# FIELDS AND PARTICLES

Broadly speaking, a field is a collection of properties ascribed to regions of space (one might also speak of the region itself as being "the field"); if the properties are quantifiable then the field is a mathematical function of spatial coordinates, Φ(x, y, z ). Examples include the temperature at each point of a room, the velocity at each point of a fluid, the gravitational potential, and the electromagnetic field. In contrastand broadly speakingparticles are entities of which positions are ascribed (and which lack any relevant internal structure). While these will do as broad characterizations, they are inadequate in a number of ways.

## Classical Fields

For instance, one could say that a field theory ascribes positions (and field strengths) to the parts of a field, as a particle theory treats particles. Worse, one can reformulate particle theories (e.g., Isaac Newton, 16421727, and Immanuel Kant, 17241804) as theories that ascribe mobile particle-sized regions of repulsion to space: as a field theory according to the intuitive distinction. Hence a useful formal characterization adopts the practice of physicists and takes the difference between field and particle theories to be that the former associates infinitely many "degrees of freedom" (kinematically independent variablesthe values of Φ at each point) with finite regions of space, but the latter only finitely many (the positions and momenta of a finite number of particles in a finite region).

The problem with the broad and formal characterizations of the field is that they ignore historically important distinctions. For instance, Aristotle's (384322 BCE) plenum (i.e., space full of matter, with no vacuum) ascribes different propertiesgravity here, levity thereto regions of space, but one would like to distinguish the modern concept of a field from the ancient plenum. Newton's (1687) gravitational field ascribes to every point of space a quantitative disposition for bodies to move (absent other bodies, if a body were at a point a distance r from a body of mass M then it would have acceleration proportional to M/r2 ), which distinguishes it from the ancient plenum. But understood literally, Newtonian gravity is a force that acts at a distance without mediation, hence Newton took it as a purely mathematical, "effective" description of some unknown underlying physics (which he sought in vain; in the early twenty-first century it is believed to be general relativity). Indeed, arguably the modern conception of the field is of something physical that mediates the long-range interactions between bodies. If so, Michael Faraday's (mid-nineteenth-century) arguments for the reality of the electromagnetic field are crucial. For instance, he distinguished physical from merely mathematical fields according to whether changes propagate at a finite speed or not (i.e., "through" the medium or not).

The atomists (especially Democritus, 460370 BCE) rejected the plenum, arguing that the physical world could be understood in terms of atoms moving in the void. However, general rejection of the vacuum meant that this idea did not become the foundation of useful science until Descartes (15961650); and while he believed in the plenum, he envisioned it to be composed of particles of varying sizes. Although Descartes failed to derive quantitative consequences from atomism, his successors, up to the present, have found it one of the most fruitful ideas in physics.

## Quantum Fields and Quantum Particles

In the twentieth century, quantum field theory (henceforth "QFT") was developed, and experimentally tested with unprecedented accuracy, particularly in particle accelerators. Classical fields can be decomposed into a sum of waves of different amplitudes (as a chord can be decomposed into different notes), which means, intuitively speaking, that quantum fields can be decomposed into a sum of waves with quantized (i.e, whole number) amplitudes. In quantum mechanics a wave(-function) represents a particle (its probability distribution in space), so there is a natural equivalence of a quantum field with a system of quantum particles, with the whole number amplitude of a wave in the decomposition representing the number of particles with that wavefunction. Thus because amplitudes become quantized, QFT is the best theory of both fields and subatomic particles.

However, the particle interpretation is only approximate: The field-particle distinction does not really dissolve in QFT. First, quantum mechanical superposition means that a quantum field may contain an indeterminate number of particles (e.g., two with some probability and three with another), which conflicts with the intuitive idea of a particle. Second, an accelerating observer will decompose a field into waves differently from a nonaccelerating observer; in particular, when the nonaccelerating observer says the field contains no particles, the accelerating observer will say that it does (these are known as "Rindler" particles). There is no contradiction, because if the accelerating observer captures a particle, he or she thereby changes the field to a state that all observers agree contains particles. All the same, the concept of a "particle" does not allow for the absence or presence of particles to be frame-dependent. Finally, there is a theorem that in relativistic QFT it is impossible to localize particles to any finite region; if so, they don't fit the intuitive idea of a particle at all.

According to formal definitions, QFT is a field, not particle theory, because it involves infinitely many degrees of freedoma fact with profound consequences in quantum mechanics, which are obscured by the particle interpretation. Infinite degrees of freedom mean that there are many quantum versions of a field, some of which may not allow a particle decomposition at all (technically, there are unitarily inequivalent representations of the canonical commutation relations). One might think that observations of particles in the world show that the particle version is the correct one, but because of the Haag-Hall-Whiteman theorem there are reasons to think that realistic fields have no particle formulation (technically, there may be no Fock representation of an interacting field). If so, the appearance of particles is presumably explained by the correct version suitably approximating a system of particles. Specifically, there are field states arbitrarily close to states of particles infinitely far apart.

Quantum mechanics can also treat a system of particles, which is (modulo the previous discussion) a field for which the particle content is always definite. Beyond the fact that quantum particles are represented by wavefunctions, there are important differences in the "identities" of classical and quantum particles that the following analogies illustrate. Classical particles are like badges with different pictures on them; the pictures make them distinguishable entities. Some quantum particlesbosonsare like money in the bank: Nothing distinguishes two of the dollars in an account from each other. Other particlesfermionsare like memberships in a particular club: Like money, one membership isn't any different from another; but unlike dollars, one can only have a single membership. (Technically, fermions satisfy the "exclusion principle": there can be at most one particle in any state.) To distinguish bosons and fermions, they are called "quanta"; however, these analogies fail to reveal that quantum mechanics allows other kinds of particles"quarticles"that differ from both quanta and classical particles.

## Bibliography

Earman, John, and Doreen Fraser. "Haag's Theorem and Its Implications for the Foundations of Quantum Field Theory." Forthcoming.

Hesse, Mary. Forces and Fields. London: Greenwood, 1962.

Huggett, Nick. "Philosophical Foundations of Quantum Field Theory." British Journal for the Philosophy of Science 51 (2000): 617638. Reprinted in Philosophy of Science Today, edited by Peter Clark and Katherine Hawley. Oxford: Oxford University Press, 2003.

Lange, Marc. An Introduction to the Philosophy of Physics: Locality, Fields, Energy and Mass. Oxford: Blackwell Publishers, 2002.

McMullin, Ernan. "The Origins of the Field Concept in Physics." Physics in Perspective 4 (2002): 1339.

Pullman, Bernard. The Atom in the History of Human Thought. Translated by Axel R. Reisinger. Oxford: Oxford University Press, 2002.

Teller, Paul. An Interpretive Introduction to Quantum Field Theory. Princeton, NJ: Princeton University Press, 1995.

Nick Huggett (2005)

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