# Ladder operators in electron field and electron's charge

• DOTDO
The creation and annihilation operators for electrons must satisfy anti-commutation relations. In summary, S. Weinberg explains in his book, "The Quantum Theory of Fields Volume I", that the field for electrons should not mix annihilation and creation operators due to their charge. Instead, it can be written as a sum of orthonormal plane-wave solutions with a label for 3-momentum, spin, and energy. This is because electrons cannot be created or destroyed in isolation due to their electric charge and the need to satisfy anti-commutation relations.
DOTDO
S. Weinberg says in his book, "The Quantum Theory of Fields Volume I", that

Since electrons carry a charge, we would not like to mix annihilation and creation operators, so we might try to write the field as $$\psi(x)=\sum_{k}u_k (x)e^{-i\omega_k t}a_k$$
where ##u_k (x)e^{-i\omega_k t}## are a complete set of orthonormal plane-wave solutions of the Dirac equation with ##k## labelling the 3-momentum, spin, and sign of the energy.
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At the first time, I thought it's because ##\sum_{k}## already involves the negative ##\omega_k## so that ##b_k## and ##b^\dagger _k## in ##\psi(x)=\sum_{k}u_k (x)e^{-i\omega_k t}b_k+u_k (x)e^{i\omega_k t}b^\dagger _k## can be merged into ##a_k##.

But he says it's because of the electron's charge and explains no more. Can someone explain, please?

Thank you.

Last edited:
I don't have the book, but electrons cannot be created or destroyed in isolation - their electric charge is one reason, the lepton number is another.

## 1. What are ladder operators in electron field?

Ladder operators are mathematical operators that are used to describe the behavior of electrons in a quantum mechanical system. They are used to create and destroy electrons in different energy states, and can be thought of as "ladders" that electrons can climb up or down on.

## 2. How do ladder operators relate to the electron's charge?

Ladder operators do not directly relate to the electron's charge. Instead, they describe the energy states and behavior of electrons within a system. However, the charge of an electron can be affected by the energy states it occupies.

## 3. What is the significance of ladder operators in quantum mechanics?

Ladder operators are important in quantum mechanics because they help us understand the behavior of electrons and other particles in a quantum system. They allow us to predict the energy states and interactions of these particles, and are essential in many quantum mechanical calculations and models.

## 4. Can ladder operators be used to manipulate the charge of an electron?

No, ladder operators do not directly manipulate the charge of an electron. They describe the energy states and behavior of electrons, but do not directly affect their charge. The charge of an electron is a fundamental property that cannot be changed or manipulated in this way.

## 5. Are ladder operators specific to electrons or can they be applied to other particles?

Ladder operators can be applied to other particles besides electrons. They are a general mathematical concept that can be used to describe the behavior of any quantum particle, such as protons, neutrons, or even photons. However, the specific equations and calculations may differ depending on the properties and interactions of the particle in question.

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