# Complex / real representations of the Lorentz group

In Michele Maggiore's book "A Modern Introduction to QFT" he describes the spinorial representations of the Lorentz group as

The representations are in general complex.

I always thought the difference between real and complex representations to be fundamental, because it is determined by the vector space: if the representation acts on a complex vector space, then the rep is complex. The real case can involve complex numbers, for example $\mathbb{C}$ can be treated as a real vector space - as long as the scalar multiplication is limited to real numbers, the rep is real.

Is that correct? If yes, then why can a representation be "in general" complex? The complexity is a fundamental property, not something like "irreducibility" which is sometimes satisfied, sometimes not.

• I'm pretty sure the author means that most of the representations he's considering are complex representations. You're right that a representation is either complex or real - by saying "the representations are in general complex," he means "In general, the representations of the Lorentz group are complex" -- that is, most of them (or most of the interesting ones) are complex representations. – user_35 Sep 2 '15 at 17:14
• Well, the meaning you describe is exactly what puzzles me. Whether one analyzes real or complex reps is a completely different thing. It's not "We're analyzing reps and see which ones are real and which ones are complex", but it is "We're analyzing real reps, and when we're done we're analyzing the complex ones." So why can reps be treated as if "it turns out that the rep is complex"? – Bass Sep 2 '15 at 17:20
• Note that the typical treatment of "group theory" or "Lie algebras" found in physics books is non-rigorous, schematic and often confusing to many students (myself included): Maybe the book is simply bad (in this aspect). – Danu Sep 2 '15 at 20:44

I think you successfully answered your own question, but I'll elaborate a bit.

Usually, the Lorentz Group refers to SO(3,1). The fundamental representation of these group elements is by 4 x 4 matrices with real numbers in them. Since all higher dimensional matrix representations can be built using an outer product of fundamental reps, all the higher dimensional rep matrices only have real numbers in them. These matrices will only cause linear combinations with real coefficients of the basis vectors in the reps carrier space...therefore I guess you could call it a real vector space.

These SO(3,1) irreps are able to rotate/boost many (but not all) objects found in the physical world. These irreps are only good for integer spin objects. None of these irreps can rotate half integer spin objects (spinors)(like electrons).

However,there is another group SL(2,C) for which there are 3 complex Lie group parameters that specify each group element. For SO(3,1) there are 6 real Lie group parameters (3 rotation angles, 3 boost parameters) that specify each group element. You can stuff the 6 real parameters into the 3 complex parameters using the imaginary $i$. SL(2,C) is called the "covering group" of SO(3,1). The generators of the two groups can be put in a correspondence with each other such that both sets of generators have the same commutation relations. The group elements of the fundamental rep of SL(2,C) are 2 x 2 matrices with complex numbers in them. All the higher dimensional irreps are built of outer products of these complex 2 x 2 matrices. Some of these higher dim matrices still have complex numbers. Others end up with just real numbers and duplicate the real irreps of O(3,1). So, many of the SL(2,C) irreps will cause linear combinations with complex coefficients of the basis vectors in the irreps carrier space...therefore I guess you could call it a complex vector space.

The SL(2,C) irreps are able to rotate/boosts all objects found in the physical world (ie: both the half integer and integer spin particles).

The pedantically correct opening lines of your question should be:

In Michele Maggiore's book "A Modern Introduction to QFT" he describes the spinorial representations of SL(2,C), the covering group of the Lorentz group as

The representations are in general complex.

Well I just realized that my notion of a real/complex representation was wrong. The property of a representation of being real or complex is not as fundamental as I thought.

For example $SU(2)$ has the real irreps of integer spin, those that are also irreps of $SO(3)$. The other irreps, the ones with non-integer spin, are complex.

So, the analysis of Lie group irreps really is like

We're analyzing reps and see which ones are real and which ones are complex.