The Road to Reality Study Notes: Difference between revisions

To explore the process of pursuing mathematical truth, Penrose outlines a few proofs of the [https://en.wikipedia.org/wiki/Pythagorean_theorem Pythagorean theorem]. The theorem can be stated as such, "For any right-angled triangle, the squared length of the hypotenuse $$c$$ is the sum of the squared lengths of the other two sides $$a$$ and $$b$$ or in mathematical notation $$ a^2 + b^2 = c^2. $$

Euclid's fifth postulate, also known as the [https://en.wikipedia.org/wiki/Parallel_postulate parallel postulate], was more troublesome. In Penrose words "it asserts that if two straight line segments $$a$$ and $$b$$ in a plane both intersect another straight line $$c$$ (so that $$c$$ is what is called a ''transversal'' of $$a$$ and $$b$$) such that the sum of the interior angles on the same side of $$c$$ is less than two right angles, then $$a$$ and $$b$$, when extended far enough on that side of $$c$$, will intersect somewhere". One can see that the formulation of the fifth postulate is more complicated than the rest which lead to speculation of it's validity. With the fifth postulate one can go on to properly build a square and begin to explore the world of [https://en.wikipedia.org/wiki/Euclidean_geometry Euclidean geometry].

Again Penrose asks us to revisit our assumptions and examine which of Euclid's postulates were needed. Particularly our claim that the sum of the angles in a triangle add up to the same value of 180° (or $$\pi$$ [https://en.wikipedia.org/wiki/Radian radians]). One must use the parallel postulate to show that this is true. Penrose asks us to consider what would it mean for the parallel postulate to be false? What would that imply? Would that make any sense? With these questions in mind we begin to explore a different kind of geometry.

‘Counting’ numbers from 1 to $$\infty$$.

:<math>Ax^{2} + Bx + C = 0</math>

:<math>-\frac{B}{2A}\sqrt{\left(\frac{B}{2A}\right)^2}+\frac{C}{A}, \quad -\frac{B}{2A}\sqrt{\left(\frac{B}{2A}\right)^2}-\frac{C}{A}</math>

$$1-x^2+x^4+\cdots$$

converges to the function $$1/(1+x²)$$ only when $$|x|<1$$, despite the fact that the function doesn't seem to have "singular" behavior anywhere on the real line. This can be explained by switching to the complex number system using $$z=x+iy$$ whereby $$1/(1+z²)$$ can be examined to have singularities at $$x=i,-i$$.   

With this, Penrose introduces us to the idea of the [https://mathworld.wolfram.com/RadiusofConvergence.html circle of convergence] as a circle in the complex plane centered at 0 with poles/singularities of $$f(z)$$ defining the circle radius.  The series is convergent for any point z inside of this circle.

Finally, the [https://en.wikipedia.org/wiki/Mandelbrot_set Mandelbrot set] is defined as the set of all points $$c$$ in the complex plane so that repeated applications of the transformation mapping $$z$$ to $$z^2+c$$, starting with $$z=0$$, do not escape to infinity.

Rather than just ‘adding’ and ‘multiplying’ these can be viewed as ‘translation’ and ‘rotation’ within the complex plane. For example, multiply a real number by the complex number $$i$$ rotates the point in the complex plane π/2 and viewing the parallelogram and similar-triangle laws as translation and rotation:

Penrose further introduces the concept of polar coordinates where $$r$$ is the distance from the origin and $$θ$$ is the angle made from the real axis in an anticlockwise direction.

This idea is fundamental in the use of logarithms.  We first start with the expression $$b^{m+n} = b^m \times b^n$$, which represents the idea of [https://en.wikipedia.org/wiki/Exponentiation exponentiation]. This is easy to grasp for $$m$$ and $$n$$ being positive integers, as each side just represents $$m+n$$ instances of the number $$b$$, all multiplied together.  If $$b$$ is positive, this law is then showed to hold for exponents that are positive integers, values of 0, negative, and fractions.  If $$b$$ is negative, we require further expansion into the complex plane.  

We would need a definition of $$b^p$$ for all complex numbers $$p,q,b$$ such that $$b^{p+q} = b^p \times b^q$$.  If we then define the [https://en.wikipedia.org/wiki/Logarithm logarithm to the base b] as the inverse of the function $$f(z) = b^z$$ such that <math>z=log_bw</math> for $$w=b^z$$ then we should expect <math>z=log_b(p \times q) = log_bp + log_bq</math>.  This would then convert multiplication into addition and allow for exponentiation in the complex plane.

We need to be careful with the above assertion of the logarithm, mainly since $$b^z$$ and <math>log_bw</math> are ‘many valued’.  Solving the equations would require a particular choice for $$b$$ to isolate the solution.  With this, the ‘base of natural logarithms’ is introduced as the [https://en.wikipedia.org/wiki/E_(mathematical_constant) number e], whose definition is the power series <math>1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+…</math>. This power series converges for all values of z which then makes for an interesting choice to solve the ambiguity problem above.  Thus we can rephrase the problem above with the natural logarithm, <math>z=logw</math> if $$w=e^z$$.

However, even with this natural logarithm we run into multi-valuedness ambiguity from above.  Namely that $$z$$ still has many values that lead to the same solution with $$z+2πin$$, where $$n$$ is any integer we care to choose.  This represents a full rotation of $$2π$$ in the complex plane with all multiples of $$n$$ achieving the same point, $$z$$.

Penrose goes further in representing $$z$$ with polar coordinates showing <math>z=logr+iθ</math>, then <math>e^z=re^{iθ}</math>.  This formulation shows us that when we multiply two complex numbers, we take the product of their moduli and the sum of their arguments (using the addition to multiplication formula introduced in 5.2).

Rounding out the chapter, Penrose gives us another further representation of assuming $$r=1$$, such that we recover the ‘unit circle’ in the complex plane with $$w=e^{iθ}$$.  We can 'encapsulate the essentials of trigonometry in the much simpler properties of complex exponential functions' on this circle by showing <math>e^{iθ}=cos(θ) + isin(θ)</math>.   

* $$e^{i\theta}$$ is helpful notation for understanding rotating  

* $$e^{i\theta} = cos \theta + i sin \theta$$

Returning to the ambiguity problem of multi-valuedness, it seems the best way to avoid issues is when a particular choice of $$logw$$ has been specified.  As an example, $$w^z$$ with $$z=\frac{1}{2}$$.  We can specify a rotation for $$logw$$ to achieve $$+w^\frac{1}{2}$$, then another rotation of $$logw$$ to achieve $$-w^\frac{1}{2}$$.  The sign change is achieved because of the Euler formula <math>e^{πi}=-1</math>. Note the process:

<math>w^z=e^{zlogw}=e^{zre^{iθ}}=e^{ze^{iθ}}</math>, then specifying rotations for theta allows us to achieve either $$+w^\frac{1}{2}$$ or $$-w^\frac{1}{2}$$.

Penrose notes an interesting curiosity for the quantity $$i^i$$.  We can specify <math>logi=\frac{1}{2}πi</math> because of the general relationship <math>logw=logr+iθ</math>.  If $$w=i$$, then its easy to see <math>logi=\frac{1}{2}πi</math> from noting that y is on the vertical axis in the complex plane (rotation of $$\frac{π}{2}$$).  This specification, and all rotations, amazingly achieve real number values for $$i^i$$.

We end the section with the idea of finite multiplicative groups, or [https://en.wikipedia.org/wiki/Cyclic_group cyclic groups] Z_n, which contain $$n$$ quantities ([https://en.wikipedia.org/wiki/Root_of_unity#:~:text=The%20nth%20roots%20of%20unity%20are%2C%20by%20definition%2C%20the,and%20often%20denoted%20%CE%A6n. nth roots of unity if around the unit circle]) with the property that any two can be multiplied together to get another member of the group.  As an example, Penrose gives us <math>w^z=e^{ze^{i(θ+2πin)}}</math> with $$n=3$$, leading to three elements $$1, ω, ω^2$$ with <math>ω=e^\frac{2πi}{3}</math>. Note $$ω^3=1$$ and $$ω^-1=ω^2$$.  These form a cyclic group Z₃ and in the complex plane, represent vertices of an equilateral triangle.  Multiplication by ω rotates the triangle through $$\frac{2}{3}π$$ anticlockwise and multiplication by $$ω^2$$ turns it through $$\frac{2}{3}π$$ clockwise. The cyclic group is graphically shown below:

The notion of [https://en.wikipedia.org/wiki/Parity_(physics) parity] is introduced as approximately a multiplicative quantum number with n=2, and an example is the family of particles called [https://en.wikipedia.org/wiki/Boson bosons].  Penrose notes that [https://en.wikipedia.org/wiki/Fermion fermions] could also be considered a parity group but it is not the normal convention.  The distinction between these two particles are that bosons are completely restored to their original states under a $$2π$$ rotation, whereas fermions require $$4π$$ (two rotations).  Thus a multiplicative quantum number of $$-1$$ can be assigned to a fermion and $$+1$$ to a boson.

An example of a multiplicative quantum number with $$n=3$$ relates to quarks, which have values for electric charge that are not integer multiples of the electron’s charge, but in fact $$\frac{1}{3}$$ multiples.  If $$q$$ is the value of electric charge with respect to an electron ($$q=-1$$ for electron charge), then quarks have q=$$\frac{2}{3}$$ or $$-\frac{1}{3}$$ and antiquarks q=$$\frac{1}{3}$$ or $$-\frac{2}{3}$$.  If we take the multiplicative quantum number <math>e^{-2qπi}</math>, then we find the values $$1,ω,ω^2$$ from section 5p4, which constitute the cyclic group Z₃.

Differentiation is concerned with and calculates the rates that things change, or ‘slopes’ of these things.  For the curves given in section 6p1 above, two of the three do not have unique slopes at the origin and are said to be not ''differentiable'' at the origin, or not ''smooth'' there.  Further, the curve of theta(x) has a jump at the origin which is to say that it is discontinuous there, whereas $$|x|$$ and $$x^2$$ were continuous everywhere.

Taking differentiation a step further, Penrose shows us two plots which look very similar, but are represented by different functions, $$x^3$$ and $$x|x|$$.  Each are differentiable and continuous, but the difference has to do with the curvature ([https://en.wikipedia.org/wiki/Second_derivative second derivative]) at the origin.  $$x|x|$$ does not have a well-defined curvature here and is said to not be ''twice differentiable''.

=== 6.3 Higher derivatives; $$C^\infty$$-smooth functions ===

Looking closer at the concept of two derivatives of the same function (second derivative, or curvature), Penrose shows us the functions from 6p2 and their first and second derivatives.  Note that the first derivative of $$f(x)$$, written $$f’(x)$$, meets the x-axis at places of local minima or maxima and the second derivative of $$f(x)$$, written $$f’’(x)$$, meets the x-axis where the curvature goes to $$0$$ and is said to be a point of inflection.

In general, a function can be smooth for many derivatives and the mathematical terminology for general smoothness is to say that $$f(x)$$ is $$C^n$$-smooth.  It can be seen that $$x|x|$$ is $$C^1$$-smooth but not $$C^2$$-smooth due to the discontinuity at the origin in the derivative. In general $$x^n|x|$$ is $$C^n$$-smooth but not $$C^{n+1}$$-smooth.  In fact, a function can be $$C^\infty$$-smooth if it is smooth for every positive integer.  Note that negative integers for $$x^n$$ immediately are not smooth for $$x^{-1}$$ (discontinuous at the origin).   

Penrose notes that Euler would have required $$C^\infty$$-smooth functions to be defined as functions, and then gives the function:

:<math>h(x) = \begin{cases}

as an example of a $$C^\infty$$-smooth function but one that Euler would still not be happy with since it is two functions stuck together.

How, then do we define the notion of a ‘Eulerian’ function?  This can be accomplished in two ways.  The first using complex numbers and is incredibly simple.  If we extend $$f(x)$$ to $$f(z)$$ in the complex plane, then all we require is for $$f(z)$$ to be once differentiable (a kind of $$C^1$$-smooth function).  That’s it, magically. We will see that this can be stated with $$f(x)$$ being an [https://en.wikipedia.org/wiki/Analytic_function analytic function].

For the second method, the power series of $$f(x)$$ is introduced, <math>f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + …</math> For this series to exist then it must be $$C^\infty$$-smooth.  We must take and evaluate derivatives $$f(x)$$ to find the coefficients, thus an infinite number of derivatives (positive integers) must exist for the power series to exist.  If we evaluate $$f(x)$$ at the origin, we call this a power series expansion about the origin.  About any other point $$p$$ would be considered a power series expansion about $$p$$. (Maclaurin Series about origin, see also [https://en.wikipedia.org/wiki/Taylor_series Taylor Series] for the general case)

The power series is considered analytic if it encompasses the power series about point $$p$$. If it is analytic at all points of its domain, we call it an analytic function or, equivalently, a $$C^ω$$-smooth function.  Euler would be pleased with this notion of an analytic function, which is ‘smoothier’ than the set of $$C^\infty$$-smooth functions ($$h(x)$$ from 6p3 is $$C^\infty$$-smooth but not $$C^ω$$-smooth).  

Penrose ends the chapter noting that there are approaches which enable the process of differentiation to be continued indefinitely, even if the function is not differentiable.  One example is the [https://en.wikipedia.org/wiki/Dirac_delta_function Dirac Delta Function] which is of ‘considerable importance in quantum mechanics’.  This extends our notion of $$C^n$$-functions into the negative integer space ($$C^{-1},C^{-2},...$$) and will be discussed later with complex numbers.  Penrose notes that this leads us further away from the ‘Eulerian’ functions, but complex numbers provide us with an irony that expresses one of their finest magical feats of all.

As will be stated in 7.3, instead of directly providing the definition of holomorphic functions, Penrose chooses to demonstrate the argument with the ingredients in order to show a "wonderful example of the way that mathematicians can often obtain their results. Neither the premise ($$f(z)$$ is complex-smooth) nor the conclusion ($$f(z)$$ is analytic) contains a hint of the notion of contour integration or the multivaluedness of a complex logarithm.  Yet these ingredients provide the essential clues to the true route to finding the answer".

In the real number sense, integrals are taken from a single point $$a$$ to another point $$b$$ along the real number line.  Usually the horizontal axis, and there is only one way to travel along this line (moving positive and negative along the axis).  However, in the complex plane points involve two dimensions, and therefore have many such routes that allow us to get from a complex point $$a$$ to $$b$$.

The [https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations Cauchy-Riemann equations] (to be formally introduced later in chapter 10) allow us to narrow our focus to find a path-specific answer, where the value of the integral on this path is the same for any other path that can be formed from the first by continuous deformation in its domain.  Note that the function <math>\frac{1}{z}</math> has a hole in the domain at the origin, which can prevent a continuous deformation thereby allowing for different answers for the value of the integral depending on the path taken.

Continuous deformation in this sense is defined as [https://en.wikipedia.org/wiki/Homology_(mathematics)#:~:text=When%20two%20cycles%20can%20be,in%20the%20same%20homology%20class. homologous deformation], where parts of the path can cancel each other out if they are traversed in opposite directions. This contrasts with homotopic paths where you may not cancel parts. As a visualization we take the function of <math>\frac{1}{z}</math> with a homologous path:

The amazing result here is that a general contour from $$a$$ to $$b$$ for the function <math>\frac{1}{z}</math> has be rephrased and shown to be equal to the result for a ''closed contour'' that loops around the point of non-analyticity (the origin in this case), regardless of where the points $$a$$ and $$b$$ (or the point of non-analyticity) lie in the complex plane.  Note that since $$logz$$ is multi-valued, we need to specify the actual closed contour being used (if we looped twice rather than once, then the answer is different).

The example in section 7p2 is a particular case for the well-known [https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula Cauchy Integral Formula], which allows us to know what the function is doing at the origin (or another general point $$p$$) by what it is doing at a set of points surrounding the origin or the general point $$p$$.  

:<math>\frac{1}{2πi}\oint\frac{f(z)}{z-p}dz=f(p)</math>

A 'higher-order' version of this formula allows us to inspect $$n$$ number of derivatives with the same relationship.

:<math>\frac{n!}{2πi}\oint\frac{f(z)}{(z-p)^{n+1}}dz=f^{(n)}(p)</math>

If we use this to provide the definition of a derivative at a point, we can then construct a Maclaurin formula (if using the origin, otherwise the more general [https://en.wikipedia.org/wiki/Taylor_series Taylor series]) for $$f(z)$$ using the derivatives in the coefficients of the terms.

:<math> \sum_{n=0} ^ {\infty} \frac {f^{(n)}(p)}{n!} (z-p)^{n} </math>

This can be shown to sum to $$f(z)$$, thereby showing the function has an actual $$n$$th derivative at the origin or general point $$p$$.  This concludes the argument showing that complex smoothness in a region surrounding the origin or point implies that the function is also holomorphic. Penrose notes that neither the premise ($$f(z)$$ is complex-smooth) nor the conclusion ($$f(z)$$ is analytic) contains contour integration or multivaluedness of a complex logarithm, yet these ingredients are essential for finding the route to the answer and that this is a ‘wonderful example of the way that mathematicians can often obtain their results’.

For example, if there is no singularity in the function, the region can be thought of as a circle of infinite radius. Taking <math>f(z)=\frac{1}{z}</math> however forces an infinite number of circles centered at any point with boundary radii passing through the origin (noting that an open region does not contain the boundary) to construct the domain.

Now we consider the question, given a function $$f(z)$$ holomorphic in domain $$D$$, can we extend the domain to a larger $$D’$$ so that $$f(z)$$ also extends holomorphically?  A procedure is formed in which we use a succession of power series about a sequence of points, forming a path where the circles of convergence overlap.  This then results in a function that is uniquely determined by the values in the initial region as well as the path along which it was continued.  Penrose notes this [https://en.wikipedia.org/wiki/Analytic_continuation analytic continuation] as a remarkable ‘rigidity’ about holomorphic functions.

An example of this rigidity and path dependence is ‘our old friend’ $$logz$$.  There is no power series expansion about the origin due to the singularity there but depending on the path chosen of points around the origin (clockwise or anti-clockwise) the function extends or subtracts in value by $$2πi$$.  See chapter 5 and the euler formula (<math>e^{πi}=-1</math>) for a refresher.