Two-dimensional illustrations related to the
many-sheeted space-time concept
Matti Pitkänen (January 20, 2003)
Department of Physical Sciences, High Energy Physics Division, PL 64, FIN-00014,
University of Helsinki, Finland.
Kadermonkatu 16, 10900, Hanko, Finland
A. TGD based spacetime concept
The starting point of TGD is the 'energy problem' of General Relativity. By
Noether's theorem, conservation laws are in one-one correspondence with
symmetries. In particular, translational invariance of the empty Minkowski space
M^4 implies energy and momentum conservation in Special Relativity. By the basic
postulate of General Relativity matter makes spacetime curved. This means that
the symmetries of the empty Minkowski space are lost as are lost also the
corresponding conservation laws, in particular the conservation of energy.
The basic idea of TGD is to assume that spacetime is representable as a
surface of some higher dimensional space H=M^4xS and that translational
symmetries, and more generally, Poincare invariance correspond to the symmetries
of M^4-factor of this higher-dimensional space rather than those of spacetime
itself. Hence a fusion of Special and General Relativities in a well defined
sense is in question. In fact mathematical and physical reasons force to replace
empty Minkowski space M^4 with its light cone M^4_+. Future light cone
corresponds to empty Robertson Walker cosmology and TGD inspired cosmology has
subcritical mass density as a consequence. There is small cosmological breaking
of Poincare invariance since M^4 is replaced by its lightcone.
Fig. 1. Matter makes spacetime curved and spoils translational invariance.
By physical constraints (elementary particle spectrum) the space S must be
CP_2, the complex projective space of two complex (four real) dimensions. The
size of CP_2 is about 10^4 Planck lengths (roughly 10^(-30) meters). [It took
long time to realize that the original assumption about size of order Planck
length was not correct].
Fig. 2. Future light cone of Minkowski space.
Fig. 3. CP2 is obtained by identifying all points of C^3, space having 3
complex dimensions, which differ by a complex scaling Lambda: z is identified
CP2 can also be regarded as a coset space SU(3)/U(2), U(2)=SU(2)xU(1). What
this means is that one starts from the 8-dimensional group SU(3) of unitary 3x3
matrices of determinant one and identifies all matrices which differ by a left
multiplication by an element of the 4-dimensional subgroup U(2). One can also
say that each point of SU(3) is obtained from CP2 by replacing each point of CP2
with the group U(2). U(2) in turn can be regarded as S^3xS^1, that is as the
space obtained by replacing each 2-dimensional disk giving a cross section of
the ordinary torus (doughnut) with a 3-dimensional sphere S^3. By this
construction CP2 is so called symmetric space, whose all points are equivalent
metrically (like those of Euclidian space) and has color group SU(3) as its
group of distance preserving transformations, isometries.
Fig. 4. H= M^4_+xCP_2 is obtained by replacing each point of the future light
cone with the 4-dimensional compact space CP_2 of size R of order 10^4 Planck
lengths (10^(-30) meters).
The second manner to end up with TGD is to start from the old fashioned
string model, which also served as a starting point of super string models,
which have been in fashion during the last ten years.
Mesons are strongly interacting particles and string model description was in
terms of a string with quark and antiquark attached to the ends of the string. A
problem was encountered in an attempt to generalize this description to apply to
baryons which consist of three quarks. One cannot put 3 quarks to the ends of
the string since it has only two ends.
Fig. 5. The transition from hadronic string model to TGD.
The solution of the problem is simple. Replace one-dimensional strings with
small 3-dimensional surfaces. Since the ends of the string correspond to the
boundaries of a one-dimensional manifold they correspond in 3-dimensional case
boundaries of small holes drilled in 3-dimensional space. Put quarks on these
boundaries. In 3-dimensional case one can drill arbitrary number of these holes
so that also baryons can be described in this kind of model.
TGD based spacetime concept differs in many crucial aspects from the
conventional one. In the following this difference is visualized by replacing
3-dimensional space (now surface in H) with two-dimensional surface whereas
8-dimensional imbedding space is replaced with 3-dimensional slab of thickness
of order 10^4 Planck lengths. This simplification makes it possible to
illustrate the most essential aspects of the generalization easily and at least
geometrically/topologically oriented reader can guess the rest.
Below is a general view of what many-sheeted 3-space would look like if it
Fig. 6. This is what 3-space would look if it were a 2-dimensional surface in
3-dimensional slab of thickness of order 10^4 Planck lengths.
B. Elementary particles as 3-surfaces of size of order R=10^4 Planck
lengths: CP2 extremals.
Elementary particles have geometric representation as so called CP2 type
extremals. Instead of standard imbedding of CP2 as a surface of M^4+xCP2
obtained by putting Minkowski coordinates m^k constant
one considers 'warped' imbedding
m^k =f^k(u) u is arbitrary function of CP2 coordinates with the property that
the M4_+ projection of the surface is random light like curve:
mkl dm^k/du dm^l/du =0, mkl is flat M4 metric. (A)
The condition implies that induced metric is just CP2 metric, which is
Euclidian! The curve is random and therefore one has classical nondetermism:
this makes sense since the solution is vacuum extremal.
Fig. 7. The projection of CP2 type extremal to M4+ is lightlike curve.
Elementary particles correspond to CP2 type extremals with holes: the
intersection of bound with m^0=const hyperplane is sphere, torus, sphere with
two handles, etc...: shortly a surface with genus g=0,1,2,... . Different
fermion families correspond to different genera. Bosons are also predicted to
have family replication phenomenon.
Fig. 8. Different fermion families correspond to different genera for the
boundary component of CP2 type extremal.
Feynmann diagrams correspond to topological sums of CP2 type extremals: the
lines of diagram being thickened to CP2 type extremals:
Fig. 9. Feynmann diagrams correspond to connected sums of CP2 type extremals:
each line of Feynmann diagram is thickened to CP2 type extremal.
The quantum version of the condition (A) stating that the M4_+ projection is
light like curve leads to Super Virasoro conditions and it turns out that
elementary particles together with their 10^(-4) Planck mass excitations belong
to representation of p-adic Super Virasoro and Kac Moody. The p-adic mass
calculations lead to excellent predictions for particle masses.
C. Induced gauge field concept implies radical generalization of spacetime
The concept of connection geometrizes the concept of the parallel translation
appearing already in elementary geometry. Parallel translation can be performed
for vectors, tensors, spinors,... In Euclidian case parallel translation is just
what one would imagine it to be and the parallel translation around a closed
curve brings the vector back without any change in its direction. The formal
definition of the parallel translation along a geodesic line (the counterpart of
a straight line) in a more general context requires that the angle between the
vector and geodesic line is preserved. Sphere is a simple example of a situation
in which parallel translation around a closed curve changes the direction of the
vector. One says that sphere is curved and curvature is locally measured by the
amount of change in the direction of a vector for very small geodesic triangle.
Fig. 10. Parallel translation on sphere and on plane.
In General Relativity, the so called Riemann connection defining the parallel
translation in spacetime leads to a beautiful geometrization of the
gravitational interaction. The presence of matter makes spacetime curved and
geodesics are not straight lines anymore. The advent of the gauge theories led
to a partial geometrization of the boson fields. The components of the gauge
potentials can be regarded as components of a connection defining parallel
The problem is however that there is not direct geometric interpretation for
this parallel translation and here TGD provides the final geometrization of
classical gauge field concept. The components of electroweak gauge potentials
are obtained as projections of the spinor connection of CP2 to spacetime
Amu = A_k partialmu h^k
(mu is the coordinate index for spacetime coordinates, k for imbedding space
or geometrically; parallel translation on spacetime surface is performed
using spinor connection of the imbedding space.
Fig. 11. Classical electroweak gauge potentials at spacetime surface are
obtained as projections of the components of CP2 spinor connection.
Classical color gauge potentials are identified as projections of Killing
vector fields of su(3) to spacetime surface (very much like in Kaluza-Klein
theories). The requirement that standard model electroweak gauge group allows
only M^4_+xCP2 as imbedding space. Also standard model quantum numbers are
geometrized in terms of CP2 geometry and topology of the boundary component of
CP2 type extremal. The special features of CP2, in particular the fact that it
does not allow standard spinor structure, are crucial for obtaining the coupling
structure of standard model.
The induced gauge field concept differs from the ordinary one. The PRIMARY
dynamical variables are the four CP2 coordinates and this implies strong
constraints among classical gauge fields: for instance classical electric field
is often accompanied by classical Z0 field even in macroscopic length scales.
There is rather precise metaphor making possible to understand the concept of
induced gauge field intuitively. The shadow (projection!) of a nondynamical solid
object ( < --> metric and spinor connection of H) with time-independent
size and shape to a surface (<--> 3-surface) changing its size and shape
is dynamical .
Fig. 12. Classical electroweak gauge potentials at spacetime surface are
obtained as projections of the components of CP2 spinor connection.
Even more importantly: gauge potentials are determined by the image of map
whereas ordinary gauge potentials are determined by the map
X^4 --> TM^4,
where TM^4 is the space of field values at given point of spacetime and
isomorphic to tangent space of M4. CP2 is COMPACT whereas TM^4 is NONCOMPACT: a
This implies that general electromagnetic gauge potentials are imbeddable
only in some open region surrounding given point of spacetime and that the
imbeddability fails at the boundary of this region.
Fig. 13. The Maxwell field associated with a given charge distribution is
representable as induced gauge field only in a finite region of spacetime. This
implies the presence of boundaries. Two-dimensional illustration.
The failure of imbeddability leads to generation of spacetime BOUNDARIES in
all lengths scales. At the boundary spacetime simply ends.
There is following problem on the boundaries: Kähler electric gauge flux
must be conserved on the boundary. Since the 3-space ends at boundary there is
no other manner to cope with situation than to introduce second, larger,
spacetime sheet parallel to the first one and allow the gauge flux to run on
this spacetime sheet via tiny wormholes connecting the two sheets. Wormhole is
constructed by drilling tiny spherical holes inside the two parallel spacetime
sheets and connecting the boundaries S^2 of the holes with a cylinder S^2 xI
having two ends with S^2 topology. The figure below illustrates the situation if
3-space were two-dimensional.
Fig. 14. Charged wormholes feed the electromagnetic gauge flux to the 'lower'
By adding these wormholes on the boundary of 3-surfaces the gauge flux can
flow to the lower spacetime sheet. An interesting possibility is that wormholes
are itself slightly deformed pieces of CP2 type extremals.
The throats of wormhole behave as classical charges -Q and Q , where Q is the
electric gauge flux flowing to the wormhole at upper spacetime sheet and out of
it at lower spacetime sheet. Thus they serve as currents and sources (of
opposite sign) of classical gauge fields at the two spacetime sheets.
Fig. 15. The two throats of wormhole behave as classical charges of opposite
Wormholes couple to the DIFFERENCE OF CLASSICAL GAUGE POTENTIALS associated
with the two spacetime sheets since the classical charges are opposite.
It seems safe to assume that photons (the extremely small CP2 type extremals!)
see wormholes from wider perspective that is extremely small dipoles formed by
the throats. The distance between charges is of order 10^4 Planck lengths and
the direction of dipole is transversal to spacetime surface so that polarization
vector has very small projection in M^4, where polarization vector of photon is.
Thus the coupling to photons should be negligible (dipole moment satisfies p
> Q*R, R the size of CP_2, and thus also dissipation effects.
Fig. 16. As for as coupling to photons is considered wormoholes are expected
to behave as extremely tiny dipoles.
This suggests very stronly that wormholes behaves much like conduction
electrons and are thus localized to the boundaries of spatime surface. If
wormholes are light (as they turn out to be) they obey d'Alembert type equation
and there is large energy gap between ground state and excited states. Thus
wormholes become suffer BE condensation to ground state: charged wormholes
behave thus much like super-conductors.
Wormholes can have several topologies: in general one can drill holes of
genus g (sphere, torus, etc...) on two spacetime sheets and connect them using
cylinder I+genus g surface. In this they resemble ordinary elementary particles
which have also several genera (family replication phenomenon).
The gauge flux conservation problem is encountered also in the lower
spacetime sheet and one must introduce third, fourth, etc. spacetime sheet and
in general one has hierarchy of spacetime sheets with increasing sizes.
Fig. 17. Many sheeted spacetime structure results from the requirement of
gauge flux conservation.
The conclusion is that induced gauge field concept leads unavoidably to the
concept of many sheeted spacetime. This has radical consequences for the
structure of physical theory: one must replace thermodynamics, hydrodynamics,
etc with a hierachy of dynamics of various types, one for each spacetime sheet
in the hierarchy. This replacement must be performed in ALL length scales.
D. Matter as topology
Since many sheetedness is encountered in all length scales a very attractive
manner to reinterpret our visual experience about world suggests itself.
Material objects having macroscopic boundaries correspond actually to sheets of
3-space and 3-space literally ends at the boundary of object. The 3-space
outside the object corresponds to the 'lower' spacetime sheet. Actually we can
see this wild 3-topology every moment!! The following 2-dimensional illustration
should make clear what the generalization really means.
Fig. 18. Matter as topology
E. Join along boundaries contacts and join along boundaries condensate
The receipe for constructing many-sheeted 3-space is simple. Take 3-surfaces
with boundaries, glue them by topological sum to larger 3-surfaces, glue these
3-surfaces in turn on even larger 3-surfaces, etc.. The smallest 3-surfaces
correspond to CP2 type extremals that is elementary particles and they are at
the top of hierachy. In this manner You get quarks, hadrons, nuclei, atoms,
molecules,... cells, organs, ..., stars, ..,galaxies, etc...
Besides this one can also glue different 3-surfaces together by tubes
connecting their BOUNDARIES : this is just connected sum operation for
boundaries. Take disks D^2 on the boundaries of two objects and connect these
disks by cylinder D^2xD1 having D^2:s as its ends. Or more concretely: let the
two 3-surfaces just touch each other.
Fig. 19. Join along boundaries bond a): in two dimensions and b): in
3-dimensions for solid balls.
Depending on the scale join along boundariers bonds are identified as color
flux tubes connecting quarks, bonds giving rise to strong binding between
nucleons inside nuclei, bonds connecting neutrons inside neutron star, chemical
bonds between atoms and molecules, gap junctions connecting cells, the bond
which is formed when You touch table with Your finger, etc.
One can construct from a group of nearby disjoint 3-surfaces so called join
along boundaries condensate by allowing them to touch each other here and there.
Fig. 20. Join along boundaries condensate in 2 dimensions.
The formation of join along boundaries condensates creates clearly strong
correlation between two quantum systems and it is assumed that the formation of
join along boundaries condensate is necessary prequisite for the formation of
MACROSCOPIC QUANTUM SYSTEMS.
F. p-Adic numbers and vacuum degeneracy
p-Adic length scale hypothesis derives from the analogy between SPIN GLASS
and TGD. Kähler action allows enormous VACUUM DEGENERACY: ANY spacetime
surface, which belongs to M^4_+xY^2, where Y^2 is so called Legendre submanifold
of CP2 is vacuum due to the vanishing of induced Kähler form (recall that Kähler
action is just Maxwell action for induced Kähler form which can be regarded as
U(1) gauge field).
Legendre submanifolds can be written in the canonical coordinates P^i,Q^i, i=1,2
for CP2 as
f^i =partial_i f(Q^1,Q^2)
where partial_i means partial derivative with respect to Q^i. f is arbitrary
function of Q^i! Legendre submanifolds are 2-dimensional. The topology of vacuum
space time is restricted only by the imbeddability requirement. Vacuum
spacetimes can have also finite extend in time direction(!!): charge
conservation does not force infinite duration.
Fig. 21. Vacuum extremals can have finite time duration.
This enormous vacuum degeneracy resembles the infinite ground state
degeneracy of spin glasses. In case of spin glasses the space of free energy
minima obeys ultrametric topology. This raises the question whether the
effective topology of the real spacetime sheets could be also ultrametric in
some length scale range so that the distance function would satisfy
d(x,y) <= Max(d(x),d(y)) rather than d(x)+d(y)
p-Adic topologies are ultrametric and there is p-adic topology for each prime
p=2,3,5,7,... The classical non-determinism of the vacuum extremals implies also
classical non-determinism of field equations (but not complete randomness of
p-Adic differential equations are also inherently non-deterministic. This
suggests that the non-determinism of Kähler action is effectively like p-adic
non-determinism in some length scale range, so that that the topology of the
real space-time sheet is effectively p-adic for some value p. The lower cutoff
length scale could be CP_2 length scale. Of course, cutoff length scales could
Standard representation of p-adic number is defined as generalization of
x= SUM_(n>=n0) x_np^n
p-Adic norm reads as
N(x)_p = p^(-n0) ,
and clearly depends on the lowest pinary digit only and is thus very rough:
for reals norm is same only for x and -x. Note that integers which are infinite
as real numbers are finite as p-adic numbers: p-adic norm of any integer is at
Essential element is the so called CANONICAL CORRESPONDENCE between p-adics
x= SUM_(n>=n0) x_np^n
to real number
y = SUM_(n>=n0) x_np^(-n)
Note that only the signs of powers of p are changed.
Second natural correspondence between p-adics and reals is based on the fact
that both reals and p-adics are completions of rational numbers. Hence rational
numbers can be regarded as common to both p-adic and real numbers. This defines
a correspondence in the set of rationals. Allowing algebraic extensions of p-adic
numbers, one can regard also algebraic numbers as common to reals and algebraic
extensions of p-adics. p-Adic and real transcendentals do not have anything in
common. Note that rationals have pinary expansion in powers of p, which becomes
periodic for high pinary digits (predictability) whereas transcendentals have
non-periodic pinary expansions (non-predictability). One could say that the
numbers common to reals and p-adics are like islands of order in the middle of
real and p-adic seas of chaos. Both correspondences are important in the recent
formulation of p-adic physics.
G. p-Adic length scale hypothesis
p-Adic mass calculations force to conclude that the length scale below which
p-adic effective topology is satisfied is given
L_p simeq R*sqrt(p), R= 10^4*sqrt(G) (CP_2 length scale).
('simeq' means 'in good approximation'). One has also good reasons to guess
that p-Adic effective topology makes sense only above CP_2 length scale. One can
also define n-ary p-adic length scales
It is very natural to assume that the spacetime sheets of increasing size
have typical sizes not too much larger than L_p(n). The following figure
illustrates the situation.
Fig. 22. p-Adic length scale hierarchy
The obvious question is 'Are there some physically favoured p-adic primes?'.
p-Adic mass calculations encourage the following hypothesis
The most interesting p-adic primes p correspond to primes near prime powers
p simeq 2^k, k prime
Especially important are physically Mersenne primes M(k) for which this
condition is optimally satisfied p= 2^k-1 Examples: M(127)= 2^127 -1, M(107) =
2^107 -1, M(89)= 2^89 -1: electron, hadrons, intermediate gauge bosons.
A real mathematical justification for this hypothesis is still lacking:
probably the p-adic dynamics depends sensitively of p and this selects certain
p-adic primes via some kind of 'natural selection'.
H. Generalization of spacetime concept
One can wonder whether p-adic topology is only an effective topology or
whether one could speak about a decomposition of the space-time surface to real
and genuinely p-adic regions, and what might be the interpretation of the p-adic
regions (note that also real space-time regions would still be characterized by
some prime characterizing their effective topology).
The development of TGD inspired theory of consciousness led finally to what
seems to be a definite answer to this question. p-Adic physics is physics of
cognition and intention. p-Adic non-determinism is the classical space-time
correlate for the non-determinism of imagination and cognition. p-Adic spacetime
sheets represent intentions and quantum jump in which p-adic space-time sheet is
transformed to real one can be seen as a transformation of intention to action.
This forces to generalize the notion of the imbedding space. The basic idea
is that rational numbers are in a well-defined sense common to both real number
field R and all p-adic number fields R_p. The generalized imbedding space
results when the real H and all p-adic versions H_p of the imbedding space are
glued together along rational points. One can visualize real and p-adic
imbedding spaces as planes, which intersect along a common axis representing
rational points of H. Real and p-adic spacetime region are glued together along
the boundaries of the real spacetime sheet at rational points.
The construction of p-adic quantum physics and the fusion of real physics and
p-adic physics for various primes to a larger scheme is quite a fascinating
challenge. For instance, a new number theoretic view about information emerges.
p-Adic entropy can be negative, which means that system carries genuine
information rather than entropy.