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The inevitability of uncertainty in measurement

There are a few more topics I’d hoped to cover in my digging through the archives of 1920s probability, but they require more reading and introspection before I can summarize them on the blog. I want to better understand Keynes’ thinking about probability as a subbranch of logic. I want to dig into how Wiener and others thought about stochastic processes before the rigor revolution of the 1930s. I’ll hope to come back to them later.

And one other can of worms that could consume weeks on here is quantum mechanics. Woo boy were people confused about quantum mechanics in the 1920s. The most influential interpretations, Born’s ensemble interpretation and Bohr and Heisenberg’s Copenhagen interpretation, were both proposed in mid-1920s. I want to read more of the original scholarly dialog about these interpretations before saying anything about the history. 

No topic causes more probabilistic confusion than quantum mechanics. Schrodinger's Cats, God Playing Dice, Uncountable Universes. Endless Nova specials. Michio Kaku. Sheesh. But my hot take is that this confusion would also arise in classical mechanics if physicists were forced to better explain measurement. 

In undergraduate classical mechanics, you model the world deterministically and see deterministic outcomes. In quantum mechanics, such determinism is impossible. All measurements are inherently probabilistic. But what is measurement here? In order to measure a system, you somehow have to “prepare” a classical device, a measurement device, and a quantum system to be in an initial state where you know everything. These systems are all physical and interact according to the laws of physics. A measurement begins by letting this joint system “evolve” using the laws of quantum mechanics. The measurement ends at some point, collapsing the wave function or whatever, and the classical device displays some probabilistic outcome. If you start to pick apart each of these steps, you see how measurement requires all sorts of engineering and modeling, and the outcome predictions always require some logical leaps.

This is no different in classical mechanics. If you want to precisely measure the current flow from a resistor, you would need to prepare a resistor in a classically deterministic ground state. This would require knowing the exact location of a practically infinite number of tiny classical subsystems. If you knew this configuration and simulated reversible physical laws forward, the current out of the resistor would be zero. But knowing this initial state requires measurement too, and such measurement is not possible without destroying the resistor. Since we can’t observe the internal guts of the resistor, we could assume an ensemble of potential initial conditions. When we do this, theory now predicts we’ll see Johnson Noise. I’m not the first to observe this, but a lot of the “weird behavior” in quantum mechanics reappears in classical systems when you try to come up with an interface between macroscopic measurement of microscopic systems in classical mechanics.¹

Once you get down to the fine details, it’s impossible to design a “classical” measurement system without some kind of “measurement back action” that happens in quantum mechanics. Classical measurement devices are also physical systems coupled to the experimental apparatus. The instrument and the system necessarily interact. The measurement device necessarily changes the state of the system. As you try to measure more delicate details, the measurement device has more impact on the system state. We all know this: The laws of classical mechanics suggest you can build a perpetual motion machine, but the fact that the measurement device is itself a physical system and that its measurement of the machine must consume energy shows this is impossible. 

In all parts of physics, when you start digging into the details of what a measurement is, you will find endless sources of uncertainty. Though we often associate our inability to predict the future as our prime source of probabilistic thinking, our inability to measure is just as essential.

The impossibility of measurement highlights the dirty secret of physics. While the naive view is that physics provides reductionist, fundamental laws, these laws have limited scope. If you design an experiment that requires a computer larger than our universe, is the outcome actually consistent with the laws of physics? (I’m looking at you, my quantum computing friends.) 

I’ve blogged about her writing before, but this is where I think Nancy Cartwright has Physics dead to rights. Physics had tremendous success predicting all sorts of different phenomena. But when you dig into the details, you realize there are always transitions between theories that are far less rigorous than they should be. Many of these transitions occur because there are physical states that are so uncertain that they are unknowable. 

Anyway, whenever I start writing about this stuff, I feel like a weirdo. Just because the laws of physics only hold locally doesn’t mean that they are bad.  Experimental physicists all know that there are some parts of physical theory that are a bit ad hoc. They also know that no experimental measurement is possible without uncertainty. But they don’t let this get in the way of them not doing new experiments to improve our understanding. Physics undeniably enables the most wondrous discoveries and technologies. In this way, physics is quite similar to probability.

1. If you want more details, Here are a couple of fun links from the control theorists: (1) Mitter and Newton (2) Sanberg et al. See those papers and the associated references.