# Unitary Representations of Lorentz/Poincare Group

• LarryS
In summary, Unitary Representations of Lorentz/Poincare Group refer to mathematical models used to describe the symmetries of physical systems in special and general relativity. These representations are based on the Lorentz group and account for the effects of time dilation and length contraction. They are important in understanding the behavior of elementary particles and the fundamental laws of physics, and have applications in fields such as quantum mechanics and cosmology.
LarryS
Gold Member
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.

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).

George Jones, andresB, dextercioby and 1 other person
referframe said:
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.

Which lectures are you watching?

PeroK said:
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.

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

LarryS
referframe said:
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.

LarryS and vanhees71
vanhees71 said:
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

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.

dextercioby
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

dextercioby and vanhees71
That's a nice shortcut to derive this result in the spirit of "canonical quantization".

Exactly :)

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

vanhees71 and weirdoguy
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).

vanhees71
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.

dextercioby and vanhees71
vanhees71 said:
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.

referframe said:
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.

vanhees71 said:
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.

Which paper? Journal, volume, page, year?

vanhees71 said:
Which paper? Journal, volume, page, year?
This paper

vanhees71
LarryS said:
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?

Last edited:
Delta2

## 1. What is a unitary representation of the Lorentz/Poincare group?

A unitary representation of the Lorentz/Poincare group is a mathematical description of the symmetries of spacetime. It describes how objects and physical laws behave under transformations of space and time, such as rotations and boosts. Unitary representations are important in quantum field theory and help us understand the fundamental principles of the universe.

## 2. How are unitary representations of the Lorentz/Poincare group used in physics?

Unitary representations of the Lorentz/Poincare group are used in physics to study the behavior of particles and fields in spacetime. They help us understand the symmetries and properties of physical systems, and are essential in the development of theories such as quantum mechanics and relativity.

## 3. What is the difference between a unitary representation and a non-unitary representation?

A unitary representation preserves the inner product of vectors, meaning that the length and angle between vectors remain the same after the transformation. In contrast, a non-unitary representation does not preserve the inner product and can change the length and angle between vectors. In physics, unitary representations are preferred because they preserve the physical properties of systems.

## 4. Can all symmetries of the Lorentz/Poincare group be represented unitarily?

No, not all symmetries of the Lorentz/Poincare group can be represented unitarily. Some symmetries, such as translations and rotations, can be represented unitarily, while others, such as boosts, require a more complex representation. This is because boosts involve a change in the time coordinate, which can affect the inner product of vectors.

## 5. How do unitary representations of the Lorentz/Poincare group relate to special relativity?

Unitary representations of the Lorentz/Poincare group are closely related to special relativity. Special relativity is based on the principle that the laws of physics are the same for all observers in uniform motion. Unitary representations of the Lorentz/Poincare group mathematically describe this principle and help us understand the symmetries of spacetime that are essential in special relativity.

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