[“Quantum foundations” series]
I encountered a Wired reprint of a Quanta Magazine article which I’d flagged awhile ago; so, it’s time to write a new post.
While reconciling Quantum Theory and General Relativity is an ongoing topic, I found this Quanta Magazine article interesting for its discussion of unitarity and Hilbert space. Although based on really arcane math.
• Wiki > Unitarity
(quote) In quantum physics, unitarity is the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of quantum mechanics, while generalizations of or departures from unitarity are part of speculations about theories that may go beyond quantum mechanics. A unitarity bound is any inequality that follows from the unitarity of the evolution operator, i.e. from the statement that time evolution preserves inner products in Hilbert space.
• Wiki > Hilbert space (Note the caveat in this Wiki article that “for a general system, states are typically not pure.”)
(quote) In mathematics, Hilbert spaces … allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. … Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space.
An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis, in analogy with Cartesian coordinates in classical geometry. When this basis is countably infinite, it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable.
Wood’s opening sentence gave me pause – that quantum theory “portrays subatomic particles as probabilistic waves.”
I’d prefer something like: In quantum (field) theory so-called elementary particles are localized (wave-like) excitations in fields. And mathematical models treat their properties as probabilistic. The Schrödinger wave equation often is used to predict these properties – to predict “how a quantum system evolves in time” – such as position and momentum. In practice, mathematical squares – known as amplitudes – are used to predict (quantify) the likely state of a particle.
His article contains some useful diagrams. For example, showing how unitarity requires (the complex mathematical description of) a particle’s state to stay on a circle in Hilbert space where probabilities add to 1.
• Quanta Magazine > “Physicists Rewrite a Quantum Rule That Clashes With Our Universe” by Charlie Wood, Staff Writer (September 26, 2022) – The expansion of space spells trouble for quantum mechanics, by presenting particles with a growing smorgasbord of options for where to be.
Unitarity, as the principle is called, says that something always happens [the sum of the squares of all the amplitudes must equal one]. … It also ensures that change is a two-way street: Any imaginable event at the quantum scale can be undone, at least on paper.
“Unitarity in quantum gravity is a very open question,” said Bianca Dittrich, a theorist at the Perimeter Institute for Theoretical Physics in Waterloo, Canada.
[Cosmic] expansion is well described by general relativity. But it means that the future of the cosmos looks totally different from its past, while unitarity demands a tidy symmetry between past and future on the quantum level.
The question is: What sort of alternative framework could accommodate both cosmic expansion and the rigid mathematics of quantum theory?
An isometric change resembles a unitary one with added flexibility [at the cost of the Hilbert space gaining additional dimensions].
Strominger and Cotler’s path integral [Feynman’s path integral formulation] spit out a matrix encapsulating the growth of the toy cosmos, and it was indeed an isometric matrix rather than a unitary one.
For his part, however, Arkani-Hamed doubts that swapping in isometry for unitarity goes far enough. He is one of the leaders of a research program that is trying to break free of many fundamental assumptions in quantum theory and general relativity, not just unitarity.
[perhaps a] purely geometric formulation of particle physics …
And another related article. Constructor theory – “a kind of meta-theory more fundamental than even our most foundational physics.”
• Quanta magazine > “How to Rewrite the Laws of Physics in the Language of Impossibility” by Amanda Gefter, Contributing Writer (April 29, 2021) – “Declaring something impossible leads to more things being possible …” (physicist Chiara Marletto).
(quote) The goal of constructor theory is to rewrite the laws of physics in terms of general principles that take the form of counterfactuals — statements, that is, about what’s possible and what’s impossible. … [Such as Einstein’s] It’s impossible to exceed the speed of light; it’s impossible to tell the difference between gravity and acceleration.
[From interview with Marletto]
Marletto: there are some phenomena in nature that you can’t quite capture in terms of trajectories … A transformation is possible when you have a “constructor” that can perform a task and then retain the capacity to perform it again. In biology, we call that a catalyst, but more generally we can call it a constructor. In the current approach to physics, some laws already have this counterfactual structure — the conservation of energy, for example, is the statement that it is impossible to have a perpetual motion machine.
… constructor theory provides us with a robust and general theoretical foundation for an experimental test that would prove that gravity must be quantum. … a constructor-theoretic principle called “interoperability,” which implies that if entanglement can be generated locally through a medium, that medium has to be quantum.
What makes quantum information different [from classical physics] is that it has two additional counterfactual properties.
First, it has at least two information variables — for example, position and velocity — for which it’s impossible to copy both simultaneously with arbitrarily high accuracy. Second, it must be possible to reverse any transformations on those variables.
Initially I thought quantum information is just some quirky application of quantum physics to computer science — but it’s not true. It’s the best tool we have to understand quantum theory itself.
Gefter: So the technology that could come out of constructor theory would be something like an all-purpose 3D printer, capable of constructing any physical object, including a copy of itself. That’s a universal constructor.