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

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

- Dressed propagator

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- Spectral representationFor a stable particle, the

spectral density is a

Dirac delta

Spectral representation and unitarity

- 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

- 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)

- 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)

- 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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