Ta

Pei Cheng
talk based on …
Oxford Univ Press
(2/
2013)
Einstein’s Physics
Atoms, Quanta, and Relativity

Derived,
Explained, and Appraised
ATOMIC NATURE OF MATTER
1. Molecular size from classical fluids
2.
The Brownian motion
WALKING IN EINSTEIN’S STEPS
16. Internal symmetry and gauge interactions
17.
The Kaluza
–
Klein theory and extra dimensions
SPECIAL RELATIVITY
9.
Prelude to special
relativity
10. The new kinematics and
E
=
mc
2
11. Geometric formulation of relativity
GENERAL RELATIVITY
12.
Towards a general theory of
relativity
13. Curved spacetime as a gravitational field
14. The Einstein field equation
15. Cosmology
3. Blackbody radiation: From Kirchhoff to
Planck
4. Einstein’s proposal of light quanta
5. Quantum theory of specific heat
6.
Waves, particles, and quantum jumps
7. Bose
–
Einstein statistics and condensation
8. Local reality and the Einstein
–
Bohr debate
QUANTUM THEORY
2TOC
Today’s talk
provides
without
math details
some highlights
in
historical context
The book
explains his Physics
in equations
Albert Einstein
1879
–
1955
Molecular size & Avogadro’s number
classical liquids with suspended particles
3Atoms
(4/1905)
U Zurich
doctoral thesis:
“
On the determination of molecular dimensions
”
→ 2 equations relating
P
&
N
A
to viscosity and diffusion coefficients
Hydrodynamics
Navier

Stokes equation,
balance of
osmotic and viscous forces
E’s most cited publication!
A careful measurement of this
zigzag motion through
a simple microscope would
allow us to deduce the
Avogadro number!
(11 days later)
the Brownian motion paper:
While
thermal forces change the direction and
magnitude
of the
velocity of a suspended particle on such a small
time

scale
that
it cannot
be measured,
the
overall
drift
of
such
a particle is observable quantity.
Fluctuation of a particle system
random walk
as the prototype of discrete system
P
t
N
RT
Dt
x
A
6
2
2
2
N
k
2
Jean Perrin
It finally convinced everyone, even the skeptics, of
the reality of molecules & atoms.
Blackbody Radiation
(
rad
in thermal equilibrium) = cavity radiation
4Quanta
1
Einstein, like Planck, arrived at the quantum hypothesis thru BBR
Kirchhoff (1860)
densities
= universal functions
2
nd
law
)
,
(
&
)
(
T
T
u
0
)
,
(
)
(
d
T
T
u
Maxwell EM radiation = a collection of oscillators
u = E
2
, B
2
~
oscillator
energy
kx
2
The ratio of
oscillating energy
to
frequency
is an
adiabatic
invariant
;
p = u/3
Stefan (1878) Boltzmann (1884):
Wien’s
displacement law
(1893):
4
)
(
aT
T
u
)
/
(
)
,
(
3
T
f
T
Wien’s distribution
(1896)
:
fits
data
well….
until IR
Planck’s distribution
(1900)
:
T
e
T
/
3
)
,
(
1
)
,
(
/
3
T
e
T
key: Wien 2

>1
var
Excellent fit
of all the data
Wien = high
ν
of Planck
What is the physics?
Planck found a relation
What microstate counting
W
that
can lead to this
S
via
Boltzmann’s principle
S=k
ln
W
?
Planck was “compelled” to make the hypothesis of
energy quantization
h
T
dU
dS
U
c
/
entropy
8
2
3
Einstein’s 1905 proposal of light quanta
was
not
a direct follow

up
of Planck’s
h
Rayleigh

Jeans = the low frequency limit of the successful Planck’s distribution
5Quanta 2
Einstein
used
Planck’s calculation
and
invoked the
equipartition theorem
of
stat
mech
to derive the
Rayleigh

Jeans law
noted its
solid theoretical foundation
and
the problem of
ultraviolet catastrophe
T
ν
2
0
d
u
U
c
2
3
8
kT
U
2
1
showing
BBR = clear challenge to classical physics
The high frequency limit (
Wien’s distribution
)
=
new physics
Statistical study of (BBR)
wien
entropy change due to volume change: (BBR)
wien
~
ideal gas
→
(BBR)
wien
=
a gas of light quanta
with energy of
Einstein arrived at energy quantization independently

cited Planck only in 2 places
h
concentrate on
The history of Rayleigh
–
Jeans law
:
•
June 1900
,
Rayleigh
, applying the equipartition theorem to radiation, he
obtained the result of
C
1
ν
2
T
.
Only a limit law? Intro cutoff
ρ = C
1
ν
2
T
exp(

C
2
ν
/T)
•
October
–
December 1900
, The
Planck
spectrum distribution was discovered;
energy quantization proposed two months later
•
March 1905
,
Einstein
correctly derived the R

J law
noted its
solid theoretical foundation
and the problem of
ultraviolet catastrophe
•
May 1905
,
Rayleigh
returned
with
a
derivation of
C
1
. But
missed a factor of
8
•
June 1905
,
James Jeans
corrected Rayleigh’s error…
But, explained away the incompatibility with experimental results by insisting that
the observed radiation was somehow out of thermal equilibrium
.
•
A.Pais: “It should really be called
Rayleigh

Einstein

Jeans law
”
.
6Quanta 2
kT
ν
2
3
c
8
An historical aside:
“Planck’s fortunate failure”?
The quantum idea
Einstein
vs
Planck
7Quanta 3
1906 Einstein
came in agreement with Planck’s.
Also
,
gave a new derivation of Planck’s law
I
t clearly explained why energy quantization
can cure ultraviolet catastrophe
The new physics must be applicable beyond
BBR
:
quantum theory of specific heat
W
h
K
max
Einstein
1905
:
as P’s
W

calculation unreliable…
E’s quantum “in
opposition” to
P’s quantum
Einstein
:
the
quantum idea must represent
new physics
;
proposed
photoelectric effect
as test.
Einstein’s photon idea was strongly resisted by the physics community for many years
because it conflicted with the known evidence for the
wave nature
of light
(Millikan 1916):
“
Einstein’s photoelectric equation . . . cannot in my judgment be looked upon at
present as resting upon any sort of a satisfactory theoretical foundation
”, even though
“
it actually represents very accurately the behavior” of the photoelectric effect
”.
Planck did not accept Einstein’s photon for at least 10 years
Planck
1900
:
is only a formal
relation, not physical (radiation not
inherently quantum: only during
transmission, packets of energy, somehow)
h
(1909)
Light quanta = particles ?
uh
Δu
h
E
E
u
Δu
d
v
E
c
E
W
RJ
2
2
2
2
2
2
3
2
:
particles
~
s
Wein'
:
waves
~
ˆ
8
Jeans'

Rayleigh
two!
the
of
fusion"
"
a
but
particles,
as
just
or
waves
as
just
Neither
on
distributi
s
Planck'
2
2
2
RJ
W
Planck
ΔE
ΔE
ΔE
8Quanta 4
T
kT
v
ΔE
v
E
T
E
kT
E
E
ΔE
2
2
2
2
2
ˆ
that
so
ˆ
ons
distributi
radiation
to
:
1904
of
n theory
fluctuatio
his
applied
Einstein
1st time stated
: quanta carried
by point

like particles
“
point of view of
Newtonian emission theory
”
Photon carries energy + momentum
particle
wave
factor
conversion
as
/
h
h
p
h
Wave

Particle Duality: a deep riddle
9 Quanta 5
1916
–
17, Einstein used Bohr’s quantum jump idea to construct a
microscopic theory of radiation
–
matter interaction:
absorption and
emission of photons
(
A and B coefficients
);
he showed how Planck’s
spectral distribution followed.
The central novelty and lasting feature is
the introduction of probability in quantum dynamics
Modern quantum mechanics :
states
= vectors in Hilbert space
(superposition)
observables
= operators
(commutation relations)
Classical radiation field = collection of oscillators
Quantum radiation field = collection of
quantum
oscillators
h
n
E
n
)
(
2
1
•
A
firm mathematical
foundation for Einstein’s photon idea
•
Quantum jumps
naturally accounted for by
ladder operators
ˆ
,
ˆ
1
~
ˆ
behavior
particle
h
a
a
n
n
a
Looking beyond Einstein:
His discoveries in quantum theory:
Wave/particle nature of light and quantum jumps
can all be accounted for in the framework of
quantum field theory
The picture of interactions broadened
QFT description:
Interaction
can change not only motion
,
but also allows for
emission and absorption of radiation
→
creation and annihilation of particles
“
three

man paper
” of
(Born, Heisenberg, and
Jordan
1926):
The same calculation of fluctuation of a system of waves,
but replacing classical field by operators
The riddle of wave
–
particle duality in radiation fluctuation
elegantly resolved in QFT
10Quanta
6
uh
u
Δu
a
h
a
a
e
a
e
a
Ae
jk
k
j
i
j
i
j
i
j
j
j
2
2
with
term
extra
an
about
bring
ˆ
ng
Noncommuti
ˆ
,
ˆ
ˆ
ˆ
Alas, Einstein never accepted this beautiful resolution
as he never accepted the new framework of quantum mechanics
forgotten history
Local reality & the Einstein

Bohr debate
•
Bell’s theorem
(1964) : these seemingly philosophical questions could lead to
observable results. The experimental vindication of the orthodox interpretation has
sharpened our appreciation of the nonlocal features of quantum mechanics. Einstein’s
criticism allowed a better understanding of the meaning of QM.
•
Nevertheless, the
counter

intuitive picture of objective reality
as offered by QM
still troubles many, leaving one to wonder whether quantum mechanics is ultimately
a complete theory
11Quanta 7
The orthodox view (measurement actually produces an object’s property)
the measurement of one part of an entangled quantum state would instantaneously
produce the value of another part, no matter how far the two parts have been separated.
Einstein, Podolsky
&
Rosen (1935)
: a thought experiment highlighting this
“
spooky action

at

a

distance
” feature ; the discussion and debate of “EPR paradox”
have illuminated some of the fundamental issues related to the meaning of QM
Orthodox interpretation of QM
(
Niels Bohr
& co): the attributes of a physical object
(position, momentum, spin, etc.) can be assigned only when they have been measured.
Local realist viewpoint of reality
(
Einstein
,…): a physical object has definite attributes
whether they have been measured or not. …. QM is an incomplete theory
Special Relativity
Maxwell’s equations: EM
wave
–
c
Contradict relativity? 2 inertial frames
x’ = x

vt
get velocity
add’n
rule
u’ = u

v
The then

accepted interpretation:
Max
eqns
valid only in the rest

frame of
ether
12SR 1
Q:
How should EM
be described for sources and observers
moving with respect to the ether

frame?
“The
electrodynamics of a moving
body”
Einstein’s very different approach ..
1895 Lorentz’s theory (a particular
dynamics
theory
of ether/matter
)
could account all
observation
stellar aberration, Fizeau’s
expt
…
to O(
v/c
)
[
+ a math construct
‘local time
’]
Michelson

Morley null result @ O(
v
2
/c
2
)
length contraction
Lorentz transformation
Maxwell ‘covariant’ to all orders (1904)
vt
x
x
'
x
c
v
t
t
2
'
Special Relativity
13SR 2
Case I
: moving charge in
B
(ether frame)
Lorentz force
(per unit charge)
Case II
: changing
B
induces an
E
via
Faraday’s law, resulting exactly the
same force.
yet such diff descriptions
B
v
e
f
▪
Invoke
the
principle of relativity
This equality can be understood naturally
as two cases have the same relative motion
▪
Dispense with
ether
The magnet

conductor thought
expt
c
onstructive theory
vs
theory of principle
Einstein’s very different approach ..
Relativity = a symmetry in physics
Physics unchanged under some transformation
How to reconcile (Galilean) relativity
u’ = u

v
with the constancy of
c
?
Resolution:
simultaneity is relative
Time is not absolute, but frame dependent
Relation among
inertial frames
Correctly given by Lorentz transformation,
with
Galilean transformation
as low
v/c
approx
t
t
'
The
new kinematics
allows for an simple derivation of the Lorentz transformation.
All
unfamiliar features follow from .
time dilation
,
length contraction, etc.
14SR 3
Special Relativity
1905
From “no absolute time” to the complete theory in five weeks
10yr
t
t
'
Transformation rule for EM fields, radiation energy,..
Lorentz force law from Max field equations
Work

energy theorem
to
mass

energy equivalence
E = mc
2
Even simpler perspective
Hermann Minkowski
(1907)
E
ssence of
SR:
time
is on an equal footing as
space
.
To bring out this, unite them in a single math structure,
spacetime
Geometric formulation
Emphasizes the
invariance
of the theory:
c → s
s
= a spacetime length
(c as the conversion factor)
Lorentz

transformation = rotation → SR features
4

tensor equations are automatically relativistic
1
1
1
1
2
2
2
2
2
2
g
metric
x
g
x
z
y
x
t
c
s
Special Relativity
Einstein was initially not impressed
,
calling it
“
superfluous
learnedness
”
15SR 4
SR:
The arena of physics is the
4D
spacetime
.. until he tried to formulate
General relativity
(non

inertial frames)
= Field theory of gravitation
Gravity = structure of
spacetime
SR = flat spacetime
GR = curved spacetime
The Equivalence Principle (1907)
played a key role in the formulation of
general theory of relativity
16GR 1
Why does GR principle automatically
bring gravity into consideration?
How is gravity related to spacetime?
starting from Galileo
Remarkable empirical observation
All objects fall with the same acceleration
“
Gravity disappears in a free fall frame
”
a ↑ = ↓ g
From mechanics
to electromagnetism… →
light deflection by gravity, time dilation
with such considerations...
Einstein proposed
a geometric theory of gravitation
in 1912
gravitational field = warped spacetime
Note: A curved space being locally flat, EP incorporated in GR gravity theory in a
fundamental way.
accelerated frame = inertial frame w/ gravity
EP as the handle of going from SR to GR
Einstein: “
My happiest thought
”
Source particle Field Test particle
Field
eqn
Eqn
of
motion
Source particle Curved spacetime Test particle
Einstein
field
eqn
Geodesic
Eqn
gravitational field = warped spacetime
metric tensor
[
g
μν
]
=
rela
.
grav
. potential
T
energy momentum tensor
N
g
Newton’s constant
1915
G
curvature tensor = nonlinear 2
nd
derivatives of
[
g
μν
]
Metric =
gravi
pot
Curvature = tidal forces
The
Einstein equation
10 coupled PDEs
s
olution =
[
g
μν
]
T
g
G
N
17GR 2
In the limit of test particles moving
with
non

relativistic velocity
in
a static
and
weak
grav
field
Einstein → Newton (
1/r
2
law explained!)
ie
new realms of gravity
In relativity, space

dep
→ time

dep
,
GR → gravitational wave
Indirect, but convincing, evidence thru decade

long observation of
Hulse

Taylor binary pulse system
3 classical tests
Grav
redshift
Bending of light
Precession of planet orbit
Black Holes
=
full power and glory of GR
Gravity so strong that even light cannot escape
Role of space and time is reversed
:
lightcones
tip over across the horizon
Alas, Einstein
never believed
the reality of BH
GR = field theory of gravitation
18GR 3
19
cosmo
(Einstein 1917)
The 1
st
paper on modern cosmology
The universe =
a phys system
the constituent elements being galaxies
Gravity the only relevant interaction
GR = natural framework for cosmology
Spatial homogeneity & isotropy
(the
cosmological principle
) →
Robertson

Walker metric :
k, a(t)
In order to produce a
static universe
he found
a way to introduce a
grav
repulsion in the
form of the cosmology constant
Λ
Easier to interpret it as a vacuum energy:
constant density and negative pressure →
repulsion that increases w/ distance
.
–
significant only on cosmological scale
T
g
g
G
N
Λ
= a great discovery
key ingredient of modern cosmology
Inflation
theory of the big bang: a large
Λ→
the universe underwent an explosive
superluminal expansion in the earliest mo
Λ
=
dark energy
→ the U’s expansion to
accelerate
in the present epoch
The concordant
Λ
CDM cosmology
Cosmology
Einstein equation
derivatives
Expanding Universe
0
)
(
t
a
GR
provide the
framework !
Still,
Einstein missed the chance of its
prediction before the discovery in late 1920’s
20 sym
Einstein and the symmetry principle
Before Einstein
, symmetries were generally regarded
as mathematical curiosities of great value to
crystallographers, but hardly worthy to be included
among the fundamental laws of physics.
We now
understand
that a symmetry principle is
not only an organizational device,
but also
a method to discover new dynamics
.
0
ˆ
0
'
'
'
0
ˆ
'
n
nsformatio
vector tra
a
m
F
R
a
m
F
a
m
F
A
R
A
A
Rotation symmetry
Tensors have def transf property
Tensor equations are
automatically
rotational symmetric.
Spacetime

independent
Global symmetry
R
ˆ
Special relativity
=
Lorentz transformation
4

tensor
eqns
are auto relativistic
R
ˆ
General relativity
curved spacetime with moving basis vectors
spacetime
–
dependent
metric [
g
] = [
g(x)
]
general coord transf = spacetime dependent
Local symmetry
)
(
ˆ
ˆ
x
R
R
Differentiation results in a non

tensor
0
ˆ
d
ˆ
'
R
dA
R
dA
Must replace by
covariant differentiation
ˆ
'
g
DA
R
DA
d
D
d
.
.
in
brought
is
gravity
with
D
d
GR
SR
symmetry → dynamics
21gauge1
Einstein & unified field theory
the last 30 years of his life , strong conviction:
GR + ED → solving the
quantum
mystery?
Was not directly fruitful, but his insight had
fundamental influence on effort by others:
Gauge theories
and
KK unification
, etc.
But both made sense only in modern QM
Gauge invariance
of electrodynamics
E, B → A,
Φ
invariant under
in quantum mechanics must +
wf
transf
U(1)
local
transformation
t
r
t
r
A
A
t
,
'
,
'
)
,
(
)
,
(
'
)
,
(
t
r
e
t
r
t
r
i
Transformation in the internal charge space
“changing particle label”
Such
local symmetry
in a
charge space
is
now called
gauge symmetry
Gauge principle
:
Regard
ψ
transf as more basic,
as it can be gotten
by changing
U(1)
from
global
to be
local
.
brings in the compensating field A ,
the gauge field
A
d
D
d
Given A , Maxwell derived by
SR+gauge
ie
the simplest
Electrodynamics as a gauge interaction
Gauge principle
can be used to extend
consideration to other interactions
History
: Inspired by Einstein’s geometric GR
1919
H Weyl
attempt GR+ED unification
via
Local scale symmetry
[g’ (x)]=
λ
(x)[g(x)]
Calling it
eichinvarianz
1926
V
Fock
, after the advent of QM,
discovered phase transf of
ψ
(x)
F London
: drop “
i
” is just Weyl transf
Weyl
still kept the name: gauge transf
Particle physics
Special relativity, photons, & Bose

Einstein statistics = key elements
But Einstein did not work
directly on any particle phys theory.
Yet,
the influence of his ideas had
been of paramount importance
to
the successful creation of
the Standard Model
of particle physics
Symmetry principle
as the guiding light.
The Standard Model is a good example of
a theory of principle
:
the gauge symmetry principle → dynamics,
as well as
a constructive theory
: discoveries of
* quar
ks and leptons,
* the sym groups of SU(2)
xU
(1) & SU(3)
follow from trial

and

error theoretical prepositions
and experimental checks
ED is a gauge interaction based on
abelian
(commutative) transf.
1954 CN Yang + R Mills extend it to
non

abelian
(non

commutative)
Much richer, nonlinear theory, can
describe
strong
&
weak interactions
22gauge2
Quantization
and
renormalization
of Yang

Mills
th
extremely difficult. Furthermore, the truly
relevant
degrees of freedom
for strongly interacting particle
are
hidden
(
quark confinement
).
The applicability of gauge sym to weak
int
was
doubted because the
symmetry itself is
hidden
(
spontaneous sym breaking
due to Higgs
mech
)
1970’s renaissance of QFT → SM’s triumph
Straightforward extension of QED ?
SM is formulated in the framework of QM
Holy grail of modern unification = [GR + QM]
23KK
Kaluza

Klein theory
u
nification
of
GR+Maxwell
1919
Th
Kaluza : 5D GR
extra dimension w/ a particular geometry
[g]
kk
GR
5
kk
= GR
4
+ ED
4
The Kaluza

Klein miracle!
In physics , even a miracle requires an explanation
1926
O Klein
explained in
modern QM
*Gauge transf = coord transf
in
extra D
Internal charge space = extra D
Foreshadowed
modern unification theories.
GR + SM
t
he compactified space =
multi

dimensional
Einstein’s
influence lives on!
*Compactified extra D → a tower of KK states
the decoupling of heavy particles
simplifies the metric to
[g]
kk
Q:
What is the charge space?
What’s the origin of gauge symmetry?
24
h

c

g
N
form an unit system of
mass/length/time
s
c
g
t
cm
c
g
l
GeV
g
c
c
M
N
P
N
P
N
P
44
5
33
3
19
5
2
10
4
.
5
10
6
.
1
10
2
.
1
Natural
units, not human construct
Dimensions of a
fundamental theory
i.e
.
quantum gravity
(GR + QM)
The fundamental nature of
Einstein’s contribution
illustrated by
Planck unit system
Summary
Summary
of a summary
Fundamental nature of these constants
shown as
conversion factors
connecting disparate phenomena
All due to
Einstein
’s
e
ssential contribution !
h
:
Wave & Particle
c
: Space & Time
g
N
: Mass/energy & Geometry
(QT)
(SR)
(GR)
These PowerPoint slides are posted @
www.umsl.edu/~chengt/einstein.html
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