What is a particle? – Physics and Reality

Classically, a particle is envisioned as a point-like object characterized by a position and a velocity, or equivalently a momentum. At any given moment, both can in principle be specified with arbitrary precision, and together they determine how the particle will subsequently move.

In quantum mechanics, this picture changes. Position and momentum become quantum observables, and they cannot simultaneously take arbitrarily precise values. The more sharply the position of a particle is specified, the less sharply its momentum can be specified, and vice versa. This is not simply a limitation of our ability to measure the particle. It is a property of the quantum state itself.

This is one example of a more general feature of quantum mechanics. Some quantum observables are represented by quantities that do not commute with each other, which prevents them from having simultaneously definite values. Position and momentum provide the most familiar example, while the more general meaning of noncommuting observables will be discussed separately.

There is another interesting aspect of the quantum-mechanical description. Space and time no longer enter the theory in quite the same way. Time remains a parameter that labels the evolution of the system, much as it does in classical mechanics, whereas position becomes a quantum observable. A quantum state determines the probabilities for the possible outcomes of measurements of position, momentum, and other observables. A particle therefore no longer follows a trajectory specified by a precise position and momentum at every instant in the classical sense.

This description works remarkably well when the number and identity of particles can be regarded as fixed. At sufficiently high energies, however, particles can be created and destroyed. A photon can produce a particle-antiparticle pair under appropriate circumstances, while a particle and its antiparticle can annihilate into other particles. A theory in which particles themselves can appear and disappear requires a different starting point.

This brings us to quantum field theory.

In relativistic quantum field theory, space and time once again appear together as coordinates on which quantum fields depend. Rather than assigning a position operator to each particle and following its evolution, the fundamental objects are quantum fields extending throughout space. What we call particles arise as particular excitations of those fields.

Momentum and energy remain particularly natural quantities in this description. A one-particle state can have a precisely defined momentum. Such a state, however, is completely delocalized: it extends throughout space rather than describing a particle sitting at one particular point.

There is nothing uniquely quantum about the way localization is constructed from different momentum modes. Consider a radio wave. A wave with one perfectly defined frequency and wavelength would extend indefinitely through space and would therefore have no definite location. To produce a localized radio pulse, we instead have to combine waves with a range of frequencies and wavelengths so that they reinforce one another within a certain region and largely cancel elsewhere. The more localized we want the pulse to be, the broader the range of wavelengths needed to construct it. Since wavelength is directly related to momentum in quantum theory, the same idea carries over to a particle.

A localized particle can therefore be described by combining states with different momenta into a wavepacket. By arranging these momentum components appropriately, the resulting state can be concentrated around a particular region of space. Its momentum, however, is no longer precisely defined.

The familiar position-momentum uncertainty therefore survives, but its interpretation becomes somewhat different. Instead of imagining a tiny object whose position is somehow blurred, we construct a localized particle state by superposing momentum modes. A state with one exact momentum is spread throughout space, while localization requires uncertainty in momentum.

Strictly speaking, the notion of a particle position in relativistic quantum field theory is also more subtle than in ordinary quantum mechanics. There is no universal fundamental position operator playing the same role as the position operator of nonrelativistic quantum mechanics. Nevertheless, localized particle states can be constructed, and in regimes where particle creation and relativistic effects are unimportant, the familiar quantum-mechanical description in terms of particle positions emerges as an excellent approximation.

This also changes what we should mean when we say that a particle is “point-like.” An electron being point-like does not mean that its quantum state must be concentrated at an infinitely precise point. Rather, it means that as far as experiments have been able to probe, the electron shows no detectable internal structure. Its quantum state can still be spread over space, just as any other quantum state can.

From the perspective of quantum field theory, then, a particle is not fundamentally a tiny object moving through space. The underlying entity is the quantum field, while a particle is a particular excitation of that field that can carry definite quantities such as energy, momentum, charge, and spin. Such an excitation can be completely delocalized or combined with other momentum modes to form a localized wavepacket.

The classical picture of a point moving along a trajectory is therefore not entirely wrong. It is an approximation that emerges when a quantum wavepacket is sufficiently localized and remains so over the scales relevant to the problem. At a more fundamental level, however, what we call a particle is a quantum state of a field, and the familiar little object with a position and momentum is only one limit of that description.