− ηα˙ , (1.7) of a Majorana particle, P = 1 and mation L → → the transformation of spinors under− P is different than given here. We will define a Majorana particle later. Lorentz boost α α 1 L = L ξ Lξ ηα˙ L− ηα˙ , (1.8) 1 1 † → → (L†)− = L∗− .

α α space inversion P ξ η ηα˙ ξ . (1.9) → α˙ → Suppose that, there is, described in a particular reference frame, a spin 1/2 particle with a 4-momentup pµ represented via the eqn 1.4 by pαβ˙ . Following our non-relativistic intuition gained from using Pauli 11˙ 0 3 α p = p22˙ = p + p spinors, we want to represent the spin of that particle by ξ . In an ˙ 13 p22 = p = p0 p3 attempt to write a covariant eqn, we can tray to contract undotted 11˙ − ˙ index α but that would lead to something like this p12 = p = p1 ip2 − 21˙ − 21˙ 1 2 p = p ˙ = p + ip α − 12 pαβ˙ ξ = mηβ˙ αβ˙ βα˙ p is identical with p ; it is not α transpose operation. where ηβ˙ is a dotted spinor diﬀerent from ξ coming from not contracted 14 13 dotted index, and m is a dimensional scalar parameter appearing there Here and in the rest of the RQM, a µ covariant means covariant with respect because of the energy dimensionality of p . So that equation is covariant to Lorentz transformations. only when m = 0 because we don’t have any dotted spinor in hands to 14From now on, a scalar means a scalar put on the right hand side of the eqn. Similar outcome is obtained α with respect to Lorentz transforma- having only ηβ˙ instead of ξ . Those attempts lead to two independent tions. Lorentz-invariant Weyl eqns:

α 0 p ˙ ξ = 0 (p p σ)ξ = 0 (1.10) αβ − · and

˙ In quantum mechanics, a helicity oper- αβ 0 p ηβ˙ = 0 (p + p σ)η = 0. (1.11) ator representing a projection of parti- · cle’s spin on the direction of its momen- In the context of RQM, eqn 1.10 represents an eqn of motion for a free tum is defined as massless spin 1/2 particle with a positive helicity and eqn 1.11, an eqn p σ · . p of motion for a different, free massless spin 1/2 particle with a negative helicity. Each equation is not space inversion-covariant and violates For a massless particle, this is equivalent to p σ · . p0 1.1 Special Relativity 7 parity because the space inversion, eqn 1.9, sends each spinor beyond the formalism; only one type of spinor is present in each formalism. At present there are no known particles which could be described by any of the Weyl eqns. If the electron neutrino would have no mass, it would be described by eqn 1.11 and hypotetically massles and different electron anti-neutrino would be described by eqn 1.10. αβ˙ α Suppose now that we have p and two different spinors: ξ and ηβ˙ to describe a spin 1/2 particle. We can then contract undotted index α αβ˙ α αβ˙ µ α getting pαβ˙ ξ = mηβ˙ . Acting with p on ηβ˙ from that eqn, gives mξ p pγβ˙ = p pµδγ 2 µ under the condition that m = p pµ. The result is a covariant set of equations

αβ˙ α 0 p ηβ˙ = mξ (p + p σ)η = mξ α 0 · p ˙ ξ = mη ˙ (p p σ)ξ = mη (1.12) αβ β − ·

α Requiring that under the space inversion ξ and ηβ˙ are transformed into each other as in eqn 1.9, makes the set of eqns 1.12 space inversion αβ˙ invariant as, simultaneously, p and pαβ˙ are also transformed into each α other. Spinors ξ and ηβ˙ are combined into a four component bispinor, called Dirac spinor and two eqns become one eqn 1.12 called the Dirac equation. In the context of RQM, the Dirac equation describes a spin 1/2 particle like the electron. Paul A.M. Dirac In order to have more insight on the origin of the Dirac equation, we will consider the Lorentz boost, eqns 1.5 and 1.8, from the rest frame, momentum p = 0, to the frame in which the particle has the energy E and the momentum p. The relevant spinors transform as15 15 E + m ξ(p) = (cosh(ρ/2) n σ sinh(ρ/2))ξ(0) (1.13) cosh(ρ/2) = − ρ · (2m(E + m))1/2 p η(p) = (cosh(ρ/2) + n σ sinh(ρ/2))η(0) (1.14) sinh(ρ/2) = ρ · (2m(E + m))1/2 which can be written as E + m + p σ ξ(p) = · ξ(0) (1.15) (2m(E + m))1/2

E + m p σ η(p) = − · η(0). (1.16) (2m(E + m))1/2 In the particle’s rest frame and in all frames moving with respect to it slowly enough, such that the Lorentz boost can be approximated by the Galilean transformation (not affecting time component) when transforming between those frames, any differences between spinors with dotted or undotted indexes disappear; in that case, spinors effectively live in 3 real dimensions. This can be illustrated by a simple cartoon in fig. 1.3. α Thus, at rest, both Weyl spinors, ξ and ηβ˙ , become identical Pauli α spinors and we can write ξ (0) = ηβ˙ (0). This allows, after some algebra, to remove p = 0 spinors from the eqns 1.15 and 1.16 and to obtain the Dirac equation 1.12.

Fig. 1.3 At the limit of negligible hight h, a triangle and a rectangular of the same base, live eﬀectively in 1 dimension where they are identical 8 Relativistic Quantum Mechanics

So the Dirac equation 1.12 is equivalent to the Lorentz boost. This should be expected. Once an object, like a bispinor, is found to represent a particle in its rest frame, everything else left to do is to boost it to another frame as needed.

1.2 One-particle states

The fact that quantum states of free relativistic particles are fully defined by the Lorentz transformation supplemented by the space-time Eugen Wigner. translation was discovered by Wigner. Here we will follow his idea in a qualitative way just to get the main concept across. First, we note that Lorentz transformations are not able to transform µ µ a given arbitrary 4-momentum p to every possible p . Instead, the vector space of 4-momenta is divided into sub-spaces of 4-momenta which can be Lorentz transformed into each other. Three of those sub-spaces represent experimentally known states. The simplest, at this stage, is µ µ the vacuum state given by the conditions p = 0 and p pµ = 0. There is no Lorentz transformation which would transform a 4-momentum not satisfying these conditions to the one which does and vice-versa. We will not study vacuum states in this course and therefore, without any further considerations about them, we will move on to consider two other possibilities. A 4-momentum sub-space related to massive particles, like the elec- µ tron, is given by the condition p pµ > 0. In addition to a 4-momentum pµ, what are other degrees of freedom present and which geometrical object represent them? To answer this question, we can consider Lorentz 16The group of such transformations is transformations which leave pµ invariant16. To see what they are, we known as the little group. µ µ can transform p to the particle rest frame where p = (mass, 0, 0, 0), µ find the largest subset of the Lorentz transformations leaving p invariant, and then transform back to the same pµ. It turns out, as intuitively expected, that the looked for transformations are space rotations acting on 2s + 1 spinors representing 2s + 1 spin projections of spin s particle. Thus, the electron, s = 1/2, is represented by two Dirac spinors. In fact, by two Dirac spinors multiplied be a dimensionless scalar. To get the scalar, we add space-time translations. Looking for a theory which is space-time translations invariant, meaning the energy and momentum conservation, we are looking for the energy and momentum eigenstates (free particles) which in the position representation, would µ lead to exp( ip xµ) scalar. The third− 4-momentum sub-space is defined by the conditions pµ = 0 µ  and p pµ = 0. Photons belong to this class. The question is again to find the largest sub-set of Lorentz transformations leaving pµ invariant. There is no rest frame in this case and therefore and instead, we trans- µ µ form an arbitrary p to the frame where p = (ω, 0, 0, ω). We can see 17 17 µ For a discussion of some subtle issues that the largest sub-set of the Lorentz transformations leaving p in- see (6). variant are the rotations in the x1x2 plane. As a result, spin s massless particle is represented by only one state, helicity eigenstate, and not by 1.2 One-particle states 9

2s + 1 states as in the massive case. This is an important diﬀerence. In order to get parity conserving electromagnetism with photons having either helicity + or helicity – states, one puts those two, in principle diﬀerent, helicity states into one theory.

1.2.1 Fields and probability amplitudes We have now all what is needed to develop RQM and to describe fundamental particles and their interactions. But before we move on, we will pose for a short moment to look at a larger picture of which RQM is only a part. The Dirac equation, for example, can be studied in the context of classical fields (CF) or RQF or RQM. The algebra would be often identical but basic objects and the interpretation different. The most natural way to progress from where we are right now, would be to study CF. The model for that would be the electro-magnetism, or classical tensor field F µν or 4-vector potential field Aµ;

F µν = ∂µAν ∂ν Aµ − Then such a field, for example a classical electron field, i.e. a classical Dirac spinor field Ψ, would be quantized, promoting Ψ of CF to a relativistic quantum field operator Ψ of RQF. Probability amplitudes would be obtained by Ψ of RQF acting on particle states living in a suitably constructed space. In RQM, Ψ of RQM represents a particle state and it isn’t an operator. Describing it in a basis of eigenstates of a Hermitian operator representing a physical observable one gets a probability amplitude. A Ψ of RQM looks like a Ψ of CF but they are different. Algebraically they are the same but a large modulus of Ψ18 of CF would correspond to many electrons in a considered small 18To be defined shortly; one can think volume (similarly, a large electric field created by a focused laser light about something like Ψ 2 in non- | | corresponds to many photons) and if we would multiply that field by a relativistic quantum mechanics big number, the number of electrons would increase. A corresponding quantity in RQM would represent a large probability to find an electron in that volume. Multiplying Ψ by a big number would not change anything because in comparison, there is an extra requirement of the normalization. Also a question about how to measure a physical quantity leads to different considerations and a different answer in each case. For example one can measure a phase of a classical electric field but one can’t measure a phase of Ψ in RQM. As mentioned earlier, in few lectures, we would not be able to study CF and RQF; a one year course at a basic level would be needed for that alone. Fortunately, considering the most important, aspects of physics being studied in this course, RQF would be giving the same results as those which we will obtain in RQM. Differences will be in details beyond leading effects. The one important exception is that we will be missing the idea of a vacuum state. In RQF, a vacuum is not a nothingness, although particles are absent. For example, QCD vacuum is a very complicated state. 10 Relativistic Quantum Mechanics

1.3 The Klein–Gordon equation

RQM of spin 0 particles was considered by Schroedinger first, before he published his famous equation for the non-relativistic case. He aban- doned RQM because of formal difficulties which were understood many years later. Here, we will see what they are and we will define an area of RQM applicability. As argued earlier, spin 0 particles with

µ 2 p pµ = m > 0, (1.17)

in the position representation, would be described by a scalar wave function exp( ipµx ). Replacing the energy by i ∂ and the momentum ∼ − µ ∂t 19Or the Klein–Gordon–Fock equation. by i in eqn 1.17, one gets the Klein–Gordon (K–G) equation19 of RQM− ∇ in position representation:

( + m2)Ψ(t, x) = 0, (1.18)

µ ∂2 2 where  = ∂ ∂µ = ∂t2 . For a particle at rest, i Ψ(t, x) = 0, pµ = i∂µ only time (the proper time− ∇τ) derivative would be present− in∇ eqn 1.18 and ∂ ∂ p0 = i = i there would be two independent solutions: Ψ±(τ, x) = exp( imτ)Ψ±(0, 0). 0 ∂x ∂t Therefore in a frame in which the particle has a momentum∓p and the en- i i ∂ p = i∂ = i∂i = i 2 2 − − ∂xi ergy Ep = + p + m > 0 (the subscript p for the plus sign; indicating 20 . that Ep is positive), we could expect 20 µ  mτ = p xµ Ψ+(t, x) = Nexp( ip x) = Nexp( iE t + ip x) (1.19) − · − p · Instead of boosting the other solution we take the complex conjugate of 21Why one is doing that should become Ψ+(t, x) to get21 clear after reading section 1.3.1 Ψ−(t, x) = Nexp(+ip x) = Nexp(+iE t ip x) (1.20) · p − · where N is a normalization constant which will be defined shortly. By a direct substitution, one can check that a general solution of eqn 1.18 + is indeed a linear combination of Ψ (t, x) and Ψ−(t, x). + We obtained, as expected, Ψ (t, x) but in addition, we also got Ψ−(t, x). This is the first puzzle of RQM, the nature of which will be becoming clearer when we progress a little further. Both solutions of the K–G eqn are eigenfunctions of the energy oper- ∂ + ator i ∂t ;Ψ−(t, x) with an eigenvalue of Ep and Ψ (t, x) with Ep, a negative energy for a free particle! − In exactly the same way as in the case of the non-relativistic Schroedinger eqn, one can derive the continuity equation for a probability density ρ and a probability current j: ∂ρ + j = 0 (1.21) ∂t ∇ · where ∂Ψ ∂Ψ∗ ρ = i(Ψ∗ Ψ ) (1.22) ∂t − ∂t 1.3 The Klein–Gordon equation 11 and j = i(Ψ∗ Ψ Ψ Ψ∗). (1.23) − ∇ − ∇ The probability current comes out to be given by the same expression as the non-relativistic one but the probability density is different, although nicely symmetric to the current and we can define a 4-vector current µ µ µ j (ρ, j) = i(Ψ∗∂ Ψ Ψ∂ Ψ∗). The continuity eqn 1.21 can be then ≡ µ − written as ∂µj = 0. The corresponding conserved quantity is the total probability which one gets integrating j0 = ρ over the 3D space. The underling symmetry is the invariance with respect to the multiplication by a global phase factor: physics described by Ψ is identical to physics described by exp(iθ)Ψ for any fixed real parameter θ22. 22One can prove that considering Substituting Ψ+(t, x) from eqn 1.19 into 1.22 and 1.23 we obtain the Klein–Gordon Lagrangian and the Noether theorem. + 2 + 2 ρ = 2 N Ep and j = 2 N p. (1.24) | | | | Now we can fix the normalization N. In the non-relativistic quantum mechanics, a volume integral of the probability density was a constant; 1 for one particle in the whole space. This doesn’t work in RQM because there is the Lorentz contraction which modifies the volume, contracting one side of a cube, parallel to the Lorentz boost, by the Lorentz factor γ. To keep the integral independent of the Lorentz transformation, the probability density, should grow by the same γ23. So putting N = 1 23No problem, as ρ is the time-like would do the job, as any other constant would do it as well. The choice of component of a 4-vector N = 1 is called the covariant normalization and corresponds to 2Ep particles in a unit volume. Another popular choice is N = 1/ (2m) which at the non-relativistic limit, E m makes ρ Ψ Ψ and j velocity p ∗  approaching expressions of the non-relativistic→ → quantum mechanics.→ For Ψ−(t, x), eqn 1.24 becomes

2 2 ρ− = 2 N Ep and j− = 2 N p. (1.25) − | | − | | Summarizing, Ψ+(t, x) and related observables, the energy, probability density and the probability current come out as expected and behave nicely at the non-relativistic limit. In contrast to Ψ+(t, x) an unexpected additional wave function Ψ−(t, x), describes a free particle with negative energy, negative probability density and with the probability current ﬂowing in the opposite direction to the particle’s momentum; all properties unexpected and diﬃcult to accept.

1.3.1 The Feynman–Stueckelberg interpretation of negative energy states Adapt from the Feynman lecture; the 1986 Dirac memorial lecture. In summary, negative energy solutions of the Klein–Gordon eqn represent antiparticles with positive energy. The probability density represent the charge density and can be either negative or positive. Similarly for the probability current, representing the charge current; the number of charges passing through the unit area per unit time. 12 Relativistic Quantum Mechanics

Inclusion of interactions via a potential Now, we will introduce interactions using a potential and see what happens. Following the way a potential V was introduced into the 24For a proper discussion on interac- non-relativistic Schroedinger eqn, we will modify the energy operator24: tions one needs to wait until the section i ∂ i ∂ V transforming eqn 1.18 into 1.5 ∂t → ∂t − ∂ (i V )2Ψ = ( 2 + m2)Ψ ∂t − −∇ which for time independent, time-like potential V and for energy eigen- 25 25 states with the energy Ep, and in 1-dimension s, becomes ∂ i Ψ = EpΨ ∂t 2 2 ∂ 2 Ψ(t, s) = ψ(s)exp( iEpt) (Ep V (s)) ψ(s) = ( + m )ψ(s). (1.26) − − −∂s2 Let’s consider a time-like potential barrier of fixed hight V > 0 for s 0, as shown in fig. 1.4 The wave function ψ(s) consists of incident, Iexp≥ (ips), reflected, Rexp( ips), and transmitted, T exp(iks) waves: − ψ (s) = Iexp(ips) + Rexp( ips) L −

ψR(s) = T exp(iks),

where ψ(s) = ψL(s) for s < 0 and ψ(s) = ψR(s) for s 0. Substituting 2 2 2 ≥ 2 2 2 ψL and ψR into eqn 1.26, one gets E = p +m and (E V ) = k +m leading to p = (E2 m2) and k = ((E V )2 − m2). In both ± − ± − − cases one chooses + sign in front of () to match expected propagation   directions as in ﬁg. 1.4.  dψ(s) From the continuity condition at s = 0 for ψ(s) and ds one gets

I + R = T pI pR = kT −

2p p k Fig. 1.4 A time-like potential barrier of and T = I R = − I. (1.27) hight V. Incoming, reflected and trans- p + k p + k mitted waves are also indicated. Probability currents along s for s < 0 and s 0 are ≥ p 2 2 k 2 j = ( I R ) jR = T (1.28) L m | | − | | m | | We will keep the energy E fixed and consider three different regions of the potential strength. The first region is of a weak potential, E > V + m where k is real and k < p. Probability densities in two regions are E E V ρ = ψ 2 > 0 ρ = − ψ 2 > 0 L m | L| R m | R| and this case looks like the non-relativistic one; nothing special, a small fraction of the incoming wave is reflected and the rest is transmitted. 1.3 The Klein–Gordon equation 13

In the second region, the potential is of a moderate strength, V m < E < V + m and k = i (m2 (E V )2)) = iκ is pure imaginary− (κ real) and − −  p iκ R = − R = I and j = 0. p + iκ ⇒ | | | | L The incoming wave is totally reﬂected and the probability density in the barrier shows expected exponential decay E V E V ρ = − ψ 2 = − exp( 2κs) R m | R| m − like in the non-relativistic case. The situation is however not identical because growing potential V changes the sign of the probability density from positive (ρR was > 0 in the ﬁrst case; weak potential) to negative:

E > V ⇒ ρR > 0 but if

E < V ⇒ ρR < 0. (1.29) We will come back to this after discussing now the case of the strong potential, E < V m, when k becomes pure real and k2 > p2. The probability, which− became negative when the potential became strong enough in the previous case, stays negative and the probability current becomes real inside the barrier: E V k ρ = − T 2 < 0 and j = T 2 . R m | | R m | | We will consider now, previously not physical case of k < 0, because, in a counter-intuitive way, this case now corresponds to a particle moving to the right. To see that we will calculate the group velocity ∂E k = = > 0. Vg ∂k E V − A consequence of that is that T and R given by eqn 1.27 can be arbitrary large, possibly making the reflected current bigger than the incoming one; the Klein paradox, another puzzle of RQM. The reason for this is that the very strong potential provides the energy needed to produce particle-antiparticle pairs. The probability current and the probability density within the barrier are negative because created antiparticles (see eqn 1.25) are attracted to the barrier; moving to the right ( g > 0). Pro- duced particles are repelled by the barrier, moving to theV left, increas- ing the reflected probability current. Such a situation can be created by focussing light from a high power laser, making very strong electric field which in turn produces electron-positron pairs from the vacuum. One can also think about the Hawking radiation in a neighbourhood of a black hole. The fact that the probability density became negative already in the case of a moderate strength potential, corresponds 14 Relativistic Quantum Mechanics

to the vacuum polarization by creation of virtual particle-antiparticle pairs which didn’t influence probability currents because there was not enough energy in the system to promote them to become real particles. Looking for an analogy, one can probably think about the Lamb shift in atomic physics where the vacuum polarizations affects energy levels. The problem of RQM is now clear: the formalism describes one particle (or a fixed number of particles) but physics needs many particles; the number of which can’t be fixed; particles can be created and particles can be annihilated. One needs RQF to describe such physics. RQM can be used, as long as the number of particles is fixed. Considering the limit of the uncertainty relation p s ¯h, we see that the pair creation which starts at p mc sets  the∼ limit on the s ¯h/(mc). So as long as we are studying ∼ physics at a scale bigger than h/ ¯∼(mc), known as the Compton length, RQM can be applied. Atomic physics is an example when this condition is fulfilled. But RQM can also be applied for many processes of high energy particle physics. A representative example could be the electron-positron annihilation producing hadrons; many + hadrons. A fundamental process in that case is e + e− q + ¯q. The number of particles is 2 and is fixed and the change from→ 2 leptons to 2 quarks can be handled by RQM. A fragmentation of quarks to hadrons is taking place on a different, much slower time scale and therefore can + be separated from the fundamental process of the e + e− annihilation. Is there any limit on p? How well can one measure momentum? In the non-relativistic quantum mechanics, momentum can be measured with any precision but due to the limit speed < c, ¯h p t , References   ∼ c see the Introduction chapter in (5). Infinite precision p 0 requires infinite measurement time t .  → → ∞ 1 Steane A.M., Relativity1.4 made The relatively Dirac easy, equation to be published. 2 Schutz B.F., A First Course in General Relativity, Cambridge Univer- sity Press (1985). This is the main section of the book on RQM. First, we will consider 3 Misner C.W., Thornedifferent K.S. representationsand Wheeler J.A., of theGravitation, Dirac eqn, W.H. a probability Free- current and bi- man and Companylinear (1995). covariants. Then, we will find wave functions describing free spin , neutron interference1/2 particles experiment. like electrons, discuss chirality and helicity operators before applying them to describe the Standard Model (SM) interactions at the Landau course, Bierestecki W.B., Lifshitz E.M. and Pitajewski L.P., limit when masses can be neglected. Electromagnetic interactions will Relativistic Quantum Theory part I. 5 be introduced via, so called, minimal coupling and the non-relativistic Maggiore M., A Modernlimit will Introduction be obtained. to Quantum Field Theory, Ox- 6 ford University Press (2005). ξ Combining ξα and η into one Dirac spinor Ψ, Ψ = , the Dirac β˙ η Halzen F. and Martin A.D., Quarks & Leptons: An Introductory   7 Course in Moderneqn Particle 1.12 canPhysics, be written Wiley (1984).as Aitchison I.J.R. and Hey A.J.G., Gauge Theories0 p in+ Particlepσ Physics, 0 Ψ = mΨ. (1.30) 8 3rd edition, Institute of Physics Publishingp (2003).pσ 0  0 −  Perkins D.H., IntroductionInstead of to Ψ, High we could Energy use Physics, Ψ = UΨ, 3rd where edition,U is a unitary operator. Addison–Wesley (1987).In the new basis, eqn 1.30 would look different. So, in general, the Dirac Perkins D.H., Introduction to High Energy Physics, 4th edition, Cam- bridge University Press (2000).