comments on physics and theology

Fields

by Posted on November 10, 2012

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 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!

About Aron Wall

I am a postdoctoral researcher studying quantum gravity and black hole thermodynamics at UC Santa Barbara. Before that, I studied the Great Books program at St. John's college Santa Fe, and got my Ph.D. in physics from U Maryland.
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