tachyonic and localy equivalent canonical lagrangians - the polynomial case -

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


Outline

• Introduction and Motivation

• Tachyons and Non-standard Lagrangians

• Canonical Transformation

• Few examples of Tachyonic Potentials

• Conclusion


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


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


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


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


„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


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



• 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



• 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



• Equation of motion coresponds to a standard type of

Lagrangian:

• Where:

21( , )

2quadL x x x W

2

1

2 ( )W

V F


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


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


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



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



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



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.


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.


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


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


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

tachyonic and localy equivalent canonical lagrangians - the polynomial case -

Science

x x e g x e

t x t v x dx v x dx2

ltach x

x b x dxz

tach t t v t g t t

dt v t x dt

homogenous field

given equation of motion