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Complex Numbers

The complex number, a + ib, is defined as an ordered pair (a, b) of real numbers, obeying certain rules of addition and multiplication, such that i ² = −1, so that we can write i = √−1. a is called the real part and ib is the imaginary part. The terminology is misleading. Imaginary numbers are no more and no less “real” than real numbers defined using infinite series, in the sense that both are defined rigorously in mathematics (including the axiom of infinity), and neither have empirical existence.

A complex number is often represented on a graph, known as the complex plane, with real parts on the horizontal axis and imaginary parts on the vertical axis. Complex numbers obey the same rules as two dimensional vectors, and can be regarded as vectors in the complex plane. The conjugate of a complex number is its mirror image in the real axis. The magnitude of a complex number is found by multiplying by its complex conjugate and using Pythagoras’ theorem.

e to the i

The exponential function, exp(x), is defined by an infinite series. An infinite series does not mean that the series is really infinite. It just means that we can take as many terms as we like until we realise that the result of taking more makes so little difference that we no longer care. It is then possible to show that there exists a real number, e, such that exp(x) = e^x. Using i² = −1,
The series expansions for sin(x) and cos(x) are
and
So, we find that


This is probably the most useful and important identity in the whole of mathematics, certainly as so far as quantum theory is concerned. It means that stacks of stuff in trigonometry can be shown using the laws of indices which are really easy to apply, and it is the mathematical reason why quantum interference patterns behave just like wave interference patterns, even though there is no physical wave in quantum mechanics. In fact all the general features of quantum theory will boil down to mathematical trickery based on this identity. It follows at once that:

1.  e^ix traces out a unit circle in the complex plane.
2.  e^−ix = 1/e^ix is the complex conjugate of e^ix.
3.  e^πi = −1.
4.  e^2πi = 1.
5.  Any complex number, a + ib, can be expressed in the form re^ix where r = |a + ib| is magnitude, and x = atan2(b, a) is phase.
6.  The product of two complex numbers, re^ix and se^iy, can be found by multiplying their magnitudes and adding their arguments, re^ix se^iy = (rs)e^i(x +y).

Sums and Integrals

Mathematics is a language full of short hand notations. Because there is so much adding up, we use ∑ to denote a sum of many terms. For example a shorthand way of writing the exponential function is
A particular type of infinite sum is known as an integral. An integral is used when all the terms get smaller and smaller (tending to zero), when at the same time the number of terms in the sum gets larger and larger in such a way that the result of the sum tends to the value we actually want. An integral is often used to find the area under a graph, but it has many other uses. To find an approximate area under a graph, we draw a lot of thin rectangular strips of width δx, then add up the areas under the strips. This gives
By taking more, and narrower, strips we get closer to finding the actual area. When we have taken so many strips that taking more makes so little difference to the answer that we no longer care, we call this sum an integral, and write it like this:

Analysis

In this presentation I am just seeking to convey general ideas, but in reality we need to be very careful when talking about things like infinite sums of small quantities. Small errors in each small quantity can lead to large, or infinite, overall errors. Everything is fine when we can show that the result gets as close as we like to the answer we want, but sometimes this is not true. Consider an infinite sum of zeros,
So, 0 = 1. Obviously this isn’t true. The mistake lies in abusing the meaning of an infinite sum. An infinite sum really just means a finite sum with more terms than we need to count. In this particular sum there is always a −1 at the end, and it is necessary to count all the terms. The study of exactly what we can or can’t do with infinite series is analysis and requires extreme care. For most of the infinite series used in physics, everything is fine, but one area of modern physics is famous for being beset with problems of infinity, quantum electrodynamics, or qed. Discrete quantum electrodynamics shows how these problems can be avoided if one recognises that there is no such thing as a mathematical continuum in Nature. The maths of qed becomes complicated, and the errors are well buried, but, underneath it all, the problems are really no different from those in the paradoxes of Zeno.

The Axiom of Infinity

Modern mathematics is usually formulated to include the axiom of infinity. This enables us to talk about structures like infinite dimensional vector space and a mathematical continuum. This is dangerous. Theorems in finite dimensional space do not always hold in infinite dimensions. Subtle errors can slip in, much harder to spot than the error in showing 0 = 1. Relational quantum gravity avoids that kind of error by basing physics on empirical fact. This allows us to use only finite structures to model physics. We are still able to use mathematical structures incorporating infinity, just as we are able to use abstract mathematical ideas like the complex numbers, but statements about reality do not involve these structures.

The Relationship of Mathematics to Physics

It used to be thought that mathematics was abstracted from physical reality, but in modern mathematics that is not the case. Physics makes statements about the real world, while mathematics consists of mathematical structures defined from axioms. Pure mathematics proceeds by strict deduction from axioms, while making no comment on meaning or applicability. A statement is true within the context of a mathematical structure if it can be deduced from the axioms of that structure. A mathematical structure is said to exist if no contradiction can be deduced from its axioms.

In order to apply a mathematical structure in the study of Nature, we have to be able to interpret a subset of sentences of that structure as statements about reality. A statement in mathematics becomes a statement about Nature, when it is interpreted as such, but it is not the case that all statements of mathematics can be interpreted as statements about Nature. In my view, understanding how and when one should interpret mathematical statements is the most subtle, and interesting, aspect of the application of mathematics in any field. The validity of mathematics as a tool for scientific study depends upon correct interpretation.

In logic, our justification for using mathematical structures which don’t model reality goes like this. Let N be a set of statements about Nature. Let M be a set of mathematical statements which (with suitable interpretation) includes N, NM, and is such that any true statement in N is also true in M. Suppose we can prove that a statement, p, is true in M, and suppose p is in N. Then it follows that p is true in N. (Otherwise, ~p (not p) would be true in N. So, ~p would be true in M, giving a contradiction). This argument applies to the use of both real numbers and complex numbers in the mathematics of empirical science.

Kurt Gödel famously showed that there are undecidable statements in mathematics, and hence that mathematics can never be complete. His proof relies on the axiom of infinity. We cannot justify the axiom of infinity on empirical grounds. If we don’t allow its use in statements about Nature, then undecidable statements in mathematics are also excluded. It does not follow from Gödel’s theorem that science cannot be complete.

Geometric Progression

A geometric progression is a sequence of numbers, each one equal to the last multiplied by the same value, known as the common ratio, e.g. 2, 4, 8, 16, 32. The general form of a geometric progression is a, ar, ar², ... , ar^N − 1, where the common ratio is r and the initial value is a. A geometric series is the sum of a geometric progression.
We can calculate a geometric series, like this,
So,
If |r| < 1 then for large enough N, r^N gets so small that we can forget it. Then the sum of an infinite geometric progression is

The Dirac Delta Function

Dirac’s delta function, δ(x), is essentially a narrow spike at x = 0 with an area under its graph equal to 1, and narrow enough that making it any narrower makes so little difference to answers that we don’t care. Probably the simplest delta function is a rectangle with width δx and height 1/δx, where δx is small. The delta function can be used to express (almost) any useful function as sum of spikes each of height equal the value of the function at a point, like this
(Check this by replacing the integral with a sum).

Representations of the Delta Function

There are any number of (pseudo)functions with the property of the delta function, known as representations. A particularly useful representation of the delta function in quantum theory is
Strictly, this is not a function, because it is infinite when x = 0 (because e⁰ = 1). This is perfectly all right, so long as we remember that an integral is not really an infinite sum, but is a finite sum of enough terms that taking any more makes so little difference to the answer that we don’t care. There is no such answer for a representation of the delta function, but this is not important because we are only interested in finding answers after the delta function is placed inside another integral. If you want, you can skip the demonstration that this is a delta function, and go straight to the next section, but for those who want to see some detail, here it is.

Position, x, is usually treated using real numbers. Actually we never measure a real number, but only a terminating decimal in a finite range dictated by the means of measurement. A choice of units makes no difference to fundamental physical law. For convenience, measure x as a fraction of the range of measurement. So, we may write x = m/N, where 1/N is less than the resolution of measurement and m is an integer with −N < m < N. Consider the sum
If m = 0, then all the terms in the sum are 1. So,
If m ≠ 0, S(m) is a geometric series with 2N terms, initial value a = e^−πim = (−1)^m, and common ratio r = e^πim/N = (−1)^m. So,
So, S has the property of a delta function when the strip width is 1/N. For large N, let q = m/N and write S as an integral with strip width dq = 1/N.
Substitute x = m/N and p = πqN.
Then, for large N,

As always with infinite sums, care is required. The theory based on finite sums leads to the use of the Discrete Fourier Transform in quantum theory. Discrete transforms have nice properties and allow simple and precise definitions of plane wave states and position states. Standard quantum theory uses continuous transforms, introducing mathematical difficulties concerning the definability of these states. Relational quantum gravity mathematically embeds the discrete theory into a continuous model in which plane wave states and position states are permitted, but we must keep sight of the fact that integrals are really finite sums if we are to avoid mathematical subtleties, as well as the infinite quantities which plague standard quantum electrodynamics.

Delta Function of a Function

Proof: Let u = x ⁄ |α| in the integral,


Proof: Let A_i be a set of disjoint intervals of the real line such that x_i is in A_i and such that g(x) is monotonic and differentiable in each A_i. Let
Let u = g(x) in the integral, observing that the limits of integration are reversed if g is decreasing, i.e. if g' < 0,
Clearly,
as required.

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