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princeton118
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What is the rotation transformation generator?
What is the Lorentz group generator?
What is the Lorentz group generator?
princeton118 said:What is the rotation transformation generator?
The set of all rotations:
[tex]\bar{r} = R( \theta ) {} r[/tex]
that keeps [itex]r^{2}[/itex] invariant ( i.e., [itex]\bar{r}^{2} = r^{2}[/itex] ) form a group called the rotation group SO(3) (= Special Orthogonal group in 3 dimension).
For a very small (infinitesimal) rotations, the matrix [itex]R[/itex] will differ from the unit matrix [itex]I_{3}[/itex] by an infinitesimal 3x3 matrix [itex]\omega ( \theta )[/itex] [infinitesimal matrix is a matrix whose elements are infinitesimal numbers].
Thus;
[tex]\bar{r} = \left[ I + \omega ( \theta ) \right] {} r[/tex]
or, in terms of the components [itex](x_{1},x_{2},x_{3})[/itex] of the vector r,
[tex]\bar{x}_{i} = \left[ \delta_{ij} + \omega_{ij}( \theta ) \right] {} x_{j}[/tex]
Now, the invariance condition;
[tex]\bar{x}_{i} \bar{x}_{i} = x_{j}x_{j}[/tex]
forces the matrix [itex]\omega[/itex] to be antisymmetric;
[tex]\omega_{ij} = - \omega_{ji}[/tex]
Therefore, there are only three independent elements in [itex]\omega[/itex]. So, we may write
[tex]
\omega = \left( \begin{array}{ccc} 0 & - \theta_{3} & \theta_{2} \\ \theta_{3} & 0 & - \theta_{1} \\ - \theta_{2} & \theta_{1} & 0 \end{array} \right)
[/tex]
We rewrite this as
[tex]
\omega = i \theta_{1} J_{1} + i \theta_{2} J_{2} + i \theta_{3} J_{3} \equiv i \theta_{k}J_{k}
[/tex]
where
[tex]
J_{1} = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & i \\ 0 & -i & 0 \end{array} \right)
[/tex]
[tex]
J_{2} = \left( \begin{array}{ccc} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{array} \right)
[/tex]
and
[tex]
J_{3} = \left( \begin{array}{ccc} 0 & i & 0 \\ -i & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)
[/tex]
Now it is easy to see that these J's satisfy the following commutation relations
[tex]\left[ J_{1}, J_{2} \right] \equiv J_{1}J_{2} - J_{2}J_{1} = i J_{3}[/tex]
[tex]\left[ J_{3}, J_{1} \right] = i J_{2}[/tex]
and
[tex] \left[ J_{2}, J_{3} \right] = i J_{1}[/tex]
we write these in the compact form;
[tex] [ J_{i} , J_{j} ] = i \epsilon_{ijk} J_{k}[/tex]
In quantum mechanics, the components of the angular momentum operator satisfy the same commutation relations. Therefore, one speaks of the angular momentum operator as the generator of the rotation group SO(3).
Mathematically, the matrix [itex]\omega[/itex] is an element of 3-dimensional vector soace [the Lie algebra of SO(3)] whose basis vectors,[itex]J_{i}[/itex], satisfy the above-derived commutation relations [Lie brackets]. The components [itex]\theta_{i}[/itex] of the Lie algebra element [itex]\omega[/itex], play the role of local coordinates of the group manifold. All elements of the group SO(3) can be represented as exponentials of Lie algebra elements, i.e., the matrix [itex]R( \theta )[/itex] can now be written as
[tex]R = \exp (i \theta_{k} J_{k} )[/tex]
What is the Lorentz group generator?
Doing the same thing with Lotentz transformations leads to six 4x4 matrices (generators).
Try it yourself.
regards
sam
samalkhaiat said:princeton118 said:The set of all rotations:
[tex]\bar{r} = R( \theta ) {} r[/tex]
that keeps [itex]r^{2}[/itex] invariant ( i.e., [itex]\bar{r}^{2} = r^{2}[/itex] ) form a group called the rotation group SO(3) (= Special Orthogonal group in 3 dimension).
For a very small (infinitesimal) rotations, the matrix [itex]R[/itex] will differ from the unit matrix [itex]I_{3}[/itex] by an infinitesimal 3x3 matrix [itex]\omega ( \theta )[/itex] [infinitesimal matrix is a matrix whose elements are infinitesimal numbers].
Thus;
[tex]\bar{r} = \left[ I + \omega ( \theta ) \right] {} r[/tex]
or, in terms of the components [itex](x_{1},x_{2},x_{3})[/itex] of the vector r,
[tex]\bar{x}_{i} = \left[ \delta_{ij} + \omega_{ij}( \theta ) \right] {} x_{j}[/tex]
Now, the invariance condition;
[tex]\bar{x}_{i} \bar{x}_{i} = x_{j}x_{j}[/tex]
forces the matrix [itex]\omega[/itex] to be antisymmetric;
[tex]\omega_{ij} = - \omega_{ji}[/tex]
Therefore, there are only three independent elements in [itex]\omega[/itex]. So, we may write
[tex]
\omega = \left( \begin{array}{ccc} 0 & - \theta_{3} & \theta_{2} \\ \theta_{3} & 0 & - \theta_{1} \\ - \theta_{2} & \theta_{1} & 0 \end{array} \right)
[/tex]
We rewrite this as
[tex]
\omega = i \theta_{1} J_{1} + i \theta_{2} J_{2} + i \theta_{3} J_{3} \equiv i \theta_{k}J_{k}
[/tex]
where
[tex]
J_{1} = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & i \\ 0 & -i & 0 \end{array} \right)
[/tex]
[tex]
J_{2} = \left( \begin{array}{ccc} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{array} \right)
[/tex]
and
[tex]
J_{3} = \left( \begin{array}{ccc} 0 & i & 0 \\ -i & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)
[/tex]
Now it is easy to see that these J's satisfy the following commutation relations
[tex]\left[ J_{1}, J_{2} \right] \equiv J_{1}J_{2} - J_{2}J_{1} = i J_{3}[/tex]
[tex]\left[ J_{3}, J_{1} \right] = i J_{2}[/tex]
and
[tex] \left[ J_{2}, J_{3} \right] = i J_{1}[/tex]
we write these in the compact form;
[tex] [ J_{i} , J_{j} ] = i \epsilon_{ijk} J_{k}[/tex]
In quantum mechanics, the components of the angular momentum operator satisfy the same commutation relations. Therefore, one speaks of the angular momentum operator as the generator of the rotation group SO(3).
Mathematically, the matrix [itex]\omega[/itex] is an element of 3-dimensional vector soace [the Lie algebra of SO(3)] whose basis vectors,[itex]J_{i}[/itex], satisfy the above-derived commutation relations [Lie brackets]. The components [itex]\theta_{i}[/itex] of the Lie algebra element [itex]\omega[/itex], play the role of local coordinates of the group manifold. All elements of the group SO(3) can be represented as exponentials of Lie algebra elements, i.e., the matrix [itex]R( \theta )[/itex] can now be written as
[tex]R = \exp (i \theta_{k} J_{k} )[/tex]
Doing the same thing with Lotentz transformations leads to six 4x4 matrices (generators).
Try it yourself.
regards
sam
This should be framed. You just tied a lot of things together which were popping about.
The Lorentz group generator is a mathematical tool used to describe the symmetries of spacetime in special relativity. It is a 4x4 matrix that represents the transformations of space and time coordinates between two inertial reference frames.
The Lorentz group generator is important because it helps us understand the fundamental laws of physics in the context of special relativity. It is used in various fields of physics, such as particle physics and cosmology, to study the behavior of particles and the universe.
The Lorentz group generator is the mathematical representation of the Lie algebra of the Lorentz group. This means that it describes the infinitesimal transformations of the Lorentz group, which is a group of transformations that preserve the fundamental laws of physics in special relativity.
The Lorentz group generator has several important properties, including being anti-symmetric, traceless, and Hermitian. It also satisfies the commutation relations of the Lorentz group, which are essential for understanding the symmetries of spacetime in special relativity.
The Lorentz group generator is used in practical applications, such as in the calculation of scattering amplitudes in particle physics experiments. It is also used in the study of relativistic quantum mechanics and in the development of theories that unify gravity and quantum mechanics.