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NOTES ON ULTRAMETRIC
EXTRA DIMENSIONS
Goran S. Djordjevic
Faculty of Science
University of Nis, Serbia
In collaboration with
D. Dimitrijevic and Lj. Nesic
 The old idea that macroscopic 4-dimensional universe in which
we exist, is product of multidimensional space-time attract
many attention nowadays. In such models compactification
1
extra dimensions play the key role and in the some of them
leads to the period of accelerated expansion of universe.
 For this model we construct corresponding p-adic model and
show existence of p-adic ground state wave functions. In
addition we explore quantum evolution of the model and
possibility for its adelic generalization. It is necessary for the
investigation of space-time discreteness at very short
distances.
P-ADIC NUMBERS AND ADELES
- Concept (notion) of distance.
- We measure distances (between real and rational numbers)
using the absolute value on the real numbers, (natural metric on
the field of the real numbers R).
- How did the real numbers enter analysis?
- The field of real numbers R is the result of completing the field
of rationals Q with the respect to the usual absolute value |.|.
- The field Q is Causchi incomplete with respect to the usual
absolute value |.|
{1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, …}
(A sequence {xn } is called a Caushy sequence with the respect to the norm
|.| if it satisfies the following property: Given any a>0, there exists some N
such that m,n > N implies | xm - xn | < a. Basically, a sequence is Caushy if its
terms became “arbitrarily close” with respect to the norm |.|)
- A map |.| from the rationals to the non-negative reals is called a
norm (or valuations) if it satisfies the three following conditions:
(1) | x | 0 if and only if x  0
(2) x, y we have | xy || x || y | (norm of product is product of
norms)
(3) x, y we have | x  y || x |  | y | (the triangle inequality)
2
- The usual absolute value |.| clearly satisfies these properties,
but what other kinds of norms can exist?
- There is trivial norm |x|=1 for all rationals x except 0, with |0|=0.
- Besides the usual completion of field of rationals (which leads
to the R) there is a non-obvious completion of rationals . These
are the p-adic fields Qp where p is some fixed prime number
(Kurt Hensel in 1902).
- Each p-adic field Qp is defined by completing Q with respect to
the “new” norm |.|p which is defined as follows:
-
m
where m and n are nonzero integers,
n
neither divisible by the prime p, and k is an integer
0  x Q , x  pk
| x | p  p k .
- If we further define |0|p=0, then that |.|p satisfies the necessary
conditions above to be a norm on Q.
- The set of equivalence classes of Caushy (with respect to |.|p )
sequences has a natural field structure. It is a completion of Q
which we call the field of p-adic numbers
3
NONARCHIMEDEAN NORM
- There is a stronger inequality for an absolute value |.| than
triangle inequality which is known as the ultrametric inequality
or strong triangle inequality:
| x  y | max{| x |, | y |}
- Any norm |.| satisfying this is called nonarchimedean or
ultrametric.
- The usual norm on the real line is clearly archimedean – in fact
there is an archimedean axiom:
“Any given large segment of a straight line can be surpassed by
successive addition of small segments along the same line.”
- A more formal statement of the axiom would be that if
0  | x |  | y | then there is some positive integer n such that
| nx || y | .
Can we measure distances as small as we want?
- It is very difficult to imagine a situation where this axiom does
not hold, but the archimedean axiom breaks down at the Planck
scale, (10-33 m, 10-44 s)
- .
The properties of space and time:
- continuous,
- homogeneous,
- infinitely divisible quantities,
- Below this scale, distances and durations cannot scaled up in
order to produce macroscopic distances and durations.
4
- In other words, we must abandon the archimedean axiom at
very small distances.
- How can one construct a physical theory corresponding to a
non-archimedean geometry?
geometry  number system
- What should be used instead of real numbers?
- In computations in everyday life, in scientific experiments and
on computers we are dealing with integers and fractions, that is
with rational numbers and we newer have dealings with
irrational numbers.
- Results of any practical action we can express only in terms of
rational numbers which are considered to have been given to
us by God. What norms do exist on Q?
- There is a remarkable Ostrowski theorem describing all norms
on Q. According to this theorem: any nontrivial norm on Q is
equivalent to either ordinary absolute value or p-adic norm for
some fixed prime number p.
- p-adic fields are nonarchimedean, it is natural to consider
formulations of physics in terms of Qp rather than R
-
“Which p do we use?”
- The adeles constitute a locally compact topological ring AQ,
individually taking the form
(a ; a2 , a3 , a5 , )
In all but a finite number of cases a p is a p-adic integer.
- There are primary two kinds of analyses on Qp : Qp  Qp
(class.) and Qp  C (quant.).
5
p-ADIC FUNCTIONS AND INTEGRATION
- Usual complex valued functions of p-adic variable, which are
employed in mathematical physics, are :
(i)
an additive character  p ( x)  exp 2i{x} p , where {x} p is the
fractional part of x  Q p ,
(ii)
multiplicative character  s ( x) | x |sp , where s  C , and
(iii)
locally constant functions with compact support, like
(| x | p ) , where
1 | x | p  1
.
(| x | p )  
0
|
x
|

1
p

- There is well defined Haar measure and integration. Important
integrals are
1
 Q  p (ayx)dx   p (ay) | a | p  p ( y), a  0
p
 Q  p (x
p
2
 x)dx   p ( ) | 2
|p1 / 2
 2 
,   0 ,
 p  

 4 
- The number theoretic function  p (x) is a map  p : Q*p  C
defined as follows:

m2j
p2
 1,
 x
 p ( x)   p , m  2 j  1 p  1(mod 4) ,
 x
i p , m  2 j  1 p  3(mod 4)


0
0


2 ( x)  


1
2
1
[1  (1) x1 i]
2
x1  x 2
x1
(1)
m2j
[1  (1) i] m  2 j  1
,
6
x  p m ( x0  x1 p  x2 p 2  ) , and m, j  Z .
x0
- ( p ) is the Legandre symbol defined by
 1 if
a 
    1 if
 p 
 0 if
a  y 2 (mod p)
a  y 2 (mod p) .
a  0(mod p )
Properties of  p -function
 p (a 2 x)   p ( x),  p ( x) p ( x)  1,  p ( x) p ( y)   p ( x  y) p ( x 1  y 1 )
|  p ( x) |  1, a  0.
Real analogues of integrals
1
 Q   (ayx)dx    (ay) | a |   ( y), a  0

 Q   (x

2
 x)dx   ( ) | 2
Q  R,
1 / 2
|
 2 
,   0 ,
   

4



  ( x)  exp( 2ix)
 ( x) 
sign x
i
7
QUANTUM MECHANICS ON ADELIC SPACES
Reasons to use p-adic numbers and adeles:

the field of rational numbers Q , which contains all
observational and experimental numerical data, is a dense
subfield not only in R but also in the fields of p-adic numbers Qp

there is an analysis within and over Qp like that one related to R

general mathematical methods and fundamental physical laws
should be invariant [I.V. Volovich, Number theory as
the ultimate physical theory, CERN preprint,
CERN-TH.4781/87 (July 1987)] under an interchange of
the number fields R and Qp

there is a quantum gravity uncertainty ( x  l0  G c ), when
measures distances around the Planck length l0 , which restricts
priority of Archimedean geometry based on the real numbers
and gives rise to employment of non-Archimedean geometry
related to p-adic numbers
3

it seems to be quite reasonable to extend standard Feynman's
path integral method to non-Archimedean spaces,

adelic quantum mechanics [B. Dragovich, Adelic Model
of Harmonic Oscillator, Theor,. Math. Phys, 101
(1994) 1404-1412] is consistent with all the above
assertions.
Adelic quantum mechanics
( L2 ( A), W ( z ), U (t ))
 L2 ( A) - adelic Hilbert space,
8
 W (z ) - Weyl quantization of complex-valued functions on
adelic classical phase space,
 U (t ) - unitary representation of an adelic evolution operator
on L2 ( A)
Dynamics of p-adic quantum model?
U p (t ) ( p) ( x)   Q Kt( p) ( x, y) ( p) ( y)dy
p
p-adic quantum mechanics is given by a triple
( L2 (Qp ),Wp ( z p ),U p (t p ))
Adelic evolution operator U (t ) is defined by
U (t ) ( x)  A Kt ( x, y) ( y)dy 

v  , 2,3,..., p,...
Q
v
Ktv ( xv , yv ) (v) ( yv )dyv
The eigenvalue problem for U (t ) reads
U (t )  ( x)   ( E t )  ( x)
The main problem in our approach is computation of p-adic
propagator K p ( x" , t"; x' , t ' ) in Feynman's path integral method, i.e.
( x", t ")
t"
K p ( x" , t"; x' , t ' )   ( x ',t ')  p ( 1h  t ' L(q , q, t )) Dq
Exact general expression for propagator
K p ( x" , y" , t" ; x' , y' , t ' )   p ( 21h x'Sx" ) 
2
1/ 2
1 2S
p
h x ' x" p
 S ( x" , t"; x' , t ' )
9
   1  S  1  S  1  S 
  2 x 'x"
2 x 'y"
2 x 'z "  
K p ( x" , y" , z" , t"; x' , y ' , z ' , t ' )   p  det  12 y 'Sx"  12 y 'Sy"  12 y 'Sz"  
 

   1  S  1  S  1  S 
2 z ' y"
2 z ' z "  
  2 z 'x"
2
2
2
2
2
2
2
2
2
1/ 2
  S   S   S 
x ' y"
x ' z " 
 x 'x"
 S
 S

 det  y 'x"  y 'y"  y 'Sz"   p  S ( x" , y" , z" , t"; x' , y ' , z ' t ' ) 


  S   S   S 
z 'y"
z 'z " 
 z 'x"
p
2
2
2
2
2
2
2
2
2
where S ( x", y", z", t"; x' , y' , z' t ' ) is the action for classical trajectory.
Illustration of p-adic and adelic quantum mechanical models,
 a free particle and harmonic oscillator [VVZ, Dragovich]
 a particle in a constant field, [Djordjevic, Dragovich]
 a free relativistic particle[Djordjevic,
Nesic]
Dragovich,
 a harmonic oscillator with time-dependent
[Djordjevic, Dragovich]
frequency
Resume of AQM: AQM takes in account ordinary as well as p-adic
effects and may me regarded as a starting point for construction of
more complete quantum cosmology and string/M theory. In the low
energy limit AQM effectively becomes the ordinary one.
An adelic wave function has form
    ( x )   pM  p ( x p )   pM (| x | p )
10
ADELIC QUANTUM COSMOLOGY
- The main task of AQC is to describe the very early stage in the
evolution of the Universe.
- At this stage, the Universe was in a quantum state, which
should be described by a wave function (complex valued and
depends on some real parameters).
-
But, QC is related to Planck scale phenomena - it is natural to
reconsider its foundations.
- We maintain here the standard point of view that the wave
function takes complex values, but we treat its arguments in a
more complete way!
- We regard space-time coordinates, gravitational and matter
fields to be adelic, i.e. they have real as well as p-adic
properties simultaneously.
- There is no Schroedinger and Wheeler-De Witt equation for
cosmological models.
- Feynman’s path integral method was exploited [Drag. Nes.] and
minisuperspace cosmological models are investigated as a
model of adelic quantum mechanics [Drag. Djordj. Nes. Vol.].
- Adelic minisuperspace quantum cosmology is an application of
adelic quantum mechanics to the cosmological models.
- Path integral approach to standard quantum cosmology, the
starting point is Feynman’s path integral method
 hij'' ,  '' , '' | hij' ,  ' , '    D( g  ) D( )   (S [ g  ,  ])
The standard 3+1 decomposition
ds2  g  dx dx  ( N 2  Ni N i )dt 2  2 Ni dxi dt  hij dxi dx j
p-adic complex valued cosmological amplitude
 hij'' ,  '' , '' | hij' ,  ' , '  p   D( g  ) p D( ) p  p (S p [ g  ,  ])
11
(4+D)-DIMENSIONAL COSMOLOGICAL MODELS OVER THE
FIELD OF REAL NUMBERS
4+D – dimensional Kaluza-Klein cosmology with RW type metric
having two scale factors a and R corresponding to a D-dimensional
internal space and four dimensional universe {t , r i ,  a } . r i - space
coordinates,  a  internal space coordinates.
dri dr i
da d a
~2 2
2
2
ds   N dt  R (t )
 a (t )
2 2
2 2
kr
1

k
'

1 4
2




This model is describing an accelerating universe with dynamical
compactification of extra dimensions.
 The form of the energy-momentum tensor is
TAB  diag (  , p, p, p, pD, pD, ..., pD ),
If we want the matter is to be confined to the four-dimensional
universe, we set all pD  0 .
We assume the energy-momentum tensor of spacetime to be an
exotic  fluid with the equation of state
m 
p    1  
3 
Dimensionally extended Einstein-Hilbert action (without a higherdimensional cosmological term) is
~
S    g R dt d 3R d D   Sm    dtL
1
D( D  1) 3 D  2 2 D 2 D 1 
D 2
a  ~ R a Ra
~ Ra R 
~ R a
2N
12 N
2N
1 ~
1~
 kNRa D  N  R3a D
2
6
For closed universe ( k  1), continuity equation
L
12
  R  3( p     ) R  0
energy density
m
R 
  ( R)    ( R0 ) 0  .
 R
If we define     (R)
1
D( D  1) 3 D  2 2 D 2 D 1 
L  ~ Ra D R 2 
a  ~ R a Ra
~ R a
2N
12 N
2N
1~
1~
 NRa D  NR3a D
2
6
Growth of the scaling factor R , leads to the decrease of the
cosmological constant
( R)  ( R0 )( R0 / R) m .
(the universe evolves form its small to large size, the large initial
value of  decays to small values)
~
2
If we take m  2 , ( R0 ) R0  3 , N (t )  R (t )a (t ) N ,
3
D
1 R 2 D( D  1) a 2
D R a
L


2N R2
12 N a 2 2 N Ra
Classical solutions
R(t )  Ae t , a(t )  Be  t
For R(0)  a(0)  lP
R(t )  lP et , a(t )  lP e t
For D  1
R(t )  lP e Ht , a(t )  lPe Ht , H  R / R  Huble
parameter
accelerating (de Sitter) universe and a contracting internal space with
exactly same rates.
13
for D  1
R(t )  lPe Ht , a(t )  lP
R(t )  lP
2 Ht
[ 1 1 2 / 3(11 / D ) ]1
e D
Dt
[ 1 1 2 / 3(11 / D ) ]
2
e
,
a(t )  lPe t ,
Quantum solutions
Wheeler-DeWitt equation
H ( R, a)  0
X  ln R , Y  ln a

2
6 2
  
(
D

1
)


6

 ( X , Y )  0
2
2
D

X

Y

X

Y


xX
3
D
X Y
Y
, y
D3
D3
D3

2 D  2 2 
 ( x, y )  0
 3 2 
2
D

x

y


with four possible solutions
 D ( x, y)  Ae
 / 3x  D /( D  2) y
 D ( x, y)  B e
 / 3x  D /( D  2) y
,
,
14
(4+D)-DIMENSIONAL MODEL OVER THE FIELD OF P-ADIC
NUMBERS
Lagrangian of the model
L
1  2 D( D  1)  2 D  
X 
Y 
XY
2N
12 N
2N
Solutions
X  C1t  C2 , Y (t )  C3 t  C4
Classical action
S ( X ", Y ", N ; X ' , Y ' ,0)
1
D( D  1)
D

( X " X ' ) 2 
(Y "Y ' ) 2 
( X " X ' )(Y "Y ' )
2N
12 N
2N
Kernel of p-adic operator of evolution
K p ( X ", Y ", N ; X ' , Y ' ,0)
 D( D  2)  D( D  2)
 p 

 48N 2  12 N 2
change x  X
1/ 2
 p ( S ( X ", Y ", N ; X ' , Y ' ,0))
p
3
D
X Y
Y
, y
D3
D3
D3
S ( x", y", N ; x' , y ' ,0)
1
D( D  5)
1

(1 
)( x" x' ) 2 
D( D  3)( y" y ' ) 2
2N
6
2N
 6  D( D  5)   D( D  3) 
K p ( x", y", N ; x' , y ' ,0)   p 
 p  

12 N
2N 

 
1/ 2
D( D  3)  D( D  5) 

1 
  p ( S ).
2
6

p
2N
15
p-adic ground state wave function in (x,y) minisuperspace
 p ( x, y)  (| x | p )(| y | p )
Conditions
| N | p | 1  D( D  5) / 6 | p
| N | p | D( D  3) | p , p  2.
p-adic ground state wave function in (X,Y) minisuperspace
 p ( X , Y )   | (1 

D
D
  X Y 
)X 
Y | p  |
|p .
D3
D3
D

3
 

16
CONCLUDING REMARKS
It is shown that there is possibility of construction adequate p-adic
(4+D)-Kaluza-Klein model.
It is shown that there exist adelic ground states of the form
 ( x, y)   D , ( x , y ) pM  p ( x p , y p ) pM (| x p | p )(| y p | p )
- Adopting the usual probability interpretation of the wave
function, we have
2

2
2
|  ( x, y) |
|  D
, ( x , y ) |  pM |  p ( x p , y p ) |  pM (| x p | p )(| y p | p )
((| u | p )) 2  (| u | p )
- At the rational points x and y and for M  

2
|  D
( x, y) |
2 
,

|  ( x, y ) |  


0
x, y  Z
x, y  Q \ Z
- Discretization of minisuperspace coordinates
17