Anti-Hydrogen Spectrum My question is: What is the spectrum of anti-hydrogen?
Could it be the same as regular hydrogen? They both have a core (a proton or an anti-proton) and an electron or a positron so they should have the same orbits   (the positron or electron) and so could emit they the same spectral lines? Am I right?
 A: Fundamental interactions are assumed to satisfy a fundamental symmetry called CPT which says that the laws of physics must be invariant under charge conjugation, parity transformation and time reversal. Whether this is an exact or only an approximate symmetry is still a subject of research. In particular a CPT violation implies into a Lorentz violation.
In order to satisfy CPT theorem it is necessary that particles and antiparticles have same masses and lifetimes and same magnitude (but opposite sign) of electric charge and magnetic moment. In particular, the spectrum of bound states of matter and their corresponding antimatter have to be the same. So a difference in the antihydrogen spectrum relative to hydrogen spectrum would show a CPT as well as a Lorentz symmetry violations. However, as answered by Countto10, recent experiments show no difference in the spectra.
A: I'd say the only possible differences could arise from mass differences between proton and antiproton, or between electron and positron, and charge differences of the same kind.
According to the Particle Data Book, both mass and charge relative differences (i.e., $|m_1-m_2|/m_1$ or $|q_1-q_2|/q_1$) are lower than $10^{-9}$ at $90\,\%$ C.L., so you could say that hydrogen and anti-hydrogen are the same at a really high confidence level.
A: As explained by others, there's a fundamental symmetry called Charge-Parity-Time invariance that is thought to hold for the laws of physics. Simply, it implies that the laws of physics are identical under charge conjugation, parity reversal and time reversal.
If CPT invariance holds, we expect that there will be no difference between matter and matter spectra. However, there are some outstanding questions in physics, such as the reason for the dominance of matter over antimatter and how gravity integrates with quantum mechanics that could require CPT breaking.
Presently, there are some experiments that are examining CPT invariance, and some of them are working towards measuring antihydrogen's spectrum. The ALPHA experiment recently announced a first measurement of the 1S-2S transition. There has been previous work on the ground-state hyperfine transitions, that are a different part of the spectrum.
These experiments are very difficult, because only a very very small amount of antimatter can be produced and trapped. Though the results published so far don't show any difference between matter and antimatter, the results are still fairly imprecise as far as atomic spectroscopy goes. We expect better, more precise results in the next few years that will allow better comparisons to be made, perhaps revealing very small differences. 
A: So far, we haven't measured the spectrum very well, only one line of antihydrogen, rather imprecisely.
It is worth emphasizing just how new the ALPHA results are - the first experimental detection of an antihydrogen spectrum is just over a month old. At present it can be obtained with modest accuracy (I seem to remember ~6 significant figures, but that may be wrong), rather short of the $10^{−15}$ that can be achieved with frequency combs on hydrogen, but future work should significantly reduce that gap relatively quickly
My thanks to Gremlin and Emilio Pisanty respectively for these corrections/comments.
We do expect that, on further study, the antimatter spectrum will be identical to the spectrum of  matter hydrogen. It would be (very) nice if  was different, but to date no new physics has emerged  from studying  it.
 Antihydrogen Laser Physics Apparatus (ALPHA)  has a good description of the experiment.
Could it be the same as regular hydrogen?  Basically, yes it should be, but the point of these experiments is to verify this idea.
From the above link:

Today’s ALPHA result is the first observation of a spectral line in an antihydrogen atom, allowing the light spectrum of matter and antimatter to be compared for the first time. Within experimental limits, the result shows no difference compared to the equivalent spectral line in hydrogen. This is consistent with the Standard Model of particle physics, the theory that best describes particles and the forces at work between them, which predicts that hydrogen and antihydrogen should have identical spectroscopic characteristics.

From  Wikipedia ALPHA 

On 26 April 2011, ALPHA announced that they had trapped 309 antihydrogen atoms, some for as long as 1,000 seconds (about 17 minutes). This was longer than neutral antimatter had ever been trapped before. ALPHA has used these trapped atoms to initiate research into the spectral properties of the antihydrogen.
The biggest limiting factor in the large-scale production of antimatter is the availability of antiprotons. Recent data released by CERN states that, when fully operational, their facilities are capable of producing ten million antiprotons per minute. Assuming a 100% conversion of antiprotons to antihydrogen, it would take 100 billion years to produce 1 gram or 1 mole of antihydrogen (approximately $6.02×10^{23} $ atoms of anti-hydrogen).

Of course, I would love to put a spectrum of antihydrogen in this answer, but it would pointless.
A: CPT violation would not result in the spectrum from being different from regular hydrogen. That is because antihydrogen and hydrogen are similar and light spectrum has almost nothing to do with charge.
