Note: Started 3 February 2026
Patrick Reany
Note: This chat I had with Copilot is likely edited for relevance and brevity. I often do not take the time to highlight all the words that Copilot highlights. Further, in the interest of being didactic, I might arbitrarily highlight an occasional word or phrase that Copilot did not. (Naturally, I get to correct my spelling, grammar, and other minor errors for free.)
I report as faithfully as I can what LLMs tell me, but I can't guarantee that they are correct on every factual matter. If the subject should touch on physical or mental self-care, regard the content merely for information sake, rather than medical advice.
I have discussed quantum mechanics (QM) in its various forms with Copilot for a couple years now, so why this emphasis on QM here? Simple. I want to have Copilot explain QM from the viewpoint of the 'Great Attractor Program' (GAP), which it has been developing over the last couple months. In time, I want to have a discussion/presentation of all the major aspects of QM available through this index page.
The GAP theory has been presented many times in the various chats I had with Copilot on Entropy and elsewhere (i.e., previously). See those chats for a foundation on the GAP approach to organization/description of physical theories: GAP/Copilot regarding Entropy.
The following list is the plain index to the topics of QM chats ready for download. Following the list are short teasers to each chat I had with Copilot on QM from the GAP perspective.
This first chat I had with Copilot specifically on Quantum Mechanics from the GAP perspective will cover at length the Heisenberg-Schrodinger-Born presentation of QM. (For the time being, we'll skip the Bohr model.) Some highlights include:
Patrick:
You said for the first two keepers:
2–3. Heisenberg / Schrödinger‑BornOkay, I'm ready to see your GAP synthesis.Keep both, but treat them as:
- Dual presentations of non‑commutative observable algebras
- With entropy defined relative to commuting subalgebras
- Born rule as a maximum entropy extension under spectral constraints
They are foundational, but GAP will compress them into a single conceptual block.
Copilot:
GAP synthesis for Heisenberg and Schrödinger–Born
GAP’s move is to treat “Heisenberg vs Schrödinger” as a gauge choice (picture choice) inside a deeper object: an observable algebra plus an information interface. The foundational question becomes:
What can an agent stably ask about a system, and how do those questions compose under coarse-graining?That’s where entropy enters as a bookkeeping device for lost distinguishability.
The kernel object
Observable algebra
Take an (abstract) unital \(C^{*}\)-algebra \(\mathcal A\) as “the system’s observables.” In finite dimensions you can read this as \(\mathcal A = \mathsf{B}(\mathcal H)\). The key split is:
States as probability rules
A state \(\omega\) is a positive normalized linear functional on \(\mathcal A\): it assigns expectation values \(\omega(A)\). In Hilbert-space language \(\omega(A)=\mathrm{tr}(\rho A)\).
So the primitive data are \((\mathcal A, \omega)\), not \(\psi(x,t)\) nor matrices per se.
Measurement as a two-step map
GAP separates measurement into two conceptually different operations.
Step 1: Decohering projection
You apply a coarse-graining that discards relative phases between the sectors defined by \(\mathcal C\). In the simplest case of projectors \(\{P_i\}\) generating \(\mathcal C\), the map looks like: \[ \mathcal D(\rho)=\sum_i P_i \rho P_i. \] This is not “collapse to an outcome.” It’s collapse of the description to the interface.
Entropy content: \(\rho \mapsto \mathcal D(\rho)\) typically increases entropy because it discards distinctions that were only encoded in noncommuting observables.
Step 2: Conditioning on an outcome
If you read an outcome \(i\), you condition: \[ \rho \mapsto \rho_i=\frac{P_i \rho P_i}{\mathrm{tr}(P_i\rho)}. \] This step is epistemic: it updates your state given a registered event. GAP treats it like Bayesian conditioning—but crucially, conditioning is only defined once you’ve fixed the commuting interface \(\mathcal C\).
Why GAP insists on this language
GAP elevates “observable algebra” because it is the minimal structure needed to talk about information loss.
Here’s the crucial point:
Entropy is not a property of states alone.You cannot even define entropy without first saying:
It is a property of states relative to an observable algebra.
Wavefunctions hide this choice. Algebras expose it.
Copilot presents its version of a QM axiomatic system on which (supposedly) everything it clains in this first chat can be proved thereby.
My first chat with Copilot specifically regarding QM from the GAP perspective.
This second chat I had with Copilot specifically regarding Bohmian Mechanics from the GAP perspective:
Patrick:
In our last chat, we finished our introduction to the Heisenberg-Schrodinger-Born theory, our first presentation on how GAP views the various subdisciplines within the larger Quantum Mechanics picture. This time it's GAP's view on Bohmian Mechanics. As a reminder from the introduction (in the first chat):
GAP has something important to say here:
Copilot:
GAP on Bohmian Mechanics
Entropy, Hidden Variables, and Epistemic Instability
Bohmian mechanics is often presented as a restoration project: classical realism rescued from quantum indeterminacy by adding hidden variables—particle positions guided by a deterministic law. GAP treats this move very differently. Not as wrong, but as epistemically unstable once entropy and coarse‑graining are taken seriously.
What Bohmian Mechanics Is Really Doing
At its core, Bohmian mechanics makes three commitments:
This is usually framed as “determinism underneath, probability on top.” GAP reframes it more sharply:
Bohmian mechanics externalizes entropy by relocating epistemic uncertainty into inaccessible micro‑variables.The randomness hasn’t disappeared. It has been pushed behind the curtain.
The GAP verdict on Bohmian Standard Model ambitions
This is not a condemnation. It’s a classification.
The deeper lesson (and why this matters)
Bohmian mechanics works best when:
The Standard Model is the opposite of that environment.
So GAP predicts—correctly, in hindsight—that Bohmian mechanics would:
That prediction does not rely on taste. It follows from how gauge theories and RG actually work.
Conclusion
Bohmian mechanics succeeds as a conceptual completion of quantum mechanics but fails to become the organizing language of modern quantum field theory. Its classical-particle ontology delivers early clarity at the cost of long-term robustness under coarse-graining, renormalization, and universality.
From the GAP perspective, this is not a refutation but a classification:
The overarching lesson is sharp: In quantum field theory, realism that does not align with projection-stable structures becomes technical debt.
My second chat with Copilot specifically regarding Bohmian Mechanics from the GAP perspective.
Restrict to the position-only algebra and compute the reduced kernel explicitly
The “spin-blind” observable algebra consists of operators of the form \[ A = A_{\text{pos}}\otimes I_4, \] so the induced bookkeeping state is \[ \rho_{\text{pos}}=\mathrm{Tr}_{\text{spin}}(\rho). \] In position representation, the reduced density operator is conveniently expressed by its integral kernel: \[ \rho_{\text{pos}}(\mathbf x,\mathbf y) = \sum_{\alpha=1}^4 \psi_\alpha(\mathbf x)\,\psi_\alpha(\mathbf y)^* = \psi(\mathbf x)^\dagger \psi(\mathbf y). \] Now plug in \(\psi(\mathbf x)\). You get four terms: \[ \rho_{\text{pos}}(\mathbf x,\mathbf y) = p\,\varphi_1(\mathbf x)\varphi_1(\mathbf y)^*\,\tilde u^{(1)\dagger}\tilde u^{(1)} + (1-p)\,\varphi_2(\mathbf x)\varphi_2(\mathbf y)^*\,\tilde u^{(2)\dagger}\tilde u^{(2)} \] \[ \quad +\sqrt{p(1-p)}\,\varphi_1(\mathbf x)\varphi_2(\mathbf y)^*\,\tilde u^{(1)\dagger}\tilde u^{(2)} +\sqrt{p(1-p)}\,\varphi_2(\mathbf x)\varphi_1(\mathbf y)^*\,\tilde u^{(2)\dagger}\tilde u^{(1)}. \] Using \(\tilde u^{(s)\dagger}\tilde u^{(s')}=\delta_{ss'}\), the cross terms vanish and \(\tilde u^{(1)\dagger}\tilde u^{(1)}=\tilde u^{(2)\dagger}\tilde u^{(2)}=1\), so \[ \rho_{\text{pos}}(\mathbf x,\mathbf y) = p\,\varphi_1(\mathbf x)\varphi_1(\mathbf y)^* + (1-p)\,\varphi_2(\mathbf x)\varphi_2(\mathbf y)^*. \] That is exactly the mixed state \[ \rho_{\text{pos}} = p\,|\varphi_1\rangle\langle \varphi_1| + (1-p)\,|\varphi_2\rangle\langle \varphi_2| \] derived earlier—but now the “orthogonal internal labels” are explicitly Dirac spin states.
GAP reframing: the positron as forced duality
From the GAP perspective, the positron is the first clear instance of a general principle:
When an algebra admits a symmetry, the ontology must realize it — or the theory collapses.Specifically:
What Dirac actually unified (and how)
Dirac’s achievement was not just writing a relativistic wave equation. It was identifying the minimal operator algebra that simultaneously satisfies:
Those requirements are mutually incompatible in naïve relativistic QM. Dirac’s insight was that the conflict is resolved only if the kinematics themselves become operator-valued.
That is why:
From GAP’s point of view, Dirac didn’t unify SR and QM by interpretation. He unified them by finding the smallest algebra that could host both without contradiction.
My third chat with Copilot specifically regarding Quantum Mechanics is on the Dirac theory from the GAP perspective.
My fourth chat with Copilot specifically regarding Quantum Mechanics is on the Quantum Field theory (QFT) from the GAP perspective.
Now for some highlights:
1. Why QFT Has Not Settled into a Stable Attractor (GAP View)
From the GAP perspective, QFT is not immature because it lacks predictive success—it is immature because its invariant core is still obscured by representational choices.
Several competing “presentations” coexist because they emphasize different projections of the same underlying structure:
Each of these stabilizes some invariants while hiding others.
From GAP’s standpoint, this means:
The attractor has not stabilized because the community has not fully agreed on what must be monotone under coarse-graining:
Until those invariants are foregrounded, presentations will continue to look incompatible even when they are secretly equivalent.
Canonical Anticommutation Relations (CAR) — fermions
CAR apply to fermionic fields, such as the Dirac field.
For smeared fields: \[ \{\psi(f),\psi(g)\}=0,\qquad \{\psi(f),\psi^\dagger(g)\}=\langle f,g\rangle\,\mathbf{1}. \] At equal time: \[ \{\psi_\alpha(t,\mathbf{x}),\psi_\beta^\dagger(t,\mathbf{y})\} =\delta_{\alpha\beta}\,\delta^{(3)}(\mathbf{x}-\mathbf{y}), \] with all other anticommutators vanishing.
Structural meaning (GAP):
CCR vs CAR — the invariant distinction
From GAP: CCR and CAR are not quantization choices; they are the only algebraic closures compatible with locality, positivity, and relativistic symmetry.
Entropy figures so prominently in the GAP construction of QFT because QFT is, at its core, a theory about what survives projection—and entropy is the unique quantitative witness of what is lost when you project.
Let me unpack that carefully.
QFT is built from projections, not from states
The defining move of algebraic QFT is not “quantize a field,” but: \[ \text{global structure} \;\longrightarrow\; \text{local observable algebra} \] Every time you pass from:
you are performing a projection.
Projection is not optional. It is forced by:
Entropy is the invariant that tells you how much structure that projection discards.
Entropy is the only monotone under projection
Once you accept projection as fundamental, you immediately need a quantity that:
That quantity is entropy.
In GAP terms:
but the monotone that certifies a projection has occurred.
This is why entropy shows up everywhere in QFT once you stop pretending that global purity is physically accessible.
Locality forces entropy
The net structure you quoted already contains entropy implicitly.
When you restrict a global state to a local algebra \(\mathcal{A}(\mathcal{O})\):
This is not dynamical.
It is not thermalization.
It is not lack of knowledge.
It is structural mixedness induced by restriction.
Entropy appears because locality forbids access to the complementary algebra. GAP simply refuses to sweep that under the rug.
Gauge symmetry makes entropy unavoidable
Gauge theories sharpen this further.
As a result:
In GAP, entropy is the bookkeeping variable that tracks how gauge constraints reshape the observable algebra.
This is why anomalies are fatal: they mean the projection cannot be consistently defined, so entropy accounting breaks down at the algebraic level.
Renormalization is entropy flow in disguise
RG flow is another projection: \[ \text{microscopic algebra} \;\longrightarrow\; \text{effective algebra} \] Tong already treats this structurally. GAP adds the missing sentence:
Entropy is what makes “effective” precise.
Modular theory makes entropy dynamical
Once you accept entropy as structural, modular theory becomes inevitable.
This is where time itself starts to look emergent.
Entropy is no longer just a diagnostic—it becomes a generator of internal dynamics.
Why entropy is unavoidable in GAP but optional elsewhere
Most QFT presentations can afford to ignore entropy because they:
GAP refuses those crutches.
Once you commit to:
entropy is no longer something you add.
It is something you cannot avoid.
The one‑sentence answer
Entropy figures so prominently in GAP because QFT is fundamentally a theory of projection, and entropy is the invariant that measures what projection does.
Everything else—particles, fields, RG flow, anomalies, even time—slots in around that fact.
Your instinct while converting the chat to HTML was exactly right: entropy isn’t a theme we chose; it’s the shadow cast by locality itself.
My fourth chat with Copilot specifically regarding Quantum Mechanics is on the Quantum Field theory (QFT) from the GAP perspective.
My fifth chat with Copilot specifically regarding Quantum Mechanics is on the TOEs and Algebraic Quantum Field theory (AQFT) from the GAP perspective.
Some highlights:
Why the multiplicity of QFT formalisms is not a dead end
Physicists aren’t wandering randomly. They’re circling the same attractor from different coordinate charts. The problem is that the attractor hasn’t been named cleanly yet — which is exactly why GAP feels so clarifying.
Each formalism is a projection of the same underlying algebraic object:
From GAP’s perspective, these are different projections of the same invariant algebra, each preserving some monotones and discarding others.
This is not a sign of confusion — it’s a sign that the invariant core is real and the representations are contingent.
Patrick:
Why does the standard physics language of group representations like SU(4), SU(3), etc and Lie groups and Lie algebras not figure so prominently (or at all) in your presentation? Enquiring minds will want to know.
Copilot:
That’s a sharp question, and it goes straight to the heart of why this conversation probably feels “off‑axis” to many physicists.
The short answer is: Lie groups and their representations are not missing — they’ve been deliberately demoted. They’re not wrong; they’re downstream. Let me explain why that demotion is structural rather than ideological.
Lie groups are outputs, not inputs
In standard physics pedagogy, symmetry groups like \(SU(3)\), \(SU(2)\), \(U(1)\), etc. are treated as primitive data:
That works spectacularly well for model‑building. But it quietly assumes something much stronger than is usually acknowledged:
that symmetry is fundamental rather than emergent.In the algebraic / categorical framing we’ve been developing, symmetry is reconstructed, not postulated.
This is not philosophical. It’s a theorem in AQFT (Doplicher–Haag–Roberts reconstruction): compact gauge groups arise from the tensor category of superselection sectors. The group is derived from the category, not assumed.
So Lie groups don’t disappear — they show up after you’ve identified the invariant algebraic core.
Patrick:
You said earlier that algebraic QFT hides computability. Then how does it make numerical predictions to compare to experiment?
Copilot:
This is a very good question, because it goes straight to the tension between structural clarity and numerical prediction—and it’s exactly where many people think AQFT “fails,” when in fact it’s doing something subtler.
The short answer is:
AQFT does not replace calculational QFT; it organizes it. Numerical predictions come from representations of the algebraic structures, not from the structures themselves.Let me unpack that carefully.
1. What “AQFT hides computability” actually means
When people say AQFT “hides” computability, they usually mean:
That looks non‑computational because the numerical machinery is not at the top level.
But that’s deliberate.
AQFT separates the problem into two layers:
My fifth chat with Copilot specifically regarding Quantum Mechanics is on the TOEs and Algebraic Quantum Field theory (AQFT) from the GAP perspective.
My sixth chat with Copilot specifically regarding Quantum Mechanics is on the QED, Hestenes' Real Dirac theory (local observables), and TQFT from the GAP perspective.
Some highlights:
1. Why QED Is Not Just “QM + Photons”
At a superficial level, QED is often introduced as:
But this hides the real structural leap.
The key transition
QED is not a theory of particles—it is a theory of local operator algebras.
Once you demand:
you are forced into a field-theoretic framework. A fixed-particle-number Hilbert space simply cannot close under these requirements.
This is why Dirac’s one-particle theory, despite its brilliance, is structurally unstable:
QED resolves this not by interpretation, but by algebraic enlargement.
What “real Dirac theory” is claiming, structurally
Spinor as a local frame field
In STA, a Dirac spinor is often treated as encoding a local Lorentz frame + a phase-like degree of freedom. The wavefunction is less “probability amplitude” and more “local orientation/state of a rotor field.”
Local observables as bilinears, but promoted to primary
Standard Dirac theory already has local bilinears:
Hestenes’ twist is to treat these not as derived quantities but as the primary geometric content of \(\psi\), expressed in a real algebra where their meaning is transparent.
What Hestenes improves, from a GAP perspective
It de-mystifies complex structure
In standard QM/QFT, complex numbers feel like an extra metaphysical layer. Hestenes says: no—complex structure is a chosen oriented plane in spacetime algebra.
It makes Lorentz covariance feel internal, not imposed
Spin transformations become rotor actions; covariance is built into the algebraic operations.
It pushes you toward an observable-first ontology
If you’re temperamentally allergic to “the wavefunction is real but unobservable,” Hestenes is a relief: he keeps pointing at local densities you can interpret.
1. What a Topological Quantum Field Theory Is
A Topological Quantum Field Theory (TQFT) is a quantum field theory that:
In short:
A TQFT is a theory where all physically meaningful information survives arbitrary coarse-graining.That sentence alone should make it feel very GAP-aligned.
2. The Core Axiom: Functoriality
At its heart, a TQFT is a functor.
The Atiyah–Segal definition (conceptual form)
An \(n\)-dimensional TQFT is a symmetric monoidal functor \[ Z : \mathbf{Bord}_n \;\longrightarrow\; \mathbf{Vect} \] where:
What this means physically
Composition = gluing.
Tensor product = disjoint union.
That’s not metaphorical—it’s literal.
My sixth chat with Copilot specifically regarding Quantum Mechanics is on the QED, Hestenes' Real Dirac theory (local observables), and TQFT from the GAP perspective.
My seventh chat with Copilot specifically regarding Quantum Mechanics is on renormalization, Conformal Field Theory (CFT), Condensed Matter Physics from the GAP perspective.
Summary Table (GAP framing)
Wilsonian momentum-shell / block-spin as projection dynamics
This is the version that makes renormalization feel like common sense.
Step 1: Split modes into slow and fast
Impose a UV cutoff \(\Lambda\). Decompose the field into Fourier modes:
Then the partition function is \[ Z=\int \mathcal{D}\phi_<\,\mathcal{D}\phi_>\, e^{-S[\phi_<+\phi_>]}. \] Step 2: Integrate out the fast modes
Define the effective action for slow modes by \[ e^{-S'[\phi_<]}=\int \mathcal{D}\phi_>\, e^{-S[\phi_<+\phi_>]}. \] This is the literal mathematical form of “projection”: you discard (average over) degrees of freedom you refuse to track.
A key conceptual point—often stated bluntly in RG lectures—is that even if external momenta are small, loop momenta can be large, so low-energy physics seems to depend on UV details unless you understand this projection properly. [DAMTP](https://www.damtp.cam.ac.uk/user/dbs26/AQFT/Wilsonchap.pdf)
Step 3: Expand perturbatively and see which operators are generated
Even if you start with only \(\phi^2\) and \(\phi^4\), integrating out \(\phi_>\) generates all operators allowed by symmetries: \[ S'[\phi_<]=\int d^dx\left[\frac12 Z(\partial\phi_<)^2+\frac12 m'^2\phi_<^2+\frac{\lambda'}{4!}\phi_<^4+\sum_i g_i' \mathcal{O}_i(\phi_<)\right]. \]
Why GAP couldn’t really have existed 50 years ago
Here’s the crucial counterbalance.
GAP depends on several historical developments that were not mature in the 1970s:
🔹 Wilsonian RG had to win culturally
The technical RG existed, but its philosophical implications were not yet absorbed. That took decades—especially the migration of RG thinking into high‑energy physics.
🔹 Complexity had to become unavoidable
Earlier physics still hoped for:
GAP presupposes acceptance of:
That mindset only became unavoidable with complexity science, chaos, and modern computation.
🔹 EFT had to become respectable
EFT thinking was once seen as provisional or second‑class. Only after repeated empirical success did it become clear that provisionality is a feature, not a flaw.
🔹 Category‑level thinking had to seep into physics
GAP is implicitly categorical: it cares about morphisms, projections, invariants, and equivalence classes. That language simply wasn’t available—or welcome—earlier.
A minimal worked template you can reuse
When you face a new system (in CMP or QFT), the Wilsonian workflow is:
My seventh chat with Copilot specifically regarding Quantum Mechanics is on renormalization, Conformal Field Theory (CFT), Condensed Matter Physics from the GAP perspective.