# Non-relativistic limit of Dirac bilinear

• Andrea M.
In summary, in the non-relativistic limit, the vector and axial currents take on simplified forms, as mentioned in multiple articles, such as Gondolo's "Phenomenological Introduction to Direct Dark Matter Detection." This is due to the fact that in this limit, the first two components of the Dirac spinor are large while the last two are small. This results in the off-diagonal components of the gamma matrices coupling the small and large components, while the diagonal components couple the large components to themselves and the small components to themselves. The opposite is true for the gamma5 case, where the diagonal components become off-diagonal and vice versa. This concept is also discussed in Peskin's "Coulomb potential" in
Andrea M.
Hi,
I'm studying direct detection techniques for dark matter and in almost all the articles I read (e.g.
Gondolo, P. (1996, May 13). Phenomenological Introduction to Direct Dark Matter Detection. arXiv.org.) the authors say that in the non-relativistic limit the vector and axial currents take the following forms: $$\bar{\nu}\gamma_{\mu}\nu\rightarrow\nu^{\dagger}\nu$$ $$\bar{\nu}\gamma_{\mu}\gamma_5\nu\rightarrow\bar{\nu}\vec{\gamma}\gamma_5\nu$$
can anyone explain me why?

One fast answer, in the non-relativistic limit, the first two components of the Dirac spinor are large and the last two are small...
That means that when you take the products of the type: ## \bar{\psi} \gamma_\mu \psi## the ##\gamma_i## which are off-diagonal will couple your small component to the large... while the ##\gamma_0## will couple your large to large and small to small. That's why the ##\bar{\psi} \gamma_i \psi## can be neglected versus the ##\bar{\psi} \gamma_0 \psi = \psi^\dagger \psi##.

For the ##\gamma_5## case the thing is the opposite because gamma5 reverses them (the gamma0 becomes off-diagonal in block form, while the gamma_i become diagonal).

You can also have a look in Peskin, Ch4.8 Coulomb potential, where he gives $\bar{u} \gamma_0 u \approx 2m \xi^\dagger \xi$ while he also mentions that the other can be neglected for small momenta... The thing is that again the 1st two go with ##m## for ##p \rightarrow 0## while the other two go with ##p##.

Thank you so much, I'm having little hard time trying to understand the physics behind the direct detection techniques.

## 1. What is the non-relativistic limit of Dirac bilinear?

The non-relativistic limit of Dirac bilinear is a mathematical concept used in quantum mechanics to describe the behavior of particles at low speeds, where the effects of relativity can be ignored. It involves taking the limit of the Dirac equation, which describes the behavior of relativistic particles, as the speed of the particle approaches zero.

## 2. Why is the non-relativistic limit of Dirac bilinear important?

The non-relativistic limit of Dirac bilinear is important because it allows for the description of particles at low speeds, where the effects of relativity are negligible. This is useful in many practical applications, such as in the study of atoms and molecules, as well as in the development of technologies such as transistors and lasers.

## 3. How is the non-relativistic limit of Dirac bilinear calculated?

The non-relativistic limit of Dirac bilinear is calculated by taking the limit of the Dirac equation as the speed of the particle approaches zero. This involves simplifying the equation and neglecting terms that are negligible at low speeds. The resulting equation is known as the non-relativistic limit of the Dirac equation.

## 4. What are some applications of the non-relativistic limit of Dirac bilinear?

Some applications of the non-relativistic limit of Dirac bilinear include the study of atoms and molecules, the development of technologies such as transistors and lasers, and the calculation of energy levels and transition probabilities in quantum systems. It is also used in the development of quantum field theories and in the study of condensed matter systems.

## 5. Are there any limitations to the non-relativistic limit of Dirac bilinear?

While the non-relativistic limit of Dirac bilinear is useful in many practical applications, it does have its limitations. It is only applicable at low speeds, where the effects of relativity can be ignored. At high speeds, the full Dirac equation must be used to accurately describe the behavior of particles. Additionally, the non-relativistic limit does not take into account the spin of particles, which may be important in some systems.

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