Understanding the Principle of Covariance: A Comprehensive Guide

[IFNT]

Can anyone explain to me the meaning of " the Principle of Covariance"? I find it hard to understand the wikipedia explanation.

 

[CompuChip]

I am probably going to miss a lot of subleties and give a crude explanation here, but here's the idea.

So you know that physicists usually measure scalars (real numbers) and vectors ("arrows" with a magnitude direction), and in general some more complicated objects.

Now of course, all physicists are different. I mean, that in general they are at different locations, can set up different directions for their measurement coordinates (i.e. choose different x-, y-, z-axis) and - in special relativity - can have some velocity with respect to each other. Now of course, if I know exactly how you move with respect to me, I can use that to correlate my measurement results to yours. For example, if I know that some vector I measure points along my z-axis, and I know exactly how you chose your coordinate system, then I can tell you what coordinates the vector will have when you measure it (assuming that it still physically represents the same vector).

All this is mathematically expressed with something called covariance. Basically, something like a vector is called covariant, if it transforms in some specific way under a coordinate transformation. So if I know what coordinate transformation I have to do to go from my lab to yours, I can translate my mathematical description of a vector (my x, y, z-coordinates) to yours (your x', y', z'-coordinates). So usually, when (theoretical) physicists talk about a "vector", they don't just mean any set of three (or however many needed) numbers, but a set of three numbers which transforms in the right way.

The "principle of covariance", as far as I can see it, simply states that physical quantities should transform covariantly. In the physical terms I used before, that simply states that whenever we can measure some physical quantity in one observers' frame, and we know how that observers' frame relates to another observers' frame, we can mathematically calculate what that other observer should get when he measures the same physical quantity - and that this agrees with experiment (i.e. if the other observer actually performs the measurement, he does get that result).

The example given on the Wikipedia page, is
$$m\frac{d\vec v}{dt} = \vec F.$$
The fact that this is covariant (and in fact, invariant) means that if you take another inertial observer ("inertial observers" are the Newtonian way of specifying which "transformations" are allowed, e.g. if you go from a stationary to a rotating observer it won't work) who measures the velocity $\vec v'$ and force $\vec F'$, he will find that the values found by him satisfy
$$m\frac{d\vec v'}{dt} = \vec F'.$$

 

[IFNT]

I am probably going to miss a lot of subleties and give a crude explanation here, but here's the idea.

So you know that physicists usually measure scalars (real numbers) and vectors ("arrows" with a magnitude direction), and in general some more complicated objects.

Now of course, all physicists are different. I mean, that in general they are at different locations, can set up different directions for their measurement coordinates (i.e. choose different x-, y-, z-axis) and - in special relativity - can have some velocity with respect to each other. Now of course, if I know exactly how you move with respect to me, I can use that to correlate my measurement results to yours. For example, if I know that some vector I measure points along my z-axis, and I know exactly how you chose your coordinate system, then I can tell you what coordinates the vector will have when you measure it (assuming that it still physically represents the same vector).

All this is mathematically expressed with something called covariance. Basically, something like a vector is called covariant, if it transforms in some specific way under a coordinate transformation. So if I know what coordinate transformation I have to do to go from my lab to yours, I can translate my mathematical description of a vector (my x, y, z-coordinates) to yours (your x', y', z'-coordinates). So usually, when (theoretical) physicists talk about a "vector", they don't just mean any set of three (or however many needed) numbers, but a set of three numbers which transforms in the right way.

The "principle of covariance", as far as I can see it, simply states that physical quantities should transform covariantly. In the physical terms I used before, that simply states that whenever we can measure some physical quantity in one observers' frame, and we know how that observers' frame relates to another observers' frame, we can mathematically calculate what that other observer should get when he measures the same physical quantity - and that this agrees with experiment (i.e. if the other observer actually performs the measurement, he does get that result).

The example given on the Wikipedia page, is
$$m\frac{d\vec v}{dt} = \vec F.$$
The fact that this is covariant (and in fact, invariant) means that if you take another inertial observer ("inertial observers" are the Newtonian way of specifying which "transformations" are allowed, e.g. if you go from a stationary to a rotating observer it won't work) who measures the velocity $\vec v'$ and force $\vec F'$, he will find that the values found by him satisfy
$$m\frac{d\vec v'}{dt} = \vec F'.$$

Thank you for your thorough reply Compuchip. You are much better at explaining these things than my teacher in quantum field theory.
How do you check if some physical quantity or equation is covariant? Suppose that I know the coordinate transformation matrix and I want to check if an equation is covariant. Do I have to make experiments in different inertial frames to ensure that the equation is covariant. My lecturer sometimes derives an equation and say: "This equation is covariant, you will show it in an exercise."
I have just started a seven-week-course on quantum field theory and the notation and physical meaning of the equations my teacher derives are hard to understand.
I have borrowed the book "Quantum field theory in a nutshell" written by A. Zee and I find it amusing and well written.

 

[CompuChip]

Personally I like Zee's book because of his humour, while being (relatively) rigorous.

Mathematically, the way to check that an equation is covariant, is to check that it satisfies the proper transformation rule. For example, a vector is covariant if you can take a matrix $A^\mu{}_\nu$ from the transformation group (for example, if you are talking about frames which are at the same point but look in different directions, it may be a matrix that represents an element from SO(3) or SO(3, 1)) and the components of the vector satisfy
$$v^\mu' = A^{\mu'}{}_{\nu} v^\nu$$
where the primed components refer to the new coordinate system (to which A transforms) and the unprimed components to the original coordinate system.

If you can find it, I can recommend Sean Carroll's General Relativity text. They are printed as a book, but there is also a free copy of the lecture notes the book is based on which is available on the internet. The first part is all about this kind of things, and I started really understanding them when I thouroughly read his notes.