Physics from Symmetry

The idea of this roadmap is, in some sense, to turn the traditional roadmap upside down. Instead of starting with only approximately correct theories (Classical mechanics, Classical electrodynamics) and then slowly moving toward the more correct ones (Quantum Mechanics, Quantum Field Theory), here we start with the best theories of nature that we have.

The modern theories can be derived straight-forwardly from the principles of special relativity and using symmetry arguments. The older classical theories can then be understood as approximations of the modern theories.

One advantage of this approach is that less time is spent on only approximately correct theories, and more time on the best theories of nature that we have. In addition, this approach shows the connection between the various theories clearer than the traditional approach.

Here is the general outline:

The journey begins with the fundamental postulates of special relativity. These can be used to derive the Minkwoski metric, which is crucial for everything that follows.

Afterwards, the basic notions of group theory are introduced. Using basic examples, like $SU(2)$ and $SO(3)$, basic concepts like Lie algebras, representation theory can be straight-forwardly understood. A solid understand of both groups $SU(2)$ and $SO(3)$ and their common Lie algebra is essential for the rest of the roadmap.

After the basic groups are understood, the Lorentz group can be tackled. This is the group that encodes the symmetry of special relativity. It consists of all transformations that leave the Minkowksi metric invariant. The most important goal is to understand the representations of the Lorentz group. The lowest-dimensional representations of the Lorentz group, called scalar, spinor and vector representation, are the tools that we need to describe elementary particles. Scalars describe spin $0$ particles, spinors spin $1/2$ particles and vectors spin $1$ particles.

Then, when these are sufficiently understood, it's time to get familiar with the Lagrangian framework. The basic idea is that we get the fundamental equations of nature, by minimizing something. This idea is motivated by Fermat's principle. We can actually derive what this something is by using symmetry arguments. Only fundamental equations that hold in all inertial frames of reference, are good fundamental equations. Hence, the fundamental something that we use to derive these equations, must be the same in all inertial frames of reference. In mathematical terms this means it must be invariant under all Lorentz transformations, i.e. it must be a scalar. The fundamental something is called the action. To actually derive the fundamental equations from a given action, we need to understand the basic idea of variational calculus. A crucial thing to understand is Noether's Theorem, which tells us that there is a conserved quantity for every symmetry of the system in question.

Then, we can start to actually understand the various fundamental equations of nature: Klein-Gordon equation, Dirac equation, Maxwell equations, Proca equations. These describe non-interaction particles. Moreover, we can derive the canonical commutation relations by using the basic message of Noether's Theorem. All we have to do is identity the conserved quantities, like momentum or angular momentum, with the generators that generate the corresponding symmetries. For example, the symmetry that is responsible for the conservation of momentum, is invariance under translations. The generator of this symmetry is $i \partial_x$ and hence we make the identification $p \to i \partial_x$. The leads us automatically to the canonical commutation relations. The same is true for the canonical commutation relations of quantum field theory. There we only need to consider invariance under shifts of the field itself. This symmetry leads us to a conserved quantity called "conjugate momentum". When we identify the conjugate momentum with the generator of "field shifts", we get the correct canonical commutation relations of quantum field theory.

Then, to derive the correct theories that describe interacting particles, we can make use of gauge symmetry.

Now we have everything we need to actually start describing nature. A good way to get familiar with the modern way of describing nature, i.e. quantum theory, is to start with quantum mechanics. Starting with the Schrödinger equation several simple examples should be understood: a free particle, a particle in a box, the double slit experiment. In addition, it's crucial to get familiar with the Dirac notation. All examples mentioned here should be understood in terms of wave functions and in terms of the Dirac notation.

Now with a basic understanding of the tools used in quantum theory, quantum field theory can be tackled. Starting with a Fourier expansion of the fields with different spin and the canonical commutation relations, we can derive the commutation relations for the Fourier coefficients. We can then understand that their commutation relations imply that they generate and destroy particles. That's how we describe particles in a field theory. Afterwards, it makes sense to get familiar with some scattering theory, because that's what quantum field theory is mostly about.

Lastly, we can start to understand the classical theories. These arise as approximations of the correct theories, in the limit of large object that move slowly.

Crucial puzzle pieces that are missing in the above roadmap are mathematical tools. These are best learned when they are actually needed. This means, whenever you are trying to understand some topic like, for example, quantum mechanics, and a mathematical concept that you don't know is used, simply learned it then.

Here is a (incomplete) list of mathematical tools that are crucial for the topics listed above:

• Vector calculus: basis vectors, change of coordinate systems, matrix multiplication, scalars, right-handed and left-handed coordinate systems
• Calculus: product rule, integration by parts, Taylor series expansion, series, index notion, Kronecker symbol, Levi-Civita symbol
• Linear Algebra: conjugation and transposition of matrices, matrix exponential function, determinants, eigenvalues and eigenvectors, diagonalization of matrices.
• Misc: Fourier transform, Delta distribution

There is a book called "Physics from Symmetry", that tries to implement the roadmap outlined above. However, of course, additional books are needed to understand each topic mentioned here fully.

Here are some concrete further reading recommendations