TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Tachyonic and Localy Equivalent
Canonical Lagrangians
– The Polynomial Case -
Dragoljub D. Dimitrijevic,
Goran S. Djordjevic and
Milan Milosevic
Department of Physics, Faculty of Science and Mathematics,
University of Nis, Serbia
TIM14 Physics Conference – Physics without frontiers
SEENET-MTP Workshop
20-22 November, 2014, Timisoara, Romania
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Outline
• Introduction and Motivation
• Tachyons and Non-standard Lagrangians
• Canonical Transformation
• Few examples of Tachyonic Potentials
• Conclusion
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Introduction and Motivation
• Quantum cosmology: to describe the evolution of the
universe in a very early stage.
• One of the most challenging period of the evolution of
the Universe - inflation period
• Tachyonic inflation, in particular on non-archimedean (p-
adic) spaces is a very interesting and actual topic
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Tachyons and Non-standard
Lagrangians
• The field theory of tachyon matter proposed by A. Sen
• We study non-standard Lagrangian of DBI type
– It contains potential as a multiplicative factor, and a term with
derivatives:
– T - tachyonic scalar field; V(T) – potential; - components of the
metric tensor
• Zero-dimensional version:
– Simple mechanical analog of the field theory of tachyon, suggested by
S. Kar
– a model of a particle moving in a constant external field with quadratic
"damping"-like term, however Hamiltonian is conserved
– dozen of interesting classical - toy models
( , ) ( ) 1tach T T V T g T T
L L
g
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Tachyons and Non-standard
Lagrangians
• Zero-dimensional version: 𝑥𝑖 𝑡, 𝑇 𝑥, 𝑉(𝑇) 𝑉(𝑥)
• Lagrangian for spatially homogenous field:
• Equation of motion for spatially homogenous field:
• Action:
• The task to quantize the system is a non-trivial one
2( , ) ( ) 1tach x x V x x L
1 1( ) ( )
( ) ( )
dV dVx t x t
V x dx V x dx
2( ) 1c tach V x tS t tL d d
“toy” model
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Non-Uniqueness of Lagrangian
• Classical mechanics
– Different Lagrangians lead to the same equation of motion
• Quantum mechanics
– Very old problem
– We should be concerned, but...
• we are going to apply our model for a very short period
of time, beginning of inflation, where a "local
equivalence„ of Lagrangians should be a reasonable
assumption
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
„Old“ method
• Z. E. Musielak gave an algorithm for writing a standard
Lagrangian for a given equation of motion
• For example, equation of motion is given as
• It can be obtained from the standard type Lagrangian
• Where
2( ) ( ) ( ) ( ) 0x t b x x t g x
2 2 ( ) 2 ( )1( , ) ( )
2
x
I x I x
stL x x x e g x e dx
( ) ( )
x
I x b x dx
Z. E. Musielak J. Phys. A: Math. Theor. 41 (2008) 055205
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Canonical Transformations
• Classical canonical transformation is a change of the phase
space pair of variables to a new pair
– Preserves the Poisson bracket:
– Jacobian of the canonical transformation
• Unitary transformations of field 𝑥 and conjugate momenta 𝑝
• Do not change a form of Hamiltonian equations:
( , )px( , )x p{ , } { , } 1xx p p
1J
, ,x p x p
( , ) ( , )tachtach x p x pH H
( , ) ( , )tachtach x p x px x
p p
H H
( , ) ( , )tachtach x p x pp p
x x
H H
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Canonical Transformations
• Generating function of canonical transformation:
• 𝐹( 𝑥) arbitrary (for now) function of a new field.
• Connections:
Old coordinates 𝑥 Old and new variables
and new momenta 𝑝
( , ) ( )G x p pF x
( )
( )
Gx F x
p
G dF xp p
x dx
1( ) ( )
'
1 ( ), '
x F F xx x
dF
xF
xp p F
d
Inverse function
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Canonical Transformations
• Equation of motion is transformed to:
• We are free to choose 𝐹 𝑥 , so
• Equation of motion simplifies to
2ln ( ) 1 ln ( )0
F d V F d V Fx F x
F dF F dF
0
1( )( )
x
x
dxF x
V x
1 ln ( )0
d V Fx
F dF
Arbitrary
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Canonical Transformations
• Equation of motion coresponds to a standard type of
Lagrangian:
• Where:
21( , )
2quadL x x x W
2
1
2 ( )W
V F
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
A few interesting potentials
𝑽 𝒙 = 𝒆−𝜶𝒙𝟏
𝒄𝒐𝒔𝒉(𝜷𝒙)
𝐹−1 𝑥 =1
𝛼𝒆𝜶𝒙
1
𝛽sinh(𝛽𝑥)
𝐹 𝑥 =1
𝛼ln(𝛼 𝑥)
1
𝛽arcsinh(𝛽 𝑥)
𝐺 𝑥, 𝑃 = −𝑃𝐹 𝑥 = −𝑃
𝛼ln(𝛼 𝑥) −
𝑃
𝛽arcsinh(𝛽 𝑥)
Equation of motion 𝑥 − 𝛼2 𝑥 = 0 𝑥 − 𝛽2 𝑥 = 0
ℒ𝑞𝑢𝑎𝑑 𝑥, 𝑥 =1
2 𝑥2 +
1
2𝛼2 𝑥
1
2 𝑥2 +
1
2𝛽2 𝑥
Inverse harmonic oscillator
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Polinomial potential 𝑉 𝑥 = 𝑥−𝑛
• In general case 𝑛 ∈ 𝑅, however only 𝑛 > 0 is tachyonic
• We choose
• Full generating function:
• Equation of motion:
• Unfortunately solution include elimptic integral and
hypergeometric function
11( )
1
nxF x
n
1
1( , ) ( ) (1 ) nG x p pF x p n x
1
1( 1) 0n
nx n n x
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Potential 𝑉 𝑥 =1
𝑥
• Special case 𝑛 = 1 𝑉 𝑥 =1
𝑥
• Equation of motion for tacyonic Lagrangian
𝐿 = −𝑉 𝑥 1 − 𝑥2 is
• Solution:
• Tachyonic Lagrangian
2 ( ) 3 ( ) 4( ) ( ) 0
( ) 1 ( )
x t x tx t x t
x t x t
1
2
2
( )C
eC t
Cx t
t
1 2 1 2
2
1 2 2 1 22 2
T T x T x T x xL
t T x x tT T x Tx T x
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Potential 𝑉 𝑥 =1
𝑥
Canonical transformation
• We choose:
• Full generating function:
• EoM is reduced to:
• Quadratic Lagrangian:
1 21( )
( ) 2
xdx
F x xV x
( , ) ( ) 2G x p pF x p x
( ) 1 0x t
21( , )
2quadL x x x x
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Potential 𝑉 𝑥 =1
𝑥
• Solution:
• For initial and final conditions: 𝑥 0 = 𝑥1 and 𝑥 𝑇 = 𝑥2solution is:
2
2 1 1
1( )
2
tx t t tT x x x
T
2
2 1
1( )
2x t t C t C
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Potential 𝑉 𝑥 =1
𝑥
• The action:
• Quantisation - the propagator:
0
32
1 2 1 2
1
2 2 24
T
cl quad
cl
S L dt
T TS y y y y
T
2
2 1
1 2
2
24
1 2 1 2
1
2
1, ; ,0
2
, ;12 12
exp2
,4
02
clSi
cl
i
SK y T y e
i y y
KT T
yy y y yi
T TT y
C. Morette:
Phys. Rev. 81
(1951), 848.
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Conclusion
• We started with the tachonic (DBI type) Lagrangian - highly
nonlinear and not suitable for quantization
• It has been showed it is possible to find a locally equivalent
standard/canonic Lagrangian applying (local) canonical
transformation
• We made a review of two well-known tachyonic models potential,
discussed polynomial ones, and calculate the unique analyticaly
solvable
• We calculate the corresponding propagator. P-adic and adleic
generalization were done – will be presented elsewhere
• We proceed with application of this results towards their application
in quantum cosmology and their FRW limits.
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Acknowledgement
• SEENET-MTP Project PRJ-09 “Cosmology and Strings”
• Serbian Ministry for Education, Science and
Technological Development under projects No 176021,
No 174020 and No 43011.
• We are thankfull to the organizers of TIM14 for
hospitality
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Reference
1. G.S. Djordjevic and Lj. NesicTACHYON-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 and Lj. NesicCOSMOLOGY - FROM CLASSICAL TO QUANTUM
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 TopicsVol. 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
TIM14 Physics Conference – Physics without frontiers, 20-22 November, 2014, Timisoara, Romania
Reference
7. G. S. Djordjevic, Lj. Nesic and D RadovancevicA 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