# The Lorentz group generator

1. Jan 21, 2008

### princeton118

What is the rotation transformation generator?
What is the Lorentz group generator?

2. Jan 22, 2008

### Fredrik

Staff Emeritus
I kind of suck at the Lie Algebra stuff, so I'll just mention some of what Steven Weinberg said in The quantum theory of fields, volume 1, chapter 2. (Great chapter, I highly recommend it even if you're not going to read the rest of the book).

A rotation is a linear transformation

$$x\mapsto Rx : \mathbb{R}^3\rightarrow\mathbb{R}^3$$

where R is orthogonal. We can let R be a function of three parameters, e.g. Euler angles.

In quantum mechanics, there must be an operator U(R) that acts on state vectors to give us the vector that a rotated observer would use to describe the same physical system. If we take this to be a function of the parameters and Taylor expand it, the result can be expressed as

$$1-i\vec\theta\cdot\vec{J}+\cdots$$

where the components of theta are real numbers, the components of J are Hermitean operators, and terms of second and higher order in the parameters have been omitted. The components of J are called spin operators, and they are said to be generators of rotations, because the operators in the higher order terms can be constructed from the operators in the first order terms, so the generators are all that's needed except values of the parameters to construct a particular rotation operator.

It's possible to show that the components of J must satisfy the commutation relations

$$[J_i,J_j]=i\varepsilon_{ijk}J_k$$

Most QM books start the discussion about spin with these commutation relations, and don't say much about how they are generators of rotations.

So there are three generators of rotation, called spin. If we do the same with (inhomogeneous) Lorentz transformations instead, we get ten generators instead of three (because there are ten parameters: 3 boost parameters, 3 Euler angles and 4 translation parameters). They are called boost operators, spin operators and components of four-momentum. The zeroth component of the four-momentum is the Hamiltonian. It's the generator of translations in time, and it's eigenvalues are the possible energy levels of the physical system.

The Lie algebra stuff that I suck at, goes something like this: The PoincarĂ© group is a Lie group (meaning it's both a group and a manifold). The tangent space at the identity element can be given the structure of a Lie algebra (a vector space with a way to "multiply" its members, similar to a cross product or a commutator) called the PoincarĂ© algebra. The members of the PoincarĂ© group are Lorentz transformations by definition, and the members of the PoincarĂ© algebra are generators of Lorentz transformations.

Last edited: Jan 22, 2008
3. Jan 28, 2008

### samalkhaiat

Last edited: Jan 28, 2008
4. Jan 21, 2009