Can the universe be fully deterministic on a macro scale but not on a micro scale?

Question

Suppose you have a dice. The “probability” of a dice landing on 1 is defined to be 1/6. However, many say that this is a function of ignorance. If we knew everything about the initial conditions, we could predict with certainty whether or not it will land on 1. Given this knowledge, the “probability” now changes to 0 or 1. Probability disappears as a concept completely.

However, on a quantum scale, atleast according to some standard theories, quantum events happen probabilistically. It is claimed that even if one were to know everything about certain conditions, you could not predict quantum events with certainty.

But quantum events add up to macro events, no? How then can we determine with certainty where a dice will roll but not quantum events?

Is the probability that a dice will roll on 1, given knowledge of all initial conditions, 1/0 or merely close to it given the unpredictability of quantum events? And does this change depending on when you have this knowledge? For example, is the probability of the dice landing on 1 given all knowledge of conditions of the world 3 seconds before the dice roll the same as having all knowledge 10 seconds before?

This is what I’m having trouble wrapping my head around.

Answer 1

This was the same idea that stumped Einstein. The truth of the matter is that Quantum Mechanics is still deterministic, just not in the way you think. The probabilistic description of Quantum Mechanics comes arises from the fundamental difference between states and observation.

Consider the state of a quantum system-for instance spin along some direction-to be described by $$\mid\psi_z\rangle=\mid\psi\rangle$$ So far, so good. Now, if we wish to measure the spin along the $z$ direction, then we just slap the state in front of the spin operator. $$spin_r=\sigma_r\mid\psi\rangle$$ This measures the spin state of the system in the $r$ direction. Now this measurement gives us a new state, that is composed of a linear combination of the two eigenvectors which are $\mid u\rangle$ and $\mid d\rangle$. This is the new state vector, but what does it mean?

This operation gives us the probabilities associated with the two states-$\mid u\rangle$ and $\mid d\rangle$.

Because once the observation is made, the state is either $\mid u\rangle$ or $\mid d\rangle$, and their corresponding factors gives us a measure of the probability of the same.

Now, that is for a single system. When we get to an ensemble of a vast number of such systems, a new quantity becomes important-the average behavior of the system under such circumstances. This is what makes Classical Mechanics deterministic-it is the average behavior of the many systems that come together to form the macroscopic objects of our day-to-day life.

Answer 2

Physical models, frameworks and theories have a domain of validity, and they all consider different things to be important while neglecting others.

When you say 'suppose you have a dice', I take it as a classical description of a dice. Then its probability to land on a specific side is 1/6, with no caveat at all.

Now if you want to consider the dice and its environment in a reductionist manner, as an interacting ensemble of a myriad particles and forces, then you can wonder where the 1/6 probability comes from. It's not obvious to answer, because by changing the model describing the system (the dice) we lost track of the simple symmetry that appears in the macro description. But it is not lost, and could theoretically be retrieved by averaging a huge mass of quantities, in effect moving back to a macro description.

So if there is no randomness at the bottom, how can there be randomness at the top?

First, we do not know how to translate the micro description into the macro one. What we have are just two descriptions, and the micro one does not feature the notion of 'landing probability'. If we could go smoothly from a micro model to a macro one, there would be just one model, not two. In fact the micro representation you are imagining is just a thought experiment, not an actual model, because nobody has ever described precisely, in a perfectly reductionist manner, a macro objects from the atoms that constitute it.

And so, there is in fact no 0 or 1 probability coming from the micro description, and the problem simply does not exists. Only in the macro description there is a landing probability, and it is 1/6.

As for quantum mechanics, it is irrelevant here, because the question of 'if there is randomness at the bottom, how can there be no randomness at the top' is trivial to answer: averaging. Two bottles of water can be described identically while having water molecules that are all different in positions and motions.

In short: reductionism is a philosophical stance - you may believe in it or not (I don't), but it will not give you actual tools to do physics with. To compute the landing probability of a dice, you have no choice but use a macro description.