and the related infrared physics Sotaro Sugishita (Osaka Univ.) - - PowerPoint PPT Presentation
Memory effects and the related infrared physics Sotaro Sugishita (Osaka Univ.) Strings and Fields 2018 YITP, July 30, 2018 Based on: Yuta Hamada & SS, JHEP 1711 (2017) 203 [arXiv:1709.05018] Yuta Hamada & SS, JHEP 1807 (2018)
Memory effects and the related infrared physics
Sotaro Sugishita (Osaka Univ.)
Strings and Fields 2018 YITP, July 30, 2018
Based on:
Yuta Hamada & SS, JHEP 1711 (2017) 203 [arXiv:1709.05018] Yuta Hamada & SS, JHEP 1807 (2018) 017 [arXiv:1803.00738] Hayato Hirai & SS, JHEP 1807 (2018) 122 [arXiv:1805.05651]
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Gravitational Waves Detected!
GW150914
https://www.ligo.c altech.edu/image/ ligo20160211a
from Nobelprize.org
[Nature 327, 123–125 (1987)]
from Physics-Uspekhi 55(1)109-110 (2012)
from Nobelprize.org
[Nature 327, 123–125 (1987)]
from Physics-Uspekhi 55(1)109-110 (2012)
[Zel’dovich & Polnarev (1974)]
Gravitational memory effect
GW
detector source
before after GW passes
Gravitational wave burst w/ finite duration
[Zel’dovich & Polnarev (1974)]
Gravitational memory effect
GW
detector source
before after GW passes
Gravitational wave burst w/ finite duration permanent displacement of test particles
Memory effect
flat spacetime.
Physics in asymptotically flat space
GW
detector source
Detector is far away from the source.
related to soft theorem asymptotic symmetry
Isolated source.
Memory Effect Soft Theorem Asymptotic Symmetry
IR Triangular Relation
[Strominger (2013), … ]
Each corner is not new. Soft Photon Theorem & Soft Graviton Theorem Asymptotic Symmetry (BMS symmetry) Gravitational Memory Effect
Low (1954), Gell-Mann & Goldberger (1954), Weinberg (1965) Bondi, van der Burg & Metzner (1962), Sachs (1962) Zel’dovich & Polnarev (1974)
The relations among them are somewhat new.
[Strominger (2013), … ]
Massless radiation
[cf. Nonlinear memory, Christodoulou (1991),Thorne (1992)]
Other massless radiations satisfy almost the same eq.
Massless radiation
[cf. Nonlinear memory, Christodoulou (1991),Thorne (1992)]
Other massless radiations satisfy almost the same eq.
Massless radiation
EM memory effect Scalar memory effect
[Bieri & Garfinkle (2013), Tolish & Wald (2014), Susskind (2015)] [Tolish & Wald (2014), Hamada & SS (2017)]
[cf. Nonlinear memory, Christodoulou (1991),Thorne (1992)]
Electromagnetism is probably simpler than gravity. Maxwell’s eq. is easier than Einstein’s eq. Let’s consider the electromagnetism for a while, though I was asked to give a talk about gravitational memory.
Excuse me
Electromagnetism is probably simpler than gravity. Maxwell’s eq. is easier than Einstein’s eq. Let’s consider the electromagnetism for a while, though I was asked to give a talk about gravitational memory. Klein-Gordon is much easier.
Excuse me
Electromagnetism is probably simpler than gravity. Maxwell’s eq. is easier than Einstein’s eq. Let’s consider the electromagnetism for a while, though I was asked to give a talk about gravitational memory. Klein-Gordon is much easier. But, the relation of scalar memory to symmetry is not so clear….
[Campiglia, Coito & Mizera (2017)]
Excuse me
Outline
Introduction Memory effect in electromagnetism Relation to soft theorem Some comments
Memory effect as charge conservation
Electromagnetic memory effect EM radiation with the change of the asymptotic behaviors
This change is the memory, which is related to (or memorize) the change of the configurations of the source (charged
like
We derive the electromagnetic memory effect from the conservation laws associated with the asymptotic symmetry for classical systems.
Memory effect as charge conservation
Electromagnetic memory effect EM radiation with the change of the asymptotic behaviors
This change is the memory, which is related to (or memorize) the change of the configurations of the source (charged
like
[Hirai & SS (2018)]
Assume that charged particles behave as free particles except for a small scattering region.
scattering of charged particles
Consider the solution from large-𝑠 region.
Maxwell’s eq.
Setup
Gauge charge
Initially and finally, charged particles move at constant velocities (in our assumption).
Gauge charge
arbitrary function
If = const., Initially and finally, charged particles move at constant velocities (in our assumption).
Gauge charge
Take the surface as two-sphere with radius 𝑆 at past or future infinity, and set the radius 𝑆 → ∞.
(The precise definition will be given later.)
initial final
Gauge charge conservation?
If = const., from the total electric charge conservation,
initial final
For general ,
This change implies the EM radiation with a memory.
future light cone
Easy example
The retarded potential:
A static charge at Ԧ 𝑦0 suddenly moves with velocity Ԧ 𝑤 at time 𝑢0.
future light cone
Easy example
In the Lorenz gauge,
We use the retarded coordinates to see this shift in large 𝑠.
shift
Easy example
Future null infinity retarded coordinates
Easy example
Future null infinity retarded coordinates
two angular components. DOF of EM waves. (cf. TT comp. of GW)
memory
This memory knows information of charges.
Memory formula
: arbitrary function on 𝑇2 It satisfies
This memory knows information of charges.
Memory formula
: arbitrary function on 𝑇2 It satisfies
The formula holds for more general scatterings.
This memory knows information of charges.
Memory formula
: arbitrary function on 𝑇2 It satisfies
The formula holds for more general scatterings. It can be derived from the charge conservation of the large gauge transformation.
This memory knows information of charges.
Memory formula
: arbitrary function on 𝑇2 It satisfies
The formula holds for more general scatterings. It can be derived from the charge conservation of the large gauge transformation.
e.g.
global U(1) trsf. = const., total electric charge consv.
“Large gauge transformations” in this talk mean residual gauge trsfs in the Lorenz gauge, which satisfy and behave as
Large gauge transformation
near . We use the Lorenz gauge
“Large gauge transformations” in this talk mean residual gauge trsfs in the Lorenz gauge, which satisfy and behave as
Large gauge transformation
near . We use the Lorenz gauge Such gauge parameters are given by
[Campiglia & Laddha (2015)]
near
is the antipodal pt of
Gauge current conservation
Gauge current conservation
𝑊 is surrounded by 5 surfaces.
Gauge current conservation
𝑊 is surrounded by 5 surfaces.
Gauge current conservation
𝑊 is surrounded by 5 surfaces. Take the limit 𝑈 → ∞ memory formula
Charge on null surface
Let’s first consider the surface on null surface:
Charge on null surface
Let’s first consider the surface on null surface:
Charge on null surface
Let’s first consider the surface on null surface:
L.H.S of memory formula
No initial radiation
In our setup, there is no initial radiation
Charge on timelike infinity
Gauge charges in balls Σ𝑔/𝑗 for matter currents and their Lienard-Wiecherd potentials.
Large gauge current conservation: So far,
Large gauge current conservation: So far,
Large gauge current conservation: So far,
Large gauge current conservation: So far,
Charge on spacelike infinity
electric fields created by charged particles before scatterings
Charge on spacelike infinity
electric fields created by charged particles before scatterings
The integrand does not decay for the large gauge parameter.
Charge on spacelike infinity
electric fields created by charged particles before scatterings
The integrand does not decay for the large gauge parameter.
antipodal
Antipodal matching
Electric fields also satisfy the antipodal matching condition.
[He, Mitra, Porfyriadis & Strominger (2014)]
Antipodal matching
Electric fields also satisfy the antipodal matching condition.
[He, Mitra, Porfyriadis & Strominger (2014)]
Finite memory formula
[Hirai & SS (2018)]
Nontrivial change of the gauge charge implies the existence of the radiation with a memory. Infinite number of conservation laws.
: arbitrary function on 𝑇2
Indep. of the details of scatterings.
Asymptotic conservation laws
Same for the cases with initial radiations antipodal matching
Asymptotic conservation laws
Same for the cases with initial radiations antipodal matching Take the limit 𝑉 → ∞.
charge at 𝑢 = ∞ charge at 𝑢 = −∞
Outline
Introduction Memory effect in electromagnetism Relation to soft theorem Some comments
charge conservation.
Soft theorem is charge conservation
Soft theorems
[Strominger (2013), …]
Leading soft photon theorem
Strominger (2013), He, Mitra, Porfyriadis & Strominger (2014) Campiglia & Laddha (2015), Kapec, Pate & Strominger (2015)
Leading soft graviton theorem
He, Lysov, Mitra & Strominger (2014)
Campiglia & Laddha (2015)
Subleading soft photon theorem
Lysov, Pasterski & Strominger (2014), Campiglia & Laddha (2016), Conde & Mao (2016) Hirai & SS (2018)
Subleading and sub-subleading soft graviton theorem
Kapec, Lysov, Pasterski & Strominger (2014), Campiglia & Laddha (2016)× 2, Conde & Mao (2016)
All orders including non-universal parts
Hamada & Shiu (2018)
= (singular factor) ×
Soft photon theorem
scattering amplitude with a soft photon
momentum
[Low (1958), Weinberg (1965)]
Soft factor is memory
This soft factor naturally appears in the classical radiation.
We can always take the basis of the polarization as
invariant under
[Yennie, Frautschi & Suura (1961)]
Memory for this scattering takes the following form
Soft photon theorem as WT id.
[He, Mitra, Porfyriadis & Strominger (2014), Campiglia & Laddha (2015)]
Leading soft photon theorem
soft charge hard charge
Ward-Takahashi identity for large gauge trsfs Review the equivalence for massive hard particles.
Soft charge
free radiation
Soft charge creates a soft photon.
Hard charge
Assume no interactions at the asymptotic regions.
(Comments will be given later.)
Hard charge
Assume no interactions at the asymptotic regions.
(Comments will be given later.)
Hard charge acts on the Fock space of massive particles. After some computations (done by introducing hyperbolic foliations) [See Campiglia & Laddha (2015), or Hirai & SS (2018)]
Equivalence
This eq. holds for any
Soft photon theorem WT id. for LGT
Outline
Introduction Memory effect in electromagnetism Relation to soft theorem Some comments
We have seen that leads to
Subleading corrections
the conservation laws. The next corrections in large 𝑈 exp. also lead to the similar conservation laws.
Subleading conservation laws [Hirai & SS (2018)]
No subsubleading
At the subsubleading order, we should take account of the spacelike infinity charge . It might be related to the fact that there is no universal subsubleading soft photon theorem. No conservation laws b/w 𝑢 = −∞ and 𝑢 = ∞.
[Hirai & SS (2018)]
Dressed states
To show the equivalence of soft theorem and WT id for LGT, we have assumed that hard particles can be regarded as free at the asymptotic regions, i.e., adopt the usual Fock space approach. If there is a charged particle, there is the Coulomb potential around it (Gauss’s law). It looks better to use the asymptotic states including the effects of the Coulomb potential. Such a formalism was already investigated to resolve the infrared divergences in QED.
[Chung (1965), Kibble (1968), Kulish & Faddeev (1970)]
FK formalism state with a cloud of photons
Dressed states and LGT
[Chung (1965), Kibble (1968), Kulish & Faddeev (1970)]
[Mirbabayi & Porrati (2016), Gabai & Sever (2016), Kapec, Perry, Raclariu & Strominger (2017)]
(In usual approach, we need to sum up all amplitudes with soft emission.)
and the relation to asympt. sym.
[Ware, Saotome & Akhoury (2013)] [Choi, Kol & Akhoury (2017), Choi & Akhoury (2017)]
hard graviton emission
Gravitational memory
Linearized gravitational memory can be obtained by solving Nonlinear analysis was done by Christodoulou (1991).
[Zel’dovich & Polnarev (1974)]
Thorne (1992) pointed out that it can be obtained by Strominger & Zhiboedov (2014) showed the relation to the supertranslation.
BMS symmetry
[Bondi, van der Burg, Metzner (1962), Sachs (1962)]
Asymptotically flat spacetime
BMS sym = Diffeo preserving this structure.
Original BMS = supertranslation + Lorentz
~ angle dependent 𝑣-translation
Supertranslation sym.
[Strominger & Zhiboedov (2014)]
Gravitational memory
Summary
Memory exists to compensate the change of the hard LGT charges.
formalism (dressed states formalism).
Dresses are quantum realization of memories.
IR physics in asymptotically flat spacetime needs much more investigations.
Subleading soft charge
with
agrees with [Lysov, Pasterski, Strominger (2014), Campiglia,
Laddha (2016), Conde, Mao (2016)]
It creates subleading soft photons
LGT parameter It satisfies Expanded as
Large gauge parameters
Subleading soft photon theorem
[Low (1954)] It deletes the leading singularity .
This soft theorem is also related to the asymptotic symmetry.
[Lysov, Pasterski, Strominger (2014), Campiglia, Laddha (2016), Conde, Mao (2016)]
Subleading soft factor
Subleading charge contains the integral
[Hamada & SS (2018)]
Subleading soft factor naturally appears in the classical scatterings like the leading factor. Subleading soft factor
Gravitational memory
Gravitational memory formula
Bondi gauge
: parameter 𝑔 is an arbitrary function on 𝑇2. roughly angle-dependent u-translation, called supertranslation.
where 𝑍𝐵 is a conformal Killing vector (CKV) on 𝑇2.
If we require that 𝑍𝐵 be globally well-defined as in the original BMS papers, the trsf is just the Lorentz trsf 𝑇𝑀(2, ℂ) in asympt region.
Supertranslation and superrotation
𝑍𝐵 may be local CKVs. 𝑨 ∈ ℂ parametrized 𝑇2
superrotation
[Barnich & Troessaert (2009)]
Extended BMS
Detection of EM memory
integration proportional to memory 1/𝑠 coeffs of 𝐹 or 𝐶
memory
[Bieri & Garfinkle (2013), Tolish & Wald (2014)]
memory
[Hamada & SS (2018)]
Spin also couples to gravity.
probe charge probe spin
detection of grav memory by change of spin