-
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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