EINSTEIN’S PHYSICSA modern understanding

Ta-Pei Cheng talk based on …

Oxford Univ Press (2/ 2013)

Einstein’s PhysicsAtoms, Quanta, and Relativity --- Derived,

Explained, and Appraised

ATOMIC NATURE OF MATTER1. Molecular size from classical fluids2. The Brownian motion

WALKING IN EINSTEIN’S STEPS16. Internal symmetry and gauge interactions17. The Kaluza–Klein theory and extra dimensions

SPECIAL RELATIVITY9. Prelude to special relativity10. The new kinematics and E = mc2

11. Geometric formulation of relativity

GENERAL RELATIVITY12. Towards a general theory of relativity13. Curved spacetime as a gravitational field14. The Einstein field equation15. Cosmology

3. Blackbody radiation: From Kirchhoff to Planck 4. Einstein’s proposal of light quanta5. Quantum theory of specific heat6. Waves, particles, and quantum jumps7. Bose–Einstein statistics and condensation8. Local reality and the Einstein–Bohr debate

QUANTUM THEORY

2TOC

Today’s talk provideswithout

math details

some highlights in

historical context

The book explains his Physics

in equations

Albert Einstein1879 – 1955

Molecular size & Avogadro’s numberclassical liquids with suspended particles

3Atoms

(4/1905) U Zurich doctoral thesis: “On the determination of molecular dimensions”

→ 2 equations relating P & NA to viscosity and diffusion coefficientsHydrodynamics 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

RTDtx

A 6

222

Nk 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 2nd law ),( & )( TTu

0

),()( dTTuMaxwell EM radiation = a collection of oscillators u = E2, B2 ~ oscillator energy kx2

The ratio of oscillating energy to frequency is an adiabatic invariant ; p = u/3

Stefan (1878) Boltzmann (1884):

Wien’s displacement law (1893):

4)( aTTu

)/(),( 3 TfT

Wien’s distribution (1896): fits data well…. until IR

Planck’s distribution (1900):

TeT /3),(

1),(

/

3

TeT

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 lnW ? Planck was “compelled” to make the hypothesis of energy quantization h

TdUdSUc /entropy 8 23

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

Tν2

0

du

Uc 238 kTU

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 C1ν2T . Only a limit law? Intro cutoff ρ = C1ν2T exp(-C2ν/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 C1. 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

kTν2-3c8

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

It clearly explained why energy quantization

can cure ultraviolet catastrophe

The new physics must be applicable beyond BBR: quantum theory of specific heat

W hK 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 ?

uhΔuhEE

uΔudv

EcE

W

RJ

22

222

2

32

:particles ~ sWein'

: waves~ ˆ8

Jeans'-Rayleigh

two! theof fusion"" abut particles, asjust or wavesasjust Neither

ondistributi sPlanck' 222

RJWPlanckΔEΔEΔE

8Quanta 4

TkTvΔEvE

T

EkTEEΔE

22

222

ˆ that so ˆ onsdistributiradiation to

:1904 ofn theory fluctuatio his appliedEinstein

1st time stated: quanta carried by point-like particles

“point of view of Newtonian emission theory”

Photon carries energy + momentum

particlewavefactor conversion as

/

h

hph 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 hnEn )(2

1

• A firm mathematical foundation for Einstein’s photon idea• Quantum jumps naturally accounted for by ladder operators

ˆ ,ˆ

1~ˆ

behaviorparticle

haa

nna

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 fluctuationelegantly resolved in QFT

10Quanta 6

uhuΔua

haaeaeaAe jkkji

ji

ji jjj

22

with

termextraan about bring ˆ ngNoncommuti

ˆ ,ˆ ˆˆ

WAS EINSTEIN JUST TOO SET-IN-HIS-WAYS TO APPRECIATE THE NEW ADVANCES ?

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 RelativityMaxwell’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(v2/c2) length contraction

Lorentz transformation Maxwell ‘covariant’ to all orders (1904)

vtxx ' xc

vtt

2'

Special Relativity13SR 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

Bve

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

constructive 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

tt '

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

tt '

Transformation rule for EM fields, radiation energy,..

Lorentz force law from Max field equations

Work-energy theorem to mass-energy equivalence E = mc2

Even simpler perspectiveHermann Minkowski (1907)

Essence 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

222222

gmetric

xgxzyxtcs

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 formulateGeneral relativity (non-inertial frames)

= Field theory of gravitationGravity = 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 observationAll 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 spacetimeNote: 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

GeodesicEqn

gravitational field = warped spacetimemetric tensor [gμν] = rela. grav. potential

T energy momentum tensor

Ng Newton’s constant

1915

G curvature tensor = nonlinear 2nd derivatives of [gμν] Metric = gravi potCurvature = tidal forces

The Einstein equation10 coupled PDEssolution = [gμν]

TgG 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 waveIndirect, but convincing, evidence thru decade-long observation of

Hulse-Taylor binary pulse system

3 classical testsGrav redshift

Bending of lightPrecession of planet

orbit

Black Holes = full power and glory of GRGravity 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 1st paper on modern cosmology

The universe = a phys system the constituent elements being galaxies

Gravity the only relevant interactionGR = 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

TggG N

Λ = a great discoverykey 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 ΛCDM cosmology

Cosmology

Einstein equation

derivatives

Expanding Universe

0)( ta

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 principleBefore 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.

0ˆ

0''' 0

ˆ'n nsformatiovector tra

amFR

amFamF

ARAA

Rotation symmetry

Tensors have def transf propertyTensor equations are automatically

rotational symmetric.

Spacetime-independent

Global symmetry R̂

Special relativity = Lorentz transformation

4-tensor eqns are auto relativistic

R̂

General relativitycurved spacetime with moving basis vectors

spacetime –dependent metric [g] = [g(x)] general coord transf = spacetime dependent

Local symmetry )(ˆˆ xRR

Differentiation results in a non-tensor

0ˆd ˆ' RdARdA Must replace by covariant differentiation

ˆ'

g

DARDA

dDd

.

. inbrought isgravity

with DdGRSR

symmetry → dynamics

21gauge1

Einstein & unified field theorythe 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 electrodynamicsE, B → A, Φ invariant under

in quantum mechanics must + wf transf

U(1) local transformation

trtrAA t ,' ,'

),(),(' ),( tretr tri

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

AdDd

Given A , Maxwell derived by SR+gauge ie the simplest

Electrodynamics as a gauge interactionGauge principle can be used to extend

consideration to other interactions

History: Inspired by Einstein’s geometric GR1919 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 transfWeyl 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

* quarks 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 QMHoly grail of modern unification = [GR + QM]

23KK

Kaluza-Klein theoryunification of GR+Maxwell

1919 Th Kaluza : 5D GR extra dimension w/ a particular geometry [g]kk

GR5kk

= GR4 + ED4

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

the compactified space = multi-dimensional

Einstein’s influence lives on!

*Compactified extra D → a tower of KK statesthe 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 -- gN

form an unit system of mass/length/time

sc

gt

cmc

gl

GeVg

ccM

NP

NP

NP

445

333

195

2

104.5

106.1

102.1

Natural units, not human constructDimensions of a fundamental theory

i.e. quantum gravity (GR + QM)

The fundamental nature of Einstein’s contribution

illustrated by Planck unit system SummarySummary of a summary

Fundamental nature of these constants shown as conversion factors

connecting disparate phenomena

All due to Einstein’s essential contribution !

h: Wave & Particle c: Space & Time gN: Mass/energy & Geometry

(QT) (SR) (GR)

These PowerPoint slides are posted @

www.umsl.edu/~chengt/einstein.html