Wednesday, April 20, 2005

Feynman diagrams as higher level particles and their scattering as dynamics of self consciousness

The hierarchy of imbeddings of hyper-finite factors of II_1 as counterpart for many-sheeted space-time lead inevitably to the idea that this hierarchy corresponds to a hierarchy of generalized Feynman diagrams for which Feynman diagrams at a given level become particles at the next level. Accepting this idea, one is led to ask what kind of quantum states these Feynman diagrams correspond, how one could describe interactions of these higher level particles, what is the interpretation for these higher level states, and whether they can be detected. In the following M_n denotes a II_1 factor in the hierarchy of Jones inclusions M_0 subset M_1 subset.... (for the notations and background see the earlier postings).

1. Higher level Feynman diagrams

The lines of Feynman diagram in M_{n+1} are geodesic lines representing orbits of M_n and this kind of lines meet at vertex and scatter. The evolution along lines is determined by Delta_{M_{n+1}}. These lines contain within themselves M_n Feynman diagrams with similar structure and the hierarchy continues down to the lowest level at which ordinary elementary particles are encountered. For instance, the generalized Feynman diagrams at the second level are ribbon diagrams obtained by thickening the ordinary diagrams in the new time direction. The interpretation as ribbon diagrams crucial for topological quantum computation and suggested to be realizable in terms of zero energy states in is natural. At each level a new time parameter is introduced so that the dimension of the diagram can be arbitrarily high. The dynamics is not that of ordinary surfaces but the dynamics induced by the Delta_{M_n}.

2. Quantum states defined by higher level Feynman diagrams

The intuitive picture is that higher level quantum states corresponds to the self reflective aspect of existence and must provide representations for the quantum dynamics of lower levels in their own structure. This dynamics is characterized by S-matrix whose elements have representation in terms of Feynman diagrams.
  • These states correspond to zero energy states in which initial states have "positive energies" and final states have "negative energies". The net conserved quantum numbers of initial and final state partons compensate each other. Gravitational energies, and more generally gravitational quantum numbers defined as absolute values of the net quantum numbers of initial and final states do not vanish. One can say that thoughts have gravitational mass but no inertial mass.
  • States in sub-spaces of positive and negative energy states are entangled with entanglement coefficients given by S-matrix at the level below.
To make this more concrete, consider first the simplest non-trivial case. In this case the particles can be characterized as ordinary Feynman diagrams, or more precisely as scattering events so that the state is characterized by S_1== P_{in}SP_{out}, where S is S-matrix and P_{in} resp. P_{out} is the projection to a subspace of initial resp. final states. An entangled state with the projection of S-matrix giving the entanglement coefficients is in question. The larger the domains of projectors P_{in} and P_{out}, the higher the representative capacity of the state. The norm of the non-normalized state hat{S} is Trace(S_1 S_1^dagger) and is smaller or equal to one for II_1 factors, and equals to one at the limit S_1=S. Hence, by II_1 property, the state always entangles infinite number of states, and can in principle code the entire S-matrix to entanglement coefficients.

3. The interaction of M_n Feyman diagrams at the second level of hierarchy

What constraints can one pose to the higher level reactions? How Feynman diagrams interact? Consider first the scattering at the second level of hierarchy (M_1), the first level M_0 being assigned to the interactions of the ordinary matter.
  • Conservation laws pose constraints on the scattering at level M_1. The Feyman diagrams can transform to new Feynman diagrams only in such a manner that the net quantum numbers are conserved separately for the initial positive energy states and final negative energy states of the diagram. The simplest assumption is that positive energy matter and negative energy matter know nothing about each other and effectively live in separate worlds. The scattering matrix form Feynman diagram like states would thus be simply the tensor product SxS^{\dagger}, where S is the S-matrix characterizing the lowest level interactions and x denotes tensor product. Reductionism would be realized in the sense that, apart from the new elements brought in by Delta_{M_n} defining single particle free dynamics, the lowest level would determine in principle everything occurring at the higher level providing representations about representations about... for what occurs at the basic level. The lowest level would represent the physical world and higher levels the theory about it.
  • The description of hadronic reactions in terms of partons serves as a guide line when one tries to understand higher level Feynman diagrams. The fusion of hadronic space-time sheets corresponds to the vertices M_1. In the vertex the analog of parton plasma is formed by a process known as parton fragmentation. This means that the partonic Feynman diagrams belonging to disjoint copies of M_0 find themselves inside the same copy of M_0. The standard description would apply to the scattering of the initial resp. final state partons.
  • After the scattering of partons hadronization takes place. The analog of hadronization in the recent case is the organization of the initial and final state partons to groups I_i and F_i such that the net conserved quantum numbers are same for I_i and F_i. These conditions can be satisfied if the interactions in the plasma phase occur only between particles belonging to the clusters labelled by the index i. Otherwise only single particle states in M_1 would be produced in the reactions in the generic case. The cluster decomposition of S-matrix to a direct sum of terms corresponding to partitions of the initial state particles to clusters which do not interact with each other obviously corresponds to the "hadronization". Therefore no new dynamics need to be introduced.
  • One cannot avoid the question whether the parton picture about hadrons indeed corresponds to a higher level physics of this kind. This would require that hadronic space-time sheets carry the net quantum numbers of hadrons. The net quantum numbers associated with the initial state partons would be naturally identical with the net quantum numbers of hadron. Partons and they negative energy conjugates would provide in this picture a representation of hadron about hadron. This kind of interpretation of partons would make understandable why they cannot be observed directly. A possible objection is that the net gravitational mass of hadron would be three times the gravitational mass deduced from the inertial mass of hadron if partons feed their gravitational fluxes to the space-time sheet carrying Earth's gravitational field.
  • This picture could also relate to the suggested duality between string and parton pictures. In parton picture hadron is formed from partons represented by space-like 2-surfaces X^2_i connected by join along boundaries bonds. In string picture partonic 2-surfaces are replaced with string orbits. If one puts positive and negative energy particles at the ends of string diagram one indeed obtains a higher level representation of hadron. If these pictures are dual then also in parton picture positive and negative energies should compensate each other. Interestingly, light-like 3-D causal determinants identified as orbits of partons could be interpreted as orbits of light like string word sheets with "time" coordinate varying in space-like direction.

4. Scattering of Feynman diagrams at the higher levels of hierarchy

This picture generalizes to the description of higher level Feynman diagrams.
  • Assume that higher level vertices have recursive structure allowing to reduce the Feynman diagrams to ordinary Feynman diagrams by a procedure consisting of finite steps.
  • The lines of diagrams are classified as incoming or outgoing lines according to whether the time orientation of the line is positive or negative. The time orientation is associated with the time parameter t_n characterizing the automorphism Delta_{M_n}^{it_n}. The incoming and outgoing net quantum numbers compensate each other. These quantum numbers are basically the quantum numbers of the state at the lowest level of the hierarchy.
  • In the vertices the M_{n+1} particles fuse and M_n particles form the analog of quark gluon plasma. The initial and final state particles of M_n Feynman diagram scatter independently and the S-matrix S_{n+1} describing the process is tensor product S_nxS_n^{\dagger} (x denotes tensor product). By the clustering property of S-matrix, this scattering occurs only for groups formed by partons formed by the incoming and outgoing particles M_n particles and each outgoing M_{n+1} line contains and irreducible M_n diagram. By continuing the recursion one finally ends down with ordinary Feynman diagrams.

5. A connection with TGD inspired theory of consciousness

The implications of this picture TGD inspired theory of consciousness are rather breathtaking.
  • The hierarchy of self representations and the reduction of their quantum dynamics to the dynamics of the material world apart from the effects brought in by the automorphisms Delta_{M_n} determining the free propagation of thoughts, would mean a concrete calculable theory for the quantum dynamics of cognition. My sincere hope is however that no one would ever christen these states "particles of self consciousness". These states are not conscious, consciousness would be in the quantum jump between these states.
  • Cognitive representations would possess "gravitational" charges, in particular gravitational mass, so that thoughts could be put into "gravitational scale". I have proposed that "gravitational" charges correspond to classical charges characterizing the systems at space-time level as opposed to quantum charges.
  • As found, even hadrons could form self representations usually assigned with human brain. This is certainly something that neuroscientist would not propose but conforms with the basic prediction of TGD inspired theory of consciousness about infinite self hierarchy involving cognitive representations at all levels of the hierarchy (see for instance the chapter Time, Space-time, and Consciousness of "Genes,Memes, Qualia,...".
  • The TGD inspired model of topological quantum computation in terms of zero energy cognitive states inspired the proposal that the appearance of a representation and its negative energy conjugate could relate very intimately to the fact that DNA appears as double helices of a strand and its conjugate. This could also relate to the fact that binary structures are common in living matter.
  • One is forced to consider a stronger characterization of the dark matter as a matter at higher levels of the hierarchy with vanishing net inertial quantum numbers but with non-vanishing "gravitational" quantum numbers. We would detect dark matter via its "gravitational" charges. We would also experience it directly since our thoughts would be dark matter! The cosmological estimates for the proportion of dark matter and dark energy would give also estimate for the gravitational mass of thoughts in the Universe: if this interpretation is correct the encounters with UFOs and aliens cease to be material for news!
For more details see the new chapter "Was von Neumann Right After All?" of TGD. Matti Pitkanen

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