Guide To Theoretical Physics

MONDAY, 18TH JANUARY 2010 - Michael Halls-Moore - 1 Comment


The majestic spiral galaxy NGC 4414

Recently a few friends of mine have asked me how to begin learning about some of the cutting edge theories being developed in Theoretical Physics. Having investigated the situation at length for my own learning process I decided to formulate a comprehensive work plan which I have outlined below.

There are a multitude of popular science texts that encompass new theories in Physics and Mathematics and I will list them at the end of the article. I do believe however that it is difficult, if not impossible to grasp the significance of the results without a mathematical background and treatment. As such, the course outline below will contain a significant mathematical component, sufficient to undertake self-study in Theoretical Physics.

The plan requires that one is comfortable with GCSE Mathematics or the international equivalent. It will attempt to replicate the knowledge gained via an extensive undergraduate Masters course in Mathematical Physics at a top University but the article is aimed at the enthusiastic amateur and will not be as comprehensive as a full-time University study course. If one is keen enough, then one should consider application to a full-time course at University.

I will emphasise now that a full understanding will only be gained by a significant investment of time spent learning, as well as working through questions and examples. Mathematics and Physics are not "spectator sports". Self-study is not akin to the "red cross" culture of school Mathematics. Gaining incorrect answers (at least initially!) is something to be encouraged. Physics is fundamentally about experimentation. Try something out and see what happens. It will no doubt be surprising.

Mathematical Basics

Any reasonable course on Physics necessitates a solid understanding of A-Level Mathematics (or international equivalent). In particular the Pure component will provide a basic introduction to algebra, geometry and calculus. I am personally familiar with the Edexcel (London) examinations board. However the OCR board is also highly recommended. It will be necessary to learn all four of the Pure Core Mathematics texts:

Depending upon the amount of time available it is strongly recommended that the Pure component of the Edexcel Further Mathematics A-Level be studied:

A solid grounding in Newtonian Mechanics is also a prerequesite for undergraduate study. The Edexcel Mechanics components are suitable:

Statistics is an important subject in many areas of study, including Physics. However, it is not as much a prerequesite as Pure Mathematics or Mechanics. However, for completeness, the Edexcel texts are presented:

This concludes the basic grounding in Mathematics that will prepare you for an undergraduate course on Mathematical Physics or Theoretical Physics.

Early Undergraduate

At this stage it will be necessary to learn the extremely important tools of Linear Algebra, Differential Equations, Real Analysis and Vector Analysis. These subjects will allow you to tackle Electromagnetism, Classical Mechanics, Special Relativity and Quantum Mechanics in a straightforward manner.

It will also be necessary to gain an understanding into how University Mathematics is carried out, as any graduate text on Theoretical Physics will invariably utilise some difficult Mathematics. Thus, it is wise to read up on Mathematical Foundations:

Later Undergraduate

Once the basic mathematical tools have been mastered the next step will be to progress onto the physical applications. Electromagnetism has a heavy field-based component and so will be difficult without an understanding in Vector Calculus. Classical Mechanics will make extensive use of Differential Equations. Special Relativity will rely on both your vector intuition and your Linear Algebra. Finally, Quantum Mechanics will require Complex Numbers, Linear Algebra and Vector Calculus.

An absolutely indispensible tool for physical problems is that of the Partial Differential Equation. PDE encompass areas as significant as Heat Transfer, Fluid Flow, Quantum Mechanics, Electromagnetics and General Relativity. A good grounding in the subject, as well as their numerical solution, is a prerequesite for most Physics graduate and Masters level courses.

Masters Level

Once the basic undergraduate material has been assimilated it is necessary to gain a deeper understanding of Differential Geometry in order to progress to the more advanced classical physics courses (such as General Relativity) and for the unification theories (such as String Theory).

The first stage is to gain confidence in Tensor Analysis. Tensors are the natural tool for describing abstract geometrical situations and are a definite prerequesite for later courses. Once Tensors, Quantum Mechanics and Special Relativity have been studied a course on Quantum Field Theory can be taken. However, this can be taken alongside a course on General Relativity. Both are needed for a treatment on String Theory.

This is by no means a fully comprehensive treatment of Theoretical Physics. There are many courses I have glossed over or ignored entirely. Below I present some optional courses which will partially "fill in the blanks" as well as provide additional interest.

Physical Options

I am slightly biased in favour of Cosmology and Astrophysics. Once Quantum Mechanics and the Big Bang Theory were developed Cosmology and Astrophysics were brought into the modern world. Cosmology attempts to describe the large scale structural evolution of the Universe in time - a very bold project. Astrophysics encompasses the nuclear processes in stellar objects as well solar and galactic formation. There is a significant degree of overlap between Cosmology, Astrophysics and Relativity. Some interesting texts to consider include:

Although I am not as well versed in the area of Particle Physics as I would like to be (my speciality is Applied Mathematics) I do have friends who have informed me that these texts are useful:

Particle Physics leads one onto the road towards The Standard Model, which is the area of Quantum Field Theory. You may feel more comfortable attempting QFT with a solid background in Particles.

Other areas to look into include Plasma Electrodynamics and Fusion, Statistical Physics, Fluid Dynamics, Optics and Solid State Physics.

Mathematical Excursions

Since I am a mathematician by training I would be remiss in neglecting to mention some additional fascinating and highly applicable areas of Mathematics. The most obvious subject which has been neglected is that of Group Theory which provides a robust framework for describing the concept of symmetry. Groups (and their more advanced-structured friends Rings and Fields) appear in many areas of Physics, in particular Quantum Mechanics, Relativity and String Theory. A basic grounding in Group Theory as well as Lie Groups, Lie Algebras and Manifolds will aid efforts into learning String Theory:

Another area of Mathematics that I have neglected to mention is that of Topology, which can be regarded as an extension of Geometry where the notions of distance are gradually abstracted from metric spaces to topological spaces. It is a subject of vital importance in String Theory, but it is fascinating to study in its own right. Some good introductory texts are:

Other areas to look into include Measure Theory, Complex Analysis, Fractal Geometry, Knot Theory, Numerical Analysis, Dynamical Systems, Stochastic Calculus and Fourier Analysis.

Resources

Here a list of additional resources which you may find useful. Some of them are free or open and I have indicated this via an asterisk where appropriate:

Popular Science Texts

If the above mathematical content looks daunting upon first impression then reading some of the following texts may bring about the necessary motivation. I have listed them in an approximate order of complexity. Asterisks denote the extensive presence of mathematics. Admittedly I have not read all of the following works but have extensively "flicked through" each!

I personally found "Quantum" and "The Universe in a Nutshell" to be fascinating reads. However, I do hope that the popular science texts above motivate you to begin the mathematics as I believe a much deeper enjoyment is gained from a fuller understanding of the material.

Good luck with your scientific endeavours!

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Learning

TUESDAY, 17TH NOVEMBER 2009 - Michael Halls-Moore - 1 Comment

As I type this I have just finished working through the problems in Chapter 1 of Schaum's Outlines of Tensor Calculus. I have always had a healthy fear of Tensor Analysis ever since I was introduced to it via a course on Continuum Mechanics at Warwick. It is however a prerequesite for understanding General Relativity, itself a precursor to gaining insight into String Theory, the current theory-du-jour of how the Universe works at the most fundamental level. One of my life goals is to be able to grasp the concepts in String Theory and it is certainly an ambitious one.

String Theory is a complicated beast. From what I understand at this stage it is an entirely theoretical framework. The predictions it currently makes are few and those which it does admit are even harder to test. It requires a substantial grounding in multiple prerequesites including, but not limited to, Relativity (Special and General), Elementary Particle Physics, Quantum Mechanics, Quantum Electrodynamics, Lie Groups (in particular Matrix Groups) and Quantum Field Theory. Personally, I am not even sure if this hierarchy of subjects is correct or remotely exhaustive.


A Calabi-Yau Manifold

A relatively accessible (and I use that word in the loosest sense) book on the subject is by Zweibach. I will consider my life goal complete when I can understand and solve the problems in that text. However, I do not possess that level of mathematical or physical sophistication as of yet. Hence my journey must begin at General Relativity, as I do possess a reasonable understanding of Quantum Mechanics. I managed to battle through a good introductory text on Special Relativity at Imperial College and decided as a PhD submission "present" I would attempt to begin the sequel book on General Relativity. I rapidly became stuck - primarily based on my limited understanding of tensors.

General Relativity is almost entirely built around the mathematical theory of tensor analysis. It describes the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. Einstein sums it up very well with his famous field equations. In order to describe this bending and warping of space and time, it is necessary to use more sophisticated geometric tools than most are acquainted with. This is where tensors come in.

As previously stated, in the past I found tensor analysis extremely difficult. The first obstacle is that the equations use Einstein Summation to avoid the extensive use of sigma summation signs in order to make the equations logistically manageable. However, one very quickly loses geometric insight unless one is well versed in tensor algebraic manipulation. The second issue is that once arbitrary tensors (of higher orders) are brought in, it is challenging to relate these rules to any visual representation that might have helped out in the lower orders.

I am extremely motivated with this goal, however. In my current circumstances I have a reasonable amount of free time. I have decided to spend some of it re-attempting my goal to grasp String Theory. There is no fear of external consequences or deadlines, nobody to answer to except my own perceived limitations and nobody to compete with except my own discipline. Working through the problems today was an exceedingly enjoyable experience. I managed to grasp basic Einstein Summation and all of the problems I attempted were correct. Mathematics is a set of rules built on other rules. If they are followed consistently, concepts become straightforward. I had forgotten how fascinating it is to learn and fully grasp a concept which one previously considered too complicated to understand. I can think of very little else that can compare with that feeling.

I will never tire or become bored of learning new material. I feel like I am exploring the Universe in my own private sanctuary and on my own terms. There is almost nothing that compares to the clarity and purity of acquiring new knowledge. I do hope that certain elements of society do not relegate continual learning to a minor subset of the population. Humanity should not be an amalgam of reality TV, uninspiring careers and package holidays. Rather, it should be about unbounded curiosity, exploration and overcoming the hardest challenges.

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Passive Income Generation Part 1 - Theory

SUNDAY, 4TH OCTOBER 2009 - Michael Halls-Moore - 0 Comments

The concept of a continual source of income trickling into a bank account for little or no expenditure of time is the dream of many. Yet most people do not realise that it can be done, let alone possess knowledge of how to achieve this dream. Is it possible for one to augment, or even replace, their current income source without exchanging time for money? I believe so. In this article series, I will outline how it can be done.

The first major hurdle to overcome is believing that a business owned by an individual can generate income independent of that individual's time contribution to the business. This concept is generally rather alien to the freelancer or contractor as this type of business requires a continual time investment to find and produce work for clients. This article will discuss the theoretical basis for forming multiple income streams which produce money passively, that is, without continual time investment.

In order to explain the concept of the compounding effect of multiple income streams, I will borrow an example from one of my favourite video game franchises, the Command & Conquer series. The C&C games allow one to practice real-time strategy on a small scale with various factions. One can construct vehicles, structures and infantry in order to defeat the opponent. A critical component of the war effort involves resource management, including the strategic management of cash flow.

Some factions possess particular structures which pay out a fixed amount of cash (a virtual "dividend") over a certain time period. For instance the "Supply Drop Zone" structures have a construction cost of $2500, pay $600* every 2 minutes and take approximately 2 minutes to construct. This $600 is guaranteed to arrive** and does not require any additional investment of time or income to generate. However, in order to simply recoup the initial investment one must wait at least 12 minutes. This is a huge risk, as the cash could arguably be better spent on a new group of tanks or a set of fighter aircraft to defend the base.

Investing the dividend paid by the first supply pad into the construction of a second yields $1200 every two minutes and thus the ability to construct the 3rd pad in approximately 6 minutes. This continual re-investment of the supply pad dividend allows construction of successive pads at a much faster rate. It can be seen that the majority of the risk occurs initially as there is a larger gap between pad construction. This risk rapidly reduces as more pads are created.

Although constructing 4 pads leaves the player with little interim finance, this is more than offset by the $50,000 difference gained compared to the purchase of only a single pad in the 80 minute time period. Please note that this whole scenario is a simplification. Even within the framework of the game, there will be additional sources of income such as supply collecting from the battlefield that will slightly alter the in-game economics. However, the compounding effect is still the same.

So how does this discussion of video game resource management relate to generating a passive income that can potentially alleviate the need for a full-time job? Suppose one were able to construct a set of real-life analogues of a "supply pad" that can mimic their virtual counterparts, then the same compounding effect would occur and the money generated would be independent of the time spent on the pad.

This is not as academic as it sounds given that the vast majority of successful companies essential function on a more elaborate version of this concept. The discussion thus reduces to how an individual can achieve the same effect without a vast outlay of time or money.

In the next article I will identify business types that lend themselves to passive income generation.

* I'm not sure that it is exactly $600, but for argument's sake, it will be taken to be.

** Unless your opponent manages to down your C-130 Hercules transport before it drops financial supplies at the pad!

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Welcome

THURSDAY, 24TH SEPTEMBER 2009 - Michael Halls-Moore - 0 Comments

Welcome to the website of Michael Halls-Moore. To get started please read About Me. The Press section will give you an idea of both the popularity and reach of my previous projects. If you wish to Contact Me, please do so. Note: I respond extremely well to both praise and constructive criticism, but not to spam, trolling or other derogatory communication.

I recently parted ways with PopJam, the investor-backed startup I co-founded and ran as Chief Technical Officer (CTO), in order to pursue other ventures. One of the motivations for developing this site is to analyse and highlight some of the issues we faced within subsequent articles so that myself and others may learn from our actions.

My philosophy is based around a jack-of-all-trades, master-of-some approach in life. I do not subscribe to the notion of specialisation. I seek to continually improve myself everyday in multiple areas with the goal of becoming a polymath. As such, subsequent articles here will span many topics. I want to give you a reason to continue reading so there will be many how-to articles as well as updates about my current projects.

I will also discuss the entire formation of running a business as I plan to start another in the near future. The blog is also here to keep me interacting with other entrepreneurs and advocates of alternative lifestyles so that I may gather thoughts, flesh out ideas and receive encouragement along the way.

Thank you for taking the time to read the blog - I look forward to hearing from you.

Michael Halls-Moore.

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