Quantum Physics, Quantum Spacetime and Quantum Geometry Sergio Doplicher Universit`adi Roma \Sapienza" February 16, 2016 Colloquium "Levi - Civita", 17 Febbraio 2016 1. Why Quantum 2. Why Quantum Spacetime 3. Quantum Minkowski Space 4. Quantum Geometry 5. QFT on QST 6. Comments on QST and Cosmology 1 Why Quantum 2 In a physical theory, the observable quantities are described by the selfad- joint elements in a C* Algebra A (a Banach * algebra s.t. jjA∗Ajj = jjAjj2; equivalently a uniformly closed * algebra of linear bounded operators on a Hilbert space); the physical states are described by states of that alge- bra (i.e. positive normalized linear functionals) and the expectation value of the observable A in the state ! is given by !(A). 3 Thus the uncertainty of A in ! is (∆A)2 =< (A− < A > I)2 > : If AB = BA, then A and B are compatible; otherwise, in any state of the system, ∆A∆B & (1=2)j < AB − BA > j: (Elementary consequence of Schwarz's inequality for the positive semidefinite sesquilinear form assigning to A; B the number !(A∗B)). If the C* Algebra generated by the observables is com- mutative, it can be identified with C0(Ω) for some lo- cally compact space Ω, the phase space of a Classical Theory. Noncommutativity of A is equivalent to the presence of any (hence of all) the following properties: 1. There is a pair of observables which cannot be mea- sured simultaneously. 2. There is an observable A and a pure state where A does not take a precise value (Quantum fluctuations). 3. There are pairs of distinct pure states which can be superposed, giving rise to a pure state distinct from both. 4. there exists a pair of distinct pure states with a nonzero transition probability from each to the other. 5. While time evolution in deterministic, measuring some observable in some pure state produces a state which is no longer pure. Thus commutativity characterizes Classical Theories, Noncommutatity is the sole responsible of the unintu- itive features of Quantum Mechanics. Is this choice a "capriccio" or even a "stravaganza"? NO. It was IMPOSED on us by Nature, after nearly 30 years of misunderstandings of what Nature was telling us. Is QM relevant only at the atomic scale?NO: - Superconductors (LHC); - Specific heat of metals, ambient temperatures; - The largest system we can think of, CMB, is a Quan- tum System. A word on the "Black body problem" is here in order. 4 (S.M.McMurry: Quantum Mechanics, Addison - Wesley, 1994). 6 Thus classically equilibrium matter - radiation was im- possible! Great failure of Classical Physics at the end of the XIX Century. CMB = Black Body radiation at T = 2:725 degrees Kelvin, temperature fluctuations 0.0002 degree, angu- lar size = that of a full Moon. Horizon Problem Solved by inflation hypothesis, but also by Quantum Spacetime. How did the Quantum picture of the World emerge? ** To explain the experimental curve of the spectrum of the black body radiation we had to assume that (Planck, 1900): the energy of a one dimensional harmonic oscillator of frequency ν can only take values which are integral mul- tiples of hν (h = Planck constant). ** To explain the photoelectric effect we had to assume that (Einstein, 1905): Electromagnetic Radiation of frequency ν consists of an integer number of indivisible units (photons) of energy hν. ** To explain why matter is stable and with a discrete spectrum of spectral lines (while the Planetary Model of atoms, confirmed by the Rutherford experiment (1911), would be instable, with continuos emission / absorption spectra according to Classical Physics!) we had to as- sume that: Bohr (1911 - 1924): Atomic bound states have (as Planck's oscillator) a discrete set of energy levels, the conservation of energy is respected in jumps by emission / absorption of a single photon: En − Em = hνnm This would explain the discrete nature of the observed emission / absorption spectra (hence would explain why we see colors!), and stability of atoms. BUT WHY? Dramatic wreck in Classical Physics. The answer was somehow implicit in the famous three 1905 papers of Einstein! The photon is indivisible like a particle, but is governed by a wave equation (Maxwell Equations)! If the electron, as the photon, is also governed by a wave equation, discrete energy levels arise as the dis- crete serie of frequencies produced by a vibrating string. Schroedinger equation. But: a bullet shot through an open window does not feel the window; while a wave gets diffracted going through an opening in a dam. Thus precision in one component of the coordinates im- plies imprecision in the same component of the speed. And, for the electron too, the precision in one compo- nent of the coordinates would imply imprecision in the same component of the momentum. Ideed: Heisenberg (1927): assigning a point in phase space to an electron has NO operational meaning: in any mea- surement process which is actually at least conceptually possible, we must have: ∆q∆p & ~ (~ = Planck constant per unit radiant). (The three 1905 papers by Einstein included that on Special Relativity. Key point: Operational definition of physical quantities to criticize the notion of simultane- ity. Heisenberg analysis took up that lesson 22 years later!) Can this fact be implemented by a precise mathematical scheme? Yes: pq − qp = −i~ (Born; Born and Jordan; Born, Heisenberg and Jordan, 1925; gives precise mathematical shape to Quantum Conditions invented by Heisenberg). Dirac (1926 - 1930): (Heisenberg - Born - Jordan and Schroedinger approaches are just special representa- tions of a new picture, where:) Observables are de- scribed by Operators (selfadjoint operators acting on a Hilbert space); spectrum = possible values; discrete spectrum and continuos spectrum; for the Energy Op- erator (Hamiltonian) the discrete spectrum describes BOUND STATES. The assumptions of Planck and of Bohr, stability of atoms, discrete emission - absorption spectra, . explained! The lesson that Nature was whispering in those nearly 30 years eventually understood. Modern view: the observables are selfadjoint elements of a C* algebra, Hilbert spaces arise as representation spaces (e.g. determined by a state, GNS construction). Heart of QM: Uncertainty Relations as a shadow of Noncommutativity. Probabilistic aspects are a consequence. Copernicus; Darwin; Freud; Quantum Mechanics! LAWS OF NATURE; Naturalism; Ethics and primacy of Culture. Why QST QM finitely many d. o. f. ∆q∆p & ~ positions = observables, dual to momentum; in QFT, local observables: A 2 A(O); O (double cones) - spacetime specifications, in terms of coordinates - accessible through measurements of local observables. Allows to formulate LOCALITY: 7 AB = BA whenever A 2 A(O1);B 2 A(O2; ); and O1; O2 are spacelike separated. "Unreasonable effectiveness”. OK at all accessible scales (down to 10−17cm:, LHC at CERN); in QFT at all scales, if we neglect GRAVITA- TIONAL FORCES BETWEEN ELEMENTARY PAR- TICLES. 8 If we DON'T: Heisenberg: localizing an event in a small region costs energy (QM); Einstein: energy generates a gravitational field (CGR). A strong gravitational field can generate "closed trapped surfaces". QM + CGR: PRINCIPLE OF Gravitational Stability against localization of events [DFR, 1994, 95]: The gravitational field generated by the concentra- tion of energy required by the Heisenberg Uncer- tainty Principle to localize an event in spacetime should not be so strong to hide the event itself to any distant observer - distant compared to the Planck scale. Spherically symmetric localization in space with accu- racy a: an uncontrollable energy E of order 1=a has to be transferred; Schwarzschild radius R ' E (in universal units where ~ = c = G = 1). 9 Hence we must have that a & R ' 1=a; so that a & 1; i.e. in CGS units −33 a & λP ' 1:6 · 10 cm: (1) (Planck length). (FOLKLORE). But elaborations in two significant directions are sur- prisingly recent. First, if we consider the energy content of a generic quantum state where the location measurement is per- formed, the bounds on the uncertainties should also depend upon that energy content. Second, if we consider generic uncertainties, the argu- ment above suggests that they ought to be limited by uncertainty relations. To the first point: background state: spherically symmetric distribution, total energy E within a sphere of radius R, with E < R. If we localize, in a spherically symmetric way, an event at the origin with space accuracy a, due to the Heisen- berg Principle the total energy will be of the order 1=a + E. Reasoning as above, we must then have 1 + E < R: a Thus, if R − E is much smaller than 1, the \minimal distance" will be much larger than 1. Indication that Quantum Spacetime solves the Horizon Problem. To the second point: The condition that closed trapped surfaces do not arise implies that the ∆qµ must satisfy UNCERTAINTY RELATIONS. Should be implemented by commutation relations, making spacetime into a QUANTUM SPACETIME. Dependence of Uncertainty Relations, hence of Com- mutators between coordinates, upon background quan- tum state i.e. upon metric tensor. Hard problem. CGR: Geometry ∼ Dynamics QG: Algebra ∼ Dynamics WARNING: Size of the known Universe: 1028cm. Planck scale: much further away from us in log scale. QM: no contradiction from the size of the Universe (CMB is a quantum system!) down to scales explored by LHC. Below that scale no direct experimental knowledge. GR: no contradiction from the size of the Universe down to scales of order 10−2cm. Below, nothing is known. The (Planck length), 10−33cm, is very far. Quantum Minkowski Space The Principle of Gravitational Stability against localization of events implies : 3 X X ∆q0 · ∆qj & 1; ∆qj∆qk & 1: (2) j=1 1≤j<k≤3 [DFR 1994 - 95]; see also [TV 2012] [DMP 2013] Must be implemented by SPACETIME commutation relations 10 2 [qµ; qν] = iλP Qµν (3) imposing Quantum Conditions on the Qµν.