UNIVERSE   AS  INFORMATIONAL SUPERSPACE

 

Valentin Koulikov

 

Copyright (C) 1992

 

 

     Physical world  is a system of the interacting elementary events.  As a matter of fact,  the quantum field theory from certain point of view could be considered as an information  exchange  theory,  so  every  event  we may consider here as a fundamental sign, material unit carrying the information about itself (this particular event) [3-5]. All the other signs, symbols,etc.(in  this  paper particularly) are taken as a simple redesignation. So, Universe as  a  system  of the interacting symbols,  becomes the living Fundamental Language, transforming physical laws to the laws of grammar.

   Main principle of quantum theory tells that investigation of  any object transforms it (wave packet reduction).  One can observe  the  microwave “relic”  radiation   today,   consequently   he interacts  with  the  Universe  birth  process.  These facts  allow us to use  concepts  of  reference  experimental device or subject from the very moment of Big Bang [5-6].

    Already known object, by definition,  does not give us any information about itself,so the Man - Nature dialogue  is  the dialogue  between Subject and Unknown.  Unknown,  which exists now in  all phenomena. So, Universe  becomes  the  Space  of Language, Space of Dialogue between Intelligence and Unknown [5].

  Informational quantum, named informon have the meaning, determined by the interaction with all quantums in Universe, by the  whole  context  of  Informational  Universe, such  as  the meaning of a word in space of language.

   Let us consider something almost unknown, Universe before Big  Bang, Vacuum of information. By definition,  this Vacuum main property is  an  absolute absence of any information. Vacuum  itself is a fundamental sign,symbol of this absence, at the same time. In other words, one knows, and Vacuum, as a real phenomenon, becomes the symbol,containing this very knowledge, that there are no any information in it.  This "paradox" means that the language,constructed   with the help of such fundamental signs must be essentially recursive,the signs, words itself - self-acting, and all the situation,  in terms of physics, becomes nonlinear [7-9].

   This problem can be solved effectively by the means of finite  groups  theory.  Real  physical  objects,  processes, events,  being the fundamental signs at  the  same  time,  we consider  as a self-acting,  self-interacting group operators. Let us designate (redesignate!) Vacuum by symbol  "e".  This operator, being the sole object, formes the group of one element, namely, the unit of the group e . The cyclic group {e} has a very simple "algebra":  e• e = e .

  We have to remind that operator is acting on the object field of  Unknown, and one of the unknown objects is itself, designating Unknown as a single whole.  From this  very moment  of self-acting (e)e, Vacuum-Vacuum interacting, the absolute symmetry of  Unknown is destroyed , forming a new symmetry, symmetry  between  known (differentiated) and unknown (integrated).

   Now we have a group of order 2 with the group unit e as a symbol of unknown and (e)e - the sign of “all known”. Let redesignate elements to e0 and, e1 correspondingly, so the group becomes {e0, e1}. By the main finite group properties this group is cyclic, with e1 being a generator:  e1•e1= e0.

   The example of the group unit self-division is very important, showing the essential nonlinearity of the process, which in one or another way takes place on  each level of symmetry hierarhy. Because of the element chains degeneration,existing in cyclic groups, this process may be named a spontaneous symmetry destroying, too.

    Now we  can  consider a new object field for the group of information   transformations/operators, which includes not Vacuum only, but e1- operator of "known" too. The sign e1, really is a single sign, differentiated from  "unknown" itself, sign which accumulates now all the knowledge about "nothing", "unknown". As a material sign it is a single existing device, or a single subject we have now.

   To transform the "unknown" to the "known", e0 to e1 we must multiply e0 by e1:  e0•e1= e1. This operator "•e1=" we designate as a regular pair, a translation vector (e0, e1).  It is the sign of information producing, the sign of "answer". The inverse vector (e1, e0) appears to be the same  "•e1= " in this group, so the sign of "question" coincides with the sign "answer". Thus, the "•e1="  operator is the operator of projecting (interacting) Vacuum e0 upon base (device) e1, having e1 as a projection.  From  another  hand  we  can  consider  it  as  a projecting operator with base e0 and projection e1.  We  can say that it is an operator of cognition. The cognition here is a non-interrupting exchange of roles, playing by e0 and e1, where the continious multiplaying by e1 represents the time flow.

  This material designating cognition process is a sign of itself, differentiated from e0 and e1  . We can redesignate it as e2.  Now we have a new group of order 3: { e0, e1, e2}  . Due to finite groups properties, e2 is an inverse element to e1 , so "answer"  ( e0, e1)  as  "•e1=" and "question" (e1, e0)   as "•e2=" are different signs now.  However,  we can distinguish  the directions  of  the  information  flow  only,  and  because of continious role  exchange  (•e1= )  we  have e1•e1= e2 and e2•e2= e1 ,  that  is  to  say,  answer  is  question  from  the partner's point of view and vice versa.

     Our group time cycle is the follows:

 

            e1•e2= e1•e1•e1= e0                                                      (1)

 

   The base designating operator, as a group generator, is a sign creation operator, with e2 being a sign destruction, or a sign using operator  (we can see here, that sign is used as a real material device, as a tool).  These operators together formes the life of sign, its birth e1 and death e2. Under  vector designations we have for  "•e1= " operator:

            (e0 , e1 ) = i     - translation from present (e0 ) to past(e1 );

            (e2 , e0  ) = -j    - translation from  future (e2 ) to  present;     (2)

            (e1 , e2 ) = k     - translation from past to future;

   Here we must remind, that for the sign (subject!) itself its birth is in the past, and death is in the future, in present we have e0, only.  We have  made some natural redesignations, so:

            (e0 , e0 ) = (e1 , e1 ) = (e2 , e2 ) = 1

                                                                                                (3)

            (e0 , e2 ) = - ( e2 , e0 ), ...

where the formal unit designates the whole time/life cycle, and the sign "minus" means the inverted time. Natural multiplication rule for these  vectors leads to well-known quaternion algebra with i, j, k,  as imaginary units:

 

            i•j =(e0,e1) • (e1,e2)= (e0,e2)= k

                                                                                           (4)

             j•k = i  , k•i = j  , i•j•k = - 1 .

 

  Real unit  1, as a whole time circle, is a discrete analogue of  topological charge,  so time units count here is the count of cycles, count of charges. This count allows us to define naturally discrete  addition/subtraction  operations.Thus, i + i  = 2i  is a double count of the time cycles, started from i :

 

            2i = i • [j• (-k)• i]• [j• (-k)• i],         where  j• (-k)• i = 1

 

(is one of the possible forms of unit ). In general, for example,

 

             Nj =j• [(-k)• i• j]•...•[(-k)• i• j]     ( N times )   (5)

 

    We can see now, that time/count starting point (i , j, k, or 1, may be) is of great importance.  Using wave analogy, we may call it the phase of discrete time wave. ( Here discrete signs designate  the  time  flow,  which  can  be  not  discrete  in reality). Due to cyclic group properties there  is  the degeneration, identity of units/cycles.  So, the number of cycles is just  formal number here.  Degenerated cycles are absolutely independent and we may consider all cycles with any phases as a quaternion:

 

            Q =  ai  +  bi  +  ck ,             where a,b,c - integer numbers    (6)

 

  Time count, as we have already mentioned, is the exchange (transformation, change) of signs, reference systems, personal roles ( You / I ) of  the  subject,  too.  Difference between  the  inverse operators becomes very important in this context.  Thus,  the sign creation operator:  - i =  (e1,e0), considering as a reper vector, makes a projection of the whole time cycle vector ( 1 = (e1,e1) ) to be (e0,e1) = jk   = i , the inverse of (-i )  .  This  operator  is  the sign destruction operator,  being  an object creating  (transforming) operator at the same time. So, (-i ) transformes device and     i - object (the well-known  passive and  active form  of  one  operator), they form the time cycle  unit  vector  altogether.  Taking  a widely  used conception of co- and contravariant components, we have to redesignate reper vectors as:

 

            (i , j, k ) =(e1,e2, e3),                                           ( 7 )

            ( -i ,-j ,-k  ) = ( e1 , e2 , e3 )

 

   Thus, the   whole   mumber  of  time  cycles  (  a  whole topological charge ) one can calculate as a scalar  product of two ( co- and contravariant ) vectors (quaternions) :

 

            (P,Q) = P•Qc = piqi                                                   (8)

                                                                       

 where index ‘c’ marks quaternion  conjugation: Qc = - q1i - q2j - q3k.

  As we have seen, the material cognition process forms in material  signs  elementary conceptions of time flow and time tenses: past, present, future, gathered in topological charge, whole time cycle. To describe this process more completely, lets introduce a new isomorphic representation of  our cyclic groups,  a multiplicative groups of discrete abstract numbers.

   At first, we must notice that the transition  from  one finite group  to  a  group  of next order always occurs with formal multiplaying unit of the group (cycle)  by e -  group generator (cycle lengthening).In vector designation it appears to be multiplaying by

(e1, e0 ), or (-i ) in quaternion group. We have, actually,

 

          {e}• e =  {e , e • e} =  {e1, e0}                    - 1st transition     (9).

 

     In multiplicative representation of cycle transformation,

 

            [1•1= 1]• (-i ) --> 1• (-i )=(-i )] --> [ i • (-i )= 1]                                 (10)

 

            {e1, e0}• e =  {e , e • e, e • e • e } = {e1, e2, e3}    - 2nd transition.  (11)

                                                                    

   As a transition from  e • e  cycle to  e • e • e  cycle, we have in vector designations a new sign for e: ( e2 , e0 ) =  - k , according to group of order 3. So, in multiplicative form:

 

            [i• (-i) = 1]• (-k )  --->  [ i• (ik) = (-k )]  ---> [ i• j• (-k ) = 1]              (12)

      

                       where we designate naturally   j = - ik  .

 

Going further on this way, lets multiply this unit cycle by a new imaginary unit g,        (g•g = -1), which commutate with any other operator (easy to see). New cycle is the follows, where we use the identity  g• g •g = - g :

 

            [ij(-k )= 1]• g ---> [( gi)( gj )(- gk ) • g = 1] ---> [s1•s2•s3•g =1]   (13)

 

                     where   s1 = - gi,   s2 = - gj,   s3 = - gk

 

  One can see, that  s1•s1 = s2•s2 = s3•s3 = 1  and  i, j, k may be represented [10, 11] as

 

             i = gs1 ,  j = gs2,   k =  gs3                                                        (14).

 

   These operators forms a complex quaternion (biquaternion) finite  algebra,  where  plays  the  role  of commutative imaginary unit.  Taking into account cycles, starting from all types of reper operators, we define a biquaternion,

 

            R = (a+b g )s1 + ( c+ dg )s2 + ( e + fg )s3                                                   (15),

                        where   a, b, c, d, e, f  - integer numbers.

 

    We have  to  understand  now,  what is the meaning of new real units: s1,s2,s3. Semantically, they were created by forming an operator, inversed to the time cycle operator. This antitime, time destruction operator must have the  meaning  of memory operator, only. Mathematically, the biquaternion algebra corresponds to well-known space-time transformations  (Lorentz boosts and space rotations). Thus, time count flow leads to the cyclic  acceleration/braking  movement  and  rotation in space. The latter is known as a quantum spin [12]. We can see now, that s1,s2,s3   form  the  reper base in space (frame  of reference), while i, j, k  describes "the dimensions of time" - the base tenses.

  Fundamental space-time symmetry here exists due to symmetry destruction operator, imaginary unit g, being one of the  forms  of  group  generator e.  We know it as a duality operator, which appears to describe one of the most fundamental relations in our conception [10, 11, 14, 15].

   Because of the element chain degeneration in finite groups,  our  space-time have the unified both time and space. We may say,  that  is  only  one space-time event, or many absolutely  undifferentiated events. We have a quantum events condensate.

   Up to this time we have no information now, even the space-time is discrete or continuous.  The duality operator g and  its inversed (-g ), being the transformations from time to space and vice versa, introduce  the    difference "discrete/continuous" for the first time. Namely, the identity of the whole space-time cycles is of two different types. They are  closely connected with the two types of reference devices existing  in  space-time  duality  group  (where  g  is   the generator).  Thus,  multiplicative  unit 1  ,  taken as a reper vector,  formes  the  cycle  chains  of  almoust  degenerated, non-ordered bosonic type:

 

             1 • 1 • 1 • ... • 1 • 1 • ...                                         (16)

 

                                        where   1 =  g (-g)

 

  When we want to start from another reper vector, to  take another cycle wave phase, namely g  , we must take into account, that on the very next step the self-acting of imaginary unit g  will generate  new  anticommutative quaternion units g1, g2, g3  just as it have happened with imaginary  i and   i, j, k  .  The cycle chain will be as follows:

 

            g1• g2•(-g3) • g1• g2 • (-g3) • ... • g1• g2 •(-g3) •...             (17)

 

                                   where        g1• g2 •(-g3)  = 1

   It is easy to see,  that this one chain (space-time path, hystory,  world line) is  streamlined,  ordered.  In fact, the frame of reference,  the device with g  as a reper vector is of anticommutative, fermionic type.

   So, the choice of device shows Universe from the discrete or continuous side. This symmetry is well-known under the name "supersymmetry",  so the duality operator becomes the operator of supersymmetry at the same time.[ 1, 2].

   The space-time cycle operator designates the elementary portion  of  movement (acceleration-braking) with rotation (spin) in space-time [12].  This translation vector in both physical and linguistical meanings, this  space-time topological charge, or instanton we must call an informational quantum, informon.  The count of informons is  the  quantative count of bits, the elementary units of information. To get the information, by  definition,  is  to  make  a  choice  between absolutely identical  objects,  introducing some order into undifferentiated chaos [7-9]. The superspin operator  of  duality  g plays this one role.

  By informon  producing, g destroys the bosonic chaos, deleting antiinformon, which may be called entropon. And vice versa, the creation  of  chaos  quantum  entropon  is  the informon's destruction. Fermionic time order is the well-known property - time causality.  Bosonic space chaos,cycle identity is the main cause of quantum ocassionality,  the cause of such a fundamental fact, that the sign of a space-time cycle is not the sign of real translation,  real shift,  but is the sign of translation  chance,  only.  Informon superspace-time does not need any "quantization", it is quantum from its own birth.

  Due to degeneration of the all possible  cycles (for all  our  finite  groups),  we have a lot of chains for one and the same translation biquaternion  R:

 

            R = gs1•s3•s2 •gs2 gs3• ...             (any quaternion unit)                          (18)

 

  All this chains are of equal possibility, consequently - of equal  probability, so we have to consider some average chain, or the average between all possible chains-histories (well-known path or   history  integrating). This  averaged biquaternion   showes the density of chances to  form  one  or another reper  frame  sign.  The conjugated  biquaternion Rc showes the chance density of the frame  sign  destructing,  or the device

operation, action. Their multiplication R • Rc is the density of probability to registrate the quantum by  this  one device (registrating  means  the  both sign forming and device operating).

     To translate,  to  shift  the  informon  we must multiply biquaternion by any chain,  simply by gs3 , for example. We may consider this  shift  as  infinitezimal,  because,  as we have seen, one and the same reality may be discrete and  continious at the  same time.  So,  to describe this shift we may use the differential shift operator (which one knows under the name of momentum operator  p  (h = 1 ), taken  in the self frame of reference. Thus, due to equality of these operator forms, we have:

 

            pR  =  s0/t R = R gs3,                                         (19)

 

                                                where   s0 = 1.

   Here we have taken for  t  (self time) the count of pure time cycles ( ijk - cycles) without any tense  differentiation, such as it is taken in common use.  The simple way to take into  consideration  an  arbitrary laboratory frame   of   reference   lay  in  introducing  (new self-acting!) a new formal unit g0  . This leads to the creation of new imaginary units  g1 , g2 , g3 anticommutative with g  and g0   . The new cycle is:

 

            g1 • g2 • g3 • g • (-g0 ) =  1 ,                                                 (20)

                        where: si = g0gi  ,     gi = g0 si ,        g0 g0 = 1

 

    Using these units (  Dyrac numbers  ), we  have  in laboratory frame:

             

             g0gi /xi R = R gs3 ,                                                   (21)

 

This is the biquaternion form of Dyrac equation for the quantum of unit mass, where coordinates xi mean the numbers of cycles in different dimensions: s1(g1), s2(g2), s3(g3) and  s0(g0) correspondingly. Time dimensions (tenses) are not taken into account,  so we have only one time dimension here (g0).

     Taking into  consideration non-unit topological charge m (the number of cycles in a single  self-time  cycle),  we  get Dyrac equation for a massive quantum [10, 11]:

               

            g0gi /xi R = mR gs3 ,                                                        (22)

 

     To derive the  space-time  equations  we  must  define  a covariant (gauge) derivative, being the same shift operator, where the local  reper  frame  transformation  is  taken  into account:

 

            DR = gi(/xi + Ai) R,                                                   (23)

                                                     

  Here the  vector-biquaternions Ai describe the informon field  (space-time) - the  informational  interaction  between informons, between two local reper frames (devices, subjects).  When the  informon  field  is  present  (in  informon  or entropon forms), the particle equations becomes [11] :

 

            s3 giDi R = mR gs3                                                        (24)

 

                             where   Di= /xi + Ai

 

     Remembering that all  gi are the supersymmetry  operators, which transformes  space  to time (or bozon to fermion) we may describe the  whole  space-time  cycle  of  informon  movement (discrete or  infinitezimal  -  without any difference) as two translations, two shifts, inversed to each other:

 

            (gi Di ) • ( gj Dj )c                                                             (25)

 

                where index  c means the sign change in all gi

     This symmetry  between  two  shift   forms   (time-   and space-like)   is   the  same  dual  space-time  sypersymmetry, represented here as Dyrac operator  g . Taking into account the obvious  invariance of the space-time shift to the double-dual transformation, we can write:

      

            g [(gi Di )•(gj Dj )c ] ( -g) = (gi Di ) •( gj Dj )c                       (26)

 

This equation may be taken as a space-time  equation.  It leads to double-selfdual curvature tenzor. The main solutions of such equation are well-known space-time  instantons  (Kerr and Shwartzshield metrics,  for example).  We can say now that instantons describe the  space-time  structure  of  informons, restricted in space and in time (cyclically).

 The author  gratefully acknowledge useful and stimulating conversations with my dear friends and colleagues Dr.S.Elkin and D.Gavrilov.

 

 

 

 

 

 

 

 

 

 

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