How Do Lorentz Group Commutation Relations Apply to Spin Matrices?

In summary: And so on. So you can calculate ##[S_{\mu \nu}, S_{\rho \sigma}]## for all possible values of ##\mu, \nu, \rho, \sigma##. You'll need to pay attention to the order of the indices in the Levi-Civita symbols, but it should work out.In summary, the sets ##(S_{\mu\nu})_L## and ##(S_{kl})_R## satisfy the commutation relation of the Lorentz group, namely $$\left[M_{\mu \nu}, M_{\rho \sigma}\right]=-\eta_{\mu \rho} M_{\nu \sigma}+\eta_{\mu
  • #1
Markus Kahn
112
14

Homework Statement


Prove that the sets ##(S_{\mu\nu})_L## and ##(S_{kl})_R##, where
$$
\left( S _ { k \ell } \right) _ { L } = \frac { 1 } { 2 } \varepsilon _ { j k \ell } \sigma _ { j } = \left( S _ { k \ell } \right) _ { R } \quad\text{and}\quad \left( S _ { 0 k } \right) _ { L } = \frac { 1 } { 2 } i \sigma _ { k } = ( S ^ { 0 k }) _ { R }
$$
satisfy the commutation relation of the Lorentz group, namely
$$
\left[M_{\mu \nu}, M_{\rho \sigma}\right]=-\eta_{\mu \rho} M_{\nu \sigma}+\eta_{\mu \sigma} M_{\nu \rho}-\eta_{\nu \sigma} M_{\mu \rho}+\eta_{\nu \rho} M_{\mu \sigma}.
$$

The Attempt at a Solution


My attempt was straight forward
$$
\begin{align*}
[(S_{kl})_L, (S_{bc})_L]
&= \frac{1}{4}\varepsilon_{jkl}\varepsilon_{abc}[\sigma_j,\sigma_a] = \frac{1}{4}\varepsilon_{jkl}\varepsilon_{abc} (2i \varepsilon_{jau}\sigma_u) = \frac{i}{2}\varepsilon_{jkl}\varepsilon_{abc} \varepsilon_{jau}\sigma_u\\
&=\frac{i}{2} (\delta_{ka}\delta_{lu}-\delta_{ku}\delta_{al})\varepsilon_{abc}\sigma_u = \frac{i}{2}\varepsilon_{kbc}\sigma_l -\frac{i}{2} \varepsilon_{lbc}\sigma_k
\end{align*}
$$
but this seems to lead to nowhere. One of my problems here is that ##(S_{kl})_L## is only defined for ##k,l\in\{1,2,3\}## but ##M_{\mu\nu}## is defined for ##\mu,\nu\in\{0,\dots,3\}##... I'm not sure how to make sense of this but I honestly also don't know where I made a mistake in the above calculation...
 
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  • #2
Markus Kahn said:
My attempt was straight forward
$$
\begin{align*}
[(S_{kl})_L, (S_{bc})_L]
&= \frac{1}{4}\varepsilon_{jkl}\varepsilon_{abc}[\sigma_j,\sigma_a] = \frac{1}{4}\varepsilon_{jkl}\varepsilon_{abc} (2i \varepsilon_{jau}\sigma_u) = \frac{i}{2}\varepsilon_{jkl}\varepsilon_{abc} \varepsilon_{jau}\sigma_u\\
&=\frac{i}{2} (\delta_{ka}\delta_{lu}-\delta_{ku}\delta_{al})\varepsilon_{abc}\sigma_u = \frac{i}{2}\varepsilon_{kbc}\sigma_l -\frac{i}{2} \varepsilon_{lbc}\sigma_k
\end{align*}
$$

I find working with the Levi-Civita symbols extremely tedious and error-prone, but I think that's correct. You can check for a couple of special cases:

##[(S_{xy})_L, (S_{yz})_L] = [i/2 \sigma_z, i/2 \sigma_x] = (-1/4) (- 2 i ) \sigma_y = (i/2) \sigma_y##

Your formula gives:
##(i/2) \varepsilon_{xyz} \sigma_y - (i/2) \varepsilon_{yyz} \sigma_x##

which simplifies to the same thing.

but this seems to lead to nowhere. One of my problems here is that ##(S_{kl})_L## is only defined for ##k,l\in\{1,2,3\}## but ##M_{\mu\nu}## is defined for ##\mu,\nu\in\{0,\dots,3\}##... I'm not sure how to make sense of this but I honestly also don't know where I made a mistake in the above calculation...

##S_{\mu \nu}## is defined for all ##\mu## and ##\nu##. If ##\mu = \nu = 0##, then it's zero. If ##\mu = j ## and ##\nu = k## are both 1,2 or 3, then it's ##S_{jk}##. If ##\mu = 0## and ##\nu = j##, then it's ##S_{0j} = (i/2) \sigma_j##
 

Related to How Do Lorentz Group Commutation Relations Apply to Spin Matrices?

What is a commutation relation?

A commutation relation is a mathematical relationship between two operators that describes how they behave when applied to a system in different orders. It is an important concept in quantum mechanics and is used to understand the behavior of particles at the microscopic level.

Why is proving commutation relation important?

Proving commutation relation is important because it helps us understand the fundamental properties of a physical system and how it behaves under different conditions. It also allows us to make predictions and calculations about the system using mathematical models.

How is commutation relation proven?

Commutation relation is proven using mathematical techniques such as operator algebra and matrix manipulation. It involves showing that the operators in question satisfy certain conditions, such as the commutator being equal to zero or a constant value.

What are some examples of commutation relation?

Some examples of commutation relation include the position and momentum operators in quantum mechanics, which have a non-zero commutator, and the angular momentum operators, which have a commutator that depends on the system's properties.

How does commutation relation relate to uncertainty principle?

Commutation relation is closely related to the uncertainty principle, which states that certain pairs of physical quantities, such as position and momentum, cannot be known simultaneously with perfect accuracy. This is because the commutator between these quantities is non-zero, indicating that they do not commute and cannot be measured simultaneously.

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