Theory Group at Dep. Phys. "Enrico Fermi"
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Congressino Dip Fis 11/4/2011
Theory Group at Dep. Phys. “Enrico Fermi”
• “Theories of the fundamental interactions towards 2020”
D. Anselmi, A. Strumia, L. Bracci, G. Cicogna, F. Bigazzi, E. Meggiolaro,
G. Paffuti, C. Giannessi, S. Servadio, P. Christillen, K. Konishi;
A. Di Giacomo, L. Picasso, T. Elze, G. Morchio, E. D’Emilio, T. Fujimori, Y. Jiang,
M. Cipriani, A. Michelini, D. Dorigoni, S. Giacomelli, M. Taiuti, E. Ciuffoli,
C. Bonati, R. Torre, P. Giardino,
• “A survey of quantum field theory and applications”
E. Vicari, P. Rossi, E. Guadagnini, M. Campostrini, M.Mintchev,
B. Alles, P. Calabrese;
P. Menotti, C. Torrero, G. Paoletti, G. Ceccarelli, E. Profumo, M. Fagotti
• “Theoretical nuclear physics”
I. Bombaci, A. Bonaccorso, M. Viviani, A. Kievsky, L. Marcucci;
S. Rosati, R. Kumar
The Speakers
Tuesday, April 12, 2011
Fundamental Problems in Physics Today
Three “melodies” of the 20th C Theoretical Physics: (C.N.Yang 2002) Anomaly
cancellations !
“Quantization, Symmetries and Phase Factors”
“Standard Model” of the fundamental interactions
Local, renormalizable gauge theory
(of pointlike objects - the elementary particles)
➩ SU(3)QCD x (SU(2)xU(1))GWS
Quant. chromodynamics Glashow-Weinberg-Salam’s
’70-’74
Nuclear forces Electroweak theory
Orig
BUT !! Why MW / MPlanck ~
“naturalness/hierarchy” problem
10-17
η pr
mas in of
oble
s?
mν ≪
≪ m me , mu
c ≪
m ? mt ?
• Higgs? Supersymmetry ? GUTS? Origin of
• Quantum gravity? Black-hole entropy?
LHC μ problem? the universe? ν?
• New principles? New paradigm? Holographic principle? AdS/CFT? Maldacena ’97
◦ String theory?
◦ Lorentz Invariance Violation at short distances?
BIG BANG / Inflation
◦ Susy breaking? Extra dimensions?
Standard Observational Cosmology
• Cosmology Cosmology and Astroparticle physics
Ωm =0.26±0.02; ΩΛ =0.74±0.02; Ωb =0.04;
dark matter? dark energy? GRB, UHECR COBE, WMAP, SDSS, ...
• Quark Confinement (non-Abelian strong gauge dynamics) ? FERMI/LAT , AMS...
σtot ∿ log2 s ? Quark-gluon plasma, Color superconductivity?
• Quantum mechanics : fundamental aspects. Time? Schrödinger’s cat
Pn = |!n|ψ"|2 ?
Entanglement/Quantum computing
We are perhaps at the pre-dawn of a new scientific revolution
Tuesday, April 12, 2011
This presentation
Damiano
Enrico
Alessandro
Adriano and Com.
Thomas
Luc, Lui, Giam
Gianni
Francesco (B):
Ken & Com.
Daniele
Giampiero & Ken
END
Tuesday, April 12, 2011
Tuesday, April 12, 2011
Tuesday, April 12, 2011
Tuesday, April 12, 2011
Tuesday, April 12, 2011
arXiv:1012.4515 and 1009.0224
We found that electroweak corrections are relevant if DM is heavier than the
weak scale, and included them in a public code.
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Tuesday, April 12, 2011
Lorentz violating renormalizable Standard Model
Damiano Anselmi
+ Emilio Ciuffoli, Martina Taiuti and now Dario Buttazzo and Diego Redigolo
Idea: use the violation of Lorentz symmetry to renormalize interactions
that are normally nonrenormalizable
Higher powers of momenta in dispersion relations and propagators make the
integrands of Feynman diagrams more convergent in the UV
A modified power counting criterion, which assigns different weights to space and
time, controls the UV behavior and the renormalizability of the theory
Apart from violating Lorentz symmetry, the theory remains renormalizable, local,
polynomial, unitary and causal (with causality defined according to
Bogoliubov,, which only needs past and future, no light cones)
No counterterms with higher time derivatives are generated by renormalization, so
(perturbative) unitarity is safe
Since the purpose is to cure the UV behavior of otherwise nonrenormalizable
interactions, Lorentz symmetry can be recovered in the IR by a fine tuning
of parameters. It is possible to have agreement with data
Tuesday, April 12, 2011
Consider the free theory (a hat denotes time,
time a bar denotes space)
Its propagator is
and the dispersion relation reads
The improved ultraviolet behavior allows us to renormalize otherwise non-
non
renormalizable vertices. They can be classified using a weighted power counting
Tuesday, April 12, 2011An example of nonrenormalizable vertex that becomes renormalizable is
which gives neutrinos Majorana masses after symmetry breaking.
Other examples are the four-fermion vertices
at the fundamental level.
Four-fermion vertices are bounded by existing limits on proton decay.
Both vertices are compatible with a scale of Lorentz violation
(10-28 - 10-29 cm )
which agrees with present data (possibly apart from the still mysterious ultrahigh-
energy cosmic rays), if Lorentz symmetry is violated but CPT is preserved (or
broken at much larger energies)
Tuesday, April 12, 2011We can build a Standard Model extension without elementary scalars
where
The model contains four fermion interactions at the fundamental level. It is possible to
describe the known low-energy
energy physics in the Nambu—Jona-Lasinio spirit, which gives
masses to fermions and gauge bosons dynamically. The Higgs field is a composite
field and arises as a low-energy effect
An interesting low-energy
energy prediction is the formula
which is in perfect agreement with data for
Tuesday, April 12, 2011Summary of research topics and recent papers
Formulation of Lorentz Violating Stardard Model (LVSM):
D.A., Weighted power counting, neutrino masses and Lorentz violating extensions of
the Standard Model, Phys.. Rev. D 79 (2009) 025017 and arXiv:0808.3475 [hep-ph]
Scalarless LVSM and its phenomenology:
I build a version with no fundamental scalar and analyse its phenomenology
D.A., Standard Model Without Elementary Scalars And High Energy Lorentz Violation,
Violation Eur.
Phys. J. C 65 (2010) 523 and arXiv:0904.1849 [hep-ph]
[
Detailed analysis of low-energy phenomenology of scalarless LVSM:
We show that we can find agreement with all data, within theoretical errors
D.A. and E. Ciuffoli, Low-energy
energy phenomenology of scalarless Standard Model extensions
with high-energy Lorentz violation, Phys. Rev. D 83 (2011) 056005 and
arXiv:1101.2014 [hep-ph]
Experimental limits and theoretical analysis on the scale of Lorentz violation:
Here we show that is consistent with all data (at preserved CPT).
We claim that in Nature Lorentz symmetry may be broken well below the Planck scale
D.A. and M. Taiuti, Vacuum Cherenkov radiation in quantum electrodynamics with high-
energy Lorentz violation, PRD in print and arXiv:1101.2019 [hep-ph]
[
Tuesday, April 12, 2011Attività di ricerca di ENRICO MEGGIOLARO
Diffusione “soffice” ad alta energia in QCD
Diffusione “soffice” ad alta energia in QCD
Usando un approccio basato sull’integrale funzionale, le ampiezze
di diffusione elastica adrone–adrone (e.g.,! mesone–mesone), ad
√
alta energia ( s " 1 GeV) e “soffici” ( |t| ! 1 GeV), vengono
ricostruite da certe funzioni di correlazione di due “loop di Wilson”
nello spazio–tempo di Minkowski (ampiezze dipolo–dipolo).
Tuesday, April 12, 2011Attività di ricerca di ENRICO MEGGIOLARO
Diffusione “soffice” ad alta energia in QCD
In [M. Giordano, E. Meggiolaro, Phys. Rev. D 78 (2008) 074510;
Phys. Rev. D 81 (2010) 074022] il problema è stato affrontato
(per la prima volta) dal punto di vista della QCD su reticolo,
mediante un calcolo diretto (utilizzando l’infrastruttura GRID
dell’I.N.F.N.), con simulazioni Monte Carlo nella teoria di pura
gauge SU(3), della funzione di correlazione Euclidea di due
loop di Wilson, da cui l’ampiezza di diffusione mesone–mesone può
essere ricostruita mediante continuazione analitica.
[M. Giordano, E. Meggiolaro, Phys. Lett. B 675 (2009) 123-132;
M. Giordano, Tesi di Dottorato, Pisa, 20/10/2009; relatore: E. M.]
Questo è attualmente l’UNICO approccio al problema della
------------------
diffusione “soffice” adrone–adrone ad alta energia da principi primi
(QCD) e non–perturbativo.
----------------
--------------------------
=⇒ I modelli analitici testati (SVM, ILM, AdS/CFT) risultano
insoddisfacenti. Si cercano nuove forme funzionali che fittino
meglio i dati su reticolo . . .
Tuesday, April 12, 2011Attività di ricerca di ENRICO MEGGIOLARO
Diffusione “soffice” ad alta energia in QCD
[E. Meggiolaro, M. Giordano, “High–energy hadron–hadron
(dipole–dipole) scattering on the lattice”; E–print: arXiv:1010.0914
[hep–lat]; presentato da E. Meggiolaro al simposio della conferenza
HESI 2010, 10–13 agosto 2010, Kyoto, Giappone.]
. . . La speranza è quella di riuscire a spiegare il comportamento
(universale?) ad alta energia delle sezioni d’urto adrone–adrone
a partire dall’ampiezza (fondamentale) dipolo–dipolo, calcolata
nell’Euclideo: alcuni risultati preliminari sembrano condurre a
σtot (s) ∼ (ln s)2 , in accordo coi dati sperimentali (e con il
limite di Froissart) . . . [Work in progress]
Tuesday, April 12, 2011Attività di ricerca di ENRICO MEGGIOLARO
Simmetrie chirali e topologia in QCD (anche per T > 0)
Simmetrie chirali e topologia in QCD (anche per T > 0)
Si studia un modello di Lagrangiana Chirale Efficace che include
(oltre all’usuale condensato chirale !q̄q" e all’anomalia) anche un
certo condensato U(1) assiale (irriducibile) del tipo:
CU(1) ∼ ![det(q̄sR qtL ) + det(q̄sL qtR )]",
st st
che agisce come parametro d’ordine per la sola simmetria U(1)
assiale e resta diverso da zero attraverso la transizione chirale a
Tch $ 170 MeV, fino a una certa temperatura TU(1) > Tch .
=⇒ implicazioni fenomenologiche, per esempio (per T < Tch ):
i) nei decadimenti radiativi η, η ! → γγ
[M. Marchi, E. Meggiolaro, Nucl. Phys. B 665 (2003) 425;
E. Meggiolaro, Phys. Rev. D 69 (2004) 074017.]
ii) nei decadimenti forti η, η ! → 3π, η ! → ηππ
[E. Meggiolaro, E–print: arXiv:1010.1140 [hep–ph]; Phys. Rev. D
(2011), in stampa.]
Tuesday, April 12, 2011Confinement in QCD
nonperturbative methods on and off the lattice
People
Pisa: C. Bonati, A. Di Giacomo
Active collaboration: M. D’Elia, P. Incardona (Genova),
F. Sanfilippo (Roma), G. Cossu (KEK, Japan)
Starting collaboration: APE group (Roma), M. Caselle (Torino)
Main interests and recent works:
Mechanism of color confinement Nucl. Phys. B 828, 390 (2010),
vacuum dual superconductivity Phys. Rev. D 81, 085022 (2010),
through monopole condensation Phys. Rev. D 82, 094509 (2010),
JHEP 0907, 048 (2009),
QCD phase diagram
Phys. Rev. D. 82 114515 (2010),
critical points & universality arXiv:1011.4515 [hep-lat]
classes (accepted in PRD)
Tuesday, April 12, 2011Dual superconductivity & monopoles
Continuum Lattice
The gauge independence of ! The gauge dependence of the
the monopole definition was
monopole detection was
established.
3
clarified.
SU(2) gauge theory 4xNs
Wu-Yang monopole of chage 4 ! A revised version of the
0.01 monopole operator was
0 introduced.
~ )/N 1/"
-0.01
s
1. The problems of the previous
b
Ns=16
~-#
-0.02 Ns=20
implementation are solved.
(#
Ns=24
-0.03
2. Good scaling at deconfinement
-0.04
transition.
-6 -4 -2 0 2 4 6 8 10
1/"
(!-!c)Ns
Perspectives: the revised order parameter can now be
used to investigate confinement in real QCD and in
other confining theories (e.g. G2 gauge theory)
Tuesday, April 12, 2011QCD phase diagram
Nf = 2 chiral transition
∞ Study of the structure of the QCD phase
O(4)
T
1st diagram at finite temperature and density,
Z2
with particular emphasis on those aspects
cr
o
sso
Z2
of the phase diagram related to known
ve
ms
r
1st
symmetries of QCD (i.e. chiral symmetry)
0 mu ∞
Main focus: determination of the order Nf = 2 chiral transition.
Previous studies of the group, Phys. Rev. D 72, 114510 (2005),
indicated the first order nature of the transition, which is usually
believed to be 2nd order.
Huge computational resources needed!
Tuesday, April 12, 2011Computational tools
The video game market developments compelled graphic cards
manufacturers to increase the floating point calculation
performance of their products ⇒ New architecture for
computations: Graphic Processing Units (GPUs)
Need to rewrite all codes and some care is needed in optimizations
(see arXiv:1010.5433) but TOTALLY WORTH IT!
With our current implementation
1 GPU ⇐⇒ 1 − 3 apeNEXT crates
possible present alternatives (e.g. CPU clusters) lose a
factor 3 in price and 6 in power consumption
Ongoing developments:
! (short-term) parallelize the work between several GPUs
(in collaboration with the APE group)
! (long-term) fermions with improved chiral properties
(still more computationally demanding!)
Tuesday, April 12, 2011Tuesday, April 12, 2011
Tuesday, April 12, 2011
--------------------------- Tuesday, April 12, 2011
L. Bracci e L. E. Picasso: rappresentazioni algebra di
Weyl
! !
In Rn : U (α) ≡ e−i αip̂i V (β) ≡ e−i βiq̂i , U (α)V (β) = eiα·β V (β)U (α)
von Neumann: RI equivalenti; rappres. completamente riducibili.
Spazio semilimitato U (α) semigruppo isometrie, V (β) gruppo ⇒
σ(q̂) = [x0, ∞), R I con dato x0 equivalenti1. Z ≡ Centro = {λI} ⇒
H = ⊕iHi, Hi irriducibili, stesso x0. σ(q̂) è omogeneo in [x0, ∞) 4.
In generale: integrale di Hi con diversi x0 4.
Irriducibilità ⇔ irrid. rispetto agli U (α) 1. L’algebra per R, per spazio
semilimitato e quella generata da {U (α)} sono identiche 5.
Segmento U (α) isometrie parziali, U (0) = I, U (1) = 0 ⇒
σ(q̂) = [x0, x0 + 1], RI con lo stesso x0 equivalenti 1,3.
Z = {λI}
{λI} ⇒ rappresentazioni completamente
⇒ rappresentazioni riducibili33. .
completamenteriducibili
Se U!! (α), !! (n)
(α), VV (n) unitari
unitari che
che obbediscono
obbediscono Weyl,
Weyl, perper RI RIèèU!(1)
!U eiφ
(1)==eiφ I.I.
con dato
RI con dato φφ sono
sono equivalenti.
equivalenti. Se
Se U!(1)
!U =eeiφiφII èèZZ=
(1)= {λI}33. .
={λI}
Sfera
Sfera Algebra
Algebra A A generata
generata da da $n ee J$J$ èè EE33 (gruppo
n$ (gruppo euclideo
euclideo3-dim.)
3-dim.)
Le RIRI sonosono le
le RIRI (l
(l00,,0)
0) di so(3,1).
di so(3, Casimir J$J$· ·$
1). Casimir n ≡≡σσ ==±l
n$ ±l
00. . AAèè
sottoalgebra
sottoalgebra di di A ASS generata
generata da da $ $L,
n,, L,
n$ $S,
$ S, $ ⇒ ⇒perperparticella
particelladidispin
spinS, S,HH
irriducibile
irriducibile sotto
sotto A ASS,, èè H
H= =⊕ ⊕σ=S
σ=S H , H sede della RI (|σ|, 0) di A
σ=−S Hσσ , Hσσ sede della RI (|σ|, 0) di A
σ=−S
con JJ$$ ·· $
n=
n
$ = σσ 66..
Tuesday, April 12, 2011Spazio non semplicemente connesso RI non equivalenti. Nel piano
bucato, {$ q , −i∇ + f$($
q , −i∇} e {$ r)}, ∇ ∧ f$ = 0, non equivalenti se
"
$ r) · d$
γ f ($ r '= 0 ⇒ re-interpretazione effetto Aharonov-Bohm: H =
$2
p
Hlibera = 2m , ma
Φ !
! r), f = 2π!r2 (−x2, x1), γ f!d!
! = −i∇ + f (!
p ! r=Φ .
Φ #= 2nπ ⇒ effetto Aharonov-Bohm. Quindi A-B segue dall’esistenza
di RI non equivalenti. È l’osservazione che determina quale Φ (quale
rappresentazione) scegliere 2.
1) Journal Math. Phys. 47 112102 (2006)
2) American J. Phys. 75 268 (2007)
3) Bull. London Math. Soc. 39 791 (2007)
4) Lett. Math. Phys. 89 277 (2009)
5) Lett. Math. Phys. 93 267 (2010)
6) Eur. Phys. J. Plus 126 4 (2011l
G. Cicogna:
Studio analitico e algebrico di equazioni differenziali non lineari di interesse fisico.
Speciale attenzione e' dedicata alla introduzione di opportune generalizzazioni
della nozione di algebra di Lie delle simmetrie. Le applicazioni includono:
problemi nella fisica dei plasmi, fenomeni di biforcazione, comparsa di soluzioni
periodiche e/o complesse, tecniche di riduzione e di integrazione, leggi di conservazione
generalizzate.
Tuesday, April 12, 20111. Algebre di Poisson non commutative, invarianza per diffeomorfismi e
quantizzazione
Risultati:
A) MQ sulle varietà differenziabili More about it
B) Derivazione della quantizzazione di Dirac senza inconsistenze
2. Estensione delle previsioni della MQ e dise-
guaglianze di Boole-Bell
3. La matrice di scattering in QED: esistenza,
costruzione non perturbativa. Possibile costruzione dei campi carichi asintoti-
ci attraverso correzioni di stringa che superano lʼostruzione data dallʼassenza di
stati carichi a massa definita.
4. Risultati esatti su identità di Ward, topologia e simmetria chirale in QCD:
Tuesday, April 12, 2011Progetti:
- Gradi di libertà interni e diffeomorfismi
- Covarianza e invarianza per diffeomorfisimi in gravità quantistica
- Possibilità di una descrizione completa di matrice S in QED via LSZ modificato
- Implicazioni dei modelli e della costruzione LSZ generalizzata sulla localizzabilit`a e la
classificazione degli stati carichi in QED
- Implicazioni della struttura delle osservabili locali sulle identit`a di Ward del problema
U(1) e sul problema CP forte
- Parametri e gerarchie di massa in supercon-
duttivit`a oltre il BCS
Tuesday, April 12, 20113(1*-/'-)&M4+1NN4 >OH3H? !"#$%&'()* !"#$%&'"()*+,%&-)#.,/$&0#1*"#2&34/,$&56/)(4/'&7%&2/1*'&)8&&6),)+(1.64-&$#1,'9& 4:/:&;/1-)*84*/2/*"9&21''&+1.9&F#1(
Tuesday, April 12, 2011
K. Konishi and
Fujimori, Jiang, Dorigoni,
Michelini, Giacomelli, Cipriani
+ Carlino, Murayama, Spanu, Grena, Auzzi,Yung, Bolognesi,
Ferretti, Nitta, Ookouchi, Ohashi,Yokoi, Marmorini, Vinci, Eto, Gudnason, Evslin
(Armenia-Italy-Japan-USA-Russia-Denmark-China Collaboration)
• Nonabelian vortices: (non-Abelian monopoles and confinement ) ’03-’11
• “Almost conformal” vacua for confinement Auzzi, Grena, Konishi, ’03 Giacomelli
Evslin, Giacomelli
• Faddeev-Niemi decomposition for Yang-Mills theories + ’10-’11
Michelini, Konishi
• Large N, dimensionally reduced SU(N) SYM Dorigoni, Veneziano,
Wosiek
’10
Tuesday, April 12, 2011Nonabelian vortex, monopole and quark confinement
• Dirac’s quantization condition ( ’31 -- But he no longer believed it ’80) e · g = n/2, n= ±1,±2,...
• Vortex in Landau-Ginzburg theory (Abrikosov ’52, Nielsen, Olesen’74)
• ’t Hooft-Polyakov monopoles (’74) GUT? ➟ Inflation
• Confinement by monopole condensation (dual Meissner effect) (Mandelstam, ‘t Hooft ’80)
But no evidence of dynamical abelianization
Seiberg-Witten
exact solns N=2
• Nonabelian monopoles? Quantum mechanical nonabelian monopoles do appear ’94
in N=2 susy theories (Carlino, Konishi, Murayama, 2000)
• Nonabelian vortices: discovered by the Pisa group in 2003
’03-’11
➪ Rich and deep physics results Pisa,
Minnesota,
◦ 4D gauge dynamics = 2D sigma model Cambridge,
Tokyo,
◦ Vortex effective world sheet action ➭ GNO duality ... ...
◦ Vortices in high-density QCD; multicomponent superconductivity
◦ Fractional vortices
◦ Role of the Global symmetry in dual gauge group
Attracting the interest of
◦ Monopole-vortex complex soliton mathematics communitiy
Tuesday, April 12, 2011Some References
S.B. Gudnason, Y. Jiang, K. Konishi,
"Non-Abelian vortex dynamics: Effective world-sheet action".
JHEP 1008:012, 2010. (2010) e-Print: arXiv:1007.2116 [hep-th].
M. Eto, T. Fujimori, S.B. Gudnason, Y. Jiang, K. Konishi, M. Nitta, K. Ohashi,
"Group Theory of Non-Abelian Vortices".
JHEP 1011:042, (2010). e-Print: arXiv:1009.4794 [hep-th].
M. Eto, J. Evslin, K. Konishi, G. Marmorini, M. Nitta, K. Ohashi, W. Vinci, N. Yokoi (2007),
"On the moduli space of semilocal strings and lumps",
Phys. Rev. D76:105002, (2007), arXiv:0704.2218 [hep-th].
K. Konishi "The Magnetic Monopoles Seventy-Five Years Later",
Lecture Notes in Physics, (vol. 1, pp. 473-532). (2007). ISBN-10: 3540742328: Springer.
M. Eto, K. Konishi, G. Marmorini, M. Nitta, K. Ohashi, W. Vinci, N. Yokoi,
"Non-Abelian Vortices of Higher Winding Numbers",
Phys. Rev. D, vol. D74, 065021, (2006).
K. Konishi, R. Auzzi, S. Bolognesi, J. Evslin,
"NonAbelian monopoles and the vortices that confine them",
Nucl. Phys. B686, 119 (2004).
K. Konishi, R. Auzzi, S. Bolognesi, A. Yung, J. Evslin,
"Nonabelian superconductors: vortices and confinement in N=2 SQCD",
Nucl. Phys. B673, 187 (2003). e-Print: hep-th/0307287.
G. Carlino, K. Konishi, H. Murayama,
"Dynamical symmetry breaking in supersymmetric SU(n(c)) and USp(2n(c)) gauge theories",
Nucl. Phys. B 608, 51 (2001) e-Print: hep-th/0005076.
Tuesday, April 12, 201120 0
Figure 3: The four complex profile functions
20
10
0
“Fractional vortex” Eto et. al. ’09
!10
l e - Vo rtex
o
Monop plex
com origon
i,
u d n a son, D ni ’11 !20
i, G eli
Ciprian Konishi, Mich !30 !20 !10 0 10
ri,
Fujimo
Gauge profile function f ! Ρ,z" Figure 4: The behaviour of the magnetic field in the complex
Gauge profile function l! Ρ,z"
!1.0
!1.5
1.0
15
20
0.5
!2.0
20 15 !50
0.0
10 0
10
0
10
0
5 20
!10 50
30
!20 0 40
Scalar profile function s! Ρ,z"
Quark profile function q! Ρ,z"
Fig. 5: The energy (the left-most and the 2nd left panels) and the magnetic flux (the 2nd right panels) density,
1.0
20
1.0
Vortex orientational zeromodes
0.5
0.5
40
10
together with the boundary values (A, B) (the right-most panel) for the
0.00
5
minimal 0lump of the first type in the
0.0
50
20
30
10
0
strong gauge coupling limit. The moduli parameters are fixed as a1 = 0, a2 = 1, b1 = −1 in Eq. (4.18). The red
!10
10
15
!50
!20
20 0
dots are zeros of A and Figure 3:the black
The four onefunctions
complex profile is the zero of B. ξ = 1. The last figures illustrates the minimum lump
√
defined at exactly the orbifold point (see Eq. (4.20)) with Avev = 1/ 2, and with b = 0.8.
Tuesday, April 12, 2011
20Dorigoni
Dorigoniinincollaboration
collaborationwith
withVeneziano,
Veneziano,WosiekWosiek
IDEA:
IDEA:
!! Studying
StudyingQCD-Like
QCD-Liketheories
theoriesspectra
spectraininthe theLarge-N
Large-Nlimit,
limit,
!! Volume
Volumeindependence
independence++Discretized
DiscretizedLight-Cone
Light-Conequantization
quantization
⇒⇒reduces
reducescomputation
computationtotoquantum
quantummechanics
mechanicsproblem.
problem.
Model
ModelStudied:
Studied:SYM reducedtotoNN==(2,
SYM4 4reduced (2,2)2)inindd==22
Observations:
Observations:
!! String-like spectrumMMn n""TT#∆x$
String-likespectrum #∆x$n ,n ,
!! Quantized
Quantizeddistance
distancebetween partons#∆x$
betweenpartons #∆x$n .n .
|Wavefunctions|2 2inincoordinate
|Wavefunctions| coordinatespace
spacefor
fortwo twoand
andthree
threepartons:
partons:
1.0 1.0 0.5 0.5 0.5 0.5
0.8 0.8 0.4 0.4 0.4 0.4
0.6 0.6 0.3 0.3 0.3 0.3
0.4 0.4 0.2 0.2 0.2 0.2
0.2 0.2 0.1 0.1 0.1 0.1
0.0 0.0 0.0 0.0 0.0 0.0
!100!100!50 !50 0 0 50 50 100 100
!100!100!50 !50 0 0 50 50 100 100
!100!100!50 !50 0 0 50 50 100 100
0.5 0.5 0.5 0.5 0.5 0.5
0.4 0.4 0.4 0.4 0.4 0.4
0.3 0.3 0.3 0.3 0.3 0.3
0.2 0.2 0.2 0.2 0.2 0.2
0.1 0.1 0.1 0.1 0.1 0.1
0.0 0.0 0.0 0.0 0.0 0.0
!100!100!50 !50 0 0 50 50 100 100
!100!100!50 !50 0 0 50 50 100 100 !50 !50 0 0 50 50 100 100
Tuesday, April 12, 2011Giampiero Paffuti and Ken Konishi’s hobby:
Quantum Mechanics
• Generalized uncertainty relations (string theory) ’90
∆x = /∆p + ℷ ∆p ➯ Minimum physical length in Nature !
• Cyclic oscillator theorem ’06 : Microscopic QM systems cannot act as engines
• New Quantum Mechanics book (800 p. + CD), Oxford Univ. Press (’09)
Tuesday, April 12, 2011Physics of 2020
Be open minded
Tuesday, April 12, 2011Table: Gauge boson masses
Table: Quark masses
photon ugluons
(MeV) Wc±(GeV)
(GeV) Z (GeV) d (MeV)
t (GeV) s (MeV) b (GeV)
24.1 Mathematical appendices 761
0 0 80.425 ± 0.038 91.1876 ± 0.0021
1.5 − 4 1.15 − 1.35 174.3 ± 5.1 4−8 80 − 130 4.1 − 4.4
Table: Lepton masses
Table 24.10
Table 24.8
Table: Lepton masses
νe (eV)
Table: Neutrino masses νµ (MeV) ντ ( MeV)Remarks
• We are basically made of
p ∼ uud; n ∼ udd; e; γ
i.e., of u, d, e, γ, gluons
• Nevertheless, baryogenesis (CKM quark mixing, CP violation, B-violation)
➩ all quarks, leptons and gauge bosons of the Table B.T.W.
fundamental contributions
indispensable by the experimental
HE groups of Pisa
for us to be here today • the top quark discovery
• CP in K
• CP in B
Tuesday, April 12, 2011G.Morchio, F.Strocchi, C.Budroni (dottorando a Siviglia) Fondamenti della MQ e effetti non perturbativi in teorie di gauge
1. Algebre di Poisson non commutative, invarianza per diffeomorfismi e quantizzazione
Risultati: A) MQ sulle varietà differenziabili: Per ogni varietà M esiste unʼunica ∗ algebra A(M), generata dalle
funzioni f su M e dalle traslazioni infinitesime Tv lungo tutti i campi vettoriali v, con le relazioni di commutazione
di Lie tra funzioni e campi vettoriali e le relazioni di Lie-Rinehart
Tf v = 1/2(f Tv + Tv f ) .
A(M) `e invariante per diffeomorfismi. Le relazioni di Lie-Rinehart sono essenziali per la non proliferazione
dei gradi di libert`a (associati allʼalgebra di Lie infinito dimensionale dei diffeomorfismi)
Le rappresentazioni di A(M) sono tutte localmente Schroedinger (in generale con molteplici-
t`a) e sono classificate dal primo gruppo di omotopia π1(M ), che in generale non `e commutativo
e d`a perci`o origine a “fasi non abeliane”.
B) Derivazione della quantizzazione di Dirac senza inconsistenze:
A ogni varieta `e associata lʼalgebra di Poisson delle funzioni e dei campi vettoriali, con le relazioni di
Lie-Rinehart, senza altri vincoli su prodotti o commutatori. Tale algebra contiene una variabile centrale
Z , che commuta e ha Poisson nullo con tutti gli el-
ementi e che mette in relazione Poisson e commutatori:
Tv f (x) − f (x)Tv = Z {Tv , f (x)}
Z = −Z ∗. Nelle rappresentazioni irriducibili
Soli risultati possibili:
- c = 0: Meccanica Classica lagrangiana
- Meccanica quantistica come sopra con c = i
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