Notes
Sunday 7 March 2021 - Saturday 13 March 2021
[Notebook: DB 86: Hilbert / Minkowski]
[page 117]
Sunday 7 March 2021
Wheeler: Search for Links: Leibniz: '. . . time and space are not things but orders of things.' [80] Einstein: 'Time and space are orders by which we think and not the conditions in which we live.' [81] Nevertheless, time and space are physical realities / fundamental symmetries. John Archibald Wheeler: Information, Physics, Quantum: The Search for Links
Rovelli reinforced my ancient belief that quantum mechanics is an information processing system which I first learnt from Feynman and Zurek. How does this fit my program to turn the initial singularity into a universe? It all revolves around making copies of god which began with the Trinity and probably Eastern beliefs about fertility and emanation [the Trinity is a structure created by real relationships]. Carlo Rovelli: Relational Quantum Mechanics, Aquinas, Summa I, 28, 1: Are there real relations in God?
[page 218]
Computer networks implement the Aristotelian definition of continuity "ends in common". To send a message from one part of a computer to another the two have to read to and write from the same memory location. The same happens in nature with massless gauge particles like the photon which exist outside time. One particle writes a message on a photon (a phase) which is read by another particle, the photon acting as a shared memory. This is the role of bosons. Massive bosons are also eternal, so long as they last, and are also able to carry SU(2) and SU(3) more complex versions of phase, just as this piece of paper is in effect carrying phase in a high entropy space, the written english language.
The Catholic Church has a good memory and has been able to sustain the same delusions for 2000 years. We may think the same of military and poltiical history, but would like to think that generation after generation are revising these delusions. But the delusion of continuity is older than the Church and is deeply embedded in physics. How much damage is it causing? cf Wheeler [ref above]
Monday 8 March 2021
Physicists are concerned to get the numbers right but they have incorporated so many adjustable parameters into their more fundamental models that one wonders whether different models might not lead more plausibility to the same result. I find it very hard to understand what they are doing much of the time and so I feel a need to understand it all in my own terms by reading alternative pictures
[page 119]
like that developed by Francis.
Francis [from page 102-3, 28 February 2021]
Not much help here, so we go to Charles Francis: Charles Francis: A construction of full QED using finite dimensional Hilbert space
page 3: 'In standard approaches to quantum field theory one starts with a classical field and then quantises it. Space is thus a fundamental physical
[page 103]
concept on which the theory is built. Covariance requires that space must be a continuum and hence that if a lattice is used the limit of small lattice spacing must be taken. In the present treatment quantum properties are understood to arise precisely because space does not appear as a fundamental physical concept [I like this, of course].
'The interpretation here follows Dirac and Von Neumann, but goes further than either. . . . It is found that this formulation of quantum mechanics allows a complete construction of qed in which Maxwell's equations and the Lorentz force law are derived in the classical correspondence.
Francis page 6: Space-time coordinate lattice.
page 8: Quantum logic: Birkoff and Von Neumann. Quantum logic - Wikipedia, Garrett Birkoff & John Von Neumann: The Logic of Quantum Mechanics
'Kets are interpreted as formal conditional clauses. The dual space [bras] consists of corresponding consequent causes. The inner product combines clauses to generate formal propositions in the subjunctive mood, showing that the language is a consistent and intuitive extension of two-valued logic and classical probability theory. The principle of superposition is simply logical disjunction in formal language. There is no suggestion of an ontological magnitude |< x | f >| associated with a particular particle.' [tbc, 7 March]
Francis [from page 111 , 28 February 2021]
Now back to Charles Francis [from page 102-3, ]
Francis page 3: This paper reviews the Fock space formulation of quantum electrodynamics in the context of an axiomatic formulation of quantum theory using a finite Hilbert space based on the principle described by, eg Rovelli, that all measured quantities are relational, not just velocity as in relativity. Fock space - Wikipedia
Francis page 4: 'Here Hilbert space is abstracted from the formal statement of sentences in ordinary language. The principle of superposition is not assumed as part of the structure of Hilbert space, but is exhibited as a property of conditional and consequent causes in a formal language describing the meaurement of results.
It will be shown that although the theory is superficially not covariant, a new form of covariance, quantum covariance, is obeyed (section 3.2).
It is found that this formulation of quantum mechanics allows a complete construction of qed in which Maxwell's equations and the Lorentz force law are derived in classical correspondence. The fundamental physical concepts are particles, and Feynman diagrams have a natural interpretation in terms of interactions between particles in the absence of a space-time background. The predictions or perturbative qed are unaltered.
page 5: 'Here measurement is distinguished from a simple count of a number of objects and is defined to mean a count of units of a measured quantity, where the definition of the unit of measurement involves comparison between some aspect of the subject of measurement and a property of the reference matter used to define the unit of measurement.'
page 6: 'We are particularly interested in measurement of time and position.'
[page 120]
Lattice Cartesian coordinates represented by ν ∈ N. Knowledge of a ket is a triplet of ± integers with respect to the lattice. Times is measured on the lattice too, so c = 1.
Francis page 7: 2.3 Particles
'An elementary particle is one which cannot, even in principle, be subdivided into particles for which separate positions can be measured.
2.4 Many valued logic
Boolean {0, 1}. Bayesian [0. 1], value given to proposition in the future tense.
page 8: R(x): 'In a measurement of position, the result would be x.
'Kets are interpreted as formal conditional clauses, rather than propositions. The dual space consists of the corresponding consequent clauses.
2.5 Formal language: "+ = or, in a| f > + b| g > = weighed disjunction'.
Tuesday 9 March 2021
Francis page 9: Braket <x | y>: 'If a measured position at time t were y then in a second measurement at t the position measured would be x.' Can only be true (per impossibile) if x = ypage 10: Truth value of <x | y> is Kronecker delta δxy.
'With linearity and complex conjugation, this defines an
[page 121]
inner product between two kets | f >, | g > ∈ H1(t). Note the overloading the the notation such that <f | g> is both a statement and its truth value. Thus H1(t) is a Hilbert space, the basic conditional clauses of rule I are an orthonormal basis and the space of bras is the dual space.
' for any complex number a the clause |f > means exactly the same as a|f >. When not part of a larger construction containing +, a has the role of a redundant word.
page 11: Postulate; 'The space of any number of particles of the same type, γ is Hγ = ⊕nHn.
page 12: 'multiparticle space is Fock space F ≡ ⊕SHn where S means that groups of tensor indices referring to the same types of particles are symmetrized for Bosons and anti-symmetrized for Fermions.'
Hilbert space is made of statements, ie quanta of action with truth values when connected to one another. The logical rather than the numerical picture is slowly seeping in. Characterizing particles by spacelike position seems a bit retro - what seems more interesting is their role as snippets of code.
The Piano tuner of Earthquakes The Piano Tuner of Earthquakes - Wikipedia
[page 122]
Biting off more than I can chew is a way forward that makes it impossible to turn back, making progress inevitable, I hope.
Droz (?) 'Music the most irrational rationality of all.'
The aim is to give theology a new heart, the true god.
The exponent in all quantum mechanical equations has the dimension of action reduced to a scalar by dividing by h.
Wednesday 10 March 2021
The more I read of quantum field theory the more it gnaws at me that it is unnecessarily complicated. Complication arises from permutation and combination. Although the hydrogen atom is exceedingly simple, comprising an electron and a proton, it nevertheless has a countably infinite number of discrete electronic states coded by the principal and subsidiary quantum numbers which refer to the multiple channels of communication between the electron and the proton associated with different values of angular momentum (action) and [consequently] energy.
The complexity of the nucleons has a similar explanation although the starting point here is a set of eight gluons and six quarks in both sets of particles and antiparticles. The question arises: are there simpler states from which these
[page 123]
states are constructed? . . . There is evidence for an affirmative answer in the gravitational prediction of the initial singularity. We can break atoms into nucleons and electrons and detect photons that bind electrons to their nucleons, but we have now learnt that there is evidence for constituent parts in protons, for instance, but they cannot be observed as free particles because their binding energy is so great that the effort to separate them simply creates more nucleons and gives us a clue to the original emergence of the vast numbers of fundamental particles to be found in the universe. So are there deeper layers of confined states which cannot be made manifest as free particles moving in space-time in some way, not capable of existing alone in spacetime but must remain in close union with their confreres? If we imagine each of these particles as snippets of code we may see them as a system like a cell which needs the cooperation of all of its subsystems to exist in spacetime as an independent entity.
The idea here is that we can make structures of states in Hilbert space which are subject to a form of invisibility theorem arising from the fact that they are too simple to be observed because to be observed is to communicate and if all your energy is taken up by the task of survival you may have got nothing left over to explain what you are doing to a potential observer [which hidden states are necessary to carry us from the initial singularity to the structure of protons and related particles]. Jeffrey Nicholls: Is the whole truth inacessible? A case for invincible ignorance
The fundamental issue in the music of the spheres is timing, ie phase. Digitized phase means complete quanta of action which we might call 1/1 time, overlaying frequencies / energies / bosons / fermions / colours.
[page 124]
So the question arises: how do particles actually interact? From a frequency [/ energy] point of view it is a matter of harmony, as with photons and electrons. Why does light shine through glass? Because there are no electrons with the frequency to capture it. This can work prior to space, action, time, energy, freuency only. From action to potential and kinetic energy, form and energy. Thinking in terms of traffic time division multiplexed by traffic lights.
Thursday 11 March 2021
We learn from the Trinity that reality is created by relationships which are created by communication and the initial god, like the initial singularity, is effectively nothing until it duplicates itself like the father creates the son and enters into a relationship with it which is represented in the Thomistic tradition by the spirit. From a quantum mechanical point of view all Hamiltonians are interaction hamiltonians arranged in a hierarchy so that, for instance, the interactions that create atoms are the symmetries that underlie the interactions that create molecules. Going back the other way we need to examine the interactions that create nucleons and other particles like electrons and protons [to see whether they] are built on more primitive particles and their relationships
A concept of a Fock space of set of identical particles does not make much sense since independence is equivalent to orthogonality and we can only conceive of orthogonality by the inner product of two vectors, so independence can only be understood through communication which is the contrary of independence [an instance of the via negativa?].
[page 125]
In short, communication is everything, everything is communication. So perhaps we can work out how bosons (inhabitants of Hilbert space) become fermions (inhabitants of space-time) and clearly it has something to do with spin which is another name for action. What is the real difference between spin 1 and spin ½ [and what would a mathematical explanation add to our understanding over and above the fact that one is not the other and they behave differently when permuted, and how does the wave function know this and behave accordingly]?
Friday 12 March 2021
Aquinas invokes the via negativa to discuss the structure of god, arriving at the conclusion that god has no structure, ie that god is altogether simple (omnino simplex. This is in effect an example of logical confinement. Given two possibilities |a> and |a> (propositions in the Francis sense) if one is impossible the other must be true. Here we are trying to understand why the initial singularity would want to change into a universe. The standard story is that it is endowed with infinite energy which implies an infinite frequency of quantum fluctuation [which may just make sense if we interpret infinite to mean indeterminate, without boundary]. Here we are trying to reconcile the initial singularity with the more primitive quantum of action becasue it seems hard to imagine the notion of infinite energy before the emergence of time and therefore the emergence of spacetime. This might be just a quibble, but the other motivation is to start from a point of harmony with the ancient view of god as actus purus, the realization of all possibility. This can be reconciled with the Thomistic view of god as omnino simplex if we assume that there is only one possibility for god, to be god and nothing else. But then, of course, along comes the Trinity which we see as the first step in the differentiation of god into the universe, introducing, in modern terms, a threesome of bosons and fermions.
[page 126]
Does the turn to quantum theory assist my theology project? We have to look behind the observations appearing in the classical world at the thermodynamic limit to see what quantum mechanics has to offer : a) an important point may be entanglement, which serves to create a quantum network which we may see as underlying and explaining the classical spacetime network offered in Prolegomenon; b) quantum theory seems well adapted to communication and computation which are the backbone of network theology; and c) quantum theory is well adapted to the creation and annihilation as the fundamental mechanisms of change in the universe. Jeffrey Nicholls (2019): Prolegomenon to Scientific Theology
My attempts at simplifying quantum [field] theory will not carry much weight unless they introduce significant simplifications into the calculations necessary for physics and engineering [nevertheless, before we can apply the mathematics we need a model to apply it to].
Perturbation in quantum theory: in the network model every source is being perturbed by all the messages it receives from other sources. The 'coupling constants' in the perturbations may vary with the content of the received message, so in the treatment of perturbation at the human scale, news of the death of my child may perturb me much more than a flat tyre on my car [and in physics the receipt of a graviton may have a very different effect that the receipt of a photon].
Francis page 47: '. . . Feynman diagrams describe the fundamental structure of a particulate relational model in which only particles exist and in which other properties, including spacetime geometry, emerge from interaction between particles.' Really? You have already introduced spacetime geometry by assuming that energy-momentum is four dimensional.
So how do we get from the quanta of action in Hilbert space to the properties of 4D Minkowski space / 4D energy momentum?
[page 127]
Saturday 13 March 2020
Birkoff & von Neumann Garrett Birkoff & John Von Neumann: The Logic of Quantum Mechanics [ref above]
Birkoff page 823: 'The object of the present paper is to discover what logical structure one may hope to find in physical theories which, like quantum mechanics, do not conform to classical logic. Our main conclusion, based on admittedly heuristic arguments, is that one can reasonably expect to find a calculus of propositions which is formally indistinguishable from the calculus of linear subspaces with respect to set products, linear sums, and orthogonal complements - and resembles the usual calculus of propositions with respect to and, or, and not.'
page 824: Measurements μi yield values xi which lie in a subset S of xi space. xi spaces are "observation spaces" and subsets of S are "experimental propositions" concerning S.
"mathematical causation" describes time evolution of points in phase space.
'In any case the law of propagation may be imagined as introducing a steady fluid motion in the phase space.'
'. . . in many important cases of classical dynamics. this flow conserves volumes. . . . in quantum mechanics the flow conserves distances [in Hilbert space] ie all the equations are "unitary" [rotations at constant scale?]
page 825: '4. Propositions are subsets of phase space. . . . before a phase space can be imbued with reality its elements and subsets must be correlated in some
[page 128]
way with "experimental propositions".
Birkoff page 826: 'In quantum theory . . . the possibility of predicting in general the readings from measurements on a physical system S from knowledge of the "state" is denied; only statistical predictions are always possible.
'. . . a more subtle idea is involved. The central idea is that physical quantities are related but are not all computable from a number of independent basic quantities.
'We shall show in §12 that this situation has an exact algebraic analogue in the calculus of propositions
'5. Propositional calculi in classical dynamics
'. . . uncritical acceptance of the ideas of classical mechanics leads one to identify each subset of phase space with an experimental proposition . . . and conversely.'
'. . . at least in statistics it seems best to assume that it is the Lebesque measurable subsets of phase space which correspond to experimental propositions . . . " Lebesgue integration - Wikipedia
page 826: 'The experimental propositions concerning any system in classical mechanics correspond to a "field" of subsets of its phase space.'
'6. A propositional calculus for quantum mechanics.
'(1) . . . the mathematical representative of any experimenta proposition is a closed linear subspace of Hilbert space (2) since all operators of hilbert space are Hermitian, . . .
[page 129]
the mathematical representation of the negative of any experiemental proposition is the "orthogonal complement" of the mathematical representative of the proposition itself . . .'
Birkoff page 827: 'Postulate: The set theoretical product of any two representatives of experimental propositions concerning a quantum mechanical system is itself a mathematical representative of an empirical proposition [superposition?].
' This postulate . . . would even be implied by the conjecture that those operators which correspond to observations coincide with the Hermitian symmetric elements of a suitable operator ring M' (ref Murray and Newman). Von Neumann algebra - Wikipedia
Here (I suspect) we see how natural selection works at the quantum mechanical level, selecting "observables" from all other possibilities.
'II. Algebraic Analysis
7. Implication as partial ordering'
page 828: '. . . we see that the properties of logical implication are indistinguishable from those of set inclusion, and that therefore it is algebraically reasonable to try to correlate physical quantities with subsets of phase space.
So 'our first postulate concerning propositional calculi: . . . physical properties attributable to any physical system form a partially ordered system.'
Two additional propositions: system exists; and system does not exist, ie is identically false, absurd or self contradictory.
[page 129]
'8. Lattices
Birkoff: page 829: 'Def: A partially ordered system has a greatest lower bound (intersection) and a least upper bound (union). Corresponding to and and or.
In quantum mechanics intersection and union constitute an experimental proposition only when experimental propositions commute [which is true of classical experimental propositions].
page 839: '9. Complemented lattices' (not): not(not-p) = p. [In the quantum mechanical world this need not be true, since not (not vector x) may well turn out to be vector y since spectra of operators are not necessarily binary.]
'10. Distributive identity' - formal feature which distinguished quantum from classical logic.'L6. a ⋃ (b ⋂ c) = (a ⋃ b) ⋂ (a ⋃ c) classical but not quantum.
page 831: 'L6. is a consequence of the compatibility of observables a, b and c.
'. . . if a denotes the observation of a wave-packet ψ on one side of a plane in ordinary space, a' ψ observed on the other side and b the observation of a state symmetrical about the plane we have b ⋂ (a ⋃ a' ) = T, ie a = a' ?.
II. Modular Identity
Moving from Hilbert to Minkowski space we may think in terms of transduction - my enormously complex interior, mapped onto Hilbert space, communicates by sensors and muscles with my environment using complex [internal] processing for the externally simple large scale process of navigating in space.
[page 131]
Birkoff page 832: 'L5: if a ⊃ c then a ⋃ (b ⋂ c) = (a ⋃ b) ⋂ c.
In Hilbert space we can find counterexamples to L5.
page 833: Relation to abstract projective geometries.
'the propositional calculus of quantum mechanics has the same structure as an abstract projective geometry' [where the space projected into may have many dimensions, ie spectral elements]
page 586: 'One conclusion which can be drawn from the preceding algebraic considerations, is that one can construct many different models for s propositional calculus in quantum mechanics. which cannot be differentiated by known criteria' [hidden variables?].
'16. The logical coherence of quantum mechanics. '. . . The above heuristic considerations suggest in particular that the physically significant statements in quantum mechanics actually constitute a sort of projective geometry while the physically significant statements in classical dynamics constitute a Boolean algebra.'
'they suggest more strongly that whereas in classical mechanics any propositional calculus involving more than two propositions can be decomposed into independent constituents (direct sums in the sense of modern algebra), quantum theory involves irreducible propositional calculi of unbounded complexity. This indicates that quantum mechanics has greater logical coherence than classical mechanics - a conclusion corroborated by the impossibility in general of measuring different quantities independently.'
