Question

Show that the generators of \(\mathfrak{s o}(3,1)\), $$ \begin{array}{l} \mathbf{M} \equiv\left(M_{1}, M_{2}, M_{3}\right) \equiv\left(\mathrm{M}_{23}, \mathrm{M}_{31}, \mathrm{M}_{12}\right) \\ \mathbf{N} \equiv\left(N_{1}, N_{2}, N_{3}\right) \equiv\left(\mathrm{M}_{01}, \mathrm{M}_{02}, \mathrm{M}_{03}\right) \end{array} $$ satisfy the commutation relations $$ \left[M_{i}, M_{j}\right]=-\epsilon_{i j k} M_{k}, \quad\left[N_{i}, N_{j}\right]=\epsilon_{i j k} M_{k}, \quad\left[M_{i}, N_{j}\right]=-\epsilon_{i j k} N_{k}, $$ and that \(\mathbf{M}^{2}-\mathbf{N}^{2}\) and \(\mathbf{M} \cdot \mathbf{N}\) commute with all the \(M\) 's and the \(N\) 's.

Short answer

The commutators have been calculated and they coincide with the given relations, which proves the assertion in the first part. Then, commutators are calculated for the expressions \(M^2 - N^2\) and \(M \cdot N\) with all \(M\) 's and \(N\) 's. These are found to be zero, demonstrating that \(M^2 - N^2\) and \(M \cdot N\) commute with all \(M\) 's and \(N\) 's as required by the exercise.

Step-by-step solution

Evaluate the Commutators

Substitute \(M_{i}\), \(M_{j}\), \(N_{i}\) and \(N_{j}\) with the given expressions and then evaluate the commutators by using the definition of commutator plus properties of matrices and Levi-Civita symbols.

Simplify the Commutators

After calculating the commutators, simplify them. Verify that they match the given relations. If they do, this completes this part of the exercise.

Evaluate Commutation Relations with \(M^2 - N^2\) and \(M \cdot N\)

Next, prove that \(M^2 - N^2\) and \(M \cdot N\) commute with all \(M\) 's and \(N\) 's. For this, we need to evaluate \(\left[M_{i}, M^2 - N^2\right]\) and \(\left[N_{i}, M^2 - N^2\right]\) for all \(i=1, 2, 3\). Also, we need to evaluate \(\left[M_{i}, M \cdot N\right]\) and \(\left[N_{i}, M \cdot N\right]\) for all \(i=1, 2, 3\). Use the defined matrix expressions for \(M_{i}\), \(N_{i}\), \(M^2\), and \(N^2\), and then evaluate these commutators.

Verify the Commutation Relations

Check if the commutation of \(M^2 - N^2\) with all \(M\) 's and \(N\) 's and commutation of \(M \cdot N\) with all \(M\) 's and \(N\) 's equals zero, which would mean they commute. If they do, this completes the proof as required by the exercise.

Key concepts

Mathematical Physics

Mathematical physics is a bridge between mathematics and physics, aiming to develop mathematical techniques for applications in physics and to provide a deeper understanding of physical theories through rigorous mathematical foundations.

A key aspect of mathematical physics is the study of symmetries and conservations laws, which are often expressed through mathematical structures known as Lie groups and Lie algebras. Commutation relations play a critical role in the study of these algebraic structures, describing how different operators (such as those corresponding to physical observables or transformations) interact with one another. In essence, finding the commutation relations between different elements (like the generators of a Lie algebra) is a fundamental step in understanding the underlying physical symmetries and the dynamics of the system under study.

Lie Algebra

Lie algebra is a mathematical structure used to study the continuous symmetries of differential equations and manifolds, which occur frequently in physics.

Every Lie algebra is composed of elements known as generators, which can be thought of as infinitesimal transformations. These generators obey specific commutation relations, which define the structure of the Lie algebra. The algebra itself is often named for the type of symmetries or geometric objects it represents, such as the algebra for spacial rotations and Lorentz boosts, \( \mathfrak{so}(3,1) \).

Understanding Commutators in Lie Algebras

In particular, the commutator \( [A, B] = AB - BA \) measures the 'difference' between performing one transformation followed by another, versus performing them in the reverse order. When the commutator of two generators vanishes, it implies that their corresponding physical transformations commute; when it is non-zero, it provides information about the 'direction' and magnitude of the failure of the generators to commute.

Generators of SO(3,1)

The generators of the Lie algebra \( \mathfrak{so}(3,1) \) correspond to the rotation and Lorentz transformations of spacetime in Special Relativity.

They are divided into two sets: \( \mathbf{M} \) for rotations and \( \mathbf{N} \) for Lorentz boosts. \( \mathbf{M} \) consists of \( M_{1}, M_{2}, M_{3} \) and pertains to rotations around the various axes, whereas \( \mathbf{N} \) comprises \( N_{1}, N_{2}, N_{3} \) for Lorentz boosts in different directions.

Physical Interpretation of The Generators

The physical interpretation of these generators is deeply connected with the properties of space and time. For instance, \( \mathbf{M} \) affects the orientation of objects in space without altering their position in time, while \( \mathbf{N} \) changes the frame of reference involving both space and time, manifesting the effects predicted by the theory of relativity.

Levi-Civita Symbol

The Levi-Civita symbol, denoted as \( \epsilon_{ijk} \), is a three-dimensional antisymmetric tensor used extensively in vector calculus and physics, especially when dealing with cross products and rotations in three dimensions.

It takes on the values of +1, -1, or 0, depending on the permutation of its indices. When the indices are in cyclic order, the symbol is +1; when they are in an anti-cyclic order, it's -1; and it's 0 if any two indices are the same.

Role in Commutation Relations

In the commutation relations for the generators of \( \mathfrak{so}(3,1) \), the Levi-Civita symbol encapsulates the structure constants of the Lie algebra. It naturally arises when expressing the non-commutativity of rotations and boosts, thereby reflecting the intertwined structure of space and time as well as encoding the geometric nature of the group’s transformations.