Can vectors in physics be represented by complex numbers and can they be divided? [closed]

Question

Below is attached for reference, but the question is simply about whether vectors used in physics in a vector space can be represented by complex numbers and whether they can be divided.

---

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic number fields, p-adic fields, and so forth.

Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. The theory of field extensions (including Galois theory) involves the roots of polynomials with coefficients in a field; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel–Ruffini theorem on the algebraic insolubility of quintic equations. In modern mathematics, the theory of fields (or field theory) plays an essential role in number theory and algebraic geometry.

In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the value of the scalar field at the same point in space (or spacetime). Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.

In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it is an algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as an algebra, this multiplication must satisfy certain compatibility axioms with the given vector space structure, such as distributivity. In other words, an algebra over a field is a set together with operations of multiplication, addition, and scalar multiplication by elements of the field.

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex numbers, rational numbers, or even more general fields instead. The operations of vector addition and scalar multiplication have to satisfy certain requirements, called axioms,... An example of a vector space is that of Euclidean vectors which are often used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real factor is another force vector. In the same vein, but in more geometric parlance, vectors representing displacements in the plane or in three-dimensional space also form vector spaces.

In classical mechanics as in physics, the field is not real, but merely a model describing the effects of gravity. The field can be determined using Newton's law of universal gravitation. Determined in this way, the gravitational field around a single particle is a vector field consisting at every point of a vector pointing directly towards the particle. The magnitude of the field at every point is calculated applying the universal law, and represents the force per unit mass on any object at that point in space. The field around multiple particles is merely the vector sum of the fields around each individual particle. An object in such a field will experience a force that equals the vector sum of the forces it would feel in these individual fields.

Because the force field is conservative, there is a scalar potential energy per unit mass at each point in space associated with the force fields, this is called gravitational potential.

Gauss' law for gravity is mathematically equivalent to Newton's law of universal gravitation, but is stated directly as vector calculus properties of the gravitational field.

Answer 1

Can vectors in physics be represented by complex numbers?

Absolutely. There exists a direct isomorphism between the 2D Euclidean vector space and the Argand plane, for a start.

In fact, it is possible to talk of mathematical objects called quaternions and use quaternion algebra analogously to vector algebra. Historically quaternions were used to represent geometrical operations and transformations in 3D space - in the days before vector algebra. (They still are used, especially in areas such as computer graphics where they offer one or two advantages over the simpler world of vectors.) In any case, the relationship to vector algebra is a very close one.

Can vectors in physics be divided?

In general, no, vector-vector division is not a well-defined operation. At least, not within the bounds of linear algebra. i.e. There exist none or multiple solutions to the equation $\vec{y} = \mathbb{A} \vec{x}$. (See the Wolfram page.)

Saying this, the concept of vector division has an interesting relationship with your first question. If we map vectors to complex numbers (or quaternions in > two dimensions), we can use complex division or quaternion algebra respectively to define an analogous "vector division" operation. Note that quaternions can in fact be extended to higher dimensions, which allows for interesting possibilities. This study of this falls under the area of Clifford algebras.

(Note: vector-scalar division is of course well-defined, as is point-wise division of equal-length vectors, but I presume you are not referring to that.)

Answer 2

In a certain sense, 3-d vectors can be divided in physics. To define the division, you begin by defining multiplication using the cross product. I.e. if $\vec{u} \times \vec{v} = \vec{w}$ then $\vec{w}/\vec{v} = \vec{u}$. This is a definition of division that is non commutative. It is also not well defined (as written) because you can always add multiples of $\vec{v}$ to $\vec{u}$ and still get a product of $\vec{w}$. That is, $\vec{u} \times \vec{v} = \vec{w}$ implies that $(\vec{u}+k\vec{v})\times\vec{v} = \vec{w}$. To make the division well defined you can require that $\vec{u}\cdot\vec{v}=0$ by subtracting off any component of $\vec{u}$ in the $\vec{v}$ direction.

Before you have division, you might want to have inverses. That is, for a vector $\vec{u}$, we need to define a vector $\vec{v}$ such that $\vec{u}\vec{v} = 1$. In the usual approach to physics this is nonsense because "1" is not a vector. But it is possible to make sense out of this and a group of physicists have done just that. To get a "1" from a vector product you use the dot product. Then the inverse of a real vector is just the same vector, divided by its squared magnitude.

In actual practice, it's probably clear that you need both the dot product and the cross product to fully define vector multiplication. The discussion I've given here is intended only to whet the interest of the reader, an adequate understanding will require a lot of work. For example, in the first paragraph, w is always perpendicular to both u and v; how do you define division when w and u aren't perpendicular?

Not very many physicists are interested in this sort of thing. It's studied by groups led by David Hestenes and makes up a part of what is called the "Geometric Algebra". As you can see from the above, it requires that vectors be generalized. For example, since addition works with vectors, one must allow constructions such as the sum of a vector and a scalar such as $\vec{u} + 3$.

As you would expect, the geometric algebra also implies a geometric calculus (which includes all the usual vector calculus relations you're familiar with such as Stoke's theorem). To learn more about it will likely require a great deal of time. To start you can click around on his website:
http://geocalc.clas.asu.edu/html/Intro.html

The geometric algebra is used in standard physics under a different name; the "Dirac algebra" or "gamma matrices" (so important to our modeling of the electron) are the geometric algebra in 3+1 dimensions. In this algebra, the usual spacetime or 4-vectors are generalized to include scalars, tensors, psuedo-vectors, and a pseudo-scalar.

Answer 3

A complex number is an ordered pair of real numbers (a,b), that is also exactly the definiton of vectors in a two-dimensional real vector space. Most vectors in physics are not two dimensional, and in QM they need not even be real and can have infinite dimensions. Any vector can be divided by a scalar quantity. Vector division by vectors is not a well-defined concept, probably because it is useless. If you have an ordered set of complex numbers you can represent any vector you like as complex numbers, you just map one coordinate as (a,b), where b is 1 if a is to be taken as real, and 0 otherwise, this is a trivial concept and reflects the asinine nature of the question.