Functional Integrals in Affine Quantum Gravity

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---- Functional Integrals in ----
----
Affine Quantum Gravity ----

• John R. Klauder
• University of Florida
• Dresden, September, 2007

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Affine Quantum Gravity

• Kinematical variables
• Metric positivity
• Initial representation
• Without constraints
• Quantum constraints
• Projection operator method
• Imposition of constraints
• Functional integral representation

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Kinematical variables

• Affine commutation relations

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Initial representation

• Affine coherent states

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Quantum constraints

• Enforcing the constraints

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Quantum constraints (2)

• Non-standard constraints

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Quantum constraints (3)

• Observables

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Imposition of Constraints

• Path integral

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Initial representation

• Affine coherent states

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Imposition of Constraints (2)

• Functional integral

The Heart of the Matter
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Further Issues

• Nonrenormalizability?
• Hard-core interpretation
• Appearance of Time?
• Soluble examples
• Removal of Cutoffs?
• Choose proper representation
• Reduce as necessary

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Hard-core interaction

• Free and pseudo-free theories

Theory
DISCONTINUOUS
PF
F
CONTINUOUS
Coupling constant
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Hard-core interaction (2)

• Idealized functional integral

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Hard-core interaction (3)

• Unconventional model analysis

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Summary AQG

• Preserve metric positivity
• Affine kinematical variables
• Gravitational anomaly
• Projection operator method
• Functional integral formalism
• Continuous-time regularization
• Nonrenormalizable theory
• Hard-core interaction

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• Thank you !