Gastão KreinInstituto de Física Teórica, São Paulo

STOCHASTIC QUANTIZATION — From QCD to semi-classical gravity

XXXIX Encontro Nacional de Física de Partículas e Campos

24 a 28 de setembro de 2018 Campos do Jordão, SP

This talk is on solving nonperturbatively

field theories

using stochastic quantization

Show an application to quantum fields

in curved spacetimes

Talk based on:

— EA Reis, TP Neto, GK & IL Shapiro, arXiv:1804.04569

— GK, J Phys Conf Ser 378, 012032 (2012)

— NC Cassol-Seewald, RLS Farias, ES Fraga, GK & RO Ramos, Physica A 391, 4088 (2012)

Observables in field theory— in general, can be expressed in terms of correlation functions

Cannot be solved exactly using analytical methodsbeyond perturbation theory

Take QCD as an example

Take QCD as an example

— asymptotic freedom allows use of perturbation theory, predictions confirmed experimentally

Take QCD as an example

— asymptotic freedom allows use of perturbation theory, predictions confirmed experimentally

— but is it is really THE theory of the strong interactions?

Take QCD as an example

— asymptotic freedom allows use of perturbation theory, predictions confirmed experimentally

— but is it is really THE theory of the strong interactions?

— we believe so.

Take QCD as an example

— asymptotic freedom allows use of perturbation theory, predictions confirmed experimentally

— but is it is really THE theory of the strong interactions?

— we believe so. Why?

Take QCD as an example

— asymptotic freedom allows use of perturbation theory, predictions confirmed experimentally

— but is it is really THE theory of the strong interactions?

— we believe so. Why?

2008

2015

2018

Great, Impressive …

See talk by Cristina Aguilar

Great, Impressive …

BUT, we do not know how precisely those

numbers come out from

the QCD Lagrangian

See talk by Cristina Aguilar

Great, Impressive …

BUT, we do not know how precisely those

numbers come out from

the QCD Lagrangian

Our ignorance is profound!

See talk by Cristina Aguilar

Computation of the masses

hadron interpolating field, e.g. ⇡+(x) = u(x)�5d(x)h(x) :

hh(x)h(x+ T )i =R ⇥

D Aµ

⇤h(x)h(x+ T ) e�

Rd

4xLQCD

R ⇥D A

µ

⇤e

�Rd

4xLQCD

limT!1

hh(x)h(x+ T )i ⇠ e

�Mh T

~ = 1 = c<latexit sha1_base64="CoRfOeXF692oUOGr7WBiMIBP0JM=">AAACGXicbVA9SwNBEN3zM8avU0ubxSCIRbhLo40QtLGMYD4gOcLcZpIs2ds7d/eEcORv2PhXbCwUsdTKf+Pmo4iJDwYe780wMy9MBNfG836cldW19Y3N3FZ+e2d3b989OKzpOFUMqywWsWqEoFFwiVXDjcBGohCiUGA9HNyM/fojKs1jeW+GCQYR9CTvcgbGSm3Xa4XY4zLDBwlKwfB8lG/1Q1D0ivq2GM23UHbm7LZb8IreBHSZ+DNSIDNU2u5XqxOzNEJpmACtm76XmCADZTgTaNelGhNgA+hh01IJEeogm3w2oqdW6dBurGxJQyfq/EQGkdbDKLSdEZi+XvTG4n9eMzXdyyDjMkkNSjZd1E0FNTEdx0Q7XCEzYmgJMMXtrZT1QQEzNsy8DcFffHmZ1EpF3yv6d6VC+XoWR44ckxNyRnxyQcrkllRIlTDyRF7IG3l3np1X58P5nLauOLOZI/IHzvcvZEufQg==</latexit><latexit sha1_base64="CoRfOeXF692oUOGr7WBiMIBP0JM=">AAACGXicbVA9SwNBEN3zM8avU0ubxSCIRbhLo40QtLGMYD4gOcLcZpIs2ds7d/eEcORv2PhXbCwUsdTKf+Pmo4iJDwYe780wMy9MBNfG836cldW19Y3N3FZ+e2d3b989OKzpOFUMqywWsWqEoFFwiVXDjcBGohCiUGA9HNyM/fojKs1jeW+GCQYR9CTvcgbGSm3Xa4XY4zLDBwlKwfB8lG/1Q1D0ivq2GM23UHbm7LZb8IreBHSZ+DNSIDNU2u5XqxOzNEJpmACtm76XmCADZTgTaNelGhNgA+hh01IJEeogm3w2oqdW6dBurGxJQyfq/EQGkdbDKLSdEZi+XvTG4n9eMzXdyyDjMkkNSjZd1E0FNTEdx0Q7XCEzYmgJMMXtrZT1QQEzNsy8DcFffHmZ1EpF3yv6d6VC+XoWR44ckxNyRnxyQcrkllRIlTDyRF7IG3l3np1X58P5nLauOLOZI/IHzvcvZEufQg==</latexit><latexit sha1_base64="CoRfOeXF692oUOGr7WBiMIBP0JM=">AAACGXicbVA9SwNBEN3zM8avU0ubxSCIRbhLo40QtLGMYD4gOcLcZpIs2ds7d/eEcORv2PhXbCwUsdTKf+Pmo4iJDwYe780wMy9MBNfG836cldW19Y3N3FZ+e2d3b989OKzpOFUMqywWsWqEoFFwiVXDjcBGohCiUGA9HNyM/fojKs1jeW+GCQYR9CTvcgbGSm3Xa4XY4zLDBwlKwfB8lG/1Q1D0ivq2GM23UHbm7LZb8IreBHSZ+DNSIDNU2u5XqxOzNEJpmACtm76XmCADZTgTaNelGhNgA+hh01IJEeogm3w2oqdW6dBurGxJQyfq/EQGkdbDKLSdEZi+XvTG4n9eMzXdyyDjMkkNSjZd1E0FNTEdx0Q7XCEzYmgJMMXtrZT1QQEzNsy8DcFffHmZ1EpF3yv6d6VC+XoWR44ckxNyRnxyQcrkllRIlTDyRF7IG3l3np1X58P5nLauOLOZI/IHzvcvZEufQg==</latexit><latexit sha1_base64="CoRfOeXF692oUOGr7WBiMIBP0JM=">AAACGXicbVA9SwNBEN3zM8avU0ubxSCIRbhLo40QtLGMYD4gOcLcZpIs2ds7d/eEcORv2PhXbCwUsdTKf+Pmo4iJDwYe780wMy9MBNfG836cldW19Y3N3FZ+e2d3b989OKzpOFUMqywWsWqEoFFwiVXDjcBGohCiUGA9HNyM/fojKs1jeW+GCQYR9CTvcgbGSm3Xa4XY4zLDBwlKwfB8lG/1Q1D0ivq2GM23UHbm7LZb8IreBHSZ+DNSIDNU2u5XqxOzNEJpmACtm76XmCADZTgTaNelGhNgA+hh01IJEeogm3w2oqdW6dBurGxJQyfq/EQGkdbDKLSdEZi+XvTG4n9eMzXdyyDjMkkNSjZd1E0FNTEdx0Q7XCEzYmgJMMXtrZT1QQEzNsy8DcFffHmZ1EpF3yv6d6VC+XoWR44ckxNyRnxyQcrkllRIlTDyRF7IG3l3np1X58P5nLauOLOZI/IHzvcvZEufQg==</latexit>

We work in Euclidean space

Computation of that path integral— use Monte Carlo method

Computation of that path integral— use Monte Carlo method

But there is a problem:— Monte Carlo not always applicable

It relies on

being positive

e�Rd

4xL(') = e�S[']

<latexit sha1_base64="Le+E7TNjD4CowTPr/CrtoLJoSzM=">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</latexit><latexit sha1_base64="Le+E7TNjD4CowTPr/CrtoLJoSzM=">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</latexit><latexit sha1_base64="Le+E7TNjD4CowTPr/CrtoLJoSzM=">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</latexit><latexit sha1_base64="Le+E7TNjD4CowTPr/CrtoLJoSzM=">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</latexit>

Importance sampling

But there is a problem:— Monte Carlo not always applicable

It relies on

being positive

e�Rd

4xL(') = e�S[']

<latexit sha1_base64="Le+E7TNjD4CowTPr/CrtoLJoSzM=">AAACTHicbVBBaxNBFJ6NWmO0bapHLw9DoZY27BahvQilXjx4iGjaQmYb3s6+JENnZ7czs8Ww7A/sxYM3f4UXD4oITtI91LYfDHx833tv3vuSQknrwvB70Hrw8NHK4/aTztNnq2vr3Y3nxzYvjaChyFVuThO0pKSmoZNO0WlhCLNE0Uly/m7hn1ySsTLXn928oDjDqZYTKdB5adwVPKGp1BVdaDQG59t1h/MOnVW7wKV2kJ69+QJ8ByouUMGHeotfoilm8nUNbwF8HezCp1EjxjXAop2TTm9MHHd7YT9cAu6SqCE91mAw7n7jaS7KjLQTCq0dRWHh4gqNk0KR37C0VKA4xymNPNWYkY2rZRg1bHolhUlu/PMHLNWbHRVm1s6zxFdm6Gb2trcQ7/NGpZscxJXURelIi+uPJqUCl8MiWUilIeHU3BMURvpdQczQoHA+/44PIbp98l1yvNePwn70ca93eNTE0WYv2Su2xSK2zw7ZezZgQybYFfvBfrHfwdfgZ/An+Htd2gqanhfsP7RW/gGQw7FA</latexit><latexit sha1_base64="Le+E7TNjD4CowTPr/CrtoLJoSzM=">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</latexit><latexit sha1_base64="Le+E7TNjD4CowTPr/CrtoLJoSzM=">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</latexit><latexit sha1_base64="Le+E7TNjD4CowTPr/CrtoLJoSzM=">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</latexit>

S[']<latexit sha1_base64="rhbIG0KHFsAdN1uTxPUNPOcuBJY=">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</latexit><latexit sha1_base64="rhbIG0KHFsAdN1uTxPUNPOcuBJY=">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</latexit><latexit sha1_base64="rhbIG0KHFsAdN1uTxPUNPOcuBJY=">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</latexit><latexit sha1_base64="rhbIG0KHFsAdN1uTxPUNPOcuBJY=">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</latexit>

Importance sampling

But there is a problem:— Monte Carlo not always applicable

It relies on

being positive

e�Rd

4xL(') = e�S[']

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can be complex in multi-particle systems with net baryon number

(baryon chemical potential)

S[']<latexit sha1_base64="rhbIG0KHFsAdN1uTxPUNPOcuBJY=">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</latexit><latexit sha1_base64="rhbIG0KHFsAdN1uTxPUNPOcuBJY=">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</latexit><latexit sha1_base64="rhbIG0KHFsAdN1uTxPUNPOcuBJY=">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</latexit><latexit sha1_base64="rhbIG0KHFsAdN1uTxPUNPOcuBJY=">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</latexit>

Importance sampling

But there is a problem:— Monte Carlo not always applicable

It relies on

being positive

e�Rd

4xL(') = e�S[']

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can be complex in multi-particle systems with net baryon number

(baryon chemical potential)

S[']<latexit sha1_base64="rhbIG0KHFsAdN1uTxPUNPOcuBJY=">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</latexit><latexit sha1_base64="rhbIG0KHFsAdN1uTxPUNPOcuBJY=">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</latexit><latexit sha1_base64="rhbIG0KHFsAdN1uTxPUNPOcuBJY=">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</latexit><latexit sha1_base64="rhbIG0KHFsAdN1uTxPUNPOcuBJY=">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</latexit>

Sign problem

Importance sampling

QCD dense matter— phenomenological input from gravitational waves observations

QCD dense matter— phenomenological input from gravitational waves observations

Sign problem is the major impediment for progress in using QCD directly to study dense matter

Sign problems— not exclusive do QCD

Charged multiboson systems have a sign problem too:

S(µ) =

Zd

4x

⇥(@µ')

2 + (m2 � µ

2)|'|2 + µ ('⇤@4'� @4'

⇤')

⇤<latexit sha1_base64="EErOMSuq9Z3Uisv2eyFA3aRzJsI=">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</latexit><latexit sha1_base64="EErOMSuq9Z3Uisv2eyFA3aRzJsI=">AAACsXicbVFNb9NAEF2br2I+GuDIZUSE5LQhskMluCBVcOFYRNMWZe2w3oyTVdZrs7uuiNz+P87c+DesU6sqKSOt9unNm53ZN1klhbFR9Mfz79y9d//BzsPg0eMnT3d7z56fmLLWHCe8lKU+y5hBKRROrLASzyqNrMgknmarT23+9By1EaU6tusKk4ItlMgFZ9ZRs94vmuFCqAZ/KKY1W+9dBpQGX0Na1AP4AFQoC/P04CfQIVCJuZ2GtGLaCiZnTgP0nOlqKQbpGPYhLNz1Bhyfjgdw0eUuHBnsw0Y9hLBj0z24fuig49raba7VDa/bUC0WS5uAm5Gimt8Ye9brR6NoE3AbxB3oky6OZr3fdF7yukBluWTGTOOosknTtucSnQ21wYrxFVvg1EHFCjRJs3H8El47Zg55qd1xDm3YmxUNK4xZF5lTFswuzXauJf+Xm9Y2f580QlW1RcWvGuW1BFtCuz6YC43cyrUDjGvhZgW+ZJpx65YcOBPi7S/fBifjURyN4i/j/uHHzo4d8pK8IiGJyTtySD6TIzIh3Bt5x17ipf5b/5v/3c+upL7X1bwg/4S/+gv8Ec+T</latexit><latexit sha1_base64="EErOMSuq9Z3Uisv2eyFA3aRzJsI=">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</latexit><latexit sha1_base64="EErOMSuq9Z3Uisv2eyFA3aRzJsI=">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</latexit>

Z = Tr e��(H�µN) =

ZD'D'⇤ e�S(µ)

<latexit sha1_base64="vAwj0P5D50majhVi7Nf/p1u0tDc=">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</latexit><latexit sha1_base64="vAwj0P5D50majhVi7Nf/p1u0tDc=">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</latexit><latexit sha1_base64="vAwj0P5D50majhVi7Nf/p1u0tDc=">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</latexit><latexit sha1_base64="vAwj0P5D50majhVi7Nf/p1u0tDc=">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</latexit>

Sign problems— not exclusive do QCD

Charged multiboson systems have a sign problem too:

S(µ) =

Zd

4x

⇥(@µ')

2 + (m2 � µ

2)|'|2 + µ ('⇤@4'� @4'

⇤')

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Z = Tr e��(H�µN) =

ZD'D'⇤ e�S(µ)

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S⇤(µ) = S(�µ) 6= S(µ)<latexit sha1_base64="TU3qQQlGtpyiGsbjRnsXCwEiJsA=">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</latexit><latexit sha1_base64="TU3qQQlGtpyiGsbjRnsXCwEiJsA=">AAACTnicbVFLSwMxEM7WV62vqkcvwSJU0bLbi16EohePldoHdNeSTac1NJvdJlmhLP2FXsSbP8OLB0U03Vaoj4GBL9/M92Uy8SPOlLbtZyuzsLi0vJJdza2tb2xu5bd3GiqMJYU6DXkoWz5RwJmAumaaQyuSQAKfQ9MfXE7qzXuQioXiRo8i8ALSF6zHKNGG6uTB9aHPRAJDQaQko6NxznVziZs6t2Xf9xK7ZB9P0inZaYxrt0dFN4gP8TmuFU9S5AoYmkOKx8bABdGd8+zkC/ZMjf+Cb+MCmkW1k39yuyGNAxCacqJU27Ej7SVEakY5mBljBRGhA9KHtoGCBKC8JB16jA8M08W9UJoUGqfsvCIhgVKjwDedAdF36ndtQv5Xa8e6d+YlTESxBkGnF/VijnWIJ7vFXSaBaj4ygFDJzKyY3hFJqDY/kDNLcH4/+S9olEuOXXKuy4XKxWwdWbSH9lEROegUVdAVqqI6ougBvaA39G49Wq/Wh/U5bc1YM80u+hGZ7BdMbrBR</latexit><latexit sha1_base64="TU3qQQlGtpyiGsbjRnsXCwEiJsA=">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</latexit><latexit sha1_base64="TU3qQQlGtpyiGsbjRnsXCwEiJsA=">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</latexit>

Way out?!

Stochastic quantization, perhaps(does not rely on importance sampling)

Stochastic quantizationAn old idea— derive Schrödinger equation from Brownian motion (Langevin equation)

Failed, because use of areal-time stochastic process

Field theory in D spacetime dimensions is obtained as the equilibrium limit of a classical theory in D+1 dimensions

coupled to a random noise source of strength

What was missing?— fictitious time (quantum axis)*

* Parisi & Wu, Sic Sin 24, 483 (1981)

— the classical stochastic process happens in an extra dimension

— the quantum state is the equilibrium state of the fictitious classical stochastic process

~<latexit sha1_base64="amodbagaxhRuR49vavgcT8pfbEM=">AAACKnicbVDLSsNAFJ34rPUVdekmWAQRCUld6LLqxmUFW4UklJvpbTs4mcSZiVBCv8eNv+KmC6W49UOc1iy0emDgcO65jzlxxpnSnjexFhaXlldWK2vV9Y3NrW17Z7et0lxSbNGUp/I+BoWcCWxppjneZxIhiTnexQ9X0/rdE0rFUnGrhxlGCfQF6zEK2kgd+yKMsc9EgY8CpITh8ahahLOxgezHUeG53onnnp74rjcKBzHIUTVE0f3h79g1Y5rB+Uv8ktRIiWbHHofdlOYJCk05KBX4XqajAqRmlKOZnyvMgD5AHwNDBSSoomJ208g5NErX6aXSPKGdmfqzo4BEqWESG2cCeqDma1Pxv1qQ6955VDCR5RoF/V7Uy7mjU2eam9NlEqnmQ0OASmZudegAJFBt0q2aEPz5L/8l7brre65/U681Lss4KmSfHJAj4pMz0iDXpElahJJn8kreyLv1Yo2tifXxbV2wyp498gvW5xd1c6ai</latexit><latexit sha1_base64="amodbagaxhRuR49vavgcT8pfbEM=">AAACKnicbVDLSsNAFJ34rPUVdekmWAQRCUld6LLqxmUFW4UklJvpbTs4mcSZiVBCv8eNv+KmC6W49UOc1iy0emDgcO65jzlxxpnSnjexFhaXlldWK2vV9Y3NrW17Z7et0lxSbNGUp/I+BoWcCWxppjneZxIhiTnexQ9X0/rdE0rFUnGrhxlGCfQF6zEK2kgd+yKMsc9EgY8CpITh8ahahLOxgezHUeG53onnnp74rjcKBzHIUTVE0f3h79g1Y5rB+Uv8ktRIiWbHHofdlOYJCk05KBX4XqajAqRmlKOZnyvMgD5AHwNDBSSoomJ208g5NErX6aXSPKGdmfqzo4BEqWESG2cCeqDma1Pxv1qQ6955VDCR5RoF/V7Uy7mjU2eam9NlEqnmQ0OASmZudegAJFBt0q2aEPz5L/8l7brre65/U681Lss4KmSfHJAj4pMz0iDXpElahJJn8kreyLv1Yo2tifXxbV2wyp498gvW5xd1c6ai</latexit><latexit sha1_base64="amodbagaxhRuR49vavgcT8pfbEM=">AAACKnicbVDLSsNAFJ34rPUVdekmWAQRCUld6LLqxmUFW4UklJvpbTs4mcSZiVBCv8eNv+KmC6W49UOc1iy0emDgcO65jzlxxpnSnjexFhaXlldWK2vV9Y3NrW17Z7et0lxSbNGUp/I+BoWcCWxppjneZxIhiTnexQ9X0/rdE0rFUnGrhxlGCfQF6zEK2kgd+yKMsc9EgY8CpITh8ahahLOxgezHUeG53onnnp74rjcKBzHIUTVE0f3h79g1Y5rB+Uv8ktRIiWbHHofdlOYJCk05KBX4XqajAqRmlKOZnyvMgD5AHwNDBSSoomJ208g5NErX6aXSPKGdmfqzo4BEqWESG2cCeqDma1Pxv1qQ6955VDCR5RoF/V7Uy7mjU2eam9NlEqnmQ0OASmZudegAJFBt0q2aEPz5L/8l7brre65/U681Lss4KmSfHJAj4pMz0iDXpElahJJn8kreyLv1Yo2tifXxbV2wyp498gvW5xd1c6ai</latexit><latexit sha1_base64="amodbagaxhRuR49vavgcT8pfbEM=">AAACKnicbVDLSsNAFJ34rPUVdekmWAQRCUld6LLqxmUFW4UklJvpbTs4mcSZiVBCv8eNv+KmC6W49UOc1iy0emDgcO65jzlxxpnSnjexFhaXlldWK2vV9Y3NrW17Z7et0lxSbNGUp/I+BoWcCWxppjneZxIhiTnexQ9X0/rdE0rFUnGrhxlGCfQF6zEK2kgd+yKMsc9EgY8CpITh8ahahLOxgezHUeG53onnnp74rjcKBzHIUTVE0f3h79g1Y5rB+Uv8ktRIiWbHHofdlOYJCk05KBX4XqajAqRmlKOZnyvMgD5AHwNDBSSoomJ208g5NErX6aXSPKGdmfqzo4BEqWESG2cCeqDma1Pxv1qQ6955VDCR5RoF/V7Uy7mjU2eam9NlEqnmQ0OASmZudegAJFBt0q2aEPz5L/8l7brre65/U681Lss4KmSfHJAj4pMz0iDXpElahJJn8kreyLv1Yo2tifXxbV2wyp498gvW5xd1c6ai</latexit>

1) Introduce a fictitious time variable

Stochastic quantization

'(x) ! '(x, ⌧)<latexit sha1_base64="DrBzUk+gacHrxnaDI3PleZtUk4w=">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</latexit><latexit sha1_base64="DrBzUk+gacHrxnaDI3PleZtUk4w=">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</latexit><latexit sha1_base64="DrBzUk+gacHrxnaDI3PleZtUk4w=">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</latexit><latexit sha1_base64="DrBzUk+gacHrxnaDI3PleZtUk4w=">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</latexit>

2) Postulate a Langevin equation

where

3) Theorem (nonperturbatively valid):

where

4) Proof of the theorem:

— associated with the Langevin equation, there is a Fokker-Planck equation for the probability distribution of field variables, such that

with

Fokker-Planck

— from this last equation, one sees that the equilibrium distribution function*

* normalization can be fixed by an appropriate initial condition

Conclusion: — quantum vacuum expectation values can be calculated as equilibrium stochastic averages

Observations:— there is nothing special about white noise beyond simplicity, can use coloured noise as well (see e.g. Menezes & Svaiter)

— renormalization: multiplicatively renormalizable in the continuum and on the lattice

— gauge theories: no need to fix the gauge

— fermions: difficult, for quadratic actions integrate and exponentiate the determinant using boson fields

— complex actions: duplicate field variables. Watch for runaway solutions, convergence to wrong solutions (Aarts et al.)

V (')

'

Tunneling

Fluctuations

TrueVacuum

MetastableVacuum

0

Application— potential relevance in vacuum instability by quantum fluctuations

Figure from T Markkanen, A Rajantie & A StopyraarXiv: 1809:06923

Example in flat space

Can go beyond this 1-loop result using RG improvement,

BUT no fully nonperturbative results exist

T Markkanen, A Rajantie & A StopyraarXiv: 1809:06923

Quantum scalar field— coupled to classical gravity

Euclidean action

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V (')<latexit sha1_base64="B1DsxJrKIrzppBmivHyXQPaQRKg=">AAACFXicbVDLSgNBEJyNr7i+oh69LAYhioTdXPQY9OIxgnlANoTeSScZMju7zswGwpKf8OKvePGgiFfBm3/j5HGIiQUNRVU33V1BzJnSrvtjZdbWNza3stv2zu7e/kHu8KimokRSrNKIR7IRgELOBFY10xwbsUQIA471YHA78etDlIpF4kGPYmyF0BOsyyhoI7Vzl36APSZSfBQgJYwuxnat4A9Bxn12bvsoOgtWO5d3i+4Uzirx5iRP5qi0c99+J6JJiEJTDko1PTfWrRSkZpTj2PYThTHQAfSwaaiAEFUrnX41ds6M0nG6kTQltDNVFydSCJUahYHpDEH31bI3Ef/zmonuXrdSJuJEo6CzRd2EOzpyJhE5HSaRaj4yBKhk5laH9kEC1SZI24TgLb+8SmqloucWvftSvnwzjyNLTsgpKRCPXJEyuSMVUiWUPJEX8kberWfr1fqwPmetGWs+c0z+wPr6BX4AnwA=</latexit><latexit sha1_base64="B1DsxJrKIrzppBmivHyXQPaQRKg=">AAACFXicbVDLSgNBEJyNr7i+oh69LAYhioTdXPQY9OIxgnlANoTeSScZMju7zswGwpKf8OKvePGgiFfBm3/j5HGIiQUNRVU33V1BzJnSrvtjZdbWNza3stv2zu7e/kHu8KimokRSrNKIR7IRgELOBFY10xwbsUQIA471YHA78etDlIpF4kGPYmyF0BOsyyhoI7Vzl36APSZSfBQgJYwuxnat4A9Bxn12bvsoOgtWO5d3i+4Uzirx5iRP5qi0c99+J6JJiEJTDko1PTfWrRSkZpTj2PYThTHQAfSwaaiAEFUrnX41ds6M0nG6kTQltDNVFydSCJUahYHpDEH31bI3Ef/zmonuXrdSJuJEo6CzRd2EOzpyJhE5HSaRaj4yBKhk5laH9kEC1SZI24TgLb+8SmqloucWvftSvnwzjyNLTsgpKRCPXJEyuSMVUiWUPJEX8kberWfr1fqwPmetGWs+c0z+wPr6BX4AnwA=</latexit><latexit sha1_base64="B1DsxJrKIrzppBmivHyXQPaQRKg=">AAACFXicbVDLSgNBEJyNr7i+oh69LAYhioTdXPQY9OIxgnlANoTeSScZMju7zswGwpKf8OKvePGgiFfBm3/j5HGIiQUNRVU33V1BzJnSrvtjZdbWNza3stv2zu7e/kHu8KimokRSrNKIR7IRgELOBFY10xwbsUQIA471YHA78etDlIpF4kGPYmyF0BOsyyhoI7Vzl36APSZSfBQgJYwuxnat4A9Bxn12bvsoOgtWO5d3i+4Uzirx5iRP5qi0c99+J6JJiEJTDko1PTfWrRSkZpTj2PYThTHQAfSwaaiAEFUrnX41ds6M0nG6kTQltDNVFydSCJUahYHpDEH31bI3Ef/zmonuXrdSJuJEo6CzRd2EOzpyJhE5HSaRaj4yBKhk5laH9kEC1SZI24TgLb+8SmqloucWvftSvnwzjyNLTsgpKRCPXJEyuSMVUiWUPJEX8kberWfr1fqwPmetGWs+c0z+wPr6BX4AnwA=</latexit><latexit sha1_base64="B1DsxJrKIrzppBmivHyXQPaQRKg=">AAACFXicbVDLSgNBEJyNr7i+oh69LAYhioTdXPQY9OIxgnlANoTeSScZMju7zswGwpKf8OKvePGgiFfBm3/j5HGIiQUNRVU33V1BzJnSrvtjZdbWNza3stv2zu7e/kHu8KimokRSrNKIR7IRgELOBFY10xwbsUQIA471YHA78etDlIpF4kGPYmyF0BOsyyhoI7Vzl36APSZSfBQgJYwuxnat4A9Bxn12bvsoOgtWO5d3i+4Uzirx5iRP5qi0c99+J6JJiEJTDko1PTfWrRSkZpTj2PYThTHQAfSwaaiAEFUrnX41ds6M0nG6kTQltDNVFydSCJUahYHpDEH31bI3Ef/zmonuXrdSJuJEo6CzRd2EOzpyJhE5HSaRaj4yBKhk5laH9kEC1SZI24TgLb+8SmqloucWvftSvnwzjyNLTsgpKRCPXJEyuSMVUiWUPJEX8kberWfr1fqwPmetGWs+c0z+wPr6BX4AnwA=</latexit>

For numerical results

V (') =1

4�'4

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In collaboration with- Eduardo Antonio dos Reis- Tibério de Paula Neto- Ilya Shapiro

Langevin equation

Zd

Dy

pg(y) f(y) �Dc (x, y) = f(x)

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�

Dc (x, y)

<latexit sha1_base64="Q/DUuyGW8U3PDP1FemGNmX/BC5k=">AAACG3icbVA9SwNBFNyL3/Hr1NJmMQhRJNyl0TKohWUEE4Ukhnd7L3HJ3t65uyceh//Dxr9iY6GIlWDhv3ETU6hxYGGYmcfbN0EiuDae9+kUpqZnZufmF4qLS8srq+7aelPHqWLYYLGI1UUAGgWX2DDcCLxIFEIUCDwPBkdD//wGleaxPDNZgp0I+pL3OANjpa5bbQfY5zLHawlKQbZ7V2yHKAxcHncZLd/uZTvFNsrwR6DrlryKNwKdJP6YlMgY9a773g5jlkYoDROgdcv3EtPJQRnOBNqFqcYE2AD62LJUQoS6k49uu6PbVglpL1b2SUNH6s+JHCKtsyiwyQjMlf7rDcX/vFZqegednMskNSjZ96JeKqiJ6bAoGnKFzIjMEmCK279SdgUKmLF1Fm0J/t+TJ0mzWvG9in9aLdUOx3XMk02yRcrEJ/ukRk5InTQII/fkkTyTF+fBeXJenbfvaMEZz2yQX3A+vgB4TaEH</latexit><latexit sha1_base64="Q/DUuyGW8U3PDP1FemGNmX/BC5k=">AAACG3icbVA9SwNBFNyL3/Hr1NJmMQhRJNyl0TKohWUEE4Ukhnd7L3HJ3t65uyceh//Dxr9iY6GIlWDhv3ETU6hxYGGYmcfbN0EiuDae9+kUpqZnZufmF4qLS8srq+7aelPHqWLYYLGI1UUAGgWX2DDcCLxIFEIUCDwPBkdD//wGleaxPDNZgp0I+pL3OANjpa5bbQfY5zLHawlKQbZ7V2yHKAxcHncZLd/uZTvFNsrwR6DrlryKNwKdJP6YlMgY9a773g5jlkYoDROgdcv3EtPJQRnOBNqFqcYE2AD62LJUQoS6k49uu6PbVglpL1b2SUNH6s+JHCKtsyiwyQjMlf7rDcX/vFZqegednMskNSjZ96JeKqiJ6bAoGnKFzIjMEmCK279SdgUKmLF1Fm0J/t+TJ0mzWvG9in9aLdUOx3XMk02yRcrEJ/ukRk5InTQII/fkkTyTF+fBeXJenbfvaMEZz2yQX3A+vgB4TaEH</latexit><latexit sha1_base64="Q/DUuyGW8U3PDP1FemGNmX/BC5k=">AAACG3icbVA9SwNBFNyL3/Hr1NJmMQhRJNyl0TKohWUEE4Ukhnd7L3HJ3t65uyceh//Dxr9iY6GIlWDhv3ETU6hxYGGYmcfbN0EiuDae9+kUpqZnZufmF4qLS8srq+7aelPHqWLYYLGI1UUAGgWX2DDcCLxIFEIUCDwPBkdD//wGleaxPDNZgp0I+pL3OANjpa5bbQfY5zLHawlKQbZ7V2yHKAxcHncZLd/uZTvFNsrwR6DrlryKNwKdJP6YlMgY9a773g5jlkYoDROgdcv3EtPJQRnOBNqFqcYE2AD62LJUQoS6k49uu6PbVglpL1b2SUNH6s+JHCKtsyiwyQjMlf7rDcX/vFZqegednMskNSjZ96JeKqiJ6bAoGnKFzIjMEmCK279SdgUKmLF1Fm0J/t+TJ0mzWvG9in9aLdUOx3XMk02yRcrEJ/ukRk5InTQII/fkkTyTF+fBeXJenbfvaMEZz2yQX3A+vgB4TaEH</latexit><latexit sha1_base64="Q/DUuyGW8U3PDP1FemGNmX/BC5k=">AAACG3icbVA9SwNBFNyL3/Hr1NJmMQhRJNyl0TKohWUEE4Ukhnd7L3HJ3t65uyceh//Dxr9iY6GIlWDhv3ETU6hxYGGYmcfbN0EiuDae9+kUpqZnZufmF4qLS8srq+7aelPHqWLYYLGI1UUAGgWX2DDcCLxIFEIUCDwPBkdD//wGleaxPDNZgp0I+pL3OANjpa5bbQfY5zLHawlKQbZ7V2yHKAxcHncZLd/uZTvFNsrwR6DrlryKNwKdJP6YlMgY9a773g5jlkYoDROgdcv3EtPJQRnOBNqFqcYE2AD62LJUQoS6k49uu6PbVglpL1b2SUNH6s+JHCKtsyiwyQjMlf7rDcX/vFZqegednMskNSjZ96JeKqiJ6bAoGnKFzIjMEmCK279SdgUKmLF1Fm0J/t+TJ0mzWvG9in9aLdUOx3XMk02yRcrEJ/ukRk5InTQII/fkkTyTF+fBeXJenbfvaMEZz2yQX3A+vgB4TaEH</latexit> : covariant delta function

Use of Riemann normal coordinates:

— allows to use flat-space methods (e.g. momentum representation)

— expand relevant quantities at a point P(x) in powers of the curvature tensor and its covariant derivatives

— details, see poster P022 (and Ilya!)

For local quantities— like effective potential, field fluctuations, v.e.v. of fields

Relevant formulae: The metric and all relevant quantitiesare expanded near a point P(x’)

at which the metric is supposed to be flat

y

µ = x

µ � x

0µ<latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit>

Relevant formulae:

Redefinition of field variables:

The metric and all relevant quantitiesare expanded near a point P(x’)

at which the metric is supposed to be flat

y

µ = x

µ � x

0µ<latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit>

Relevant formulae:

Redefinition of field variables:

(

<latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit><latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit><latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit><latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit>

The metric and all relevant quantitiesare expanded near a point P(x’)

at which the metric is supposed to be flat

y

µ = x

µ � x

0µ<latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit>

Relevant formulae:

Redefinition of field variables:

(

<latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit><latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit><latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit><latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit>

The metric and all relevant quantitiesare expanded near a point P(x’)

at which the metric is supposed to be flat

y

µ = x

µ � x

0µ<latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit>

Relevant formulae:

Redefinition of field variables:

(

<latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit><latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit><latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit><latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit>

The metric and all relevant quantitiesare expanded near a point P(x’)

at which the metric is supposed to be flat

y

µ = x

µ � x

0µ<latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit>

Relevant formulae:

Redefinition of field variables:

(

<latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit><latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit><latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit><latexit sha1_base64="95/OY/K6lije/wjfPcMDLu/Cy1Y=">AAACPHicbVDLSsNAFJ34rPEVdekmWAQRKUk3uix147KifUASymR6mw6dTOLMRCihH+bGj3Dnyo0LRdy6dvqC2npg4HDuvefeOWHKqFSO82qsrK6tb2wWtsztnd29fevgsCGTTBCok4QlohViCYxyqCuqGLRSATgOGTTD/vWo3nwEIWnC79UghSDGEaddSrDSUtu680OIKM/hgWMh8OB8aOb+2NYTURjkbskZ48JZJEO/SqOI+fnQ9IF35hzaVnHWZC+TmWMRTVFrWy9+JyFZDFwRhqX0XCdVQY6FooSB9s8kpJj0cQSephzHIIN8fOXQPtVKx+4mQj+u7LE6P5HjWMpBHOrOGKueXKyNxP9qXqa6V0FOeZop4GSyqJsxWyX2KEm7QwUQxQaaYCKovtUmPSwwUTpvU4fgLn55mTTKJdcpubflYqU6jaOAjtEJOkMuukQVdINqqI4IekJv6AN9Gs/Gu/FlfE9aV4zpzBH6A+PnF4X7q0I=</latexit>

The metric and all relevant quantitiesare expanded near a point P(x’)

at which the metric is supposed to be flat

y

µ = x

µ � x

0µ<latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit><latexit sha1_base64="4dWVitpLT/hqQGlJFUwQdZX2NXA=">AAACInicbVDLSgMxFM34rPU16tJNsIgiWGa6URdC0Y3LCvYBnVoy6W0bmsmMSUY6DP0WN/6KGxeKuhL8GNPHorYeCDmccy7JPX7EmdKO820tLC4tr6xm1rLrG5tb2/bObkWFsaRQpiEPZc0nCjgTUNZMc6hFEkjgc6j6veuhX30EqVgo7nQSQSMgHcHajBJtpKZ94fnQYSKFB0GkJMnJIJvce0GML3F/dJ/itH80GFIPRGsq17RzTt4ZAc8Td0JyaIJS0/70WiGNAxCacqJU3XUi3UiJ1IxyGGS9WEFEaI90oG6oIAGoRjpacYAPjdLC7VCaIzQeqdMTKQmUSgLfJAOiu2rWG4r/efVYt88bKRNRrEHQ8UPtmGMd4mFfuMUkUM0TQwiVzPwV0y6RhGrTataU4M6uPE8qhbzr5N3bQq54Nakjg/bRATpGLjpDRXSDSqiMKHpCL+gNvVvP1qv1YX2NowvWZGYP/YH18wtpBaQx</latexit>

Restrict discussion tofirst order in R only

Expand field as

where

and replace this in the full Langevin equation

Flat spacetime

Flat spacetime

Flat spacetime

First order correction in the curvature

Flat spacetime

First order correction in the curvature

Fully nonperturbative— no expansion in (or )�

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~<latexit sha1_base64="xy/KU6/0MDtZ/g2Qa9l3VjWZlUk=">AAACEHicbVC7TsNAEDzzDOYVoKSxiBCIIrLTQBlBQxkk8pDiKFqf18kp57O5OyNFVj6Bhl+hoQAhWko6/obLowgJI600mtnV7k6Qcqa06/5YK6tr6xubhS17e2d3b794cNhQSSYp1mnCE9kKQCFnAuuaaY6tVCLEAcdmMLgZ+81HlIol4l4PU+zE0BMsYhS0kbrFMz/AHhM5PgiQEoYXI9vvByBtH0U4p3aLJbfsTuAsE29GSmSGWrf47YcJzWIUmnJQqu25qe7kIDWjHM2WTGEKdAA9bBsqIEbVyScPjZxTo4ROlEhTQjsTdX4ih1ipYRyYzhh0Xy16Y/E/r53p6KqTM5FmGgWdLooy7ujEGafjhEwi1XxoCFDJzK0O7YMEqk2GtgnBW3x5mTQqZc8te3eVUvV6FkeBHJMTck48ckmq5JbUSJ1Q8kReyBt5t56tV+vD+py2rlizmSPyB9bXLxb9nTo=</latexit><latexit sha1_base64="xy/KU6/0MDtZ/g2Qa9l3VjWZlUk=">AAACEHicbVC7TsNAEDzzDOYVoKSxiBCIIrLTQBlBQxkk8pDiKFqf18kp57O5OyNFVj6Bhl+hoQAhWko6/obLowgJI600mtnV7k6Qcqa06/5YK6tr6xubhS17e2d3b794cNhQSSYp1mnCE9kKQCFnAuuaaY6tVCLEAcdmMLgZ+81HlIol4l4PU+zE0BMsYhS0kbrFMz/AHhM5PgiQEoYXI9vvByBtH0U4p3aLJbfsTuAsE29GSmSGWrf47YcJzWIUmnJQqu25qe7kIDWjHM2WTGEKdAA9bBsqIEbVyScPjZxTo4ROlEhTQjsTdX4ih1ipYRyYzhh0Xy16Y/E/r53p6KqTM5FmGgWdLooy7ujEGafjhEwi1XxoCFDJzK0O7YMEqk2GtgnBW3x5mTQqZc8te3eVUvV6FkeBHJMTck48ckmq5JbUSJ1Q8kReyBt5t56tV+vD+py2rlizmSPyB9bXLxb9nTo=</latexit><latexit sha1_base64="xy/KU6/0MDtZ/g2Qa9l3VjWZlUk=">AAACEHicbVC7TsNAEDzzDOYVoKSxiBCIIrLTQBlBQxkk8pDiKFqf18kp57O5OyNFVj6Bhl+hoQAhWko6/obLowgJI600mtnV7k6Qcqa06/5YK6tr6xubhS17e2d3b794cNhQSSYp1mnCE9kKQCFnAuuaaY6tVCLEAcdmMLgZ+81HlIol4l4PU+zE0BMsYhS0kbrFMz/AHhM5PgiQEoYXI9vvByBtH0U4p3aLJbfsTuAsE29GSmSGWrf47YcJzWIUmnJQqu25qe7kIDWjHM2WTGEKdAA9bBsqIEbVyScPjZxTo4ROlEhTQjsTdX4ih1ipYRyYzhh0Xy16Y/E/r53p6KqTM5FmGgWdLooy7ujEGafjhEwi1XxoCFDJzK0O7YMEqk2GtgnBW3x5mTQqZc8te3eVUvV6FkeBHJMTck48ckmq5JbUSJ1Q8kReyBt5t56tV+vD+py2rlizmSPyB9bXLxb9nTo=</latexit><latexit sha1_base64="xy/KU6/0MDtZ/g2Qa9l3VjWZlUk=">AAACEHicbVC7TsNAEDzzDOYVoKSxiBCIIrLTQBlBQxkk8pDiKFqf18kp57O5OyNFVj6Bhl+hoQAhWko6/obLowgJI600mtnV7k6Qcqa06/5YK6tr6xubhS17e2d3b794cNhQSSYp1mnCE9kKQCFnAuuaaY6tVCLEAcdmMLgZ+81HlIol4l4PU+zE0BMsYhS0kbrFMz/AHhM5PgiQEoYXI9vvByBtH0U4p3aLJbfsTuAsE29GSmSGWrf47YcJzWIUmnJQqu25qe7kIDWjHM2WTGEKdAA9bBsqIEbVyScPjZxTo4ROlEhTQjsTdX4ih1ipYRyYzhh0Xy16Y/E/r53p6KqTM5FmGgWdLooy7ujEGafjhEwi1XxoCFDJzK0O7YMEqk2GtgnBW3x5mTQqZc8te3eVUvV6FkeBHJMTck48ckmq5JbUSJ1Q8kReyBt5t56tV+vD+py2rlizmSPyB9bXLxb9nTo=</latexit>

Equations can not be solved analytically— discretize on a lattice, integrate numerically

— need to compare with existing literature

— results available at one and two loops only

— formulate the loop expansion, compare with what is available

— solve equations nonperturbatively and see how good the loop expansion is

Loop expansion in SQ— expansion in ~

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From the noise correlation function

one sees that

Therefore, loop expansion is equivalent to expansion in the noise field

Noise perturbation theory (NPT)KE Kunze, Martin & MussoCassoll-Seewald et al.

Noise perturbation theory

Expand flat-space and O(R) fields as

replace these in the Langevin equations and count

)⇠ O(~n/2)

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Classical, flat spacetime

All those equations can be

solved analytically

when classical background is constant

Here, the use of Riemann normal coordinates pays the price — equations can be solved in momentum space

Explicit solutions— see poster P022 for details

Obtain the one-loop two-point correlation function:

At equilibrium— i.e. ⌧ ! 1

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Agrees with:

Bunch & Parker, Phys Rev D 20, 2499 (1979)

Effective potential— one-loop

Flat space

First order in R

For D = 4, agrees with:

Sobreira, Ribeiro & Shapiro, Phys Lett B 705, 273 (2011)

Numerical simulations— loop expansion & fully nonperturbative

Restrict here to simpler case of D = 3

— discretize on a lattice with spacing a

— discretize Langevin time in steps

As usual, all dimensionful quantities are rescaled by the lattice spacing

Langevin equations become (Itô calculus)

The analytical one-loop correlation functionon the lattice is (R-part only)

Show results obtained with:

— 16 x 16 x 16 lattice

—

—

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

�14.0

�12.0

�10.0

�8.0

�6.0

�4.0

�2.0

0.0

⇠ � 1/6 = 1

⇠ � 1/6 = 3

⇠ � 1/6 = 0.5

m = 1 � = 1 R = 0.1

⌧

�R(x,x|⌧)⇥10

3

Dashed: analyticalDotted: numerical

Analytical x numerical

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Flat spacetime

⇠ � 1/6 = 10

⇠ � 1/6 = 5

m = 1

ˆR = 0.1

ˆ�

hˆ �i/

ˆ �cl

Nonpertubative

O(h)

O(h2)

1- and 2-loop x nonperturbative

Summary & PerspectivesRiemann normal coordinates & SQ — multi-loop corrections — and also fully nonperturbative results— this in flat and curved spacetimes

Next steps— move to D = 4 (renormalization)— higher order in R— include fermions and gauge bosons

Challenges— students and post-docs (that like computers)— computer power

Funding