Undivided Looking

What is the world made out of?  In the most usual formulations of our current best theories of physics, the answer is fields.  What are those?

Well, if you know what a function is, you're already most of the way there.  A function, you will recall, is a gadget where, for any number you input, you can get a number out as an output.  We can write $f(x)$ where $x$ is the number you input, and $f(x)$ is the number you output.  The function $f$ itself is the rule for going from one to the other, e.g.  For example $f(x) = \sqrt{\sin x^2 + 1}$.

Now, nothing stops you from having a function that depends on multiple numbers as input; for example the function $f(x,\,y) = xy^2 + x^3y$ depends on two input variables, $x$, and $y$.  If there are $D$ input numbers, then the $D$-dimensional space of possible combinations of input numbers is called the domain of the function.

Also nothing stops you from having the output be a set of several numbers.  In this case we would need some sort of subscript $i$ to refer to the different possible output numbers.  For example, if we had a function with one input number $x$ and three output numbers $y$, then we could write $f_i(x)$, where $i$ takes the values 1, 2, or 3.  Then $f_i(x)$ would really be just a package of three different functions: $f_1(x)$, $f_2(x)$, and $f_3(x)$.  So if you specify the input $x$, you get three output numbers $(f_1, f_2, f_3)$.  If there are $T$ different output numbers, the $T$ dimensional space of possible outputs is called the target space.

Now a field is just a function whose domain is the points of spacetime.  For example, the air temperature in a room may vary from place to place, and it may also change with time.  So if you imagine checking all possible points of space in the room at all possible times, you could describe this with a temperature field $T(t, x, y, z)$.  However, the temperature field isn't a fundamental entity that exists on its own.  It subsists in a medium (air) and describes its motion.  When the air molecules are moving around quickly in a random way, we say it's hot, and when they start to move around slower, we say it's getting chilly.  An example of a field which actually is fundamental (as far as we know) would be the electromagnetic field.  This has 6 output numbers, since the electric field can point in any of the 3 spatial directions, and the magnetic field also has 3 numbers.

For a while in the 19th century, scientists were confused about this.  They thought that electromagnetic waves had to be some sort of excitation of some sort of stuff, which they called the aether.  That's because they were assuming (based on physical intuitions filtered through Newtonian mechanics) that matter is something solid and massy, which interacts by striking or making contact with other things.  The 20th century scientific advances partly came from realizing that its okay to describe things with abstract math.  Any kind of mathematical object you write down satisfying logically consistent equations is OK, as long as it matches experiment.  So electromagnetic waves don't have to be made out of anything.  They just are, and other things are (partly) made out of them.

In our current best theory of particle physics, the Standard Model, there are a few dozen different kinds of fields, and all matter is explained as configurations of these fields.  I can't tell you exactly how many fields there are, because it depends on how you count them.  Not counting the gravitational field, there are 52 different output numbers corresponding to bosons, and 192 different output numbers corresponding to fermions (Don't worry about what these terms mean yet).  So you could say that there are 244 different fields in Nature, each with one output number.

That sounds awfully complicated.  But there's also a lot of symmetries in the Standard Model which relate these output numbers to each other.  This includes not only the Poincaré group of spacetime symmetries, but also various internal symmetries related to the dynamics of the strong, weak, and electromagnetic forces.  They are called internal because they don't move the points of spacetime around.  Instead they just mutate the different kinds of output numbers into each other.

So normally, particle physicists just package the output numbers into sets, such that the numbers in each set are related by the various kinds of symmetry.  (For example, the 6 different numbers of the electromagnetic field are related by rotations and Lorentz boosts.)  Each of these sets is called a field.  In future posts I'll give more details about the different kinds of fields.  As always, questions are welcome.

UPDATE: I forgot to include the 4 vector components of the spin-1 gauge bosons, so the numbers of degrees of freedom of the bosons were wrong before.  Note to Experts: These are the "off-shell" degrees of freedom before taking into consideration constraints or gauge symmetry.  Note to Non-Experts: the numbers in this post are just for flavor, in order to give you the sense that there are a LOT of different fields in Nature.  You won't need to understand how I got these numbers in order to enjoy future posts!

Previously, I described the main formula of Special Relativity:

$$s^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 - (\Delta t)^2.$$

This formula tells us the amount of distance squared between two points (if $s^2 > 0$) or the amount of duration squared (if $s^2 < 0$).  (By using some trigonometry we can also use this formula to figure out the size of angles, so this encodes everything about the geometry).  All the crazy time dilation and distance contraction effects you've probably heard about are encoded in this formula.

Today I want to talk about the symmetries of spacetime.  What I mean by a symmetry is this: a way to change the coordinates $(t,\,x,\,y,\,z)$ of spacetime in a way that leaves the laws of physics the same.  Now I haven't told you what the laws of physics are, but the important thing is that they depend on the geometry of spacetime.  So that means that we need to check in what ways we can change the coordinates of spacetime without changing the formula for $s^2$.

The first kind of symmetry is called a translation.  This consists of simply shifting the coordinate system e.g. one meter to the right, or one second to the future.  This doesn't affect the formula for $s^2$ since it only depends on the coordinate differences $\Delta t$, $\Delta x$ etc.  We can write a time translation like this:

$$t^\prime = t + a,$$

i.e. the new time parameter $t^\prime$ equals the old one plus some number $a$.  Similarly, the three possible kinds of spatial translations are:

$$x^\prime = x + b; \\ y^\prime = y + c; \\ z^\prime = z + d.$$

By choosing the numbers a, b, c, d, arbitrarily, one obtains a four dimensional space of possible translation symmetries.

The second kind of symmetry is more complicated, but you've certainly heard of it before—it's called a rotation.  If we have two spatial coordinates, then we can rotate them by some angle $\theta$ (measured in radians), which leaves all the distances the same.  The algebraic formula for a rotation looks like this:

$$x^\prime = \phantom{-}\cos (\theta) \, x + \sin (\theta) \,y; \\ y^\prime = -\sin (\theta) \,x + \cos (\theta) \,y.$$

That involves some trigonometry, but things look a bit simpler if we take the angle $\theta$ to be a really tiny parameter $\epsilon$, and just consider the resulting infinitesimal coordinate changes $\delta x \approx (x^\prime - x)$:

$$\delta x = \phantom{-}\epsilon\,y; \\ \delta y = -\epsilon\,x.$$

Translated into English, that says that if you rotate the y-axis of your coordinate chart a little bit towards the x-axis, you have to rotate the x-axis a little bit away from the y-axis (or vice versa if $\epsilon$ is negative).  I'm too lazy to draw this, but if for some reason you can't visualize it, a little bit of fidgeting with any rigid flat object should convince you.

Now actually we have three different spatial coordinates: x, y, and z.  That means that you can actually rotate in 3 different ways: along the x-y plane, the y-z plane, and the z-x plane.  Of course there are other angles you can rotate at as well, but they are all just combinations of those three; in other words the space of possible rotations is 3-dimensional.

But now, what about the time direction?  It would feel terribly lonely if it were left out, and in fact it is also possible to rotate spacetime about the t-x plane, the t-y plane, and the t-z plane.  However, remember how time is not quite the same as space?  Instead, it's just like space except for a funny minus sign.  So not surprisingly, the formula for a rotation also has a funny minus sign—or rather, a funny absence of a minus sign:

$$\delta t = \epsilon\,x; \\ \delta x = \epsilon\,t.$$

So if you rotate the t-axis towards the x-axis (which corresponds to changing your coordinate system so that you are travelling at a constant speed), then the x-axis has to rotate towards the t-axis (which means that your notion of simultaneity has to change as well).  If you know how to integrate this with calculus, you can get the effects of a finite "rotation" in space (called a Lorentz boost) through an "angle" $\chi$:

$$t^\prime = \cosh (\chi) \, t + \sinh (\chi) \,x; \\ x^\prime = \sinh (\chi) \, t + \cosh (\chi) \,x.$$

In the above, cosh and sinh are functions similar to cosine and sine but defined using hyperbolas instead of circles.

So this rotation has some wierd properties: It describes a crazy world (ours!) in which things rotate in hyperbolas instead of circles.  That's because of the minus sign in the formula for $s^2$ above, which makes it so the points of equal distance (or duration) correspond to hyperbolas instead of circles.  This has some additional consequences: 1) Because hyperbolas are infinitely long, the "hyperbolic angle" $\chi$ ranges from $-\infty$ to $+\infty$, unlike circular angles which come back to where you started after you rotate through $2\pi$ radians.  2) Because the two axes both move towards (or both move away) from each other, when you do a really big rotation it scrunches everything up towards $t = x$ or $t = -x$.  What this means is that when you accelerate objects more and more, they don't go arbitrarily fast.  Instead they just get closer and closer to the speed of light.

In conclusion, spacetime has 10 kinds of symmetry: 4 kinds of translations and 6 kinds of rotations.  The space of possible symmetries is 10 dimensional.  It is called the Poincaré group.

P.S. In this whole discussion I have ignored the possibility of reflection symmetries such as $t \to -t$ or $x \to -x$.  These are also symmetries of the formula for $s^2$, but they are discrete rather than continuous—there's no such thing as a "small" reflection the way you can have a small rotation.  Adding these in doesn't change the fact that the Poincare group is 10 dimensional.  However, these transformations are actually NOT symmetries of Nature.  They are violated by our theory of the weak force.  The only discrete symmetry like this which is preserved by the weak force is CPT: the combination of time reflection, space reflection, and switching matter and antimatter.