Dr. Jonathan Kenigson, FRSA

NEW YORK, NY. Vectors are merely things that have a magnitude and direction, like wind speed and heading, the bearing of an aircraft, the velocity of a dart, or the orbital mechanics of a spacecraft. A vector field is a mathematical concept that associates a vector to every point in a given space, usually a multidimensional space like the plane or the 3-dimensional world. Abstract vector fields are used to model physical quantities that have both magnitude and direction, such as velocity, force, or electric and magnetic phenomena. In essence, a vector field can be thought of as a continuous ("smooth") distribution of vectors. There are two main classes of elementary operations we may perform on such fields: Those that "operate" on fields and return numbers, and those which operate upon fields and return other fields in-turn. The first sort of operation is quite intuitive. Given two vector fields, you can compute the "dot product" at each point, which is a measure of the similarity in direction and magnitude between the two vector fields. The divergence of a vector field is a scalar field that measures the rate at which the field is spreading out or converging at each point.

The second sort of operation - from fields and back to fields - makes a great number of abstract physical concepts mathematically explicable, but is also more complex. First, let us inspect several common operations of this type. Two vector fields can be added together by simply adding the corresponding vectors at each point in the space. The result is a new vector field. In three-dimensional space (R), the cross product between two vector fields can be computed at each point, resulting in a new vector field. The cross product is a vector that is orthogonal (perpendicular) to both input vectors. A vector field can be multiplied by a scalar (a single real number) by multiplying each vector in the field by that scalar. In three-dimensional space, the curl of a vector field is another vector field that measures the rate of rotation or circulation of the field around each point. A tensor is a mathematical object that is a generalization of scalars, vectors, and matrices. It can be thought of as a multidimensional array of numbers, where the number of dimensions (or "ranks") determines the type of tensor. A scalar (regular number) is a rank-0 tensor, a vector is a rank-1 tensor, and a matrix is a rank-2 tensor. Tensors can have higher ranks as well, depending on the complexity of the problem being modeled. A tensor can describe linear transformations, geometric transformations, and relationships between different vector spaces. The difference between a tensor field and a vector field is that a vector field assigns a vector (a rank-1 tensor) to each point in space, while a tensor field assigns a tensor of a particular rank to each point. A vector field is a special case of a tensor field, where the associated tensors are rank-1.

Classical vector fields are continuous and can take any values in a given range, whereas quantum fields are quantized and exhibit both wave-like and particle-like properties due to the principles of quantum mechanics. In Quantum Field Theory (QFT), fields are quantized, meaning they can only take certain discrete values. Particles are conceived as excitations or quanta of these underlying fields. In quantum electrodynamics (QED), the electromagnetic field is described by the quantized version of the classical electromagnetic field. Photons are the quanta of this field and mediate the electromagnetic force between charged particles. In attempts to develop a quantum theory of gravity, the gravitational field is treated as a quantum field. However, a complete and consistent quantum gravity theory has not been established as of 2023. One popular and historical approach in the 20th century was loop quantum gravity. Another approach is string theory, which proposes that fundamental particles are not point-like but rather one-dimensional vibrating strings, and gravity arises from the interactions of these strings. Spinor Fields and Gauge Fields arise in Unified Field Theories. These are examples of fields in QFT and Yang-Mills theories, which both attempt to describe particles with different spins and internal symmetries. In Yang-Mills theories, non-Abelian gauge fields mediate the strong and weak nuclear forces via the exchange of gluons and electroweak bosons (W and Z bosons), respectively. Field coupling refers to the interaction between different quantum fields through their mutual influence on each other. In the context of particle physics, coupling constants determine the strength of the interaction between particles mediated by the exchange of other particles or quanta. For example, the electromagnetic coupling constant determines the strength of the electromagnetic interaction between charged particles, while the strong coupling constant determines the strength of the strong nuclear force between quarks mediated by gluons.

Non-Abelian physical fields like those arising in QFT are almost always represented by tensors that break in a fundamental way: They do not behave as ordinary numbers in which the order of multiplication does not matter. QFT complicates the picture substantially, in that some elements multiply "as they should" (e.g. as regular numbers where order doesn't matter), but many more elements vary wildly or are even undefined or infinite if the order of multiplication is reversed. Physically, it is from the context of rank-2 tensors (matrices) that algebraic fields arise. In abstract algebra, just as in physics, a non-Abelian ring is a structure in which the multiplication operation is not commutative, meaning that the order of multiplication "breaks" for some elements a and b in R. A canonical example of a noncommutative ring is the ring of n n matrices with real or complex entries, where matrix multiplication is not commutative. Because tensors are represented as matrices, this behavior can be seen as a problem with the underlying algebraic rings that is severe enough to warrant parameters that minimize the damage to both the algebraic and geometric structures involved. An algebraic field is the hero we seek: It is a special type of ring in which every nonzero element has a multiplicative inverse (e.g. can be made a fraction). Examples of fields include the rational numbers (Q), the real numbers (R), and the complex numbers (C). If a field is too small for our physical or mathematical needs, we can enlarge it in an agreed-upon way. For the pure algebraist, the results are Galois Theory and Splitting Fields, from which all of traditional school algebra derives. For the physicist, the result is new quanta and the ability to represent quanta as having imaginary mass or energy. For the algebraist, more formally, if F and K are fields such that F is a subfield of K (meaning F is a subset of K and the operations of F are the same as those in K), then K is said to be an extension of F, and it is denoted as K/F. Field extensions are used to study the algebraic properties and relationships between different fields.