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Aspects of non-perturbative unitarityin Quantum Field Theory
Alessia Platania
Heidelberg University
Quantum Gravity & Matter
IWH Heidelberg12.09.2019
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Motivation
- Einstein-Hilbert gravity: unitary, but perturbatively non-renormalizable
- Quadratic gravity is a renormalizable theory, but has one massive spin-2 ghost
Stelle, PRD 16 (1977) 953-969
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Motivation
- Wilsonian renormalization group:
→ Perturbative approaches can fail in the description of the UV behavior of a theory
→ Strong indications that gravity could be renormalizable from a non-perturbative point of view
- Einstein-Hilbert gravity: unitary, but perturbatively non-renormalizable
- Quadratic gravity is a renormalizable theory, but has one massive spin-2 ghost
Stelle, PRD 16 (1977) 953-969
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Motivation
- Wilsonian renormalization group:
→ Perturbative approaches can fail in the description of the UV behavior of a theory
→ Strong indications that gravity could be renormalizable from a non-perturbative point of view
- Einstein-Hilbert gravity: unitary, but perturbatively non-renormalizable
- Quadratic gravity is a renormalizable theory, but has one massive spin-2 ghost
Non-perturbative effects could be important for the understanding of fundamental properties of quantum field theories, such as renormalizability and unitarity
Stelle, PRD 16 (1977) 953-969
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Motivation
- Wilsonian renormalization group:
→ Perturbative approaches can fail in the description of the UV behavior of a theory
→ Strong indications that gravity could be renormalizable from a non-perturbative point of view
- Einstein-Hilbert gravity: unitary, but perturbatively non-renormalizable
- Quadratic gravity is a renormalizable theory, but has one massive spin-2 ghost
Non-perturbative effects could be important for the understanding of fundamental properties of quantum field theories, such as renormalizability and unitarity
Stelle, PRD 16 (1977) 953-969
Non-perturbative unitarity: Although Stelle gravity has a ghost, quantum effects could make the ghost unstable, thus restoring unitarity
Salam, Strathdee (1978)E. S. Fradkin, A. A. Tseytlin (1981)
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- Unitarity condition
Unitarity
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- Unitarity condition
- Optical theorem
Unitarity
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- Unitarity condition
- Optical theorem
Unitarity
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- Unitarity condition
- Optical theorem
Unitarity
Cutting rulesT’Hooft, Veltman (1973)
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- Unitarity condition
- Optical theorem
Unitarity
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- Unitarity condition
- Optical theorem
Unitarity
If the space of asymptotic states contains ghosts
⇒ Loss of physical unitarity
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- Spectral representationFor a stable particle, the
spectral density is a
Dirac delta
Spectral representation and unitarity
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- Dressed propagator
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- Spectral representationFor a stable particle, the
spectral density is a
Dirac delta
Spectral representation and unitarity
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- Dressed propagator
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- Spectral representationFor a stable particle, the
spectral density is a
Dirac delta
Complex polesUnstable particlesReal poles (stable particles)
Spectral representation and unitarity
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- Dressed propagator
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- Spectral representationFor a stable particle, the
spectral density is a
Dirac delta
Complex polesUnstable particlesReal poles (stable particles)
Spectral representation and unitarity
Salam, Strathdee (1978), Fradkin, Tseytlin (1981) Donoghue, Menezes (2019)
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- Dressed propagator
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- Spectral representationFor a stable particle, the
spectral density is a
Dirac delta
Complex polesUnstable particlesReal poles (stable particles)
Spectral representation and unitarity
Bare propagator
Salam, Strathdee (1978), Fradkin, Tseytlin (1981) Donoghue, Menezes (2019)
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- Dressed propagator
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- Spectral representationFor a stable particle, the
spectral density is a
Dirac delta
Complex polesUnstable particlesReal poles (stable particles)
Spectral representation and unitarity
Bare propagator
self-energyDressed propagator
Salam, Strathdee (1978), Fradkin, Tseytlin (1981) Donoghue, Menezes (2019)
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Functional Renormalization Group
C. Wetterich. Phys. Lett. B 301:90 (1993)M. Reuter. Phys. Rev. D. 57 (2): 971 (1998)
Solving the quantum theory is equivalent to solve the renormalization group equation
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Functional Renormalization Group
C. Wetterich. Phys. Lett. B 301:90 (1993)M. Reuter. Phys. Rev. D. 57 (2): 971 (1998)
Fundamental (bare) action, k→∞
Ordinary effective action, k→0
Effective action at the energy scale k
Fast fluctuating modes are integrated out
Solving the quantum theory is equivalent to solve the renormalization group equation
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Functional Renormalization Group
C. Wetterich. Phys. Lett. B 301:90 (1993)M. Reuter. Phys. Rev. D. 57 (2): 971 (1998)
Fundamental (bare) action, k→∞
Ordinary effective action, k→0
Effective action at the energy scale k
Fast fluctuating modes are integrated out
Solving the quantum theory is equivalent to solve the renormalization group equation
All terms compatible with symmetry and field content of the theory are generated
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Functional Renormalization Group
C. Wetterich. Phys. Lett. B 301:90 (1993)M. Reuter. Phys. Rev. D. 57 (2): 971 (1998)
Fundamental (bare) action, k→∞
Ordinary effective action, k→0
Effective action at the energy scale k
Fast fluctuating modes are integrated out
Solving the quantum theory is equivalent to solve the renormalization group equation
All terms compatible with symmetry and field content of the theory are generated
All quantum fluctuations are integrated out → non-localityIncorporates all quantum effects → fully-dressed quantitiesThis is the object to use to check unitarity!
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Functional Renormalization Group
C. Wetterich. Phys. Lett. B 301:90 (1993)M. Reuter. Phys. Rev. D. 57 (2): 971 (1998)
Fundamental (bare) action, k→∞
Ordinary effective action, k→0
Effective action at the energy scale k
Fast fluctuating modes are integrated out
Solving the quantum theory is equivalent to solve the renormalization group equation
All terms compatible with symmetry and field content of the theory are generated
All quantum fluctuations are integrated out → non-localityIncorporates all quantum effects → fully-dressed quantitiesThis is the object to use to check unitarity!
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Functional Renormalization Group
C. Wetterich. Phys. Lett. B 301:90 (1993)M. Reuter. Phys. Rev. D. 57 (2): 971 (1998)
Fundamental (bare) action, k→∞
Ordinary effective action, k→0
Effective action at the energy scale k
Fast fluctuating modes are integrated out
Solving the quantum theory is equivalent to solve the renormalization group equation
All terms compatible with symmetry and field content of the theory are generated
All quantum fluctuations are integrated out → non-localityIncorporates all quantum effects → fully-dressed quantitiesThis is the object to use to check unitarity!
Problem: Need to work within truncation ⇒ higher-derivatives ⇒ Poles
Questions:- What is the nature of these poles?- Are these poles removed by quantum effects? - Connection between poles in finite truncation and
poles in the effective action?- How do we understand, within truncation, if these
poles are dangerous for unitarity?
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Unitarity in QED
Take the one-loop effective action as a toy model for the full effective action
Boulware, Gross (1984)
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Unitarity in QED
Take the one-loop effective action as a toy model for the full effective action
In this case the propagator has one massless pole and one massive ghost pole
Boulware, Gross (1984)
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Unitarity in QED
Take the one-loop effective action as a toy model for the full effective action
In this case the propagator has one massless pole and one massive ghost pole
Boulware, Gross (1984)
Absorptive part of the propagator
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Unitarity in QED
Take the one-loop effective action as a toy model for the full effective action
In this case the propagator has one massless pole and one massive ghost pole
Boulware, Gross (1984)
Absorptive part of the propagator
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Unitarity in QED
Take the one-loop effective action as a toy model for the full effective action
In this case the propagator has one massless pole and one massive ghost pole
Boulware, Gross (1984)
Absorptive part of the propagator
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What happens within truncations?
If all terms are
included
Truncation of the action N (derivative expansion of the action)
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● Real poles● Complex poles
What happens within truncations?
Persistent ghost pole at
It is a pole for N odd
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What happens within truncations?
Persistent ghost pole at
It is a pole for N odd
The apparent ghost pole is
generated by the convergence
properties of the function P(z)
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What happens within truncations?
Persistent ghost pole at
It is a pole for N odd
The apparent ghost pole is
generated by the convergence
properties of the function P(z)
Unstable ghost lives in the branch cut (cannot be seen in any perturbative expansion)
Fake ghost living in the principal branch of the Log (non appearing in the full theory)
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What happens within truncations?
Persistent ghost pole at
It is a pole for N odd
The apparent ghost pole is
generated by the convergence
properties of the function P(z)
Unstable ghost lives in the branch cut (cannot be seen in any perturbative expansion)
Fake ghost living in the principal branch of the Log (non appearing in the full theory)
How to determine whether a pole appearing in truncation is a genuine degree of freedom of the full theory?
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What happens within truncations?
Persistent ghost pole at
It is a pole for N odd
The apparent ghost pole is
generated by the convergence
properties of the function P(z)
The answer lies in the residue!
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What happens if the full theory has a stable ghost?
Flipping the sign of the log generates a stable
ghost, living in the principal branch of the Log
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Flipping the sign of the log generates a stable
ghost, living in the principal branch of the Log
● Real poles● Complex poles
Two persistent ghost poles!
What happens if the full theory has a stable ghost?
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Flipping the sign of the log generates a stable
ghost, living in the principal branch of the Log
Ghost of the full theory → persistent negative residue
Fake ghost (generated by convergence properties of P(z))→ residue approaches zero
What happens if the full theory has a stable ghost?
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Summary● We discussed unitarity from the point of view of the Functional Renormalization Group
● Including all quantum fluctuations is crucial for unitarity: it determines what states appear in the sum over states in the optical theorem
● Truncations / derivative expansion of the action ⇒ fictitious poles
● The fictitious pole is however a fake ghost: its residue approaches zero when a sufficiently large number of terms in the action are included.
● Stable ghosts in the full theory are instead characterized by a persistent negative residue.
● Most reliable instrument to check unitarity in full glory: fully-quantum effective action
Outlook:
● Case of entire functions (expectation: fake poles are moved to infinity by increasing truncation order)
● Quantum Gravity
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