Chapter 9: Fundamental Physics

Section 16: Quantum Phenomena

Quantum effects

Over the years, many suggested effects have been thought to be characteristic of quantum systems:

• Basic quantization (1913): mechanical properties of particles in effectively bounded systems are discrete;

• Wave-particle duality (1923): objects like electrons and photons can be described as either waves or particles;

• Spin (1925): particles can have intrinsic angular momentum even if they are of zero size;

• Non-commuting measurements (1926): one can get different results doing measurements in different orders;

• Complex amplitudes (1926): processes are described by complex probability amplitudes;

• Probabilism (1926): outcomes are random, though probabilities for them can be computed;

• Amplitude superposition (1926): there is a linear superposition principle for probability amplitudes;

• State superposition (1926): quantum systems can occur in superpositions of measurable states;

• Exclusion principle (1926): amplitudes cancel for fermions like electrons to go in the same state;

• Interference (1927): probability amplitudes for particles can interfere, potentially destructively;

• Uncertainty principle (1927): quantities like position and momenta have related measurement uncertainties;

• Hilbert space (1927): states of systems are represented by vectors of amplitudes rather than individual variables;

• Field quantization (1927): only discrete numbers of any particular kind of particle can in effect ever exist;

• Quantum tunnelling (1928): particles have amplitudes to go where no classical motion would take them;

• Virtual particles (1932): particles can occur for short times without their usual energy-momentum relation;

• Spinors (1930s): fermions show rotational invariance under SU(2) rather than SO(3);

• Entanglement (1935): separated parts of a system often inevitably behave in irreducibly correlated ways;

• Quantum logic (1936): relations between events do not follow ordinary laws of logic;

• Path integrals (1941): probabilities for behavior are obtained by summing contributions from many paths;

• Imaginary time (1947): statistical mechanics is like quantum mechanics in imaginary time;

• Vacuum fluctuations (1948): there are continual random field fluctuations even in the vacuum;

• Aharonov–Bohm effect (1959): magnetic fields can affect particles even in regions where they have zero strength;

• Bell's inequalities (1964): correlations between events can be larger than in any ordinary probabilistic system;

• Anomalies (1969): virtual particles can have effects that violate the original symmetries of a system;

• Delayed choice experiments (1978): whether particle or wave features are seen can be determined after an event;

• Quantum computing (1980s): there is the potential for fundamental parallelism in computations.

All of these effects are implied by the standard mathematical formalism of quantum theory. But it has never been entirely clear which of them are in a sense true defining features of quantum phenomena, and which are somehow just details. It does not help that most of the effects—at least individually—can be reproduced by mechanisms that seem to have little to do with the usual structure of quantum theory. So for example there will tend to be quantization whenever the underlying elements of a system are discrete. Similarly, features like the uncertainty principle and path integrals tend to be seen whenever things like waves are involved. And probabilistic effects can arise from any of the mechanisms for randomness discussed in Chapter 7. Complex amplitudes can be thought of just as vector quantities. And it is straightforward to set up rules that will for example reproduce the detailed evolution of amplitudes according say to the Schrödinger equation (see note below). It is somewhat more difficult to set up a system in which such amplitudes will somehow directly determine probabilities. And indeed in recent times consequences of this—such as violations of Bell's inequalities—are what have probably most often been quoted as the most unique features of quantum systems. It is however notable that the vast majority of traditional applications of quantum theory do not seem to have anything to do with such effects. And in fact I do not consider it at all clear just what is really essential about them, and what is in the end just a consequence of the extreme limits that seem to need to be taken to get explicit versions of them.

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From Stephen Wolfram: A New Kind of Science [citation]