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Most of classical and quantum physical phenomena are fundamentally described and explained in terms of fields, such as the electromagnetic and gravitational fields. These physical entities are not localized objects or particles, they generally vary in time, and they are defined at every point in space. They represent the influence, or force, an object or particle would experience at each point in space, at the time indicated. These fields are represented mathematically by functions of space and time. Field theories are the systematic theoretical mathematical-physical descriptions and elaborations of these fields, including their generation, detection, behavior, and relationships with one another and with other physical entities, such as particles. Generally, field theories are expressed in terms of partial differential equations that describe the relation of the fields to the entities that cause, or source, the fields.

There are also energy and momentum conservation equations that further constrain the fields, as well as the closely related equations of motion, which describe how particles or objects move at every point under the influence of the fields. All of these equations are generally derivable from a very special function, called a Lagrangian, which gives the kinetic energy minus the potential energy of the entire system. The behavior of the system described by the field equations and the equations of motion is always given by an extremum (maximum or minimum) of the Lagrangian action.

In physics there are four basic interactions, or forces: electromagnetism, the strong nuclear interaction, the weak nuclear interaction, and gravity. All of these are represented by fields, and their description, generation, behavior, and associated phenomena are treated and explained by field theories. As just indicated, these fields are generated by sources. For example, an electric field has charge as its source, and a magnetic field has a magnetized object or a current as its source. A gravitational field is generated by anything that has mass-energy. As also mentioned, these fields can be time-varying and they can propagate. Time-varying, propagating fields are often referred to as waves, or radiation. Thus, we have electromagnetic radiation—light, X-rays, radio-waves—which are really time varying electromagnetic fields. Similarly, a gravity wave or gravitational radiation is a time-varying gravitational field, or the time-varying, propagating changes of curvature through space. Fields also have particles associated with them, both those that function as sources and those that are quanta of their waves (photons for electro-magnetic waves and gravitons for gravitational waves). These bosons (integer-spin particles) are the force carriers of their respective fields between other particles, or distributions of particles, usually those that constitute matter (half-integer spin particles, like quarks, protons, neutrons, and electrons—the fermions).

Finally, at the highest energies, these four fundamental interaction fields probably undergo unification. That is, they become indistinguishable from one another at very high energies. At a relatively low energy, equivalent to a temperature of 10¹⁵ K (kelvin), the electromagnetic and the weak nuclear interaction become indistinguishable—the electroweak interaction.

Electroweak unification has been securely demonstrated and described; this theoretical achievement is due to Stephen Weinberg and Abdus Salam. At a much higher energy (equivalent to 10²⁷ K) the electroweak interaction and strong interaction unify in the Grand Unified Theory (GUT) interaction, and this in turn probably unifies with gravity at an energy equivalent to 10³² K, above which is the realm of quantum gravity and quantum cosmology. As of 2002, there was no theory adequately describing these last two levels of unification, nor were there experiments and observations unequivocally requiring them. However, there has been very promising theoretical and experimental progress made on both fronts. If and when total unification of all four fundamental interactions is attained this will complete the unification program that began with James Clerk Maxwell's brilliant unification of the electric and magnetic fields in 1864.

The early history of field theory

The concept of a field first appeared in hydrodynamics in the treatment of continuous media, such as fluids. Many mathematical physicists of the seventeenth and eighteenth centuries treated fluids or continuous bodies by dividing them up into small volumes or elements, but it was John Bernoulli who in 1732 first wrote down the equations of motion for these elements, considering them as point particles. Using this approach, Leonard Euler fashioned hydrodynamics into a field theory by modeling the motion of a fluid by giving its velocity at each point in the fluid, and using partial differential equations of these velocity components as functions of time and of the spatial coordinates. In doing this, the molecular structure of the fluid was neglected and it was treated as a continuum, with key parameters being determined at every point. This enabled researchers to describe the transmission of effects through the fluids. Somewhat later the more challenging problem of the propagation of displacements in solids, where elastic forces are prominently involved, was tackled. This was adequately solved by George Stokes in 1845.

Thus, field theory first emerged to describe the behavior of a continuous medium. There was at that time a different set of important physical phenomena (gravity, and electric and magnetic attraction and repulsion), which seemed to involve action at a distance. In these cases, such as in Isaac Newton's theory of gravity, it was assumed that there was no medium transmitting these interactions and that their effects occurred instantaneously. During the nineteenth century, due to the work of Joseph-Louis Lagrange, Pierre-Simon Laplace, and Siméon-Denis Poisson, the action at a distance in these cases began to be considered somewhat like a field, but without the presence of a fluid or a medium. In the case of gravity, for instance, the force of attraction at any location outside of a massive body can be designated in terms of what a point test mass would experience at each of those coordinate positions. This can be expressed in terms of the gravitational potential V at each position, which satisfies the well-known second-order differential equations of Laplace (empty space) or Poisson (nonempty space). Thus, both the gravitational force and the gravitational potential are fields. As such, they are no longer properties of discernable matter, but of empty space itself.

Despite its representation by a potential field, gravity continued to be considered an action at a distance throughout the nineteenth century because the gravitational potential could not be associated with any discernable medium. The propagation of gravitational effects was not affected by any material changes in the intervening space; it appeared instantaneous, no mechanical model for the action of a medium could be conceived, nor could any energy be located between the gravitationally interacting bodies. It was only in the early twentieth century, with the advent of Albert Einstein's General Relativity (his theory of gravity), that gravitation became recognized as a field theory in the strict physical sense. Electromagnetism, in contrast, began to be considered as a genuine field-theory in the nineteenth century, precisely because it clearly fulfilled these same criteria.

Classical field theory

It was Michael Faraday who in the mid-nineteenth century first showed that action at a distance provides an inadequate description of electric and magnetic phenomena. His studies convinced him that electrical and magnetic influences propagated through a medium and not at a distance. The basic idea is that of a "continuous action" of forces filling space, not of a continuous mechanical action. Faraday's diagrams of lines of force originating in and returning to conductors and magnets stimulated Baron Kelvin (William Thomson) and Maxwell to formulate electromagnetic behavior in terms of fields. In comparing gravitational forces with electromagnetic forces, however, Faraday was unable to extend to gravity his arguments for propagation through a medium. Thus, Newtonian gravity continued to be considered by most, from a physical point of view, to be an "action at a distance" theory.

Faraday's key insights concerning electromagnetism were confirmed by Kelvin, who in the 1840s was able to show that the same mathematical formulae could be used to describe fluid and heat flow, electrical and magnetic behavior, and elastic behavior. Kelvin thus established the important analogies among all five classes of phenomena, as well as that representing electric and magnetic phenomena by lines of force was consistent with their inverse-square falloff. Both Kelvin and Maxwell were careful not to draw conclusions about the reality of physical media from these detailed mathematical analogies. However, once Maxwell had formulated his highly successful electromagnetic field equations, which really provide a detailed quantitative unified description of electrical and magnetic phenomena, he and other physicists began to interpret these fields as a form of matter, so much so that matter in the usual sense gradually came to be looked upon in terms of fields, rather than vice versa. This was especially true once it was clear from Maxwell's theory that propagation of electro-magnetic fields is not instantaneous and that electromagnetic energy, which can be transformed into other forms of energy, is contained in the fields themselves. Maxwell also succeeded in associating momentum with the electromagnetic field, and the physicist John Henry Poynting later developed the concept of energy-flux, and showed that this applied in a concrete way to electromagnetic fields and electromagnetic radiation. These developments have all contributed to supporting the conception of electromagnetic fields as a genuine form of matter, and they presaged the discoveries in Special Relativity that mass and energy are equivalent, and later in relativistic quantum theory that all forms of matter are fundamentally interacting fields. Maxwell's theory was the first fully successful and complete field theory, and remains the best example of a classical field theory.

The influence of Special and General Relativity

Along with Maxell's electromagnetic theory, Special and General Relativity strongly reinforced the usefulness and strength of the field-theory perspective, and even the realistic physical interpretations given to fields. The formulation and confirmation of Special Relativity have been especially influential in effecting this. Besides the discovery of mass-energy equivalence, mentioned above, perhaps the most influential event was the 1887 experiment by Albert Michelson and Edward Morley, the most compelling interpretation of which held that "the ether" does not exist and that therefore the velocity of light is constant with respect to any inertial frame (any coordinate system moving at a constant velocity). Thus, there is no absolute standard of rest. Moreover, there is no medium needed for electro-magnetic fields to propagate. The fields themselves are fundamental and are, in a sense, their own media. Furthermore, since nothing propagates instantaneously, there are no perfectly rigid bodies or incompressible fluids, as envisioned in Newtonian mechanics. These are idealizations that, strictly speaking, are never realized. What is most impressive is that Maxwell's electromagnetic field theory turns out to be completely consistent with Special Relativity and can be explicitly formulated as such (in Lorentz-invariant fashion) in a natural and straightforward way. This confirms the insights that fields are a basic form of matter and that they are integral and indivisible.

Newton's theory of gravity was not generally looked upon as a field theory in the same way electromagnetism was, but rather as an "action at a distance" theory. Einstein changed that. In formulating his theory of gravitation, he fundamentally conceived space and time as fields that obey field equations, connecting space and time with the mass-energy distribution that they "contain." These space and time fields are the components of the metric tensor that makes space-time measurements possible. They are like, and in fact replace, the gravitational potential of Newton, but they are not defined in a pre-existing background space-time. They are the space-time. And this space-time is, in general, not flat but curved, depending on the density and pressure of the mass-energy on the spacetime manifold, including all nongravitational (e.g., electromagnetic) fields. As a result, light rays (electromagnetic radiation) and freely moving particles follow the geodesics in curved space-time. Gravity is no longer conceived as a force, strictly speaking, but rather as the curvature of space-time. And light is affected by this curvature, unlike in Newtonian gravitational theory. This is consistent also from the point of view that light possesses energy, which is equivalent to mass, according to Special Relativity. Through observations of the bending and the red-shifting of light rays in gravitational fields, as well as through other observations (including the evidence for the existence of black holes), General Relativity has been impressively confirmed. General Relativity also predicts the existence of gravitational radiation—the propagation, at the speed of light, of variations in the curvature of space-time. This has been indirectly detected. And there is a massive effort to detect these gravity waves directly.

Quantum mechanics and quantum field theory

One of the great accomplishments of twentieth-century physics was the development and experimental confirmation of quantum theory. This began with failures of classical physics to account for the stability of atoms, the photoelectric effect, the explanation of the Planck blackbody spectrum, wave-particle duality, and the intrinsic uncertainties in certain types of measurements. Essentially, it became clear that physical reality, at its most fundamental level, could not be modeled in a continuous way, but only in terms of discrete quanta of energy, angular momentum, spin, and so on. Furthermore, any measurement of a system automatically affects that system in some way, with the Un-certainty Principle always applying. In any quantum measurement, the outcome is never precisely predictable. The theory gives probabilities that any one of a set of possible outcomes will result from a given measurement. All of these issues have been more or less satisfactorily incorporated into quantum mechanics by Erwin Schrödinger, Werner Heisenberg, and others. Paul Dirac properly formulated quantum mechanics within the framework of Special Relativity, yielding relativistic quantum mechanics. As such, in both these formulations, quantum mechanics is not a field theory, but rather a quantum theory of discrete bodies and individual particles in their interactions with one another.

Relativistic quantum mechanics, however, is plagued by a serious problem: it allows for negative-energy states, which would seem to predict an infinite series of decays. It turns out that this problem can be solved only by moving from consideration of single particles to indefinitely many particles. This automatically leads us to consider quantum fields as fundamental, with the particles being localized realizations (modes or quanta) associated with these fields. The result is the development of the extraordinarily successful quantum field theory. The fundamental structure of physical reality has come to be understood in terms of the interaction of these quantum fields, some of which are bosonic, or force-carrying, and some which are fermionic, or particle-constituting. As mentioned at the beginning, there is strong evidence that at higher and higher energies or temperatures the four fundamental field interactions (electromagnetism, the strong and weak nuclear interactions, and gravity) unify step by step, and become indistinguishable. There are still many unknown details and challenges in constructing a completely adequate unified field theory and in explaining some of the features that physical reality manifests, particularly with respect to the quantum connections between gravity and space-time, as well as gravity and the other three interactions. But quantum field theory as it is understood in the early twenty-first century provides an impressive and reliable, though provisional and incomplete, description and guide to how reality at its most fundamental levels is constituted and behaves.

Relevance to the religion-science dialogue

The principal relevance of field theory to the religion-science dialogue is that it gives a reliable, well tested, and nearly comprehensive account of how reality is put together at its most fundamental levels. It also ultimately sheds light, through its applications in cosmology, on how the universe evolved from an extremely hot, homogeneous, simple, and undifferentiated quantum-dominated state to its present cool, lumpy, complex, and highly differentiated state. This strongly constrains theology in speaking about how creation occurred and about how God acts in creating and in sustaining what has been created. The relationships, processes, interactions, and regularities described by field theory—the laws of nature and physics—must be acknowledged to play a key role as channels of God's creative ordering power in reality. The concept of dynamic interacting fields, along with the auxiliary concepts and phenomena connected with them, can also provide analogies that can be employed in constructive theological programs.

See also FIELD; GRAND UNIFIED THEORY; GRAVITATION; PHYSICS, QUANTUM; RELATIVITY, GENERAL THEORY OF; RELATIVITY, SPECIAL THEORY OF

WILLIAM R. STOEGER