Non-spatial resolution of quanta (split off from Local realism ruled out? )

[jambaugh]

Non-spatial resolution of quanta (split off from "Local realism ruled out?")

I have taken the liberty of creating a new thread since this sub-discussion in
https://www.physicsforums.com/showthread.php?p=2561841&posted=1" had diverged a bit from the OP.

Moving to charge doesn't help -- you need "some thing" to "possess" the charge, even if you attribute it to the detectors. So, again, how do you distinquish two such otherwise identical "things" without space?

You distinguish two isomorphic things by distinct values of their observables, or in the case of coherent bosonic composites, you needn't distinguish them to count them.

It comes down to whether all observables are space-time observables which I point out is not the case. We can consider lepton-number or weak isospin or strong color. I can for example discuss a pair creation event and distinguish either by the one going this-away vs going that-away (invoking spatial degrees of freedom) or I can distinguish the one with positive charge vs the one with negative charge. Yes eventually I must observe that charge with a device possessing spatio-temporal qualities. That isn't the issue. The issue is this:

In factoring the system of e.g. anti-correlated electron-positron pair. I can speak of the electron and likewise the positron each of which being in a superposition of "this-away" vs. "that-away" spatial motion. OR I can speak of the "this-away" particle as in a superposition of being an electron vs a positron and likewise the "that-away" particle.

I may use either momentum or charge as the observable I use to speak of and distinguish components using "the". Neither of these (nor the continuum of admixtures of ways in between) is more or less valid and each is a different "reality model" of the composite as a pair of components.

Remember "superposition" isn't a system property, it is a frame relationship. We choose one basis (eigen-basis of a given non-degenerate observable) and then prepare a system by assuring a distinct value of that observable. Choosing a different basis yields that "non-superposition" system now as a superposition.

That then carries into resolving components of a composite system via various distinct values of common observables, said observables needn't necessarily be position, or momentum. Reality is relative in QM and the choice of how to factor is no different from a choice of inertial frame (time axis) for observers in SR.

I struggle here in part because e.g. QM is fundamentally time based, as relativistic QFT is fundamentally space-time based. Indeed there is the classic fiber-bundle structures in each case and "based" has literal meaning. But fiber-bundles indicate non-stable non-semi-simple choices of description. SR transformed the space fibers over time base into a unified space-time. To unify gauge and space and time we must go beyond field theories which are inherently fibrated. To say what I am trying to say more clearly probably will require a different language than QM or QFT,... say a language of quantized events (not necessarily localized in space-time) rather than of quantum systems (persisting over time). Heck, Feynman diagrams are almost such a language already.

 

[RUTA]

As long as you have different quantum numbers, you're ok. But, you have to use SOMETHING to distinquish two particles with the same quantum numbers, otherwise your formalism violates the principle of identity and it will be impossible for anyone to follow it. Multiplicity iff discernibility.

No matter how you approach the pregeometry (topological versions of space & time & stuff), you must recover trans-temporal objects which, other than spatial location, are identical. That's an empirical fact in accord with current physics and your theory must make correspondence with current successful theory. I don't yet see how these qualities (spatiality and temporality) emerge from anything you've proposed without being put in a priori. You've conceded that maybe time has to be put in topologically, but then I'm curious to see how you get Lorentz invariance because, as you said, if you're using pregeometric time but not a pregeometric space, you'll need to mix these notions at the geometric level without a basis for that mixing at the topological level. We solved that problem by using graphical relations to co-define topologically what we mean by space, time and sources in a way that leads to Lorentz invariance at the geometric level, so I'm curious to see if it can be done with one less item, i.e., without a topological notion of space, as you propose.

Fundamentally it strikes me as impossible, because I can't see how to avoid identity and differentiation in the construct of "things," plural, no matter what the "things" are. Again, multiplicity iff discernibility. But, as you said, that's simply a statement of ignorance.

 

[jambaugh]

As long as you have different quantum numbers, you're ok. But, you have to use SOMETHING to distinquish two particles with the same quantum numbers, otherwise your formalism violates the principle of identity and it will be impossible for anyone to follow it. Multiplicity iff discernibility.

Quite correct. But if the quanta are fermions then that isn't a problem as the statistics dictates their distinctness. But again why need they be necessarily distinguishable? One can factor a system into independent isomorphic components which sometimes have identical q-numbers and thus are not distinguishable, but other times can be given distinct q-numbers one distinct one of which may be used to index them.

No matter how you approach the pregeometry (topological versions of space & time & stuff), you must recover trans-temporal objects which, other than spatial location, are identical.

That's an empirical fact in accord with current physics and your theory must make correspondence with current successful theory.

One would hope so. And it is the recovery of such which is the ultimate goal. But in the process one would hope to also get some prediction as to how this recovery may manifest, especially (dare one hope) a dynamics of the variable geometry a.k.a. quantum gravity.

I don't yet see how these qualities (spatiality and temporality) emerge from anything you've proposed without being put in a priori. You've conceded that maybe time has to be put in topologically, but then I'm curious to see how you get Lorentz invariance because, as you said, if you're using pregeometric time but not a pregeometric space, you'll need to mix these notions at the geometric level without a basis for that mixing at the topological level.

Note I don't even have what could be remotely called a theory. Only a notion of an approach. But note Lorentz invariance was already there in Maxwell's theory before it was recognized and invoked as a principle and the theory reformulated with manifest Lorentz covariance.

Also note that, e.g. in the aggregate water, consisting of anisotropic components, is none the less isotropic. The lack of SO(3) symmetry of the components doesn't interfere with the SO(3) symmetry of the composite. One similliarly might manifest Lorentz symmetry. What's more, the symmetry is there only in the vacuum and indeed defines the vacuum.
It's really Lorentz covariance one needs to invoke.

We solved that problem by using graphical relations to co-define topologically what we mean by space, time and sources in a way that leads to Lorentz invariance at the geometric level, so I'm curious to see if it can be done with one less item, i.e., without a topological notion of space, as you propose.

Fundamentally it strikes me as impossible, because I can't see how to avoid identity and differentiation in the construct of "things," plural, no matter what the "things" are. Again, multiplicity iff discernibility. But, as you said, that's simply a statement of ignorance.

Not impossible. You just are not considering all the degrees of freedom, some of which would eventually manifest as location.

As I've said I don't have a theory yet but consider a simple 2-dimensional model of spin-1/2/ particles. Firstly we need to compactify our space so let our 2-D universe be a 2-sphere with SO(3) symmetry. We can then identify the spherical harmonics on this 2-sphere with the wave-function of a given particle. But that spherical harmonic is also an element of an irreducible representation of SO(3) representable as a totally symmetrized tensor product of an even number of fundamental su(2) irreps. We can then in essence treat the single scalar particle as a composite of say N "spinons". Whats more we can add singlet pairs of spinons to pad the total number to fit some universal maximum for our toy universe.

Picture a young diagram of N boxes (#) of the form:

$...$$...#$...#

supposing there are k boxes in the second row and thus 2k of the N total component irreps are in "singlet" pairs contributing 0 to the total SO(3) observables. The remainder form a tensor irrep equivalent to a given set of spherical harmonics:
\{ Y^m_\ell: \ell =N-2k, |m|\le \ell\}

Thus this set of N su(2) "spinons" models a single particle in 2-dimensional spherical space. The tensor product of them (letting them have Maxwell Boltzmann statistics) separates into the irreps of the above form defining a full set of modes up to some maximum total momentum for our single 2-d "particle".

Note also the Hamiltonian (in the absence of any other "things" in our 2-space to interact with) is the trivial one, H is the Casimir operator.

Now this is of course, just math, but the math that shows how one may construct a non-relativistic space from pre-geometric entities these "spinons" are just "qubits".
I am also at this stage putting the geometry in by hand by picking a specific connection between the various partons' internal su(2) groups. But to me that's a pre-local gauge condition. It suffices to pick some random connection, an arbitrary SO(3) subgroup of the SO(3)^N Lie group for the N components.

The components themselves are not distinguishable in the sense you're insist upon. They are isomorphic parts of the whole with abstract degrees of freedom. But again the components themselves are not the "things" which are to emerge as having full spatial degrees of freedom.

BTW I can extend the above construction to incorporate Poincare (or deSitter) composite symmetry plus additional gauge degrees of freedom. This also gives the composite system a "unified field of many quanta" rather than single particle interpretation as well.

 

[RUTA]

Quite correct. But if the quanta are fermions then that isn't a problem as the statistics dictates their distinctness. But again why need they be necessarily distinguishable? One can factor a system into independent isomorphic components which sometimes have identical q-numbers and thus are not distinguishable, but other times can be given distinct q-numbers one distinct one of which may be used to index them.

One would hope so. And it is the recovery of such which is the ultimate goal. But in the process one would hope to also get some prediction as to how this recovery may manifest, especially (dare one hope) a dynamics of the variable geometry a.k.a. quantum gravity.


Note I don't even have what could be remotely called a theory. Only a notion of an approach. But note Lorentz invariance was already there in Maxwell's theory before it was recognized and invoked as a principle and the theory reformulated with manifest Lorentz covariance.

Also note that, e.g. in the aggregate water, consisting of anisotropic components, is none the less isotropic. The lack of SO(3) symmetry of the components doesn't interfere with the SO(3) symmetry of the composite. One similliarly might manifest Lorentz symmetry. What's more, the symmetry is there only in the vacuum and indeed defines the vacuum.
It's really Lorentz covariance one needs to invoke.

Not impossible. You just are not considering all the degrees of freedom, some of which would eventually manifest as location.

As I've said I don't have a theory yet but consider a simple 2-dimensional model of spin-1/2/ particles. Firstly we need to compactify our space so let our 2-D universe be a 2-sphere with SO(3) symmetry. We can then identify the spherical harmonics on this 2-sphere with the wave-function of a given particle. But that spherical harmonic is also an element of an irreducible representation of SO(3) representable as a totally symmetrized tensor product of an even number of fundamental su(2) irreps. We can then in essence treat the single scalar particle as a composite of say N "spinons". Whats more we can add singlet pairs of spinons to pad the total number to fit some universal maximum for our toy universe.

Picture a young diagram of N boxes (#) of the form:

$...$$...#$...#

supposing there are k boxes in the second row and thus 2k of the N total component irreps are in "singlet" pairs contributing 0 to the total SO(3) observables. The remainder form a tensor irrep equivalent to a given set of spherical harmonics:
\{ Y^m_\ell: \ell =N-2k, |m|\le \ell\}

Thus this set of N su(2) "spinons" models a single particle in 2-dimensional spherical space. The tensor product of them (letting them have Maxwell Boltzmann statistics) separates into the irreps of the above form defining a full set of modes up to some maximum total momentum for our single 2-d "particle".

Note also the Hamiltonian (in the absence of any other "things" in our 2-space to interact with) is the trivial one, H is the Casimir operator.

Now this is of course, just math, but the math that shows how one may construct a non-relativistic space from pre-geometric entities these "spinons" are just "qubits".
I am also at this stage putting the geometry in by hand by picking a specific connection between the various partons' internal su(2) groups. But to me that's a pre-local gauge condition. It suffices to pick some random connection, an arbitrary SO(3) subgroup of the SO(3)^N Lie group for the N components.

The components themselves are not distinguishable in the sense you're insist upon. They are isomorphic parts of the whole with abstract degrees of freedom. But again the components themselves are not the "things" which are to emerge as having full spatial degrees of freedom.

BTW I can extend the above construction to incorporate Poincare (or deSitter) composite symmetry plus additional gauge degrees of freedom. This also gives the composite system a "unified field of many quanta" rather than single particle interpretation as well.

You brought spatiality in the back door in two respects. First, you started with a 2 sphere and ended up with a 2 sphere space. You did identify points on the 2 sphere with the symmetry group SO(3), so you can use that association to distinguish otherwise identical points. If you don't do it that way, you can use the atlas, as part of the definition of a differentiable manifold, to provide mathematical rescue. Often people say, "Imagine a 2 sphere whose points are indistinguishable." This relates to the second means by which you tacitly used spatiality to create multiplicity, i.e., you want to believe you have more than one # without any mathematical baggage (such as elements of SO(3) or the atlas in the case of S2). Do you truly have multiplicity without discernibility? Of course not,. The way you communicated the notion of multiple # was to make rows and columns on the page, i.e., you used the spatiality of the page. You have not avoided multiplicity iff discernibility with this example.

The distinction between Lorentz invariance and covariance is only relevant to algebraic approaches where one has state vectors (invariance) and operators (covariance). In our graphical approach, for example, there is no such distinction.

 

[jambaugh]

You brought spatiality in the back door in two respects.

You aren't paying attention. It's not sneaking "space" in or not. It's constructing the locality of space-time from a pre-local format. As I said we're arguing diagonal to each other.

First, you started with a 2 sphere and ended up with a 2 sphere space.

First I started with the relativity group of a quantum 2-dimensional system.

You did identify points on the 2 sphere with the symmetry group SO(3), so you can use that association to distinguish otherwise identical points.

The group is the group. That's just mathematical isomorphism. The structure of the irreducible su(2),SO(3) representations is representable as, and thus able to model scalar and spinor wave-functions on that 2-sphere. Clearly I can't model space without modeling space. The proper observation to make at this stage is that this mathematical sphere is not yet a representation of a space on which object interact locally. It is simply a starting point.

For example, in Galilean relativity we have for example the group ISO(3) of spatial translations and rotations and we have a distinct ISO(3) group of velocity frame translations and rotations. The mathematical isomorphism is not physical identification.

If you don't do it that way, you can use the atlas, as part of the definition of a differentiable manifold, to provide mathematical rescue.

Often people say, "Imagine a 2 sphere whose points are indistinguishable." This relates to the second means by which you tacitly used spatiality to create multiplicity, i.e., you want to believe you have more than one # without any mathematical baggage (such as elements of SO(3) or the atlas in the case of S2).

Are you saying I should be constructing spatial points or space-time points? They don't exist. I'm trying to show you how to construct a model of composite physical objects where by virtue of their composite nature they manifest the degrees of freedom reflecting spatial relationships.

I do not have to distinguish space-time points. They are not distinguishable. Space-time events, and spatial objects are distinguishable by virtue of not having identical values for their observables. Even indistinguishable quanta (such as the photons in a laser beam) are yet quantifiable.

Do you truly have multiplicity without discernibility? Of course not,.

Tell it to lasers!

The way you communicated the notion of multiple # was to make rows and columns on the page, i.e., you used the spatiality of the page. You have not avoided multiplicity iff discernibility with this example.

You are just being argumentative here. The #'s represent a Young diagram, a standard notation of irreducible representations of the unitary groups. Next you'll be telling me I'm sneaking space in because the symbols I use have spatial extent.

The distinction between Lorentz invariance and covariance is only relevant to algebraic approaches where one has state vectors (invariance) and operators (covariance). In our graphical approach, for example, there is no such distinction.

Huh? Execute an active Lorentz transformation on an electron and it is no longer in the same state. It is not invariant.

Here again I think I see the difference in our perspectives. Please withhold judgment about which approach is best for the moment and just let me know if I have correctly expressed your half of it.

As I see it space-time itself (GR not withstanding) is not physically real. This is not to say that it is any less physically meaningful but just that spatial, and space-time points are not themselves physical "things". Recall the idea of six degrees of separation which is to say on average a chain of acquaintanceship exists between any two people not longer than six links long. Now you could sit down and pick a random person, construct a weighted network with closest associates having closest links. You could estimate the dimension and topology of this network based on how fast the number of associates grew as you moved outward from anyone node. You would have then a simple spatial model of relationships. The physically real entities are the people, i.e. the nodes. The relationships are not the people. Carry this further, let the relationships reflect e.g. how long it has been since two people last physically met or communicated with each other and then allow that model to evolve over time. Replace people with elementary quanta and you have what I envision as spatial structure defined as the model describing the structure of the relationships between interacting quanta. The quanta themselves may be isomorphic but are distinguished by their relationships to other quanta. In some cases the distinct elements may not be spatially distinguishable (like a close knit family each member sharing the same associations) like quarks in a nucleon. As I then see it it is sufficient to model these physical quanta in such a way that the relationships may be defined but needn't manifest as physical links, only definitional ones as e.g. rate of intimate communication and numbers of links in causal interaction.

Now as I'm sensing you see things, and as field theories generally model physics, one considers a spatial array of systems which are "real in the model" but such that the physical objects we study in nature are aggregates of excitations of the components of the array. I.e. bosons are excitations of an array of simple harmonic oscillators (like phonons in a crystal.) Thus in particular you are concerned with the identification and distinguishability of each component system of the model. They in effect are the spatial points.

Is that a clear qualification of your position?

Now my position is now well formulated. As I've repeatedly said it is at present a notion not a theory or model. It really is only half a notion at that as the real physics is in the dynamics not the kinematic model. The principle problem I have it to explain the manifestation of dynamic locality in this picture. As it starts out pre-local there is not yet a reason to forbid every object from interacting intimately with every other object. As I am starting basically with "just mathematics" I probably am going to have to introduce the actual space-time structure, by hand as selection rules restricting causal interactions.

Now as to distinquishability. Consider an electron. We cannot truly separate the electron from the electro-magnetic field around it. As we see when we renormalize the mass of that electron is a manifestion of itself interaction. As we see in semi-conductor theory the electrons and holes conducting in the crystal are different, having different effective masses than free electrons principally because each particle's induced e-m field behaves differently in the distinct environments. We nonetheless can quantify charge and count the quantized charges which is what leads us to describe these electronic charge carriers.

From another vantage, we could enumerate types of elementary particles in different ways. We may speak of an electron while meaning a class of electrons with various spins or we may be more specific and speak of an L-electron or a R-electron. We can be less specific and speak of a lepton without being specific about its weak isospin charge, treating electrons and e-neutrinos as just different cases of the same lepton system.

In each case we write down e.g. a wave-function representing the system in question. More generally I would specify its Hilbert space and possibly give a density operator representing partial knowledge about it. Implicit in all of this is the space-time degrees of freedom for "the particle". There again we can be more or less specific as the application dictates.

Now when it comes to making distinctions, consider if I speak of two leptons, they are isomorphic and not distinguishable in that context. However if I speak of an electron, and an e-neutrino I have a.) reduced the context by b.) implying an observation has been made, and thus distinguished them. When it comes to describing a single electron I also describe its spatial degrees of freedom, its momentum or position or superpositions thereof. Typically in a lab we may describe isomorphic cases of e.g. an electron in a cavity with our cavities distinct (e.g. in different labs) and thus we are picking two subspaces of the big Hilbert space for an electron in the universe.

Now I wish to stay finite so I am considering theories where space is compact, e.g. a 3-sphere. That plus an upper cutoff on momentum let's me approximate the description in a finite Hilbert space. It is still of very large dimension. In that very large Hilbert space the electron in your cavity of your lab is distinguishable from my electron in my isomorphic cavity in my lab via distinct values for observables, i.e. we have observed different values for observables sufficiently to project into distinct though isomorphic subspace of the grand electron hilbert space. I say this to be sure that you know that I know this is the case. Clearly no matter what I do I'm going to have to distinguish, distinct particles by what will be spatial degrees of freedom.

• I've made the point that spatial degrees of freedom are not the only means to distinguish particle, e.g. we can look at isospin.
• I've made the point that we can construct the large degrees of freedom isomorphic to spatial ones, from atomic components, (my "spinons") Naturally the component group will be isomorphic to the spatial group I seek to model. Firstly because I want it to and secondly because the number of low rank compact groups is quite limited and coincidence will occur (and I'm counting on it).
• This however does not mean I am "sneaking space-time in the back door" by any means. Firstly the example I gave doesn't even have the true dynamic structure of space yet. As I said I haven't begun to construct the dynamics in that example. It was meant to show how the kinematic structure could be constructed without invoking an array of spatial point systems as you're doing with your discrete field theory.

Now in that "spinon" model I did put a bit of spatial geometry in by hand in a sense. I think I mentioned that I implied a connection by which N component partons manifest a representation of SO(3). Let me elaborate a bit on that.

The composite of those N qubits each has a unitary group of su(2) ~ SO(3), or more properly u(2)~SO(3)xU(1) (this is the group of possible dynamic evolutions, i.e.:
iH \in \mathfrak{so}(3)\oplus \mathfrak{u}(1).

Considered as a (maxwell boltzman) composite the total has a unitary group:
U(2^N)
A specific factorization (turning off any interactions) gives the sub-group with Lie algebra:
N\cdot \mathfrak{u}(2)=\bigoplus_{k=1}^N \mathfrak{u}(2)
The connection of which I spoke is an isomorphism mapping from the SO(3) group which will manifest as spatial transformations to each copy of the N U(2) groups. Without that this the geometry I construct is meaningless, and which irreducible or reducible representation of SO(3) this aggregate of partons manifest is likewise meaningless. My notion is that the selection of such, and even the factorization itself must evolve from the dynamics of the system. The dynamics itself is some generator of the big U(2^N) group. For an arbitrary factorization as I described I can write that generator as a sum of U(2) dynamics for each component plus interaction terms between components. Since the factorization, and dynamic, are at this point arbitrary, doing so is pretty meaningless at this point. (Again I'm still really only playing with the math here.)

However I can consider how varying the choice of factorization for a given dynamic guided by some meaningful principle could lead to a relatively unique class of factorizations... wherein the dynamics of the composite, now expressed in terms of treating that SO(3) group as a coordinate transformation group, leads to meaningful localization of interactions w.r.t. coordinates, then I will have demonstrated a means to manifest the spatial structure from the causal structure (dynamics). In short I suppose one could find how the choice of global dynamic leads to a manifestation of what we perceive as localized object interacting in space.

Now I don't think it can work here because, for one, we don't see a 2-dim universe. My thoughts are to a more involved model with, hopefully the manifestation of certain gauge degrees of freedom are necessary to get the sought-after locality. (Really it is a delocalization i.e. the distancing of objects manifest by weakened interaction.)
The real golden apple would be if such a program actually manifested the standard model plus gravity but that's really wishing hard on little substance. The realistic hope is the usual alternative paradigm for expressing observed nature with less put in by hand than is currently done with QFT and the standard model.

And let me also state that I whipped out this example to show how one might go about deriving spatial structure from the causal structure of the dynamics. It needn't be the only way or the best way.

Finally let me recall my original position which prompted this discussion. I claim that we should treat causal structure as primary and space-time geometry as derivative from that. I reiterate that in the case of my example which lacks as yet any dynamic thus any causal structure and thus its large 2-sphere may be as easily model spatial degrees of freedom as it may model a single particle with a very very large isospin. This is why I called it "prelocal" without the dynamics it doesn't yet have causal structure to make it local.

Where you object to me "sneaking space in the back-door" I would point out I haven't as I haven't yet manifested the causal structure of space-time. Many systems may have isomorphic group structure without being identical or even similar dynamically. Where you object that I must use space to distinguish things I say I first must distinguish things to relate them spatially and define coordinates. I would point out that a computer can encode a cube as a series of numbers in memory. It is not the numbers which make the cube but the computer code and hardware an how it treats those numbers. Imagine if the entire universe was "virtual" and then ask yourself what is the difference? The meaning is in the interactions which manifest our observations. We don't observe space, we observe objects in space.

 

[RUTA]

You have some interesting ideas which I’m eager to discuss. However, I need to clarify a problem I’ve created by my violation of the principle of identity (A = A).

I’ve been using the word “spatiality/space” to mean differentiation generally and the “space” of spacetime specifically. It’s a problem inherited from math, e.g., vector space, Hilbert space, etc., but in this context I have to be careful. Confusion of this sort is evidence of what happens when we violate a principle of logic. Now let me explain why the same will obtain if we violate multiplicity iff discernibility (MiD). I’ll begin by showing you how it applies in standard set theory.

Let the domain of discourse be that of proper names.

You claim to have a set of order unity. I ask you to show me the set and you give me {Bob, Tim}. I conclude that the order of your set exceeds unity – discernibility is sufficient for multiplicity.

You claim to have a set whose order exceeds unity. I ask you to show me the set and you give me {Bob}. I conclude that the order of your set does not exceed unity – discernibility is necessary for multiplicity. [This is the basis for A = A of standard logic.]

Like any principle of logic, a violation of MiD can cause confusion. Suppose, for example, that you introduce the set {Bob} in a discussion. Anyone seeing that set will assume it is order unity, so if you want it to represent “5 indiscernibles” (akin to your example of photons), you’ll have to add a qualifier (which then creates discernibility). I didn’t qualify (provide discernibility for) my use of the word “spatiality” in what were intended to be different contexts and it caused confusion.

Given MiD, I define “topological” bases for time and space as follows:

Identification (basis for time) – the construct of apparent identity from true multiplicity. “Me at 2 years old” is discernible from “me today,” yet we consider “me” to be “one thing.”

Differentiation (basis for space of spacetime) – the construct of true multiplicity from apparent identity. I have two hydrogen atoms in this jar.

So, from now on I will try to use the word “space” to mean that subset of spacetime, rather than “differentiation” in general. Of course, we’ll have problems if we want to use “differentiation” to mean “taking derivatives,” but thus far we’ve been safe in that respect. Let me pause here and see if you’re on board. If so, let’s move on to your ideas.