20211021 Quantum Mechanics II

Special Relativity Preparatory Course

Teaching Assistant: Oz Davidi

October 18-19, 2020

Disclaimer: These notes should not replace a course in special relativity, but should serve

as a reminder. If some of the topics here are unfamiliar, it is recommended to read one of the

references below or any other relevant literature.

Notations and Conventions

1. We use τ as a short for 2π.1

References

There exist lots of references to this subject. Many books about general relativity include

good explanations in their first chapters. Other sources are advanced books on mechanics and

electromagnetism. Here is a list of some examples which cover the subject from those different

points of view. In each of them, look for the relevant chapters.

1. Classical Mechanics, H. Goldstein.

2. Classical Electrodynamics, J. D. Jackson.

3. A First Course in General Relativity, B. F. Schutz.

4. Gravitation and Cosmology, S. Weinberg.

1See https://tauday.com/tau-manifesto for further reading.

1

https://tauday.com/tau-manifesto

2 INDEX NOTATION

1 Motivation

One of the main topics of the Quantum Mechanics II course is to develop a (special) relativistic

treatment of quantum mechanics, which is done in the framework of quantum field theory. We

will learn how to quantize (relativistic) scalar and fermionic fields, and about their interactions.

For this end, a basic knowledge in special relativity is needed.

2 Index Notation

We will find that index notation is the most convenient way to deal with vectors, matrices, and

tensors in general. Let us focus on tensors of rank 2 and below.

• For a vector ~v =(v1 v2 · · · vn

)T, we denote the i’s component by vi.

• For a matrix M =

m11 m12 · · ·m21 m22 · · ·

......

. . .

, we denote the [ij]’s entry by Mij.Notice: In general, Mij 6= Mji, but Mji =

[MT

]ij

.

• When we multiply a vector by a matrix from the left, we get a new vector ~u = M~v. Thei’s component of the new vector is given by ui = [M~v]i =

∑jMijvj.

From now on, we will use Einstein’s Summation Convention:

1. If an index appears twice, we sum over it

Mijvj ≡∑j

Mijvj . (2.1)

2. An index will NEVER appear more then twice!

• What about multiplying by a matrix from the right, ~vTM? Again, we get a new vector~wT = ~vTM . In index notation

[~wT]i

= [~w]i = wi =[~vTM

]i

=[~vT]jMji = vjMji.

Here is an example why this is so useful:

2

3 FAST INTRODUCTION TO SPECIAL RELATIVITY

Exercise 2.1. Prove that the cross product of two three dimensional vectors ~v and ~w can be

written as

[~v × ~w]i = �ijkvjwk , (2.2)

where �ijk is the fully anti-symmetric Levi-Civita tensor, with �123 = 1.

Exercise 2.2. Prove the following identity

�ijk�lmk = δilδjm − δimδjl . (2.3)

where δij is the Kronecker delta.

Example 2.1. Prove that

~u×(~v × ~w) = ~v (~u · ~w)− (~u · ~v) ~w . (2.4)

Proof. By using index notation

[~u×(~v × ~w)]i = �ijkuj [~v × ~w]k= �ijk�klmujvlwm

= �ijk�lmkujvlwm

= (δilδjm − δimδjl)ujvlwm= viujwj − ujvjwi= [~v (~u · ~w)− (~u · ~v) ~w]i .

3 Fast Introduction to Special Relativity

3.1 Defining Special Relativity (B. F. Schutz: 1.1, 1.2)

At first, Einstein’s theory of special relativity was understood algebraically, as a set of (Lorentz)

transformations that move us from one inertial observer’s system to another. Special relativity

can be deduced from two fundamental postulates:

1. Principle of Relativity (Galileo): No experiment can measure the absolute velocity of an

observer; the results of any experiment performed by an observer do not depend on his

speed relative to other observers who are not involved in the experiment.

3

3.2 Transformation Rules 3 FAST INTRODUCTION TO SPECIAL RELATIVITY

2. Universality of the Speed of Light (Einstein): The speed of light relative to any unacceler-

ated (inertial) observer is 3×108 m/s, regardless of the motion of the light’s source relativeto the observer. Let us be quite clear about this postulate’s meaning: two different iner-

tial observers measuring the speed of the same photon will each find it to be moving at

c = 3× 108 m/s relative to themselves, regardless of their state of motion relative to eachother.

But what is an “inertial observer”? An inertial observer is simply a coordinate system

for spacetime, which makes an observation by recording the location(x y z

)and time

(t)

of any event. This coordinate system must satisfy the following three properties to be called

inertial :

1. The distance between point P1 =(x1 y1 z1

)and point P2 =

(x2 y2 z2

)is indepen-

dent of time.

2. The clocks that sit at every point, ticking off the time coordinate t, are synchronized and

all run at the same rate.

3. The geometry of space at any constant time t is Euclidean.

3.2 Transformation Rules

Let us derive the transformation rules of special relativity in 1 + 1 dimensions (1 space and 1

time dimensions).

• Imagine two systems (observers), O and O′, with respective velocity v between them.

• We choose that at t = t′ = 0, the origins of the two observers coincide.

• The position of a wave-front in system O is measured to be

x = ct . (3.1)

• We would like to see how this wave form is seen (parametrized) in system O′. We take thetransformation to be linear x′ = ax + bt, where a (which is dimensionless) and b (which

has dimensions of velocity) will be found below. The physical reason is that we want

O −→T1O′ −→

T2O′′ to be identical to O −−−−−→

T1“+”T2O′′ (you can compare it to rotations).2

2We will make this statement more precise once we study group theory.

4

3.2 Transformation Rules 3 FAST INTRODUCTION TO SPECIAL RELATIVITY

• The origin of O′ in the O system is given by x = vt, hence

0 = (av + b) t =⇒ b = −av =⇒ x′ = a(x− vt) . (3.2)

• The inverse transformation is given by changing the sign of the velocity, namely

x = a(x′ + vt′) . (3.3)

• Plugging x′ into x gives

t′ = at+(1− a2)x

av. (3.4)

• Now, we demand that a wave-front in O, i.e. x = ct, is also a wave-front in O′, i.e.x′ = ct′ (here, we demand that the speed of light is the same for all observers). By using

the expression for x′, Eq. (3.2), and the expression for t′, Eq. (3.4), and substituting

x = ct, we get

a(c− v) = ca+ (1− a2) c2

av. (3.5)

Solving for a, one gets

a ≡ γ = 1√1− v2

c2

. (3.6)

To summarize, the transformation rules are

x′ = γ(x− vt) , (3.7)

t′ = γ(t− v

c2x). (3.8)

As an important side note: Always check the dimensions of the quantities you look for.

Indeed, a turned out to be dimensionless, and b has the dimensions of velocity.

An immediate result is that the time coordinate is not universal! This is depicted in Fig. 1.

In classical mechanics, an event A =(tA xA yA zA

)shares the same time with an infinite

number of events B =(tA xB yB zB

). They all have the same time, meaning that events

that happen simultaneously at one inertial system, also happen at the same time in another.

On the other hand, the same event A, under special relativity, has a unique “now”. Otherevents have their own “now”, hence different observers may not agree on the relative time

between events.

A trajectory x(t), of a particle for example, is called a world line. A world line must cross a

5

3.2 Transformation Rules 3 FAST INTRODUCTION TO SPECIAL RELATIVITY

Figure 1: Spacetime structure in classical mechanics (top) and special relativity (bottom). In classical me-chanics, a universal time slice exists, while in special relativity, each event defines a light-cone. (B. F. Schutz:1.6)

constant time slice once (and only once), but the crossing point can be at any point, depending

on the observer. The slope of a world line is the velocity reciprocal, v−1 = ẋ−1. Because the

velocity is bounded from above by the same constant value c in all reference frames, at each

point of the trajectory x(t), one can draw a light-cone, and all inertial observers will agree that

the trajectory is within this light-cone.

We will sometime use β = v/c, and from now on, we set the speed of light to 1

c = 1 . (3.9)

Using the general 3 + 1 dimensional transformation rules, one can show that while differ-

ent inertial observers determine different world-lines for the same particle, they agree on the

distance

(∆t)2 − (∆x)2 = (∆t′)2 − (∆x′)2 . (3.10)

6

4 THE METRIC

We take the infinitesimal limit, and define the interval ds2

ds2 ≡ dt2 − dx2 = dt′2 − dx′2 . (3.11)

The interval is a Lorentz scalar - it is invariant under Lorentz transformations (to be discussed

in Sec. 5).

4 The Metric

Minkowski pointed out that space (~x) and time (t) should be treated all as coordinates of a four-

dimensional space, which we now call spacetime. We thus define the spacetime four-vector

xµ =(t ~x

). The index µ ∈ { 0, 1, 2, 3 } is called the Lorentz index.

The Minkowski spacetime is not Euclidean. In order to measure distances, we define the

metric as a symmetric function which maps two four-vectors to R

g(v1, v2) = g(v2, v1) ∈ R . (4.1)

Note that we did not define g to be positive definite. We parametrize the metric by the rank-2

tensor ηµν = diag(1,−1,−1,−1).3 The inverse metric, ηµν , defined by

3∑ρ=0

ηµρηρν = δµν , (4.2)

is ηµν = diag(1,−1,−1,−1).

Einstein’s Summation Convention (Revisited):

1. If an index appears twice, once as a lower and once as an upper index, we sum over

it.

2. An index will NEVER appear twice as a lower/an upper index!

3. An index will NEVER appear more then twice!

3This is the choice in the Quantum Mechanics II course. Some other sources use ηµν = diag(−1, 1, 1, 1), soyou need to pay attention to the convention used in the book you read!

7

5 LORENTZ TRANSFORMATIONS

We raise and lower indices using the metric

vµ = ηµνvν , vµ = ηµνvν . (4.3)

Example 4.1. What is xµ?

xµ = ηµνxν =

(t −~x

). (4.4)

Exercise 4.1. What is ηµν?

Exercise 4.2. What is ηµµ?

The interval, Eq. (3.11), can be now defined as

ds2 = ηµνdxµdxν . (4.5)

The modern way to define special relativity is the following. We start from the interval/metric,

and ask what are the symmetries of the system. Namely, what are the transformation rules

between different frames of reference that leave the interval invariant.

5 Lorentz Transformations

The allowed transformations, dxµ → dx′µ = Λµνdxν , are those that leave the interval ds2

invariant

ηµνdxµdxν → ηµνdx′µdx′ν = ηµνΛµρdxρΛνσdxσ =

[ΛTηΛ

]ρσdxρdxσ

!= ηρσdx

ρdxσ . (5.1)

We see that Lorentz transformations are given by all transformations that preserve the metric.4

Comment: Note that Λµν 6=(ΛT) µν

! Tensors do not behave as simple matrices. We use

the matrix notation to make expressions more familiar and intuitive, but the position of the

indices dictates the identity of the tensor! We will go back to this when we learn about Lorentz

transformations in the course. When dealing with tensors, pay special attention.

Let us count degrees of freedom:

• Λµν has 16 parameters (µ, ν ∈ { 0, 1, 2, 3 }).4In the second part of the Quantum Mechanics II course, we will define the Lorentz transformations as the

symmetries of spacetime (with the flat metric ηµν), and using group theory, we will gain a lot of informationwhich is not evident when looking at Eqs. (3.7,3.8).

8

6 HYPARBOLIC STRUCTURE OF SPACETIME

• We have the condition ηµν = ηρσΛρµΛσν −→

µ = ν gives 4 equationsµ 6= ν gives 6 equations .• 16− 4− 6 = 6 −→ in general Λµν has 6 continuous parameters.

How do we interpret them?

• Three boosts, e.g. Λµν =

cosh(ηx) sinh(ηx) 0 0

sinh(ηx) cosh(ηx) 0 0

0 0 1 0

0 0 0 1

.

• Three rotations, e.g. Λµν =

1 0 0 0

0 1 0 0

0 0 cos(θx) sin(θx)

0 0 − sin(θx) cos(θx)

.This gives Λ(ηi, θi). We will come back to that when we arrive to the subject of group theory.

Defining arctanh(ηx) =vxc

, we get the same transformation between two inertial observers,

that we developed in the old-fashion point of view in Eqs. (3.7,3.8).

6 Hyparbolic Structure of Spacetime

Recall that ds2 = dt2 − d~x2, and that all inertial observers measure the same value of ds2. Wecan plot curves of constant values of ds2. A curve determines the same event as observed by

different inertial observers which have the same origin. We distinguish between three different

scenarios:

• Light-Like: For light, ds2 = 0 ⇒ dxdt

= ±1. This describes the dashed curves of Fig. 2.All inertial observers with same origin will determine the same curve.

• Time-Like: ds2 > 0. This describes the top and bottom parts of Fig. 2. Here, causalorder is well defined, i.e. all observers will agree which event happened before another

event.

• Space-Like: ds2 < 0. This describes the right and left parts of Fig. 2. Here, causal orderis not well defined, namely events which are space-like cannot affect each other.

9

7 THE POLE, THE BARN, AND SCHRÖDINGER’S CAT

Figure 2: Minkowski spacetime. The curves describe constant value of the interval ds2 = dt2 − d~x2.

As an example, consider Rocket Raccoon and Groot, which are sitting at rest on the x axis:

Rocket at x = 1, and Groot at x = 2. They synchronized their clocks in advance, and decided

to push a button at t = 0.5 Drax the Destroyer and Mantis, sitting at rest at the origin, will

say that Rocket and Groot pushed the button at the same time, and will mark the blue squares

of Fig. 2.

Star-Lord on the Milano, flying to the right direction (and passes through x = 0 at t = 0),

will say that Rocket (x = 1 in the rest frame) pushed the button after Groot (x = 2 in the rest

frame). He will mark the red squares of Fig. 2.

Finally, Gamora on the Benatar, flying to the left direction (and passes through x = 0 at

t = 0), will say that Rocket (x = 1 in the rest frame) pushed the button before Groot (x = 2

in the rest frame). She will mark the green squares of Fig. 2.

7 The Pole, the Barn, and Schrödinger’s Cat

Einstein sits at rest inside his barn. His rest frame is denoted by O, and the length of his barnin this frame is Lbarn = 10 m. Einstein’s barn has two doors. The left one is open, and the

5Hopefully, Groot pushes the right button https://youtu.be/Hrimfgjf4k8.

10

https://youtu.be/Hrimfgjf4k8

7 THE POLE, THE BARN, AND SCHRÖDINGER’S CAT

Figure 3: The pole and the barn paradox - Schrödinger’s cat version. Heisenberg (H) and Schrödinger (S)sit on a pole that moves to the right with velocity β. Einstein (E) sits inside a barn. wµ and w′µ are thecoordinates of the event of measuring the pole’s tail position, and yµ and y′µ are the coordinates of the eventof measuring the pole’s head position, in the barn frame O and pole frame O′ respectively. zµ and z′µ are thecoordinates of the event of Heisenberg entering the barn.

right one is closed.

Meanwhile, Heisenberg and Schrödinger are sitting on a horizontal pole which arrives from

the left of the barn with a velocity β =√

3/2. Heisenberg sits on the tail of the pole (the

leftmost point of the pole), while Schrödinger sits on the head of the pole, and he holds a box

with a quantum cat inside. The rest frame of Heisenberg and Schrödinger is denoted by O′,and the length of the pole in this frame is Lpole = 10 m.

1. Einstein has a smart barn. When Schrödinger reaches the right door, it opens automat-

ically. Because he doesn’t like to keep all doors open, as soon as Heisenberg enters the

barn, the left door automatically closes.

2. Schrödinger told Einstein that when he will exit his barn, he is going to check whether

11

7 THE POLE, THE BARN, AND SCHRÖDINGER’S CAT

the cat is dead or alive. He doesn’t know that Heisenberg turned his box into a classical

system, by connecting it to a button. If Heisenberg pushes the button, the cat will

die, otherwise it lives. Heisenberg decides to push the death button when he enters the

barn. Schrödinger and Heisenberg do not know what will Schrödinger find inside the box.

Einstein, on the other hand, does know.

The first question is: how do we measure the length of the pole in the barn rest frame O?The way we measure length is by determining the x coordinate of the head and tail of the pole

at the same time.

• We choose the origin of O and O′ to determine the head of the pole and the entrance ofthe barn at t = t′ = 0. This is the point y = y′ =

(0 0

)in Fig. 3.

• The point wµ (see figure) is the length of the pole in the barn rest frame. By definition,its time component is set to zero, so wµ =

(0 −`pole

). Here, `pole stands for the yet

unknown length of the pole in the barn rest frame.

• Now, we determine the same point in the pole rest frame. Since both systems sharethe same origin, by definition, the space component of the tail of the pole is minus

its length, Lpole (recall that the pole does not move within its rest frame). Therefore,

w′µ =(T ′ −10

). Note that the time is yet unknown.

• β =√

3/2 ⇒ γ = (1− β2)−1/2 = 2. Using our transformation rules, Eqs. (3.7,3.8), andtheir inverse, we getw

µ =(

0 −`pole)

w′µ =(T ′ −10

) =⇒t = 0 = 2

(T ′ −

√32

10)

= γ(t′ + βx′)

x = −`pole = 2(−10 +

√32T ′)

= γ(x′ + βt′). (7.1)

The solutions are T ′ = 5√

3 and `pole = 5.

We see that the pole is shorter in the barn system by a factor of two. It means that in the barn

rest frame, O, at time tH = `pole/β = 10/√

3, Heisenberg enters the barn, and Schrödinger is still

inside. Both of the barn doors are closed, and the pole is locked inside the barn. Furthermore,

Heisenberg pushes the button, killing the cat before Schrödinger checks to see if it is alive or

not. When Schrödinger checks the box, the cat is dead.

The Pole and The Barn Paradox - We can repeat the exercise in the pole rest frame, O′,and find that the barn is twice shorter in this frame. How come that the pole can enter into a

barn which is half of its size?

12

8 LORENTZ SCALARS AND MORE FOUR-VECTORS

Solution - In the pole frame, the time it takes for Schrödinger to arrive to the right door

of the barn is (one can use Lorentz transformations, but we can derive everything using simple

kinematics) t′S = 5/β = 10/√

3. The time it takes for Heisenberg to arrive to the left door

is given by t′H = 10/β = 20/√

3 = 2t′S. According to Heisenberg’s (and Schrödinger’s) clock,

Heisenberg enters the barn after Schrödinger had already exited, with a time difference of t′S.

The left door closes only after the right door opens.

Schrödinger’s Cat Paradox - In the pole frame, Schrödinger checks the cat’s status before

Heisenberg pushes the button. This means that the cat is alive! But it is dead in the barn

frame...

Solution - In order to affect the cat’s condition before Schrödinger exits the barn in the

barn frame, Heisenberg must send a signal which is faster than light. The interval between the

events (Heisenberg sending a signal and Schrödinger checking the box) is space-like - the events

are causally disconnected! Because Heisenberg can’t control the fate of the cat, and because

cats usually have more than one soul anyway, the cat is alive, and can be found outside of the

faculty building.

8 Lorentz Scalars and More Four-Vectors

In order to get Lorentz scalars, i.e. objects that do not transform under Lorentz transforma-

tions, we contract all Lorentz indices using the metric.

A ·B ≡ ηµνAµBν = AµBµ , (8.1)

A ·B −→ A′ ·B′ = ηµνΛµρΛνσAρBσ = ηρσAρBσ = A ·B . (8.2)

Let us consider few important examples related to four-vectors.

8.1 Four-Momentum

The four momentum is given by

pµ =(E ~p

), (8.3)

where E is the energy, and ~p is the three momentum. The scalar which is obtained by taking

the four-momentum square, is the particle’s rest mass squared

p2 ≡ p · p = pµpµ = m2 . (8.4)

13

8.2 Derivatives, Currents and Electromagnetism (J. D. Jackson: 11.6, 11.9)8 LORENTZ SCALARS AND MORE FOUR-VECTORS

The four-momentum is related to the four-velocity by pµ = muµ, where uµ ≡ dxµ√ds2

= γ(

1 ~v)

.

Note that for light, γ →∞ but m→ 0, so the four-momentum is well defined, and E = |~p|.

8.2 Derivatives, Currents and Electromagnetism (J. D. Jackson:

11.6, 11.9)

Another important four-vector is the four-derivative ∂µ ≡ ∂∂xµ =(∂t ~∇

). It transforms as

∂′µ =∂

∂x′µ= Λ νµ ∂ν , as expected (see Jackson). Notice that while x

µ or pµ were defined with an

upper index (contravariant vectors), the four-derivative is defined with a lower index (covariant

vector).

The four-dimensional Laplacian, the d’Alembertian � ≡ ∂µ∂µ = ∂2t − ~∇2, is a Lorentzscalar, and hence also the wave equation.

Recall the continuity equation ∂ρ∂t

+ ~∇ · ~J = 0, which must hold at any frame of reference.We can define the four-current Jµ =

(ρ ~J

). Using this definition, the continuity equation is

clearly satisfied at all frames, and can be written as ∂µJµ = 0.

8.2.1 Electromagnetism

In Lorentz gauge, ∂tφ+ ~∇ · ~A = 0, the wave equations for the scalar and vector potentials are

∂2 ~A

∂t2− ~∇2 ~A = 2τ ~J , (8.5)

∂2φ

∂t2− ~∇2φ = 2τρ . (8.6)

It is just natural to define the four-potential Aµ =(φ ~A

). Then, the gauge can be written as

∂µAµ = 0, and the wave equations are

�Aµ = 2τJµ . (8.7)

You can check that the electromagnetic tensor is given by

Fµν = ∂µAν − ∂νAµ =

0 Ex Ey Ez

−Ex 0 −Bz By−Ey Bz 0 −Bx−Ez −By Bx 0

, (8.8)

where we used ~E = −∂ ~A∂t− ~∇φ and ~B = ~∇× ~A.

14

9 FOURIER TRANSFORM

Exercise 8.1. Find F µν = ηµρηνσFρσ.

How do Fµν and Fµν transform? Each index transforms as a vector, i.e.

F ′µν = Λρµ Λ

σν Fρσ , F

′µν = ΛµρΛνσF

ρσ . (8.9)

Exercise 8.2. Convince yourself that F µνFµν is a Lorentz scalar.

Exercise 8.3. Find the transformation rules of ~E and ~B by using Eq. (8.9).

Most of the time in relativistic theories, we use Aµ and Fµν , and not ~E and ~B.

9 Fourier transform

As a last comment, we will make use of the relativistic Fourier transform

f̃(p) =

∫d4x f(x) eiτ

x·ph , f(x) =

∫d4p

h4f̃(p) e−iτ

x·ph , (9.1)

where x · p is another useful Lorentz scalar.We welcome aboard Planck’s constant. This is the first place where we see an integration

between special relativity and quantum mechanics.

15

MotivationIndex NotationFast Introduction to Special RelativityDefining Special Relativity (B. F. Schutz: 1.1, 1.2)Transformation Rules

The MetricLorentz TransformationsHyparbolic Structure of SpacetimeThe Pole, the Barn, and Schrödinger's CatLorentz Scalars and More Four-VectorsFour-MomentumDerivatives, Currents and Electromagnetism (J. D. Jackson: 11.6, 11.9)Electromagnetism

Fourier transform