Quantum Mechanics and Quantum Mathematics

Quantum Mathematics (QM) suggests that the basic structures of Quantum Mechanics (QM) might reduce to fundamental mathematical and metamathematical structures, and that one even consider the possibility that Quantum Mechanics reduces to Quantum Mathematics with mathematician included or expressing it in a concice manner: QM=QM!

The notes below were stimulated by an observation raising a question about a possible connection between multiverse interpretation of quantum mechanics and quantum mathematics. The heuristic idea of multiverse interpretation is that quantum state repeatedly branches to quantum states which in turn branch again. The possible outcomes of the state function reduction would correspond to different branches of the multiverse so that one could save keep quantum mechanics deterministic if one can give a well-defined mathematical meaning to the branching. Could quantum mathematics allow to somehow realize the idea about repeated branching of the quantum universe? Or at least to identify some analog for it? The second question concerns the identification of the preferred state basis in which the branching occurs.

Quantum Mathematics briefly

Quantum Mathematics replaces numbers with Hilbert spaces and arithmetic operations + and × with direct sum ⊕ and tensor product ⊗.

1. The original motivation comes from quantum TGD where direct sum and tensor product are naturally assigned with the two basic vertices analogous to stringy 3-vertex and 3-vertex of Feynman graph. This suggests that generalized Feynman graphs could be analogous to sequences of arithmetic operations allowing also co-operations of ⊕ and ⊗.
2. One can assign to natural numbers, integers, rationals, algebraic numbers, transcendentals and their p-adic counterparts for various prime p Hilbert spaces with formal dimension given by the number in question. Typically the dimension of these Hilbert spaces in the ordinary sense is infinite. Von Neuman algebras known as hyper-finite factors of type II₁ assume as a convention that the dimension of basic Hilbert space is one although it is infinite in the standard sense of the word. Therefore this Hilbert space has sub-spaces with dimension which can be any number in the unit interval. Now however also negative and even complex, quaternionic and octonionic values of Hilbert space dimension become possible.
3. The decomposition to a direct sum matters unlike for abstract Hilbert space as it does also in the case of physical systems where the decomposition to a direct sum of representations of symmetries is standard procedure with deep physical significance. Therefore abstract Hilbert space is replaced with a more structured objects. For instance, the expansion ∑_n x_npⁿ of a p-adic number in powers of p defines decomposition of infinite-dimensional Hilbert space to a direct sum ⊕_n x_n⊗ pⁿ of the tensor products x_n⊗ pⁿ. It seems that one must modify the notion of General Coordinate Invariance since number theoretic anatomy distinguishes between the representations of space-time point in various coordinates. The interpretation would be in terms of cognition. For instance, the representation of Neper number requires infinite number of pinary digits whereas finite integer requires onlya finite number of them so that at the level of cognitive representations general coordinate invariance is broken.

   Note that the number of elements of the state basis in pⁿ factor is pⁿ and m∈ {0,...,p-1} in the factor x_n. Therefore the Hilbert space with dimension pⁿ>x_n is analogous to the Hilbert space of a large effectively classical system entangled with the microscopic system characterized by x_n. p-Adicity of this Hilbert space in this example is for the purpose of simplicity but raises the question whether the state function reduction is directly related to cognition.

4. On can generalize the concept of real numbers, the notions of manifold, matrix group, etc... by replacing points with Hilbert spaces. For instance, the point (x₁,..,x_n) of Eⁿ is replaced with Cartesian product of corresponding Hilbert spaces. What is of utmost importance for the idea about possible connection with the multiverse idea is that also this process can be also repeated indefinitely. This process is analogous to a repeated second quantization since intuitively the replacement means replacing Hilbert space with Hilbert space of wave functions in Hilbert space. The finite dimension and its continuity as function of space-time point must mean that there are strong constraints on these wave functions. What does this decomposition to a direct sum mean at the level of states? Does one have super-selection rules stating that quantum inteference is possible only inside the direct summands?
5. Could one find a number theoretical counterpart for state function reduction and preparation and unitary time evolution? Could zero energy ontology have a formulation at the level of the number theory as earlier experience with infinite primes suggest? The proposal was that zero energy states correspond to ratios of infinite integers which as real numbers reduce to real unit. Could zero energy states correspond to states in the tensor product of Hilbert spaces for which formal dimensions are inverses of each other so that the total space has dimension 1?

The minimal view about unitary process and state function reduction is provided by ZEO.

1. Zero energy states correspond to a superposition of pairs of positive and negative energy states. The M-matrix defining the entanglement coefficients is product of Hermitian square root of density matrix and unitary S-matrix, and various M-matrices are orthogonal and form rows of a unitary U-matrix. Quantum theory is square root of thermodynamics. This is true even at single particle level. The square root of the density matrix could be also interpreted in terms of finite measurement resolution.
2. It is natural to assume that zero energy states have well-defined single particle quantum numbers at the either end of CD as in particle physics experiment. This means that state preparation has taken place and the prepared end represents the initial state of a physical event. Since either end of CD can be in question, both arrows of geometric time identifiable as the Minkowski time defined by the tips of CD are possible.
3. The simplest identification of the U-matrix is as the unitary U-matrix relating to each other the state basis for which M-matrices correspond to prepared states at two opposite ends of CD. Let us assume that the preparation has taken place at the "lower" end, the initial state. State function reduction for the final state means that one measures the single particle observables for the "upper" end of CD. This necessarily induces the loss of this property at the "lower" end. Next preparation in turn induces localization in the "lower" end. One has a kind of time flip-flop and the breaking of time reversal invariance would be absolutely essential for the non-triviality of the process.

Note that direct sum, tensor product, and the counterpart of second quantization for Hilbert spaces in the proposed sense would be quantum mathematics counterpart for set theoretic operations, Cartesian product and formation of the power set in set theory.

ZEO, state function reduction, unitary process, and quantum mathematics

State function reduction acts in a tensor product of Hilbert spaces. In the p-adic context to be discussed n the following x_n⊗ pⁿ is the natural candidate for this tensor product. One can assign a density matrix to a given entangled state of this system and calculate the Shannon entropy. One can also assign to it a number theoretical entropy if entanglement coefficients are rationals or even algebraic numbers, and this entropy can be negative. One can apply Negentropy Maximization Principle to identify the preferred states basis as eigenstates of the density matrix. For negentropic entanglement the quantum jump does not destroy the entanglement.

Could the state function reduction take place separately for each subspace x_n⊗ pⁿ in the direct sum ⊕_n x_n⊗ pⁿ so that one would have quantum parallel state function reductions? This is an old proposal motivated by the many-sheeted space-time. The direct summands in this case would correspond to the contributions to the states localizable at various space-time sheets assigned to different powers of p defing a scale hierarhcy. The powers pⁿ would be associated with zero modes by the previous argument so that the assumption about independent reduction would reflect the super-selection rule for zero modes. Also different values of p-adic prime are present and tensor product between them is possible if the entanglement coefficients are rationals or even algebraics. In the formulation using adeles the needed generalization could be formulated in a straightforward manner.

How can one select the entangled states in the summands x_n⊗ pⁿ? Is there some unique choice? How do unitary process and state function reduction relate to this choice? Could the dynamics of Quantum Mathematics be a structural analog for a sequence of state function reductions taking place at the opposite ends of CD with unitary matrix U relating the state basis for which single particle states have well defined quantum numbers either at the upper or lower end of CD? Could the unitary process and state function reduction be identified solely from the requirement that zero energy states correspond to tensor products Hilbert spaces, which correspond to inverses of each other as numbers? Could the extension of arithmetics to include co-arithmetics make the dynamics in question unique?

What multiverse branching could mean?

Could QM allow to identify a mathematical counterpart for the branching of quantum states to quantum states corresponding to preferred basis? Could one can imagine that a superposition of states ∑ c_nΨ_n in a direct summand x_n⊗ pⁿ is replaced by a state for which Ψ_n belong to different direct summands and that branching to non-intefering sub-universes is induced by the proposed super-selection rule or perhaps even induces state function reduction? These two options seem to be equivalent experimentally. Could this decoherence process perhaps correspond to the replacement of the original Hilbert space characterized by number x with a new Hilbert space corresponding to number y inducing the splitting of x_n⊗ pⁿ? Could the interpretation of finite integers x_n and pⁿ as p-adic numbers p₁≠ p induce the decoherence?

This kind of situation is encountered also in symmetry breaking. The irreducible representation of a symmetry group reduces to a direct sum of representations of a sub-group and one has in practice super-selection rule: one does not talk about superpositions of photon and Z⁰. In quantum measurement the classical external fields indeed induce symmetry breaking by giving different energies for the components of the state. In the case of the factor x_n⊗ pⁿ the entanglement coefficients define the density matrix characterizing the preferred state basis. It would seem that the process of branching decomposes this state space to a direct sum 1-D state spaces associated with the eigenstates of the density matrix. In symmetry breaking superposition principle holds true and instead of quantum superposition for different orientations of "Higgs field" or magnetic field a localization selecting single orientation of the "Higgs field" takes place.

Could state function reduction be analogous process? Could non-quantum fluctuating zero modes of WCW metric apper as analogs of "Higgs fields". In this picture quantum superposition of states with different values of zero modes would not be possible, and state function reduction might take place only for entanglement between zero modes and non-zero modes.

The replacement of a point of Hilbert space with Hilbert space as a second quantization

The fractal character of the Quantum Mathematics is what makes it a good candidate for understanding the self-referentiality of consciousness. The replacement of the Hilbert space with the direct sum of Hilbert spaces defined by its points would be the basic step and could be repeated endlessly corresponding to a hierarchy of statements about statements or hierarchy of n^th order logics. The construction of infinite primes leads to a similar structure.

What about the step leading to a deeper level in hierarchy and involving the replacement of each point of Hilbert space with Hilbert space characterizing it number theoretically? What could it correspond at the level of states?

1. Suppose that state function reduction selects one point for each Hilbert space x_n⊗ pⁿ. The key step is to replace this direct sum of points of these Hilbert spaces with direct sum of Hilbert spaces defined by the points of these Hilbert spaces. After this one would select point from this very big Hilbert space. Could this point be in some sense the image of the Hilbert space state at previous level? Should one imbed Hilbert space x_n⊗ pⁿ isometrically to the Hilbert space defined by the preferred state x_n⊗ pⁿ so that one would have a realization of holography: part would represent the whole at the new level. It seems that there is a canonical manner to achieve this. The interpretation as the analog of second quantization suggest the identification of the imbedding map as the identification of the many particle states of previous level as single particle states of the new level.
2. Could topological condensation be the counterpart of this process in many-sheeted spacetime of TGD? The states of previous level would be assigned to the space-time sheets topologically condensed to a larger space-time sheet representing the new level and the many-particle states of previous level would be the elementary particles of the new level.
3. If this vision is correct, second quantization performed by theoreticians would not be a mere theoretical operation but a fundamental physical process necessary for cognition! The above proposed unitary imbedding would imbed the states of the previous level as single particle states to the new level. It would seem that the process of second quantization, which is indeed very much like self-reference, is completely independent from state function reduction and unitary process. This picture would conform with the fact that in TGD Universe the theory about the Universe is the Universe and mathematician is in the quantum jumps between different solutions of this theory.

For background see the chapter Quantum Adeles.