🦶TOESNAIL-MODE🐌

Theory of Everything - Some Novel Approach Including Love ❤️ – Math on Demand Edition

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About

Doesn’t Theory of Everything (TOE) sound just great? So that’s the general idea. A theory explaining life, the universe and everything. And since I was looking for an unused acronym starting with TOE that is sufficiently easy to pronounce I thought of TOENAIL. Then I thought better of potential internet search results and added the 🐌S. And since all is full of love and all you need is love, that’s obviously what a theory of everything must include. Well, that’s the theory at least, let’s see where we really end up…

https://xkcd.com/55 (CC BY-NC 2.5)

So the idea I’ve been carrying around is a bit different from the typical textbook approaches I know of. In Physics there’s usually the eternal cycle of observation, theoretical explanation, extrapolation (to new potential observations) and experimentation leading to both confirmed or new observation. This Scientific Method is just marvellous and really should be applied more in our everyday lives instead of hearsay, fake news and alternative facts. And that’s what I love about Theoretical Physics and Pure Mathematics – you cannot argue with $1+1=2$. It’s pure theory, and no experiment can be rigged in order to favour a desired outcome, nor can measurement errors yield wrong conclusions. That is not to say it’s perfect let alone trivial, and what’s the point of a great theory if there’s no reality to apply it to? So ultimately the goal of this project is to describe the actual world, and yes, even including love despite it’s incredible capability to escape most logical explanations 😅

Math on Demand Edition?

Not everybody likes mathematics^{[citation needed]}. Crazy, I know 🤪 No, let me be honest: I like mathematics iff it is useful. So maybe Pure Mathematics is not actually my preference. As an example back when studying I saw no purpose in learning about Eigenvalues in Linear Algebra lectures until much later when in Physics they were applied to e.g. Quantum Mechanics. So in this project I try a different approach called “math on demand”. There will be mathematics(https://en.wikipedia.org/wiki/Group_theory) as an example), but only when it’s actually needed and motivated. Anything we don’t need here I won’t bother to explain – think YAGNI, or even better, Occam’s razor, which boils down to “don’t overthink it (unless you have to)”, and good old KISS.

The Shape of Things to Come – What to expect, and when?

Let’s see where this leads to… Consider yourself lucky I woke up way too early tonight so I felt like finally starting to write this, but be aware that I’m an expert in getting things not-so-done aka procrastination, so don’t get your hopes up too much. Then again, feel free to motivate me gently.

Getting started

Name it to tame it

First of all, always writing something like “life, the universe and everything” is quite tedious. It is very convenient to use single-letter¹ variables as abbreviations instead, such as $t$ for a time. For various reasons, not only Latin letters are used, but sometimes also Greek ones. In order determine the state of a system, the letter Psi $\Psi$ (or it’s lowercase version $\psi$) is a very typical choice. And in order to indicate that it’s a state, it is surrounded by a bar $|$ and an angle bracket $\rangle$ as $\ket\Psi$. Since this is using the right half of an angle bracket pair, this is called a ket vector. And yes, we’ll later also encounter the left bracket counterpart $\bra\Psi$, called a bra vector². Such a ket-vector can describe things as complicated as “life, the universe and everything” and as simple as a tossed coin. The entirety of all possible ket vectors is called a vector space. Now if multiple states are considered, some options occur:

• Just use different letters, such as Phi $\Phi$ (lowercase $\phi$ or sometime $\varphi$ though that usually rather denotes an angle),
• Add indices as subscripts, e.g. $\ket{\Psi_{\text{in}}}$,
• Explicit labels, such as $\ket{42}$ or $\ket{\text{coin=heads}}$.

Observation: (Insert witty headline here)

So describing the entirety of existence as $\ket{42}$ is all fun and games until you try to actually work with it. To get more serious, reality and theory must meet somewhere.

Subsystems: Divide and Conquer!

If $\ket{42}$ is known, you’re omniscient and don’t need to read on (since you already know everything including this very text. #TODO: try and proof whether omniscience is impossible?). Most mortals however don’t, and it is infeasible to try and know everything about the entire universe at all times. Let us therefore split $\ket{42}$ into smaller parts that ideally don’t interact too strongly with one another, e.g.

\[\ket{42} = \ket{\text{coin}, \text{everything else}}.\]

There are multiple kinds of notation for this, another form omits the comma, and sometimes one encounters the form

\[\ket{42} = \ket{\text{coin}} \otimes \ket{\text{everything else}}\]

where the $\otimes$ denotes a so-called tensor product. Don’t worry about the details for now, it’s just mathematical notation for the composition. The $\otimes$ makes it more clear that the state is considered to combine to partial states.

As mentioned before, the split should be made in such a way that $\ket{\text{everything else}}$ does not interfere much, at least for a while. Now, when we say “a while” so nonchalantly, we are very much ignoring that we have not even defined the passage of time so far, which is something we’ll fix later.

Projection: What you perceive is what you believe

In the coin example, the outcome can be either³ $\ket{\text{heads}}$ or $\ket{\text{tails}}$. If the probability of $\ket{\text{heads}}$ is $p$ (ideally 50% for a fair coin tossed fairly), then the probability of $\ket{\text{tails}}$ is $1-p$. Before the result of the toss is checked, or “measured”, we hence have

\[\ket{\text{tossed coin}} \propto p\cdot\ket{\text{heads}} + (1-p)\cdot\ket{\text{tails}} \ltag{t1}.\]

Here the symbol $\propto$ means “proportional to”, i.e. there can be a linear factor scaling between the sides, e.g. $a = c\cdot b$ implies $a\propto b$ but looses information about $c$. We’re also adding a tag $\eqref{t1}$ so we can more easily refer to this formula later.

Note how we multiply multiple ket vectors with real numbers and add those together - just like you’d say “three apples and two bananas”. It’s not witchcraft, but if you haven’t seen this yet make sure you accept this crucial concept. This kind of combination of states is calles linear combination.

Bra-vectors

Before we talk about actual measurement, let’s finally properly introduce a very convenient tool: The bra vector $\bra{\Psi}$, which is the dual vector to the ket vector $\ket{\Psi}$. The term “duality” indicates that for each ket vector there exists precisely one unique “partnering” bra vector and vice versa - a very important concept. Since the entirety of ket vectors is called vector space, its dual counterpart (the entirety of bra vectors) is called a dual (vector) space. What is it good for? Just like real numbers, we can also combine bra vectors with ket vectors, e.g. $\bra{\text{heads}}\cdot\ket{\text{tails}}$, or more concise as a so-called bra-ket $\braket{\text{heads}\vert\text{tails}}$. The bra vectors are constructed by definition such that a bra-ket term is a so-called inner product, which yields a scalar number.

Inner product - and complex numbers

The inner product $\braket{\Psi\vert\Phi}$ of two vectors $\ket\Psi$ and $\ket\Phi$ (the greek letter Phi) can be any number, not just also a negative one but even a so-called complex number. For some first details please refer to the footnote.[^complex-numbers]

[^complex-numbers]{

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Complex numbers

For now, let’s just say complex numbers are a pair of two real numbers, one denoting a positive length and the other an angle for a direction. The “usual” real numbers are either positive, which means an angle of 0°, or negative, which means an angle of 180° (but still a positive length!).

The angles of 90° and 270° yield imaginary numbers, which have the peculiar property of squaring to negative numbers - something deemed impossible for real numbers, hence the name. Just like $1$ denotes the positive unit of real numbers, i.e. a number of length one and angle 0°, and $-1$ the negative unit, we use the letter $i$ to denote the imaginary unit of length one and angle 90° (and similarly $-i$ has the angle 270°, which is actually just a combination $(-1)\cdot i$). Complex numbers with arbitrary angles can be considered a linear combination of a real and an imaginary number, such as $3+4i$, which has the length $\sqrt{3^2+4^2}=5$ and angle slightly below 37°. Complex numbers have many important properties and benefits compared to “mere” real numbers, which we’ll learn more about later. For now it should suffice to say that there is also one significant drawback to complex numbers: They can’t be ordered! There is no unique way to determine which of two complex numbers is “greater” or “smaller” than the other, e.g. $i \not>1$ but $i \not<1$ as well.

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The inner product fulfills the following properties:

\[\begin{align*} \braket{\Phi\vert\Psi} &= \overline{\braket{\Psi\vert\Phi}} \ltag{conjugate symmetry} \\ \braket{\Xi\vert a\Psi+b\Phi} &= a\braket{\Xi\vert\Psi} + b\braket{\Xi\vert\Phi} \ltag{linearity} \\ \forall\ket\Psi\neq\ket0: \|\Psi\|^2 := \braket{\Psi\vert\Psi} &> 0 \ltag{positive-definiteness} \end{align*}\]

Aside from having introduced another greek letter Xi $\Xi$ (lowercase $\xi$) in $\eqref{positive-definiteness}$ we’ve used the forall-notation $\forall$ and the definition notation “$:=$” - spelled out this reads:

For each $\ket\Psi$ which is not the zero-Vector $\ket0$, the following holds: $|\Psi|^2$, which we define as $\braket{\Psi\vert\Psi}$, is always positive ($> 0$).

Clearly $\forall$ is shorter though it takes some getting used to, especially when we later combine it with the “there exists” symbol $\exists$ 😉. Completely in passing we also introduced the zero-Vector $\ket0$, which however should be straight-forward to understand and gives us a direct opportunity to use $\forall$ again:

\[\begin{align*} \forall \ket\Psi &: \ket\Psi + \ket0 = \ket\Psi, \\\forall \ket\Psi &: 0\cdot\ket\Psi = \ket0 \end{align*}\]

In other words, the zero-vector $\ket0$ basically behaves like the normal number zero 0 by not changing anything it is added to.

For two unit vectors, the length of the inner product is always $\le1$ and only exactly equal to one of the vectors are parallel or anti-parallel.

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Norms

More specifically, the bra-ket of a vector with itself, $\braket{\Psi\vert\Psi} =: \Vert\Psi\Vert^2 \ge 0$ is a positive number or zero and its (positive) square-root $\sqrt{\braket{\Psi\vert\Psi}}=\Vert\Psi\Vert=:\Psi$ called its norm. We have also introduced some mathematical notations here:

• $=:$ means the term right to the colon is defined to be equal the term left of the equals sign. The other direction $:=$ is also possible. This is a helpful notation to introduce abbreviations or new definitions
• $\ge$ means “greater $>$ or equal $=$ to”. If a number is $\ge0$ that means it either zero or positive, which is also called non-negative. Similarly there’s $\le$ denoting “less $<$ or equal to”
• It is very common in Physics to use the same variable name for different things that should be clear from context and notation. Here both the vector $\ket\Psi$ and its norm $\Psi$ use the greek letter $\Psi$. We’ll later also encounter e.g. operators $\hat\Psi$ and eigenvalues $\Psi_k$, but don’t worry about those for now and just keep in mind that the notation around the letter is relevant as well.

It is convention to use $\ket 0$ to denote the⁴ “smallest” vector with vanishing norm $\Vert0\Vert=0$. Vectors $\ket e$ with a norm $\Vert e\Vert=1$ of one are called unit vectors. All vectors $\ket\Psi$ with a finite norm $\Psi=\Vert\Psi\Vert<\infty$ (where $\infty$ denotes infinity) can be normalized to a unit vector via division $\ket{e_\Psi} := \frac1{\Psi}\ket\Psi$. Since that unit vector only has to be multiplied by the original norm in order to obtain the original vector, it can be considered “pointing into $\ket\Psi$’s direction”. Note that there can exist vectors of infinite norm which are also called non-normalizable.

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The inner product is a measure of the similarity of two vectors. If it is zero the two vectors have nothing in common and are called orthogonal or perpendicular.

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Let’s continue with a roughly commented outline, consider this a sneak preview.

Transition: Time keeps on slippin’

Basically $\ket{42}$ encodes everything forever. But being beings that perceive time in a linear fashion we’re typically interested in transitions between states, such as the question how a state evolves over time. Interactions between systems are of interest as well. Let’s take the coin example again:

$\ket{\text{coin}} \to \ket{\text{tossed coin}}$

The arrow $\to$ just denotes this transition. #TODO: get to operators…

Observe

Measurement via operators, but not the Eigenvalue-Approach, rather some product space or such. Eigenvalues of course, too, but not as “measurement values” but rather to justify bases and such.

Symmetry / Noether / Gauge

Gently get from dynamics via operators to group theory, conservation thanks to Noether’s Theroem, and the connection between particle fields and gauge theory. Probably something about symmetry breaking à la Higgs, but I’ll have to read up on that again first…