# Equations in this field

1. Mar 19, 2006

### Gluonium

Well i always read about them, but does anyone know where i can look at the actual equations? Some examples might be Einsteins Theory of Relativity (not mass, but space and time) or Dirac's complete equation describing the electron; the one which earned him the Nobel Prize.

I dont understand why these are never in books, i could assume that they are too long and complicated, but i dont know. I would be very interested in seeing some of the math, instead of just reading about it. I made a thread a while ago about books that are intermediate in this field but also have some math involved. THanks!

2. Mar 19, 2006

### Staff: Mentor

3. Mar 19, 2006

### chroot

Staff Emeritus
Well, there are actually many books that cover the actual math in enormous detail. They're called.... textbooks. Perhaps you should visit your local university bookstore or library and look for some of the more widely-known physics textbooks.

Note that most of these books are graduate-level, and require a pretty good understanding of the math and physics involved in undergraduate physics.

If you're interested not in textbooks but in popular treatments, I offer you this: There's really not much reason to print the math in a book intended for people who can't actually understand it yet, is there?

- Warren

4. Mar 19, 2006

### Gluonium

True, but i want to be able to understand it which is why i asked if anyone knew of any books that kind of introduced things. Like if you take a look at the dirac equation (thanks for that BTW jtbell), at first glance i have no idea what it is, but i will read it and try to gain some sort of understanding. I wanted to know if there were any books out there that introduced it. Something like an intro textbook i suppose, if they exist.

Thanks!

5. Mar 19, 2006

### chroot

Staff Emeritus
The only way to truly understand advanced physics is to first understand basic physics, then work your way up in sophistication. It can take years of study to really be able to understand the general theory of relativity. There really is no "short-cut," or way for the subject to be made clear to people with no understanding of the underlying material.

- Warren

6. Mar 20, 2006

### Gluonium

If anyone can recomend any books on either particle and/or nuclear physics that'd be great! I understand i sound too eager, but i am not going to lie, i am. Can't wait for college and graduate work. I just want to learn more about the field, cant explain it, i just love it :D Thanks.

7. Mar 20, 2006

### chroot

Staff Emeritus
I think you should pick up an undergraduate Modern Physics textbook; they cover a wide variety of subjects fairly informally.

- Warren

8. Mar 21, 2006

### Staff: Mentor

Last edited by a moderator: May 2, 2017
9. Mar 21, 2006

### Perturbation

I’ll give you a brief over-view of the Dirac equation:

$$\sum_{\mu}\left(i\hbar\gamma^{\mu}\partial_{\mu}-mc)\psi =0$$

$\hbar$ is Plank's constant over two pi, $\partial_{\mu}$ a partial derivative (a derivative of a function of more than one variable with respect to a single variable) with respect to the subscripted variable ($\mu =t, x, y, z$), m the mass, c the speed of light and $i=\sqrt{-1}$. $\psi$ is a Dirac/Weyl spinor that is a more complicated variety of wave-function than that found in simple non-relativistic quantum mechanics, it's a bit like a vector but they transform under SU(2) rather than SO(N), meaning they're a vector in a complex (involving roots of negatives) vector space. $\gamma^{\mu}$ is the mu-th component of a set of matrices that transform as a vector. They form a Clifford algebra with the following anti-commutation relation in Minkowski space-time (the space-time of special relativity with metric diag(1 -1 -1 -1))

$$\left\{\gamma^{\mu}, \gamma^{\nu}\right\} =2\eta^{\mu\nu}\times\mathbf{1}_{n\times n}$$

$\eta^{\mu\nu}$ is the symmetric (same under interchange of the indices mu and nu) Minkowskian metric tensor, the braces with a comma denotes the commutator $\{a, b\}=ab+ba$ and $\mathbf{1}_{n\times n}$ is the identity operator with dimension n equal to the representation of the algebra.

You'll find a discussion of the Dirac equation and its results in any text that covers relativistic quantum mechanics and/or field theory. Its use in quantum field theory is of great importance, as it accurately describes the dynamics of fermions that are both free of interactions and involved in some form of interaction (with the appropriate coupling to the interaction and other terms that come from those in the Lagrangian that are gauge invariant under the symmetry group of the field theory).

To get the best results from the Dirac theory we need to apply a method of quantisation to the Dirac field $\psi$ (particle can only take on set, discrete amounts of energy). If we don't use quantum field theory and minimally couple the Dirac equation (whack in a gauge connection term that accounts for the interaction's coupling to the field) the Dirac equation can only, at best, account for first order interactions, meaning it can only make calculations from the simplest Feynman diagrams we can draw (known as tree-level, which are absent of virtual particle interactions/bubbles).

The full equation for an electron interacting with an electromagnetic field (Quantum Electrodynamics, QED) is

$$\mathcal{L}=\sum_{\mu,\nu}\left[\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi-\tfrac{1}{4}F^{\mu\nu}F_{\mu\nu}-e\bar{\psi}\gamma^{\mu}A_{\mu}\psi\right]$$

Here the Dirac field with the bar over it represents the anti-particle form of the fermion field $\psi$, e is the charge on the electron, F is the field-strength tensor of classical electrodynamics ($=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$) and A is the electromagnetic potential (satisfying $\mathbf{E}=\nabla A_0$, $\mathbf{B}=\nabla\times\mathbf{A}$). This quantity is known as a Lagrangian density $\cal{L}$ or simply Lagrangian, which by applying the principle of least action, also known as Hamilton's principle, will give one the equations of motion for the electron/photon. The QED Lagrangian accurately describes all electrodynamic interactions between fermions.

There aren't really any complicated equations from special relativity. The most important equation from general relativity is

$$G_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}$$

which are a set of 10 second order differential equations in the metric tensor (or connection in Einstein-Cartan theory) known as the Einstein field equations (in my opinion they should really be called the Einstein-Hilbert equations, but nevermind).

Last edited: Mar 21, 2006