I have seldom in my life felt so astounded as when I first typed the words
Topological Geometrodynamics into Google and followed the path of links
deeper and deeper into the stupendous intellectual abyss that this phrase leads
to. The only adequate analogue must certainly be Alices venture into the
depths of the Rabbit Hole! This particular hole features eighteen on-line books
and over ten thousand pages of beautiful and highly original mathematics and
Matti Pitkanen has during many long years looked deeper into the secrets
of the universe than any other person that I have known. His study is
systematic and meticulous, yet awe-inspiring by the all-encompassing width
of his mathematical and physical treasure chest that features such proverbial
beasts as zero energy ontology, infinite primes and p-Adic space-time. Yet
Matti Pitkanen himself is the first person to acknowledge that his journey is
still incomplete: there is no world equation or other forms of a closed formulation,
let alone solutions to such systems of equations. Matti Pitkanen
humbly describes himself as the scribe of the universe that faithfully records
the beauty of the symmetries that he perceives through his equations and operators
with a deep physical meaning. Symmetry is indeed the cornerstone
of Topological Geometrodynamics, or TGD. On one hand, TGD is a proper
generalization of John Archibald Wheelers eponymous theory. On the other,
it is a generalized M-theory where particles are represented by 3-surfaces in
an eight-dimensional manifold. In the former case, Matti Pitkanens worldsheet
is parameterized by a Cartesian product of the 4-dimensional Minkowski
space and a compact two-dimensional complex projective sphere. In the second
case, this same manifold is conformally symmetric in the sense that it
must possess an infinite-dimensional Kähler geometry. This requirement leads
to the necessity of infinite-dimensional groups of isometries to exist.
This extremely simple requirement of symmetry results in a number of
astonishing deviations from other M-theories with hadronic strings. For example,
world sheet diagrams do not describe particle decays, but instead the propagation of particles by different routes. Particle reactions are described
by generalized Feynman diagrams where 3-dimensional light-like surfaces are
identified with particles. The ensuing four-dimensional space-time surfaces
that now replace the vertices of Feynman diagrams are therefore singular, just
like Feynman diagrams are as one-dimensional manifolds. The equivalence
between the two interpretations of TGD implies that TGD necessarily unifies
quantum mechanics with the General Theory of Relativity by purely geometric
If you find visions like this strange and counter-intuitive, you have picked
the right book! Matti Pitkanens monograph on TGD leads you gently through
the beautiful symmetric geometry that features such unusual structures and
connections between them, and lets you yourself be the judge of their merit.
Please join me on this journey maybe the most courageous intellectual Odyssey
that mankind has ever embarked upon!
Joensuu, October 23, 2014, Finland
Associate professor of Mathematics,
Lappeenranta University of Technology.
President, Arbonaut Ltd.
This book belongs to a series of online books summarizing the recent state
Topological Geometrodynamics (TGD) and its applications. TGD can be regarded
as a unified theory of fundamental interactions but is not the kind of
unified theory as so called GUTs constructed by graduate students at seventies
and eighties using detailed recipes for how to reduce everything to group
theory. Nowadays this activity has been completely computerized and it probably
takes only a few hours to print out the predictions of this kind of unified
theory as an article in the desired format. TGD is something different and I am not ashamed to confess that I have devoted the last 37 years of my life to
this enterprise and am still unable to write The Rules.
If I remember correctly, I got the basic idea of Topological Geometrodynamics
(TGD) during autumn 1977, perhaps it was October. What I realized was that
the representability of physical space-times as 4-dimensional surfaces of some
higher-dimensional space-time obtained by replacing the points of Minkowski
space with some very small compact internal space could resolve the conceptual
difficulties of general relativity related to the definition of the notion of energy.
This belief was too optimistic and only with the advent of what I call zero
energy ontology the understanding of the notion of Poincare invariance has
become satisfactory. This required also the understanding of the relationship
to General Relativity.
It soon became clear that the approach leads to a generalization of the notion of
space-time with particles being represented by space-time surfaces with finite
size so that TGD could be also seen as a generalization of the string model.
Much later it became clear that this generalization is consistent with conformal
invariance only if space-time is 4-dimensional and the Minkowski space factor
of imbedding space is 4-dimensional. During last year it became clear that
4-D Minkowski space and 4-D complex projective space CP2 are completely
unique in the sense that they allow twistor space with Kahler structure.
It took some time to discover that also the geometrization of also gauge interactions and elementary particle quantum numbers could be possible in this
framework: it took two years to find the unique internal space (CP2) providing
this geometrization involving also the realization that family replication phenomenon
for fermions has a natural topological explanation in TGD framework
and that the symmetries of the standard model symmetries are much
more profound than pragmatic TOE builders have believed them to be. If
TGD is correct, main stream particle physics chose the wrong track leading to
the recent deep crisis when people decided that quarks and leptons belong to
same multiplet of the gauge group implying instability of proton.
There have been also longstanding problems.
Gravitational energy is well-defined in cosmological models but is not
conserved. Hence the conservation of the inertial energy does not seem to
be consistent with the Equivalence Principle. Furthermore, the imbeddings
of Robertson-Walker cosmologies turned out to be vacuum extremals
with respect to the inertial energy. About 25 years was needed
to realize that the sign of the inertial energy can be also negative and
in cosmological scales the density of inertial energy vanishes: physically
acceptable universes are creatable from vacuum. Eventually this led to
the notion of zero energy ontology (ZEO) which deviates dramatically
from the standard ontology being however consistent with the crossing
symmetry of quantum field theories. In this framework the quantum
numbers are assigned with zero energy states located at the boundaries
of so called causal diamonds defined as intersections of future and past
directed light-cones. The notion of energy-momentum becomes length
scale dependent since one has a scale hierarchy for causal diamonds. This
allows to understand the non-conservation of energy as apparent.
Equivalence Principle as it is expressed by Einstein's equations follows
from Poincare invariance once it is realized that GRT space-time is obtained
from the many-sheeted space-time of TGD by lumping together
the space-time sheets to a regionof Minkowski space and endowing it
with an effective metric given as a sum of Minkowski metric and deviations
of the metrices of space-time sheets from Minkowski metric.
Similar description relates classical gauge potentials identified as components
of induced spinor connection to Yang-Mills gauge potentials in
GRT space-time. Various topological inhomogenities below resolution
scale identified as particles are described using energy momentum tensor
and gauge currents.
From the beginning it was clear that the theory predicts the presence of
long ranged classical electro-weak and color gauge fields and that these fields necessarily accompany classical electromagnetic fields.
It took about 26 years to gain the maturity to admit the obvious: these fields are classical correlates for long range color and weak interactions
assignable to dark matter. The only possible conclusion is that TGD
physics is a fractal consisting of an entire hierarchy of fractal copies of
standard model physics. Also the understanding of electro-weak massivation
and screening of weak charges has been a long standing problem,
and 32 years was needed to discover that what I call weak form of
electric-magnetic duality gives a satisfactory solution of the problem and
provides also surprisingly powerful insights to the mathematical structure
of quantum TGD.
The latest development was the realization that the well- definedness of
electromagnetic charge as quantum number for the modes of the induced
spinors field requires that the CP2 projection of the region in which
they are non-vanishing carries vanishing W boson field and is 2-D. This
implies in the generic case their localization to 2-D surfaces: string world
sheets and possibly also partonic 2-surfaces. This localization applies to
all modes except covariantly constant right handed neutrino generating
supersymmetry and mplies that string model in 4-D space-time is part of
TGD. Localization is possible only for Kahler-Dirac assigned with Kahler
action defining the dynamics of space-time surfaces. One must however
leave open the question whether W field might vanish for the space-time
of GRT if related to many-sheeted space-time in the proposed manner
even when they do not vanish for space-time sheets.
I started the serious attempts to construct quantum TGD after my thesis
around 1982. The original optimistic hope was that path integral formalism
or canonical quantization might be enough to construct the quantum theory
but the first discovery made already during first year of TGD was that these
formalisms might be useless due to the extreme non-linearity and enormous
vacuum degeneracy of the theory. This turned out to be the case.
It took some years to discover that the only working approach is based
on the generalization of Einstein's program. Quantum physics involves
the geometrization of the infinite-dimensional \world of classical worlds"
(WCW) identified as 3-dimensional surfaces. Still few years had to pass
before I understood that general coordinate invariance leads to a more
or less unique solution of the problem and in positive energyontology
implies that space-time surfaces are analogous to Bohr orbits. This in
positive energy ontology in which space-like 3-surface is basic object. It
is not clear whether Bohr orbitology is necessary also in ZEO in which space-time surfaces connect space-like 3-surfaces at the light-like boundaries
of causal diamond CD obtained as intersection of future and past
directed light-cones (with CP2 factor included). The reason is that the
pair of 3-surfaces replaces the boundary conditions at single 3-surface
involving also time derivatives. If one assumes Bohr orbitology then
strong correlations between the 3-surfaces at the ends of CD follow. Still
a couple of years and I discovered that quantum states of the Universe
can be identified as classical spinor fields in WCW. Only quantum jump
remains the genuinely quantal aspect of quantum physics.
During these years TGD led to a rather profound generalization of the
space-time concept. Quite general properties of the theory led to the notion
of many-sheeted space-time with sheets representing physical subsystems
of various sizes. At the beginning of 90s I became dimly aware of
the importance of p-adic number fields and soon ended up with the idea
that p-adic thermodynamics for a conformally invariant system allows
to understand elementary particle massivation with amazingly few input
assumptions. The attempts to understand p-adicity from basic principles
led gradually to the vision about physics as a generalized number theory
as an approach complementary to the physics as an infinite-dimensional
spinor geometry of WCW approach. One of its elements was a generalization
of the number concept obtained by fusing real numbers and
various p-adic numbers along common rationals. The number theoretical
trinity involves besides p-adic number fields also quaternions and
octonions and the notion of infinite p.prime.
TGD inspired theory of consciousness entered the scheme after 1995 as
I started to write a book about consciousness. Gradually it became difficult to say where physics ends and consciousness theory begins since
consciousness theory could be seen as a generalization of quantum measurement
theory by identifying quantum jump as a moment of consciousness
and by replacing the observer with the notion of self identified as
a system which is conscious as long as it can avoid entanglement with
environment. The somewhat cryptic statement \Everything is conscious
and consciousness can be only lost" summarizes the basic philosophy
The idea about p-adic physics as physics of cognition and intentionality
emerged also rather naturally and implies perhaps the most dramatic
generalization of the space-time concept in which most points of p-adic
space-time sheets are infinite in real sense and the projection to the real
imbedding space consists of discrete set of points. One of the most fasvicinating outcomes was the observation that the entropy based on p-adic
norm can be negative. This observation led to the vision that life can
be regarded as something in the intersection of real and p-adic worlds.
Negentropic entanglement has interpretation as a correlate for various
positively colored aspects of conscious experience and means also the
possibility of strongly correlated states stable under state function reduction
and different from the conventional bound states and perhaps
playing key role in the energy metabolism of living matter.
If one requires consistency of Negentropy Mazimization Pronciple with
standard measurement theory, negentropic entanglement defined in terms of number theoretic negentropy is necessarily associated with a density
matrix proportional to unit matrix and is maximal and is characterized
by the dimension n of the unit matrix. Negentropy is positive and maximal
for a p-adic unique prime dividing n.
One of the latest threads in the evolution of ideas is not more than
nine years old. Learning about the paper of Laurent Nottale about the
possibility to identify planetary orbits as Bohr orbits with a gigantic
value of gravitational Planck constant made once again possible to see
the obvious. Dynamical quantized Planck constant is strongly suggested
by quantum classical correspondence and the fact that space-time sheets
identifiable as quantum coherence regions can have arbitrarily large sizes.
Second motivation for the hierarchy of Planck constants comes from bioelectromagnetism
suggesting that in living systems Planck constant could
have large values making macroscopic quantum coherence possible. The
interpretation of dark matter as a hierarchy of phases of ordinary matter
characterized by the value of Planck constant is very natural.
During summer 2010 several new insights about the mathematical structure
and interpretation of TGD emerged. One of these insights was the
realization that the postulated hierarchy of Planck constants might follow
from the basic structure of quantum TGD. The point is that due
to the extreme non-linearity of the classical action principle the correspondence
between canonical momentum densities and time derivatives
of the imbedding space coordinates is one-to-many and the natural description
of the situation is in terms of local singular covering spaces of
the imbedding space. One could speak about effective value of Planck
constant heff = n x h coming as a multiple of minimal value of Planck
constant. Quite recently it became clear that the non-determinism of
Kahler action is indeed the fundamental justification for the hierarchy:
the integer n can be also interpreted as the integer characterizing the dimension of unit matrix characterizing negentropic entanglement made
possible by the many-sheeted character of the space-time surface.
Due to conformal invariance acting as gauge symmetry the n degenerate
space-time sheets must be replaced with conformal equivalence classes of
space-time sheets and conformal transformations correspond to quantum
critical deformations leaving the ends of space-time surfaces invariant.
Conformal invariance would be broken: only the sub-algebra for which
conformal weights are divisible by n act as gauge symmetries. Thus deep
connections between conformal invariance related to quantum criticality,
hierarchy of Planck constants, negentropic entanglement, effective p-adic
topology, and non-determinism of Kahler action perhaps reflecting p-adic
The implications of the hierarchy of Planck constants are extremely far
reaching so that the significance of the reduction of this hierarchy to the
basic mathematical structure distinguishing between TGD and competing
theories cannot be under-estimated.
From the point of view of particle physics the ultimate goal is of course a
practical construction recipe for the S-matrix of the theory. I have myself
regarded this dream as quite too ambitious taking into account how far reaching
re-structuring and generalization of the basic mathematical structure of
quantum physics is required. It has indeed turned out that the dream about
explicit formula is unrealistic before one has understood what happens in quantum
jump. Symmetries and general physical principles have turned out to be
the proper guide line here. To give some impressions about what is required
some highlights are in order.
With the emergence of ZEO the notion of S-matrix was replaced with Mmatrix
defined between positive and negative energy parts of zero energy
states. M-matrix can be interpreted as a complex square root of density
matrix representable as a diagonal and positive square root of density
matrix and unitary S-matrix so that quantum theory in ZEO can be said
to define a square root of thermodynamics at least formally. M-matrices
in turn bombine to form the rows of unitary U-matrix defined between
zero energy states.
A decisive step was the strengthening of the General Coordinate Invariance
to the requirement that the formulations of the theory in terms of
light-like 3-surfaces identified as 3-surfaces at which the induced metric
of space-time surfaces changes its signature and in terms of space-like
3-surfaces are equivalent. This means effective 2-dimensionality in the sense that partonic 2-surfaces defined as intersections of these two kinds
of surfaces plus 4-D tangent space data at partonic 2-surfaces code for
the physics. Quantum classical correspondence requires the coding of
the quantum numbers characterizing quantum states assigned to the partonic
2-surfaces to the geometry of space-time surface. This is achieved
by adding to the modified Dirac action a measurement interaction term
assigned with light-like 3-surfaces.
The replacement of strings with light-like 3-surfaces equivalent to spacelike
3-surfaces means enormous generalization of the super conformal
symmetries of string models. A further generalization of these symmetries
to non-local Yangian symmetries generalizing the recently discovered
Yangian symmetry of N = 4 supersymmetric Yang-Mills theories
is highly suggestive. Here the replacement of point like particles
with partonic 2-surfaces means the replacement of conformal symmetry
of Minkowski space with infinite-dimensional super-conformal algebras.
Yangian symmetry provides also a further refinement to the notion of
conserved quantum numbers allowing to define them for bound states
using non-local energy conserved currents.
A further attractive idea is that quantum TGD reduces to almost topological
quantum field theory. This is possible if the Kahler action for
the preferred extremals defining WCW Kahler function reduces to a 3-D
boundary term. This takes place if the conserved currents are so called
Beltrami fields with the defining property that the coordinates associated
with flow lines extend to single global coordinate variable. This
ansatz together with the weak form of electric-magnetic duality reduces
the Kahler action to Chern-Simons term with the condition that the 3-
surfaces are extremals of Chern-Simons action subject to the constraint
force defined by the weak form of electric magnetic duality. It is the latter
constraint which prevents the trivialization of the theory to a topological
quantum field theory. Also the identification of the Kahler function
of WCW as Dirac determinant finds support as well as the description
of the scattering amplitudes in terms of braids with interpretation in
terms of finite measurement resolution coded to the basic structure of
the solutions of field equations.
In standard QFT Feynman diagrams provide the description of scattering
amplitudes. The beauty of Feynman diagrams is that they realize
unitarity automatically via the so called Cutkosky rules. In contrast to
Feynman's original beliefs, Feynman diagrams and virtual particles are
taken only as a convenient mathematical tool in quantum field theories. QFT approach is however plagued by UV and IR divergences and one
must keep mind open for the possibility that a genuine progress might
mean opening of the black box of the virtual particle.
In TGD framework this generalization of Feynman diagrams indeed emerges
unavoidably. Light-like 3-surfaces replace the lines of Feynman diagrams
and vertices are replaced by 2-D partonic 2-surfaces. Zero energy ontology
and the interpretation of parton orbits as light-like \wormhole
throats" suggests that virtual particle do not differ from on mass shell
particles only in that the four- and three- momenta of wormhole throats
fail to be parallel. The two throats of the wormhole contact defining
virtual particle would contact carry on mass shell quantum numbers but
for virtual particles the four-momenta need not be parallel and can also
have opposite signs of energy.
The localization of the nodes of induced spinor fields to 2-D string world sheets (and possibly also to partonic 2-surfaces) implies a stringy formulation
of the theory analogous to stringy variant of twistor formalism
with string world sheets having interpretation as 2-braids. In TGD
framework fermionic variant of twistor Grassmann formalism leads to a
stringy variant of twistor diagrammatics in which basic fermions can be
said to be on mass-shell but carry non-physical helicities in the internal
lines. This suggests the generalization of the Yangian symmetry to
infinite-dimensional super-conformal algebras.
What I have said above is strongly biased view about the recent situation
in quantum TGD. This vision is single man's view and doomed to contain
unrealistic elements as I know from experience. My dream is that young critical
readers could take this vision seriously enough to try to demonstrate that some
of its basic premises are wrong or to develop an alternative based on these or
better premises. I must be however honest and tell that 32 years of TGD is a
really vast bundle of thoughts and quite a challenge for anyone who is not able
to cheat himself by taking the attitude of a blind believer or a light-hearted
debunker trusting on the power of easy rhetoric tricks.
Karkkila, October, 30, Finland
List of Contributors
“The universe is fundamentally a geometric construction. This simple basic observation is used to tremendous extent in topological geometrodynamics – a novel theory that unifies general relativity with quantum field theory. Topological Geometrodynamics: An Overview explores this connection to a beautiful and coherent view of our galactic home.” – Prof. Tuomo Kauranne, Lappeenranta University of Technology.