Thursday, January 26, 2012

The anatomy of quantum jump in zero energy ontology

The understanding of the anatomy of quantum jump identified as a moment of consciousness in the framework of Zero energy ontology (ZEO) is gradually getting more detailed and the following is the summary of the recent understanding. The general vision about quantum jump in zero energy ontology generalizes the ordinary auantum measurement theory bringing in also the selection of maximal set of mutually commuting set of observables. Also the connection with the breaking of time reversal invariance at the level of zero energy states as a necessary condition for the non-triviality of U-matrix is new.

  1. Quantum jump begins with unitary process U described by unitary matrix assigning to a given zero energy state a quantum superposition of zero energy states. This would represent the creative aspect of quantum jump - generation of superposition of alternatives.

  2. The next step is a cascade of state function reductions proceeding from long to short scales. It starts from some CD and proceeds downwards to sub-CDs to their sub-CDs to ...... At a given step it induces a measurement of the quantum numbers of either positive or negative energy part of the quantum state. This step would represent the measurement aspect of quantum jump - selection among alternatives.

  3. The basic variational principle is Negentropy Maximization Principle(NMP) stating that the reduction of entanglement entropy in given quantum jump between two subsystems of CD assigned to sub-CDs is maximal. Mathematically NMP is very similar to the second law although states just the opposite but for individual quantum system rather than ensemble. NMP actually implies second law at the level of ensembles as a trivial consequence of the fact that the outcome of quantum jump is not deterministic.

    For ordinary definition of entanglement entropy this leads to a pure state resulting in the measurement of the density matrix assignable to the pair of CDs. For hyper-finite factors of type II1 (HFFs) state function reduction cannot give rise to a pure state and in this case one can speak about quantum states defined modulo finite measurement resolution and the notion of quantum spinor emerges naturally. One can assign a number theoretic entanglement entropy to entanglement characterized by rational (or even algebraic) entanglement probabilities and this entropy can be negative. Negentropic entanglement can be stable and even more negentropic entanglement can be generated in the state function reduction cascade.

The irreversibility is realized as a property of zero energy states (for ordinary positive energy ontology it is realized at the level of dynamics) and is necessary in order to obtain non-trivial U-matrix. State function reduction should involve several parts. First of all it should select the density matrix or rather its Hermitian square root. After this choice it should lead to a state which prepared either at the upper or lower boundary of CD but not both since this would be in conflict with the counterpart for the determinism of quantum time evolution.

Generalization of S-matrix

ZEO forces the generalization of S-matrix with a triplet formed by U-matrix, M-matrix, and S-matrix. The basic vision is that quantum theory is at mathematical level a complex square roots of thermodynamics. What happens in quantum jump was already discussed.

  1. U-matrix as has its rows M-matrices , which are matrices between positive and negative energy parts of the zero energy state and correspond to the ordinary S-matrix. M-matrix is a product of a hermitian square root - call it H - of density matrix ρ and universal S-matrix S commuting with H: [S,H]=0. There is infinite number of different Hermitian square roots Hi of density matrices which are assumed to define orthogonal matrices with respect to the inner product defined by the trace: Tr(HiHj)=0. Also the columns of U-matrix are orthogonal. One can interpret square roots of the density matrices as a Lie algebra acting as symmetries of the S-matrix.

  2. One can consider generalization of M-matrices so that they would be analogous to the elements of Kac-Moody algebra. These M-matrices would involve all powers of S.

    1. The orthogonality with respect to the inner product defined by < A| B> = Tr(AB) requires the conditions Tr(H1H2Sn)=0 for n≠ 0 and Hi are Hermitian matrices appearing as square root of density matrix. H1H2 is hermitian if the commutator [H1,H2] vanishes. It would be natural to assign n:th power of S to the CD for which the scale is n times the CP2 scale.

    2. Trace - possibly quantum trace for hyper-finite factors of type II1) is the analog of integration and the formula would be a non-commutative analog of the identity ∈tS1 exp(inφ) dφ=0 and pose an additional condition to the algebra of M-matrices. Since H=H1H2 commutes with S-matrix the trace can be expressed as the sum

      i,jhisj(i)= ∑i,j hi(j)sj

      of products of correspondence eigenvalues and the simplest condition is that one has either ∑j sj(i)=0 for each i or ∑i hi(j)=0 for each j.

    3. It might be that one must restrict M matrices to a Cartan algebra for a given U-matrix and also this choice would be a process analogous to state function reduction. Since density matrix becomes an observable in TGD Universe, this choice could be seen as a direct counterpart for the choice of a maximal number of commuting observables which would be now hermitian square roots of density matrices. Therefore ZEO gives good hopes of reducing basic quantum measurement theory to infinite-dimensional Lie-algebra.

Unitary process and choice of the density matrix

Consider first unitary process followed by the choice of the density matrix.

  1. There are two natural state basis for zero energy states. The states of these state basis are prepared at the upper or lower boundary of CD respectively and correspond to various M-matrices MK+ and ML-. U-process is simply a change of state basis meaning a representation of the zero energy state MK+/- in zero energy basis MK-/+ followed by a state preparation to zero energy state M+/-K with the state at second end fixed in turn followed by a reduction to ML-/+ to its time reverse, which is of same type as the initial zero energy state.

    The state function reduction to a given M-matrix MK+/- produces a state for the state is superposition of states which are prepared at either lower or upper boundary of CD. It does not yet produce a prepared state on the ordinary sense since it only selects the density matrix.

  2. The matrix elements of U-matrix are obtained by acting with the representation of identity matrix in the space of zero energy states as

    I= ∑K | K+> < K+|

    on the zero energy state | K-> (the action on | K+> is trivial!) and gives

    U+KL= Tr(M+KM+L) .

    In the similar manner one has

    U-KL=(U+†)KL= Tr(M-LM-K) = (U+LK)* .

    These matrices are Hermitian conjugates of each other as matrices between states labelled by positive or negative energy states. The interpretation is that two unitary processes are possible and are time reversals of each other. The unitary process produces a new state only if its time arrow is different from that for the initial state. The probabilities for transitions |K+> → |K-> are given by

    pmn= |Tr(MK+ ML+)|2.

State function preparation

Consider next the counterpart of the ordinary state preparation process.

  1. The ordinary state function process can act either at the upper or lower boundary of CD and its action is thus on positive or negative energy part of the zero energy state. At the lower boundary of CD this process selects one particular prepared states. At the upper boundary it selects one particular final state of the scattering process.

  2. Restrict for definiteness the consideration to the lower boundary of CD. Denote also MK by M. At the lower boundary of CD the selection of prepared state - that is preparation process- means the reduction

    m+n-M+/-m+n-| m+> | n-> → ∑n-M+/-m+n-| m+> | n-> .

    The reduction probability is given by

    pm= ∑n- | Mm+n-|2 = ρm+m+ .

    For this state the lower boundary carries a prepared state with the quantum numbers of state | m+> . For density matrix which is unit matrix (this option giving pure state might not be possible) one has pm=1.

State function reduction process

The process which is the analog of measuring the final state of the scattering process is also needed and would mean state function reduction at the upper end of CD - to state | n-> now.

  1. It is impossible to reduce to arbitrary state | m+> | n-> and the reduction must at the upper end of CD must mean a loss of preparation at the lower end of CD so that one would have kind of time flip-flop!

  2. The reduction probability for the process

    | m+ >== ∑n-Mm+n-| m+> | n-> → n->= ∑m+Mm+n-| m+> | n->

    would be

    pmn =| Mmn|2 .

    This is just what one would expect. The final outcome would be therefore a state of type | n-> and - this is very important- of the same type as the state from which the process began so that the next process is also of type U+ and one can say that a definite arrow of time prevails.

  3. Both the preparation and reduction process involves also a cascade of state function reductions leading to a choice of state basis corresponding to eigenstates of density matrices between subsystems.

Can the arrow of geometric time change?

A highly interesting question is what happens if the first state preparation leading to a state | K+> is followed by a U-process of type U- rather than by the state function reduction process |K+> → |L->. Does this mean that the arrow of geometric time changes? Could this change of the arrow of geometric time take place in living matter? Could processes like molecular self assembly be entropy producing processes but with non-standard arrow of geometric time? Or are they processes in which negentropy increases by the fusion of negentropic parts to larger ones? Could the variability relate to sleep-awake cycle and to the fact that during dreams we are often in our childhood and youth. Old people are often said to return to their childhood. Could this have more than a metaphoric meaning? Could biological death mean return to childhood at the level of conscious experience? I have explained the recent views about the arrow of time here .

For background see new chapter Construction of Quantum Theory: More About Matrices of "Towards M-matrix" .

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