p-adic and adelic gravity and cosmology - …userpage.fu-berlin.de/~aschmitt/dragovich.pdf ·...

p-ADIC AND ADELIC GRAVITY ANDCOSMOLOGY

Branko Dragovichhttp://www.phy.bg.ac.yu/˜ dragovich

[email protected]

Institute of PhysicsUniversity of Belgrade

Belgrade, Serbia

ISAAC2013: 9th International CongressSession 22: Analytic Methods in Complex Geometry

August 5-9, 2013Cracow

Cracow - 2013 B. Dragovich ISAAC2013: 9th International Congress 1/36

Contents

1 Introduction2 Numbers and their applications

p-adic numbers, adeles and their functionsrational numbers and measurementsp-adic numbers, adeles and their applications

3 p-Adic and adelic gravityp-adic and adelic string theoryp-adic and adelic Einstein gravity

4 p-Adic and adelic cosmologyp-adic and adelic quantum cosmologyp-adic origin of dark matter and dark energy ?

5 Concluding remarks


1. Introduction: p-adic mathematical physics(1987)

p-adic and adelic string theoryp-adic and adelic quantum mechanics and (quantum) fieldtheoryp-adic and adelic gravity and (quantum) cosmologyp-adic and adelic space-time structure at the Planck scalep-adic and adelic dynamical systemsp-adic stochastic processesp-adic aspects of information theoryp-adic structure of the genetic codep-adic protein dynamicsp-adic wavelets, ...


1. Introduction: international interdisciplinaryjournal “p-Adic Numbers, Ultrametric Analysis andApplications”


1. Introduction: international conferences onp-adic mathematical physics

Workshop on p-adic methods in modeling complexsystems, Bielefeld, April 2013.

2nd International Conference on p-Adic Mathematical Physics Belgrade, 2005


2. p-Adic numbers, adeles and their functions

discovered by Kurt Hensel (1861-1941) in 1897.Any p-adic number (x ∈ Qp ) has a unique canonicalrepresentation

x = pν(x)+∞∑n=0

xn pn , ν(x) ∈ Z , xn ∈ 0, 1, · · · , p − 1



p-adic numbers have not natural orderingp-adic numbers cannot be completely visualized in realEuclidean space: partial visualization by trees and fractals

2‐ADIC TREE

All triangles are isosceles. There is no partial intersection of balls. Any point of a ball is its center.

. .



Ostrowski theorem: Any nontrivial norm on Q is equivalentto usual absolute value or to p-adic norm, where p is anyprime number (p = 2,3,5,7,11, . . . ) .p-adic norm of x ∈ Q: |x |p = |pν a

b |p = p−ν , ν ∈ Z.Q is dense in Qp and R.Completion of Q with respect to p-adic distance gives thefield Qp of p-adic numbers, in analogous way toconstruction of the field R of real numbers.

N Z Q



There are mainly two kinds of analysis of p-adic variable:(i) p-adic valued functions of p-adic variable(ii) complex (real) valued functions of p-adic variable.Usual complex-valued functions of p-adic argument are:

πs(x) = |x |sp, χp(x) = exp 2πixp, Ω(|x |p) =

1, |x |p ≤ 10, |x |p > 1.

Analysis of complex (real) valued functions of p-adic (andreal) variables is necessary to connect models withmeasurements.



Real and p-adic numbers are unified by adeles. An adele αis an infinite sequence

α = (α∞, α2, α3, · · · , αp , · · · ) , α∞ ∈ R , αp ∈ Qp

where for all but a finite set P of primes p one has thatαp ∈ Zp = x ∈ Qp : |x |p ≤ 1, i.e. p-adic integers.Space of adeles

A =⋃P

A(P) , A(P) = R×∏p∈P

Qp ×∏p 6∈P

Zp .



Connection of p-adic and real properties of rational numbers

|x |∞ ×∏p∈P|x |p = 1 , if x ∈ Q×

χ∞(x)×∏p∈P

χp(x) = 1 , if x ∈ Q

χ∞(x) = exp(−2πix), χp(x) = exp 2πixp


2. Rational numbers and measurements

Any measurement can be viewed as measurement of adistance.Measurement of a distance is comparison of its length withunit length.Result of measurement is a rational number with an error.This rational number is a real, but not a p-adic, number.Measurement is related to the Archimedes axiom ingeometry.According to quantum gravity one cannot measuredistances smaller than the the Planck length:

∆x ≥ `P =√

G~c3 ∼ 10−33cm. This is limit for application of

real numbers in micro-world and an window for applicationof p-adic numbers and adeles.


2. p-Adic numbers, adeles and their applications

N Z Q

Applications must contain complex (real) valued functions ofp-adic (or adelic) argument. At a hidden level applications canalso contain p-adic (or adelic) valued functions of p-adic (oradelic) valued arguments.

At the Planck scale and string theory.Gravity theory and cosmology.In bioinformation systems – genetic code, ....In very complex systems with hierarchical structure.Some other applications.


3. p-Adic and adelic string theory

Volovich, Vladimirov, Freund, Witten, Dragovich, ...

Electron (e−) and positron (e+) scattering in quantum fieldtheory and string theory.



String amplitudes:standard crossing symmetric Veneziano amplitude

A∞(a,b) = g2∞

∫R|x |a−1∞ |1− x |b−1

∞ d∞x

= g2∞ζ(1− a)

ζ(a)

ζ(1− b)

ζ(b)

ζ(1− c)

ζ(c)

p-adic crossing symmetric Veneziano amplitude

Ap(a,b) = g2p

∫Qp

|x |a−1p |1− x |b−1

p dpx

= g2p

1− pa−1

1− p−a1− pb−1

1− p−b1− pc−1

1− p−c

where a = −s/2− 1 and a,b, c ∈ C and a + b + c = 1.



Freund-Witten product formula for adelic strings

A(a,b) = A∞(a,b)∏

p

Ap(a,b) = g2∞∏

p

g2p = const .

Amplitude for real string A∞(a,b), which is a specialfunction, can be presented as product of inverse p-adicamplitudes, which are elementary functions.



There is an effective field description of scalar open andclosed p-adic strings. The corresponding Lagrangians arevery simple and exact. They describe not only four-pointscattering amplitudes but also all higher (Koba-Nielsen)ones at the tree-level.The exact tree-level Lagrangian for effective scalar field ϕwhich describes open p-adic string tachyon is

Lp =mD

p

g2p

p2

p − 1

[− 1

2ϕp−

2m2p ϕ+

1p + 1

ϕp+1]

where p is any prime number, = −∂2t +∇2 is the

D-dimensional d’Alembertian and metric with signature(− + ...+) (Freund, Witten, Frampton, Okada, ...) .


3. p-Adic and adelic Einstein gravity

Einstein Theory of Gravity (ETG) and Quantum Theory liein foundation of modern theoretical physics.General theory of relativity is Einstein theory of gravity..At the cosmic scale there is only gravitational force(interaction). Hence gravitational force governs dynamicsof the Universe as a whole.Cosmology is a science about the Universe as a whole.There are cosmic observational results which have notgenerally accepted explanation. Two of these observationsare: accelerated expansion of the Universe and largevelocities of stars in spiral galaxies.



In ETG gravity is presented in terms of thepseudo-Riemannian geometry.In (pseudo-)Riemannian geometry distance is defined byds2 = gµν dxµ dxν with respect to some system ofcoordinates.From metric tensor (gµν) one can obtain any informationabout Riemannian space.

gµν → Γαµν → Rαβµν → Rµν → R

Christoffel symbol

Γαµν =12

gαβ(∂µ gβν + ∂ν gµβ − ∂β gµν)



Riemann-Christoffel curvature tensor

Rαβµν = ∂µΓαβν − ∂νΓαµβ + ΓαµγΓγβν − ΓανγΓγµβ

Ricci tensor and Ricci scalar

Rµν = Rαµνα , R = gµν Rµν

Einstein equations for gravity field (metric tensor)

Rµν −12

R gµν =8πGc4 Tµν − Λ gµν

Einstein-Hilbert action (g is determinant of gµν )

S =1

16πG

∫ √−g(R − 2Λ)d4x , (c = 1)



One can formally consider main ingredients of(pseudo)Riemannian geometry and Einstein theory ofgravity as p-adic valued. It has not direct physical meaning,but can be used in argument of the path integrals.We now consider Quantum Cosmology (QC). In QC weare interested in wave function of the Universe, whichcontains quantum information about the Universe at itsvery early stage of evolution.We use adelic quantum mechanics to obtain adelic wavefunction of the Universe.As a simple example we consider the de Sitter model ofthe Universe.



Adelic space-time and adelic (pseudo)Riemanniangeometry

α = (α∞, α2, α3, · · · , αp , · · · ) , α∞ ∈ R , αp ∈ Qp

where for all but a finite set P of primes p one has thatαp ∈ Zp = x ∈ Qp : |x |p ≤ 1, i.e. p-adic integers.Adele α can be space-time point or any ingredient of(pseudo)Riemannian geometry, including gravity action.Space of adeles

A =⋃P

A(P) , A(P) = R×∏p∈P

Qp ×∏p 6∈P

Zp .


4. p-Adic and adelic quantum cosmology

Suppose that at the Planck scale (10−33cm) there are realand p-adic strings, and that the space-time is adelic. Thenit means that the Universe at the very beginning was adelicand it gives rise to consider adelic quantum cosmology.Adelic quantum cosmology is application of adelicquantum mechanics to the cosmological models.Adelic quantum mechanics can be viewed as a triple(L2(A),W ,U(t)), where L2(A) is the Hilbert space on A, Wmeans Weyl quantization, and U(t) is unitary evolutionoperator on L2(A).U(t) can be expressed trough its kernel K(x

′′, t

′′; x ′, t ′)

ΨP(x′′, t

′′) =

∫K(x

′′, t

′′; x ′, t ′)ΨP(x ′, t ′)dx ′,



where

ΨP(x′′, t

′′) = ψ∞(x∞, t∞)

∏p∈P

ψp(xp, tp)∏p/∈P

Ω(|xp|p))

is adelic eigenfunction, and

K(x′′, t

′′; x ′, t ′) = K∞(x

′′∞, t

′′∞; x ′∞, t

′∞)∏

p

Kp(x′′p , t

′′p ; x ′p, t

′p)

=∏

v

Kv (x′′v , t

′′v ; x ′v , t

′v )

is adelic transition probability amplitude.



K(x′′, t

′′; x ′, t ′) is naturally realized by Feynman’s path integral

K(x′′, t

′′; x ′, t ′) =

∫χA

(−1

hSA[q]

)DAq

=∏

v

∫χv

(−1

h

∫ t′′v

t ′vL(qv ,qv ) dtv

)Dqv

Vacuum state Ω(|x ′′p |p) has property

Ω(|x ′′p |p) =

∫Qp

Kp(x′′p , t

′′p ; x ′p, t

′p)Ω(|x ′p|p)dx ′p =

∫Zp

Kp(x′′p , t

′′p ; x ′p, t

′p)dx ′p



Some interesting results of adelic quantum modeling:

Connection between adelic harmonic oscillator andRiemann zeta functionDiscreteness of space and time at the Planck scale.



The Einstein-Hilbert action for the de Sitter model of theUniverse is

S =1

16πG

∫M

d4x√−g(R − 2Λ)− 1

8πG

∫∂M

d3x√

hK

Large scale space is homogenous and isotropic,anddescribed by the FLRW metric

ds2 = σ2[− N2dt2 + a2(t)

( dr2

1− kr2 + r2(dθ2 + sin2 θdϕ2))]

ds2 = σ2[− N2

q(t)dt2 + q(t)dΩ2

3] , σ2 =2G3π



The corresponding action is

Sv [q] =12

∫ t ′′

t ′dtN

(− q2

4N2 − λq + 1), λ =

2ΛG9π

For N = 1, the equation of motion q = 2λ has solution

q(t) = λt2 +(q′′ − q′

T− λT

)t + q′ ,

which presents classical trajectory, whereq′′ = q(t ′′), q′ = q(t ′),T = t ′′ − t ′. This solution resemblesthe motion of a particle in a constant field.



The corresponding action for classical trajectory (path) is

Sv (q′′,T ; q′,0) =λ2T 3

24−[λ(q′+q′′)−2]

T4−(q′′ − q′)2

8T, v =∞,2,3, ...

The corresponding transition amplitude is

Kv (q′′,T ; q′,0) =λv (−8T )

|4T |12v

χv (−Sv (q′′,T ; q′,0))

The corresponding vacuum state Ω(|q|p) follows from∫|q′|p≤1

Kp(q′′,T ; q′,0) dq′ = Ω(|q′′|p)



The p-adic Hartle-Hawking wave function is given by

Ψp(|q|p) =

∫|T |p≤1

dTλp(−8T )

|4T |12p

χp

(− λ2T 3

24[λq − 2]

T4

+q2

8T

)and gives vacuum state Ω(|q|p) with the conditionλ = 3 · 4 · `, ` ∈ Z.Existence of the vacuum state for all (or almost all primes pis a necessary condition for a model to be adelic).Any adelic quantum model provides some discreteness. Inadelic quantum cosmology length of discreteness is thePlanck length.



Discreteness at the Planck scale

Ψ(q) = Ψ∞(q)∏

p

Ω(|q|p) =

Ψ∞(q), q ∈ Z0, q ∈ Q \ Z.

It means q = n · `P , where `P =√

~Gc3 ∼ 10−35m.


4. p-Adic and adelic origin of dark matter and darkenergy

If Einstein general theory of relativity is theory of gravity forthe entire Universe then there is only about 5% of ordinarymatter and 95% of matter in the Universe of unknownnature, i.e 95% of the Universe is dark.Dark matter (27%) should be responsible for rotationvelocities of galaxies.Dark energy (68%) was introduced in 1998 as a possibilityto explain Universe expansion acceleration.It is possible that there is some p-adic matter in theUniverse, and that dark matter and dark energy havep-adic origin, i.e. origin in p-adic strings.



In the last decade there are a lot of papers related tomodification of the Einstein theory of gravity to findalternative (without dark energy) explanation ofaccelerated expansion of the Universe.Nonlocal modified gravity is one of attractive approaches togeneralization of Einstein theory of gravity, and itsmotivation is also in p-adic string theory.


5. Concluding remarks: Main references

B O O K S1 V.S. Vladimirov, I.V. Volovich and E.I. Zelenov, p-Adic

Analysis and Mathematical Physics (World Scientific,Singapore, 1994).

2 F.G. Gouvea, p-Adic Numbers (Springer - Verlag, 1993)3 S. Katok, Real and p-Adic Analysis (2001).4 W. Schikhof, Ultrametric Calculus (Cambridge Univ. Press,

Cambridge, 1984).5 A. M. Robert, A Course in p-adic Analysis (Springer-Verlag,

2000).6 K. Mahler, p-Adic Numbers and Their Functions

(Cambridge Univ. Press, Cambridge, 1981)7 N. Koeblitz, p-Adic Numbers, p-Adic Analysis and Zeta

Functions (Springer-Verlag, 1984)8 I. M. Gelf’and, M. I. Graev and I. I. Pyatetskii-Shapiro,

Representation Theory and Automorphic Functions(Saunders, Philadelphia, i969).

9 A. Weil, Adeles and Algebraic Groups (Birkhauser, Basel,1982).


5. Concluding remarks: Main references

R E V I E W A R T I C L E S1 L. Brekke and P.G.O. Freund, “p-Adic numbers in physics,”

Physics Reports 233 (1), 1–66 (1993).2 B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and

I. V. Volovich, “On p-adic mathematical physics,” p-AdicNumbers, Ultrametric Analysis and Applications 1 (1), 1–17(2009); arXiv:0904.4205 [math-ph].

S O M E P A P E R S1 Aref’eva I. Ya., Dragovich B., Frampton P. H. and Volovich I.

V., The wave function of the Universe and p-adic gravity,Int. J. Mod. Phys. A 6, 4341–4358 (1991).

2 Dragovich B., Adelic harmonic oscillator, Int. J. Mod. Phys.A 10, 2349–2365 (1995); arXiv:hep-th/0404160.

3 Dragovich B., p-Adic and adelic quantum cosmology:p-Adic origin of dark energy and dark matter, AIP Conf.Series 826, 25–42 (2006); arXiv:hep-th/0602044.

4 Djordjevic G. S., Dragovich B., Nesic Lj. D. and Volovich I.V., p-Adic and adelic minisuperspace quantum cosmology,Int. J. Mod. Phys. A 17, 1413–1433 (2002);arXiv:gr-qc/0105050.Cracow - 2013 B. Dragovich ISAAC2013: 9th International Congress 35/36

5. Concluding remarks

Numerical results of experiments are real rational numbers– they are not p-adic numbers.There are some interesting and promising applications ofp-adic analysis in p-adic mathematical physics (stringtheory, gravity and cosmology, ...) and biology (geneticcode,...).We conjecture that p-adic numbers will play significant rolein some complex systems, like complex numbers play rolein quantum theory.


p-adic and adelic gravity and cosmology - …userpage.fu-berlin.de/~aschmitt/dragovich.pdf ·...

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