## Phase enables relativity

Physicists are experts at assigning new meanings to the terms established in common language. I mentioned the term “dual” or “duality” before. In common language dual means two. In physics dual means equivalent. Physicists use the word “phase” in different ways too. The dictionary definition of phase is: (noun) a distinct period or stage in a process of change or forming part of something’s development, (verb) to carry out something in gradual stages.

In physics phase is about a dynamic relationship between 2 quantities. The key word is dynamic. We need two numbers to define a phase. Let’s start with the classic definition. Fiqure-1

The bold arrow (vector) is a single entity. It has length L. The dotted lines represent the projection operation. The line segments AB and AC are the projections of this vector onto the horizontal and vertical axes.  In this particular coordinate system (2-dimensional Cartesian coordinates) the singular vector has 2 projections.

The lengths of the projection line segments are given by

AB = L cos(θ)
AC = L sin(θ)

The angle θ is defined as

θ = arctan(AC/AB)

We used only 2 quantities to define the angle, the lengths of the projections AC and AB. An angle is always defined by 2 numbers. The coordinate system or the nature of the space in which the vector is embedded does not matter. To illustrate this let’s embed the same vector in a 3-dimensional space. Figure-2

Now, the angle is defined by

θ = arctan(AC/AE)

If the picture of Figure (1) is static, in other words, if the vector in Figure (1) is stationary then θ is just an angle. If the vector is rotating around point A then θ is not just an angle, it is a phase.

In order to express the dynamic nature of the phase explicitly, we can replace θ with ωt where ω is the rate of change of θ, and t is the elapsed time.

Using Figure (1) as an example

AB = L cos(ωt)
AC = L sin(ωt)

Where ω = 2πf

The quantity f is the number of full rotations (cycles) per second. Remember, one full rotation is 360 degrees which is equal to  2π radians where π=3.14….  The quantity f is also known as frequency.

Let’s plot sin(ωt) and cos(ωt) to see how they vary in time. Figure-3

The two curves are essentially the same but there is a time difference between them. This difference in time is called a phase difference. The phase difference between the sin(…) and cos(…) is 90 degrees (π/2 radians). In units of cycles the difference is 0.25 cycles.

Figure (3) shows the magnitudes (in time) of the projections of a vector rotating in a spatial space. What does spatial mean? The term spatial refers to the extension in the familiar space. In physics and mathematics we use the term space in the abstract sense. We refer to the familiar space as spatial to distinguish it from other types of spaces. Some examples of the other types of spaces are: space-time, phase-space, and the complex plane.

In these abstract spaces the coordinates have different units. For example, the space-time is 4-dimensional, it has 3 spatial coordinates and 1 time coordinate. The phase-space is a plane where the horizontal axis is any variable X and the vertical axis is the rate of change of that variable.

Another type of space is the complex plane where the horizontal axis consists of real numbers and the vertical axis consists of imaginary numbers.

In Figure (4) below the bold arrow represents a complex number r = x + iy Figure-4

The components x and y are similar to the projections of Figure (1) and given by

x = r cos(φ)
y = r sin(φ)

The angle φ is also known as phase. The components (projections) x and y may or may not be functions of time t. The angle φ is known as the phase of a complex number.

φ = arctan(y/x)

Phase-space

We mentioned the phase-space before. The phase-space is a plane where the horizontal axis is any variable X; the vertical axis (X’) is the rate of change of that variable. Figure-5

The φ of Figure(5) is also known as phase. The phase-space is the key concept in Nonlinear Dynamics. Hundreds of books and thousands of scientific papers have been written on the phase-space concept. There is even a Nobel Prize. Simon van der Meer received a Nobel Prize (shared with Carlo Rubbia in 1984) for his many contributions to CERN but particularly for his invention of “stochastic cooling” which is based on phase-space concepts.

There is much more to say on the concept of phase. I want to discuss the “geometric phase,”  “quantum mechanical phase,” and “gauge (phase) invariance” separately.

Since phase describes a dynamic relationship between 2 orthogonal quantities it is fundamental. Phase enables relativity. 