Quantum Gravity Part 2

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Quantum GravityPart 2

• A presentation by J. F. Wong
• Department of Physics

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Things we learnt in Part 1

• Quantum Gravity has to be background independent,
  i.e. Space-time (in which particles move) must
  also be a consequence of the theory.
  
• Quantum Field Theory is not background
  independent.
  
• In fact, the only experimentally verified theory
  that is background independent is

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General Relativity - Summary

• 1. Laws of Physics are written down in tensorial
  equations. (General Covariance)
  
• 2. Space-time is a curved pseudo-Riemannian
  manifold with a metric of signature (- ).
  

Gravitation, Misner, Thorne, Wheeler
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General Relativity - Summary

• 3. Dynamics of physical bodies is given by

• (the Geodesics)
• 4. The relationship between the curvature of
  space-time and matter is given by (Background
  Independence)

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This suggests that

• In order to construct Quantum Gravity, we should
  fix our starting point at General Relativity.
  
• We quantize it while keeping all its principles
  intact.

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Canonical Quantization

• 1. Pick a Poisson Algebra of classical quantities.

• 2. Represent these quantities as quantum
  operators acting on a space of quantum states.

Quantum Mechanics, Eugen Merzbacher
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Canonical Quantization

• 3. Implement any constraint you may have as a
  quantum operator equation and solve for the
  physical states.

• 4. Construct an inner product on physical states.

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Canonical Quantization

• 5. Develop a semi-classical approximation and
  compute expectation values of physical
  quantities.
  

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Canonical Quantization

• If we apply Canonical Quantization to a Classical
  mechanical system, we get a Quantum mechanical
  system.
  
• If we apply it to a Classical Relativistic Field
  Theory, we get a Quantum Field Theory.

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Lagrangian Formulation of GR

• The first step is to find the Lagrangian. Is
  there a Lagrangian for General Relativity?

• Such that we can have an action principle

(For simplicity, we only consider vacuum
space-time)
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Lagrangian Formulation of GR

• Note that the theory is invariant under general
  coordinates transformation. So the Action has to
  be a scalar. But the infinitesimal volume is not
  a scalar.

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Lagrangian Formulation of GR

• Letting

• The simplest possible Action is

(Einstein-Hilbert Action), R is Ricci Scalar.
David Hilbert (1916)
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Lagrangian Formulation of GR

• Proof. First we need 3 identities (exercise).

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Lagrangian Formulation of GR

• In the last step I have used the metric
  compatibility condition.

• The last term contains a total divergence, and
  the integral of it can be converted to a boundary
  term, hence does not contribute.

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Lagrangian Formulation of GR

• So it follows

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Hamiltonian Formulation of GR

• The next step is to find the Hamiltonian.
• We need to split space-time into space and time,
  without a notion of time, there is no notion of
  evolution and therefore no Hamiltonian.

R. Arnowitt, S. Deser, C. Misner, Gravitation, an
introduction to current research, 1962
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Hamiltonian Formulation of GR

• This may seem odd at first, one of the main
  points of GR is to cast space time on the same
  footing and this approach seems to separate them
  again.
• It doesnt matter because the covariance is
  restored by certain relations that appear in the
  canonical formulation.

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Hamiltonian Formulation of GR

• Consider a slicing of space-time into a family of
  3D spacelike hypersurfaces and a time coordinate
  t. Each hypersurface corresponds to t const.

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Hamiltonian Formulation of GR

• N is called Lapse, Ni is called Shift.
• ds2 (proper distance in base hypersurface)2
• (proper time from base to upper
  hypersurface)2

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Hamiltonian Formulation of GR

• That is, we have split the metric to a spatial
  metric of hypersurface, lapse and shift.

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Hamiltonian Formulation of GR

• Lagrangian can be expressed in terms of these
  variables. (calculations skipped)

• where

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Hamiltonian Formulation of GR

• We take the configuration space to be Met(S), the
  space of all spatial metrics.
  
• We identify qij to be the canonical variable and
  proceed to find its canonical conjugate
  momentum.
  

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Hamiltonian Formulation of GR

• The dynamical variables Lapse and Shift are not
  canonical variables because their canonical
  conjugate momenta are zero.
  
• We perform Legendre Transformation, and obtain
  the Hamiltonian.

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Hamiltonian Formulation of GR

• Since N and Na are dynamical variables, we expect
  dS 0 with respect to N or Na separately should
  yield some equations that must hold.

• They are called Hamiltonian constraint and
  Diffeomorphism constraint respectively.

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Intractable Problems

• It seems that General Relativity is now ready to
  be canonically quantized.

• However, people found that a lot of problems
  appear and become intractable.

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Intractable Problems

• The first severe problem is the Operator Ordering
  Problem. There is no satisfactory way of
  promoting the constraints to operators, so that

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Intractable Problems

• We may simply pick an Ordering scheme and pretend
  that the problem is solved. But nobody has ever
  found any solution of the resulting dynamical
  equation.

• (Wheeler-DeWitt Equation)

B. DeWitt, Phys. Rev. 150, 1113 (1967)
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Intractable Problems

• Since there is no physical states found, we
  cannot construct an inner product, there is no
  Hilbert Space.
  
• Such problems had haunted the people of quantum
  gravity for decades, until Abhay Ashtekar
  published his new variables for GR in 80s, and
  the problems were finally solved.

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Loop Quantum Gravity

• He initiated an approach called Loop Quantum
  Gravity.
  
• But this will be the story told in Part 3 of my
  Quantum Gravity presentations, one year later

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Conclusion

• 1. To formulate quantum gravity one should start
  from the background-independent theory, General
  Relativity.
  
• 2. People faced many problems while attempting
  the Canonical Quantization of General
  Relativity.
  
• 3. All those problems called for new variables
  for GR, and this motivates the development of
  Loop Quantum Gravity.

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References

• 1 John Baez, Javier P. Muniain, Gauge Fields,
  Knots And Gravity.
  
• 2 Rodolfo Gambini, Jorge Pullin, Loops, Knots,
  Gauge Theories and Quantum Gravity.
  
• 3 Charles W. Misner, Kip S. Thorne, John A.
  Wheeler, Gravitation.