STOCHASTIC QUANTIZATION - [ SBFenfpc/xxxix/images/slider/GKrein...Talk based on: — EA Reis, TP...
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Gastão Krein Instituto 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
Transcript of STOCHASTIC QUANTIZATION - [ SBFenfpc/xxxix/images/slider/GKrein...Talk based on: — EA Reis, TP...
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=">AAACTHicbVBBaxNBFJ6NWmO0bapHLw9DoZY27BahvQilXjx4iGjaQmYb3s6+JENnZ7czs8Ww7A/sxYM3f4UXD4oITtI91LYfDHx833tv3vuSQknrwvB70Hrw8NHK4/aTztNnq2vr3Y3nxzYvjaChyFVuThO0pKSmoZNO0WlhCLNE0Uly/m7hn1ySsTLXn928oDjDqZYTKdB5adwVPKGp1BVdaDQG59t1h/MOnVW7wKV2kJ69+QJ8ByouUMGHeotfoilm8nUNbwF8HezCp1EjxjXAop2TTm9MHHd7YT9cAu6SqCE91mAw7n7jaS7KjLQTCq0dRWHh4gqNk0KR37C0VKA4xymNPNWYkY2rZRg1bHolhUlu/PMHLNWbHRVm1s6zxFdm6Gb2trcQ7/NGpZscxJXURelIi+uPJqUCl8MiWUilIeHU3BMURvpdQczQoHA+/44PIbp98l1yvNePwn70ca93eNTE0WYv2Su2xSK2zw7ZezZgQybYFfvBfrHfwdfgZ/An+Htd2gqanhfsP7RW/gGQw7FA</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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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=">AAACOXicbVBNS8NAEN34WetX1aOXYBFEpCS96FH04rGibYUklMl22i5uNnF3I4SQv+XFf+FN8OJBEa/+AbdpBb8eDDzezJudfWHCmdKO82jNzM7NLyxWlqrLK6tr67WNzY6KU0mxTWMey6sQFHImsK2Z5niVSIQo5NgNr0/H/e4tSsVicamzBIMIhoINGAVtpF6t5Yc4ZCLHGwFSQrZfVH2/mvvlZk8OwyB3G06JA6dRVnHh+bcgkxELCjPro+h/s/dqdWdqsP+Sr111MkWrV3vw+zFNIxSaclDKc51EBzlIzShHc06qMAF6DUP0DBUQoQry8r7C3jVK3x7E0pTQdql+d+QQKZVFoZmMQI/U795Y/K/npXpwFORMJKlGQScPDVJu69gex2j3mUSqeWYIUMnMrTYdgQSqTdhVE4L7+8t/SafZcJ2Ge96sH59M46iQbbJD9ohLDskxOSMt0iaU3JEn8kJerXvr2Xqz3iejM9bUs0V+wPr4BJ00q5c=</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=">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</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'
⇤')
⇤<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><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>
S⇤(µ) = S(�µ) 6= S(µ)<latexit sha1_base64="TU3qQQlGtpyiGsbjRnsXCwEiJsA=">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</latexit><latexit sha1_base64="TU3qQQlGtpyiGsbjRnsXCwEiJsA=">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</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
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1) Introduce a fictitious time variable
Stochastic quantization
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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
⇠R'2<latexit sha1_base64="PaGVhB7qkNcCgipkvGxkt1uRyS4=">AAACIXicbVBNSwMxEM36bf2qevQSLIKIlN1e9Ch68ahibaG7ltl02gaz2TXJFpelf8WLf8WLB0W8iX/G9ONQWwcSHu+9YWZemAiujet+O3PzC4tLyyurhbX1jc2t4vbOnY5TxbDKYhGreggaBZdYNdwIrCcKIQoF1sKHi4Fe66HSPJa3JkswiKAjeZszMJZqFk/9EDtc5vgoQSnIjvoF/4lT/5jeDD6/Byrp8vsKLfgoWxO2ZrHklt1h0VngjUGJjOuqWfzyWzFLI5SGCdC64bmJCXJQhjOBdmyqMQH2AB1sWCghQh3kwwv79MAyLdqOlX3S0CE72ZFDpHUWhdYZgenqaW1A/qc1UtM+DXIuk9SgZKNB7VRQE9NBXLTFFTIjMguAKW53pawLCpixoRZsCN70ybPgrlL23LJ3XSmdnY/jWCF7ZJ8cEo+ckDNySa5IlTDyTF7JO/lwXpw359P5GlnnnHHPLvlTzs8vwzCioA==</latexit><latexit sha1_base64="PaGVhB7qkNcCgipkvGxkt1uRyS4=">AAACIXicbVBNSwMxEM36bf2qevQSLIKIlN1e9Ch68ahibaG7ltl02gaz2TXJFpelf8WLf8WLB0W8iX/G9ONQWwcSHu+9YWZemAiujet+O3PzC4tLyyurhbX1jc2t4vbOnY5TxbDKYhGreggaBZdYNdwIrCcKIQoF1sKHi4Fe66HSPJa3JkswiKAjeZszMJZqFk/9EDtc5vgoQSnIjvoF/4lT/5jeDD6/Byrp8vsKLfgoWxO2ZrHklt1h0VngjUGJjOuqWfzyWzFLI5SGCdC64bmJCXJQhjOBdmyqMQH2AB1sWCghQh3kwwv79MAyLdqOlX3S0CE72ZFDpHUWhdYZgenqaW1A/qc1UtM+DXIuk9SgZKNB7VRQE9NBXLTFFTIjMguAKW53pawLCpixoRZsCN70ybPgrlL23LJ3XSmdnY/jWCF7ZJ8cEo+ckDNySa5IlTDyTF7JO/lwXpw359P5GlnnnHHPLvlTzs8vwzCioA==</latexit><latexit sha1_base64="PaGVhB7qkNcCgipkvGxkt1uRyS4=">AAACIXicbVBNSwMxEM36bf2qevQSLIKIlN1e9Ch68ahibaG7ltl02gaz2TXJFpelf8WLf8WLB0W8iX/G9ONQWwcSHu+9YWZemAiujet+O3PzC4tLyyurhbX1jc2t4vbOnY5TxbDKYhGreggaBZdYNdwIrCcKIQoF1sKHi4Fe66HSPJa3JkswiKAjeZszMJZqFk/9EDtc5vgoQSnIjvoF/4lT/5jeDD6/Byrp8vsKLfgoWxO2ZrHklt1h0VngjUGJjOuqWfzyWzFLI5SGCdC64bmJCXJQhjOBdmyqMQH2AB1sWCghQh3kwwv79MAyLdqOlX3S0CE72ZFDpHUWhdYZgenqaW1A/qc1UtM+DXIuk9SgZKNB7VRQE9NBXLTFFTIjMguAKW53pawLCpixoRZsCN70ybPgrlL23LJ3XSmdnY/jWCF7ZJ8cEo+ckDNySa5IlTDyTF7JO/lwXpw359P5GlnnnHHPLvlTzs8vwzCioA==</latexit><latexit sha1_base64="PaGVhB7qkNcCgipkvGxkt1uRyS4=">AAACIXicbVBNSwMxEM36bf2qevQSLIKIlN1e9Ch68ahibaG7ltl02gaz2TXJFpelf8WLf8WLB0W8iX/G9ONQWwcSHu+9YWZemAiujet+O3PzC4tLyyurhbX1jc2t4vbOnY5TxbDKYhGreggaBZdYNdwIrCcKIQoF1sKHi4Fe66HSPJa3JkswiKAjeZszMJZqFk/9EDtc5vgoQSnIjvoF/4lT/5jeDD6/Byrp8vsKLfgoWxO2ZrHklt1h0VngjUGJjOuqWfzyWzFLI5SGCdC64bmJCXJQhjOBdmyqMQH2AB1sWCghQh3kwwv79MAyLdqOlX3S0CE72ZFDpHUWhdYZgenqaW1A/qc1UtM+DXIuk9SgZKNB7VRQE9NBXLTFFTIjMguAKW53pawLCpixoRZsCN70ybPgrlL23LJ3XSmdnY/jWCF7ZJ8cEo+ckDNySa5IlTDyTF7JO/lwXpw359P5GlnnnHHPLvlTzs8vwzCioA==</latexit> : nonminimal term (renormalizability)
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
