Review of Basic Physics Background

1. Review of Basic Physics Background

2. Basic physical quantities & units • Unit prefixes • Basic quantities • Units of measurement • Planck units • Physical constants

3. Unit Prefixes • See http://www.bipm.fr/enus/3_SI/si-prefixes.htmlfor the official international standard unit prefixes. • When measuring physical things, these prefixes always stand for powers of 103 (1,000). • But, when measuring digital things (bits & bytes) they often stand for powers of 210 (1,024). • See also alternate kibi, mebi, etc. system at http://physics.nist.gov/cuu/Units/binary.html • Don’t get confused!

4. Three “fundamental” quantities

5. Some derived quantities

6. Electrical Quantities • We’ll skip magnetism & related quantities this semester.

7. Information, Entropy, Temperature • These are important physical quantities also • But, are different from other physical quantities • Based on statistical correlations • But, we’ll wait to explain them till we have a whole lecture on this topic later. • Interestingly, there have been attempts to describe all physical quantities & entities in terms of information (e.g., Frieden, Fredkin).

8. Unit definitions & conversions • See http://www.cise.ufl.edu/~mpf/physlim/units.txt for definitions of the above-mentioned units, and more. (Source: Emacs calc software.) • Many mathematics applications have built-in support for physical units, unit prefixes, unit conversions, and physical constants. • Emacs calc package (by Dave Gillespie) • Mathematica • Matlab - ? • Maple - ?

9. Fundamental physical constants • Speed of light c = 299,792,458 m/s • Planck’s constant h = 6.6260755×1034 J s • Reduced Planck’s constant  = h / 2 • h : circle ::  : radian • Newton’s gravitational constant G = 6.67259×1011 Nm2 / kg • Others: permittivity of free space, Boltzmann’s constant, Stefan-Boltzmann constant to be introduced later as we go along.

10. Physics you should already know • Basic Newtonian mechanics • Newton’s laws, motion, energy, etc. • Basic electrostatics • Ohm’s law, Kirchoff’s laws, etc. • Also helpful, but not prerequisite (we’ll introduce them as we go along): • Basic statistical mechanics & thermodynamics • Basic quantum mechanics • Basic relativity theory

11. Generalized Classical Mechanics

12. Generalized Mechanics • Classical mechanics can be expressed most generally and concisely in the Lagrangian and Hamiltonian formulations. • Based on simple functions of the system state: • Lagrangian: Kinetic minus potential energy. • Hamiltonian: Kinetic plus potential energy. • The dynamical laws can be derived from either energy function. • This framework generalizes to quantum mechanics, quantum field theories, etc.

13. Euler-Lagrange Equation Note the over-dot! Where: • L(qi, vi) is the system’s Lagrangian function. • qi :≡ Generalized position coordinate indexed i. • vi :≡ Velocity of generalized coordinate i, • (as appropriate) • t :≡ Time coordinate • In a given frame of reference. or just

14. Euler-Lagrange example • Let q = qi be the ordinary x, y, z coordinates of a point particle with mass m. • Let L = ½mvi2 − V(q). (Kinetic minus potential.) • Then, ∂L/∂qi = − ∂V/∂qi = Fi • The force component in direction i. • Meanwhile, ∂L/∂vi = ∂(½mvi2)/∂vi = mvi = pi • The momentum component in direction i. • And, • Mass times acceleration in direction i. • So we get Fi = mai (Newton’s 2nd law)

15. A.k.a.Hamilton’sprinciple Least-Action Principle • The action of an energy means the integral of that energy over time. • The trajectory specified by the Euler-Lagrange equation is one that locally extremizes the action of the Lagrangian: • Among trajectories s(t)between specified pointss(t0) and s(t1). • Infinitesimal deviations from this trajectory leave the action unchanged to 1st order.

16. Hamilton’s Equations Implicitsummationover i. • The Hamiltonian is defined as H :≡ vipi − L. • Equals Ek + Ep if L = Ek − Ep and vipi = 2Ek = mvi2. • We can then describe the dynamics of (qi, pi) states using the 1st-order Hamilton’s equations: • These are equivalent to but often easier to solve than the 2nd-order Euler-Lagrange equation. • Note that any Hamiltonian dynamics is bideterministic • Meaning, deterministic in both the forwards and reverse time directions.

17. Field Theories • Space of indexes i is continuous, thus uncountable. A topological space T, e.g., R3. • Often use φ(x) notation in place of qi. • In local field theories, the Lagrangian L(φ) is the integral of a Lagrange density function ℒ(x) over the entire space T. • This ℒ(x) depends only locally on φ, e.g., ℒ(x) = ℒ(φ(x), (∂φ/∂xi)(x), (x)) • All successful physical theories can be explicitly written down as local field theories! • There is no instantaneous action at a distance.

18. Special Relativity and the Speed-of-Light Limit

19. The Speed-of-Light Limit • No form of information (including quantum information) can propagate through space at a velocity (relative to its local surroundings) that is greater than the speed of light, c, ~3×108 m/s. • Some consequences: • No closed system can propagate faster than c. • Although you can define open systems that do by definition • No given piece of matter, energy, or momentum can propagate faster than c. • All of the fundamental forces (including gravity) propagate at (at most)c. • The probability mass that is associated with a quantum particle flows in an entirely local fashion, no faster than c.

20. Early History of the Limit • The principle of locality was anticipated by Newton • He wished to get rid of the “action at a distance” aspects of his law of gravitation. • The finiteness of the speed of light was first observed by Roemer in 1676. • The first decent speed estimate was obtained by Fizeau in 1849. • Weber & Kohlrausch derived a velocity of c from empirical electromagnetic constants in 1856. • Kirchoff pointed out the match with the speed of light in 1857. • Maxwell showed that his EM theory implied the existence of waves that always propagate at c in 1873. • Hertz later confirmed experimentally that EM waves indeed existed • Michaelson & Morley (1887) observed that the SoL was independent of the observer’s state of motion! • Maxwell’s equations apparently valid in all inertial reference frames! • Fitzgerald (1889), Lorentz (1892,1899), Larmor (1898), Poincaré (1898,1904), & Einstein (1905) explored the implications of this...

21. Relativity: Non-intuitive but True • How can the speed of something be a fundamental constant? Seemed broken... • If I’m moving at velocity v towards you, and I shoot a laser at you, what speed does the light go, relative to me, and to you?Answer: both c!(Notv+c.) • Newton’s laws were the same in all frames of reference moving at a constant velocity. • Principle of Relativity (PoR): All laws of physics are invariant under changes in velocity • Einstein’s insight: The PoR is consistent w. Maxwell’s theory! Change def. of space+time.

22. Some Consequences of Relativity • Measured lengths and time intervals in a system vary depending on the system’s velocity relative to observers. • Lengths are shortened in direction of motion. • Clocks run slower. • Sounds paradoxical, but isn’t! • Mass is amplified. • Energy and mass are the same quantity measured in different units: E=mc2. • Nothing (incl. energy, matter, information, etc.) can go faster than light! (SoL limit.)

23. Three Ways to Understand c limit • Energy of motion contributes to mass of object. • Mass approaches  as velocityc. • Infinite energy needed to reach c. • Lengths, times in a faster-than-light moving object would become imaginary numbers! • What would that mean? • Faster than light in one reference frame  Backwards in time in another reference frame • Sending info. backwards in time violates causality, leads to logical contradictions!

24. The c limit in quantum physics • Sometimes you see statements about “nonlocal” effects in quantum systems. Watch out! • Even Einstein made this mistake. • Described a quantum thought experiment that seemed to require “spooky action at a distance.” • Later it was shown that this experiment did not actually violate the speed-of-light limit for information. • These “nonlocal” effects are only illusions, emergent phenomena predicted by an entirely local underlying theory respecting SoL limit.. • Widely-separated systems can maintain quantum correlations, but that isn’t true non-locality.

25. The “Lorentz” Transformation Actually it was written down earlier; e.g., one form by Voigt in 1887 • Lorentz, Poincaré: All the laws of physics remain unchanged relative to the reference frame (x′,t′) of an object moving with constant velocity v = Δx/Δt in another reference frame (x,t) under the following conditions: Where: Note: our γ here is the reciprocal of the quantity denoted γ by other authors.

26. Consequences of Lorentz Transform • Length contraction (Fitzgerald, 1889, Lorentz 1892): • An object having length  in its rest frame appears, when measured in a relatively moving frame, to have the (shorter) length γ. (For lengths parallel to direction of motion.) • Time dilation (Poincaré, 1898): • If time interval τ is measured between two co-located events in a given frame, a larger time t = τ/γ will be measured between those events in a relatively moving frame. • Mass expansion (Einstein’s fix for Newton’s F=ma): • If an object has mass m0 in its rest frame, then it is seen to have the larger mass m = m0/γ in a relatively moving frame.

27. t′ t x′ x Lorentz Transform Visualization x′=0 Original x,t(“rest”) frame Line colors: Isochrones(space-like) t′=0 Isospatials(time-like) New x′,t′(“moving”) frame Light-like In this example: v = Δx/Δt = 3/5γ = Δt′/Δt = 4/5vT = v/γ = Δx/Δt′ = 3/4 The “tourist’s velocity.”

28. An Alternative View: Mixed Frames t′ t t′ StandardFrame #1 MixedFrame #1 t x x In this example: v = Δx/Δt = 3/5 vT = Δx/Δt′ = 3/4γ = Δt′ /Δt = 4/5 Note that (Δt)2 = (Δx)2 + (Δt′)2by the PythagoreanTheorem! (Light pathsshown ingreen here.) x t′ t x′ StandardFrame #2 MixedFrame #2 x′ x′ Note the obvious complete symmetryin the relation between the two mixed frames.

29. Relativistic Kinetic Energy • Total relativistic energy E of any object is E = mc2. • For an object at rest with mass m0, Erest = m0c2. • For a moving object, m = m0/γ • Where m0 is the object’s mass in its rest frame. • Energy of the moving object is thus Emoving = m0c2/γ. • Kinetic energy Ekin :≡ Emoving − Erest = m0c2/γ − m0c2 = m0c2(1 − γ) • Substituting γ = (1−β2)1/2 and Taylor-expanding gives: Pre-relativistic kinetic energy ½ m0v2 Higher-orderrelativistic corrections

30. Spacetime Intervals • Note that the lengths and times between two events are not invariant under Lorentz transformations. • However, the following quantity is an invariant: The spacetime interval s, where: s2 = (ct)2− xi2 • The value of s is also the proper timeτ: • The elapsed time in rest frame of object traveling on a straight line between the two events. (Same as what we were calling t′ earlier.) • The sign of s2 has a particular significance: s2 > 0 - Events are timelike separated (s is real)May be causally connected. s2 = 0 - Events are lightlike separated (s is 0) Only 0-rest-mass signals may connect them. s2 < 0 - Events are spacelike separated (s is imaginary)Not causally connected at all.

31. Relativistic Momentum • The relativistic momentum p = mv • Same as classical momentum, except that m= m0/γ. • Relativistic energy-momentum-rest-mass relation:E2 = (pc)2 + (m0c2)2If we use units where c = 1, this simplifies to just:E2 = p2 + m02 • Note that if we solve for m02, we get: m02 = E2 − p2 • This is another relativistic invariant! • Later we will show how it relates to the spacetime interval s2 = t2 − x2, and to a computational interpretation of physics.