Einstein

taupeselectionMechanics

Nov 14, 2013 (3 years and 8 months ago)

83 views

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