Pamperlitudes
The ℂ Numbers are Not Physical
I had two articles drafted for a Spaghetti Western series on geopolitics, which began with (a somewhat strained) Dirty Harry metaphor in Notes on the Other PolyVlad Crisis. Next up was going to be a Man With No Name metaphor. But I need to interrupt that trilogy with some physics, to maintain some daily composure. Leftiepols who do not care for mathematical physics may now click away.
The Amplitudes are Complex?
Wonderful conversation between Jacob Barandes and Scott Aaronson here , hosted by Curt.
But I have to register some suggestions and conceptual corrections.
@11:30 they talk about the “complex amplitudes”. Both Aaronson and Barandes emphasize the importance of complex amplitudes in QM, making it out these are necessary for interference. But they are wrong. Or should I say, not quite right. Yet it is common folk lore of quantum mechanics that the ℂ numbers are responsible for interference. That’s like saying
Without the superpositions in spacetime physics we have no interference, but would still have the “complex” amplitudes.
It is not because “the amplitudes are complex” that particles self-interfere. The ℂ numbers are only a mathematical representation, and they only represent the eigenstate of spin (compact local rotation symmetry).
The physics (hence closer to some notion of “causality”) of interference is that of superposition, which in spacetime physics means the particle trajectories can traverse nearly closed timelike curves. That also (via a different paper by Deutsch, ironically) “explains” quantum computation algorithmic complexity class (PSPACE) without the Many World assumption. In other words, quantum mechanics is gravity, but GR with non-trivial local compact topology.
Note: classical waves are collective effects in a classical medium (a classical field theory). They cannot possibly explain single particle self-interference. As Jacob explained elsewhere, that would require waves in a high dim Hilbert space, and is an unnecessary assumption in his ISQM formulation.
The reason why the ℂ numbers work well as mathematical representations for superposition is because they crudely mimic the stochastic effects of closed timelike curves (the non-classical paths in the sum-over-histories). But the ℂ amplitudes are geometric, they are Clifford rotor subalgebra structure when you consider the full Dirac theory (Schrödinger theory will throw you off by taking eigenstates of spin, which reduce the full 16-dimensional Dirac algebra down to complex algebra). That’s rotation and boost symmetry generators, not “amplitudes”. The ℂ numbers are not fundamental statistical aspects in QM in other words. I wish Aaronson knew this.
Without the superposition in spacetime physics we have no interference, but would still have the “complex” amplitudes.
Only, like I just wrote, the proper Dirac spinor (which is the only proper physical “wavefunction”) shows the amplitude is not a complex number, it is a spacetime element from the 16-dimensional spacetime algebra, and specifically the even subalgebra.
These rotor parts of the spinors are bivector, scalar and pseudoscalar components, thus they encode spacetime transformation instructions. They are not metaphysical “ℂ numbers”.
I do admit the concept of a 4D spacetime is also a mathematical abstraction. But… so what? Some abstractions are closer (in platonic form) to the real noumena than others.
Quantum Mechanics is Measurement Theory
I’ve read some pretty bad takes on quantum mechanics, but none worse than those who claim QM is reality. This leads to Many Worlds nonsense and extreme Idealism (It is all in Your Mind) nonsense. Some webinars even claim outright that “QM is not a measurement theory.” — as their main thesis. That’s Orwellian doublespeak… for a physics nerd. It is like saying Darwinism is not an evolutionary theory.
The essence of QM is propositions about two ends of a spacetime cobordism. One the measurement set-up boundary, then “weird 🅂𝖍𝗶🅃 happens” then the measurement detection boundary.
The weird 𒔼𒀂𒐕𒈦 that happens in-between is not part of quantum mechanics. Some might say, “Well, all possible paths happen in-between!” — which is pure philosophy, and cannot ever be verified. (By definition, if you shrink the time gap between measurement boundaries you just abbreviate what happens in between.)
But worse, in my own Topological 4-Geon theory framework we do not need all the histories to be realized. The Sum-over-histories (Feynman Path Integral) is a statistical accounting, not actual ontology.
The core of T4G theory is to show how the effectiveness of the superposition arises. That is, why not all histories really occur, but why we nevertheless must run the stochastic analysis as if they all might have occurred, and with interference.
Those are two separate things. Just imagining all possible histories could have occurred is only classical statistical mechanics. To get Quantum Mechanics we must allow also for interference. This means non-classical interference. These cannot be classical waves. It is interference of a particle with itself!
Think about this for a minute and you will realize this is only possible in a realist ontology if the particle can time-travel. Time Travellers can interfere with their past selves.
(And, by the way, if you take the Idealist route, you are just completely unhinged. Realism is what all science is based upon. If we give this up then postmodernism has some bite and we are all lost. We may was well just all kill ourselves then so the One Mind can go about their thinking without all us submodules causing Unified confusion.1)
No paradoxes. Since the future version of the particle has already made the time trip. No free will contradiction since human minds cannot time travel. The elementary particles can only time travel in a gravitational QM story if there are wormholes. But wormholes are not macroscopically stable. The only stable wormholes are minimal — at the Planck scale. Although the ends of the wormholes can be separated arbitrarily far apart in spacetime.
If you want to see explicitly how gravitational theory supplemented with non-trivial spacetime topology reproduces exactly the Logic of Quantum Mechanics then I suggest this chapter of T4GU.
What are Spinors then?
Spinors are mathematical transform instructions. In QM they transform a fiducial (or laboratory) frame of vectors onto the frame of a moving particle (the co-moving frame).
The particle is not described by a wavefunction. The spinor is the wavefunction, and it describes an abstract transformation between measurement frames.
In QM there really is no direct topological representation of the elementary particles. That is a missing ingredient of fundamental physics. However, as a very crude first approximation, you could say all the elementary particles of the Standard Model are wormholes, or at least have wormhole-like features. No one has ever explicitly written down such a geometric picture. The spinors are not that picture. Superstrings — although not the stuff of our cosmos —are a lot more like the picture.
I want to finish this piece by preempting the bonehead Gauge—Gravity duality folks, the AdS/CFT crowd. They’d come along and say, “Entanglement is fundamental and bulk gravity and wormholes only emerge from the boundary theory.” I would say to them,
they have absolutely no idea what they mean by “emergence”. A mathematical limit from a fantasy duality is not a physical emergence.
They are in effect telling me, the existence of unicorns in their fantasy fairy story proves horses cannot exist in the real world.
Yeah. It’s the other way around bruh. Horses like Pharlap, Secretariat and Bonecrusher exist in the real world (asymtotically de Sitter spacetime), or so I am told by people who choose to watch races instead of Cricket in the summer. These real creatures are what allow you to write a fantasy story with unicorn characters. As far as a metaphor for T4G+GR versus AdS/CFT, I think this is pretty darn good (modulo my relative ignorance about real horses).
There is more to say on the Aaronson–Barandes chat, but I’m putting it on the T4G github, so I can get back here to the Spaghetti Western Geopolitics trilogy.
I could be more sarcastic and satirical, but I don’t have the energy for it. Basically, people who say “We are All One” are mad and grossly denying the reality of the individual soul. However, my reading of the mystics is that nevertheless the individual soul finds fulfillment in uniting in harmony with others, while retaining a sense of identity. Both are good. The betterment of the individual soul is found in love and compassion for others, without any metaphysical Idealist forcing you to think you exist as a Part of a One. In the Baháʼí Faith they would say the soul is an individual, but that we should strive for Unity in Diversity. But this is our freedom to choose to unify, it is not a metaphysical necessity. If such things were necessities it would mean we’d be spiritually dead.