Unitary Representations of Lorentz/Poincare Group

[LarryS]

Summary:: Looking for best literature or online courses on projective unitary representations of the Poincare Group.

I'm watching an online course on relativistic QFT. I understand that because this theory deals with both QM and SR, there is a need to represent Lorentz transformations with unitary equivalents. The course itself points out this requirement but does not really explain it in a rigorous way. My impression is that it is fairly complicated. I'm hoping to understand it without first getting my PhD in math. I'm looking for well-written literature or good online lectures on this subject.

Thanks in advance.

 

[vanhees71]

My favorite introductory treatment is in

R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles,
Springer, Wien (2001).

This covers the unitary representations of the Poincare group.

The proof that these are all relevant for Q(F)T, i.e., that there are no additional non-equivalent unitary ray representations is in

S. Weinberg, The Quantum Theory of Fields, vol. 1,
Cambridge University Press (1995).

Note that this is different for the Galilei group: There the unitary representations don't provide a successful quantum theory but you need a non-trivial ray representation adding mass as a central charge in the corresponding Lie algebra to get standard quantum mechanics. That's very nicely covered in

L. E. Ballentine, Quantum Mechanics, World Scientific,
Singapore, New Jersey, London, Hong Kong (1998).

 

[PeroK]

Summary:: Looking for best literature or online courses on projective unitary representations of the Poincare Group.

I'm watching an online course on relativistic QFT. I understand that because this theory deals with both QM and SR, there is a need to represent Lorentz transformations with unitary equivalents. The course itself points out this requirement but does not really explain it in a rigorous way. My impression is that it is fairly complicated. I'm hoping to understand it without first getting my PhD in math. I'm looking for well-written literature or good online lectures on this subject.

Thanks in advance.

Which lectures are you watching?

 

[LarryS]

Which lectures are you watching?

The instructor is Tobias Osborne, on YouTube. He follows the lecture notes of David Tong, who is a better teacher, but it is harder to see the blackboard in David's videos. I've watched Leonard Susskind's intro to QFT a few times. Unfortunately, the QFT video lectures are not available on MIT OCW, just the lecture notes.

 

[vanhees71]

Usually the lecture notes are of much greater value than videos!

 

[PeroK]

The instructor is Tobias Osborne, on YouTube. He follows the lecture notes of David Tong, who is a better teacher, but it is harder to see the blackboard in David's videos. I've watched Leonard Susskind's intro to QFT a few times. Unfortunately, the QFT video lectures are not available on MIT OCW, just the lecture notes.

I thought that might be the lectures. I'm not sure where you'd find a more rigorous treatment outside of a mathematics text on Lie Groups and Lie Algebras. I thought that he covered the idea quite thoroughly. He spends quite a long time showing that the relevant operators do form a unitary representation of the Poincare group. I'm not sure on a QFT course he could have spent much more time on the Lie mathematics.

 

[Dr Transport]

Look at this text...

https://www.amazon.com/dp/9971966573/?tag=pfamazon01-20

 

[LarryS]

Look at this text...

https://www.amazon.com/dp/9971966573/?tag=pfamazon01-20

I've looked at a lot of books on Amazon on this subject. Based on the table of contents, this book is the most thorough treatment of this subject that I have seen. Thanks!

 

[vanhees71]

Also the original paper by Wigner is very readable:

E. P. Wigner, On Unitary Representations of the Inhomogeneous Lorentz Group, Annals of Mathematics 40, 149
(1939)
https://doi.org/10.1016/0920-5632(89)90402-7.

 

[haushofer]

My favorite introductory treatment is in

R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles,
Springer, Wien (2001).

This covers the unitary representations of the Poincare group.

The proof that these are all relevant for Q(F)T, i.e., that there are no additional non-equivalent unitary ray representations is in

S. Weinberg, The Quantum Theory of Fields, vol. 1,
Cambridge University Press (1995).

Note that this is different for the Galilei group: There the unitary representations don't provide a successful quantum theory but you need a non-trivial ray representation adding mass as a central charge in the corresponding Lie algebra to get standard quantum mechanics. That's very nicely covered in

L. E. Ballentine, Quantum Mechanics, World Scientific,
Singapore, New Jersey, London, Hong Kong (1998).

From a point particle perspective the central charge is already contained in the Poisson brackets. So the Galilei algebra, having no central charge ([B,P]=0), is the wrong place to start with in the first place :P

 

[vanhees71]

This I don't understand. There are proper unitary representations of the classical Galilei group, but they don't lead to useful dynamics in quantum theory. So for a non-relativistic quantity you need a central extension of the Galilei group or equivalently a non-trivial ray representation.

That's the content of the famous paper

E. In¨on¨u and E. P. Wigner, Representations of the Galilei
group, Il Nuovo Cimento 9, 705 (1952),
https://doi.org/10.1007/BF02782239.

 

[haushofer]

I mean that the central extension does not have a "quantum nature"; it's already there classically if you write down the action of a non-relativistic point particle, namely in the Poisson brackets. The reason for its appearance is that the Lagrangian is quasi-invariant under boosts. These brackets are isomorphic to the underlying Lie algebra.

See e.g. section 4 of

https://arxiv.org/pdf/1011.1145.pdf

 

[vanhees71]

That's a nice shortcut to derive this result in the spirit of "canonical quantization".

 

[haushofer]

Exactly :)

Often, central extensions are associated with quantum mechanics, but they already pop up classically.

 

[haushofer]

By the way, the same boosts which keep the Lagrangian only quasi-invariant (i.e. it transforms to a total derivative) change the wave function by a phase factor (i.e. the wave function is not a scalar under boosts).

 

[andresB]

It's not related to your main conversation, but since unitary representations and the Galilei group and it's central charge were mentioned, you might be interested in reading the following article

https://arxiv.org/abs/2004.08661

It's not mathematically sophisticated, but I'm certain is something new for most people.

 

[LarryS]

Also the original paper by Wigner is very readable:

E. P. Wigner, On Unitary Representations of the Inhomogeneous Lorentz Group, Annals of Mathematics 40, 149
(1939)
https://doi.org/10.1016/0920-5632(89)90402-7.

Thanks. I started reading Dirac’s Paper but he has a kind of a confusing way of explaining things.

 

[vanhees71]

Thanks. I started reading Dirac’s Paper but he has a kind of a confusing way of explaining things.

Which paper by Dirac are you referring to? Usually Dirac's papers are utmost clearly written.

 

[LarryS]

Which paper by Dirac are you referring to? Usually Dirac's papers are utmost clearly written.

The paper was written in 1944. Maybe I'll give it another shot. Thanks.

 

[vanhees71]

Which paper? Journal, volume, page, year?

 

[LarryS]

Which paper? Journal, volume, page, year?

This paper

 

[JandeWandelaar]

This paper

Hi there! Nice article. I can't see why in the preliminary part, in (2), a factor r! is introduced in defining the square of the length of the a-vector, constituting the vector of coefficients in a power expansion:

$a_0+a_1 {\xi}_1 +a_2 {{\xi}_1}^2 +...$

The square of the of the length is said to be

$$\sum r!{a_r}^2$$

Where does r! come from?