Download Towards a Deformation Quantization of Gravity

Towards a Deformation
Quantization of Gravity
Luther Rinehart
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1. Classical Mechanics = Symplectic Geometry
2. Deformation Quantization
3. Field Theory
4. Gravity
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Part 1:
Classical Mechanics = Symplectic Geometry
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Definition: A symplectic manifold is a manifold
Φ with a tensor field Ω𝑎𝑏 (symplectic form)
satisfying:
1. Antisymmetric: Ω𝑎𝑏 = −Ω𝑏𝑎
2. Non-degenerate: detΩ ≠ 0
3. Closed: 𝛻𝑎 Ω𝑏𝑐 + 𝛻𝑏 Ω𝑐𝑎 + 𝛻𝑐 Ω𝑎𝑏 = 0
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𝑎𝑏
There is a unique field Ω
𝑎𝑏
Ω Ω𝑏𝑐 =
satisfying
𝑎
−𝛿 𝑐
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Canonical form:
0
Ω=
−𝟙
𝟙
0
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Definition: Let 𝑓 and 𝑔 be scalar functions on
Φ. Their Poisson bracket is
𝑓, 𝑔 =
𝑎𝑏
Ω 𝛻𝑎 𝑓𝛻𝑏 𝑔
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Definition: A canonical transformation is a
function 𝑈: Φ → Φ that preserves Ω:
𝑎
𝑐
𝐷𝑈 𝑏 𝐷𝑈 𝑑 Ω𝑎𝑐
= Ω𝑏𝑑
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Canonical transformations are generated by
scalar functions: for η ∈ Φ,
𝑎
𝑎𝑏
η = Ω 𝛻𝑏 𝐻
where 𝐻 is some function on Φ.
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This is Hamiltonian Mechanics!
• Phase space is a symplectic manifold
• Observables are scalar functions with Poisson bracket
𝑓, 𝑔 = Ω𝑎𝑏 𝛻𝑎 𝑓𝛻𝑏 𝑔
• Dynamical evolution is a canonical transformation
• Hamilton’s equations: η𝑎 = Ω𝑎𝑏 𝛻𝑏 𝐻
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Part 2:
Deformation Quantization
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In quantum mechanics, we are interested in the
∞
algebra of observables 𝔸 = 𝐶 (Φ).
This has a commutative product given by
multiplication of functions.
How do we turn this classical theory into a
quantum theory?
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Deformation Quantization: turning 𝔸 into the
quantum algebra of observables by deforming
the product to be non-commutative.
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Specifically, we want to find a smooth curve
⋆ (ℏ) in the space of associative bilinear
products.
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Correspondence principle/Canonical
commutation:
⋆ 0 = multiplication
⋆ 0 = 𝑖 2 Poisson bracket
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The algebraic viewpoint: 𝔸 with ⋆ is sufficient
structure to formulate quantum mechanics.
• States, expectation values, spectra, and time evolution can
all be defined algebraicly.
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Example: Weyl Quantization when Φ is a
finite-dimensional vector space.
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𝑖ℏ
𝑖ℏ
𝑓 ⋆ 𝑔 = 𝑓 exp
Ω(𝛻, 𝛻) 𝑔 = 𝑓𝑔 +
2
2
𝑓, 𝑔 + 𝑂(ℏ2 )
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It can be shown that Weyl quantization is
equivalent to regular quantum mechanics.
However, it does not work for more general
symplectic manifolds because there is no
canonical derivative operator 𝛻.
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Current research: how to determine an
appropriate ⋆ on more general symplectic
manifolds.
• Perturbative expansion: Kontsevich (2003) Lett. Math. Phys. 66 157
[arxiv: q-alg/9709040].
• Geodesics?: Rinehart (2015) [arxiv: 1506.01618].
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Part 3:
Field theory
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Φ = set of solutions to the field equations
An infinite-dimensional manifold, with
tangent spaces given by the linearized field
equations.
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Example:
2
2
𝜆 3
− 𝜙
3!
2
2
𝜆 2
− 𝜙 𝜓
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Field equation: 𝛻 𝜙 + 𝑚 𝜙 =
Tangent space: 𝛻 𝜓 + 𝑚 𝜓 =
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To make this into a symplectic manifold, we
need to assign a tensor Ω𝑎𝑏 to each tangent
space.
For field equations derived from an action
principle, there is a canonical choice of Ω𝑎𝑏 .
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𝜇
𝜓1 Π2
Ω 𝜓1 , 𝜓2 =
−
𝜇
𝜓2 Π1
𝑛𝜇
Σ
𝜇
Π is the canonically conjugate momentum to 𝜓
in the linearized equation.
The integral is over a spacial slice Σ with normal
𝑛𝜇 . It is independent of the choice of Σ.
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So Φ is a symplectic manifold, and
∞
𝔸 = 𝐶 (Φ) has a Poisson bracket.
• Khavkine (2014) Int. J. Mod. Phys. A 29 1430009 [arxiv: 1402.1282].
• Khavkine and Moretti (2014) [arxiv: 1412.5945].
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Part 4:
Gravity
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For a fixed spacetime manifold 𝑀, let phase
space Φ be the space of solutions to
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𝑅𝜇ν − 𝑅𝑔𝜇ν = 8𝜋𝐺𝑇𝜇ν
2
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As before, the tangent spaces to Φ are
selected by the linearized field equation:
𝛻 𝛼 𝛻𝜇 ℎν𝛼 + 𝛻 𝛼 𝛻ν ℎ𝜇𝛼 − 𝛻 2 ℎ𝜇ν − 𝑔𝜇ν 𝛻 𝛼 𝛻𝛽 ℎ𝛼𝛽 − 𝛻𝜇 𝛻ν ℎ + 𝑔𝜇ν 𝛻 2 ℎ = 0
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The canonical momenta to ℎ𝛼𝛽 are:
Π 𝜇𝛼𝛽 = 𝛻 𝛼 ℎ𝛽𝜇 + 𝛻𝛽 ℎ𝛼𝜇 − 𝛻𝜇 ℎ𝛼𝛽 − 𝑔𝛼𝛽 𝛻ν ℎν𝜇 − 𝑔𝛼𝜇 𝛻𝛽 ℎ + 𝑔𝛼𝛽 𝛻𝜇 ℎ
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𝜇𝛼𝛽
ℎ1𝛼𝛽 Π2
Ω ℎ1 , ℎ2 =
−
𝜇𝛼𝛽
ℎ2𝛼𝛽 Π1
𝑛𝜇
Σ
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Problem: Ω is degenerate. It vanishes on any
field of the form
ℎ𝛼𝛽 = 𝛻𝛼 𝑣𝛽 + 𝛻𝛽 𝑣𝛼
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This theory has gauge freedom! Redundancy
up to diffeomorphisms of 𝑀.
ℎ𝛼𝛽 → ℎ𝛼𝛽 + 𝛻𝛼 𝑣𝛽 + 𝛻𝛽 𝑣𝛼
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“Quotient out” the gauge freedom:
Let Φ′ be the set of equivalence classes of Φ
under diffeomorphisms of 𝑀.
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Now, Ω is non-degenerate on the tangent
spaces of Φ′.
∞
′
Algebra 𝔸 = 𝐶 (Φ ). Equivalently, functions
that are constant on the equivalence classes.
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So gravity can be formulated as a symplectic
manifold.
If we can work out a satisfactory deformation
quantization of symplectic manifolds, this
would give a route to a mathematically welldefined formulation of quantum gravity.
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Final Thoughts
Physics currently lacks a mathematically sound formulation of
quantum field theory.
Failure to have a clear understanding of what we mean by
quantum field theory (or a clear understanding of what is
mathematically allowable and what is not) is a major obstacle
to understanding quantum gravity.
Deformation quantization and the algebraic formulation
provide one path to resolving this issue.
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