*[This is a vague post intended to give some intuition about how particular toy models of decoherence fit in to the much hairier question of why the macroscopic world appears classical.]*

A spatial superposition of a large object is a common model to explain the importance of decoherence in understanding the macroscopic classical world. If you take a rock and put it in a coherent superposition of two locations separated by a macroscopic distance, you find that the initial pure state of the rock is very, very, very quickly decohered into an incoherent mixture of the two positions by the combined effect of things like stray thermal photons, gas molecules, or even the cosmic microwave background.

Formally, the thing you are superposing is the center-of-mass (COM) variable of the rock. For simplicity one typically considers the internal state of the rock (i.e., all its degrees of freedom besides the COM) to be in a (possibly mixed) quantum state that is uncorrelated with the COM. This toy model then explains (with caveats) why the COM can be treated as a “classical variable”, but it doesn’t immediately explain why the *rock* as a whole can be considered classical. On might ask: what would that mean, anyways? Certainly, *parts* of the rock still have quantum aspects (e.g., its spectroscopic properties). For Schrödinger’s cat, how is the decoherence of its COM related the fact that the cat, considered holistically, is either dead or alive but not both?

Consider a macroscopic object with Avagadro’s number of particles *N*, which means it would be described classically in microscopic detail by 3*N* variables parameterizing configuration space in three dimensions. (Ignore spin.) We know at least two things immediately about the corresponding quantum system:

(1) Decoherence with the external environment prevents the system from exploring the entire Hilbert space associated with the 3*N* continuous degrees of freedom. In other words, the reduced dynamics of the system are confined to a subset of its full formal Hilbert space. Some states are allowed by the interactions with the environment (e.g., a dead cat or alive cat) but some states are not (e.g. superpostions of dead and alive).

(2) Decoherence does *not* prevent the system from accessing superpositions of at least *some* pairs of orthogonal states. In other words, it’s not true that all 3*N* variables describing the system are classical all the time. We can tell this because the system still does some macroscopic stuff that is evidence that quantum things are happening at the microscopic level (e.g., the cat breathes because chemistry with quantum effects is happening inside its body).

The explanation for this is that some of the variables are being persistently decohered by the external environment (e.g., the COM motion of the cat’s heart, which beats when it is alive and does not when it is dead) but many are not (e.g., the relative position of two electrons in a molecule responsible for respiration, which is very quantum). As a crude conceptual model, we can imagine re-parameterizing the cat into a new set of 3*N* variables where those variables are broken up into two classes: those that are fully classical and those that are fully quantum. In this model, the density matrix of the cat could be expressed as a (time-evolving) classical probability distribution over the classical variables tensored together with a *pure, unitarily* evolving quantum state for the quantum variables. The quantum (part of) the overall state would be conditional on the classical probability distribution, but it would not become entangled with the environment.

In real life the distinction is not so discrete; some variables are only sort of classical, with coherent effects that are partially suppressed but not eliminated, to varying degrees in different parts of their spectrum. Furthermore, the variables that are mostly classical are probably changing with time. Still, the crude model is conceptually useful, especially because we can express it exactly it in terms of decoherence-free subsystems.

The obvious question is: “For real-life systems, what are the classical variables?” No one has a good answer yet, but this basically would answer the question “When has the dead/alive cat decohered?”. Just about any way you tried to parameterize the macroscopic parts of the cat (e.g., if you broke it up into a bunch of mesoscopic volumes and looked at the COM movements of those, from which alive-ness and dead-ness are very obvious) you’d find that *each* of those macroscopic degrees of freedom decohered very fast, and therefore they all could be treated as classical. (Nonetheless, most of the degrees of freedom would be fairly quantum.) If, on the other hand, only the overall COM decohered, like in some toy models but contrary to the real world, then you’d have just one classical variables, and an enormous quantum system that was apparently somehow isolated from its environment.

*[I thank Sabine Hossenfelder for discussion that prompted this post.]*

You mean 6N variables parameterizing phase space? The 3N variables parameterizing configuration space all commute, so the QM observables can be understood to be classical variables at a given time. At the scale of a macroscopic object, surely at >0 Kelvin, we would have to consider a classical stochastic model. A quantum mechanical state can be understood to model the evolution of the probability distribution over configuration space over time (without allowing joint probability distributions over the configuration space at multiple times), in contrast to a classical stochastic state over phase space (which does allow joint probability distributions over the configuration space at multiple times). But, this trivia aside, decoherence rocks, and rocks decohere.

> You mean 6N variables parameterizing phase space?

Not really, although things are subtle. In the discrete case that we understand well, saying that a discrete variable has fully decohered means the “conjugate discrete variable” (defined in terms of a discrete Fourier transform) has no classical value, i.e., it is maximally uncertain. In that case, it’s not in any sense true that you have two variables, one quantum and one classical. You just have one classical variable, period.

In the continuous case, we never get a continuous variable to fully decohere. Rather, it decoheres to some scale while the conjugate variable decoheres to a different scale. The two respective scales are set by the dynamics, and multiply together to (order) hbar by the uncertainty principle. Since they are inversely related, having one variable become more classical (stronger decoherence, more exactly defined) makes the other variable less well-defined. Therefore, I think it’s sensible to call this “one classical variable”.

> we would have to consider a classical stochastic model

Yes, certainly.