Can you write factorial anonymously? Yes

 

Of course you can. Use the Y-combinator.

This would look cleaner if the author had not used Church numerals. Here's something simpler:

   (λf λn. if n=0 then 1 else n*f(n-1) ) (λf λn. if n=0 then 1 else n*f(n-1)) 6.

But this isn't quite right, since recursion stops after one step. The Y combinator is more devious even than this.

So the factorial function is: Y (λf λn. if n=0 then 1 else n*f(n-1) ).

See section 7 of this PDF for a worked example.

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"Something Deeply Hidden" - some thoughts

Amazon link

The title is a quote from Einstein, revealed at the end of the book. Carroll is on Einstein's side in the great Einstein-Bohr debates on the meaning and completeness of quantum theory. Sort of.

Carroll writes well, mostly. He's a fluent author and has the surpassing virtue of conceptual clarity. This is a book about concepts: conceptual analysis and what the equations are telling us. This works well at the beginning where he covers the material in a typical undergraduate course, falters later when he discusses quantum field theory, and stutters at the end, where he sketches the research program of deriving spacetime from Hilbert space.

Who is this book for? Not the layperson - it's too unfamiliar, too conceptually abstract and dense. It's for people who know quantum mechanics - the mathematics and the calculations - people who understand the machinery but, like everyone else, struggle to understand what it's telling us about the universe itself. The math he doesn't mention underpins the concepts he's keen to articulate and talk around.

He's persuasive on the many-worlds interpretation, mostly because it seems plausible to start with the wave-function of the universe-as-a-holistic-entity. He's at pains to point out that the MWI is really no more than that: austere quantum theory.

He's excellent on how decoherence works, giving a clear conceptual overview. You have to have covered superpositions and entangled states in your QM course - and to have thought about it - to really grasp what he's saying, though. He writes like it's pretty clear but it isn't.

Should one walk away from this book an Everettian?  Carroll makes a very strong case for this over all the other interpretations - his critiques sometimes feel like shooting fish in a barrel. He's convincing that we should conceptualise QM as if the MWI were true.

But is the universe really a state vector in a high-dimensional Hilbert space? With our familiar classical-looking spacetime something emergent? A reality emergent from entropic-entanglement (hence locality and metric) and then local sampling of that so-structured Hilbert space?

No-one knows. It might be nice .. but the research isn't in.

And then there are the lacunae. The genesis of the standard model is nowhere mentioned. It may all be quantum fields - but how did we get separate quantum fields for all the different fermions and bosons?

My conclusion: every physics undergraduate should read this book. All the questions they have about how quantum theory is put together (the map of the territory in fact) and how it all relates to the universe we experience are honestly discussed here. They won't find those issues addressed in class or in their textbooks.

They will also appreciate how much we still don't understand about the fundamentals of the theory and about reality itself. Quantum gravity is still, most likely, the holy grail. But in the absence of meaningful experiments (the collider plans don't really help) solid progress is likely to remain stalled.

“Metaphorical Worlds Interpretation” (Chad Orzel)

Amazon link

I bought this a couple of weeks ago (Kindle) but it still sits in my stack. Soon!

Peter Woit has this post today, however, where he links to a piece by Chad Orzel.

Orzel thinks there is a better way to think about the "Many Worlds Interpretation":

"The problematic aspect here is that the wavefunction of the universe has everything in complicated superposition states, but when we select out a tiny piece of it as our system of interest, we often see that system only in single states, not a superposition of multiple states. The question that’s too often un-asked, though is: What measurement would you do to demonstrate that your system is really in a superposition?

The answer to this doesn’t need to be a procedure specific enough to actually do the experiment; a general outline would be sufficient. And, in fact, we have a couple of centuries of experience at doing exactly this: When we want to show that something has been in two states at the same time, we do an interference experiment. We put our system of interest in a superposition of two states, arrange for those two states to evolve at slightly different rates for some time, and then bring them back together and measure the final state.

If a superposition exists, there will be some oscillation in the probability of a given final state that depends on the differential evolution in the middle. This takes lots of forms– if the two states of the superposition correspond to passing through spatially separated slits, it’ll show up as an interference fringe pattern in space; if they’re two states of a cesium atom in an atomic clock, it’ll show up as a varying probability of ending up in one of those states as you adjust the frequency of your microwave oscillator.

In every case, though, you’re measuring a probability. And not even a Bayesian can accurately measure a probability from a single experiment. To get a good measurement of a probability of some outcome– let alone the variation in probability that is the signature of a superposition state– you need a large number of repeated measurements. And those measurements have to be made under the same conditions every time.

That’s the key feature that lets you carve out some parts of the giant wavefunction of the universe and choose to treat them as systems in definite states, while others need to be treated as full quantum superpositions. The vast majority of the universe that we’re bracketing off as “the environment” affects the measurement conditions, which changes the probabilities you’re measuring.

If the interaction with the environment is small, though, you can ensure that the conditions are close to identical for enough trials to unambiguously see the changing probabilities that show a superposition exists. That subpart of the universal wavefunction needs to be dealt with as a fully quantum system.

If the interaction with the environment is strong and poorly controlled, though, the conditions of your measurement change enough from one repetition to the next that you’re not really doing the same measurement multiple times. If you could know the full state of the environment for a given trial, you would predict one probability, but knowing the full state of the environment for the next trial would lead you to predict a different probability.

In the absence of that knowledge, adding together repeated results just gets you junk– you won’t see a clear dependence on the different evolution of the different states in the superposition, because it’s swamped by the unknown effect of the environment. If you can’t see the interference effect, that system “looks classical,” and you can treat it as having a definite state.

That process of interaction with the changing state of an unknown environment gets the name “decoherence,” and it’s what enables the bookkeeping trick that lets us split off pieces of the wavefunction and consider them in isolation. If the piece you’re interested in is big enough and interacts with the environment strongly enough, there’s no hope of doing the interference measurement that would show it’s in a superposition state. If you can’t do a measurement that would show the existence of the other piece(s) of the superposition, you can safely treat it as being in a single definite state.

It should be emphasized, though, that this is just bookkeeping, not a real separation between “copies of the universe,” or even copies of the system of interest. There’s only one universe, in an indescribably complex superposition, and we’re choosing to carve out a tiny piece of it, and describe it in a simplified way.

It’s not even true, strictly speaking, that the results of a given experiment for a particular object are unaffected by the presence of the other parts of the superposition for that specific object. If you could do the full probability calculation for the whole wavefunction, including all of “the environment,” the probability you would predict for that experiment would include a contribution from all the various states that are superposed. In the absence of that complete knowledge, though, you can get away with ignoring them, because you’ll never be able to repeat the measurements in the way you would need to see the influence.
...
Rather than “Many-Worlds Interpretation,” I’d go with “Metaphorical Worlds Interpretation,” to reflect the fact that all the different ways of cutting up the wavefunction into sub-parts are fundamentally a matter of convenience, a choice to talk about pieces of the wavefunction as if they were separate, because the whole is too vast to comprehend."

Peter Woit likes this story. What do you think?

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What is decoherence? Read this.

The Office as Utopia

I am quite aware that a professional office in an interesting company with a solid mission is a really nice place to be.

I would not like to live in any part of a neolithic or feudal society. I'll leave it to others to hanker after Viking raiding parties.

I loathe the horsing-around mateyness of the combat-facing armed forces.

I'm repelled and disoriented by overt displays of emotion, emotional appeals which make no sense, excessive moralising on emotional grounds.

No, I am truly a psychological creature of mature-capitalist cool: the NTs bred to dwell in cupboards and back-offices: useful when needed.

Nevertheless.

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A fragment:

"They talk about the feminisation of society. How in the absence of a mortal, existential enemy the liberal world has become a supersized hearth and home. No room for the masculine virtues of the warrior, or for the disinterested seeker after truth.

Yet this is only half-true. The liberal world is not modelled after the family but from an idealised model of the modern, capitalist workplace, where:

• emotion equates to transactional sociability between strangers
• strong emotional bonds would violate rational outcomes
• interpersonal violence would undermine systemic processes.

The middle-class, professional work-environment is optimally populated by lesbian women and gay men*: personality-types combining diffuse, non-aggressive, impersonal pseudo-warmth together with loyalty to corporate-outcomes rather than individuals.

Think ESFJ.

The office as utopia."

Modern liberal ideology - woke culture - seeks to make this protean, historically-conditioned target-state the foundational theory of normative human nature. A mass exercise in self-delusion, orchestrated by the ideologues who have captured the media in all the advanced western democracies, where an entrenched capitalism has already done its selective work.

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* In MB terms, people at the centre of the F-T axis: not too feminine, not too masculine. See "Capitalism is for wimps".