Life with More Than 4: Extra Dimensions

Transcription

1 Life with More Than 4: Extra Dimensions Andrew Larkoski 4/15/09 Andrew Larkoski SASS 5

2 Outline A Simple Example: The 2D Infinite Square Well Describing Arbitrary Dimensional Spacetime Motivations for Extra Dimensions Models of Extra Dimensions 4/15/09 Andrew Larkoski SASS 6

3 2D Infinite Square Well 4/15/09 Andrew Larkoski SASS 7

4 Reminder: 1D Infinite Square Well Energy Eigenstate Wavefunctions: Energy Eigenstates: 4/15/09 Andrew Larkoski SASS 8

5 2D Infinite Square Well Strategy for solving Schrödinger Equation: Separation of Variables into complete set of states! 4/15/09 Andrew Larkoski SASS 9

6 2D Infinite Square Well Energy Eigenstate Wavefunctions: Energy Eigenstates: 4/15/09 Andrew Larkoski SASS 10

7 2D Infinite Square Well Thought Experiment: Exciting with laser light If, laser cannot excite y-modes! Can only measure energy differences, so do not know that y dimension exists! Unless is large enough... 4/15/09 Andrew Larkoski SASS 11

8 2D Infinite Square Well Spectrum: Fixed : Usual spectrum Fixed : Get a tower labeled by 4/15/09 Andrew Larkoski SASS 12

9 The Geometry of Spacetime 4/15/09 Andrew Larkoski SASS 13

10 Describing Spacetime First, describing space: A manifold is an object that locally looks like Examples: 2D plane, Looks like locally and globally Distances are well defined by Pythagorean Theorem: The object defining distances is called the metric: 4/15/09 Andrew Larkoski SASS 14

11 Curvature (My) Definition of Curvature Failure of a square to sum to 360 degrees Examples: Flat Sum to 360 Positive Sum to > 360 Deviation from 360 is a measure of curvature Negative Sum to < 360 4/15/09 Andrew Larkoski SASS 15

12 Example, The 2-Sphere, Looks like locally (tangent planes are well-defined) Does not look like globally One (major) difference: volume of is finite Distances are well defined: Curvature is constant and equal to radius (here set to 1) 4/15/09 Andrew Larkoski SASS 16

13 Non-Example: Cone The cone is not a manifold! Does not look like locally everywhere There are ways to fix this (beyond the scope of my talk) 4/15/09 Andrew Larkoski SASS 17

14 Minkowski Space Spacetime is a Lorentzian, not Riemannian, manifold Time is different! Metric is not positive definite Metric (interval) of Minkowski Space: This metric is Lorentz invariant Important: In quantum field theory, metric is fixed! Quantum field theory does not describe gravity! 4/15/09 Andrew Larkoski SASS 18

15 Spacetime in General Relativity Locally, GR looks like Minkowski Now, the metric of the spacetime manifold is arbitrary Metric depends on matter content of spacetime via Einstein s equation: kinetic piece (measures curvature) source Metric is now a dynamical field! Energy-momentum sources gravity! Higher curvature = Greater energy density 4/15/09 Andrew Larkoski SASS 19

16 Motivations for Extra Dimensions 4/15/09 Andrew Larkoski SASS 20

17 Hierarchy Problem 4/15/09 Andrew Larkoski SASS 21

18 A Problem with Fundamental Scalars Or, why a massless fermion stays massless f v = c spin f v = c spin Massless fermion Lagrangian: 4/15/09 Andrew Larkoski SASS 22

19 A Problem with Fundamental Scalars Mass term requires anti-aligned spins: Quantum Mechanics respects Lorentz invariance Velocity and spin are locked in place Such a term can never be generated! Massless fermion stays massless! 4/15/09 Andrew Larkoski SASS 23

20 A Problem with Fundamental Scalars Consider a massless, fundamental scalar: s v = c s v = c Scalar has no intrinsic direction or vector Mass term of Lagrangian is not disallowed by any symmetries: Anything that is not forbidden will happen in QM Mass term is generated quantum mechanically! Nothing prohibits the mass to be as large as possible 4/15/09 Andrew Larkoski SASS 24

21 The Hierarchy Problem In the Standard Model, Higgs boson is responsible for mass All particles have mass proportional to Higgs mass Why is the mass of the Higgs boson (~ 100 GeV) so much smaller than the energy scale at which gravity becomes strong (~ 10^18 GeV)? 4/15/09 Andrew Larkoski SASS 25

22 Extra Dimensions and Their Contents 4/15/09 Andrew Larkoski SASS 26

23 Königsberg, 1921 First Extra-Dimensional Model: Kaluza-Klein Theory 1 paper, 289 citations Solution to Hierarchy problem: Unite gravity and EM in five dimensions! Idea: 4D Spacetime 5th Dimension 4/15/09 Andrew Larkoski SASS 27

24 Königsberg, 1921 Gravity can fluctuate in all 5 dimensions Fluctuations in extended 4D: continuous spectrum Fluctuations in compact dimension: discrete spectrum Lowest energy state = massless photon! A tower of increasing mass photons also exists Issues: 5D Gravity = 4D gravity + 4D Vector + Scalar Scalar corresponds to radius of 5th dimension 4/15/09 Andrew Larkoski SASS 28

25 SLAC/Stanford, 1998 Arkani-Hamed, Dimopoulos, Dvali: Large Extra Dimensions 2 papers, 6648 total citations Solution to hierarchy problem: Large extra dimensions dilute gravity! Idea: Gravity here ~ constant Gravity here ~ 1/r Gravity becomes strong at energy defined by size of ED Very weak limits on size E.g., One ED could be as large as 1 mm! 4/15/09 Andrew Larkoski SASS 29

26 Boston, 1999 Randall, Sundrum: Warped Extra Dimensions 2 papers, 8068 total citations Solution to hierarchy problem: Take logarithm of hierarchy problem! Scales are warped down from Planck to TeV Because of warping, extra dimension can actually be infinite! 4/15/09 Andrew Larkoski SASS 30

27 Things I Didn t Discuss Other solutions to the Hierarchy problem: Supersymmetry See my SASS talk Composite Higgs models Technicolor, etc. None, there is no Hierarchy problem Split Supersymmetry, etc. 4/15/09 Andrew Larkoski SASS 31

28 Things I Didn t Discuss Other connections of Extra Dimensions String Theory Requires many extra dimensions for consistency AdS/CFT Relationship between a (supergravity) theory and the (CFT) physics at its boundary In higher dimensions, many systems become easier to analyze e.g., maximal supersymmetry in 10 D maximal supergravity in 11 D 4/15/09 Andrew Larkoski SASS 32