# Question on the 2-dim representation of the Lorentz group

• Kontilera
In summary, the conversation discusses the use of Weyl spinors in quantum field theory and their representation in the Lorentz algebra. It is mentioned that the Lorentz algebra can be identified as two su(2)'s and the representation of the Lorentz algebra can be of dimension (2s_1 + 1)(2s_2 + 1). The Weyl spinor is two dimensional, so it can be represented as either a (s_1, s_2) = (1/2, 0) or a (0,1/2) representation (i.e. left or right handed). However, it is noted that the infinitesimal transformation of a left-handed spinor involves
Kontilera
Hello! I'm currently reading some QFT and have passed the concept of Weyl spinors 2-4 times but this time it didn't make that much sense..
We can identify the Lorentz algebra as two su(2)'s. Hence from QM I'm convinced that the representation of the Lorentz algebra can be of dimension (2s_1 + 1)(2s_2 + 1).
The Weyl spinor is two dimensional so it's either a (s_1, s_2) = (1/2, 0) or a (0,1/2) representation (i.e. left or right handed).

But it then seems (since one representation of the su(2)s is the trivial) as if I only need to specify three parameters when lorentz transforming my Weyl spinors.. What happened to my choice of three rotations and three boosts?

Thanks! :)

Kontilera said:
But it then seems (since one representation of the su(2)s is the trivial) as if I only need to specify three parameters when lorentz transforming my Weyl spinors.. What happened to my choice of three rotations and three boosts?
But they are three complex parameters. The SU(2) factorization of the Lorentz group involves the complex generators J ± iK. A Weyl spinor remains invariant under one of these subgroups.

Specifically, the infinitesimal transformation of a left-handed spinor is (Peskin & Schroeder, p44)

ψL → (1 - iθ·σ/2 - β·σ/2) ψL

where θ is a rotation and β is a Lorentz boost. Individually they change ψL, and any real combination of them changes ψL, but a complex combination of them leaves ψL invariant.

Last edited:
Another way to say the same thing is to let ##\vec A## be the generators for one of the SU(2)'s and ##\vec B## be the generators for the other SU(2). Then the angular momentum generators are ##\vec J=\vec A +\vec B## and the boost generators are ##\vec K=i(\vec A-\vec B)##. For the (1/2,0) rep, ##\vec A=\vec\sigma/2## and ##\vec B=0##. Then ##\vec J =\vec\sigma/2## and ##\vec K=i\vec\sigma/2##.

Yeah, I with you. The su(3) has three generators... but the relation above gives me six generators and that is what I can work with. :)
Or equivalently, I work with three generators but they now span a complex vectorspace, hence my coefficients amount to six degrees of freedom.

Thanks.

Does it exists a Loretnz transformation such as (1, 1/2) ? In that case, have do we make A and B to same dimension?

Kontilera said:
Does it exists a Loretnz transformation such as (1, 1/2) ? In that case, have do we make A and B to same dimension?

Of course there is. It's 'half' of a Rarita-Schwinger spinor. The dimension of the spinor space is (2x1+1)(2x1/2+1) = 6.

dextercioby said:
Of course there is. It's 'half' of a Rarita-Schwinger spinor. The dimension of the spinor space is (2x1+1)(2x1/2+1) = 6.

Haha. But what about the dimension? I guess we use a reducible rep for the lower s_i? Does it matter which one we choose?

## 1. What is the Lorentz group?

The Lorentz group is a mathematical group that describes the transformation of space and time coordinates between different inertial reference frames in special relativity. It includes rotations in three-dimensional space and boosts, which represent changes in velocity.

## 2. How is the Lorentz group represented in two dimensions?

In two-dimensional space, the Lorentz group can be represented by 2x2 matrices with real entries. These matrices correspond to the rotation and boost transformations in two-dimensional spacetime.

## 3. What is the significance of the 2-dim representation of the Lorentz group?

The 2-dim representation of the Lorentz group is significant because it allows us to study the properties and behavior of the Lorentz group in a simpler and more manageable way. This representation is also closely related to the four-dimensional representation of the Lorentz group, which is used in the full theory of special relativity.

## 4. Can the 2-dim representation of the Lorentz group be extended to higher dimensions?

Yes, the 2-dim representation of the Lorentz group can be extended to higher dimensions. However, the representation will become more complex and may not be as useful for studying the properties of the Lorentz group.

## 5. How is the 2-dim representation of the Lorentz group used in physics?

The 2-dim representation of the Lorentz group is used in physics to understand and analyze the effects of special relativity, such as time dilation and length contraction. It is also used in quantum field theory to study the symmetries of particle interactions and in the study of conformal field theories.

• Differential Geometry
Replies
1
Views
2K
• Quantum Physics
Replies
13
Views
1K
• Quantum Physics
Replies
9
Views
2K
• Quantum Physics
Replies
27
Views
1K
• Quantum Physics
Replies
13
Views
3K
• Quantum Physics
Replies
2
Views
2K
• Quantum Physics
Replies
11
Views
2K
• Quantum Physics
Replies
5
Views
2K
• Quantum Physics
Replies
20
Views
1K
• Quantum Physics
Replies
6
Views
1K