“ classicalization ” vs. quantization of tachyonic dynamics
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“ Classicalization ” vs. Quantization of Tachyonic Dynamics. Goran S. Djordjević In cooperation with D. Dimitrijević and M. Milošević Department of Physics, Faculty of Science and Mathematics University of Niš Serbia. - PowerPoint PPT PresentationTRANSCRIPT
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
“Classicalization” vs. Quantization of Tachyonic Dynamics
Goran S. DjordjevićIn cooperation with D. Dimitrijević and M. Milošević
Department of Physics, Faculty of Science and Mathematics
University of NišSerbia
8th MATHEMATICAL PHYSICS MEETING:Summer School and Conference on Modern Mathematical Physics24 - 31 August 2014, Belgrade, Serbia
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
Outline
• Tachyons, introduction and motivation• p-Adic inflation, strings and cosmology
background• Tachyons – from field theory to the
classical analogue – “classicalization”• DBI and canonical Lagrangians• Classical and Quantum dynamics in a
zero-dimensional mode• Equivalency and canonical transformation• Instead of a Conclusion
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
Introduction
• Quantum cosmology - to describe the evolution of the universe in a very early stage.
• Related to the Planck scale - various geometries (nonarchimedean, noncommutative …).
• “Dark energy” effect - expansion of the Universe is accelerating.
• Different inflationary scenarios.
• Despite some evident problems such as a non-sufficiently long period of inflation, tachyon-driven scenarios remain highly interesting for study.
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
p-Adic inflation from p-strings
• p-Adic string theory was defined (Volovich, Freund, Olson (1987); Witten at al (1987,1988)) replacing integrals over R (in the expressions for various amplitudes in ordinary bosonic open string theory) by integrals over , with appropriate measure, and standard norms by the p-adic one.
• This leads to an exact action in d dimensions, , .
,1
1
2
1 142
42
22
pm
p
s
pexd
g
mS p
t
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
p-Adic inflation (from strings)
• The dimensionless scalar field describes the open string tachyon.
• is the string mass scale and • is the open string coupling constant • Note, that the theory has sense for any integer and
make sense in the limit
sm
sg
1p
1
11 2
22
p
p
gg sp
.ln
2 22
p
mm sp
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
p-Adic inflation
• Potential:• Rolling tachyons
V
-1 1
0.2
0.4
p =19
12
2
4
1
1
2
1 p
p
s
pg
mV
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
Tachyons
• String theory• A. Sen’s effective theory for tachyonic field:
• - tachyon field• - potential• Non-standard type Lagrangian
1 ( ) 1 ijnS d xV T g T Ti j
g
( )T x
)(TV
00g 1 , ,...,1 n
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
In general
• DBI Lagrangian:
• Equation of motion (EoM):
• EoM for spatially homogenous field:
( , ) ( ) 1tach T T V T g T T L L
22
1(1 ( ) )
1 ( ) ( )
T T dVg T T T
T V T dT
21 1( ) ( )
( ) ( )
dV dVT t T t
V T dT V T dT
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
In general
• Lagrangian for spatially homogenous field:
• Conjugated momentum:
• Conserved Hamiltonian:
2( , ) ( ) 1tach T T V T T L
2( )
1tach T
P V TT T
L
2 2( , ) ( )tach T P P V T H
( , )0tach Td P
dt
Hd
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
Relation with cosmology
• Lagrangian (again):
• Cosmological fluid described by the tachyonic scalar field:
• Energy density and pressure:( )
tach
V T
1 T T
p L
p w w const
2( , ) ( ) 1tach T T V T T L
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
Canonical transformation
• How to quantize the system – Archimedean vs non-Archimedean case!?
• Classical canonical transformation
• Form of the generating function:
• - new field, - old momentum
2 ( , ) ( )F T P PF T
T P
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
Canonical transformation
• Connections:
• Jacobian:
• Poisson brackets:
2
2
( )
( )
FT F T
P
F dF TP P
T dT
1( )
1
( )
T F T
P PdF TdT
2
1( , )
11( , )
0
F F PT P FJT P
F
. . . .{ , } { , } 1P B P BT P T P
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
Canonical transformation
• Hamiltons’ equations:
• EoM:
2 2 2
2 22 2 2 2
1
1 1 ( )( )
PT
F P F V
dV FP F V F P
F dFP F V
2log ( ) 1 log ( )0
F d V F d V FT F T
F dF F dF
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
(smart) Choice for
• If is integrable:
• lower limit of the integral is chosen arbitrary
• Second term in the EoM vanishes:
( )F T
1( )V T
1( )( )
T dTF T
V T
log ( )0
F d V FF
F dF
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
(smart) Choice for
• EoM:
• Two mostly used potentials:
( )F T
1 log ( )0
d V FT
F dF
( ) TV T e
1( )
cosh( )V T
T
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
Example 1
• Exponential potential: • Function becomes• Leads to• Full generating function:
• EoM:• Classically equivalent (canonical) Lagrangian:
( ) ,TV T e const 1( )F T 1 1
( ) TF T e
1
( ) ln( )F T T
2 ( , ) ( ) ln( )P
F T P PF T T
2 0T T
2 2 21 1( , )
2 2quad T T T T L
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
Example 2
• “One over cosh” potential:
• Leads to
• Full generating function:
• EoM:• Classically equivalent (canonical) Lagrangian:
1( ) ,
cosh( )V T const
T
1( ) arcsinh( )F T T
2 ( , ) ( ) arcsinh( )P
F T P PF T T
2 0T T
2 2 21 1( , )
2 2quad T T T T L
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
p-Adic case, numbers…
• p – prime number• - field of p-adic numbers
• Ostrowski: Only two nonequivalent norms over and
• Reach mathematical analysis over
pQ
QRQ ||||
pQQ p ||
||||
Q| |p
pQ
pb
ap p||
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
``Non-Archimedean`` – p-adic spaces
• Compact group of 3-adic integers Z3 (black dotes)
• The chosen elements are mapped (R)
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
p-Adic QM
• Feynman’s p-adic kernel of the evolution operator operatora
• Aditive character – • Rational part of p-adic number – • Semi-classical expression also hold in the p-adic
case
)()0,;,(0
T
pp LdtDyyTyK
)}{2exp()( pp xix
px}{
1/22 2
2 1 2 12 1 2 1
1( , ; ,0) ( , ; ,0)
2( )c c
p p p c
p
S SK y T y S y T y
y y y y
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
p-Adic QM
• Lagrangian:
• Action:
• - elapsed time• Initial and final configuration: ,
2 22 1 1 2 1 2( , , ,0) coth( ) 2 csch( )
2cS y T y y y T y y T
2 2 21 1( , )
2 2y y y y L
T1y 2y
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
p-Adic QM
• The propagator:
• Group property (evolutionary chain rule or Chapman-Kolmogorov equation) holds in general:
– (Reminder: the infinitesimal version of this expression is the celebrated Schrödinger equation).
3 3 2 2 2 2 1 1 2 3 3 1 1( , ; , ) ( , ; , ) ( , ; , )p
p p pQK y T y T K y T y T dy K y T y T
/
( , ; , )( ) ( )
coth( ) ( )
( )
( )
1 2
p 2 1 p
p
2 2p 1 2 1 2
K y T y 02sinh T sinh T
y y T 2y y csch T2
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
p-Adic QM ground state
• The necessary condition for the existence of a p-adic (adelic) quantum model is the existence of a p-adic quantum-mechanical ground (vacuum) state:
• Characteristic function of p-adic integers:
1, if | | 1(| | )
0, if | | 1p
pp
yy
y
( ) (| | )vacp py y
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
p-Adic QM ground state
• Basic properties of the propagator
• Leads to
2 1 1 1 2( , ; ,0) ( ) ( )p
vac vacp p pQ
K y T y y dy y
12 1 1 2| | 1
( , ; ,0) (| | )p
p pyK y T y dy y
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
p-Adic QM ground state
• Necessary conditions for the existence of ground states in the form of the characteristic Ω-function
• Interpretation
2 22
| | 1,( ) (| | ), for
| | 1, | | 1pvac
p pp p
Ty y
T y T
( ) ( ) ( ) (| | )a p p p pp M p M
y y y y
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
Conclusion
• Tachyonic fields can be quantized on Archimedean and non-Archimedean spaces.
• Dynamics of the systems are described via path integral approach
• Classical analogue of the tachyonic fields on homogenous spaces is inverted oscillator lake system(s), in case of exponential like potentials.
• How to calculate the wave function of the Universe with ``quantum tachyon fluid`` … ?
• ``Baby`` Universe?
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
Acknowledgement
• The financial support under the
ICTP & SEENET-MTP Network Project PRJ-09 “Cosmology and Strings” and
the Serbian Ministry for Education, Science and Technological Development projects No 176021, No 174020 and No 43011. are kindly acknowledged
• A part of this work is supported by CERN TH under a short term grant for G.S.Dj.
.
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
Reference1. G.S. Djordjevic and Lj. Nesic
TACHYON-LIKE MECHANISM IN QUANTUM COSMOLOGY AND INFLATIOin Modern trends in Strings, Cosmology and ParticlesMonographs Series: Publications of the AOB, Belgrade (2010) 75-93
2. G.S. Djordjevic, d. Dimitrijevic and M. MilosevicON TACHYON DYNAMICSunder consideration in RRP
3. D.D. Dimitrijevic, G.S. Djordjevic and Lj. NesicQUANTUM COSMOLOGY AND TACHYONSFortschritte der Physik, Spec. Vol. 56, No. 4-5 (2008) 412-417
4. G.S. Djordjevic}}, B. Dragovich and Lj.NesicADELIC PATH INTEGRALS FOR QUADRATIC ACTIONSInfinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 6, No. 2 (2003) 179-195
5. G.S. Djordjevic, B. Dragovich, Lj.Nesic and I.V. Volovichp-ADIC AND ADELIC MINISUPERSPACE QUANTUM COSMOLOGYInt. J. Mod. Phys. A17 (2002) 1413-1433
6. G.S. Djordjevic}}, B. Dragovich and Lj. Nesicp-ADIC AND ADELIC FREE RELATIVISTIC PARTICLEMod. Phys. Lett.} A14 (1999) 317-325
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
Reference7. G. S. Djordjevic, Lj. Nesic and D Radovancevic
A New Look at the Milne Universe and Its Ground State Wave FunctionsROMANIAN JOURNAL OF PHYSICS, (2013), vol. 58 br. 5-6, str. 560-572
8. D. D. Dimitrijevic and M. Milosevic: In: AIP Conf. Proc. 1472, 41 (2012).
9. G.S. Djordjevic and B. Dragovichp-ADIC PATH INTEGRALS FOR QUADRATIC ACTIONSMod. Phys. Lett. A12, No. 20 (1997) 1455-1463
10. G.S. Djordjevic, B. Dragovich and Lj. Nesicp-ADIC QUANTUM COSMOLOGY,Nucl. Phys. B Proc. Sup. 104}(2002) 197-200
11. G.S. Djordjevic and B. Dragovichp-ADIC AND ADELIC HARMONIC OSCILLATOR WITH TIME-DEPENDENT FREQUENCYTheor.Math.Phys. 124 (2000) 1059-1067
12. D. Dimitrijevic, G.S. Djordjevic and Lj. NesicON GREEN FUNCTION FOR THE FREE PARTICLEFilomat 21:2 (2007) 251-260
8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics, 24 - 31 August 2014, Belgrade, Serbia
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