17.Quantum Gravity

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17. Quantum Gravity and Spacetime
Quantum Electrodynamics (1940's) RQFT of electromagnetic force. Matter fields: leptons, quarks.
Force field: (photon). U(1) symmetry.
Electroweak theory (1960's) RQFT of EM and weak forces. Matter fields: leptons, quarks.
Force fields: , W+, W, Z. U(1) SU(2) symmetry.
Quantum Chromodynamics (1970's) RQFT of strong force. Matter fields: quarks. Force fields: gluons (8 types). SU(3) symmetry.
Theory of MatterRelativistic Quantum Field Theories (RQFTs)
Standard Model (1970's80s) RQFT of EM, strong, and weak forces. Matter fields: leptons, quarks.
Force fields: , W+, W, Z, gluons. U(1) SU(2) SU(3) symmetry.

17. Quantum Gravity and SpacetimeTheory of SpacetimeGeneral Relativity (GR) (1916)
Classical (nonquantum)theory of gravitational force.
Diff(M) symmetry.
Inconsis
tent!
Quantum Electrodynamics (1940's) RQFT of electromagnetic force. Matter fields: leptons, quarks.
Force field: (photon). U(1) symmetry.
Electroweak theory (1960's) RQFT of EM and weak forces. Matter fields: leptons, quarks.
Force fields: , W+, W, Z. U(1) SU(2) symmetry.
Quantum Chromodynamics (1970's) RQFT of strong force. Matter fields: quarks. Force fields: gluons (8 types). SU(3) symmetry.
Theory of MatterRelativistic Quantum Field Theories (RQFTs)
Standard Model (1970's80s) RQFT of EM, strong, and weak forces. Matter fields: leptons, quarks.
Force fields: , W+, W, Z, gluons. U(1) SU(2) SU(3) symmetry.

Theory of MatterRelativistic Quantum Field Theories
Flat Minkowski spacetime, unaffectedby matter.
Matter/energy and forces are quantized.
"Compact" symmetries.
Theory of SpacetimeGeneral Relativity
Curved Lorentzian spacetimes,dynamically affected by matter.
Matter/energy and forces are classical.
"Noncompact" symmetries.
Two General Approaches to Reconciliation:
(A) BackgroundDependent Approach.
(B) BackgroundIndependent Approach.
Start with a background spacetime (in RQFTs its Minkowski spacetime).
Try to construct a quantized gravitational field on it.
Problem: Standard method of quantizing classical fields, when applied tothe gravitational field, produces a "nonrenormalizable" RQFT.
Start with a spacetime with no fixed metric (as in GR).
Try to construct a quantized gravitational field on it.
Problem: Standard method of quantizing classical fields requires abackground metrical structure.

Problems:
Approach A. Example 1: String Theory
General Idea: Take QFT as a given and try to force GR into its mold.
o Gravitational force is treated on par with the other forces.
o Spacetime is represented by flat Minkowski spacetime (backgrounddependence).
Requires extra spatial dimensions, depending on the version of stringtheory. Require that these extra dimensions are "compactified":severely "rolledup" so that we can't normally experience them.
No testable predictions after ~30 years of work.
Takes no lesson from Einstein's insightful geometrization of gravity.
Procedure:
Replace 0dim point particles with 1dim strings.
o Consequence: Many "nonrenormalizable" divergencesare tamed in the standard approach to quantization.

Approach B. Example 1: Loop Quantum Gravity
General Idea: Take GR as a given and try to force QFT into its mold.
Procedure:
Try to identify the "observables" of GR: very hard to do!
Problems:
Spacetime structure is not fixed, but is determined by the quantum (loop)version of the Einstein equations.
Standard method of quantizing a classical field theoryrequires identifying thetheory's "observables" (quantities that are invariant underthe theory's symmetries).
o Because of the strange Diff(M) symmetry ofGR, its observables are strange; in particular,there are no local observables!
o Identify nonlocal "loop" observables(quantities that depend on particular closedpaths in spacetime).
Constraint equations that result from quantization have yet to be solved.
No testable predictions after ~20 years of work.
Takes no lesson from the QFT approach's insightful suggestion thatQFTs are lowenergy approximations to a more fundamental theory.

Question: What do backgrounddependent and backgroundindependentapproaches to quantum gravity suggest about the nature of spacetime?
Backgrounddependent approach
priorspacetimestructure
physical fields
Backgroundindependent approach
no priorspacetimestructure
physical (andspatiotemporal?)fields
Claim: Both approaches may be interpreted in either substantivalist orrelationist terms.
BackgroundDependence
A substantivalist may claim that the prior spatiotemporal structure isexhibited by real, substantival spacetime.
A relationist may claim that the prior spatiotemporal structure isexhibited by a real physical field (the metric field, say).

Question: What do backgrounddependent and backgroundindependentapproaches to quantum gravity suggest about the nature of spacetime?
Backgrounddependent approach Backgroundindependent approach
priorspacetimestructure
no priorspacetimestructure
physical fieldsphysical (andspatiotemporal?)fields
Claim: Both approaches may be interpreted in either substantivalist orrelationist terms.
BackgroundIndependence
A substantivalist may claim that spatiotemporal structure depends(as in GR) on physical fields, but once it is sodetermined, it existsin its own right.
A relationist may claim that spatiotemporal structure is exhibited bythe relations between physical objects, so only such objects exist.

Approach A. Example 2: Condensed Matter Approaches.
General Idea: Suppose GR and the Standard Model describe the lowenergyfluctuations of a condensed matter system (like a superfluid, or asuperconductor).
Then: The condensate would explain the origin of both spacetime and gravity(as described by GR), and matter fields and the other forces (as described bythe Standard Model).
In Fact: Some nonrelativistic condensed matter systems exhibit lowenergyfluctuations that resemble aspects of GR and the Standard Model.
Example:
o Superfluid Helium 3A. Lowenergy fluctuations behave like relativisticmassless fields coupled to an electromagnetic field.
 Tickle (nonrelativistic) superfluidHe3A with small amount of energy.
 Lowenergy ripples behave like relativistic fermions coupled to EM field.
Superfluid He3A

How can spacetime be thought of in the Condensed Matter approach?
A backgrounddependent approach:
o The fundamental condensate has prior nonrelativistic (Galilean)spatiotemporal structure.
A substantivalist may say: "The prior Galilean spatiotemporal structure isgiven by properties of real spacetime points in a Galilean spacetime. Takethe fundamental condensate out of the universe and real Galilean spacetimewould be left."
A relationist may say: "The prior Galilean spatiotemporal structure is givenby properties of the fundamental condensate. Take it out of the world andnothing would be left."
Problems:
No condensed matter system has yet been identified that reproduces GRand the Standard Model exactly in the lowenergy regime.
Many disanalogies between real condensed matter systems and theiridealized lowenergy regimes.