Ta-Pei Cheng talk based on …

Oxford Univ Press (2013) Einstein’s Physics Atoms, Quanta, and Relativity --- Derived, Explained, and Appraised 1. Atomic Nature of Matter 1879 – 1955 2. Quantum Theory The book 3. Special Relativity explains his physics 4. General Relativity in equations 5. Walking in Einstein’s Steps

Today’s talk 1. The quantum postulate: Planck vs Einstein provides w/o math details 2. Einstein & quantum some highlight 3. Relativity: Lorentz vs Einstein historical context 4. The geometric theory of gravitation & influence 5. Modern gauge theory and unification 3

Einstein, like Planck, arrived at the quantum hypothesis thru the study of BBR Blackbody Radiation cavity radiation = rad in thermal

equilibrium ∞ ∞ 2nd law ν u(uT(T) =) =∫ ρρ(T(T,ν,ν)d)νdν KirchhoffTo find the (1860) u(T) and ρ ( T, ) functions:densities u ( T ) and ρ ( T ,ν) 0∫0 Stefan (1878) Boltzmann (1884) law: u(T ) = aT 4 = universal functions rep intrinsic property of radiation + Maxwell EM (radiation pressure = u/3 ) electromagnetic radiation = a collection of oscillators u = E 2, B 2 ~ oscillator energy kx 2 oscillating energy ~ frequency : Their ratio of is an adiabatic invariant

3 What is the f function? ρ (T ,ν ) =ν f (ν / T ) Wien’s displacement law (1893)

Wien’s distribution (1896) : ρ ( T , ν ) = αν 3 e − βν / T fitted data well…. until IR

αν3 ρ(T,ν ) = eβν /T −1 4 αν3 ρ(T,ν ) = Blackbody Radiation Planck (1900): e βν / T − 1

3 2 U = (c 8/ πν )ρ and dS = dU /T → entropy S

ε = nh ν n = 2,1,0 ,... Einstein’s 1905 proposal of light quanta was not a direct follow-up of Planck’s

10 5 Einstein’s 1905 proposal of light quanta ε =hν

Einstein used Planck’s calculation U = (c3 8/ πν2 )ρ and 1 invoked the of stat mech U = 2 kT to derive the Rayleigh-Jeans law: ρ = 8πc−3kT ν2

noted its solid theoretical foundation ∞ showing BBR = clear challengebut to classical physics u = ρdν = ∞ poor accounting of the data, notably the problem of ultraviolet catastrophe ∫ 0 Rayleigh-Jeans = the low frequency limit of the successful Planck’s distribution new physics The high frequency limit ( Wien’s distribution )ρ = αν3e−βν /T Einstein undertook a statistical study of (BBR) wien : instead of W, calculate ∆S due to volume change (BBR) wien ~

→ (BBR) wien = a gas of light quanta with energy of ε = nh ν

Einstein arrived at energy quantization independently ---- cited Planck only in 2 places A year later……Einstein gave a new derivation of Planck’s distribution 6

Einstein’s discoveries in quantum theory

(1900) Planck: ε = hν is only Einstein’s(1913) Bohr’s photon quantum idea was jumps strongly describe resisted a formal relation by absorptionthe physics andcommunity emission for of manyphotons years (1905) Einstein: the quantum idea must (1916–17)because Einstein it conflicted construct with athe microscopic known represent new physics theoryevidence of radiation–matterfor the wave nature interaction: of light Millikan, Planck, … proposed photoelectric effect as test (A and B coeff ); The central novelty and beyond BBR : Q theory of specific heat (1907) lasting feature is the introduction of probability in quantum dynamics

General acceptance of the photon idea h as conversion factor particle ↔ wave Only after Compton scattering (1924)

(1924–25) Bose-Einstein statistics Einstein stated for the 1 st time : & condensation quanta carried by point-like particles “point of view of Newtonian emission theory ” Photon carries energy + momentum p = h / λ

15 7 Einstein & Modern quantum mechanics : states = vectors in Hilbert space (superposition) His discoveries in quantum theory: observables = operators (commutation relations) Wave/particle nature of light, Classical radiation field quantum jumps etc. can all be = collection of oscillators elegantly accounted for in the framework of Quantum radiation field quantum field theory = collection of quantum oscillators E = (n + 1 )hν • A firm mathematical foundation for Einstein’s photon idea n 2 • Quantum jumps naturally accounted for by ladder operators ˆ ˆ ˆ ⇒ particle [a− , a+ ]= hν , a± n ~ n ±1 behavior QFT description broadens the picture of interactions not only can alter motion, but also allows for emission and absorption of radiation → creation and annihilation of particles Alas, Einstein never accepted this neat resolution Beautiful resolution of wave-particle duality as he never accepted ikx in radiation energy fluctuation wave ~ ˆea the new framework of QM 8

Einstein & QM: The debate with Bohr 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 Einstein, Podolsky & Rosen (1935) : A thought experiment highlighting the “spooky action-at-a-distance ” feature: 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. the discussion and debate of “EPR paradox” have illuminated some of the fundamental issues related to the meaning of QM 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 deeper 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.

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12SR 1 Special Relativity Maxwell’s equations: EM wave – c Contradict relativity? 2 inertial frames x’ = x - vt & d/dt’ = d/dt → velocity addition rule u’ = u - v The then-accepted interpretation: Max eqns valid only in the rest-frame of ether Key Q: How should EM be described for sources and observers moving with respect to the ether-frame ? “The electrodynamics of a moving body”

Michelson-Morley null result @ O( v2/c 2)  length contraction −1 v  v2  x ' = x − vt t'=t − v x    x ' = γ (x − vt ) t'= γ t − 2 x  γ = 1− 2  [ + a math construct ‘local time ’] c 2  c   c  Lorentz transformation Maxwell ‘covariant’ to all orders (1904) Still in the framework of ether …Applicable only for EM A very different approach by Einstein… 10

Special Relativity Magnet-conductor thought experiment Einstein’sWith very a differentkeen sense approach of aesthetics … in physics Einstein was troubled by • Dichotomy of matter ~ particles , radiation ~ waves → Light quanta • EM singles out 1 particular frame : the ether frame Relativity is a symmetry in physics Case I : moving charge in B (ether frame) Physics unchanged under some transformation Lorentz force (per unit charge) Relativity = same physics in all coordinate frames f = v × B How to reconcile (Galilean) relativity u’ = u - v e with the constancy of c? Case II : changing B induces an E via Resolution: simultaneity is relative Faraday’s law, resulting exactly the Time is not absolute, but frame dependent t’ ≠ t same force, yet such diff descriptions From this 1905 realization to full theory in 5 weeks 10yrs Relation among inertial frames Seeking a more symmetric picture valid in all frames correctly given by Lorentz transformation , with Galilean transformation as low v/c approximation • Dispense with ether All nontrivial results follow from this new conception of time ▪ Invoke the principle of relativity The new kinematics applicable to all physics

25 11 Special Relativity Geometric formulation Even simpler perspective Hermann Minkowski (1907) Essence of SR: time is on an equal footing as space . To bring out this symmetry, unite them in a single math structure, spacetime Emphasizes the invariance of the theory: c → s SR: The arena of physics is the 4D spacetime s2 = −c2t 2 + x2 + y 2 + z 2 = [x][g][x] Einstein was initially not impressed , −1 ct     calling it  1  x  .. “untilsuperfluous he tried learnedness to formulate” = ()ct x y z = length 2    General relativity (non-inertial frames)  1  y     c as the conversion factor 1space z  ↔ time = Field theory of gravitation Gravity = structure of spacetime Lorentz-transformation = rotation R ˆ in spacetime SR = flat spacetime metric [g] → all SR features GR = curved spacetime 12

Why does GR principle automatically “Gravity disappears in a free fall frame ” bring gravity into consideration? How is gravity related to spacetime? The Equivalence Principle (1907) played a key role in the formulation of general theory of relativity

starting from Galileo Remarkable empirical observation All objects fall with the same acceleration

a ↑ = ↓ g Einstein: “My happiest thought ” EP: Motion in grav field totally independent of accelerated frame = inertial frame w/ gravity properties of the test body, attributable directly to underlying spacetime? EP as the handle of going from SR to GR Einstein proposed in 1912: SR → GR, flat → curved spacetime gravitational field = warped spacetime

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Relativity Equations written in terms of 4-tensors as a coordinate symmetry are automatically relativistic Special relativity flat spacetime Rˆ = spacetime-independent transformation Global symmetry General relativity Must replace ordinary derivative by curved spacetime with moving basis vectors covariant differentiation general coord transf = spacetime dependent [ d ]→ [D = d + Γ] Rˆ = Rˆ ( x ) Local symmetry Γ = ∂[]g compensate the moving bases

SR → GR with d → D gravity is brought in local symmetry → dynamics 14

General Relativity metric tensor [gµν ] = gravitational field = warped spacetime rela. grav. potential Field theory of gravity

SourceSource particle particle Curved Fiel spacetd ime Test Test particleparticle Field eqn GeodesicEqn of Einstein motion field eqn Eqn 1915 The Einstein equation 10 coupled PDEs g N Newton’s constant Gµν = g NTµν solution = [gµν ] Tµν energy momentum tensor Metric being gravi pot, nd Gµν curvature tensor = nonlinear 2 derivatives of [gµν ] Curvature = tidal forces

gN as the conversion factor geometry ↔ mass/energy

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General Relativity

The Einstein field equation Gµν = gNTµν In the limit of test particles moving with v « c in a static and weak grav field Einstein → Newton (the 1/r 2 law explained!) GR can depict new realms of gravity: time-dep & strong time-dep : GR → gravitational wave Hulse-Taylor binary pulse system Strong : Black Holes full power & glory of GR Role of space and time is reversed Light-cones tip over instead of → (t = ∞), → (r = 0 ) Even light cannot escape 16

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 Symmetry & Geometry fundamentally influenced effort by others: Gauge theories and KK unification , etc. But both possible only with modern QM 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 .

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Gauge Theory Relativity Gauge symmetry Transformation R ˆ Local transformation R ˆ ( x ) in coordinate space in the internal charge space “changing particle label” Gauge principle :

global SR → local GR Change from global to local transformation on QM states [][]d → D = d + Γ [d ] → [D = d + A ] Γ = ∂[]g brings in the compensating field A , compensate moving bases the gauge field Rˆ = eiθ (x) → a single A(x) SR + gauge principles → Maxwell Electrodynamics as a gauge interaction Gauge principle can be used to extend consideration to other interactions 18 Particle physics Special relativity, photons, & Bose-Einstein statistics = key elements But Einstein did not work directly on any particle physics theory Yet, the influence of his ideas had been of paramount importance to the successful creation of the Standard Model of particle physics Strong, weak & electromagnetic interactions are all gauge interactions Symmetry principle allowed us to discover the basic eqns of SM QCD, electroweak field eqns = generalization of Maxwell’s eqns ● Non-commutative transformation: ● Quantization & renormalization Multiple gauge (Yang-Mill) fields ■ truly relevant degrees of freedom ● Spontaneous symmetry breaking: for strongly interacting particles are The symmetry is hidden hidden (quark confinement )

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Q: What is the charge space? Kaluza-Klein theory What’s the origin of gauge symmetry? unification of GR +Maxwell 1919 Theodor Kaluza : GR in 5D *Gauge transf = coord transf in extra extra dimension w/ a particular geometry [g] D *Internal charge space = extra D kk kk GR 5 = GR 4 + ED 4 The Kaluza-Klein miracle! Foreshadowed modern unification In physics , a miracle requires an explanation theories 1926 Oskar Klein explained in modern GR + SM require QM compactified multi-dim extra D space Compactified extra D → a tower of KK Einstein’s states the decoupling of heavy particles influence lives on! simplifies the metric to [ g]kk 20

Last slide : summarizing A pithy description the central nature of via the fundamental constants h -- c -- g Einstein’s physics N form an unit system of Besides his legacy on geometry mass/length/time & symmetry principle, (The Planck unit system ) his fundamental contribution = Ability to connect All due to Einstein5 ’s 2 hc 19 disparate phenomena MessentialPc = contribution= 2.1 ×10 GeV g N Natural units, not human construct They are conversion factors h =h Planck’s constant Dimensions of the fundamental theory g N −33 connecting disparate phenomena lP =g = Newton’s= 6.1 × 10constantcm i.e . quantum gravity (GR + QM) N c3 (QT) h: Wave & Particle c = Einstein’sh constant ?! g N −44 c: Space & Time (SR) tP = = 4.5 ×10 s c5 gN: Energy & Geometry (GR) 21

These PowerPoint slides are posted @ www.umsl.edu/~chengt/einstein.html

Copies of the book are left in PSU Physics Dept Office for your perusal Sign-up & Check-out

US publication date = April 5, 2013 Hardback ISBN 0199669910