Guide To Theoretical Physics

MONDAY, 18TH JANUARY 2010 - Michael Halls-Moore - 2 Comments

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:

• Core Mathematics 1
• Core Mathematics 2
• Core Mathematics 3
• Core Mathematics 4

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

• Further Pure Mathematics 1
• Further Pure Mathematics 2
• Further Pure Mathematics 3

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

• Mechanics 1
• Mechanics 2
• Mechanics 3

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:

• Statistics 1
• Statistics 2

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:

• The Foundations of Mathematics
• Numbers and Functions: Steps to Analysis
• Introduction to Linear Algebra
• Schaum's Outline of Linear Algebra Fourth Edition
• An Introduction to Ordinary Differential Equations
• Vector Calculus (Springer Undergraduate Mathematics)
• Schaum's Outline of Vector Analysis, 2ed

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.

• Introduction to Electrodynamics
• Electromagnetics: Second Edition (Schaum's Outline S.)
• Classical Mechanics (5th Edition)
• Schaum's Outline of Lagrangian Dynamics:
• Special Relativity (Springer Undergraduate Mathematics)
• Special Relativity (MIT Introductory Physics)
• Quantum Mechanics
• Schaum's Outline of Quantum Mechanics

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.

• Partial Differential Equations: An Introduction
• Partial Differential Equations for Scientists and Engineers
• Schaum's Outline of Partial Differential Equations
• Numerical Solution of Partial Differential Equations: An Introduction
• Numerical Methods for Partial Differential Equations (Springer Undergraduate Mathematics Series)

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.

• Vector and Tensor Analysis with Applications
• Schaum's Outline of Tensor Calculus
• Quantum Field Theory
• Quantum Field Theory in a Nutshell
• General Relativity: An Introduction for Physicists
• General Relativity (Springer Undergraduate Mathematics)
• A First Course in String Theory
• String Theory and M-Theory: A Modern Introduction

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:

• Introduction to Cosmology
• Introduction to Astronomy and Cosmology
• An Introduction to Galaxies and Cosmology
• An Introduction to Modern Astrophysics

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 (Manchester Physics Series)
• The Ideas of Particle Physics: An Introduction for Scientists
• Introduction to Elementary Particles

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:

• Classic Algebra
• Groups - Modular Mathematics Series
• Schaum's Outline of Group Theory
• Matrix Groups: An Introduction to Lie Group Theory (Springer Undergraduate Mathematics)
• An Introduction to Manifolds

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:

• Introduction to Metric and Topological Spaces
• Basic Topology (Undergraduate Texts in Mathematics)
• Essential Topology (Springer Undergraduate Mathematics Series)
• Schaum's Outline of General Topology

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:

• Motion Mountain - The Free Physics Textbook *
• Open Culture - 250 Free Courses from Top Universities *
• University of Warwick Mathematics Department - Plan Your Degree Course Guides *
• University of Warwick Physics Department - Module Guide *
• Leonard Susskind's Complete Introduction to Modern Physics *
• Springer Undergraduate Mathematics Series

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!

• Science: The Definitive Visual Guide
• A Brief History of Time
• The Universe in a Nutshell
• Quantum: Einstein, Bohr and the Great Debate About the Nature of Reality
• The Trouble with Physics: The Rise of String Theory, the Fall of a Science and What Comes Next
• The Road to Reality: A Complete Guide to the Laws of the Universe *

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!

2 Comments

• Michael Halls-Moore

  20th January 2010

  1:37 pm

  In fact, this is the real mother lode:
  http://www.openculture.com/physics_free_courses

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