EINSTEIN’S PHYSICS
description
Transcript of EINSTEIN’S PHYSICS
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 relativityGENERAL 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
Pt
NRTDtx
A 6222
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
Te
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 lnW ? Planck was “compelled” to make the hypothesis of energy quantization h
TdUdSUc /entropy 8 23
Einstein’s 1905 proposal of light quantawas 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 quantizationcan 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
uΔudvEc
E
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:particles ~ sWein'
: waves~ ˆ8
Jeans'-Rayleigh
two! theof fusion"" abut particles, asjust or wavesasjust Neither
ondistributi sPlanck' 222RJWPlanck
ΔEΔEΔE
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TkTvΔEvE
TE
kTEEΔE
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: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 oscillatorsQuantum radiation field = collection of quantum oscillators hnEn )(
21
• 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
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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 ' xcvtt 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
Bvef
▪ 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 features4-tensor equations are automatically relativistic
11
11
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 tensorNg 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-independentGlobal symmetry
R̂
Special relativity = Lorentz transformation
4-tensor eqns are auto relativisticR̂
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 physicsSpecial relativity, photons, & Bose-
Einstein statistics = key elementsBut 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
scg
t
cmcg
l
GeVgccM
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