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Y. Kurihara (KEK)14/Dec./2017 @Nikhef
YK, Journal of Mathematical Physics 58 (9) (2017) 092502YK, arXiv:1703.05574
http://www.donquijote.co.uk/blog/don-quixote-de-la-mancha
Covariant quantization of general relativity
in four dimensional space time
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Outline▪ Introduction– List of quantum GR– Our strategy
▪ Mathematical set-up
▪ Translation symmetry in GR– What can we learn from a three-dimensional case?
▪ Canonical quantization– Nakanishi-Kugo-Ojima formalism– Physical vacuum, S-matrix, Unitarity
▪ Summary
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Introduction
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J. Mielczarek, T. TrzesniewskiarXive:1708.07445
IntroductionOutline of the taxonomy of approaches to quantum gravity
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Introduction
Conservative as mush as possibleWalk the high road
Covariant quantization
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▪ Our strategy
– Rely only on experimentally established theories:
▪ Quantum field theory & General relativity
– Do not assume any experimentally unconfirmed things:
▪ Supersymmetry → GL(1,3)/SO(1,3) +(SUc(3)SUw(2)Uy(1))
▪ Extra dimension → space time dimension = 4
▪ String / D-brane / AdS/CFT / Multiverse…
Introduction
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▪ Our strategy– Nothing other than QGR → No ToE
– Experimentally testable
▪ The experiment has already been done.
Introduction
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▪ Our strategy– Particle physicist’s view
Introduction
𝑔𝜇𝜈(𝑅)
𝑔𝜇𝜈(𝐸)
Riemannian Metric tensor:
Solution of Einstein eq.:
General Relativity: 𝑔𝜇𝜈(𝑅)
≡ 𝑔𝜇𝜈(𝐸)
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▪ Our strategy– Particle physicist’s view
Introduction
𝑔𝜇𝜈(𝑅)
𝑔𝜇𝜈(𝐸)
Riemannian Metric tensor:
Solution of Einstein eq.:
𝑔𝜇𝜈(𝑅)
=< 𝑔𝜇𝜈(𝑄)
>
Quantization
General Relativity: 𝑔𝜇𝜈(𝑅)
≡ 𝑔𝜇𝜈(𝐸)
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Mathematical set-up
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Mathematical set-up
Global Manifold 𝐺𝐿(1,3)
Local Lorentz Manifold 𝑆𝑂(1,3)
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Mathematical set-up
Global Manifold 𝐺𝐿(1,3)
Local Lorentz Manifold 𝑆𝑂(1,3)
Model of the universe
general principle of
relativity
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Mathematical set-up
Global Manifold 𝐺𝐿(1,3)
Local Lorentz Manifold 𝑆𝑂(1,3)
Einstein’s Equivalent principle!
Tangent (flat) space
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Mathematical set-up
Global Manifold 𝐺𝐿(1,3)
Local Lorentz Manifold 𝑆𝑂(1,3)
vierbein :
𝜂 =
1−1
−1−1
Line element :
Metric tensor:
𝜇, 𝜈, 𝜌,⋯ 𝑎, 𝑏, 𝑐,⋯
Einstein’s Equivalent principle!
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Mathematical set-up
Vierbein formalism
Spin connection :
GL(1,3) Covariant derivative :
Levi-Civita connection
What is parallel ?
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Mathematical set-up
Vierbein formalism
Differential forms
Vierbein form :
Spin form :
Torsion form :
Curvature form :
(Fraktur letters)
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Mathematical set-up
Vierbein formalism
Differential forms
volume form :
surface form :
(Fraktur letters)
නGL(1,4) invariant
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Mathematical set-up
Vierbein formalismSO(1,3) covariant total derivative :
Torsion form:
Bianchi identity:
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Mathematical set-up
Vierbein formalism
SO(1,3) constant tensors :
𝜂𝑎𝑏 =
1−1
−1−1
(Flat) Metric tensor :
Levi-Civita tensor : 𝜖𝑎𝑏𝑐𝑑 (𝜖0123 = +1)
Completely anti-symmetric tensor
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Mathematical set-up
Vierbein formalism
Lagrangian Action integral
Euler-Lagrange eq. of motion
Torsion-less equation Einstein equation
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Mathematical set-up
Vierbein formalism
(Fraktur letters)
Connection form :
Curvature form :
Chern classes:
Differential forms
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Mathematical set-up
Vierbein formalism
Differential forms (Fraktur letters)
Connection form :
Curvature form :
Chern classes:Chern-Wiel
theory
Topological invariant
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Translation Symmetry
YK, Journal of Mathematical Physics 58 (9) (2017) 092502
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Translation Symmetry
Rely only on experimentally established theories:QFT& GR
We need something new!
• Hint:
In (1+2) dimension, quantum general relativity exists.
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Translation Symmetry
• Hint:
In (1+2) dimension, quantum general relativity exists.
E. Witten, Nucl. Phys. B311 (1988), 46
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Translation Symmetry
E. Witten, Nucl. Phys. B311 (1988), 46
• Hint:
In (1+2) dimension, quantum general relativity exists.
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Translation Symmetry
E. Witten, Nucl. Phys. B311 (1988), 46• Hint:
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Translation Symmetry
E. Witten, Nucl. Phys. B311 (1988), 46
Poincare Symmetry• Hint:
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Translation Symmetry
• Geometrical symmetry in local QFT
• Poincaré symmetry = Lorentz transformation + Translation
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Translation Symmetry
• Geometrical symmetry in local QFT
• Poincaré symmetry = Lorentz transformation + Translation
𝑉′ 𝜇 = Λ𝜈𝜇𝑉𝜈 + 𝑎𝜇
𝐴𝜇: Potnsial𝐹𝜇𝜈: Field Strength
Physics𝐼𝑔𝑎𝑢𝑔𝑒 = −1
4න𝐹𝜇𝜈𝐹𝜇𝜈𝑑𝑥
4
𝐹𝜇𝜈 = 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇 − [𝐴𝜇, 𝐴𝜈]
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Translation Symmetry
𝐼𝑔𝑎𝑢𝑔𝑒 = −1
4න𝐹𝜇𝜈𝐹𝜇𝜈𝑑𝑥
4
𝐹𝜇𝜈 = 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇 − [𝐴𝜇, 𝐴𝜈]
Connection form :
Curvature form :
𝐴𝜇: Connection𝐹𝜇𝜈: Curvature
Geometry
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Translation Symmetry
• Geometrical symmetry in local QFT
• Poincaré symmetry = Lorentz transformation + Translation
𝐼𝑆𝑂 1,3 = 𝑆𝑂 1,3 ⋉ 𝑇4
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Translation Symmetry
• Geometrical symmetry in local QFT
• Poincaré symmetry = Lorentz transformation + Translation
𝐼𝑆𝑂 1,3 = 𝑆𝑂 1,3 ⋉ 𝑇4
• Geometrical symmetry in general relativity
• General Linear Symmetry:
𝐺𝐿(1,3) ⊃ 𝑆𝑂(1,3)
Einstein’s Equivalent principle!
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Translation Symmetry
• Geometrical symmetry in general relativity
• General coordinate transformation:
𝑉′ 𝜇 =𝑑𝑥′𝜇
𝑑𝑥𝜈(𝑥) 𝑉𝜈
𝐼𝑔𝑎𝑢𝑔𝑒 = −1
4න −𝑔𝑔𝜇1𝜇2(𝑥)𝑔𝜈1𝜈2(𝑥)𝐹
𝜇1𝜈1𝐹𝜇2𝜈2𝑑𝑥4
No translation symmetry!
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Translation Symmetry
• We need something new!
• Hint: E. Witten, Nucl. Phys. B311 (1988), 46
SO(1,d-1)
Generator : 𝐽𝑎𝑏, 𝑃𝑎
rotation translation
Lie algebra
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Translation Symmetry
• We need something new!
• Hint: E. Witten, Nucl. Phys. B311 (1988), 46
𝑊 = 𝛼 𝐽𝑎𝑏𝐽𝑎𝑏 + 𝛽𝑃𝑎𝑃
𝑎
SO(1,d-1)
Generator : 𝐽𝑎𝑏, 𝑃𝑎
rotation translation
Bi-liner Invariants :
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Translation Symmetry
• We need something new!
• Hint: E. Witten, Nucl. Phys. B311 (1988), 46
𝑊 = 𝛼 𝐽𝑎𝑏𝐽𝑎𝑏 + 𝛽𝑃𝑎𝑃
𝑎
SO(1,d-1)
Generator : 𝐽𝑎𝑏, 𝑃𝑎
𝛽 𝜂𝑎𝑏𝑃𝑎𝑃𝑏rotation translation
𝜖𝑎𝑏𝑐⋯ ?
Bi-liner Invariants :
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Translation Symmetry
• We need something new!
• Hint: E. Witten, Nucl. Phys. B311 (1988), 46
𝑊 = 𝛼 𝐽𝑎𝑏𝐽𝑎𝑏 + 𝛽𝑃𝑎𝑃
𝑎
SO(1,2)
Generator : 𝐽𝑎𝑏, 𝑃𝑎
+ 𝛾 𝜖𝑎𝑏𝑐 𝐽𝑎𝑏𝑃𝑐
Magic for three dimension
Bi-liner Invariants :
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Translation Symmetry
• We need something new!
• Hint: E. Witten, Nucl. Phys. B311 (1988), 46
𝑊 = 𝛼 𝐽𝑎𝑏𝐽𝑎𝑏 + 𝛽𝑃𝑎𝑃
𝑎
SO(1,2)
Generator : 𝐽𝑎𝑏, 𝑃𝑎
Bi-liner Invariants :
Four dimension Three dimension
+ 𝛾 𝜖𝑎𝑏𝑐 𝐽𝑎𝑏𝑃𝑐
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Translation Symmetry
• We need something new!
• Hint: E. Witten, Nucl. Phys. B311 (1988), 46
𝑊 = 𝛼 𝐽𝑎𝑏𝐽𝑎𝑏 + 𝛽𝑃𝑎𝑃
𝑎
SO(1,3)
Generator : 𝐽𝑎𝑏, 𝑃𝑎𝑏
Bi-liner Invariants : + 𝛾 𝜖𝑎𝑏𝑐𝑑 𝐽𝑎𝑏𝑃𝑐𝑑
Four dimension
𝑃𝑎𝑏 ?
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Translation Symmetry
Co-translation:
Generator :
Contraction :
Operator :
Translation
Contraction
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Translation Symmetry
Co-translation:Intuitive image:
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Translation Symmetry
Principal connection:
Principal curvature :
Four dimension
Principal connection:
Principal curvature :
Three dimension
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Translation Symmetry
What we can learn from this result:
Fundamental variables :
Lagrangian :
Generalized momentum :
Hamiltonian:
Phase space
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Translation Symmetry
What we can learn from this result:
Hamiltonian:
Phase space :
Energy-momentum tensor (three form)
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Translation Symmetry
What we can learn from this result:
Hamiltonian:
Phase space :
We found correct canonical variables!
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Translation Symmetry
What we can learn from this result:
Cosmological constant: ∈ ℝ/ℤ
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Canonical Quantization
YK, arXiv:1703.05574
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Canonical Quantization
Poisson brackets:
Canonical equation of motion:
Canonical variable OperatorPoisson bracket Commutation relation
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Canonical Quantization
Canonical equation of motion:
PB representation of EoM:
Classical equations of motion
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Canonical Quantization
Commutation relation:
Operator:
Schrödinger eq.:
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Canonical Quantization
Commutation relation:
Operator:
Whole story is not so simple.
Schrödinger eq.:
Gauge fixing!!
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Canonical Quantization
Symmetry (gauge) fixing
: set of all possible spin forms
: gauge transformed subset
e.g.
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Symmetry (gauge) fixing
Quantum Lagrangian:
Classical GravityGauge fixing
Faddeev-Popov ghost
Lagrange multiplier method
Naknishi-Kugo-Ojima covariant Quantization
Canonical Quantization
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Lagrangian:
Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
Fields:
Classical Gravity:
Gauge fixing:
Faddeev-Popov ghost:auxiliary field
ghost field
anti-ghost field
Others are zero
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Fields:
BRS symmetry:
Nilpotency:
Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
BRS symmetry
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BRS transformation:
Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
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Noether current/charge:
BRS-charge
Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
Hilbert Space:
Norm:
Integration measure
Hilbert Space
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
Hilbert Space:
Norm:
Integration measure
Hilbert Space
∎ ∎>=<0
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
Hilbert Space:
Norm: ∎ ∎>=<0
Non degenerate
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
Physical states:
K-O states:
cf.
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
Unitarity of S-matrix (Kugo-Ojima Theorem)
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
o Quantum Lagrangian
o Commutation relation
✓ Operator
o Schrödinger equation
o Physical Hilbert space
✓ BRS-transformation
✓ BRS-charge
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
o Quantum Lagrangian
o Commutation relation
✓ Operator
o Schrödinger equation
o Physical Hilbert space
✓ BRS-transformation
✓ BRS-charge
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
o Quantum Lagrangian
o Commutation relation
✓ Operator
o Schrödinger equation
o Physical Hilbert space
✓ BRS-transformation
✓ BRS-charge
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
o Quantum Lagrangian
o Commutation relation
✓ Operator
o Schrödinger equation
o Physical Hilbert space
✓ BRS-transformation
✓ BRS-charge
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
o Quantum Lagrangian
o Commutation relation
✓ Operator
o Schrödinger equation
o Physical Hilbert space
✓ BRS-transformation
✓ BRS-charge
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
o Quantum Lagrangian
o Commutation relation
✓ Operator
o Schrödinger equation
o Physical Hilbert space
✓ BRS-transformation
✓ BRS-charge
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
o Quantum Lagrangian
o Commutation relation
✓ Operator
o Schrödinger equation
o Physical Hilbert space
✓ BRS-transformation
✓ BRS-charge
Formal construction
Concrete calculationsApplications
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Canonical Quantization(Naknishi-Kugo-Ojima Quantization)
o Quantum Lagrangian
o Commutation relation
✓ Operator
o Schrödinger equation
o Physical Hilbert space
✓ BRS-transformation
✓ BRS-charge
Formal construction
Concrete calculationsApplications
Next step !!
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Canonical Quantization
Where is divergence (non-renormalizability)?
We don’t use perturbation:
We don’t treat graviton:
What can we calculate (predict) ? Next step !!
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E. Witten, Nucl. Phys. B311 (1988), 46
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E. Witten, Nucl. Phys. B311 (1988), 46
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Summary
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Summary
o Covariant & non-perturbative quantization of general relativity.✓ Closed set of Lagrangian, BRS-transformations, BRS-charges✓ Unitarity is ensured by K-O theorem.
o A co-translation invariance is found in the four dimensional Einstein-Hilbert action.✓ New topological invariance.
o Applications for cosmology Next step !!
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Back pocket
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J. Mielczarek, T. TrzesniewskiarXive:1708.07445Introduction
Outline of the taxonomy of approaches to quantum gravity
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J. Mielczarek, T. TrzesniewskiarXive:1708.07445Introduction
Outline of the taxonomy of approaches to quantum gravity
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What is a topological quantum field theory?
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Field theory which has topological invariants
Topological Field Theory
≠
What is a topological quantum field theory?
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What is a topological quantum field theory?
V.G. Ivancevic & T.T. Icancevic, arXive:0810.0344v5
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Degrees of freedom of the gravitation in diverse dimension
Little group
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WKB-approximation
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Canonical Quantization
WKB-approximation
Operators:
Ordering:
Schrödinger equation:
Solution:
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Canonical Quantization
WKB-approximation
Schrödinger equation:
Taylor expansion:
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Canonical Quantization
WKB-approximation
Order-by-order comparison:
0th order Solution:
1st order Solution:
Classical solution
First quantum correction
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Canonical Quantization
WKB-approximation
Schwarzschild black hole
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