Quantum Boolean algebra instead of Boolean algebra?
Mathematical logic relies on the notion of Boolean algebra, which has a well-known representation as the algebra of sets which in turn has in algebraic geometry a representation in terms of algebraic varieties. This is not however attractive at space-time level since the dimension of the algebraic variety is different for the intersection resp. union representing AND resp. OR so that only only finite number of ANDs can appear in the Boolean function. TGD inspired interpretation of the fermionic sector of the theory in terms of Boolean algebra inspires more concrete ideas about the the realization of Boolean algebra at both quantum level and classical space-time level and also suggests a geometric realization of the basic logical functions respecting the dimension of the representative objects.
- In TGD framework WCW spinors correspond to fermionic Fock states and an attractive interpretation for the basis of fermionic Fock states is as Boolean algebra. In zero energy ontology one consider pairs of positive and negative energy states and zero energy states could be seen as physical correlates for statements A→ B
or A↔ B with individual state pairs in the quantum superposition representing various instances of the rule A→ B or A↔ B. The breaking of time reversal invariance means that either the positive or negative energy part of the state (but not both) can correspond to a state with precisely definine number of particles with precisely defining quantum numbers such as four-momentum. At the second end one has scattered state which is a superposition of this kind of many-particle states. This would suggest that A→ B is the correct interpretation.
- In quantum group theory the notion of co-algebra is very natural and the binary algebraic operations of co-algebra are in a well-defined sense time reversals of those of algebra. Hence there is a great temptation to generalize Boolean algebra to include its co-algebra so that one might speak about quantum Boolean algebra. The vertices of generalized Feynman diagrams represent two topological binary operations for partonic two surfaces and there is a strong temptation to interpret them as representations for the operations of Boolean algebra and its co-algebra.
- The first vertex corresponds to the analog of a stringy trouser diagram in which partonic 2-surface decays to two and the reversal of this representing fusion of partonic 2-surfaces. In TGD framework this diagram does not represent classically particle decay or fusion but the propagation of particle along two paths after the decay or the reversal of this process. The Boolean analog would be logical OR (A∨B) or set theoretical union A∪B resp. its co-operation. The partonic two surfaces would represent the arguments (resp. co-arguments) A and B.
- Second one corresponds to the analog of 3-vertex for Feynman diagram: the three 3-D "lines" of generalized Feynman diagram meet at the partonic 2-surface. This vertex (co-vertex) is the analog of Boolean AND (A∧B) or intersection A∩B of two sets resp. its co-operation.
- I have already earlier ended up with the proposal that only three-vertices appear as fundamental vertices in quantum TGD (see this). The interpretation of generalized Feynman diagrams as a representation of quantum Boolean algebra would give a deeper meaning for this proposal.
These vertices could therefore have interpretation as a space-time representation for operations of Boolean algebra and its co-algebra so that the space-time surfaces could serve as classical correlates for the generalized Boolean functions defined by generalized Feynman diagrams and expressible in terms of basic operations of the quantum Boolean algebra. For this representation the dimension of the variety representing the value of Boolean function at classical level is the same as as the dimension of arguments: that is two. Hence this representation is not equivalent with the representation provided by algebraic geometry for which the dimension of the geometric variety representing A∧ B and A∨ B in general differs from that for A and B. If one however restricts the algebra to that assignable to braid strands, statements would correspond to points at partonic level, so that one would have discrete sets and the set theoretic representation of quantum Boolean algebra could make sense. Discrete sets are indeed the only possibility since otherwise the dimension of intersection and union are different if algebraic varieties are in question.
- The breaking of time reversal invariance is accompanied by a generation of entropy and loss of information. The interpretation at the level of quantum Boolean algebra would be following. The Boolean function and and OR assign to two statements a single statement: this means a gain of information and at the level of physics this is indeed the case since entropy is reduced in the process reducing the number of particles. The occurrence of co-operations of AND and OR corresponds to particle decays and uncertainty about the path along which particle travels (dispersion of wave packet) and therefore loss of information.
- The "most logical" interpretation for the situation is in conflict with the identification of the arrow of logic implication with the arrow of time: the direction of Boolean implication arrow and the arrow of geometric time would be opposite so that final state could be said to imply the initial state. The arrow of time would weaken logical equivalence to implication arrow.
- If one naively identifies the arrows of logical implication and geometric time so that initial state can be said to imply the final state, second law implies that logic becomes fuzzy. Second law would weak logical equivalence to statistical implication arrow.
- The natural question is whether just the presence of both algebra and co-algebra operations causing a loss of information in generalized Feynman diagrams could lead to what might be called fuzzy Boolean functions expressing the presence of entropic element appears at the level of Boolean cognition.
- This picture requires a duality between Boolean algebra and its co-algebra and this duality would naturally correspond to time reversal. Skeptic can argue that there is no guarantee about the existence of the extended algebra analogous to Drinfeld double that would unify Boolean algebra and its dual. Only the physical intuition suggests its existence.
These observations suggest that generalized Feynman diagrams and their space-time counterparts could have a precise interpretation in quantum Boolean algebra and that one should perhaps consider the extension of the mathematical logic to quantum logic. Alternatively, one could argue that quantum Boolean algebra is more like a model for what mathematical cognition could be in the real world.
For details and background the reader can consult either to the chapter Physics as Generalized Number Theory: Infinite Primes or to the article How infinite primes relate to other views about mathematical infinity?.