Scientific determinism and the Heisenberg's uncertainty principle

Question

I'm reading Brief Answers to the Big Questions by Stephen Hawking. In chapter 4, it discusses whether we can predict the future. As many have known that Laplace put forth that if we knew the positions and speeds of all the particles in the universe, we would be able to calculate their behaviour at any other time in the past or future. Heisenberg's uncertainty principle later undermines Laplace's position as we can't measure both position and speed of a particle simultaneously accurately. I understand this part; however, what I don't understand is that it's our problem that we can't simultaneously measure the position and speed of a particle accurately. If particles do have exact position and speed at a time, does it mean that the future is completely deterministic? Even Stephen Hawking at the end of the chapter says that:

In principle, the laws allow us to predict the future. But in practice the calculations are often too difficult.

So, if I understand him correctly, the future is actually deterministic and it's just that we can't do the calculations accurately due to the lack of information. Do I understand Stephen's answer correctly?

Update: What I see from the answers or comments is that we are focusing on the question "whether we can predict the future". At this point, we can agree that we can't. The question, I want to ask, is the future really deterministic as Hawking seems to claim?

Answer 1

Laplace wrote (Essai philosophique sur les probabilités, see also this review)

We must envisage the present state of the universe as the effect of its previous state, and as the cause of the state to come. An intelligence which for a given instant would know all the forces by which nature is animated and the respective locations of the beings which compose it, if moreover it were vast enough to submit these data to analysis, it would embrace in the same formula the motions of the largest bodies of the universe and those of the lightest atom: nothing would be uncertain for this intelligence, and the future like the past would be present to its eyes.

According to the mathematician R. Thom, in modern language Laplace's statement is basically just the theorem of existence and uniqueness for the differential equations governing physics (in particular, Newton's second law). Here, uniqueness is important (the future is uniquely determined).

At the time when Laplace wrote the Essai, numerous differential equations were known, but no theorem existed yet stating that their integration was possible for a fairly 'regular' problem. Such a theorem would not be established until some thirty years later (the Cauchy-Lipschitz existence and uniqueness theorem).

Now, it is clear that we're talking about classical physics: to integrate Newton's equations of motion (for any given system, possibly the whole universe), we need to specify the initial positions and velocities of all the particles. This is not a big deal in Laplace's view, as he explicitly speaks of a "supernatural intelligence": his point is that in principle the universe is deterministic.

Issue #1: in practice, the universe is not deterministic for two separate reasons: we do not know how to set the initial condition, and we do not know how to integrate the incredibly complex system of equations (provided that you believe that the theory you are using is the correct representation of nature, ofc).

Issue #2: does quantum mechanics challenge Laplace's view? Time evolution is given by the Schrödinger equation, which is perfectly deterministic: for a given initial state $|\psi(t=0)\rangle$ the future $|\psi(t)\rangle$ is determined uniquely. Of course, the two points in #1 still apply. First, we do not know $|\psi(t=0)\rangle$, we always make a - sometimes physically motivated - guess. Secondly, we do not know how to solve the Schrödinger equation unless the system is quite small and for short times. A possible way for quantum mechanics to challenge Laplace's view is via the measurement problem: according to the standard textbook interpretation, measurement is not described by the Schrödinger equation (it is through the Born rule that probability enters into the theory).

Considerations:

• Complex systems are often chaotic: deterministic chaos means that small departures in the initial conditions produce large departures in the following trajectories (this is, e.g., why weather forecast is difficult). This is a practical limitation, I think Laplace was well aware of this on an intuitive level (chaos theory is rather recent).

• Following Thom's mathematical definition of determinism, we may say that the uncertainty principle has nothing to do with determinism: it is a mathematical statement on the variance of non-commuting observables (or, even outside the realm of quantum mechanics, quantities related by Fourier transform, see also this video). Statements on determinism are statements on the time-evolution laws of a theory$^1$. Yes, to solve Newton's law you need the initial positions and velocities with infinite precision, but in a quantum world, you need the full information that uniquely defines a state $|\psi(t=0)\rangle$.

• The measurement problem is crucial, as it's the possible source of "true" non-determinism: see e.g. On a measurement level, is quantum mechanics a deterministic theory or a probability theory? and links therein (including Is there consensus among physicists that reality is fundamentally deterministic? and Is the universe fundamentally deterministic?). Another, closely related, question is Why was quantum mechanics regarded as a non-deterministic theory?.

• Finally, a short quote from the book mentioned in the original question (Brief Answers to the Big Questions, chapter 4):

The wave function contains all that one can know of the particle, both its position and its speed. If you know the wave function at one time, then its values at other times are determined by what is called the Schrödinger equation. Thus one still has a kind of determinism, but it is not the sort that Laplace envisaged. Instead of being able to predict the positions and speeds of particles, all we can predict is the wave function.

$^1$ For example, Langevin dynamics is non-deterministic (as it is based on stochastic differential equations rather than ordinary or partial dofferential equations). However, from a physical point of view, this example does not challenge Laplace's determinism as it is an effective, incomplete, description of a dynamical process (e.g. Brownian motion).

Answer 2

There is no conflict between quantum mechanics and determinism since we already have deterministic interpretations, like Many Worlds or Bohmian mechanics.

On the other hand it can be proven that all nondeterministic approaches to quantum mechanics must be non-local (EPR argument). Since locality is required for compatibility with Special Relativity we can conclude that nondeterminism is unlikely.

The uncertainty principle puts a limit on our knowledge but it says nothing about determinism.

Answer 3

That the actual laws of physics are different from the classical physics known to Laplace does not really affect the argument: Instead of predicting the positions and momenta of particles, we could predict the wave function or the density matrix of the Universe, by solving the relativistic Schrödinger/von Neuman equation that includes all the interactions or some more general equation encompassing all the known physics.

Let us now returning to the Hawking statement:

In principle, the laws allow us to predict the future. But in practice the calculations are often too difficult.

A more serious issue with this statement is the difference between claiming that the current state of the universe and the laws of nature determine its future vs. the laws allow us predicting the future. While the first is certainly true (at least to the extent that we believe the physics is true), the second implies an observer, i.e., relies on human ability to measure, store the information and perform calculation. Since such an observer/computer is also the part of the universe, they will affect the result of the computation and also would possibly need the computational capacities exceeding those of the Universe itself.

The situation is somewhat akin to the assumptions that we can measure instantaneously position and momentum or that we can measure microscopic objects without affecting their state. Relativity and quantum mechanics teach us that we need to be careful with such assumptions, and this is probably true when performing a computation on the scale of the whole Universe (although us affecting this computation doe snot mean that we would necessarily discover new physics.)

(I don't mean to say that Hawking was wrong - more likely, he was making a specific point and didn't want to say more than necessary to carry it through.)

Related:
Is limited computational capacity a fundamental obstacle?
Determinism vs prediction