In mathematics and in physics, the reality behind complex numbers remains mysterious. Although they are essential to solving fundamental problems in science and engineering, their true relationship with the physical world is unknow. For example, why do they have two components- referred to as real and imaginary parts? Why is the imaginary part closely linked to the square root of minus one? And most importantly, what is their relationship with physical reality. In this post, I answer those questions and unravel the mystery of complex numbers.
Defining a space requires a reference system that gives every location therein a meaningful existence—i.e., an address, as a unit or a quanta of that space, which is elementary and indivisible. In effect, it represents, the point in the space, as well as the smallest piece of information by assuming one of two states. In computer jargon it is referred to as a bit, because a domain (two-dimensional space such as display screen) is defined by bits, each of which has an address, could be in any one of two states and carries information. In quantum computing it is referred to as a Qubit and has properties that relate to quantum mechanics. Regardless of the nature of the elementary unit of a space, the smaller it is in relation to the overall space, the greater the level of space definition. A space void of such elements is an indefinable space is mathematically inert. It cannot be accessed or manipulated physically or mathematically, except as a whole. Therefore, an undefinable region within a definable space is considered a singularity, such as a black hole within the domain of space-time.
Let us assume a space definable in three dimensions by a continuum of microscopic and homogenous cubical elements. Such a space could be an entire universe! Lest us then introduce a conscious observer into it, whose physical being is constructed from clumps of those cubes. In fact, everything in that universe has to be constructed from those cubes. Else, things would be indefinable in that universe, as every location in any well defined space must be defined by the same type of element defining the rest of the space. This includes locations occupied by object that appear heterogeneous. Therefore, homogeneity of a space includes everything therein, and that is what gives any domain consistency in the mathematical logic applicable to it, including the consistency of mathematics in our universe. If for example, it were possible to define a space using a combination of say cubes and spheres as basic units, that space and the objects therein would be nonhomogeneous and as such, its laws of physics would differ from one location to another and from one object to another, depending upon the mix of elements.
Returning to our cubical universe, the observer of that universe could only distinguish large numbers of cubes when they are clustered to form much larger structures. He cannot detect individual cubes, because individually they emit no signal that he could detect and they offer no apparent resistance to his motion. Therefore, he cannot use them as units to define his space and its properties. Instead, he uses arbitrary units to make measurement and define his surroundings. Furthermore, unable to locate the boundaries of his universe, he can only reference locations relative to some object within his domain.
Now, let the cubes oscillate against each other. We shall not consider the effect of their amplitude (the maximum distance between them as they oscillate), on the observer in this post. For simplicity, we shall restrict their oscillation to one common horizontal direction, and restrict their orientation in space, so that when the observer disturbs them, they still maintain their original orientation. This requirement has direct implication on his definition of the geometry of objects in his world and on how he determine his mathematical constants therein, as we shall explain a future post.
This form of dynamics results in exposing some background vacuum. Therefore, when modelling his space, the observe has to account for two media, namely, the cubes and the exposed background vacuum. This is a fundamental requirement if he is to correctly model his space and the behaviour of the objects within it. Note that if the elements were inseparable for whatever reason, signals cannot pass through the medium and the space would be indefinable, in the same way as if there were no medium. Therefore, any medium that defines a space homogenously has to be of discrete and homogenous elements.
On account of this argument, to maintain mathematical consistency when modelling objects interacting with such a medium, as quantum level objects are, the observer has to reference elements of the medium as well as the exposed background spaces, within an interrelated framework. Since the observer in the cubical universe is dealing with a dual media, if he references the cubes positive, then he must reference the exposed background spaces negative or vice versa. In other words, if he labels a cube (+1), then an exposed space would be (−1). The sign is an arbitrary notation.
However, when modelling systems interacting with two media simultaneously and the effect of the system on one medium affects the other medium, each medium has to be account for in relation to the other within a single mathematical framework. In such cases, a simple system of referencing in positive and negative is insufficient and a more fundamental procedure that maintains the effect on each of the two media in a single framework has to be followed. However, such a procedure highlights a problem relation to keeping track of the changes in each of the two media, and this brings us to the need for complex numbers, as we shall now explain.
Because of the arbitrary nature of mathematical referencing, the square roots of any positive number could be either positive or negative. For example, [(2 x 2) = (-2 x -2) = 4]. There is no problem there. The problem arises for the observer when he has to decide on determining the square roots of the negative units in a dual media. Then, one of the roots has to be negative. Faced with negative square roots of units, the observer would be unable to trace them back to which of the media they belong. For example, (-2) as a root, could belong to [2 x -2] or to [-2 x -2]. In isolation, the source of a negative square root is impossible to identify as to which of the operations it belongs. Therefore, in order to maintain consistency of a mathematical procedure relating to dual media, the square root of the unit of one of the two media has to be assigned the negative root through out a procedure. However, rather than carrying the negative roots through out such a procedure, the negative value is assigned to a single unit, namely (√−1), and the remainder of the value could be assigned a positive value. Thus, the observer’s lack of awareness of his medium catches up with him when he attempts to model his space at the quantum level, and hence the answer to the question of how mathematics is related to the physical world.
The consistency of logic exhibited by systems within a domain is dictated by the properties of elements defining the system and the domain. When sufficient logic is acquired by a conscious mind dealing with a consistent domain, the acquired logic becomes self-supporting and the mind is then able to build on that logic without the need for further observation. This is why mathematicians have often been able to develop mathematics relating to some physical phenomena well before the discovery of those phenomena, as in the case of complex numbers and their applicability to quantum mechanics.
Returning to our topic of complex numbers and their relationship with the physical world, we need to now consider the nature of signals, for it is signals that enable the observer to observe and rationalise the world around him. Those signals are essentially quantum in nature, and the quantum world is all about signals in the form of travelling particle-waves propagating in the space fabric, as well as other secondary media. Signals are not entities. They are discrete events (isolated 3-dimensional waves) that propagate through space until they collapse on observers, hence the concept of a wave-function collapse in quantum mechanics. Each such an event (a wave) is made of two components. One component is part of one medium, which in the case of the cubical universe it is cubes, and the other is the spaces of the exposed background vacuum. The two parts are complementary and necessary for signals to materialize and propagate.
The observer in the cubical universe is unaware of the cubes bouncing against his senses or of the nature of the signals processed by his brain. He is aware only of the output in his mind in the form of images, sounds, odours, etc. Therefore, there is no way he could tell which part of a signal relates to the cubes and which relates to the background vacuum. However, to model his observations correctly, he has to take the two parts of the signals into account and keep track of them, regardless of whether or not he understands why his model is working correctly. Thus, he is able to use a double track system to model the effect on each media. One track he refers to as the real part and the other, which he tags by (√−1), he refers to as the imaginary part. To illustrate his model of a system graphically, and keep the two parts separate, he uses a complex plan in which one axis relates to the real numbers and the other axis relates to his imaginary numbers. Thus he begins to use complex numbers to model his physical reality!
Unlike the cubical universe, ours is defined by oscillating spherical elements, in which we use the letter (ί), and in some cases (j), to represent (√−1) and keep track of the imaginary part of our complex numbers in the double-entry system of our physical reality. Without such a tag to maintain consistency of our models, the two parts would become muddled and no sensible solutions could be obtained. For example, when we consider an electron translating through space, we have to account for its effect on the surrounding medium, which in this case referred to as the electric field. The electric field, as I shall explain in future posts in this blog, like all other energy fields, is composed of two components, namely elements of the fabric of space and the background vacuum. Therefore, as the electron moves, it produces an effect in two media. Furthermore, as it moves in a direction, its track could be considered a void moving in the opposite direction. Since its motion affects two media, it requires complex numbers to model its behaviour. In fact, this applies to modelling the behaviour of all quantum-level objects, including wave functions and their collapse, hence the appearance of (ί) in the Schrödinger equation.