# Field in the presence of a background electromagnetic field

• Wledig
In summary, the Klein-Gordon equation has an additional term of the form: c = -\eta^{\mu \nu} (\partial_\mu i e A_\nu + ie A_\mu \partial_\nu - eA_\mu \partial_\nu)
Wledig
Homework Statement
A field obeying the Dirac equation in the presence of a background electromagnetic field also obeys the second-order equation:$$(i\gamma^{\mu}D_{\mu}+m)(i\gamma^{\nu}D_{\nu}-m)\Psi = 0$$ Where ##D_{\mu} = (\partial_{\mu} +ieA_{\mu})## Simplify this equation by using the identity $$\gamma^{\mu}\gamma^{\nu} = \dfrac{1}{2}\{\gamma^{\mu},\gamma^{\nu}\} + \dfrac{1}{2}[\gamma^{\mu} \gamma^{\nu}]$$
a) Show that it reduces to the Klein-Gordon equation plus one extra term.

b) Simplify the new term by proving the identity $$[D_{\mu},D_{\nu}] = +ieF_{\mu \nu}$$ . Using the explicit form of the ##\gamma^{\mu}## matrices evaluate this term in a background magnetic field for which ##F_ij = \epsilon_{ijk}B^k## and ##F_{0i} = 0##.

c) Act the resulting equation on ##\Psi = \begin{pmatrix}
\xi \\
0
\end{pmatrix}
e^{-imt}##. Show that to first order in B, the energy of the state is shifted by an term of the form of ##\Delta E = \mu \cdot B##. In the expression for ##\mu##, identify g = 2.
Relevant Equations
Klein Gordon equation: ##(\partial^2 + m^2)\phi(x) = 0##

Dirac equation solution he is referring to:
##\Psi = \begin{pmatrix}
\xi \\
0
\end{pmatrix}
e^{-imt}##

Magnetic moment: ##\mu = \dfrac{-geS}{2m}##
Attempt at a solution:

$$-\gamma^{\mu}\gamma^{\nu}D_{\mu} D_{\nu} - im\gamma^{\mu} D_{\mu} + im\gamma^{\nu}D_{\nu} - m^2 =$$
$$-\gamma^{\mu}\gamma^{\nu}(\partial_{\mu} + ieA_{\mu})(\partial_{\nu}-m) - I am \gamma^{\mu}(\partial_{\mu}+ieA_{\mu})+im\gamma^{\nu}(\partial_{\nu}+ieA_{\nu}) - m^2 =$$
$$\gamma^{\mu}\gamma^{\nu}(\partial_{\mu} \partial_{\nu} - \partial_{\mu} m + ieA_{\mu} \partial_{\nu} - ieA_{\mu} m) - im\gamma^{\mu} \partial_{\mu} + emA_{\mu}\gamma^{\mu} + im\gamma^{\nu}\partial_{\nu}-em\gamma^{\nu} A_{\nu} - m^2$$

Rearranging the terms we can recover the Klein-Gordon equation with this additional term c:
$$-\gamma^{\mu}\gamma^{\nu}\partial_\mu \partial_nu + im\gamma^{\nu}\partial_\nu - im\gamma^{\mu}\partial_\mu - m^2 + c$$
$$-\eta^{\mu \nu} \partial_\mu \partial_\nu - m^2 + c$$
$$c = \gamma^{\mu}\gamma^{\nu}ieA_{\mu}\partial_{\nu} + \gamma^{\mu}\gamma^{\nu}iemA_{\mu} + eA_{\mu}\gamma^{\mu} - m\gamma^{\nu}eA_{\nu}$$

I don't see a way to incorporate the identity he suggested and still recover the Klein Gordon equation.

Just noticed a bunch of typos at my attempted solution, I apologize for that. Here's my new attempt, if we focus on the first term:
$$-\gamma^\mu \gamma^\nu D_\mu D_\nu = -\dfrac{1}{2}( \{\gamma^\mu, \gamma^\nu \} + [\gamma^\mu,\gamma^\nu]) D_\mu D_\nu =$$
$$-\dfrac{1}{2}(2\eta^{\mu \nu} + [\gamma^\mu, \gamma^\nu]) D_\mu D_\nu =$$

$$\underline{-\eta^{\mu \nu}}(\underline{\partial_\mu \partial_\nu} + \partial_\mu i e A_\nu + ieA_\mu \partial_\nu - eA_\mu A_\nu) - \dfrac{1}{2}[\gamma^\mu, \gamma^\nu] D_\mu D_\nu$$

Now we can recover the Klein-Gordon equation combining the terms underlined up there with the ##-m^2## in:

$$-\gamma^{\mu}\gamma^{\nu}D_{\mu} D_{\nu} - im\gamma^{\mu} D_{\mu} + im\gamma^{\nu}D_{\nu} - m^2$$

I still need some time to reavaluate the additional term though...

Last edited:
Ended up with an additional term of the form:
$$c = -\eta^{\mu \nu} (\partial_\mu i e A_\nu + ie A_\mu \partial_\nu - eA_\mu \partial_\nu) - \dfrac{1}{2}[\gamma^\mu, \gamma^\nu]D_\mu D_\nu - im\gamma^\mu D_\mu + I am \gamma^\nu D_\nu$$

I'll probably have to fiddle with the indices in order to make ##[D_\mu , D_\nu] ## appear, I'm just not sure how to go about doing this...

## 1. What is a background electromagnetic field?

A background electromagnetic field refers to the presence of electromagnetic radiation or energy in a given space. This can be generated by natural sources, such as the Earth's magnetic field, or by artificial sources, such as power lines or electronic devices.

## 2. How does a background electromagnetic field affect a field in its presence?

The presence of a background electromagnetic field can alter the behavior of another electromagnetic field in the same space. This can cause interference, absorption, or reflection of the field, depending on the properties of the background field and the field being affected.

## 3. What factors determine the strength of a field in the presence of a background electromagnetic field?

The strength of a field in the presence of a background electromagnetic field is determined by various factors, including the intensity and frequency of the background field, the distance between the two fields, and the properties of the materials or mediums through which the fields are passing.

## 4. How can a background electromagnetic field be measured?

A background electromagnetic field can be measured using specialized equipment, such as electromagnetic field meters or spectrum analyzers. These devices can detect the presence and strength of electromagnetic radiation in a given space.

## 5. What are the potential effects of long-term exposure to a background electromagnetic field?

The potential effects of long-term exposure to a background electromagnetic field are still being studied and debated. Some studies suggest that it may have negative health effects, while others argue that there is no significant risk. More research is needed to fully understand the potential impact of prolonged exposure to background electromagnetic fields.

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