3
$\begingroup$

I have seen in some places (especially in the context of theoretical physics) the notation $\mathcal{N}=2$, but I'm not that capable of reading and understanding these materials, thus I'm now wondering what does this notation mean? How important it is in physics? and Where can I learn things about it?

For my background: I'm an undergrad (sophomore), I have learnt some QFT this semester, and some GR. Thanks for your help!

$\endgroup$
8
  • 1
    $\begingroup$ Maybe this? Since you didn’t link to any of those places, it’s hard to know. More info. $\endgroup$
    – Ghoster
    Nov 14, 2023 at 1:10
  • $\begingroup$ How important it is in physics? That depends on how important you think string theory is. $\endgroup$
    – Ghoster
    Nov 14, 2023 at 1:14
  • 1
    $\begingroup$ Thanks! Your links are helpful! @Ghoster $\endgroup$
    – BoyanLiu
    Nov 14, 2023 at 1:16
  • $\begingroup$ Where can I learn things about it? Are you a graduate student? An undergrad? A high school student? Not studying physics but interested in it? $\endgroup$
    – Ghoster
    Nov 14, 2023 at 1:16
  • $\begingroup$ I'm an undergrad (sophomore), I have learnt some QFT this semester, and some GR @Ghoster $\endgroup$
    – BoyanLiu
    Nov 14, 2023 at 1:17

1 Answer 1

6
$\begingroup$

The notation $\mathcal{N}=2$ in the context of theoretical physics, particularly in supersymmetry and supergravity, refers to a specific kind of extended supersymmetry. In simple terms, supersymmetry is a theoretical symmetry between fermions (particles that follow the Fermi-Dirac statistics, like electrons) and bosons (particles that follow the Bose-Einstein statistics, like photons).

The "$\mathcal{N}$" in $\mathcal{N}=2$ supersymmetry denotes the number of independent supersymmetry generators in the theory. Each generator is responsible for transforming fermions into bosons and vice versa. In $\mathcal{N}=2$ supersymmetry, there are two such generators, typically denoted as $\mathbf{Q}_1$ and $\mathbf{Q}_2$. This implies a richer structure compared to $\mathcal{N}=1$ supersymmetry, which has only one supersymmetry generator.

The algebra these generators follow is known as the super-Poincaré algebra, which extends the Poincaré algebra of special relativity to include supersymmetry transformations. The super-Poincaré algebra includes the usual Poincaré algebra (commutation relations of momentum and angular momentum operators) plus additional anticommutation relations involving the supersymmetry generators.

For $\mathcal{N}=2$ supersymmetry, the anticommutation relations for the supersymmetry generators are of the form:

\begin{equation} \{ \mathbf{Q}_{\alpha}^i, \bar{\mathbf{Q}}_{\dot{\beta}j} \} = 2\sigma^\mu_{\alpha \dot{\beta}} \mathbf{P}_\mu \delta^i_j \end{equation}

Here, $\mathbf{Q}_{\alpha}^i$ and $\bar{\mathbf{Q}}_{\dot{\beta}j}$ are the supersymmetry generators (with spinor indices $\alpha, \dot{\beta}$ and $\mathcal{N}=2$ indices $i, j$), $\sigma^\mu$ are the Pauli matrices (incorporating spacetime structure into the algebra), and $\mathbf{P}_\mu$ is the four-momentum operator. The delta symbol $\delta^i_j$ ensures that the algebra closes within each supersymmetry.

$\mathcal{N}=2$ supersymmetry has significant implications in theoretical physics, particularly in string theory, where it helps in constructing more stable and less divergent models of particle physics. It also plays a crucial role in the study of supergravity and has implications in the mathematical field of topology through its connections to topological quantum field theories.

To learn more about $\mathcal{N}=2$ supersymmetry and super-Poincaré algebra, one typically needs a strong background in quantum field theory, general relativity, and group theory. Standard textbooks on quantum field theory and supersymmetry, such as "Supersymmetry and Supergravity" by Julius Wess and Jonathan Bagger, can be excellent starting points. Another great resource is Weinberg's "Quantum Theory of Fields" vol.3.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.