What is the significance of manifest Lorentz covariance for field theories?

[voltan]

My question is: what does "manifest" Lorentz covariance means for a field theory, as opposed to simply Lorentz invariance.

Thanks for the replies!

 

[dextercioby]

I guess manifest stands for "explicit", meaning the use of space-time fields which come from the finite dimensional representations of the Lorentz group (and generally SL(2,C)).
The classical theory of the electromagnetic field in vacuum is manifestly Lorentz covariant, if the Lagrangian density is written using space-time tensors and covectors, i.e. $\mathcal{L}= -\frac{1}{4} F^{\mu\nu}F_{\mu\nu}$.

 

[Nabeshin]

Another way to think about it is that a manifestly lorentz covariant theory is one formed out of only lorentz scalars (all lorentz indices are contracted), and in which time and space are treated on equal footing.

 

[samalkhaiat]

My question is: what does "manifest" Lorentz covariance means for a field theory, as opposed to simply Lorentz invariance.

Thanks for the replies!

An expression with FREE space-time indices is called covariant, all tensorial equations are covariant, for example the equation
$$\partial_{a}F^{abc} = J^{bc} + \epsilon^{bcade}F_{ade}$$
is Lorentz covariant because it is equivalent to the tensor equation $R^{bc}=0$.
If there are NO indices or NO FREE indices (i.e. all indices are summed over) then we say that the expression is scalar or Lorentz invariant, examples
$$T = \eta_{ab}T^{ab} + F^{ab}F_{ab} + \eta_{ab}\eta_{cd}T^{ac}F^{bd}$$

Sam

 

[Meir Achuz]

Maxwell's equations in 3-vector form are Lorentz invariant in that they are the same in any Lorentz system, but they are not 'manifestly' Lorentz invariant.
'Manifestly' Lorentz invariance means that the LI is obvious from the form of the equations.

 

[andrien]

Maxwell's equations in 3-vector form are Lorentz invariant in that they are the same in any Lorentz system, but they are not 'manifestly' Lorentz invariant.
'Manifestly' Lorentz invariance means that the LI is obvious from the form of the equations.

And that is what is meant by manifest,right?

 

[jim mcnamara]

manifest as an adjective is defined as: clear or obvious to the eye or mind.
manifestly is the adverb form of the word.

-- what Meir Achuz posted.

 

[voltan]

Thank you very much for all the replies!

It is still not clear to me if there could be Lorentz-covariance without it being "manifest" (or "explicit")?

Any opinion would be appreciated!

 

[dextercioby]

We generally use the term <covariance> in relation to tensor fields on a flat/curved 4D spacetime, so the 'manifest' attribute is automatic.

 

[tom.stoer]

Of course there can be Lorentz covariance w/o manifest Lorentz covariance.

As explained above manifest 'covariance' of an expression is present when it is formulated in terms of 4-tensors but w/o all indices being contracted, whereas 'invariance' means that all indices are contracted, i.e. the experession contains only Lorentz-scalars.

So the Maxwell equations formulated in terms of 4-tensors are manifest covariant. But when you split them in E- and B-fields etc. they are still covariant (why should they lose this property by rewriting them in a fully equivalent way?) but they are no longer manifest covariant (you don't see immediately that they are covariant, you have to check it explicitly).

 

[haushofer]

From a gauge-theory point of view you can make the following observation:

Take for example a photon, which has two polarizations. Naively, you would like to use some field with two degrees of freedom to describe such a photon. However, such a description is not "manifestly Lorentz covariant", because there is no representation in which fits your two degrees of freedom. The smallest representation is the vector representation, having four dof's. Gauge invariance however now cuts these four dof's down to two, which is exactly what you need.

In that sense, you could say that gauge invariance helps you to describe photons in a "manifest Lorentz-covariant way".

The subtle point is that field equations can more or less always be rewritten in "manifestly X-covariant way", where X is some symmetry group, by using the so-called Stückelberg trick. This amounts to introducing extra fields to realize the symmetry.