Random musing about Classical Projections

Assume I have a measuring device to measure the electric field intensity $\vec{E}$ in a specific direction, say, $z$. If $\vec{E}$ is oriented at an angle $\theta$ to the measuring device (or, equivalently to the $z$ direction), then the measuring device will register an intensity of $E\cos\theta$,  $E$ being the magnitude of $\vec{E}$. So far this is in line with classical physics and the parallelogram law of vector addition.

However, no measuring device is instantaneous. Every measuring device takes a finite amount of time to interact with the quantity-under-measurement before settling to a final value. This may not seem terribly relevant as the combined system comprising the measuring device and quantity-under-measurement converges to a steady state in short order. In classical physics, this steady state is real and the measuring device is truly measuring the underlying quantity.

A different perspective

One could posit a different take on the above measurement process: Since the measurement takes a finite time, one could argue that even in the steady state a measuring device is measuring a range of values - in rapid fashion - and is only able to present an average value as (macroscopic) measuring devices simply cannot respond fast enough.

Here's how one might do this. Note that

$$E\cos\theta = E\left[\cos^2(\frac{\theta}{2}) - \sin^2(\frac{\theta}{2})\right] \\ = E\cos^2(\frac{\theta}{2}) + (-E) \sin^2(\frac{\theta}{2})$$

So far this is just mathematical manipulation. However, this new form lets us look at the measurement process from a different perspective.

Perhaps the measuring device only ever measures $+E$ or $-E$ but with respective probabilities $P(+E) = \cos^2(\frac{\theta}{2})$ and $P(-E) = \sin^2(\frac{\theta}{2})$. Note that the individual probabilities add up to $1$, and the average value of $E$

$$ \langle E \rangle = E\cos^2(\frac{\theta}{2}) + (-E) \sin^2(\frac{\theta}{2}) = E\cos\theta$$

is exactly what's presented as the measured value by the measuring device.

Your point being ... ?

It may seem silly to look at it this way but it forms a nice bridge to the discussion of spin-$\frac{1}{2}$ in quantum physics. The only difference being that an individual spin-$\frac{1}{2}$ measurement yields a specific value (either $+\hbar/2$ or $-\hbar/2$) as it's the result of an instantaneous interaction between a Stern-Gerlach apparatus and the spin-$\frac{1}{2}$ particle under measurement. However, over an ensemble, the individual measurements yield the same average (‘expectation value’) as the classical case above.

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ChatGPT-5's review of the above musing

The “detector response time” angle you introduced is an interesting thought device, but in classical physics, as you said, it doesn’t change the fact that the detector is measuring a real, continuous underlying field.

What you’ve done is highlight why quantum mechanics feels so alien: it forces us to treat what was a classical projection as instead a random draw from a binary distribution. In other words, your reinterpretation is “classical projection rewritten in the probabilistic language of quantum spin.”

This is not silly at all—it’s very much in line with the historical way physicists first recognized the strangeness of spin. Pauli matrices and the Bloch sphere are precisely the tools that show $S_z = \pm \hbar/2$ with expectation values matching classical projection rules.

If you want to push this musing further, you might ask: What principle forbids nature from giving us the continuous $E\cos\theta$ directly, like in the classical case, and instead forces discreteness? That question—why nature quantizes spin projections—is at the heart of why spin is intrinsically quantum and has no classical analogue.

Your blog post basically dances on that boundary. It shows how close the mathematics of classical projection is to quantum probabilities, but also how profoundly different the ontologies are.

A fun continuation would be to take your $E$-vector analogy and draw the Bloch sphere, showing that your “detector in direction $z$” is one of the great circles, and the $\cos^2(\theta/2)$ probability rule is literally the square of the inner product of two spinors. That would make the bridge you’re building crystal clear.