/6/017 Fromm Institute for Lifelong Learning University of San Francisco Modern Physics for Frommies V Gravitation Lecture 8 Administrative Matters Suggested reading Agenda What do we mean by Quantum Gravity Quantum Gravity Unification 1 March 017 Modern Physics V Lecture 8 1 1 March 017 Modern Physics V Lecture 8 Suggested Reading Lee Smolin, Three Roads to Quantum Gravity, Basic Books, Perseus Books Group, (001) Bernard Schutz, Gravity from the ground up, Cambridge University Press What do we mean by Quantum Gravity? Quantum refers to the theory of quantum mechanics Incorporates various seemingly paradoxical properties of light and matter e.g. wave=particle duality, uncertainty principles, the probabilistic nature of measurements Gravity refers not to Newton s theory of gravitation, but to Einstein s theory of general relativity. Newton's Law of Gravitation: m m F = G r r 1 ˆ February, 1 March 017 Modern Physics V Lecture 7 3 1 March 017 Modern Physics V Lecture 8 4 1
/6/017 Special relativity is restricted to inertial reference frames. General relativity is not subject to this restriction. Einstein s great insight was the equivalence principle: Physics in a uniformly accelerating reference frame is indistinguishable from in a constant gravitational field. Allowed construction of a theory which described gravitation, not as a force acting between massive bodies, but as the manifestation of the geometry of space time surrounding any given configuration of matter. This distortion of space time is what is perceived by us to be the force of gravity. 1 March 017 Modern Physics V Lecture 8 5 Unification The primary guiding philosophy in the development of physics over the past two centuries has been the idea of unification. James Clerk Maxwell unified the theories of electricity and magnetism into a single framework, electromagnetism ρ E = ε 0 B = 0 B E = t B = µ j + µ ε 0 0 0 E t 1 March 017 Modern Physics V Lecture 8 6 In traveling wave equation 1 this is v Quantum Mechanics was also born out of the failures of classical physics. E. M. wave equation E E + µ ε = t 0 0 0 1 8 = c = 3 x 10 µ ε 0 0 = x + y m/sec + z Max Planck Albert Einstein Niels Bohr Erwin Schrödinger Werner Heisenberg Paul Dirac Special relativity reconciled Maxwell s equations with the motions of material bodies. Maxwell s equations are invariant under Lorentz transformations but are not invariant under Gallilean transformations. 1 March 017 Modern Physics V Lecture 8 7 Identification of electromagnetic fields with QM objects known as photons Non relativistic quantum mechanics ħ ψ ( x) + U ( x) ψ ( x) = Eψ ( x) m x 1 March 017 Modern Physics V Lecture 8 8
/6/017 During the period following the end of WWI and about 1950 this led to the development of Quantum Electro Dynamics (QED) which provides a unified, relativistic and unified description of electromagnetic fields in a fully quantum setting 1 March 017 Modern Physics V Lecture 8 9 Following WWII the rapid proliferation of new elementary particles being discovered (the particle zoo) led theorists to postulate the existence of two more forces of nature. To the gravitational and electromagnetic forces the weak and strong nuclear forces were added. Richard Feynman, Murray Gell-Mann, Gerard t Hooft among many others produced a unified theory of the weak force and electromagnetism, the electroweak theory. The last quarter of the 0 th century saw the establishment of the Standard Model of Elementary Particles which provides a unified - albeit, in some ways flawed description of the weak, strong and electromagnetic forces as excitations of the quantum mechanical vacuum Gravity remains outside the grasp of such unified frameworks. Current most complete formulation of physics is Standard Model + Gravity 1 March 017 Modern Physics V Lecture 8 10 Why is Gravity so Antisocial? Central feature of QM is the principle of superposition, i.e. wave functions of two different particles can overlap. Two systems described by two different wave functions ψ x, t and ψ x, t ( ) ( ) 1 Can instead be treated as a single composite system with wave function ( x, t) = ( x, t) + ( x, t) ψ ψ ψ 1 Functions must be defined on some set. In QM the set is taken to be the coordinates (x,t) of the space time in which our system is embedded. We can also write the superposition in terms of the momenta by working in the (p,t) basis with the momenta p being related to the position x by the usual Fourier transform: ipx ψ ( p, t) e ψ ( x, t) dx In either basis there is an implicit assumption of a flat (Minkowski) background geometry for which we can assign a set of C-ordinates (x,y,z,t) to each point of the space time. GR => physics should be independent of the particular coordinates describing the system. In fact, any theory which is consistent with GR must be well defined both on curved space and on flat space. The quantum field theories of the Standard Model are embedded in flat space time 1 March 017 Modern Physics V Lecture 8 11 1 March 017 Modern Physics V Lecture 8 1 3
/6/017 The case of wave functions living on curved space is tricky. The complications associated with background curvature can be resolved by resorting to sufficiently sophisticated mathematical methods The resulting framework is known as Quantum Field Theory on Curved Space Time (QFT CS). Used by Stephen Hawking to show that a black hole must emit thermal radiation at a rate inversely proportional its mass. QFT CS is however not a true theory of quantum gravity The modern conception of gravity has it arising from a nontrivial region of space time induced by some distribution of matter. 1 March 017 Modern Physics V Lecture 8 13 A true theory of quantum gravity should be able to tell us how to write down the wave function not defined on a given region of space time, but a wave function of a given region of space time,, allowing us to construct states which correspond to superpositions of different geometries, QFT CS is not a theory of quantum gravity. The curved space time merely serves as an arena on which quantum states can be defined, but there is no notion of the states of the geometry itself, rather than of the matter which moves about on that geometry. At present there are several approaches towards attempts of writing quantum states of geometry. The two most developed of these are String Theory and Loop Quantum Gravity. 1 March 017 Modern Physics V Lecture 8 14 String Theory Instead of a description of fundamental particles as point-like objects switch to a picture where the basic entities are extended one-dimensional objects called strings These strings move and interact in some background space time. This space time s dimensionality is restricted by requirements of physical and theoretical consistency to 6, 11 and 10 depending on the characteristics - fermionic, bosonic, open, closed - with which we wish to endow the strings. The excitations of a string happen to include a part corresponding to a massless spin particle, i.e. a graviton. Gravitons are often thought of as quanta of the gravitational field as photons are the quanta of the electromagnetic field, this belief is only partially correct. 1 March 017 Modern Physics V Lecture 8 15 The gravitational field is characterized by geometric attributes like lengths, areas and volumes. Quanta of the gravitational field should correspond to quantization of these attributes in same way that that a quantum of the electromagnetic field corresponds to a quantized amount of energy given by Planck s relationship E = hf The graviton picture does not predict any such relations between any fundamental geometric quantities, e.g. the area of a region of space time and the frequency of a gravitational wave passing through this region Gravitons are perturbations of the background space time which is presumed to be smooth and continuous. Studying gravitons allows studying perturbations of the gravitational field but does not give us any indication of the atoms which are its components 1 March 017 Modern Physics V Lecture 8 16 4
/6/017 Loop Quantum Gravity (LQG) Born about a decade after String Theory From the very beginning the notion of a smooth, continuous background geometry is abandoned in favor of a discrete atomic or molecular geometry, call it component geometry. These components are referred to as simplices, which is a complicated term for elementary geometric objects such as triangles and tetrahedra. A collection of simplices can be assembled to build a D or 3 D geometry respectively. Provides a framework for studying quanta of geometry and to construct superpositions of different states of geometry LQG approach has its share of problems. Two significant obstacles are: (a) The lack of a grasp on how we can obtain an (approximately) smooth, continuous space time by gluing together our elementary simplices. (b) A lack of understanding of how matter particles (fermions), such as electrons and neutrinos, should be described in terms of quanta of geometry. 1 March 017 Modern Physics V Lecture 8 17 1 March 017 Modern Physics V Lecture 8 18 String Theory and LQG each have strengths and weaknesses String Theory provides us with a description of matter in terms of extended stringy objects but does not address the question of smoothness, or lack thereof, of space time. LQG provides us with a description of space time as,, being built out of atoms or quanta of geometry but does not tackle the question of matter degrees of freedom, (1) The final theory of quantum gravity will incorporate elements of both String Theory and LQG. () Complete understanding will not be achieved by resorting only to those insights gained from research in elementary particle physics and ignoring the insights in other fields of physics such as many body phenomena ( thermodynamics emerging from statistical mechanics, condensed matter physics) or the field of quantum computation. Terry s half baked predictions for the future: It s difficult to make predictions, especially about the future. - Yogi Berra (3) Whatever happens, it ought to be a fun ride 1 March 017 Modern Physics V Lecture 8 19 1 March 017 Modern Physics V Lecture 8 0 5