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Manifest Causality in QFT with Sources and DetectorsPeter MillingtonConsortium for Fundamental Physics, University of ManchesterInstitute for Particle Physics Phenomenology, Durham University

IoP Joint HEPP and APP Group Meeting7th-9th April, 2014

+OutlineBased on R. Dickinson, J. Forshaw, PM and B. Cox, arXiv: 1312.3871.

Motivation

Observations from Thermal Field Theory

A Simple Example

Comparison with the S-matrix

Conclusions

Peter Millington (UoM & IPPP) IoP HEPP and APP, Royal Holloway, 7-9th April 2014 2/12

+Motivation

Question 1:

How are we to calculate transition amplitudes over finite space-time domains in QFT? S-matrix theory is no good.

Peter Millington (UoM & IPPP) IoP HEPP and APP, Royal Holloway, 7-9th April 2014 3/12

+Motivation

Question 2:

QFT is built to be causal, so where is the micro-causality condition in the path integral?

Peter Millington (UoM & IPPP) IoP HEPP and APP, Royal Holloway, 7-9th April 2014 4/12

+Observations

It only makes sense to talk of transition amplitudes over finite space-time domains in the presence of an observer.

Statistically, the production and scattering processes form an open subsystem held in contact with some reservoir of observers: a detector.

The relevant object is the partition function of the detector

The solution: thermal quantum field theory.

Peter Millington (UoM & IPPP) IoP HEPP and APP, Royal Holloway, 7-9th April 2014 5/12

+Encouragement

a posteriori, physical reaction rates are obtained from the absorptive parts of retarded functions.[R. Kobes, Phys. Rev. D43, 1269 (1991); M. A. van Eijck and C. G. van Weert, Phys. Lett. B278, 305 (1992)]

Three ‘anecdotal’ reasons:① This gives the correct statistical factors.

② Analytic continuation of imaginary-time self-energies gives the retarded self-energy.

③ The Breit-Wigner width of resummed equilibrium thermal propagators is obtained from the imaginary part of the retarded self-energy.

Peter Millington (UoM & IPPP) IoP HEPP and APP, Royal Holloway, 7-9th April 2014 6/12

+CTP Formalism

[J. Schwinger, J. Math. Phys. 2, 407 (1961); L. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964); E. Calzetta and B. L. Hu, Phys. Rev. D 35, 495 (1987), Phys. Rev. D 37, 2878 (1988)]

Peter Millington (UoM & IPPP) IoP HEPP and APP, Royal Holloway, 7-9th April 2014 7/12

+“Physical” Basis

By an orthogonal transformation, we can move to the so-called Keldysh or “physical” basis:[M. A. van Eijck, R. Kobes and C. G. van Weert, Phys. Rev. D50, 4097 (1994)]

Note that J couples only to the retarded propagator.

Peter Millington (UoM & IPPP) IoP HEPP and APP, Royal Holloway, 7-9th April 2014 8/12

+2 to 2 Scattering

Entanglement of Final States

Peter Millington (UoM & IPPP) IoP HEPP and APP, Royal Holloway, 7-9th April 2014 9/12

+S-matrix Limit

To compare with the S-matrix, we must project out plane waves by promoting J and K to operators in Fock space:

We can then sandwich the amplitude between in and out states to project out given initial and final states.

Peter Millington (UoM & IPPP) IoP HEPP and APP, Royal Holloway, 7-9th April 2014 10/12

+NJ to NK

Peter Millington (UoM & IPPP) IoP HEPP and APP, Royal Holloway, 7-9th April 2014 11/12

+Conclusions

Working within the paradigm of thermal field theory, one can write down a generating functional in which causality remains manifest.

Results are consistent with S-matrix amplitudes in the relevant limits, specifically if we restrict to positive energy flow forwards in time.

Analytic differences arise at the level of loops.

Potential relevance: Resummation in the presence of finite-time and volume effects, interplay of positive

and negative energy contributions. Role of CTP Effective Action QFT on non-stationary spacetimes, where Wick rotation does not always lead to a

Riemannian metric. …

Peter Millington (UoM & IPPP) IoP HEPP and APP, Royal Holloway, 7-9th April 2014 12/12