In this blog post, we will examine the meanings of determinism and indeterminism, and explore whether quantum mechanics necessarily requires us to abandon determinism by considering its various interpretations.
Determinism and Indeterminism
Determinism is the theory that, since past outcomes serve as causes for the future, everything that will happen in the future is already determined. For example, let’s consider a pendulum, and suppose that at time t, we describe the state s(t) of the pendulum solely in terms of the position of the bob. Also, for simplicity, let’s assume that time is quantized to the base unit 1. If there are no external disturbances, the pendulum’s motion will be periodic, and a specific state s(t) will occur an infinite number of times. If s(t₁) = s(t₂), does s(t₁+1) = s(t₂+1) also hold true? Since we do not know the velocity of the pendulum at the two times specified in the problem, we cannot answer this question. Therefore, the system described above is indeterministic. But what if we add other variables to describe this system in addition to the positions of the weights? If variables q₁, q₂, q₃, …, q_(n-1), q_n are given and these variables fully describe the system, then we can predict the system’s state s(q₁, q₂, q₃, …, qn-1, qn, t) at time t to predict the state s(q1, q2, q3, …, qn-1, qn, t+1) of the system at time t+1.
As in this example, if the characteristics of the system are not sufficiently described, the motion of the pendulum cannot be predicted. However, this does not necessarily mean that the system is indeterminate. If a complete description s(q1, q2, q3, …, qn-1, qn, t) of the pendulum’s state exists, then the system is deterministic. This fact implies that it is very difficult to prove that a system is non-deterministic. To prove that something is deterministic, one simply needs to ensure that the next state can be predicted from the previous state. To do this, one must establish the variables that fully describe the system—such as temperature, velocity, position, and time—along with the laws governing its motion. However, to prove that a system is non-deterministic, one must demonstrate that no set of variables and laws exists that can fully describe the system. Because of this difficulty, it is very hard to discuss whether the universe is deterministic or not, or whether a deterministic description of the universe is even possible.
If we accept quantum mechanics, must we abandon determinism?
Quantum Mechanics and Determinism
If we accept quantum mechanics as a valid description of the world—though this does not imply a “perfect” description—we must address two questions to determine whether the world is deterministic or not. First, is quantum mechanics deterministic? Second, does quantum mechanics provide a complete description of the world?
At first glance, the first question seems extremely self-evident. Most people familiar with quantum mechanics would answer, without a doubt, that quantum mechanics is non-deterministic the moment they see this question. How can we say that a theory, which provides us only with various possible outcomes and their respective probabilities, is deterministic? However, the judgment of whether quantum mechanics is deterministic or not is based on our preconceptions. Physicists seek to explain the world through their physical theories. In this process, various variables and constants that constitute the world are established and incorporated into the theory. If these variables and constants describe the world sufficiently well to predict its future, the theory is deterministic; if not, it is non-deterministic.
If we consider only the definitive results or measurements obtained through experiments to be components of reality, then current quantum mechanics is clearly non-deterministic. According to Schrödinger’s wave mechanics, all matter possesses a corresponding wave function. According to the indeterminate interpretation, solving the Schrödinger equation—which includes this wave function—yields multiple solutions, each representing the possible quantum states that the matter can occupy. In other words, before we observe the state of the matter, it is not fixed in any specific state; rather, only the possible states and the probabilities associated with each state are determined. This is described as a superposition of waves. The moment an observation is made, this superposition collapses into a single state; if observations are performed multiple times, it can be seen that the frequency with which a specific state appears matches the probability of that state obtained by solving the Schrödinger equation. In a world described by this mechanics, only the probabilities of each quantum state can be determined, making it indeterminate.
However, we do not necessarily have to understand the world solely through the precise values obtained through measurements and experiments. We could argue that the world does not correspond solely to exact values and measurable quantities, but that quantum mechanical wave functions and other mathematical descriptions constitute the fundamental building blocks of reality. Of course, this interpretation is quite far removed from our intuitive understanding of reality. Nevertheless, given that wave functions and mathematical descriptions have explained phenomena quite well, it can be accepted as a valid perspective from which to view the world. In this “wave-function world,” quantum mechanics is perfectly deterministic. This is because, in this “wave-function world,” the wave function describing the state of matter naturally changes continuously, like a wave, and provides all the elements necessary for us to predict the state s(t+) from the information about the state s(t). In other words, since the wave function itself can be predicted deterministically, if we regard the wave function as the fundamental building block of the world, we can say that the world is deterministic.
Interpretations of Quantum Mechanics
Among the interpretations of quantum mechanics, the Copenhagen interpretation—the most famous, supported by the greatest number of physicists, and featured in numerous physics textbooks—appears simple at first glance but is also ambiguous. The wave mechanics described earlier, along with the superposition and collapse of the wave function, form the core of this interpretation. This interpretation is appealing because, unlike the Many-Worlds Interpretation (MWI), it does not involve numerous parallel worlds or other complex components; unlike spring mechanics, it does not require additional equations; and unlike the GRW (Ghirardi, Rimini, and Weber) interpretation, it does not introduce new elements of randomness into the theory.
However, the Copenhagen interpretation fails to address the dilemma of what exactly constitutes a “measurement” that causes the collapse of the wave function, when it occurs, and whether to choose a continuous, deterministic wave mechanics in which no measurement takes place or a mechanics in which measurements occur that cause indeterministic collapse. Although the Copenhagen interpretation has captivated many physicists with its simplicity, this ambiguity has served as the driving force behind the emergence of new interpretations over the past several decades aimed at resolving the measurement problem. These new interpretations have primarily sought to address the question of whether measurement occurs or not, as well as the randomness of collapse.
The GRW interpretation—named after the physicists Ghirardi, Rimini, and Weber—sought to resolve the above dilemma by merging the two mechanics—one with measurement and one without—into a single framework. While wave mechanics still describes continuously varying quantum states through wave functions, it explains that the collapse of superposition is not fundamentally caused by the act of measurement. Decoherence occurs randomly and abruptly within a single particle. However, such random and abrupt decoherence has never been observed experimentally; the GRW interpretation explains this by stating that such random decoherence occurs with an extremely low probability.
But if this random decoherence occurs with such a low probability, why does it happen every time a measurement is made? To explain this, the GRW model utilizes the phenomenon of entanglement. Simply put, entanglement is a phenomenon in which two or more particles located in different positions are described by a single wave function. If a change occurs in the quantum state of one particle, the other entangled particles are immediately affected, and their quantum states change simultaneously. This bizarre phenomenon is hard to believe, but it can be observed experimentally. The GRW model explains that during measurement, the particles in the measurement apparatus become entangled with the particles being measured. Since more than 10⁸ particles become entangled at this point, the probability that at least one of them will decay becomes extremely high. If decay occurs in even a single one of the entangled particles, decay occurs in all entangled particles; thus, upon measurement, they decay and become fixed in a single state. In GRW, decay occurs randomly and abruptly, making it unpredictable and non-deterministic. However, since this interpretation is currently nearly impossible to prove or disprove, either theoretically or experimentally, it is difficult to accept it as the correct interpretation.
Another interpretation, developed by Bohm, attempted to resolve both the issues of randomness and the two-mechanics problem. To solve this problem, Bohm introduced a new formula called the guide equation. According to this interpretation, the quantum state of a particle is still described by the Schrödinger equation—which includes the wave function—as in wave mechanics, and the particle’s position is also described by a guide equation that includes the wave function. In this equation, given a wave function, knowing the particle’s position allows one to determine its velocity, and knowing the velocity allows one to determine its position; thus, the particle’s velocity and position are determined before any measurement takes place. Therefore, there is no such thing as the collapse of superposition.
However, upon even brief reflection, this interpretation appears to contradict Heisenberg’s uncertainty principle. According to the uncertainty principle, it is impossible to measure both the position and velocity of a particle with absolute precision. Bohr argued that while the uncertainty principle specifies the limits of what we can know about a particle’s position and velocity, the fact that we cannot know them does not mean that the particle’s position and velocity are actually uncertain. He believed that particles possess precise positions and velocities, but these are hidden variables that remain unmeasured by us.
Furthermore, Bohr believed that particles and waves are ultimately one and the same. For example, in the famous double-slit experiment, when a particle is fired, it passes through one of the two slits, but its wave function passes through both slits, creating an interference pattern on the opposite wall. Since the wave function guides the particle’s motion, the particle is more likely to move toward regions where the wave function’s amplitude is high, resulting in the appearance of an interference pattern on the opposite wall. From this perspective, Bormian mechanics is highly deterministic.
However, because it employs the concept of a single wave function that influences the velocity of every particle in the universe, it is nonlocal—unlike most physical theories. Opponents of Bormian mechanics criticize this aspect. However, since there is no evidence that nonlocality is a fundamental property of nature or that nonlocal theories are incorrect, the theory does not lose its validity. Nevertheless, like other interpretations, it has the weakness of not having been definitively proven to be correct, either experimentally or theoretically. If this interpretation is proven correct, we will finally be able to put an end to the debate over determinism and indeterminism and regain the conviction that the world is deterministic and predictable.
The Many-Worlds Interpretation asserts that when a measurement is made, the quantum states—which appear as solutions to Schrödinger’s equation—do not collapse, leaving only one and causing all others to vanish; rather, the universe splits into parallel universes, each possessing a different quantum state. However, to prove this interpretation, we would need to be able to observe the influence these parallel universes have on our current universe; since this is impossible with current technology, it is difficult to prove.
Conclusion
The discussion so far has shown that accepting quantum mechanics does not necessarily mean abandoning determinism. As mentioned at the beginning of this article, to prove that the world is indeterminate, one must demonstrate that no correct and complete theory describing the world exists. I believe this proof is impossible, both experimentally and theoretically.
Even if it becomes certain that quantum mechanics cannot be extended to become a deterministic theory, this does not mean that no other complete deterministic theory exists. Furthermore, if one of the existing interpretations of quantum mechanics—such as the Bohm interpretation—is proven to be correct, then the world is clearly deterministic. However, even if these interpretations are accepted, we cannot fully predict the next state of a particle when we know its initial position and state. This is because, when a measurement is made, we inevitably influence the system being measured and alter the value we are trying to measure. Nevertheless, the dynamics within it remain deterministic and assign precise positions and states to subsequent particles.
Even if we do not trust the mechanics of the spring, we can still uphold determinism. If we accept the wave function and mathematical techniques as fundamental building blocks of the physical world, and regard the measured values we perceive as merely illusions produced by the wave function, then quantum mechanics provides a complete and deterministic description of the world.
Let’s reconsider the first two questions. Regarding the first question, we have already shown that quantum mechanics can be sufficiently deterministic. And to answer the second question, while quantum mechanics can be accepted as correct, it is true that it does not yet provide a complete description of the world. Therefore, the claim that quantum mechanics is definitely non-deterministic is incorrect, and there are still many points worth discussing.
