Reality Is Not What It Seems Part 11

Smolin, Lee, Three Roads to Quantum Gravity. New York, Basic Books, 2002. An introduction to quantum gravity and its open questions.

Van Fraa.s.sen, Bas. 'Rovelli's World' in Foundations of Physics, 40 (2010), 390417. A discussion of relational quantum mechanics, by an important a.n.a.lytic philosopher.

THE BEGINNING.

Let the conversation begin ...

Follow the Penguin

Keep up-to-date with all our stories YouTube.com/penguinbooks.

Pin 'Penguin Books' to your Pinterest Like 'Penguin Books' on Facebook.com/penguinbooks Listen to Penguin at SoundCloud.com/penguin-books Find out more about the author and.

discover more stories like this at Penguin.co.uk.

ALLEN LANE.

UK USA Canada Ireland Australia.

India New Zealand South Africa.

Allen Lane is part of the Penguin Random House group of companies whose addresses can be found at global.penguinrandomhouse.com First published in Italian under the t.i.tle La realt non e come ci appare by Raffaello Cortina Editore SpA 2014 This translation first published in Great Britain by Allen Lane 2016 Copyright Raffaello Cortina Editore SpA, 2014 Translation copyright Simon Carnell and Erica Segre, 2016 The moral right of the author and translators has been a.s.serted Cover credit: Coralie Bickford-Smith ISBN: 978-0-241-25797-5.

fn1 In technical terms, there are converging infinite sums. For the example of the string, the infinite sum + + 18 + 116 ... converges to 1. Infinite convergent sums were not understood in Zeno's time. Archimedes understood them a few centuries later, and used them to calculate areas. Newton used them heavily, but not until the nineteenth century, with Bolzano and Weierstra.s.s, was conceptual clarity on these mathematical objects achieved. Aristotle, however, had already understood that this was a possible way to answer Zeno; the Aristotelian distinction between actual infinity and potential infinity already contains the key idea: the difference between the absence of a limit to divisibility, and the possibility of having already divided something an infinite number of times.

fn2 Here is the list of all of the works of Democritus, with their t.i.tles as given by Diogenes Laertius: Great Cosmology; Little Cosmology; Cosmography; On the Planets; On Nature; On Human Nature; On Intelligence; On the Senses; On the Soul; On Flavours; On Colour; On Diverse Movements of the Atoms; Of Changes in Shape; The Causes of Celestial Phenomena; The Causes of Atmospheric Phenomena; On Fire and On Things in Fire; The Causes of Acoustic Phenomena; Concerning the Magnet; The Causes of Seeds, Plants and Fruits; On Animals; A Description of the Sky; Geography; A Description of the Pole; On Geometry; Geometrical Reality; On the Tangents of the Circle and the Sphere; Numbers; On Irrational Lines and Solids; Projections; Astronomy; Astronomical Table; On Rays of Light; On Reflected Images; On Rhythm and Harmony; On Poetry; On the Beauty of Song; On Euphony and Cacophony; Concerning Homer, or on Correct Epic Diction; The Science of Medicine; On Agriculture; On Words; On Names; On Values, or On Virtue; On the Disposition which Characterizes the Wise; On Painting; A Treatise on Tactics; Circ.u.mnavigation of the Ocean; On History; The Thought of Chaldea; The Thought of the Phrygians; On the Sacred Writings of Babylon; On the Sacred Writings of Meroe; On Fevers and the Coughs Deriving from Illness; On Aporiae; Legal Questions; Pythagoras; On Logic, or Criterion of Thought; Confirmations; Points of Ethics; On Well-being. All lost ...

fn3 The bad reputation of Aristotelian physics dates back to the polemics of Galileo. Galileo had to move forward and therefore needed to be critical. He attacked Aristotle viciously, with scorn and sarcasm. But he took Aristotle's physics very seriously.

fn4 (x = a t).

fn5 The square of the period of revolution is proportional to the cube of the radius of the orbit. This law was shown to be correct not only for the planets...o...b..ting the sun (Kepler), but also for the moons of Jupiter (Huygens). Newton a.s.sumes, by induction, that it should also hold for the hypothetical little moon orbiting the Earth. The constant of proportionality depends on the body around which the orbit is made: this is why data on the lunar orbit allow us to compute the period of the little moon.

fn6 a = v/r, where v is the speed and r the radius of the orbit.

fn7 The energy released by combustion engines is chemical and therefore, ultimately, electromagnetic.

fn8 The equations fill a page in Maxwell's original treatise. Today the same equations can be written in half a line: dF = 0, d*F = J. We'll soon see why.

fn9 If you visualize the field as a vector (an arrow) at each point of s.p.a.ce, the point of the arrow indicates the direction of the Faraday lines, that is to say, the tangent of the Faraday lines, and the length of the arrow is proportional to the density of the Faraday lines.

fn10 The set of events at a s.p.a.ce-like distance from a reference event.

fn11 The astute reader will object that the halfway moment of my quarter of an hour can be considered simultaneous to your reply. The reader who has studied physics will recognize that this is 'Einstein's convention' for defining simultaneity. This definition of simultaneity depends on how I move, and consequently does not define simultaneity between two events but only a simultaneity relative to the state of movement of particular bodies. In figure 3.3 a dot is halfway between a and b, the points at which I exit from the past of the observer and enter his future. The other dot is halfway between e and d, the points at which I exit from the past of the observer and enter into his future if I move along a different trajectory. Both dots are simultaneous as regards the reader, according to this definition of 'simultaneity', but they occur in successive times. The two dots are each simultaneous to the reader, but relative to two different motions of mine. Hence the term 'relativity'.

fn12 Airplane and ball follow a geodesic in a curved s.p.a.ce. In the case of the ball, the geometry is approximately given by the metric ds = (1 2(x)) dt dx, where (x) is the Newtonian potential. The effect of the gravitational field is reduced to the dilation of time with alt.i.tude. (The reader familiar with the theory will notice the curious sign inversion: the physical trajectory maximizes proper time.) fn13 Observations of the binary system PSR B193+16 show that the two stars which revolve around one another radiate gravitational waves. These observations brought a n.o.bel Prize for Russell Hulse and Joseph Taylor in 1993.

fn14 Plutarch, Adversus colotem, 4, 1108. The word means 'nature', and includes the sense 'the nature of something'.

fn15 This term is called 'cosmological' because its effects occur only at an extremely large, or 'cosmological' distances. The constant is called the 'cosmological constant', and its value was measured at the end of the 1990s, bringing a n.o.bel Prize in 2011 for the astronomers Saul Perlmutter, Brian P. Schmidt and Adam G. Riess.

fn16 Gttingen, where Hilbert worked, was at this time the seat of the most important school of geometry.

fn17 A sphere is the set of points in R3 determined by the equation x2 + y2 + z2 = 1. The 3-sphere is the set of points in R4 determined by the equation x2 + y2 + z2 + u2 = 1.

fn18 It has been objected that Dante speaks of 'circles' and not of 'spheres'. But the objection is invalid. Brunetto Latini writes of 'a circle, like the sh.e.l.l of an egg'. The word 'circle', for Dante, as for his teacher and mentor, designates everything which is circular, including spheres.

fn19 On the surface of the Earth, for instance, the North Pole and two points on the equator can make a triangle with three sides of equal length and three right angles something which clearly cannot be done on a plane.

fn20 A Hilbert s.p.a.ce.

fn21 These are the eigenvalues of the operator a.s.sociated with the physical variable in question. The key equation is the eigenvalue equation.

fn22 This cloud is described by a mathematical object called wave function. The Austrian physicist Erwin Schrdinger has written an equation describing its evolution in time. Quantum mechanics is often mistakenly identified with this equation. Schrdinger had hopes that the 'wave' could be used to explain the oddities of quantum theory: from those of the sea to electromagnetic ones, waves are something we understand well. Even today, some physicists try to understand quantum mechanics by thinking that reality is the Schrdinger wave. But Heisenberg and Dirac understood at once that this would not do. To view Schrdinger's wave as something real is to give it too much weight it doesn't help us to understand the theory; on the contrary, it leads to greater confusion. Except for special cases, the Schrdinger wave is not in physical s.p.a.ce, and this divests it of all its intuitive character. But the main reason why Schrdinger's wave is a bad image of reality is the fact that, when a particle collides with something else, it is always at a point: it is never spread out in s.p.a.ce like a wave. If we conceive an electron as a wave, we get in trouble explaining how this wave instantly concentrates to a point at each collision. Schrdinger's wave is not a useful representation of reality: it is an aid to calculation which permits us to predict with some degree of precision where the electron will reappear. The reality of the electron is not a wave: it is how it manifests itself in interactions, like the man who appeared in the pools of lamplight while the young Heisenberg wandered pensively in the Copenhagen night.

fn23 Dirac's equation.

fn24 There is a phenomenon which seems not to be reducible to the standard model: 'dark matter'. Astrophysicists and cosmologists observe in the universe effects of matter which seems not to be the type of matter described by the standard model. Out there, there are still many things that we don't know.

fn25 I find the claim that the Higgs boson 'explains ma.s.s' exaggerated. The Higgs boson does not 'explain' anything about the origin of ma.s.s. What would 'explain' the ma.s.s of the Higgs? The point is technical: the standard model relies on certain symmetries, and these symmetries seemed to permit only particles devoid of ma.s.s. But Higgs and others realized that it is possible to have both symmetries and ma.s.s, as long as the latter enters indirectly via the interaction with the field known today as the Higgs field.

fn26 A finite region of the phase s.p.a.ce the s.p.a.ce of the possible states of a system contains an infinite number of distinguishable cla.s.sic states, but always only a finite number of orthogonal quantum states. This number is given by the volume of the region, divided by the Planck constant, raised to the number of degrees of freedom. This result is general.

fn27 Or Feynman's integral. The probability of going from A to B is the square module of the integral over all the paths of the exponential of the cla.s.sical action of the trajectory, multiplied by the imaginary unit and divided by Planck's constant.

fn28 A mechanism in the box opens the small window on the right for an instant, allowing a photon to escape at some precise time. By weighing the box, it is possible to deduce the energy of the released photon. Einstein hoped that this would create difficulties for quantum mechanics, which predicts that time and energy cannot be both precisely determined. Bohr replied, mistakenly, that the way out of the difficulty required Einstein's general relativity, and Einstein, mistakenly, accepted Bohr's reply. The correct response to Einstein, which Bohr was unable to find but that is clear today, is that the position of the escaping photon and the weight of the box remain tied to each other ('correlated'), even if the photon is already far away.

fn29 The mark on the h of Planck's constant serves only to indicate that Planck's constant is in this equation divided by 2, a rather useless and idiosyncratic addition by theoretical physicists: placing the small, hard, angular mark on the h 'makes it elegant'.

fn30 To hear this metaphor directly in his own voice, go to the site http://www.webofstories.com/play/9542?o=MS.

fn31 DeWitt replaces derivatives with derivative operators in the HamiltonJacobi equation for general relativity (written a little while earlier by Peres). That is, he does what Schrdinger had done to write his equation, in his first work: replacing derivatives with derivative operators in the HamiltonJacobi equation of a particle.

fn32 Or the 'EinsteinSchrdinger' equation.

fn33 The best known alternative to loop quantum gravity is string theory, whose main concern is not so much studying the quantum properties of s.p.a.ce and time, but rather writing a unified theory of all known fields, an objective that might be premature given current knowledge.

fn34 The eigenvalue equation for the volume operator.

fn35 Hence the quantum states of gravity are indicated with jl, vn>, where n indicates the nodes and l the links of the graph.

fn36 Imagine what a nonsensical hotchpotch the ideas of Aristotle and Plato would seem if we only had the commentaries on them written by others and were unable to access the lucidity and complexity of the original texts!

fn37 The quantum number of the states of photons in Fock's s.p.a.ce is the momentum, Fourier's transformation of position.

fn38 The operator a.s.sociated with the geometry of granular s.p.a.ce is the holonomy of the gravitational connections, or rather, in physical terms, a 'Wilson loop' of general relativity.

fn39 The gravitational potential.

fn40 x(t) = at.

fn41 Especially since he had become excited ...

fn42 The actual structure of the vertices of the spinfoam is a bit more complex than the one in figure 7.2, and resembles more closely the one shown in figure 7.4.

fn43 It's a Feynman diagram because it is a history of quanta, as in the Feynman diagrams. Except that now, the quanta are not quanta moving in s.p.a.ce, but rather quanta of s.p.a.ce. The graph they draw in their interactions is not a representation of the movement of particles in s.p.a.ce, but represents the plot of s.p.a.ce itself. But the resulting picture is also precisely a lattice like the one used in the lattice approximation, because it represents a discretized s.p.a.cetime. With the difference that it is no longer an approximation, but the real discrete structure of s.p.a.ce at a small scale.

fn44 The first defines the Hilbert s.p.a.ce of the theory. The second describes the algebra of the operators. The third describes the size of transition of each vertex, such as the one shown in figure 7.4.

fn45 '[ ...] all the different elementary particles could be reduced to one universal substance which could equally be called energy or matter, and none of the particles should be privileged and considered more fundamental. This point of view corresponds to Anaximander's doctrine, and I am convinced that in modern physics this is the correct point of view.' Werner Heisenberg, Physics and Philosophy: The Revolution in Modern Science (New York, Harper & Row, 1962).

fn46 This is an interferometer: it uses the interference between the lasers which run along the two arms to reveal the minute variations in length of these arms.

fn47 The Ecclesiasticus is considered part of the Bible by Catholics, most of the Oriental Orthodox Church and some Jews. The Lutheran churches include it in their lectionaries, and as a book proper for reading, devotion and prayer, but not in the Bible. For most Jews and.the Anglican Church the situation is similar.

fn48 A subtle point: information doesn't measure what I know but the number of possible alternatives. The information I am given when the number 3 comes up in roulette is N = 37, because there are 37 numbers; but the information I have when number 3 wins on red is N = 18, because there are 18 red numbers. How much information do we have if we learn which of the brothers Karamazov murdered their father? The answer depends on how many Karamazov brothers there are.

fn49 Boltzmann did not use the concept of information, but his work can be read in this way.

fn50 Entropy is proportional to the logarithm of the volume of the phase s.p.a.ce. The constant of proportionality, k, is Boltzmann's constant, which transforms the units of measurement for information (bits) into the units of measurement for entropy (Kelvin's joules).

fn51 In a finite region of its phase s.p.a.ce.

fn52 This is what came to be called, inappropriately, the 'collapse' of the wave function.

fn53 Here is how it works technically: a Boltzmann statistical state is described by a function on phase s.p.a.ce given by the exponential of the Hamiltonian. The Hamiltonian is the generator of time evolution. In a system in which time is not defined, there is no Hamiltonian. But if we have a statistical state, we just take its logarithm and this defines a Hamiltonian, and hence a notion of time.

end.