Invariance, covariance and symmetry

Question

Though often heard, often read, often felt being overused, I wonder what are the precise definitions of invariance and covariance. Could you please give me an example from field theory?

Answer 1

The definitions of these terms are somewhat context-dependent. In general, however, invariance in physics refers to when a certain quantity remains the same under a transformation of things out of which it is built, while covariance refers to when equations "retain the same form" after the objects in the equations are transformed in some way.

In the context of field theory, one can make these notions precise as follows. Consider a theory of fields $\phi$. Let a transformation $T$ $$ \phi \to\phi_T $$ on fields be given. Let a functional $F[\phi]$ of the fields be given (consider the action functional for example). The functional is said to be invariant under the transformation $T$ of the fields provided $$ F[\phi_T] = F[\phi] $$ for all fields $\phi$. One the other hand, the equations of motion of the theory are said to be covariant with respect to the transformation $T$ provided if the fields $\phi$ satisfy the equations, then so do the fields $\phi_T$; the form of the equations is left the same by $T$.

For example, the action of a single real Klein-Gordon scalar $\phi$ is Lorentz-invariant meaning that it doesn't change under the transformation $$ \phi(x)\to\phi_\Lambda(x) = \phi(\Lambda^{-1}x), $$ and the equations of motion of the theory are Lorentz-covariant in the sense that if $\phi$ satisfies the Klein-Gordon equation, then so does $\phi_\Lambda$.

Also, I'd imagine that you'd find this helpful.

Answer 2

It helps to remember that invariant quantities are seen as scalars to the transformation (they have no indices in the target space). In the other hand, covariant quantities are objects that transform in a certain way.

Example: Vectors in $R^{2}$, under rotation $R_{ij}$, transform covariantly since $v'_{i}=R_{ij}v_{j}$, but it's length is invariant since $|v'|^{2}=v'_{a}v'_{a}=R_{am}v_{m}R_{an}v_{n}=v_{m}R^{t}_{ma}R_{an}v_{n}=v_{m} \delta_{mn} v_{n}=v_{n}v_{n}=|v|^{2}$. This means that Newton's second Law transforms covariantly under rotations and the magnitude of the force is invariant.

Answer 3

Covariance and invariance are different expressions of the same idea. In fact, invariance is the fundamental notion and covariance is the derived notion. However, the situation requires care since we may only be able to use covariance to establish invariance. This is because we naturally use coordinate systems to describe an object and these descriptions obviously change when we change the coordinate system. In other words, operationally speaking, we may only have access to the covariant description and have to infer the invariance of the object under question.

Covariance derives from co-vary, and this suggests there are two things which vary together. For example, when we describe the coordinate of a point, a position, and hence one of the most basic of physical notions, we need to set up coordinate axes and then find the coordinates of the point. If we then change the coordinate axes, then the coordinates of the point changes. However, of course the point itself remains the same. We say invariant.

There's a lot more to say about covariance and invariance in terms of tensor analysis but the above gives the conceptual picture. In fact, this is the fundamental description since we use this notion to build manifolds. Once we have that we can build invariantly the tangent bundle over the manifold and then further, we can build invariantly the tensors of that tangent bundle and this gets us the tensors that are used in General Relativity. We can then show that the covariant description of these tensors that are built invariantly are precisely the tensors described covariantly in physics textbooks.

Finally, the symmetry group ought to be the gauge/structure group of the frame bundle over the manifold.