Download Document 8624347

 Nikolaos E. Mavromatos Physics Department King’s College London (UK) LondonCentre
for
Terauniverse
Studies
–AIG 267352
1 2 3 A PRELUDE – Playing with Light: Quantum Field Theories in non-­‐trivial Vacua and Refrac=ve index Causality & Superluminality – not necessarily incompaEble QUANTUM GRAVITY (QG) as a non-­‐trivial field theore=c Vacuum ACT I : QG AS A MEDIUM : THE SET UP ACT II : PHENOMENOLOGY OF QG FOAM VACUA (``FOAM-­‐OLOGY’’): some aspects (i)  Non-­‐trivial OpEcal properEes (refrac=ve index, birefringence(?)) -­‐ Early Theore=cal Predic=ons & suggested Tests (1996,1997) (ii) -­‐ Experimental Surprises: MAGIC (2005) et al. & OPERA (2011) ACT III : Stringy Defect QG D-­‐foam : Modified Dispersion rela=ons & =me delays due to quantum uncertain=es with…strings a0ached Beyond Local Effec=ve Field Theories (?) EPILOGUE: Reconciling MAGIC with OPERA: role of D-­‐foam, further Tests? 4 A PRELUDE Playing with Light Quantum Field Theory in non-­‐trivial Vacua & Refrac=ve Index 5 GAMES WITH LIGHT Lorentz Invariance Speed of Light in Vacuo = c (Universal constant) But in MATERIAL SYSTEMS light propagates with different speed, vlight ǂ c 6 GAMES WITH LIGHT Lorentz Invariance in MATERIAL SYSTEMS light propagates with different speed, vlight ǂ c 7 QUANTUM MECHANICS & REFRACTIVE INDEX electrons (mass m) of the medium as forced quantum oscillators, with force exerted by electromagnetc field photons interact with this background Feynman Electron area density ne = ρe Δz (ρe = volume density of electrons) e Excited-­‐atoms produced electric field e e e Distance Δz traversed by photons Speed of photons in medium c/n suppressed by refrac=ve index n causing delay Δt = (n-­‐1)Δz /c 8 QUANTUM MECHANICS & REFRACTIVE INDEX electrons (mass m) of the medium as forced quantum oscillators, with force exerted by electromagnetc field photons interact with this background Feynman Electron area density ne = ρe Δz (ρe = volume density of electrons) e Excited-­‐atoms produced electric field e e e REFRACTIVE INEX INVERSELY PROPORTIONAL TO PHOTON ENERGIES ( cf. MATTER EFFECTS) Speed of photons in medium c/n suppressed by refrac=ve index n causing delay Δt = (n-­‐1)Δz /c 9 GAMES WITH LIGHT Lorentz Invariance in MATERIAL SYSTEMS light propagates with different speed, vlight ǂ c BIREFRINGENCE 10 NEGATIVE REFRACTIVE INDICES IN METAMATERIALS Normal RefracEon 11 Courtesy Univ. of Surrey NEGATIVE REFRACTIVE INDICES IN METAMATERIALS NegaEve RefracEve Index metamaterials Normal RefracEon Courtesy Univ. of Surrey 12 NEGATIVE REFRACTIVE INDICES IN METAMATERIALS Typical Metamaterial Normal RefracEon 13 Courtesy Univ. of Surrey NEGATIVE REFRACTIVE INDICES IN METAMATERIALS Typical Metamaterial Normal RefracEon WEIRD GEOMETRIES & Boundary condiLons 14 Courtesy Univ. of Surrey Technology engineers all sorts of materials inside which Light has Strange properEes …. 15 Technology engineers all sorts of materials inside which Light has Strange properEes …. But Nature itself may also show such features… 16 Technology engineers all sorts of materials inside which Light has Strange properEes …. But Nature itself may also show such features… The Vacuum (i.e. lowest energy state) of QUANTUM SYSTEMS may also be characterized by strange properEes of light, e.g. Lorentz invariant Breaking …. 17 BOUNDARY CONDITIONS The Vacuum (i.e. lowest energy state) of QUANTUM SYSTEMS may also be characterized by strange properEes of light, e.g. Lorentz invariant Breaking …. CASIMIR VACUUM: Force due to Quantum fluctua=ons of Photons in space between neutral capacitor plates. Quantum effects lead to ``vacuum polariza=on (emission of virtual electron-­‐positron pairs as a photon propagates in this non-­‐trivial vacuum Between the plates) and to a modified group Velocity of photons, larger than c: Distance L Scharnhost (1990), Barton (1990) Ladore, Pascual, Tarrach (1995) 18 BOUNDARY CONDITIONS The Vacuum (i.e. lowest energy state) of QUANTUM SYSTEMS may also be characterized by strange properEes of light, e.g. Lorentz invariant Breaking …. CASIMIR VACUUM: Force due to Quantum fluctua=ons of Photons in space between neutral capacitor plates. Quantum effects lead to ``vacuum polariza=on (emission e+ of virtual electron-­‐positron pairs as a photon propagates in this non-­‐trivial vacuum e-­‐ plates) and to a modified group Between t
he γ γ Velocity of photons, larger than c: Distance L 19 BOUNDARY CONDITIONS The Vacuum (i.e. lowest energy state) of QUANTUM SYSTEMS may also be characterized by strange properEes of light, e.g. Lorentz invariant Breaking …. CASIMIR VACUUM: Force due to Quantum fluctua=ons of Photons in space between neutral capacitor plates. Quantum effects lead to ``vacuum polariza=on (emission of virtual electron-­‐positron pairs as a photon propagates in this non-­‐trivial vacuum Between the plates) and to a modified group Velocity of photons, larger than c: Distance L α
= e2 / 4π = Fine structure constant 20 BOUNDARY CONDITIONS The Vacuum (i.e. lowest energy state) of QUANTUM SYSTEMS may also be characterized by strange properEes of light, e.g. Lorentz invariant Breaking …. CASIMIR VACUUM: Force due to Quantum fluctua=ons of Photons in space between neutral capacitor plates. Quantum effects lead to ``vacuum polariza=on (emission of virtual electron-­‐positron pairs as a photon propagates in this non-­‐trivial vacuum Between the plates) and to a modified group Velocity of photons, larger than c: Distance L Boundary CondiLons Break Lorentz Symmetry 21 in vacuo FINITE TEMPERATURE The Vacuum (i.e. lowest energy state) of QUANTUM SYSTEMS may also be characterized by strange properEes of light, e.g. Lorentz invariant Breaking …. PLASMA VACUUM:Plasma is a state of Maker at very high temperature where Maker is ionized. Temperature T Quantum effects related to vacuum polariza=on in this non-­‐trivial vacuum lead to a modified Group Velocity for photons, larger than c (low T: analogy with Casimir Vacuum: L-­‐1 2T ) Low T modes Superluminal High T modes Subluminal 22 FINITE TEMPERATURE The Vacuum (i.e. lowest energy state) of QUANTUM SYSTEMS may also be characterized by strange properEes of light, e.g. Lorentz invariant Breaking …. PLASMA VACUUM:Plasma is a state of Maker at very high temperature where Maker is ionized. Temperature T Quantum effects related to vacuum polariza=on in this non-­‐trivial vacuum lead to a modified Group Velocity for photons, larger than c (low T: analogy with Casimir Vacuum: L-­‐1 ) 2T ) TEMPERATURE breaks Lorentz Symmetry in Vacuo Low T modes (Boundary C
ondiLons…) Superluminal High T modes Subluminal 23 FINITE TEMPERATURE-­‐NEUTRINOS The Vacuum (i.e. lowest energy state) of QUANTUM SYSTEMS may also be characterized by strange properEes of light, e.g. Lorentz invariant Breaking …. PLASMA VACUUM:Plasma is a state of Maker at very high temperature where Maker is ionized. Temperature T Quantum effects related to vacuum polariza=on in this non-­‐trivial vacuum lead to a modified Group Velocity for neutrinos, smaller than c (low T: analogy with Casimir Vacuum: L-­‐1 2T ) Low T modes Subluminal High T modes Subluminal 24 CURVED SPACE-­‐TIME BACKGROUNDS Drummond & Hathrell (1980) VACUUM POLARIZATION EFFECTS IN QED IN CURVED BACKGROUNDS EFFECTIVE HIGHER DERIVATIVE ACTION AFTER ELECTRON FIELDS ARE INTEGRATED OUT IN A PATH INTEGRAL Seff =
�
W1−loop =
�
4
√
d x −g
4
√
�
�
1
− Fµν F µν + W1−loop
4
d x −g aR Fµν F
µν
+ bRµν F
µλ
F νλ
�
+ cRµνρλ F
e+ γ µν
F
ρλ
e-­‐ + dDµ F
µν
γ Dλ F
λ
ν
�
Consequence : modified (by higher derivaEve terms) photon propagator & dispersion relaEons, Coefficients proporEonal to the product of Newton’s constant with the fine structure constant GN α & inversely proporEonal to electron mass squared me2 Example: Photon group velocity in Expanding Friedman-­‐Robertson-­‐Walker (FRW) Backgrounds Superluminal for ordinary maker, radia=on fluids ρ + p > 0 Light like for Cosmological constant (Lorentz invariant) vacua ρ + p =0 25 CURVED SPACE-­‐TIME BACKGROUNDS Drummond & Hathrell (1980) VACUUM POLARIZATION EFFECTS IN QED IN CURVED BACKGROUNDS EFFECTIVE HIGHER DERIVATIVE ACTION AFTER ELECTRON FIELDS ARE INTEGRATED OUT IN A PATH INTEGRAL Seff =
�
W1−loop =
�
4
√
d x −g
4
√
�
�
1
− Fµν F µν + W1−loop
4
d x −g aR Fµν F
µν
+ bRµν F
µλ
F νλ
�
+ cRµνρλ F
Shore, Hollowood e+ γ µν
F
ρλ
e-­‐ + dDµ F
µν
γ Dλ F
λ
ν
�
Consequence : modified (by higher derivaEve terms) photon propagator & dispersion relaEons, Coefficients proporEonal to the product of Newton’s constant with the fine structure constant GN α & inversely proporEonal to electron mass squared me2 Example: Photon group velocity in Expanding Friedman-­‐Robertson-­‐Walker (FRW) Backgrounds Superluminal for ordinary maker, radia=on fluids ρ + p > 0 Light like for Cosmological constant (Lorentz invariant) vacua ρ + p =0 26 Understanding these results Latorre, Pascual, Tarrach ρe e 2
ρ̃e e2
n=1+
=1+
2
2
2�0 me (ω0 − ω )
2�0 m2e (ω02 − ω 2 )
27 Understanding these results Latorre, Pascual, Tarrach ρe e 2
ρ̃e e2
n=1+
=1+
2
2
2�0 me (ω0 − ω )
2�0 m2e (ω02 − ω 2 )
energy density of ground state 28 Understanding these results Latorre, Pascual, Tarrach ρe e 2
ρ̃e e2
n=1+
=1+
2
2
2�0 me (ω0 − ω )
2�0 m2e (ω02 − ω 2 )
effecEve energy scale energy density of ground state 29 Understanding these results Latorre, Pascual, Tarrach ρe e 2
ρ̃e e2
n=1+
=1+
2
2
2�0 me (ω0 − ω )
2�0 m2e (ω02 − ω 2 )
ρ̃ g 2
n=1+a 4
me
e.g. FRW gravita=onal energy Energy density of non-­‐trivial vacua may be lower w.r.t. normal ρ̃ < 0
ρG = −ρ̃ < 0
30 CAUSALITY & SUPERLUMINALITY No problems with Causality if Superluminality is Observer dependent i.e. Causality paradox arise if signals travel with the same speed V > c in two different frames LiberaE, Sonego, Viser (2002) 31 CAUSALITY & SUPERLUMINALITY No problems with Causality if Superluminality is Observer dependent i.e. Causality paradox arise if signals travel with the same speed V > c in two different frames LiberaE, Sonego, Viser (2002) Previous examples are fine in this respect, the effects on photon dispersion rela=on can be represented by `` effec=ve ‘’ metrics , µ ν
p
p gµν = 0
32 CAUSALITY & SUPERLUMINALITY No problems with Causality if Superluminality is Observer dependent i.e. Causality paradox arise if signals travel with the same speed V > c in two different frames LiberaE, Sonego, Viser (2002) Previous examples are fine in this respect, the effects on photon dispersion rela=on can be represented by `` effec=ve ‘’ metrics , µ ν
p
p gµν = 0
e.g. in Casimir photons in a cavity ( nμ unit space-­‐like vector orthogonal to plates ) 11π 2 α2
ξ=
4050 L4 m2e
33 CAUSALITY & SUPERLUMINALITY No problems with Causality if Superluminality is Observer dependent i.e. Causality paradox arise if signals travel with the same speed V > c in two different frames LiberaE, Sonego, Viser (2002) Previous examples are fine in this respect, the effects on photon dispersion rela=on can be represented by `` effec=ve ‘’ metrics , µ ν
p
p gµν = 0
e.g. in Casimir photons in a cavity ( nμ unit space-­‐like vector orthogonal to plates ) 11π 2 α2
ξ=
4050 L4 m2e
As such, speed of signal depends on observer velociLes uμ relaEve to system 34 CAUSALITY & SUPERLUMINALITY No problems with Causality if Superluminality is Observer dependent LiberaE, Sonego, Viser (2002) i.e. Causality paradox arise if signals travel with the same speed V > c in two different frames Previous examples are fine in this respect, the effects on photon dispersion rela=on can be represented by `` effec=ve ‘’ metrics , µ ν
p
p gµν = 0
e.g. in Casimir photons in a cavity ( nμ unit space-­‐like vector orthogonal to plates ) µ
µ
µ
x → x + ξ (x, u)
gµν → gµν + ∂(µ ξ ν)
As such, speed of signal depends on observer velociLes uμ relaEve to system 35 36 QUANTUM GRAVITY •  Quantum Gravity (QG) is a (quantum?) theory that describes the emergence as well as the dynamical structure of space-­‐=me at Microscopic Standard
scales. Model
QuanEzaEon of GravitaEonal InteracEon is sEll a mystery 37 QUANTUM GRAVITY •  Quantum Gravity (QG) is a (quantum?) theory that describes the emergence as well as the dynamical structure of space-­‐=me at Microscopic Standard
scales. Model
QuanEzaEon of GravitaEonal InteracEon is sEll a mystery String theory is one approach to quantum gravity. The theory predicts extra dimensions But also two ``gravitaEonal scales: (i) one on our world, The Planck Mass scale, MP = 1.2 x 1019 GeV & (ii) the bulk (extra dimensional) one, which is the string mass scale characterizing string theory itself, and may be as low as a few TeV Towards Background independence via AdS/CFT 38 QUANTUM GRAVITY •  Quantum Gravity (QG) is a (quantum?) theory that describes the emergence as well as the dynamical structure of space-­‐=me at Microscopic Standard
scales. Model
QuanEzaEon of GravitaEonal InteracEon is sEll a mystery There are other approaches to QG (Loop Quantum Gravity )
with background independence built in
but not so advanced though to incorporate the Standard
Model of particle physics in their framework
39 FEATURES OF QUANTUM GRAVITY •  QUANTUM FLUCTUATIONS OF SPACE TIME Standard Model
METRIC AT PLANCK SCALES MAY RESULT IN MICROSCOPIC FOAMY SPACE-­‐TIME STRUCTURE �
Dgµν (x)D(. . . )e
1
−κ
�
d4 xR(g)+...
κ ∝ GN = MP−2
�P = MP−1 = 10−35 m
40 FEATURES OF QUANTUM GRAVITY •  QUANTUM FLUCTUATIONS OF SPACE TIME Standard Model
METRIC AT PLANCK SCALES MAY RESULT IN MICROSCOPIC FOAMY SPACE-­‐TIME STRUCTURE �
includes singular configurations
� 4
1
− κ d xR(g)+...
Dgµν (x)D(. . . )e
κ ∝ GN = MP−2
�P = MP−1 = 10−35 m
41 FEATURES OF QUANTUM GRAVITY •  QUANTUM FLUCTUATIONS OF SPACE TIME Standard Model
METRIC AT PLANCK SCALES MAY RESULT IN MICROSCOPIC FOAMY SPACE-­‐TIME STRUCTURE 42 FEATURES OF QUANTUM GRAVITY •  QUANTUM FLUCTUATIONS OF SPACE TIME Standard Model
METRIC AT PLANCK SCALES MAY RESULT IN MICROSCOPIC FOAMY SPACE-­‐TIME STRUCTURE J.A. Wheeler
10-­‐35 m Space-time Foam
43 FEATURES OF QUANTUM GRAVITY •  QUANTUM FLUCTUATIONS OF SPACE TIME Standard Model
METRIC AT PLANCK SCALES MAY RESULT IN MICROSCOPIC FOAMY SPACE-­‐TIME STRUCTURE J.A. Wheeler
10-­‐35 m Space-time Foam
44 QUANTUM GRAVITY AS A MEDIUM Space-Time at Planck
scales may have
a ``foamy structure (J.
A. Wheeler ),
with possible coordinate
non-commutativity
or Lorentz Violation at
microscopic scales
10-35 m
Quantum Gravity then may behave as a medium,
with non-trivial ``optical properties:
Vacuum Refractive Index induced by QG !
Energy dependent speed of light, effects increase
with energy of photon, due to increase in
distortion of space time. Contrast with
Matter-induced ordinary refractive indices.
45 QUANTUM GRAVITY AS A MEDIUM Space-Time at Planck
scales may have
a ``foamy structure (J.
A. Wheeler ),
with possible
10-35 m
coordinate noncommutativity
or Lorentz Violation at
microscopic scales
Quantum Gravity then may behave as a medium,
with non-trivial ``optical properties:
Vacuum Refractive Index induced by QG !
Energy dependent speed of light, effects increase
with energy of photon, due to increase in
distortion of space time. Contrast with
Matter-induced ordinary refractive indices.
46 QUANTUM GRAVITY AS A MEDIUM Space-Time at Planck
scales may have
a ``foamy structure (J.
A. Wheeler ),
with possible
10-35 m
coordinate noncommutativity
or Lorentz Violation at
microscopic scales
Quantum Gravity then may behave as a medium,
with non-trivial ``optical properties:
Vacuum Refractive Index induced by QG !
Energy dependent speed of light, effects increase
with energy of photon, due to increase in
distortion of space time. Contrast with
Matter-induced ordinary refractive indices.
.
47 48 49 Quantum-­‐Gravity Induced Modified Dispersion for Photons Modified dispersion due to QG induced space-time (metric)
distortions (c=1 units):
50 Quantum-­‐Gravity Induced Modified Dispersion for Photons Modified dispersion due to QG induced space-time (metric)
distortions (c=1 units):
Space-­‐Eme Metric describing space-­‐Eme DistorEons induced by InteracEons of Photons with space-­‐Eme defects 51 Quantum-­‐Gravity Induced Modified Dispersion for Photons Modified dispersion due to QG induced space-time (metric)
distortions (c=1 units):
52 Quantum-­‐Gravity Induced Modified Dispersion for Photons Modified dispersion due to QG induced space-time (metric)
distortions (c=1 units):
53 QUANTUM GRAVITY AS A MEDIUM 10-35 m
Quantum Gravity then may behave as a medium,
with non-trivial ``optical properties:
If Vacuum Refractive Index induced by QG non
trivial…
….then, it would be Manifested through delays
(subluminal) or advances (superluminal) in arrival
times of the the more energetic photons in a wave
packet or in general a multi frequency group of
``simultaneously’’ emitted photons.
54 Subluminal QG-induced Refractive Index: Higher energy photons arrive later
Courtesy: N. [email protected]
55 Early Theoretical Predictions Time as a Renormaliza=on Group Irreversibly flowing scale in (Liouville, non-­‐cri=cal) String Theory Ellis, NEM, Nanopoulos (1992) Time – Space different behaviour, different symmetries Lorentz Viola=on natural consequence of this microscopic approach ``Environment’’ of Quantum Gravity (QG) (stringy) d.o.f. inaccessible to low-­‐energy observer scakering experiment; Induced Decoherence and hence Microscopic Time Arrow Off-­‐shell stringy maker excita=ons due to interac=on with the QG ``environment’’ Induced Modified Dispersion Rela=ons (c(E) refrac=ve index) for photon probes due to propaga=on in QG ``medium’’ Amelino Camelia, Ellis, NEM, Mavromatos (1996/05) Early Theoretical Predictions Time as a Renormaliza=on Group Irreversibly flowing scale in (Liouville, non-­‐cri=cal) String Theory Ellis, NEM, Nanopoulos (1992) Time – Space different behaviour, different symmetries Lorentz Viola=on natural consequence of this microscopic approach ``Environment’’ of Quantum Gravity (QG) (stringy) d.o.f. inaccessible to low-­‐energy observer scakering experiment; Induced Decoherence and hence Microscopic Time Arrow Off-­‐shell stringy maker excita=ons due to interac=on with the QG ``environment’’ Induced Modified Dispersion Rela=ons (c(E) refrac=ve index) for photon probes due to propaga=on in QG ``medium’’ Amelino Camelia, Ellis, NEM, Mavromatos (1996/05) MAGIC results (2005) TeV Photons from Ac=ve Galac=c Nucleus (AGN) Mkn 501 at red-­‐shiw z = 0.03 More energe=c photons (1.2 -­‐ 10 TeV) delayed by O(1 min) compared to E < 0.6 TeV First Interes=ng result… in conflIct with Conven=onal Astrophysical accelera=on AGN Models (e.g. Crab Nebula) Interes=ngly can fit QG subluminal refrac=ve index with linear MQG suppression with MQG(1) = 0.2 x 1018 GeV or, if astrophysics at source taken into account MQG(1) > 0.2 x 1018 GeV Other Observed Photon Delays (H.E.S.S, FERMI) Ellis, NEM, Nanopoulos (2009, 2010) E
∆t =
H0−1
MQG
�
z
0
(1 + z)
dz �
ΩM (1 + z)3 + ΩΛ
OPERA RESULTS – Superluminal neutrinos 2011 : Another Surprise, more mysterious NEUTRINOS IN OPERA have been argued to propagate with superluminal velociEes v/c – 1 = ( 2.48 ± 0.28 ± 0.30 ) 10-­‐5 (but independent of the energy, at least in the range of the experiment) OPERA v2 overall significance more 6.2 σ OPERA RESULTS – Superluminal neutrinos 2011 : Another Surprise, more mysterious NEUTRINOS IN OPERA have been argued to propagate with superluminal velociEes v/c – 1 = ( 2.48 ± 0.28 ± 0.30 ) 10-­‐5 (but independent of the energy, at least in the range of the experiment) Can Subluminal Photons and Superluminal Neutrinos be consistent with Causality and models of Quantum Gravity? OPERA RESULTS – Superluminal neutrinos 2011 : Another Surprise, more mysterious NEUTRINOS IN OPERA have been argued to propagate with superluminal velociEes v/c – 1 = ( 2.48 ± 0.28 ± 0.30 ) 10-­‐5 (but independent of the energy, at least in the range of the experiment) Can Subluminal Photons and Superluminal Neutrinos be consistent with Causality and models of Quantum Gravity? YES IN A SPACE-­‐TIME FOAM MODEL INSPIRED FROM STRING THEORY 63 64 65 ``Foamy Structures’’ in String Theory •  Foamy space-­‐=me structures may also be Standard Model
provided by higher-­‐dimensional space-­‐=me ``real Defects’’ in the modern version of string theory involving branes (D-­‐Foam) Ellis, NEM, Nanopoulos, Sarben Sarkar, Szabo, Westmucked… 66 BRANE/STRING -­‐THEORY Our Universe Lorentz Invariant (Standard Model parEcles) Gravitons 67 BRANE-WORLDS with D-PARTICLE (POINT-LIKE BRANE) DEFECTS
(Standard Model
particles)
(Gravitons)
D-­‐par=cle defect J ELLIS, NEM, M. WESTMUCKETT 68 BRANE-WORLDS with D-PARTICLE (POINT-LIKE BRANE) DEFECTS
Recoil of defect (Standard Model
particles)
Our Universe
NO LONGER
Lorentz Invariant
(Gravitons)
D-­‐parEcle defect J ELLIS, NEM, M. WESTMUCKETT 69 BRANE-WORLDS with D-PARTICLE (POINT-LIKE BRANE) DEFECTS
Recoil of defect (Standard Model
particles)
Our Universe
NO LONGER
Lorentz Invariant
(Gravitons)
D-­‐parEcle defect J ELLIS, NEM, M. WESTMUCKETT 70 Recoil-­‐induced Lorentz Viola=on (locally) Defect Distribution
may be inhomogeneous
Brane Worlds
Point-like Brane defect
OUR METAUNIVERSE
Colliding Brane world model of Space-Time with point-like space-time defects
71 Two kinds of InteracLons Involve ``spontaneous’’ change of world-­‐sheet boundary condiLons for the open string 72 Two kinds of InteracLons (I) Just RECOIL of massive defect distorEon of surrounding space Eme Problem Equivalent to Strings propagaEng in Local ``electric field’’ backgrounds Time-­‐Space non-­‐commutaEvity [X i , t] ∝ F 0i (k, x) ≡ ui (k, x)
But electric field is on phase space ∆ki
u (k, x) = gs
Ms
i
D-­‐parEcle’s recoil velocity depends on momentum transfer Δki 73 Two kinds of InteracLons (I) Just RECOIL of massive defect distorEon of surrounding space Eme Problem Equivalent to Strings propagaEng in Local ``electric field’’ backgrounds Time-­‐Space non-­‐commutaEvity [X i , t] ∝ F 0i (k, x) ≡ ui (k, x)
But electric field is on phase space ∆ki
u (k, x) = gs
Ms
i
Ms / gs arbitrary D-­‐parEcle mass (bulk string scale Ms not determined, can be as low as a few TeV, gs < 1 ). D-­‐parEcle’s recoil velocity depends on momentum transfer Δki 74 Problem Equivalent to Strings propagaEng in Local ``electric field’’ backgrounds Time-­‐Space non-­‐commutaEvity [X i , t] ∝ F 0i (k, x) ≡ ui (k, x)
Induced metric depends on momenta as well as coordinates (Finsler type) : e.g. u || X1 gµν = ηµν + hµν
Explicit local breaking of SO(3,1) down to SO(2,1) rotaLon and boosts in transverse direcLons h00 = −h11 = |u1 |2
∆ki
h01 = gs
≡ u1
Ms
Local Lorentz ViolaLon due to direcLon of Defect recoil velociLes 75 Locally induced metric distorEons seen by ``open strings’’ ds2 = gµν dxµ dxν =
2
2
−(1 − |�u| )dt2 + (1 − |�u| )dx21 + dx22 + dx23 + 2�u · d�x dt
Modified photon dispersion relaEons due to locally induced metric µ ν
p p gµν = 0
�
�
1 2
µ
3
0
<
E
=
−�
p
·
�
u
+
p
1
+
|�
u
|
+
O(|�
u
|
)
p = (E, p
�) ⇒
2
(Sub)Superluminal Group photon velociEes, depending on relaEve direcEons u, p ∂E
1 2
3
vg =
= 1 − |�u|cosϑ + |�u| + O(|�u| )
∂p
2
76 Two kinds of InteracLons (II) String Spliang & (``momentary’’) Capture by the defects INTEMEDIATE STRING CREATION/EXCHANGE PURELY NON-­‐LOCAL EFFECT 77 Two kinds of InteracLons (II) String Spliang & (``momentary’’) Capture by the defects INTEMEDIATE STRING CREATION/EXCHANGE PURELY NON-­‐LOCAL EFFECT string graphs 78 CHARGE CONSERVATION MUST BE RESPECTED DURING CAPTURE (STRING SPLITTING, INTERMEDIATE STRING CREATION & STRETCHING): ONLY ELECTRICALLY NEUTRAL EXCITATIONS INTERACT VIA CAPTURE DOMINANTLY WITH FOAM DEFECT RECOIL OCCURS Time Delays due to Intermediate String Crea=on & Oscilla=ons J ELLIS, NEM, NANOPOULOS 79 Veneziano Amplitude is propor=onal to π
iπsα� −�
→e
�
sin(π sα )
Poles at s = n/α ‘ (α ‘ = Regge slope) Shiw pole prescripEon s  s + iε Delay: �
∆t ∼ α E
Time Delays due to Intermediate String Crea=on & Oscilla=ons Amplitude Pole Analysis ELLIS, NEM, NANOPOULOS, …+ LI, XIE 80 J ELLIS, NEM, DV NANOPOULOS DELAYS COMPATIBLE WITH STRINGY UNCERTAINTIES Time Delays due to Intermediate String Crea=on, growth up to length L & N Oscilla=ons  L
N
E= �+
α
L
Minimise right-­‐hand-­‐size w.r.t. L. End of intermediate string on D3-­‐brane Moves with speed of light in vacuo c=1 Hence TIME DELAY (causality) during Capture. �
∆t ∼ α E
Compa=ble with the Stringy Uncertain=es Δt Δx ≥ α’ Δt Δp ≥ 1 + α’ (Δp)2 81 J ELLIS, NEM, NANOPOULOS RECOILING DEFECT ``DRAGS THE FRAME’’ NON-­‐INERTIAL ACCELERATION DEFECT ``SUDDEN’’ RECOIL OCCURS i
∆k
i
u Θ(t) = gs
Ms
i i
≡ r k Θ(t)
82 J ELLIS, NEM, DV NANOPOULOS RECOILING DEFECT ``DRAGS THE FRAME’’ NON-­‐INERTIAL ACCELERATION NON-­‐CRITICAL STRING in Eme-­‐dependent recoil ``Electric’’ field backgrounds DEFECT ``SUDDEN’’ RECOIL OCCURS i
∆k
i
u Θ(t) = gs
Ms
i i
≡ r k Θ(t)
83 Time Delays due to Intermediate String Crea=on & Oscilla=ons increase due to recoil α� E
∆t ∼
1 − |�u|2
DEFECT ``SUDDEN’’ RECOIL OCCURS i
∆k
i
u Θ(t) = gs
Ms
i i
≡ r k Θ(t)
84 Time Delays due to Intermediate String Crea=on & Oscilla=ons increase due to recoil α� E
∆t ∼
1 − |�u|2
DEFECT ``SUDDEN’’ RECOIL OCCURS i
∆k
i
u Θ(t) = gs
Ms
i i
≡ r k Θ(t)
ANALOGY WITH FEYNMAN MODEL D-­‐PARTICLE DEFECTS PLAY ROLE OF ELECTRON OSCILLATIONS, BUT ENERGY DEPENDENCE OF DELAYS DIFFERENT FROM THAT IN MATTER 85 QUANTUM MECHANICS & REFRACTIVE INDEX electrons (mass m) of the medium as forced quantum oscillators, with force exerted by electromagnetc field photons interact with this background Feynman Electron area density ne = ρe Δz (ρe = volume density of electrons) e Excited-­‐atoms produced electric field e e e Distance Δz traversed by photons Speed of photons in medium c/n suppressed by refrac=ve index n causing delay Δt = (n-­‐1)Δz /c 86 Time Delays due to Intermediate String Crea=on & Oscilla=ons increase due to recoil α� E
∆t ∼
1 − |�u|2
Delays are Independent of photon polariza=on NO BIREFRINGENCE 87 Stringy Uncertain=es & D-­‐Foam •  D-­‐foam: popula=ons of defects encountered by probe •  D-­‐foam captures neutral probes & re-­‐emits them •  Time Delay (Causal) in each Capture: p0 = E •  Independent of photon polariza=on (no Birefringence) •  Total Delay from emission of photons =ll observa=on over a distance D (assume n* defects per string length): 88 COSMOLOGICAL D-­‐FOAM 89 COSMOLOGICAL D-­‐FOAM Eff
MQG
Ms
= �
n (z)
n* (z) can increase with z If brane moves in inhomogeneous bulk 90 Observed Photon Delays (H.E.S.S, FERMI) Ellis, NEM, Nanopoulos (2009, 2010) MQG(1) > 1.5 1019 GeV E
∆t =
H0−1
MQG
�
z
0
(1 + z)
dz �
ΩM (1 + z)3 + ΩΛ
Eff
MQG
Ms
= �
n (z)
TO RECAP: (I) RECOIL OF MASSIVE DEFECTS DISTORS SPACE TIME , MODIFIES DISPERSION RELATIONS FOR MATTER PROBES (II) CAPTURE OF MATTER BY DEFECT LEAD TO EXTRA TIME DELAYS DUE TO STRINGY NATURE/MINIMUM LENGTH OF STRINGS, WHICH ARE ENHNACED BY RECOIL �
FOR CHARGE CONSERVATION REASONS CAPTURE (HENCE DELAYS) UNDERGO ONLY ELECTRICALLY NEUTRAL PARTICLES (PHOTONS…) ∆t ∼ α E
92 Space Lme Foam situaEons – Average over both populaEons of defects & quantum fluctuaEons Isotropic & (in)homogeneous foam for a brane observer: �ui � ≡ gs �∆ki � = 0
Ms
Lorentz Invariance on Average gs2
2
�∆ki ∆kj � = σ δij Violated in flcts 2
Ms
93 Space Lme Foam situaEons – Average over both populaEons of defects & quantum fluctuaEons Isotropic & (in)homogeneous foam for a brane observer: �ui � ≡ gs �∆ki � = 0
Ms
Lorentz Invariance on Average gs2
2
�∆ki ∆kj � = σ δij Violated in flcts 2
Ms
1
1 2
2
3
vg = 1 + �|�u| � + O(|�u| ) = 1 + σ > 1
2
2
Superluminal propagaEon IF NOT CAPTURE 94 Space Lme Foam situaEons – Average over both populaEons of defects & quantum fluctuaEons Isotropic & (in)homogeneous foam for a brane observer: �ui � ≡ gs �∆ki � = 0
Ms
Lorentz Invariance on Average gs2
2
�∆ki ∆kj � = σ δij Violated in flcts 2
Ms
1
1 2
2
3
vg = 1 + �|�u| � + O(|�u| ) = 1 + σ > 1
2
2
Superluminal propagaEon IF NOT CAPTURE �
∂E
1 2
3
cf. locally vg =
= 1 − |�u|cosϑ + |�u| + O(|�u| )
∂p
2
95 �
96 97 TO RECAP: (I) RECOIL OF MASSIVE DEFECTS DISTORS SPACE TIME , MODIFIES DISPERSION RELATIONS FOR MATTER PROBES (II) CAPTURE OF MATTER BY DEFECT LEAD TO EXTRA TIME DELAYS DUE TO STRINGY NATURE/MINIMUM LENGTH OF STRINGS, WHICH ARE ENHNACED BY RECOIL �
FOR CHARGE CONSERVATION REASONS CAPTURE (HENCE DELAYS) UNDERGO ONLY ELECTRICALLY NEUTRAL PARTICLES (PHOTONS…) NEUTRINOS IN STRING THEORIES MAY HAVE ADDITIONAL ``CHARGES’’/FLUXES DUE TO EXTRA U(1)’ GROUP. IN SUCH A CASE NEUTRINOS WILL ALSO AVOID CAPTURE AND WILL THUS BE SUBJECTED ONLY TO (superluminal) GRAVITATIONAL D-­‐foam RECOIL EFFECTS ∆t ∼ α E
1
1 2
2
3
vg = 1 + �|�u| � + O(|�u| ) = 1 + σ > 1
2
2
98 TO RECAP: (I) RECOIL OF MASSIVE DEFECTS DISTORS SPACE TIME , MODIFIES DISPERSION RELATIONS FOR MATTER PROBES 2
g
s
2
�∆k
∆k
�
=
σ
δij
i
j
2
Ms
NEUTRINOS IN STRING THEORIES MAY HAVE ADDITIONAL ``CHARGES’’/FLUXES DUE TO EXTRA U(1)’ GROUP. IN SUCH A CASE NEUTRINOS WILL ALSO AVOID CAPTURE AND WILL THUS BE SUBJECTED ONLY TO (superluminal) GRAVITATIONAL D-­‐foam RECOIL EFFECTS 1
1 2
2
3
vg = 1 + �|�u| � + O(|�u| ) = 1 + σ > 1
2
2
99 POSSIBLE TO RECONCILE OPERA ``ADVANCES’’ 2
g
DUE s
2
T O R ECOIL-­‐INDUCED �∆k
∆k
�
=
σ
δij
i
j
SPACE-­‐TIME 2
DISTORTION Ms
FOR NEUTRINOS, 1
1 2
2
3
vg = 1 + �|�u| � + O(|�u| ) = 1 + σ > 1
CAPTURE BY FOAM NO 2
2
POSSIBLY DUE TO U’(1) GAUGE SYMMETRIES �
WITH ``MAGIC’’ PHOTON ``DELAYS‘’ DUE TO CAPTURE BY D-­‐FOAM ∆t ∼ α E
100 POSSIBLE TO RECONCILE OPERA ``ADVANCES’’ 2
g
DUE s
2
T O R ECOIL-­‐INDUCED �∆k
∆k
�
=
σ
δij
i
j
SPACE-­‐TIME 2
DISTORTION Ms
FOR NEUTRINOS, 1
1 2
2
3
vg = 1 + �|�u| � + O(|�u| ) = 1 + σ > 1
CAPTURE BY FOAM NO 2
2
POSSIBLY DUE TO U’(1) GAUGE SYMMETRIES �
WITH ``MAGIC’’ PHOTON ``DELAYS‘’ DUE TO CAPTURE BY D-­‐FOAM ∆t ∼ α E
ΝΟ PROBLEMS WITH CAUSALITY 101 CAUSALITY & SUPERLUMINALITY No problems with Causality if Superluminality is Observer dependent LiberaE, Sonego, Viser (2002) i.e. Causality paradox arise if signals travel with the same speed V > c in two different frames Previous examples are fine in this respect, the effects on photon dispersion rela=on can be represented by `` effec=ve ‘’ metrics , µ ν
p
p gµν = 0
e.g. in Casimir photons in a cavity ( nμ unit space-­‐like vector orthogonal to plates ) µ
µ
µ
x → x + ξ (x, u)
gµν → gµν + ∂(µ ξ ν)
As such, speed of signal depends on observer velociLes uμ relaEve to system 102 POSSIBLE TO RECONCILE OPERA ``ADVANCES’’ 2
g
DUE s
2
T O R ECOIL-­‐INDUCED �∆k
∆k
�
=
σ
δij
i
j
SPACE-­‐TIME 2
DISTORTION Ms
FOR NEUTRINOS, 1
1 2
2
3
vg = 1 + �|�u| � + O(|�u| ) = 1 + σ > 1
CAPTURE BY FOAM NO 2
2
POSSIBLY DUE TO U’(1) GAUGE SYMMETRIES �
WITH ``MAGIC’’ PHOTON ``DELAYS‘’ DUE TO CAPTURE BY D-­‐FOAM ∆t ∼ α E
ΝΟ NEUTRINO CHERENKOV RadiaLon (No Cohen-­‐Glashow pair-­‐producLon effect) 103 Alexandre, Ellis, NEM (2011) ΝΟ NEUTRINO CHERENKOV RadiaLon in D-­‐foam (No Cohen-­‐Glashow pair-­‐producLon effect) +
−
KinemaEcally allowed τ
τ
if neutrino speed is faster than c=1 in vacuo and there is a preferred frame But in D-foam there is general coordinate invariance
µ ν
built in (underlying string theory).
p
p gµν = 0
moreover superluminality is an ``effective’’
phenomenon induced in given frame by
xµ → xµ + ξ µ (x, u)
the distorted geometry due to recoil
gµν → gµν + ∂(µ ξ ν)
The latter is observer dependent,
ν →ν +e +e
However Cherenkov radiation (decay) is observer independent
one can go (within string theory) to the rest frame of neutrino,
where such a Cherenkov process is not allowed kinematically. 104 POSSIBLE TO RECONCILE OPERA ``ADVANCES’’ 2
g
DUE s
2
T O R ECOIL-­‐INDUCED �∆k
∆k
�
=
σ
δij
i
j
SPACE-­‐TIME 2
DISTORTION Ms
FOR NEUTRINOS, 1
1 2
2
3
vg = 1 + �|�u| � + O(|�u| ) = 1 + σ > 1
CAPTURE BY FOAM NO 2
2
POSSIBLY DUE TO U’(1) GAUGE SYMMETRIES WITH ``MAGIC’’ PHOTON ``DELAYS‘’ DUE TO CAPTURE BY D-­‐FOAM �
∆t ∼ α E
OTHER EXPLANATIONS (including weird effecEve geometries & boundary condiEons in the condiEons of OPERA experiment ) POSSIBLE OF COURSE…… 105 OPERA superluminal neutrino claims are not in fundamental conflict with RelaEvity or Causality if they are due to ``medium’’ or other weird background effects Even if due to QUANTUM GRAVITY strange backgrounds, there are mathemaEcally consistent theoreEcal models (that encompass relaEvity) which can provide explanaEons 106 107 108 109